45fccc5906
Replaces `@[eliminator]` with two attributes `@[induction_eliminator]` and `@[cases_eliminator]` for defining custom eliminators for the `induction` and `cases` tactics, respectively. Adds `Nat.recAux` and `Nat.casesAuxOn`, which are eliminators that are defeq to `Nat.rec` and `Nat.casesOn`, but these use `0` and `n + 1` rather than `Nat.zero` and `Nat.succ n`. For example, using `induction` to prove that the factorial function is positive now has the following goal states (thanks also to #3616 for the goal state after unfolding). ```lean example : 0 < fact x := by induction x with | zero => decide | succ x ih => /- x : Nat ih : 0 < fact x ⊢ 0 < fact (x + 1) -/ unfold fact /- ... ⊢ 0 < (x + 1) * fact x -/ simpa using ih ``` Thanks to @adamtopaz for initial work on splitting the `@[eliminator]` attribute.
39 lines
1 KiB
Text
39 lines
1 KiB
Text
namespace MWE
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universe u v w
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inductive Id {A : Type u} : A → A → Type u
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| refl {a : A} : Id a a
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attribute [induction_eliminator] Id.casesOn
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infix:50 (priority := high) " = " => Id
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inductive Unit : Type u
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| star : Unit
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attribute [induction_eliminator] Unit.casesOn
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notation "𝟏" => Unit
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notation "★" => Unit.star
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notation "ℕ" => Nat
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def vect (A : Type u) : ℕ → Type u
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| Nat.zero => 𝟏
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| Nat.succ n => A × vect A n
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def vect.const {A : Type u} (a : A) : ∀ n, vect A n
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| Nat.zero => ★
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| Nat.succ n => (a, const a n)
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def vect.map {A : Type u} {B : Type v} (f : A → B) :
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∀ {n : ℕ}, vect A n → vect B n
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| Nat.zero => λ _ => ★
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| Nat.succ n => λ v => (f v.1, map f v.2)
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def transport {A : Type u} (B : A → Type v) {a b : A} (p : a = b) : B a → B b :=
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by { induction p; apply id }
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def vect.subst {A B : Type u} (p : A = B) (f : B → A) {n : ℕ} (v : vect A n) :
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vect.map f (transport (vect · n) p v) = vect.map (f ∘ transport id p) v :=
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by { induction p; apply Id.refl }
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