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