lean2/library/logic/connectives.lean

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-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Leonardo de Moura, Jeremy Avigad
import general_notation .eq
-- and
-- ---
inductive and (a b : Prop) : Prop :=
intro : a → b → and a b
notation a /\ b := and a b
notation a ∧ b := and a b
variables {a b c d : Prop}
namespace and
theorem elim (H₁ : a ∧ b) (H₂ : a → b → c) : c :=
rec H₂ H₁
theorem elim_left (H : a ∧ b) : a :=
rec (λa b, a) H
theorem elim_right (H : a ∧ b) : b :=
rec (λa b, b) H
theorem swap (H : a ∧ b) : b ∧ a :=
intro (elim_right H) (elim_left H)
theorem not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (elim_left H) Hna
theorem not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (elim_right H) Hnb
theorem imp_and (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
elim H₁ (assume Ha : a, assume Hb : b, intro (H₂ Ha) (H₃ Hb))
theorem imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
elim H₁ (assume Ha : a, assume Hc : c, intro (H Ha) Hc)
theorem imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
elim H₁ (assume Hc : c, assume Ha : a, intro Hc (H Ha))
end and
-- or
-- --
inductive or (a b : Prop) : Prop :=
intro_left : a → or a b,
intro_right : b → or a b
notation a `\/` b := or a b
notation a b := or a b
namespace or
theorem inl (Ha : a) : a b :=
intro_left b Ha
theorem inr (Hb : b) : a b :=
intro_right a Hb
theorem elim (H₁ : a b) (H₂ : a → c) (H₃ : b → c) : c :=
rec H₂ H₃ H₁
theorem elim3 (H : a b c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
elim H Ha (assume H₂, elim H₂ Hb Hc)
theorem resolve_right (H₁ : a b) (H₂ : ¬a) : b :=
elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
theorem resolve_left (H₁ : a b) (H₂ : ¬b) : a :=
elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
theorem swap (H : a b) : b a :=
elim H (assume Ha, inr Ha) (assume Hb, inl Hb)
theorem not_intro (Hna : ¬a) (Hnb : ¬b) : ¬(a b) :=
assume H : a b, elim H
(assume Ha, absurd Ha Hna)
(assume Hb, absurd Hb Hnb)
theorem imp_or (H₁ : a b) (H₂ : a → c) (H₃ : b → d) : c d :=
elim H₁
(assume Ha : a, inl (H₂ Ha))
(assume Hb : b, inr (H₃ Hb))
theorem imp_or_left (H₁ : a c) (H : a → b) : b c :=
elim H₁
(assume H₂ : a, inl (H H₂))
(assume H₂ : c, inr H₂)
theorem imp_or_right (H₁ : c a) (H : a → b) : c b :=
elim H₁
(assume H₂ : c, inl H₂)
(assume H₂ : a, inr (H H₂))
end or
theorem not_not_em {p : Prop} : ¬¬(p ¬p) :=
assume not_em : ¬(p ¬p),
have Hnp : ¬p, from
assume Hp : p, absurd (or.inl Hp) not_em,
absurd (or.inr Hnp) not_em
-- iff
-- ---
definition iff (a b : Prop) := (a → b) ∧ (b → a)
notation a <-> b := iff a b
notation a ↔ b := iff a b
namespace iff
theorem def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
rfl
theorem intro (H₁ : a → b) (H₂ : b → a) : a ↔ b :=
and.intro H₁ H₂
theorem elim (H₁ : (a → b) → (b → a) → c) (H₂ : a ↔ b) : c :=
and.rec H₁ H₂
theorem elim_left (H : a ↔ b) : a → b :=
elim (assume H₁ H₂, H₁) H
definition mp := @elim_left
theorem elim_right (H : a ↔ b) : b → a :=
elim (assume H₁ H₂, H₂) H
theorem flip_sign (H₁ : a ↔ b) : ¬a ↔ ¬b :=
intro
(assume Hna, mt (elim_right H₁) Hna)
(assume Hnb, mt (elim_left H₁) Hnb)
theorem refl (a : Prop) : a ↔ a :=
intro (assume H, H) (assume H, H)
theorem rfl {a : Prop} : a ↔ a :=
refl a
theorem trans (H₁ : a ↔ b) (H₂ : b ↔ c) : a ↔ c :=
intro
(assume Ha, elim_left H₂ (elim_left H₁ Ha))
(assume Hc, elim_right H₁ (elim_right H₂ Hc))
theorem symm (H : a ↔ b) : b ↔ a :=
intro
(assume Hb, elim_right H Hb)
(assume Ha, elim_left H Ha)
theorem true_elim (H : a ↔ true) : a :=
mp (symm H) trivial
theorem false_elim (H : a ↔ false) : ¬a :=
assume Ha : a, mp H Ha
end iff
calc_refl iff.refl
calc_trans iff.trans
open eq.ops
theorem eq_to_iff {a b : Prop} (H : a = b) : a ↔ b :=
iff.intro (λ Ha, H ▸ Ha) (λ Hb, H⁻¹ ▸ Hb)
-- comm and assoc for and / or
-- ---------------------------
namespace and
theorem comm : a ∧ b ↔ b ∧ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff.intro
(assume H, intro
(elim_left (elim_left H))
(intro (elim_right (elim_left H)) (elim_right H)))
(assume H, intro
(intro (elim_left H) (elim_left (elim_right H)))
(elim_right (elim_right H)))
end and
namespace or
theorem comm : a b ↔ b a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a b) c ↔ a (b c) :=
iff.intro
(assume H, elim H
(assume H₁, elim H₁
(assume Ha, inl Ha)
(assume Hb, inr (inl Hb)))
(assume Hc, inr (inr Hc)))
(assume H, elim H
(assume Ha, (inl (inl Ha)))
(assume H₁, elim H₁
(assume Hb, inl (inr Hb))
(assume Hc, inr Hc)))
end or