54 lines
1.8 KiB
Text
54 lines
1.8 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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import logic.connectives logic.quantifiers logic.cast algebra.relation
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import logic.axioms.em
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open eq.ops
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theorem prop_complete (a : Prop) : a = true ∨ a = false :=
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or.elim (em a)
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(λ t, or.inl (propext (iff.intro (λ h, trivial) (λ h, t))))
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(λ f, or.inr (propext (iff.intro (λ h, absurd h f) (λ h, false.elim h))))
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definition eq_true_or_eq_false := prop_complete
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theorem cases_true_false (P : Prop → Prop) (H1 : P true) (H2 : P false) (a : Prop) : P a :=
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or.elim (prop_complete a)
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(assume Ht : a = true, Ht⁻¹ ▸ H1)
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(assume Hf : a = false, Hf⁻¹ ▸ H2)
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theorem cases_on (a : Prop) {P : Prop → Prop} (H1 : P true) (H2 : P false) : P a :=
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cases_true_false P H1 H2 a
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-- this supercedes by_cases in decidable
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definition by_cases {p q : Prop} (Hpq : p → q) (Hnpq : ¬p → q) : q :=
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or.elim (em p) (assume Hp, Hpq Hp) (assume Hnp, Hnpq Hnp)
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-- this supercedes by_contradiction in decidable
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theorem by_contradiction {p : Prop} (H : ¬p → false) : p :=
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by_cases
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(assume H1 : p, H1)
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(assume H1 : ¬p, false.rec _ (H H1))
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theorem eq_false_or_eq_true (a : Prop) : a = false ∨ a = true :=
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cases_true_false (λ x, x = false ∨ x = true)
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(or.inr rfl)
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(or.inl rfl)
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a
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theorem eq.of_iff {a b : Prop} (H : a ↔ b) : a = b :=
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iff.elim (assume H1 H2, propext (iff.intro H1 H2)) H
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theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) :=
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propext (iff.intro
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(assume H, eq.of_iff H)
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(assume H, iff.of_eq H))
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open relation
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theorem iff_congruence [instance] (P : Prop → Prop) : is_congruence iff iff P :=
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is_congruence.mk
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(take (a b : Prop),
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assume H : a ↔ b,
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show P a ↔ P b, from iff.of_eq (eq.of_iff H ▸ eq.refl (P a)))
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