/- Copyright (c) 2014 Microsoft Corporation. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Jeremy Avigad, Leonardo de Moura Useful logical identities. Since we are not using propositional extensionality, some of the calculations use the type class support provided by logic.instances. -/ import logic.connectives logic.quantifiers logic.cast open decidable theorem or.right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b := calc (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or.assoc ... ↔ a ∨ (c ∨ b) : {or.comm} ... ↔ (a ∨ c) ∨ b : iff.symm or.assoc theorem and.right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b := calc (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and.assoc ... ↔ a ∧ (c ∧ b) : {and.comm} ... ↔ (a ∧ c) ∧ b : iff.symm and.assoc theorem or_not_self_iff {a : Prop} [D : decidable a] : a ∨ ¬ a ↔ true := iff.intro (assume H, trivial) (assume H, em a) theorem not_or_self_iff {a : Prop} [D : decidable a] : ¬ a ∨ a ↔ true := iff.intro (λ H, trivial) (λ H, or.swap (em a)) theorem and_not_self_iff {a : Prop} : a ∧ ¬ a ↔ false := iff.intro (assume H, (and.right H) (and.left H)) (assume H, false.elim H) theorem not_and_self_iff {a : Prop} : ¬ a ∧ a ↔ false := iff.intro (λ H, and.elim H (by contradiction)) (λ H, false.elim H) theorem not_not_iff {a : Prop} [D : decidable a] : (¬¬a) ↔ a := iff.intro by_contradiction not_not_intro theorem not_not_elim {a : Prop} [D : decidable a] : ¬¬a → a := by_contradiction theorem not_or_iff_not_and_not {a b : Prop} : ¬(a ∨ b) ↔ ¬a ∧ ¬b := or.imp_distrib theorem not_and_iff_not_or_not {a b : Prop} [Da : decidable a] : ¬(a ∧ b) ↔ ¬a ∨ ¬b := iff.intro (λH, by_cases (λa, or.inr (not.mto (and.intro a) H)) or.inl) (or.rec (not.mto and.left) (not.mto and.right)) theorem or_iff_not_and_not {a b : Prop} [Da : decidable a] [Db : decidable b] : a ∨ b ↔ ¬ (¬a ∧ ¬b) := by rewrite [-not_or_iff_not_and_not, not_not_iff] theorem and_iff_not_or_not {a b : Prop} [Da : decidable a] [Db : decidable b] : a ∧ b ↔ ¬ (¬ a ∨ ¬ b) := by rewrite [-not_and_iff_not_or_not, not_not_iff] theorem imp_iff_not_or {a b : Prop} [Da : decidable a] : (a → b) ↔ ¬a ∨ b := iff.intro (by_cases (λHa H, or.inr (H Ha)) (λHa H, or.inl Ha)) (or.rec not.elim imp.intro) theorem not_implies_iff_and_not {a b : Prop} [Da : decidable a] : ¬(a → b) ↔ a ∧ ¬b := calc ¬(a → b) ↔ ¬(¬a ∨ b) : {imp_iff_not_or} ... ↔ ¬¬a ∧ ¬b : not_or_iff_not_and_not ... ↔ a ∧ ¬b : {not_not_iff} theorem peirce {a b : Prop} [D : decidable a] : ((a → b) → a) → a := by_cases imp.intro (imp.syl imp.mp not.elim) theorem forall_not_of_not_exists {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)] (H : ¬∃x, p x) : ∀x, ¬p x := take x, by_cases (assume Hp : p x, absurd (exists.intro x Hp) H) imp.id theorem forall_of_not_exists_not {A : Type} {p : A → Prop} [D : decidable_pred p] : ¬(∃ x, ¬p x) → ∀ x, p x := imp.syl (forall_imp_forall (λa, not_not_elim)) forall_not_of_not_exists theorem exists_not_of_not_forall {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)] [D' : decidable (∃x, ¬p x)] (H : ¬∀x, p x) : ∃x, ¬p x := by_contradiction (λH1, absurd (λx, not_not_elim (forall_not_of_not_exists H1 x)) H) theorem exists_of_not_forall_not {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)] [D' : decidable (∃x, p x)] (H : ¬∀x, ¬ p x) : ∃x, p x := by_contradiction (imp.syl H forall_not_of_not_exists)