179 lines
5.4 KiB
Text
179 lines
5.4 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: Leonardo de Moura, Jeremy Avigad
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import general_notation .eq
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-- and
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-- ---
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inductive and (a b : Prop) : Prop :=
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intro : a → b → and a b
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infixr `/\` := and
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infixr `∧` := and
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theorem and_elim {a b c : Prop} (H1 : a ∧ b) (H2 : a → b → c) : c :=
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and.rec H2 H1
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theorem and_elim_left {a b : Prop} (H : a ∧ b) : a :=
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and.rec (λa b, a) H
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theorem and_elim_right {a b : Prop} (H : a ∧ b) : b :=
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and.rec (λa b, b) H
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theorem and_swap {a b : Prop} (H : a ∧ b) : b ∧ a :=
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and.intro (and_elim_right H) (and_elim_left H)
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theorem and_not_left {a : Prop} (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (and_elim_left H) Hna
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theorem and_not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (and_elim_right H) Hnb
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theorem and_imp_and {a b c d : Prop} (H1 : a ∧ b) (H2 : a → c) (H3 : b → d) : c ∧ d :=
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and_elim H1 (assume Ha : a, assume Hb : b, and.intro (H2 Ha) (H3 Hb))
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theorem imp_and_left {a b c : Prop} (H1 : a ∧ c) (H : a → b) : b ∧ c :=
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and_elim H1 (assume Ha : a, assume Hc : c, and.intro (H Ha) Hc)
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theorem imp_and_right {a b c : Prop} (H1 : c ∧ a) (H : a → b) : c ∧ b :=
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and_elim H1 (assume Hc : c, assume Ha : a, and.intro Hc (H Ha))
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-- or
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-- --
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inductive or (a b : Prop) : Prop :=
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intro_left : a → or a b,
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intro_right : b → or a b
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infixr `\/` := or
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infixr `∨` := or
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theorem or_inl {a b : Prop} (Ha : a) : a ∨ b := or.intro_left b Ha
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theorem or_inr {a b : Prop} (Hb : b) : a ∨ b := or.intro_right a Hb
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theorem or_elim {a b c : Prop} (H1 : a ∨ b) (H2 : a → c) (H3 : b → c) : c :=
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or.rec H2 H3 H1
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theorem resolve_right {a b : Prop} (H1 : a ∨ b) (H2 : ¬a) : b :=
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or_elim H1 (assume Ha, absurd Ha H2) (assume Hb, Hb)
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theorem resolve_left {a b : Prop} (H1 : a ∨ b) (H2 : ¬b) : a :=
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or_elim H1 (assume Ha, Ha) (assume Hb, absurd Hb H2)
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theorem or_swap {a b : Prop} (H : a ∨ b) : b ∨ a :=
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or_elim H (assume Ha, or_inr Ha) (assume Hb, or_inl Hb)
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theorem or_not_intro {a b : Prop} (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
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assume H : a ∨ b, or_elim H
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(assume Ha, absurd Ha Hna)
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(assume Hb, absurd Hb Hnb)
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theorem or_imp_or {a b c d : Prop} (H1 : a ∨ b) (H2 : a → c) (H3 : b → d) : c ∨ d :=
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or_elim H1
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(assume Ha : a, or_inl (H2 Ha))
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(assume Hb : b, or_inr (H3 Hb))
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theorem or_imp_or_left {a b c : Prop} (H1 : a ∨ c) (H : a → b) : b ∨ c :=
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or_elim H1
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(assume H2 : a, or_inl (H H2))
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(assume H2 : c, or_inr H2)
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theorem or_imp_or_right {a b c : Prop} (H1 : c ∨ a) (H : a → b) : c ∨ b :=
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or_elim H1
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(assume H2 : c, or_inl H2)
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(assume H2 : a, or_inr (H H2))
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theorem not_not_em {p : Prop} : ¬¬(p ∨ ¬p) :=
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assume not_em : ¬(p ∨ ¬p),
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have Hnp : ¬p, from
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assume Hp : p, absurd (or_inl Hp) not_em,
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absurd (or_inr Hnp) not_em
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-- iff
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-- ---
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definition iff (a b : Prop) := (a → b) ∧ (b → a)
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infix `<->` := iff
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infix `↔` := iff
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theorem iff_def {a b : Prop} : (a ↔ b) = ((a → b) ∧ (b → a)) := rfl
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theorem iff_intro {a b : Prop} (H1 : a → b) (H2 : b → a) : a ↔ b := and.intro H1 H2
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theorem iff_elim {a b c : Prop} (H1 : (a → b) → (b → a) → c) (H2 : a ↔ b) : c := and.rec H1 H2
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theorem iff_elim_left {a b : Prop} (H : a ↔ b) : a → b :=
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iff_elim (assume H1 H2, H1) H
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abbreviation iff_mp := @iff_elim_left
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theorem iff_elim_right {a b : Prop} (H : a ↔ b) : b → a :=
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iff_elim (assume H1 H2, H2) H
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theorem iff_flip_sign {a b : Prop} (H1 : a ↔ b) : ¬a ↔ ¬b :=
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iff_intro
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(assume Hna, mt (iff_elim_right H1) Hna)
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(assume Hnb, mt (iff_elim_left H1) Hnb)
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theorem iff_refl (a : Prop) : a ↔ a :=
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iff_intro (assume H, H) (assume H, H)
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theorem iff_rfl {a : Prop} : a ↔ a :=
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iff_refl a
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theorem iff_trans {a b c : Prop} (H1 : a ↔ b) (H2 : b ↔ c) : a ↔ c :=
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iff_intro
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(assume Ha, iff_elim_left H2 (iff_elim_left H1 Ha))
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(assume Hc, iff_elim_right H1 (iff_elim_right H2 Hc))
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theorem iff_symm {a b : Prop} (H : a ↔ b) : b ↔ a :=
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iff_intro
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(assume Hb, iff_elim_right H Hb)
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(assume Ha, iff_elim_left H Ha)
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calc_refl iff_refl
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calc_trans iff_trans
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theorem iff_true_elim {a : Prop} (H : a ↔ true) : a :=
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iff_mp (iff_symm H) trivial
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theorem iff_false_elim {a : Prop} (H : a ↔ false) : ¬a :=
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assume Ha : a, iff_mp H Ha
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theorem eq_to_iff {a b : Prop} (H : a = b) : a ↔ b :=
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iff_intro (λ Ha, subst H Ha) (λ Hb, subst (symm H) Hb)
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-- comm and assoc for and / or
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-- ---------------------------
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theorem and_comm {a b : Prop} : a ∧ b ↔ b ∧ a :=
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iff_intro (λH, and_swap H) (λH, and_swap H)
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theorem and_assoc {a b c : Prop} : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
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iff_intro
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(assume H, and.intro
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(and_elim_left (and_elim_left H))
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(and.intro (and_elim_right (and_elim_left H)) (and_elim_right H)))
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(assume H, and.intro
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(and.intro (and_elim_left H) (and_elim_left (and_elim_right H)))
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(and_elim_right (and_elim_right H)))
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theorem or_comm {a b : Prop} : a ∨ b ↔ b ∨ a :=
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iff_intro (λH, or_swap H) (λH, or_swap H)
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theorem or_assoc {a b c : Prop} : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
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iff_intro
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(assume H, or_elim H
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(assume H1, or_elim H1
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(assume Ha, or_inl Ha)
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(assume Hb, or_inr (or_inl Hb)))
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(assume Hc, or_inr (or_inr Hc)))
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(assume H, or_elim H
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(assume Ha, (or_inl (or_inl Ha)))
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(assume H1, or_elim H1
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(assume Hb, or_inl (or_inr Hb))
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(assume Hc, or_inr Hc)))
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