2014-08-25 02:58:48 +00:00
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import logic
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2014-11-23 01:34:05 +00:00
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namespace experiment
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2014-07-04 17:10:05 +00:00
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inductive nat : Type :=
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2015-02-26 01:00:10 +00:00
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| zero : nat
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| succ : nat → nat
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2014-07-04 17:10:05 +00:00
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2014-09-04 23:36:06 +00:00
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namespace nat
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2014-09-17 21:39:05 +00:00
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definition add (a b : nat) : nat
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2014-09-04 22:03:59 +00:00
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:= nat.rec a (λ n r, succ r) b
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2014-10-21 22:27:45 +00:00
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infixl `+` := add
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2014-07-04 17:10:05 +00:00
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2014-09-17 21:39:05 +00:00
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definition one := succ zero
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2014-07-04 17:10:05 +00:00
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-- Define coercion from num -> nat
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-- By default the parser converts numerals into a binary representation num
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2014-09-17 21:39:05 +00:00
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definition pos_num_to_nat (n : pos_num) : nat
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2014-09-04 22:03:59 +00:00
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:= pos_num.rec one (λ n r, r + r) (λ n r, r + r + one) n
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2014-09-17 21:39:05 +00:00
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definition num_to_nat (n : num) : nat
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2014-09-04 22:03:59 +00:00
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:= num.rec zero (λ n, pos_num_to_nat n) n
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2015-01-25 04:23:21 +00:00
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attribute num_to_nat [coercion]
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2014-07-04 17:10:05 +00:00
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-- Now we can write 2 + 3, the coercion will be applied
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check 2 + 3
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-- Define an assump as an alias for the eassumption tactic
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2014-10-23 17:24:03 +00:00
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definition assump : tactic := tactic.eassumption
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2014-07-04 17:10:05 +00:00
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2014-07-22 16:43:18 +00:00
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theorem T1 {p : nat → Prop} {a : nat } (H : p (a+2)) : ∃ x, p (succ x)
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2014-12-16 03:05:03 +00:00
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:= by apply exists.intro; assump
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2014-07-04 17:32:01 +00:00
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2014-07-05 07:43:10 +00:00
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definition is_zero (n : nat)
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2014-09-04 22:03:59 +00:00
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:= nat.rec true (λ n r, false) n
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2014-07-04 17:32:01 +00:00
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2014-07-05 22:52:40 +00:00
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theorem T2 : ∃ a, (is_zero a) = true
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2014-12-16 03:05:03 +00:00
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:= by apply exists.intro; apply eq.refl
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2014-09-04 23:36:06 +00:00
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end nat
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2014-11-23 01:34:05 +00:00
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end experiment
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