150 lines
5.3 KiB
Text
150 lines
5.3 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.connectives.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.connectives.instances
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import logic.core.instances logic.classes.decidable logic.core.quantifiers logic.core.cast
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using 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 : (¬true) ↔ false :=
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iff_intro (assume H, H trivial) (false_elim _)
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theorem not_false : (¬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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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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-- TODO: when type class instances can use quantifiers, we can use write em
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take x, or_elim (@em _ (D 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 a 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, a_neq_a_elim H)
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(assume H, false_elim (a ≠ a) H)
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theorem eq_id {A : Type} (a : A) : (a = a) ↔ true :=
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iff_true_intro (refl a)
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theorem heq_id {A : Type} (a : A) : (a == a) ↔ true :=
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iff_true_intro (hrefl 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 (a ↔ ¬a) H)
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theorem true_eq_false : (true ↔ false) ↔ false :=
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not_true ▸ (a_iff_not_a true)
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theorem false_eq_true : (false ↔ true) ↔ false :=
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not_false ▸ (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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