Proving properties of recursive functions is usually easiest when the proof follows the structure of the function that is being verified – same case splitting, same recursion pattern. To avoid having to spell this out in every proof, the upcoming 4.8 release of Lean provides a functional induction principle for recursive functions. Used with the induction tactic, this takes care of the repetitive parts of the proof.
In this blog post we will look at examples of recursive functions where this feature can be convenient, how to use it, and where its limits lie. In particular, we look at functions with fixed parameters and overlapping patterns and mutually recursive functions, where this feature is especially useful. Because some recursive functions can cause the simp tactic to loop, as it tries to unfold the definition infinitely many times, we also examine techniques for avoiding infinite simplification.
The examples in this post are short, in order to demonstrate the feature.
If the gain does not seem to be very great, keep in mind that we are looking at particularly simple examples here. More realistic applications may have more cases or explicit termination proofs, and in these cases, functional induction can lead to shorter, more maintainable proofs.
Alternating lists, the manual way
We begin with a function that picks elements from two alternating input lists:
defalternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α:(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1)→ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1|[],ysList α=>ysList α|xα::xsList α,ysList α=>xα::alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList αtermination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList αysList α=>xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
Now let us try to prove a simple theorem about alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α, namely that the length of the resulting list is the sum of the argument lists' lengths.
The alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α function is not structurally recursive, so using induction xs is not going to be helpful (try it if you are curious how it fails). Instead, we can attempt to write a recursive theorem: we use the theorem we are in the process of proving in the proof, and then rely on Lean's termination checker to check that this is actually ok.
So a first attempt at the proof might start by unfolding the definition of alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α. This exposes a match expression, and our Lean muscle memory tells us to proceed using split. The first case is now trivial. In the second case we use the next tactic to name the new variables. Then we call the theorem we are currently proving on the smaller lists ys and xs, following the pattern of alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α itself, and after some simplification and arithmetic, this goal is closed. Lean will tell us that it cannot prove termination of the recursive call, so we help it by giving the same termination argument as we gave to alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α itself.
theoremlength_alternate1length_alternate1.{u_1} {α : Type u_1} (xs ys : List α) : (alternate xs ys).length = xs.length + ys.length(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (matchxsList α, ysList αwith
| [], ysList α => ysList α
| xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α, ysList α => xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length =
Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ ysList α.length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
h_2
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝²List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxs✝List α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
next`next => tac` focuses on the next goal and solves it using `tac`, or else fails.
`next x₁ ... xₙ => tac` additionally renames the `n` most recent hypotheses with
inaccessible names to the given names.
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ ysList α.length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
All goals completed! 🐙
next`next => tac` focuses on the next goal and solves it using `tac`, or else fails.
`next x₁ ... xₙ => tac` additionally renames the `n` most recent hypotheses with
inaccessible names to the given names.
xαxsList α
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝¹List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
x✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
All goals completed! 🐙
termination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
This works, but is not quite satisfactory. The split tactic is rather powerful, but having to name the variables with next is tedious. A more elegant variant is to use the same pattern matching in the proof as we do in the function definition:
theoremlength_alternate2length_alternate2.{u_1} {α : Type u_1} (xs ys : List α) : (alternate xs ys).length = xs.length + ys.length:(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1)→(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
|[],ysList α=>
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α [] ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
All goals completed! 🐙
|xα::xsList α,ysList α=>
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α) ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α) ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
All goals completed! 🐙
termination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList αysList α=>xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
Note that we no longer unfold the function definition, but let the simplifier reduce the function application in the respective cases.
This style brings us closer to the function to be defined, but we still have to repeat the (possibly complex) pattern match of the original function, and the (possibly complex) termination argument and proof. Moreover, we have to explicitly instantiate the recursive invocation of the theorem, or hope that simp will only use it at the arguments that we want it to.
This is not very satisfying. The repetition makes the proof brittle, as a change to the function definition may require applying the same change to every single proof. We already explained the structure of the function to Lean once, we should not have to do it again!
Alternating lists, with functional induction
This is where the functional induction theorem comes in. For every recursive function, Lean defines a functional induction principle; just append .induct to the name. In our example, it has the following type:
alternate.inductalternate.induct.{u_1} {α : Type u_1} (motive : List α → List α → Prop) (case1 : ∀ (ys : List α), motive [] ys)
(case2 : ∀ (x : α) (xs ys : List α), motive ys xs → motive (x :: xs) ys) (xs ys : List α) : motive xs ys{αType u_1:TypeA type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. _}(motiveList α → List α → Prop:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1→ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1→Prop)(case1∀ (ys : List α), motive [] ys:∀(ysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),motiveList α → List α → Prop[]ysList α)(case2∀ (x : α) (xs ys : List α), motive ys xs → motive (x :: xs) ys:∀(xα:αType u_1)(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),motiveList α → List α → PropysList αxsList α→motiveList α → List α → Prop(xα::xsList α)ysList α)(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):motiveList α → List α → PropxsList αysList α
Seeing an induction principle in its full glory can be a bit intimidating, so let us spell it out in prose:
Given a property of two lists that we want to prove (the motive),
if it holds when the first list is empty (the minor premise case1, corresponding to the first branch of alternate),
and if it holds when the first list is non-empty (the minor premise case2, corresponding to the second branch of alternate), where we may assume that it holds for the second list and the tail of the first list (the induction hypothesis, corresponding to the recursive call in alternate),
then given two arbitrary lists (the targetsxs and xy, which correspond to the varying parameters of the function)
the property holds for these lists.
An induction principle like this is used every time you use the induction tactic; for example induction on natural numbers uses Nat.recAuxNat.recAux.{u} {motive : Nat → Sort u} (zero : motive 0) (succ : (n : Nat) → motive n → motive (n + 1)) (t : Nat) :
motive tRecursor identical to `Nat.rec` but uses notations `0` for `Nat.zero` and `· + 1` for `Nat.succ`.
Used as the default `Nat` eliminator by the `induction` tactic. .
With this bespoke induction principle, we can now use the induction tactic in our proof:
theoremlength_alternate3length_alternate3.{u_1} {α : Type u_1} (xs ys : List α) : (alternate xs ys).length = xs.length + ys.length(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
inductionAssuming `x` is a variable in the local context with an inductive type,
`induction x` applies induction on `x` to the main goal,
producing one goal for each constructor of the inductive type,
in which the target is replaced by a general instance of that constructor
and an inductive hypothesis is added for each recursive argument to the constructor.
If the type of an element in the local context depends on `x`,
that element is reverted and reintroduced afterward,
so that the inductive hypothesis incorporates that hypothesis as well.
For example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,
`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,
and one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.
Here the names `a` and `ih₁` are chosen automatically and are not accessible.
You can use `with` to provide the variables names for each constructor.
- `induction e`, where `e` is an expression instead of a variable,
generalizes `e` in the goal, and then performs induction on the resulting variable.
- `induction e using r` allows the user to specify the principle of induction that should be used.
Here `r` should be a term whose result type must be of the form `C t`,
where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables
- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,
generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.
In other words, the net effect is that each inductive hypothesis is generalized.
- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`
uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.
xsList α,ysList αusingalternate.inductalternate.induct.{u_1} {α : Type u_1} (motive : List α → List α → Prop) (case1 : ∀ (ys : List α), motive [] ys)
(case2 : ∀ (x : α) (xs ys : List α), motive ys xs → motive (x :: xs) ys) (xs ys : List α) : motive xs yswithAfter `with`, there is an optional tactic that runs on all branches, and
then a list of alternatives.
|case1ysList α
case1
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α [] ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α) ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
case2
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
All goals completed! 🐙
This looks still quite similar to the previous iteration, but note the absence of a termination argument. Furthermore, just like with normal induction, we were provided the induction hypothesis already specialized at the right argument, as you can see in the proof state after case2.
With all this boilerplate out of the way, chances are good that we can now solve both cases with just proof automation tactics, leading to the following two-line proof:
theoremlength_alternate4length_alternate4.{u_1} {α : Type u_1} (xs ys : List α) : (alternate xs ys).length = xs.length + ys.length(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
case1
αType u_1
:
Type u_1
ys✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α [] ys✝List α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ys✝List α.length
case2
αType u_1
:
Type u_1
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ys✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αys✝List αxs✝List α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ys✝List α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xs✝List α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α) ys✝List α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ys✝List α.length
all_goals`all_goals tac` runs `tac` on each goal, concatenating the resulting goals, if any.
case2
αType u_1
:
Type u_1
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ys✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αys✝List αxs✝List α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ys✝List α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xs✝List α.length
⊢ ys✝List α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xs✝List α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xs✝List α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ys✝List α.length
;try`try tac` runs `tac` and succeeds even if `tac` failed.
All goals completed! 🐙
The * in simp's argument list indicates that simp should also consider local assumptions for rewriting; this way it picks up the induction hypothesis.
In terms of proof automation, this is almost as good as it can get: We state the theorem and the induction and leave the rest to Lean's automation. However, just in case you prefer explicit readable and educational proofs, we can of course also spell it out explicitly:
theoremlength_alternate5length_alternate5.{u_1} {α : Type u_1} (xs ys : List α) : (alternate xs ys).length = xs.length + ys.length(xsList αysList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αxsList αysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
inductionAssuming `x` is a variable in the local context with an inductive type,
`induction x` applies induction on `x` to the main goal,
producing one goal for each constructor of the inductive type,
in which the target is replaced by a general instance of that constructor
and an inductive hypothesis is added for each recursive argument to the constructor.
If the type of an element in the local context depends on `x`,
that element is reverted and reintroduced afterward,
so that the inductive hypothesis incorporates that hypothesis as well.
For example, given `n : Nat` and a goal with a hypothesis `h : P n` and target `Q n`,
`induction n` produces one goal with hypothesis `h : P 0` and target `Q 0`,
and one goal with hypotheses `h : P (Nat.succ a)` and `ih₁ : P a → Q a` and target `Q (Nat.succ a)`.
Here the names `a` and `ih₁` are chosen automatically and are not accessible.
You can use `with` to provide the variables names for each constructor.
- `induction e`, where `e` is an expression instead of a variable,
generalizes `e` in the goal, and then performs induction on the resulting variable.
- `induction e using r` allows the user to specify the principle of induction that should be used.
Here `r` should be a term whose result type must be of the form `C t`,
where `C` is a bound variable and `t` is a (possibly empty) sequence of bound variables
- `induction e generalizing z₁ ... zₙ`, where `z₁ ... zₙ` are variables in the local context,
generalizes over `z₁ ... zₙ` before applying the induction but then introduces them in each goal.
In other words, the net effect is that each inductive hypothesis is generalized.
- Given `x : Nat`, `induction x with | zero => tac₁ | succ x' ih => tac₂`
uses tactic `tac₁` for the `zero` case, and `tac₂` for the `succ` case.
xsList α,ysList αusingalternate.inductalternate.induct.{u_1} {α : Type u_1} (motive : List α → List α → Prop) (case1 : ∀ (ys : List α), motive [] ys)
(case2 : ∀ (x : α) (xs ys : List α), motive ys xs → motive (x :: xs) ys) (xs ys : List α) : motive xs yswithAfter `with`, there is an optional tactic that runs on all branches, and
then a list of alternatives.
|case1ysList α
case1
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α [] ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
calcStep-wise reasoning over transitive relations.
```
calc
a = b := pab
b = c := pbc
...
y = z := pyz
```
proves `a = z` from the given step-wise proofs. `=` can be replaced with any
relation implementing the typeclass `Trans`. Instead of repeating the right-
hand sides, subsequent left-hand sides can be replaced with `_`.
```
calc
a = b := pab
_ = c := pbc
...
_ = z := pyz
```
It is also possible to write the *first* relation as `<lhs>\n _ = <rhs> :=
<proof>`. This is useful for aligning relation symbols, especially on longer:
identifiers:
```
calc abc
_ = bce := pabce
_ = cef := pbcef
...
_ = xyz := pwxyz
```
`calc` works as a term, as a tactic or as a `conv` tactic.
See [Theorem Proving in Lean 4][tpil4] for more information.
[tpil4]: https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#calculational-proofs
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α[]ysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
_=ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α [] ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length
All goals completed! 🐙
_=[].lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ ysList α.length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[].length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α) ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
calcStep-wise reasoning over transitive relations.
```
calc
a = b := pab
b = c := pbc
...
y = z := pyz
```
proves `a = z` from the given step-wise proofs. `=` can be replaced with any
relation implementing the typeclass `Trans`. Instead of repeating the right-
hand sides, subsequent left-hand sides can be replaced with `_`.
```
calc
a = b := pab
_ = c := pbc
...
_ = z := pyz
```
It is also possible to write the *first* relation as `<lhs>\n _ = <rhs> :=
<proof>`. This is useful for aligning relation symbols, especially on longer:
identifiers:
```
calc abc
_ = bce := pabce
_ = cef := pbcef
...
_ = xyz := pwxyz
```
`calc` works as a term, as a tactic or as a `conv` tactic.
See [Theorem Proving in Lean 4][tpil4] for more information.
[tpil4]: https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#calculational-proofs
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α(xα::xsList α)ysList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
_=(xα::alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List α (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α) ysList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length
All goals completed! 🐙
_=(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+1:=
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
All goals completed! 🐙
_=(ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
)+1:=
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ (alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
All goals completed! 🐙
_=(xα::xsList α).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
+ysList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(xα :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList α).length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
αType u_1
:
Type u_1
xα
:
αType u_1
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
ysList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(alternatealternate.{u_1} {α : Type u_1} (xs ys : List α) : List αysList αxsList α).length = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length
⊢ ysList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. ysList α.length
;
All goals completed! 🐙
Functions with fixed parameters
The above example demonstrates how to use functional induction in the common simple case. We now look into some more tricky aspects of it.
The first arises if the the recursive function has fixed parameters, as in the following function,
which cuts up a list into sublists of length at most n:
defcutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
):(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1)→ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
(ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1)|[]=>[]|xα::xsList α=>(xα::xsList α.takeList.take.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Returns the first `n` elements of `xs`, or the whole list if `n` is too large.
* `take 0 [a, b, c, d, e] = []`
* `take 3 [a, b, c, d, e] = [a, b, c]`
* `take 6 [a, b, c, d, e] = [a, b, c, d, e]`
(nNat-1))::cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat(xsList α.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat-1))termination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList α=>xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
Note that in all the recursive calls, the parameter n remains the same, whereas the parameter xs varies. Lean distinguishes between a prefix of fixed parameters, which are identical in all recursive calls, and then the varying parameters, which can change.
Joining these lists back together should give the original list. If we try to use functional induction in the same way as before, we find that we have more goals than we expect. In addition to the expected case1 and case2, we also get a subgoal called n:
theoremcutNJoin1cutNJoin1.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : (cutN n xs).join = xs(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):List.joinList.join.{u} {α : Type u} : List (List α) → List α`O(|join L|)`. `join L` concatenates all the lists in `L` into one list.
* `join [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList α)=xsList α:=
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList α).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α
n
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
case1
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat []).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[]
case2
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(?n - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(?n - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α
unsolved goals
case n
α : Type u_1
n : Nat
⊢ Nat
case case1
α : Type u_1
n : Nat
⊢ (cutN n []).join = []
case case2
α : Type u_1
n : Nat
x✝ : α
xs✝ : List α
ih1✝ : (cutN n (List.drop (?n - 1) xs✝)).join = List.drop (?n - 1) xs✝
⊢ (cutN n (x✝ :: xs✝)).join = x✝ :: xs✝
This is not really a “case”, but the induction tactic does not know that. If we look at the type of cutN.induct, we see that – just like cutN itself - it has a parameter called n, which is hardly surprising:
cutN.inductcutN.induct.{u_1} {α : Type u_1} (n : Nat) (motive : List α → Prop) (case1 : motive [])
(case2 : ∀ (x : α) (xs : List α), motive (List.drop (n - 1) xs) → motive (x :: xs)) (xs : List α) : motive xs{αType u_1:TypeA type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. _}(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(motiveList α → Prop:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1→Prop)(case1motive []:motiveList α → Prop[])(case2∀ (x : α) (xs : List α), motive (List.drop (n - 1) xs) → motive (x :: xs):∀(xα:αType u_1)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),motiveList α → Prop(List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat-1)xsList α)→motiveList α → Prop(xα::xsList α))(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):motiveList α → PropxsList α
Note that the motive depends only on the varying parameter of cutN, namely xs, whereas the fixed parameter of cutN, namely n, did not turn into a target of the induction principle, but merely a regular parameter.
So because n is not a target, we cannot write induction n, xs. Instead, we have to instantiate this parameter in the expression passed to using, and now the proof turns out rather nice:
theoremcutNJoin2cutNJoin2.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : (cutN n xs).join = xs(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):List.joinList.join.{u} {α : Type u} : List (List α) → List α`O(|join L|)`. `join L` concatenates all the lists in `L` into one list.
* `join [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList α)=xsList α:=
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
xsList α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList α).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
xsList α
case1
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat []).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[]
case2
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α
case1
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat []).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
[]
case2
αType u_1
:
Type u_1
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
x✝α
:
αType u_1
xs✝List α
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1
(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
(nNat - HSub.hSub.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HSub α β γ] : α → β → γ`a - b` computes the difference of `a` and `b`.
The meaning of this notation is type-dependent.
* For natural numbers, this operator saturates at 0: `a - b = 0` when `a ≤ b`. 1) xs✝List α
⊢ (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat (x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α)).join = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
x✝α :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List α
All goals completed! 🐙
We also could have used the syntax cutN.induct (n := n) if we prefer named arguments.
Overlapping patterns
So far we only defined functions with non-overlapping patterns. Lean also supports defining functions with overlapping patterns, and here functional induction can be particularly helpful.
Our (contrived) example function is one that removes repeated elements from a list of Boolean values:
defdestutter_root_.destutter (xs : List Bool) : List Bool:(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
)→ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
|[]=>[]|trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xsList Bool=>destutter_root_.destutter (xs : List Bool) : List Bool(trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xsList Bool)|falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool=>destutter_root_.destutter (xs : List Bool) : List Bool(falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool)|xBool::xsList Bool=>xBool::destutter_root_.destutter (xs : List Bool) : List BoolxsList Bool
The first three patterns of the function definition do not overlap, so they can be read as theorems that hold unconditionally. Lean generates such theorems, for use with rw and simp:
destutter.eq_3destutter.eq_3 (xs : List Bool) : destutter (false :: false :: xs) = destutter (false :: xs)(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
):destutterdestutter (xs : List Bool) : List Bool(falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool)=destutterdestutter (xs : List Bool) : List Bool(falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool)
But the fourth equation is not a theorem. In general, it is not true that destutter (x :: xs) = x :: destutter xs holds. This equation is only true when the previous cases did not apply. Lean expresses this by preconditions on the equational lemma:
destutter.eq_4destutter.eq_4 (x_1 : Bool) (xs : List Bool) (x_2 : ∀ (xs_1 : List Bool), x_1 = true → xs = true :: xs_1 → False)
(x_3 : ∀ (xs_1 : List Bool), x_1 = false → xs = false :: xs_1 → False) : destutter (x_1 :: xs) = x_1 :: destutter xs(x_1Bool:BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
)(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
)(x_2∀ (xs_1 : List Bool), x_1 = true → xs = true :: xs_1 → False:∀(xs_1List Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),x_1Bool=trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. →xsList Bool=trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xs_1List Bool→FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
)(x_3∀ (xs_1 : List Bool), x_1 = false → xs = false :: xs_1 → False:∀(xs_1List Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),x_1Bool=falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. →xsList Bool=falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xs_1List Bool→FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
):destutterdestutter (xs : List Bool) : List Bool(x_1Bool::xsList Bool)=x_1Bool::destutterdestutter (xs : List Bool) : List BoolxsList Bool
One rather annoying consequence is that if we try to construct a proof about destutter by repeating the same pattern matches, we cannot make progress:
theoremlength_destutter_root_.length_destutter (xs : List Bool) : (destutter xs).length ≤ xs.length:(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
)→(destutterdestutter (xs : List Bool) : List BoolxsList Bool).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
≤xsList Bool.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
|[]=>
⊢ (destutterdestutter (xs : List Bool) : List Bool []).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` [].length
All goals completed! 🐙
|trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xsList Bool=>
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
;
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xsList Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xsList Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
;
All goals completed! 🐙
|falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool=>
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
;
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xsList Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xsList Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
;
All goals completed! 🐙
|xBool::xsList Bool=>
xBool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
⊢ (destutterdestutter (xs : List Bool) : List Bool (xBool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (xBool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
xBool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
⊢ (destutterdestutter (xs : List Bool) : List Bool (xBool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (xBool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xsList Bool).length
-- simp made no progress
In the last case of the match, Lean does not know that the previous cases did not apply, so it cannot use destutter.eq_4 and thus cannot simplify the goal.
One way out is to resort to the unfold destutter; split idiom. Functional induction also covers this case. The induction theorem produced by Lean provides the assumptions needed by the equational theorem in the appropriate case:
destutter.inductdestutter.induct (motive : List Bool → Prop) (case1 : motive [])
(case2 : ∀ (xs : List Bool), motive (true :: xs) → motive (true :: true :: xs))
(case3 : ∀ (xs : List Bool), motive (false :: xs) → motive (false :: false :: xs))
(case4 :
∀ (x : Bool) (xs : List Bool),
(∀ (xs_1 : List Bool), x = true → xs = true :: xs_1 → False) →
(∀ (xs_1 : List Bool), x = false → xs = false :: xs_1 → False) → motive xs → motive (x :: xs))
(xs : List Bool) : motive xs(motiveList Bool → Prop:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
→Prop)(case1motive []:motiveList Bool → Prop[])(case2∀ (xs : List Bool), motive (true :: xs) → motive (true :: true :: xs):∀(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),motiveList Bool → Prop(trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xsList Bool)→motiveList Bool → Prop(trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xsList Bool))(case3∀ (xs : List Bool), motive (false :: xs) → motive (false :: false :: xs):∀(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),motiveList Bool → Prop(falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool)→motiveList Bool → Prop(falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xsList Bool))(case4∀ (x : Bool) (xs : List Bool),
(∀ (xs_1 : List Bool), x = true → xs = true :: xs_1 → False) →
(∀ (xs_1 : List Bool), x = false → xs = false :: xs_1 → False) → motive xs → motive (x :: xs):∀(xBool:BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
)(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),(∀(xs_1List Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),xBool=trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. →xsList Bool=trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ::xs_1List Bool→FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
)→(∀(xs_1List Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
),xBool=falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. →xsList Bool=falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. ::xs_1List Bool→FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
)→motiveList Bool → PropxsList Bool→motiveList Bool → Prop(xBool::xsList Bool))(xsList Bool:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
):motiveList Bool → PropxsList Bool
With these assumptions in the context, simp makes progress, and the proof becomes quite short:
theoremlength_destutter2_root_.length_destutter2 {xs : List Bool} : (destutter xs).length ≤ xs.length:(destutterdestutter (xs : List Bool) : List BoolxsList Bool).lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
≤xsList Bool.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
:=
xsList Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
⊢ (destutterdestutter (xs : List Bool) : List BoolxsList Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xsList Bool.length
case1⊢ (destutterdestutter (xs : List Bool) : List Bool []).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` [].length
case2
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case3
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case4
x✝²Bool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool → FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. → xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool → FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
ih1✝(destutter xs✝).length ≤ xs✝.length
:
(destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
⊢ (destutterdestutter (xs : List Bool) : List Bool (x✝²Bool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (x✝²Bool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case1⊢ (destutterdestutter (xs : List Bool) : List Bool []).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` [].length
case2
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case3
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case4
x✝²Bool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool → FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. → xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool → FalseFalse : Prop`False` is the empty proposition. Thus, it has no introduction rules.
It represents a contradiction. `False` elimination rule, `False.rec`,
expresses the fact that anything follows from a contradiction.
This rule is sometimes called ex falso (short for ex falso sequitur quodlibet),
or the principle of explosion.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
ih1✝(destutter xs✝).length ≤ xs✝.length
:
(destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
⊢ (destutterdestutter (xs : List Bool) : List Bool (x✝²Bool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` (x✝²Bool :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool).length
case4
x✝²Bool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
ih1✝(destutter xs✝).length ≤ xs✝.length
:
(destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → ¬Not (a : Prop) : Prop`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`,
so if your goal is `¬p` you can use `intro h` to turn the goal into
`h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False`
and `(hn h).elim` will prove anything.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. → ¬Not (a : Prop) : Prop`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`,
so if your goal is `¬p` you can use `intro h` to turn the goal into
`h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False`
and `(hn h).elim` will prove anything.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool
⊢ (destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
case2
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
⊢ (destutterdestutter (xs : List Bool) : List Bool (trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
case3
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
(destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
⊢ (destutterdestutter (xs : List Bool) : List Bool (falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs✝List Bool)).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1
case4
x✝²Bool
:
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
xs✝List Bool
:
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
ih1✝(destutter xs✝).length ≤ xs✝.length
:
(destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → ¬Not (a : Prop) : Prop`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`,
so if your goal is `¬p` you can use `intro h` to turn the goal into
`h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False`
and `(hn h).elim` will prove anything.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool
∀ (xs_1List Bool : ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
), x✝²Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. → ¬Not (a : Prop) : Prop`Not p`, or `¬p`, is the negation of `p`. It is defined to be `p → False`,
so if your goal is `¬p` you can use `intro h` to turn the goal into
`h : p ⊢ False`, and if you have `hn : ¬p` and `h : p` then `hn h : False`
and `(hn h).elim` will prove anything.
For more information: [Propositional Logic](https://lean-lang.org/theorem_proving_in_lean4/propositions_and_proofs.html#propositional-logic)
xs✝List Bool = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xs_1List Bool
⊢ (destutterdestutter (xs : List Bool) : List Boolxs✝List Bool).length ≤ LE.le.{u} {α : Type u} [self : LE α] : α → α → PropThe less-equal relation: `x ≤ y` xs✝List Bool.length
All goals completed! 🐙
You might be surprised that simp [destutter] suffices here, and we did not have to explicitly write simp [destutter, *] to tell simp that it should also use facts in the local context. This is due to special handling in simp: if rewrite rules have premises that look like like the premise of a function’s equational lemma, simp will consult the local context anyways.
This special handling does not exist in rewrite and rw, so if we use these tactics to unfold the function, we will get new goals with these side conditions, and have to resolve them explicitly, for example using assumption.
Mutual recursion
Proofs about mutually defined functions can become quite tricky: following the rule that the proof should follow the structure of the function, the usual approach is to prove two mutually recursive theorems, replicating the case splits of the original functions by matching or using split. Again, functional induction can help us avoid this repetition.
We will use a minimal example of mutually defined functions, even and odd:
mutualdefeven_root_.even (n : Nat) : Bool:(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
|0=>trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. |nNat+1=>odd_root_.odd (n : Nat) : BoolnNatdefodd_root_.odd (n : Nat) : Bool:(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
|0=>falseBool.false : BoolThe boolean value `false`, not to be confused with the proposition `False`. |nNat+1=>even_root_.even (n : Nat) : BoolnNatend
The functional induction principle generated for one of these functions looks as follows:
even.inducteven.induct (motive1 motive2 : Nat → Prop) (case1 : motive1 0) (case2 : ∀ (n : Nat), motive2 n → motive1 n.succ)
(case3 : motive2 0) (case4 : ∀ (n : Nat), motive1 n → motive2 n.succ) (n : Nat) : motive1 n(motive1Nat → Propmotive2Nat → Prop:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
→Prop)(case1motive1 0:motive1Nat → Prop0)(case2∀ (n : Nat), motive2 n → motive1 n.succ:∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive2Nat → PropnNat→motive1Nat → PropnNat.succNat.succ (n : Nat) : NatThe successor function on natural numbers, `succ n = n + 1`.
This is one of the two constructors of `Nat`. )(case3motive2 0:motive2Nat → Prop0)(case4∀ (n : Nat), motive1 n → motive2 n.succ:∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive1Nat → PropnNat→motive2Nat → PropnNat.succNat.succ (n : Nat) : NatThe successor function on natural numbers, `succ n = n + 1`.
This is one of the two constructors of `Nat`. )(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
):motive1Nat → PropnNat
This theorem now has two motives, one for each function in the recursive group, and four cases, corresponding to the two cases of even and then two cases of odd.
Recall that the role of the motive is to specify the statement that we want to prove about the parameters of the recursive function. If we have two mutually recursive functions, we typically have two such statements in mind that we want to prove together, one for each of the two functions.
One of the main benefits of the induction tactic, over simply trying to apply an induction principle, is that the induction tactic constructs the motive from the current proof goal. If the proof goal is P x y then induction x, y sets (motive := fun x y => P x y). (There is a bit more to it when there are assumptions about x or y in the local context, or the generalizing clause is used.)
So if the induction principle expects multiple motives, only of them can be inferred from the goal, and we will have to instantiate the other motive explicitly. So to prove that all numbers of the form 2n are even, were we want to use that all numbers of the form 2n + 1 are odd, we can write:
theoremevenDouble_root_.evenDouble (n : Nat) : even (2 * n) = true(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
):eveneven (n : Nat) : Bool(2*nNat):=
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝odd (2 * n✝ + 1) = true
:
oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝even (2 * n✝) = true
:
eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝odd (2 * n✝ + 1) = true
:
oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝even (2 * n✝) = true
:
eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
All goals completed! 🐙
We can of course also prove the statement about odd. Now the induction tactic infers motive2 and we have to give motive1 explicitly:
theoremoddDouble_root_.oddDouble (n : Nat) : odd (2 * n + 1) = true(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
):oddodd (n : Nat) : Bool(2*nNat+1):=
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝odd (2 * n✝ + 1) = true
:
oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝even (2 * n✝) = true
:
eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝odd (2 * n✝ + 1) = true
:
oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4
n✝Nat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ih1✝even (2 * n✝) = true
:
eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. n✝Nat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
All goals completed! 🐙
If you compare even.induct and odd.induct, or inspect the proof state after induction, you will notice that we get the same proof obligations in both proofs, and we had to perform the same proofs twice to get both theorems. In the example above, where the proofs are rather trivial, this is not so bad, but in general we would be quite unsatisfied by that repetition. In that case, we can use the mutual functional induction principle:
even.mutual_inducteven.mutual_induct (motive1 motive2 : Nat → Prop) (case1 : motive1 0) (case2 : ∀ (n : Nat), motive2 n → motive1 n.succ)
(case3 : motive2 0) (case4 : ∀ (n : Nat), motive1 n → motive2 n.succ) :
(∀ (n : Nat), motive1 n) ∧ ∀ (n : Nat), motive2 n(motive1Nat → Propmotive2Nat → Prop:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
→Prop)(case1motive1 0:motive1Nat → Prop0)(case2∀ (n : Nat), motive2 n → motive1 n.succ:∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive2Nat → PropnNat→motive1Nat → PropnNat.succNat.succ (n : Nat) : NatThe successor function on natural numbers, `succ n = n + 1`.
This is one of the two constructors of `Nat`. )(case3motive2 0:motive2Nat → Prop0)(case4∀ (n : Nat), motive1 n → motive2 n.succ:∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive1Nat → PropnNat→motive2Nat → PropnNat.succNat.succ (n : Nat) : NatThe successor function on natural numbers, `succ n = n + 1`.
This is one of the two constructors of `Nat`. ):(∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive1Nat → PropnNat)∧(∀(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
),motive2Nat → PropnNat)
This induction principle proves bothmotive1 and motive2. Unfortunately, this kind of induction principle can no longer be used
by induction, so we have to manually prepare the goal to be of the right shape to be able to apply the theorem, essentially constructing both motives manually, and then project out the components:
theoremevenDouble_oddDouble_root_.evenDouble_oddDouble : (∀ (n : Nat), even (2 * n) = true) ∧ ∀ (n : Nat), odd (2 * n + 1) = true:((nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→eveneven (n : Nat) : Bool(2*nNat))∧((nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→oddodd (n : Nat) : Bool(2*nNat+1)):=
⊢ (∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. ) ∧ And (a b : Prop) : Prop`And a b`, or `a ∧ b`, is the conjunction of propositions. It can be
constructed and destructed like a pair: if `ha : a` and `hb : b` then
`⟨ha, hb⟩ : a ∧ b`, and if `h : a ∧ b` then `h.left : a` and `h.right : b`.
∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2⊢ ∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4⊢ ∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case1⊢ eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case2⊢ ∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat.succ) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case3⊢ oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. 0 + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
case4⊢ ∀ (nNat : NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
), eveneven (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. → oddodd (n : Nat) : Bool (2 * HMul.hMul.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HMul α β γ] : α → β → γ`a * b` computes the product of `a` and `b`.
The meaning of this notation is type-dependent. nNat.succ + HAdd.hAdd.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAdd α β γ] : α → β → γ`a + b` computes the sum of `a` and `b`.
The meaning of this notation is type-dependent. 1) = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`.
All goals completed! 🐙
theoremevenDouble2_root_.evenDouble2 (n : Nat) : even (2 * n) = true:(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→eveneven (n : Nat) : Bool(2*nNat):=evenDouble_oddDoubleevenDouble_oddDouble : (∀ (n : Nat), even (2 * n) = true) ∧ ∀ (n : Nat), odd (2 * n + 1) = true.1theoremoddDouble2_root_.oddDouble2 (n : Nat) : odd (2 * n + 1) = true:(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→oddodd (n : Nat) : Bool(2*nNat+1):=evenDouble_oddDoubleevenDouble_oddDouble : (∀ (n : Nat), even (2 * n) = true) ∧ ∀ (n : Nat), odd (2 * n + 1) = true.2
We plan to provide commands to perform such a joint declaration of two theorems with less repetition and more convenience in the future.
Unrelated to mutual induction, but worth pointing out at this point: in this section we were able to solve all subgoals using simp_all [even, odd]. This high level of automation means that our proofs are quite stable under changes to the function definitions, and indeed, if we changed the definition of odd in
mutualdefevenAltEvenOdd.even (n : Nat) : Bool:(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)→BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
|0=>trueBool.true : BoolThe boolean value `true`, not to be confused with the proposition `True`. |nNat+1=>oddAltEvenOdd.odd (n : Nat) : BoolnNatdefoddAltEvenOdd.odd (n : Nat) : Bool(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
):BoolBool : Type`Bool` is the type of boolean values, `true` and `false`. Classically,
this is equivalent to `Prop` (the type of propositions), but the distinction
is important for programming, because values of type `Prop` are erased in the
code generator, while `Bool` corresponds to the type called `bool` or `boolean`
in most programming languages.
:=!(evenAltEvenOdd.even (n : Nat) : BoolnNat)end
all of our proofs still go through, even though the number and shape of the cases changed!
What is a branch?
When Lean defines the functional induction principle, it has to decide what constitutes a branch of the function, and should therefore become a case of the induction principle. This is necessarily a heuristic, and is guided by the notion of tail position: Lean splits match statements and if-then-else statements that occur in the body of the function or in a case of such statements, but not when they appear, for example, in arguments to functions.
As a slight variant of cutN, consider a function that inserts a separator before, between and after groups of n:
Instead of using pattern matching as with cutN, let us try to use if-then-else:
defsepWithsepWith.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List α(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(hpos0 < n:0<nNat)(sepα:αType u_1)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1:=if`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=0then`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
[sepα]else`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
[sepα]++xsList α.takeList.take.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Returns the first `n` elements of `xs`, or the whole list if `n` is too large.
* `take 0 [a, b, c, d, e] = []`
* `take 3 [a, b, c, d, e] = [a, b, c]`
* `take 6 [a, b, c, d, e] = [a, b, c, d, e]`
nNat++sepWithsepWith.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List αnNathpos0 < nsepα(xsList α.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
nNat)termination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
Because we use if-then-else in a tail position the induction principle has two goals, one for each of the two branches:
sepWith.inductsepWith.induct.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (motive : List α → Prop)
(case1 : ∀ (x : List α), x.length = 0 → motive x)
(case2 : ∀ (x : List α), ¬x.length = 0 → motive (List.drop n x) → motive x) (xs : List α) : motive xs{αType u_1:TypeA type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. _}(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(hpos0 < n:0<nNat)(sepα:αType u_1)(motiveList α → Prop:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1→Prop)(case1∀ (x : List α), x.length = 0 → motive x:∀(xList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),xList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=0→motiveList α → PropxList α)(case2∀ (x : List α), ¬x.length = 0 → motive (List.drop n x) → motive x:∀(xList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),¬xList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=0→motiveList α → Prop(List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
nNatxList α)→motiveList α → PropxList α)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):motiveList α → PropxsList α
We could also have defined the function slightly differently, moving the if-then-else into the (sep :: ·), as follows:
defsepWith2sepWith2.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List α(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(hpos0 < n:0<nNat)(sepα:αType u_1)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1:=sepα::(if`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=0then`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
[]else`if c then t else e` is notation for `ite c t e`, "if-then-else", which decides to
return `t` or `e` depending on whether `c` is true or false. The explicit argument
`c : Prop` does not have any actual computational content, but there is an additional
`[Decidable c]` argument synthesized by typeclass inference which actually
determines how to evaluate `c` to true or false. Write `if h : c then t else e`
instead for a "dependent if-then-else" `dite`, which allows `t`/`e` to use the fact
that `c` is true/false.
xsList α.takeList.take.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Returns the first `n` elements of `xs`, or the whole list if `n` is too large.
* `take 0 [a, b, c, d, e] = []`
* `take 3 [a, b, c, d, e] = [a, b, c]`
* `take 6 [a, b, c, d, e] = [a, b, c, d, e]`
nNat++sepWith2sepWith2.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List αnNathpos0 < nsepα(xsList α.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
nNat))termination_bySpecify a termination argument for well-founded termination:
```
termination_by a - b
```
indicates that termination of the currently defined recursive function follows
because the difference between the the arguments `a` and `b`.
If the fuction takes further argument after the colon, you can name them as follows:
```
def example (a : Nat) : Nat → Nat → Nat :=
termination_by b c => a - b
```
If omitted, a termination argument will be inferred. If written as `termination_by?`,
the inferrred termination argument will be suggested.
xsList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
Now we get an induction principle with just one case:
sepWith2.inductsepWith2.induct.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (motive : List α → Prop)
(case1 : ∀ (x : List α), (¬x.length = 0 → motive (List.drop n x)) → motive x) (xs : List α) : motive xs{αType u_1:TypeA type universe. `Type ≡ Type 0`, `Type u ≡ Sort (u + 1)`. _}(nNat:NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
)(hpos0 < n:0<nNat)(sepα:αType u_1)(motiveList α → Prop:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1→Prop)(case1∀ (x : List α), (¬x.length = 0 → motive (List.drop n x)) → motive x:∀(xList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1),(¬xList α.lengthList.length.{u_1} {α : Type u_1} : List α → NatThe length of a list: `[].length = 0` and `(a :: l).length = l.length + 1`.
This function is overridden in the compiler to `lengthTR`, which uses constant
stack space, while leaving this function to use the "naive" recursion which is
easier for reasoning.
=0→motiveList α → Prop(List.dropList.drop.{u} {α : Type u} : Nat → List α → List α`O(min n |xs|)`. Removes the first `n` elements of `xs`.
* `drop 0 [a, b, c, d, e] = [a, b, c, d, e]`
* `drop 3 [a, b, c, d, e] = [d, e]`
* `drop 6 [a, b, c, d, e] = []`
nNatxList α))→motiveList α → PropxList α)(xsList α:ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
αType u_1):motiveList α → PropxsList α
Also note that the induction hypothesis in this case now has a condition ¬x.length = 0. This is to be expected: The corresponding recursive call is in the else-branch, and if x.length = 0 we may not assume that the induction hypothesis holds.
Hurdle: Looping simp
At this point we should discuss an issue with simp and certain recursive function definitions that can lead to simp [foo] looping. This is relevant because unfolding the function’s definition with simp is often the first thing we do in the case of a functional induction.
Consider the following proof that relates sepWith and cutN
theoremsepWith_cutNsepWith_cutN.{u_1} {n : Nat} {hpos : 0 < n} :
∀ {α : Type u_1} {sep : α} {xs : List α},
sepWith n hpos sep xs = (List.map (fun x => sep :: x) (cutN n xs)).join ++ [sep]:sepWithsepWith.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List αnNathpos0 < nsep?m.43519xsList ?m.43519=(cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList ?m.43519|>.mapList.map.{u, v} {α : Type u} {β : Type v} (f : α → β) : List α → List β`O(|l|)`. `map f l` applies `f` to each element of the list.
* `map f [a, b, c] = [f a, f b, f c]`
(sep?m.43519::·)|>.joinList.join.{u} {α : Type u} : List (List α) → List α`O(|join L|)`. `join L` concatenates all the lists in `L` into one list.
* `join [[a], [], [b, c], [d, e, f]] = [a, b, c, d, e, f]`
)++[sep?m.43519]:=
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. When the value is not shared, `Array α` will have better
performance because it can do destructive updates.
α✝Type u_1
⊢ sepWithsepWith.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List αnNathpos0 < nsepα✝xsList α✝ = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(List.mapList.map.{u, v} {α : Type u} {β : Type v} (f : α → β) : List α → List β`O(|l|)`. `map f l` applies `f` to each element of the list.
* `map f [a, b, c] = [f a, f b, f c]`
(funxList α✝ => sepα✝ :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xList α✝) (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNatxsList α✝)).join ++ HAppend.hAppend.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAppend α β γ] : α → β → γ`a ++ b` is the result of concatenation of `a` and `b`, usually read "append".
The meaning of this notation is type-dependent. [sepα✝]
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
⊢ sepWithsepWith.{u_1} {α : Type u_1} (n : Nat) (hpos : 0 < n) (sep : α) (xs : List α) : List αnNathpos0 < nsepα✝ [] = Eq.{u_1} {α : Sort u_1} : α → α → PropThe equality relation. It has one introduction rule, `Eq.refl`.
We use `a = b` as notation for `Eq a b`.
A fundamental property of equality is that it is an equivalence relation.
```
variable (α : Type) (a b c d : α)
variable (hab : a = b) (hcb : c = b) (hcd : c = d)
example : a = d :=
Eq.trans (Eq.trans hab (Eq.symm hcb)) hcd
```
Equality is much more than an equivalence relation, however. It has the important property that every assertion
respects the equivalence, in the sense that we can substitute equal expressions without changing the truth value.
That is, given `h1 : a = b` and `h2 : p a`, we can construct a proof for `p b` using substitution: `Eq.subst h1 h2`.
Example:
```
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
Eq.subst h1 h2
example (α : Type) (a b : α) (p : α → Prop)
(h1 : a = b) (h2 : p a) : p b :=
h1 ▸ h2
```
The triangle in the second presentation is a macro built on top of `Eq.subst` and `Eq.symm`, and you can enter it by typing `\t`.
For more information: [Equality](https://lean-lang.org/theorem_proving_in_lean4/quantifiers_and_equality.html#equality)
(List.mapList.map.{u, v} {α : Type u} {β : Type v} (f : α → β) : List α → List β`O(|l|)`. `map f l` applies `f` to each element of the list.
* `map f [a, b, c] = [f a, f b, f c]`
(funxList α✝ => sepα✝ :: List.cons.{u} {α : Type u} (head : α) (tail : List α) : List αIf `a : α` and `l : List α`, then `cons a l`, or `a :: l`, is the
list whose first element is `a` and with `l` as the rest of the list. xList α✝) (cutNcutN.{u_1} {α : Type u_1} (n : Nat) (xs : List α) : List (List α)nNat [])).join ++ HAppend.hAppend.{u, v, w} {α : Type u} {β : Type v} {γ : outParam (Type w)} [self : HAppend α β γ] : α → β → γ`a ++ b` is the result of concatenation of `a` and `b`, usually read "append".
The meaning of this notation is type-dependent. [sepα✝]
case2
nNat
:
NatNat : TypeThe type of natural numbers, starting at zero. It is defined as an
inductive type freely generated by "zero is a natural number" and
"the successor of a natural number is a natural number".
You can prove a theorem `P n` about `n : Nat` by `induction n`, which will
expect a proof of the theorem for `P 0`, and a proof of `P (succ i)` assuming
a proof of `P i`. The same method also works to define functions by recursion
on natural numbers: induction and recursion are two expressions of the same
operation from Lean's point of view.
```
open Nat
example (n : Nat) : n < succ n := by
induction n with
| zero =>
show 0 < 1
decide
| succ i ih => -- ih : i < succ i
show succ i < succ (succ i)
exact Nat.succ_lt_succ ih
```
This type is special-cased by both the kernel and the compiler:
* The type of expressions contains "`Nat` literals" as a primitive constructor,
and the kernel knows how to reduce zero/succ expressions to nat literals.
* If implemented naively, this type would represent a numeral `n` in unary as a
linked list with `n` links, which is horribly inefficient. Instead, the
runtime itself has a special representation for `Nat` which stores numbers up
to 2^63 directly and larger numbers use an arbitrary precision "bignum"
library (usually [GMP](https://gmplib.org/)).
ListList.{u} (α : Type u) : Type u`List α` is the type of ordered lists with elements of type `α`.
It is implemented as a linked list.
`List α` is isomorphic to `Array α`, but they are useful for different things:
* `List α` is easier for reasoning, and
`Array α` is modeled as a wrapper around `List α`
* `List α` works well as a persistent data structure, when many copies of the
tail are shared. Wh