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.
13 lines
504 B
Text
13 lines
504 B
Text
inductive Equality {α : Type u} : α → α → Type u
|
||
| refl {a : α} : Equality a a
|
||
|
||
open Equality
|
||
|
||
@[induction_eliminator]
|
||
def ind {α : Type u} (motive : ∀ (a b : α) (p : Equality a b), Sort v)
|
||
{a : α} (πrefl : motive a a refl) {b : α} (p : Equality a b) : motive a b p :=
|
||
@Equality.casesOn α a (λ b p => motive a a refl → motive a b p) b p
|
||
(λ (ε : motive a a refl) => ε) πrefl
|
||
|
||
def symm {α : Type u} {a b : α} (p : Equality a b) : Equality b a :=
|
||
by { induction p; apply refl }
|