2014-08-25 02:58:48 +00:00
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import logic
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2014-10-02 00:51:17 +00:00
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open eq.ops
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2014-07-25 23:00:38 +00:00
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inductive nat : Type :=
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2014-08-22 22:46:10 +00:00
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zero : nat,
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succ : nat → nat
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2014-09-04 23:36:06 +00:00
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namespace nat
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2014-09-04 22:03:59 +00:00
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definition add (x y : nat) : nat := nat.rec x (λn r, succ r) y
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2014-10-21 22:27:45 +00:00
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infixl `+` := add
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2014-09-04 22:03:59 +00:00
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definition mul (n m : nat) := nat.rec zero (fun m x, x + n) m
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2014-10-21 22:27:45 +00:00
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infixl `*` := mul
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2014-07-25 23:00:38 +00:00
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axiom mul_succ_right (n m : nat) : n * succ m = n * m + n
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2014-09-05 01:41:06 +00:00
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open eq
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2014-07-25 23:00:38 +00:00
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theorem small2 (n m l : nat) : n * succ l + m = n * l + n + m
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2014-09-04 23:36:06 +00:00
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:= subst (mul_succ_right _ _) (eq.refl (n * succ l + m))
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end nat
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