8b0dd2e835
This will collect definitions from Std.Logic --------- Co-authored-by: David Thrane Christiansen <david@davidchristiansen.dk> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>
44 lines
1.7 KiB
Text
44 lines
1.7 KiB
Text
@[inline] def decidable_of_iff'' {b : Prop} (a : Prop) (h : a ↔ b) [Decidable a] : Decidable b :=
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decidable_of_decidable_of_iff h
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theorem LE.le.lt_or_eq_dec {a b : Nat} (hab : a ≤ b) : a < b ∨ a = b :=
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by cases hab
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· exact .inr rfl
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· refine .inl ?_ ; refine Nat.succ_le_succ ?_ ; assumption
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section succeeds_using_match
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local instance decidableBallLT :
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∀ (n : Nat) (P : ∀ k, k < n → Prop) [∀ n h, Decidable (P n h)], Decidable (∀ n h, P n h)
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| 0, _, _ => isTrue fun _ => (by cases ·)
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| (n+1), P, H =>
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match decidableBallLT n (P · <| Nat.le_succ_of_le ·) with
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| isFalse h => isFalse (h fun _ _ => · _ _)
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| isTrue h =>
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match H n Nat.le.refl with
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| isFalse p => isFalse (p <| · _ _)
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| isTrue p => isTrue fun _ h' => (Nat.le_of_lt_succ h').lt_or_eq_dec.elim (h _) (· ▸ p)
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set_option maxHeartbeats 5000
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example : ∀ a, a < 9 → ∀ b, b < 9 → ∀ c, c < 9 → a ^ 2 + b ^ 2 + c ^ 2 ≠ 7 := by decide
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end succeeds_using_match
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section fails_with_timeout
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-- we change `match decidableBallLT` to `by cases decidableBallLT' ... exact`:
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local instance decidableBallLT' :
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∀ (n : Nat) (P : ∀ k, k < n → Prop) [∀ n h, Decidable (P n h)], Decidable (∀ n h, P n h)
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| 0, _, _ => isTrue fun _ => (by cases ·)
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| (n+1), P, H => by
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cases decidableBallLT' n (P · <| Nat.le_succ_of_le ·) with
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| isFalse h => exact isFalse (h fun _ _ => · _ _)
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| isTrue h =>
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exact match H n Nat.le.refl with
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| isFalse p => isFalse (p <| · _ _)
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| isTrue p => isTrue fun _ h' => (Nat.le_of_lt_succ h').lt_or_eq_dec.elim (h _) (· ▸ p)
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set_option maxHeartbeats 5000
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example : ∀ a, a < 9 → ∀ b, b < 9 → ∀ c, c < 9 → a ^ 2 + b ^ 2 + c ^ 2 ≠ 7 := by decide
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end fails_with_timeout
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