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.
27 lines
771 B
Text
27 lines
771 B
Text
namespace MWE
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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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def symm {A : Type u} {a b : A} (p : a = b) : b = a :=
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by { induction p; exact Id.refl }
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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; exact id }
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def transportconst {A B : Type u} : A = B → A → B :=
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transport id
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def transportconstInv {A B : Type u} (e : A = B) : B → A :=
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transportconst (symm e)
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def transportconstOverInv {A B : Type u} (p : A = B) :
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∀ x, transportconst (symm p) x = transportconstInv p x :=
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by { intro x; apply Id.refl }
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def transportconstInv' {A B : Type u} : A = B → B → A :=
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transportconst ∘ symm
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