axiom コマンドと CoeFun の使用例：(SKI コンビネータ計算)

[Seasawher]

inductive Lam : Type where
  | S : Lam
  | K : Lam
  | app : Lam → Lam → Lam

instance : CoeFun Lam (fun _ => Lam → Lam) where
  coe lambda := Lam.app lambda

namespace Lam

axiom K_def {a b : Lam} : K a b = a
axiom S_def {a b c : Lam} : S a b c = (a c) (b c)

theorem eta_simp {a b : Lam} : (∀(c : Lam), a c = b c) → a = b := by
  intro h
  specialize h S
  injection h

def I := S K K

theorem I_def {a : Lam} : I a = a := by rw [I, S_def, K_def]

example {a : Lam} : S K a = I := by
  apply eta_simp
  intro c
  rw [I_def, S_def, K_def]

end Lam

[Seasawher]

公理が実は間違っているという例でもある

ありがちなミスなので、言及の価値がある

example : False := by
  refine absurd @K_def ?_
  intro h
  cases @h S S