60 lines
1.8 KiB
Text
60 lines
1.8 KiB
Text
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/-
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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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Module: logic.axioms.classical
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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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open eq.ops
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axiom prop_complete (a : Prop) : a = true ∨ a = false
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definition eq_true_or_eq_false := prop_complete
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theorem cases (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 P H1 H2 a
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-- this supercedes the em in decidable
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theorem em (a : Prop) : a ∨ ¬a :=
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or.elim (prop_complete a)
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(assume Ht : a = true, or.inl (of_eq_true Ht))
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(assume Hf : a = false, or.inr (not_of_eq_false Hf))
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theorem eq_false_or_eq_true (a : Prop) : a = false ∨ a = true :=
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cases (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 propext {a b : Prop} (Hab : a → b) (Hba : b → a) : a = b :=
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or.elim (prop_complete a)
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(assume Hat, or.elim (prop_complete b)
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(assume Hbt, Hat ⬝ Hbt⁻¹)
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(assume Hbf, false.elim (Hbf ▸ (Hab (of_eq_true Hat)))))
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(assume Haf, or.elim (prop_complete b)
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(assume Hbt, false.elim (Haf ▸ (Hba (of_eq_true Hbt))))
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(assume Hbf, Haf ⬝ Hbf⁻¹))
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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 H1 H2) H
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theorem iff_eq_eq {a b : Prop} : (a ↔ b) = (a = b) :=
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propext
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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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