2014-08-01 16:37:23 +00:00
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import logic
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2014-07-25 06:43:40 +00:00
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using decidable
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inductive nat : Type :=
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| zero : nat
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| succ : nat → nat
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theorem induction_on {P : nat → Prop} (a : nat) (H1 : P zero) (H2 : ∀ (n : nat) (IH : P n), P (succ n)) : P a
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:= nat_rec H1 H2 a
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definition pred (n : nat) := nat_rec zero (fun m x, m) n
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theorem pred_zero : pred zero = zero := refl _
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theorem pred_succ (n : nat) : pred (succ n) = n := refl _
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theorem zero_or_succ (n : nat) : n = zero ∨ n = succ (pred n)
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:= induction_on n
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(or_intro_left _ (refl zero))
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(take m IH, or_intro_right _
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2014-08-20 22:49:44 +00:00
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(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
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2014-07-25 06:43:40 +00:00
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theorem zero_or_succ2 (n : nat) : n = zero ∨ n = succ (pred n)
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:= @induction_on _ n
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(or_intro_left _ (refl zero))
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(take m IH, or_intro_right _
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2014-08-20 22:49:44 +00:00
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(show succ m = succ (pred (succ m)), from congr_arg succ (symm (pred_succ m))))
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