36 lines
1,020 B
Text
36 lines
1,020 B
Text
import standard
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using num (num pos_num num_rec pos_num_rec)
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using 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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definition add [inline] (a b : nat) : nat
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:= nat_rec a (λ n r, succ r) b
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infixl `+`:65 := add
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definition one [inline] := succ zero
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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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definition pos_num_to_nat [inline] (n : pos_num) : nat
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:= pos_num_rec one (λ n r, r + r) (λ n r, r + r + one) n
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definition num_to_nat [inline] (n : num) : nat
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:= num_rec zero (λ n, pos_num_to_nat n) n
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coercion num_to_nat
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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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definition assump : tactic := eassumption
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theorem T1 {p : nat → Prop} {a : nat } (H : p (a+2)) : ∃ x, p (succ x)
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:= by apply exists_intro; assump
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definition is_zero (n : nat)
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:= nat_rec true (λ n r, false) n
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theorem T2 : ∃ a, (is_zero a) = true
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:= by apply exists_intro; apply refl
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