feat(library/logic/{connectives,identities},library/algebra/function): cleanup and some additions from Haitao Zhang

library/algebra/function.lean

@@ -37,7 +37,7 @@ definition dcompose [reducible] [unfold-f] {B : A → Type} {C : Π {x : A}, B x
   (f : Π {x : A} (y : B x), C y) (g : Πx, B x) : Πx, C (g x) :=
 λx, f (g x)
 
-definition flip [reducible] [unfold-f] {C : A → B → Type} (f : Πx y, C x y) : Πy x, C x y :=
+definition swap [reducible] [unfold-f] {C : A → B → Type} (f : Πx y, C x y) : Πy x, C x y :=
 λy x, f x y
 
 definition app [reducible] {B : A → Type} (f : Πx, B x) (x : A) : B x :=

library/logic/connectives.lean

@@ -1,7 +1,7 @@
 /-
 Copyright (c) 2014 Microsoft Corporation. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
-Authors: Jeremy Avigad, Leonardo de Moura
+Authors: Jeremy Avigad, Leonardo de Moura, Haitao Zhang
 
 The propositional connectives. See also init.datatypes and init.logic.
 -/
@@ -41,6 +41,9 @@ theorem not.intro (H : a → false) : ¬a := H
 theorem not_not_intro (Ha : a) : ¬¬a :=
 assume Hna : ¬a, absurd Ha Hna
 
+theorem not_imp_not_of_imp {a b : Prop} : (a → b) → ¬b → ¬a :=
+assume Pimp Pnb Pa, absurd (Pimp Pa) Pnb
+
 theorem not_not_of_not_implies (H : ¬(a → b)) : ¬¬a :=
 assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
 
@@ -106,6 +109,21 @@ iff.intro (assume H, and.left H) (assume H, false.elim H)
 theorem and_self (a : Prop) : a ∧ a ↔ a :=
 iff.intro (assume H, and.left H) (assume H, and.intro H H)
 
+theorem and_imp_eq (a b c : Prop) : (a ∧ b → c) = (a → b → c) :=
+propext
+  (iff.intro (λ Pl a b, Pl (and.intro a b))
+  (λ Pr Pand, Pr (and.left Pand) (and.right Pand)))
+
+theorem and_eq_right {a b : Prop} (Ha : a) : (a ∧ b) = b :=
+propext (iff.intro
+  (assume Hab, and.elim_right Hab)
+  (assume Hb, and.intro Ha Hb))
+
+theorem and_eq_left {a b : Prop} (Hb : b) : (a ∧ b) = a :=
+propext (iff.intro
+  (assume Hab, and.elim_left Hab)
+  (assume Ha, and.intro Ha Hb))
+
 /- or -/
 
 definition not_or (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=

library/logic/identities.lean

@@ -94,20 +94,31 @@ assume H, by_contradiction (assume Hna : ¬a,
   have Hnna : ¬¬a, from not_not_of_not_implies (mt H Hna),
   absurd (not_not_elim Hnna) Hna)
 
-theorem forall_not_of_not_exists {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
-    (H : ¬∃x, P x) : ∀x, ¬P x :=
-take x, or.elim (em (P x))
-  (assume Hp : P x,   absurd (exists.intro x Hp) H)
-  (assume Hn : ¬P x, Hn)
+theorem forall_not_of_not_exists {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)]
+  (H : ¬∃x, p x) : ∀x, ¬p x :=
+take x, or.elim (em (p x))
+  (assume Hp : p x,   absurd (exists.intro x Hp) H)
+  (assume Hnp : ¬p x, Hnp)
 
-theorem exists_not_of_not_forall {A : Type} {P : A → Prop} [D : ∀x, decidable (P x)]
-    [D' : decidable (∃x, ¬P x)] (H : ¬∀x, P x) :
-  ∃x, ¬P x :=
-@by_contradiction _ D' (assume H1 : ¬∃x, ¬P x,
-  have H2 : ∀x, ¬¬P x, from @forall_not_of_not_exists _ _ (take x, decidable_not) H1,
-  have H3 : ∀x, P x, from take x, @not_not_elim _ (D x) (H2 x),
+theorem forall_of_not_exists_not {A : Type} {p : A → Prop} [D : decidable_pred p] :
+  ¬(∃ x, ¬p x) → ∀ x, p x :=
+assume Hne, take x, by_contradiction (assume Hnp : ¬ p x, Hne (exists.intro x Hnp))
+
+theorem exists_not_of_not_forall {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)]
+    [D' : decidable (∃x, ¬p x)] (H : ¬∀x, p x) :
+  ∃x, ¬p x :=
+by_contradiction
+  (assume H1 : ¬∃x, ¬p x,
+    have H2 : ∀x, ¬¬p x, from forall_not_of_not_exists H1,
+    have H3 : ∀x, p x, from take x, not_not_elim (H2 x),
     absurd H3 H)
 
+theorem exists_of_not_forall_not {A : Type} {p : A → Prop} [D : ∀x, decidable (p x)]
+    [D' : decidable (∃x, ¬¬p x)] (H : ¬∀x, ¬ p x) :
+  ∃x, p x :=
+obtain x (H : ¬¬ p x), from exists_not_of_not_forall H,
+exists.intro x (not_not_elim H)
+
 theorem ne_self_iff_false {A : Type} (a : A) : (a ≠ a) ↔ false :=
 iff.intro
   (assume H, false.of_ne H)