lean2/tests/lean/run/nat_bug.lean
2015-02-11 14:09:25 -08:00

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import logic
open decidable
open eq
namespace experiment
inductive nat : Type :=
zero : nat,
succ : nat → nat
definition refl := @eq.refl
namespace nat
definition pred (n : nat) := nat.rec zero (fun m x, m) n
theorem pred_zero : pred zero = zero := refl _
theorem pred_succ (n : nat) : pred (succ n) = n := refl _
theorem zero_or_succ (n : nat) : n = zero n = succ (pred n)
:= nat.induction_on n
(or.intro_left _ (refl zero))
(take m IH, or.intro_right _
(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
theorem zero_or_succ2 (n : nat) : n = zero n = succ (pred n)
:= @nat.induction_on _ n
(or.intro_left _ (refl zero))
(take m IH, or.intro_right _
(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
end nat
end experiment