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.
34 lines
913 B
Text
34 lines
913 B
Text
namespace MWE
|
|
|
|
inductive Id {A : Type u} : A → A → Type u
|
|
| refl {a : A} : Id a a
|
|
|
|
attribute [induction_eliminator] Id.casesOn
|
|
|
|
infix:50 (priority := high) " = " => Id
|
|
|
|
def symm {A : Type u} {a b : A} (p : a = b) : b = a :=
|
|
by { induction p; exact Id.refl }
|
|
|
|
def map {A : Type u} {B : Type v} {a b : A} (f : A → B) (p : a = b) : f a = f b :=
|
|
by { induction p; apply Id.refl }
|
|
|
|
def transport {A : Type u} (B : A → Type v) {a b : A} (p : a = b) : B a → B b :=
|
|
by { induction p; exact id }
|
|
|
|
def boolToUniverse : Bool → Type
|
|
| true => Unit
|
|
| false => Empty
|
|
|
|
def ffNeqTt : false = true → Empty :=
|
|
λ p => transport boolToUniverse (symm p) ()
|
|
|
|
def isZero : Nat → Bool
|
|
| Nat.zero => true
|
|
| Nat.succ _ => false
|
|
|
|
set_option pp.funBinderTypes true
|
|
set_option pp.letVarTypes true
|
|
set_option trace.Compiler.result true
|
|
def succNeqZero (n : Nat) : Nat.succ n = 0 → Empty :=
|
|
λ h => ffNeqTt (map isZero h)
|