33 lines
917 B
Text
33 lines
917 B
Text
import data.nat.basic data.bool
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open bool nat eq.ops
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reducible nat.rec_on
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definition is_eq (a b : nat) : bool :=
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nat.rec_on a
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(λ b, nat.cases_on b tt (λb₁, ff))
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(λ a₁ r₁ b, nat.cases_on b ff (λb₁, r₁ b₁))
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b
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example : is_eq 3 3 = tt :=
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rfl
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example : is_eq 3 5 = ff :=
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rfl
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theorem eq.to_is_eq (a b : nat) (H : a = b) : is_eq a b = tt :=
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have aux : is_eq a a = tt, from
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nat.induction_on a
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rfl
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(λ (a₁ : nat) (ih : is_eq a₁ a₁ = tt), ih),
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H ▸ aux
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theorem is_eq.to_eq (a b : nat) : is_eq a b = tt → a = b :=
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nat.induction_on a
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(λb, nat.cases_on b (λh, rfl) (λb₁ H, absurd H !ff_ne_tt))
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(λa₁ (ih : ∀b, is_eq a₁ b = tt → a₁ = b) (b : nat),
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nat.cases_on b
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(λ (H : is_eq (succ a₁) zero = tt), absurd H !ff_ne_tt)
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(λb₁ (H : is_eq (succ a₁) (succ b₁) = tt),
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have aux : a₁ = b₁, from ih b₁ H,
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aux ▸ rfl))
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b
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