86 lines
2.3 KiB
Text
86 lines
2.3 KiB
Text
import standard
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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 (x y : nat)
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:= nat_rec x (λ n r, succ r) y
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infixl `+`:65 := add
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theorem add_zero_left (x : nat) : x + zero = x
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:= refl _
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theorem add_succ_left (x y : nat) : x + (succ y) = succ (x + y)
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:= refl _
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definition is_zero (x : nat)
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:= nat_rec true (λ n r, false) x
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theorem is_zero_zero : is_zero zero
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:= eqt_elim (refl _)
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theorem not_is_zero_succ (x : nat) : ¬ is_zero (succ x)
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:= eqf_elim (refl _)
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theorem dichotomy (m : nat) : m = zero ∨ (∃ n, m = succ n)
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:= nat_rec
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(or_intro_left _ (refl zero))
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(λ m H, or_intro_right _ (exists_intro m (refl (succ m))))
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m
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theorem is_zero_to_eq (x : nat) (H : is_zero x) : x = zero
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:= or_elim (dichotomy x)
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(assume Hz : x = zero, Hz)
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(assume Hs : (∃ n, x = succ n),
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exists_elim Hs (λ (w : nat) (Hw : x = succ w),
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absurd_elim _ H (subst (symm Hw) (not_is_zero_succ w))))
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theorem is_not_zero_to_eq {x : nat} (H : ¬ is_zero x) : ∃ n, x = succ n
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:= or_elim (dichotomy x)
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(assume Hz : x = zero,
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absurd_elim _ (subst (symm Hz) is_zero_zero) H)
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(assume Hs, Hs)
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theorem not_zero_add (x y : nat) (H : ¬ is_zero y) : ¬ is_zero (x + y)
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:= exists_elim (is_not_zero_to_eq H)
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(λ (w : nat) (Hw : y = succ w),
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have H1 : x + y = succ (x + w), from
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calc x + y = x + succ w : {Hw}
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... = succ (x + w) : refl _,
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have H2 : ¬ is_zero (succ (x + w)), from
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not_is_zero_succ (x+w),
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subst (symm H1) H2)
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inductive not_zero (x : nat) : Bool :=
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| not_zero_intro : ¬ is_zero x → not_zero x
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theorem not_zero_not_is_zero {x : nat} (H : not_zero x) : ¬ is_zero x
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:= not_zero_rec (λ H1, H1) H
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theorem not_zero_add_right [instance] (x y : nat) (H : not_zero y) : not_zero (x + y)
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:= not_zero_intro (not_zero_add x y (not_zero_not_is_zero H))
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theorem not_zero_succ [instance] (x : nat) : not_zero (succ x)
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:= not_zero_intro (not_is_zero_succ x)
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variable div : Π (x y : nat) {H : not_zero y}, nat
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variables a b : nat
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opaque_hint (hiding [module])
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check div a (succ b)
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check (λ H : not_zero b, div a b)
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check (succ zero)
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check a + (succ zero)
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check div a (a + (succ b))
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opaque_hint (exposing [module])
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check div a (a + (succ b))
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opaque_hint (hiding add)
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check div a (a + (succ b))
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opaque_hint (exposing add)
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check div a (a + (succ b))
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