149 lines
5.2 KiB
Text
149 lines
5.2 KiB
Text
-- 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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-- logic.identities
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-- ============================
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-- Useful logical identities. In the absence of propositional extensionality, some of the
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-- calculations use the type class support provided by logic.instances
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import logic.instances logic.quantifiers logic.cast
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open relation decidable relation.iff_ops
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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 or_left_comm (a b c : Prop) : a ∨ (b ∨ c)↔ b ∨ (a ∨ c) :=
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calc
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a ∨ (b ∨ c) ↔ (a ∨ b) ∨ c : iff.symm or.assoc
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... ↔ (b ∨ a) ∨ c : {or.comm}
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... ↔ b ∨ (a ∨ c) : 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 and_left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
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calc
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a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff.symm and.assoc
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... ↔ (b ∧ a) ∧ c : {and.comm}
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... ↔ b ∧ (a ∧ c) : and.assoc
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theorem not_not_iff {a : Prop} [D : decidable a] : (¬¬a) ↔ a :=
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iff.intro
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(assume H : ¬¬a,
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by_cases (assume H' : a, H') (assume H' : ¬a, absurd H' H))
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(assume H : a, assume H', H' H)
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theorem not_not_elim {a : Prop} [D : decidable a] (H : ¬¬a) : a :=
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iff.mp not_not_iff H
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theorem not_true_iff_false : (¬true) ↔ false :=
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iff.intro (assume H, H trivial) false_elim
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theorem not_false_iff_true : (¬false) ↔ true :=
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iff.intro (assume H, trivial) (assume H H', H')
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theorem not_or {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a ∨ b)) ↔ (¬a ∧ ¬b) :=
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iff.intro
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(assume H, or.elim (em a)
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(assume Ha, absurd (or.inl Ha) H)
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(assume Hna, or.elim (em b)
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(assume Hb, absurd (or.inr Hb) H)
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(assume Hnb, and.intro Hna Hnb)))
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(assume (H : ¬a ∧ ¬b) (N : a ∨ b),
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or.elim N
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(assume Ha, absurd Ha (and.elim_left H))
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(assume Hb, absurd Hb (and.elim_right H)))
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theorem not_and {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a ∧ b)) ↔ (¬a ∨ ¬b) :=
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iff.intro
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(assume H, or.elim (em a)
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(assume Ha, or.elim (em b)
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(assume Hb, absurd (and.intro Ha Hb) H)
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(assume Hnb, or.inr Hnb))
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(assume Hna, or.inl Hna))
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(assume (H : ¬a ∨ ¬b) (N : a ∧ b),
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or.elim H
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(assume Hna, absurd (and.elim_left N) Hna)
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(assume Hnb, absurd (and.elim_right N) Hnb))
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theorem imp_or {a b : Prop} [Da : decidable a] : (a → b) ↔ (¬a ∨ b) :=
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iff.intro
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(assume H : a → b, (or.elim (em a)
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(assume Ha : a, or.inr (H Ha))
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(assume Hna : ¬a, or.inl Hna)))
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(assume (H : ¬a ∨ b) (Ha : a),
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or.resolve_right H (not_not_iff⁻¹ ▸ Ha))
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theorem not_implies {a b : Prop} [Da : decidable a] [Db : decidable b] : (¬(a → b)) ↔ (a ∧ ¬b) :=
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calc (¬(a → b)) ↔ (¬(¬a ∨ b)) : {imp_or}
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... ↔ (¬¬a ∧ ¬b) : not_or
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... ↔ (a ∧ ¬b) : {not_not_iff}
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theorem peirce {a b : Prop} [D : decidable a] : ((a → b) → a) → a :=
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assume H, by_contradiction (assume Hna : ¬a,
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have Hnna : ¬¬a, from not_implies_left (mt H Hna),
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absurd (not_not_elim Hnna) Hna)
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theorem not_exists_forall {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, or.elim (em (P x))
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(assume Hp : P x, absurd (exists_intro x Hp) H)
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(assume Hn : ¬P x, Hn)
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theorem not_forall_exists {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 _ D' (assume H1 : ¬∃x, ¬P x,
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have H2 : ∀x, ¬¬P x, from @not_exists_forall _ _ (take x, not.decidable (D x)) H1,
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have H3 : ∀x, P x, from take x, @not_not_elim _ (D x) (H2 x),
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absurd H3 H)
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theorem iff_true_intro {a : Prop} (H : a) : a ↔ true :=
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iff.intro
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(assume H1 : a, trivial)
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(assume H2 : true, H)
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theorem iff_false_intro {a : Prop} (H : ¬a) : a ↔ false :=
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iff.intro
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(assume H1 : a, absurd H1 H)
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(assume H2 : false, false_elim H2)
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theorem a_neq_a {A : Type} (a : A) : (a ≠ a) ↔ false :=
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iff.intro
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(assume H, false.of_ne H)
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(assume H, false_elim H)
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theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
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iff_true_intro rfl
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theorem heq_id {A : Type} (a : A) : (a == a) ↔ true :=
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iff_true_intro (heq.refl a)
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theorem a_iff_not_a (a : Prop) : (a ↔ ¬a) ↔ false :=
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iff.intro
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(assume H,
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have H' : ¬a, from assume Ha, (H ▸ Ha) Ha,
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H' (H⁻¹ ▸ H'))
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(assume H, false_elim H)
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theorem true_eq_false : (true ↔ false) ↔ false :=
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not_true_iff_false ▸ (a_iff_not_a true)
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theorem false_eq_true : (false ↔ true) ↔ false :=
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not_false_iff_true ▸ (a_iff_not_a false)
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theorem a_eq_true (a : Prop) : (a ↔ true) ↔ a :=
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iff.intro (assume H, iff.true_elim H) (assume H, iff_true_intro H)
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theorem a_eq_false (a : Prop) : (a ↔ false) ↔ ¬a :=
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iff.intro (assume H, iff.false_elim H) (assume H, iff_false_intro H)
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