lean2/tests/lean/run/nat_bug.lean
Leonardo de Moura 68d9bef860 refactor(library): add 'eq' and 'ne' namespaces
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2014-09-04 18:41:06 -07:00

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import logic
open decidable
open eq
inductive nat : Type :=
zero : nat,
succ : nat → nat
abbreviation refl := @eq.refl
namespace nat
theorem induction_on {P : nat → Prop} (a : nat) (H1 : P zero) (H2 : ∀ (n : nat) (IH : P n), P (succ n)) : P a
:= nat.rec H1 H2 a
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)
:= 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)
:= @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