331 lines
11 KiB
Text
331 lines
11 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, Haitao Zhang
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The propositional connectives. See also init.datatypes and init.logic.
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-/
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open eq.ops
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variables {a b c d : Prop}
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/- false -/
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theorem false.elim {c : Prop} (H : false) : c :=
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false.rec c H
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/- implies -/
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definition imp (a b : Prop) : Prop := a → b
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theorem mt (H1 : a → b) (H2 : ¬b) : ¬a :=
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assume Ha : a, absurd (H1 Ha) H2
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theorem imp_true (a : Prop) : (a → true) ↔ true :=
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iff.intro (assume H, trivial) (assume H H1, trivial)
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theorem true_imp (a : Prop) : (true → a) ↔ a :=
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iff.intro (assume H, H trivial) (assume H H1, H)
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theorem imp_false (a : Prop) : (a → false) ↔ ¬ a := iff.rfl
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theorem false_imp (a : Prop) : (false → a) ↔ true :=
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iff.intro (assume H, trivial) (assume H H1, false.elim H1)
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theorem imp_iff_imp (H1 : a ↔ c) (H2 : b ↔ d) : (a → b) ↔ (c → d) :=
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iff.intro
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(λHab Hc, iff.elim_left H2 (Hab (iff.elim_right H1 Hc)))
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(λHcd Ha, iff.elim_right H2 (Hcd (iff.elim_left H1 Ha)))
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/- not -/
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theorem not.elim (H1 : ¬a) (H2 : a) : false := H1 H2
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theorem not.intro (H : a → false) : ¬a := H
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theorem not_not_intro (Ha : a) : ¬¬a :=
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assume Hna : ¬a, absurd Ha Hna
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theorem not_imp_not_of_imp {a b : Prop} : (a → b) → ¬b → ¬a :=
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assume Pimp Pnb Pa, absurd (Pimp Pa) Pnb
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theorem not_not_of_not_implies (H : ¬(a → b)) : ¬¬a :=
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assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
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theorem not_of_not_implies (H : ¬(a → b)) : ¬b :=
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assume Hb : b, absurd (assume Ha : a, Hb) H
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theorem not_not_em : ¬¬(a ∨ ¬a) :=
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assume not_em : ¬(a ∨ ¬a),
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have Hnp : ¬a, from
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assume Hp : a, absurd (or.inl Hp) not_em,
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absurd (or.inr Hnp) not_em
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theorem not_true [rewrite] : ¬ true ↔ false :=
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iff.intro (assume H, H trivial) (assume H, false.elim H)
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theorem not_false_iff [rewrite] : ¬ false ↔ true :=
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iff.intro (assume H, trivial) (assume H H1, H1)
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theorem not_iff_not (H : a ↔ b) : ¬a ↔ ¬b :=
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iff.intro
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(λHna Hb, Hna (iff.elim_right H Hb))
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(λHnb Ha, Hnb (iff.elim_left H Ha))
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/- and -/
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definition not_and_of_not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
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assume H : a ∧ b, absurd (and.elim_left H) Hna
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definition not_and_of_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.swap (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_of_and_of_imp_of_imp (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 and_of_and_of_imp_left (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 and_of_and_of_imp_right (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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theorem and.comm : 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 ↔ a ∧ (b ∧ c) :=
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iff.intro
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(assume H,
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obtain [Ha Hb] Hc, from H,
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and.intro Ha (and.intro Hb Hc))
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(assume H,
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obtain Ha Hb Hc, from H,
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and.intro (and.intro Ha Hb) Hc)
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theorem and_true [rewrite] (a : Prop) : a ∧ true ↔ a :=
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iff.intro (assume H, and.left H) (assume H, and.intro H trivial)
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theorem true_and [rewrite] (a : Prop) : true ∧ a ↔ a :=
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iff.intro (assume H, and.right H) (assume H, and.intro trivial H)
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theorem and_false [rewrite] (a : Prop) : a ∧ false ↔ false :=
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iff.intro (assume H, and.right H) (assume H, false.elim H)
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theorem false_and [rewrite] (a : Prop) : false ∧ a ↔ false :=
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iff.intro (assume H, and.left H) (assume H, false.elim H)
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theorem and_self [rewrite] (a : Prop) : a ∧ a ↔ a :=
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iff.intro (assume H, and.left H) (assume H, and.intro H H)
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theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
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propext
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(iff.intro (λ Pl a b, Pl (and.intro a b))
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(λ Pr Pand, Pr (and.left Pand) (and.right Pand)))
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theorem and_iff_right {a b : Prop} (Ha : a) : (a ∧ b) ↔ b :=
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iff.intro
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(assume Hab, and.elim_right Hab)
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(assume Hb, and.intro Ha Hb)
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theorem and_iff_left {a b : Prop} (Hb : b) : (a ∧ b) ↔ a :=
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iff.intro
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(assume Hab, and.elim_left Hab)
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(assume Ha, and.intro Ha Hb)
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theorem and_iff_and (H1 : a ↔ c) (H2 : b ↔ d) : (a ∧ b) ↔ (c ∧ d) :=
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iff.intro
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(assume Hab, and.intro (iff.elim_left H1 (and.left Hab)) (iff.elim_left H2 (and.right Hab)))
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(assume Hcd, and.intro (iff.elim_right H1 (and.left Hcd)) (iff.elim_right H2 (and.right Hcd)))
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/- or -/
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definition not_or (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
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assume H : a ∨ b, or.rec_on 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_of_or_of_imp_of_imp (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_of_or_of_imp_left (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_of_or_of_imp_right (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 or.elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
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or.elim H Ha (assume H₂, or.elim H₂ Hb Hc)
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theorem or_resolve_right (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 or_resolve_left (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 (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.comm : 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 ↔ 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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theorem or_true [rewrite] (a : Prop) : a ∨ true ↔ true :=
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iff.intro (assume H, trivial) (assume H, or.inr H)
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theorem true_or [rewrite] (a : Prop) : true ∨ a ↔ true :=
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iff.intro (assume H, trivial) (assume H, or.inl H)
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theorem or_false [rewrite] (a : Prop) : a ∨ false ↔ a :=
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iff.intro
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(assume H, or.elim H (assume H1 : a, H1) (assume H1 : false, false.elim H1))
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(assume H, or.inl H)
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theorem false_or [rewrite] (a : Prop) : false ∨ a ↔ a :=
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iff.intro
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(assume H, or.elim H (assume H1 : false, false.elim H1) (assume H1 : a, H1))
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(assume H, or.inr H)
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theorem or_self (a : Prop) : a ∨ a ↔ a :=
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iff.intro
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(assume H, or.elim H (assume H1, H1) (assume H1, H1))
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(assume H, or.inl H)
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theorem or_iff_or (H1 : a ↔ c) (H2 : b ↔ d) : (a ∨ b) ↔ (c ∨ d) :=
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iff.intro
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(λHab, or.elim Hab (λHa, or.inl (iff.elim_left H1 Ha)) (λHb, or.inr (iff.elim_left H2 Hb)))
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(λHcd, or.elim Hcd (λHc, or.inl (iff.elim_right H1 Hc)) (λHd, or.inr (iff.elim_right H2 Hd)))
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/- distributivity -/
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theorem and.distrib_left (a b c : Prop) : a ∧ (b ∨ c) ↔ (a ∧ b) ∨ (a ∧ c) :=
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iff.intro
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(assume H, or.elim (and.right H)
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(assume Hb : b, or.inl (and.intro (and.left H) Hb))
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(assume Hc : c, or.inr (and.intro (and.left H) Hc)))
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(assume H, or.elim H
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(assume Hab, and.intro (and.left Hab) (or.inl (and.right Hab)))
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(assume Hac, and.intro (and.left Hac) (or.inr (and.right Hac))))
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theorem and.distrib_right (a b c : Prop) : (a ∨ b) ∧ c ↔ (a ∧ c) ∨ (b ∧ c) :=
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propext (!and.comm) ▸ propext (!and.comm) ▸ propext (!and.comm) ▸ !and.distrib_left
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theorem or.distrib_left (a b c : Prop) : a ∨ (b ∧ c) ↔ (a ∨ b) ∧ (a ∨ c) :=
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iff.intro
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(assume H, or.elim H
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(assume Ha, and.intro (or.inl Ha) (or.inl Ha))
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(assume Hbc, and.intro (or.inr (and.left Hbc)) (or.inr (and.right Hbc))))
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(assume H, or.elim (and.left H)
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(assume Ha, or.inl Ha)
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(assume Hb, or.elim (and.right H)
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(assume Ha, or.inl Ha)
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(assume Hc, or.inr (and.intro Hb Hc))))
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theorem or.distrib_right (a b c : Prop) : (a ∧ b) ∨ c ↔ (a ∨ c) ∧ (b ∨ c) :=
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propext (!or.comm) ▸ propext (!or.comm) ▸ propext (!or.comm) ▸ !or.distrib_left
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/- iff -/
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definition iff.def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
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!eq.refl
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theorem iff_true [rewrite] (a : Prop) : (a ↔ true) ↔ a :=
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iff.intro
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(assume H, iff.mp' H trivial)
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(assume H, iff.intro (assume H1, trivial) (assume H1, H))
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theorem true_iff [rewrite] (a : Prop) : (true ↔ a) ↔ a :=
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iff.intro
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(assume H, iff.mp H trivial)
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(assume H, iff.intro (assume H1, H) (assume H1, trivial))
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theorem iff_false [rewrite] (a : Prop) : (a ↔ false) ↔ ¬ a :=
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iff.intro
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(assume H, and.left H)
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(assume H, and.intro H (assume H1, false.elim H1))
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theorem false_iff [rewrite] (a : Prop) : (false ↔ a) ↔ ¬ a :=
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iff.intro
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(assume H, and.right H)
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(assume H, and.intro (assume H1, false.elim H1) H)
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theorem iff_true_of_self (Ha : a) : a ↔ true :=
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iff.intro (assume H, trivial) (assume H, Ha)
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theorem iff_self [rewrite] (a : Prop) : (a ↔ a) ↔ true :=
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iff_true_of_self !iff.refl
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theorem forall_iff_forall {A : Type} {P Q : A → Prop} (H : ∀a, (P a ↔ Q a)) : (∀a, P a) ↔ ∀a, Q a :=
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iff.intro (λp a, iff.elim_left (H a) (p a)) (λq a, iff.elim_right (H a) (q a))
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theorem imp_iff {P : Prop} (Q : Prop) (p : P) : (P → Q) ↔ Q :=
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iff.intro (λf, f p) (λq p, q)
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theorem iff_iff_iff (H1 : a ↔ c) (H2 : b ↔ d) : (a ↔ b) ↔ (c ↔ d) :=
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and_iff_and (imp_iff_imp H1 H2) (imp_iff_imp H2 H1)
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/- if-then-else -/
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section
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open eq.ops
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variables {A : Type} {c₁ c₂ : Prop}
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definition if_true [rewrite] (t e : A) : (if true then t else e) = t :=
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if_pos trivial
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definition if_false [rewrite] (t e : A) : (if false then t else e) = e :=
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if_neg not_false
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end
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/- congruences -/
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theorem congr_not {a b : Prop} (H : a ↔ b) : ¬a ↔ ¬b :=
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iff.intro
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(assume H₁ : ¬a, assume H₂ : b, H₁ (iff.elim_right H H₂))
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(assume H₁ : ¬b, assume H₂ : a, H₁ (iff.elim_left H H₂))
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section
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variables {a₁ b₁ a₂ b₂ : Prop}
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variables (H₁ : a₁ ↔ b₁) (H₂ : a₂ ↔ b₂)
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theorem congr_and : a₁ ∧ a₂ ↔ b₁ ∧ b₂ :=
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iff.intro
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(assume H₃ : a₁ ∧ a₂, and_of_and_of_imp_of_imp H₃ (iff.elim_left H₁) (iff.elim_left H₂))
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(assume H₃ : b₁ ∧ b₂, and_of_and_of_imp_of_imp H₃ (iff.elim_right H₁) (iff.elim_right H₂))
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theorem congr_or : a₁ ∨ a₂ ↔ b₁ ∨ b₂ :=
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iff.intro
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(assume H₃ : a₁ ∨ a₂, or_of_or_of_imp_of_imp H₃ (iff.elim_left H₁) (iff.elim_left H₂))
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(assume H₃ : b₁ ∨ b₂, or_of_or_of_imp_of_imp H₃ (iff.elim_right H₁) (iff.elim_right H₂))
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theorem congr_imp : (a₁ → a₂) ↔ (b₁ → b₂) :=
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iff.intro
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(assume H₃ : a₁ → a₂, assume Hb₁ : b₁, iff.elim_left H₂ (H₃ ((iff.elim_right H₁) Hb₁)))
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(assume H₃ : b₁ → b₂, assume Ha₁ : a₁, iff.elim_right H₂ (H₃ ((iff.elim_left H₁) Ha₁)))
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theorem congr_iff : (a₁ ↔ a₂) ↔ (b₁ ↔ b₂) :=
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iff.intro
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(assume H₃ : a₁ ↔ a₂, iff.trans (iff.symm H₁) (iff.trans H₃ H₂))
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(assume H₃ : b₁ ↔ b₂, iff.trans H₁ (iff.trans H₃ (iff.symm H₂)))
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end
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