update in standard library

[helm.git] / matita / matita / lib / basics / logic.ma

diff --git a/matita/matita/lib/basics/logic.ma b/matita/matita/lib/basics/logic.ma
index 6723cf2cc4bbca3d4a0797d50b8d97c8495d0fb2..4b90f75181300bdaced20673e08e5d6420f4fbc1 100644 (file)
--- a/matita/matita/lib/basics/logic.ma
+++ b/matita/matita/lib/basics/logic.ma
@@ -1,4 +1,4 @@
-(*
+ (*
     ||M||  This file is part of HELM, an Hypertextual, Electronic        
     ||A||  Library of Mathematics, developed at the Computer Science     
     ||T||  Department of the University of Bologna, Italy.                     
@@ -14,31 +14,47 @@ include "hints_declaration.ma".
 
 (* propositional equality *)
 
-inductive eq (A:Type[1]) (x:A) : A → Prop ≝
+inductive eq (A:Type[2]) (x:A) : A → Prop ≝
     refl: eq A x x. 
     
 interpretation "leibnitz's equality" 'eq t x y = (eq t x y).
+interpretation "leibniz reflexivity" 'refl = refl.
 
 lemma eq_rect_r:
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → P x p.
+ ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → P x p.
  #A #a #x #p (cases p) // qed.
 
 lemma eq_ind_r :
  ∀A.∀a.∀P: ∀x:A. x = a → Prop. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
  #A #a #P #p #x0 #p0; @(eq_rect_r ? ? ? p0) //; qed.
 
+lemma eq_rect_Type0_r:
+  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  #A #a #P #H #x #p lapply H lapply P
+  cases p; //; qed.
+
+lemma eq_rect_Type1_r:
+  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[1]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  #A #a #P #H #x #p lapply H lapply P
+  cases p; //; qed.
+
 lemma eq_rect_Type2_r:
   ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[2]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
-  #A #a #P #H #x #p (generalize in match H) (generalize in match P)
+  #A #a #P #H #x #p lapply H lapply P
   cases p; //; qed.
 
-theorem rewrite_l: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. x = y → P y.
+lemma eq_rect_Type3_r:
+  ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[3]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
+  #A #a #P #H #x #p lapply H lapply P
+  cases p; //; qed.
+
+theorem rewrite_l: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. x = y → P y.
 #A #x #P #Hx #y #Heq (cases Heq); //; qed.
 
 theorem sym_eq: ∀A.∀x,y:A. x = y → y = x.
-#A #x #y #Heq @(rewrite_l A x (λz.z=x)); //; qed.
+#A #x #y #Heq @(rewrite_l A x (λz.z=x)) // qed.
 
-theorem rewrite_r: ∀A:Type[1].∀x.∀P:A → Type[1]. P x → ∀y. y = x → P y.
+theorem rewrite_r: ∀A:Type[2].∀x.∀P:A → Type[2]. P x → ∀y. y = x → P y.
 #A #x #P #Hx #y #Heq (cases (sym_eq ? ? ? Heq)); //; qed.
 
 theorem eq_coerc: ∀A,B:Type[0].A→(A=B)→B.
@@ -131,13 +147,40 @@ inductive ex (A:Type[0]) (P:A → Prop) : Prop ≝
 interpretation "exists" 'exists x = (ex ? x).
 
 inductive ex2 (A:Type[0]) (P,Q:A →Prop) : Prop ≝
-    ex_intro2: ∀ x:A. P x → Q x → ex2 A P Q.
+    ex2_intro: ∀ x:A. P x → Q x → ex2 A P Q.
+
+interpretation "exists on two predicates" 'exists2 x1 x2 = (ex2 ? x1 x2).
+
+lemma ex2_commute: ∀A0. ∀P0,P1:A0→Prop. (∃∃x0. P0 x0 & P1 x0) → ∃∃x0. P1 x0 & P0 x0.
+#A0 #P0 #P1 * /2 width=3/
+qed-.
 
 (* iff *)
 definition iff :=
  λ A,B. (A → B) ∧ (B → A).
 
-interpretation "iff" 'iff a b = (iff a b).  
+interpretation "iff" 'iff a b = (iff a b).
+
+lemma iff_sym: ∀A,B. A ↔ B → B ↔ A.
+#A #B * /3/ qed.
+
+lemma iff_trans:∀A,B,C. A ↔ B → B ↔ C → A ↔ C.
+#A #B #C * #H1 #H2 * #H3 #H4 % /3/ qed.
+
+lemma iff_not: ∀A,B. A ↔ B → ¬A ↔ ¬B.
+#A #B * #H1 #H2 % /3/ qed.
+
+lemma iff_and_l: ∀A,B,C. A ↔ B → C ∧ A ↔ C ∧ B.
+#A #B #C * #H1 #H2 % * /3/ qed.  
+
+lemma iff_and_r: ∀A,B,C. A ↔ B → A ∧ C ↔ B ∧ C.
+#A #B #C * #H1 #H2 % * /3/ qed.  
+
+lemma iff_or_l: ∀A,B,C. A ↔ B → C ∨ A ↔ C ∨ B.
+#A #B #C * #H1 #H2 % * /3/ qed.  
+
+lemma iff_or_r: ∀A,B,C. A ↔ B → A ∨ C ↔ B ∨ C.
+#A #B #C * #H1 #H2 % * /3/ qed.  
 
 (* cose per destruct: da rivedere *) 
 
@@ -222,5 +265,14 @@ definition R4 :
 @a4 
 qed.
 
-(* TODO concrete definition by means of proof irrelevance *)
-axiom streicherK : ∀T:Type[1].∀t:T.∀P:t = t → Type[2].P (refl ? t) → ∀p.P p.
+definition eqProp ≝ λA:Prop.eq A.
+
+(* Example to avoid indexing and the consequential creation of ill typed
+   terms during paramodulation *)
+lemma lemmaK : ∀A.∀x:A.∀h:x=x. eqProp ? h (refl A x).
+#A #x #h @(refl ? h: eqProp ? ? ?).
+qed-.
+
+theorem streicherK : ∀T:Type[2].∀t:T.∀P:t = t → Type[3].P (refl ? t) → ∀p.P p.
+ #T #t #P #H #p >(lemmaK T t p) @H
+qed-.