25 lines
856 B
Text
25 lines
856 B
Text
----------------------------------------------------------------------------------------------------
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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
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-- Released under Apache 2.0 license as described in the file LICENSE.
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-- Author: Floris van Doorn
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----------------------------------------------------------------------------------------------------
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import logic
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open tactic
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inductive nat : Type :=
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zero : nat,
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succ : nat → nat
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notation `ℕ`:max := nat
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namespace nat
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definition plus (x y : ℕ) : ℕ
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:= nat.rec x (λ n r, succ r) y
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definition to_nat [coercion] (n : num) : ℕ
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:= num.rec zero (λ n, pos_num.rec (succ zero) (λ n r, plus r (plus r (succ zero))) (λ n r, plus r r) n) n
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print "=================="
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theorem nat_rec_zero {P : ℕ → Type} (x : P 0) (f : ∀m, P m → P (succ m)) : nat.rec x f 0 = x :=
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eq.refl _
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end nat
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