X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fmatita%2Flibrary%2Flogic%2Fequality.ma;h=b87dc6c95656dcf5de3c3ceb00e9ed105067c6b8;hb=4167cea65ca58897d1a3dbb81ff95de5074700cc;hp=0be69f9b7e3054e93cc5fc72270739a46ebcd06d;hpb=633474751ddf1074947ff0d324fb1aca2293eff8;p=helm.git

diff --git a/helm/matita/library/logic/equality.ma b/helm/matita/library/logic/equality.ma
index 0be69f9b7..b87dc6c95 100644
--- a/helm/matita/library/logic/equality.ma
+++ b/helm/matita/library/logic/equality.ma
@@ -19,24 +19,36 @@ include "higher_order_defs/relations.ma".
 inductive eq (A:Type) (x:A) : A \to Prop \def
     refl_eq : eq A x x.
 
+(*CSC: the URI must disappear: there is a bug now *)
 interpretation "leibnitz's equality"
    'eq x y = (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y).
-alias symbol "eq" (instance 0) = "leibnitz's equality".
+(*CSC: the URI must disappear: there is a bug now *)
+interpretation "leibnitz's non-equality"
+  'neq x y = (cic:/matita/logic/connectives/Not.con
+    (cic:/matita/logic/equality/eq.ind#xpointer(1/1) _ x y)).
 
-    
+theorem eq_ind':
+ \forall A. \forall x:A. \forall P: \forall y:A. x=y \to Prop.
+  P ? (refl_eq ? x) \to \forall y:A. \forall p:x=y. P y p.
+ intros.
+ exact
+  (match p return \lambda y. \lambda p.P y p with
+    [refl_eq \Rightarrow H]).
+qed.
+ 
 theorem reflexive_eq : \forall A:Type. reflexive A (eq A).
 simplify.intros.apply refl_eq.
 qed.
     
 theorem symmetric_eq: \forall A:Type. symmetric A (eq A).
-simplify.intros.elim H. apply refl_eq.
+unfold symmetric.intros.elim H. apply refl_eq.
 qed.
 
 theorem sym_eq : \forall A:Type.\forall x,y:A. x=y  \to y=x
 \def symmetric_eq.
 
 theorem transitive_eq : \forall A:Type. transitive A (eq A).
-simplify.intros.elim H1.assumption.
+unfold transitive.intros.elim H1.assumption.
 qed.
 
 theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
@@ -45,7 +57,7 @@ theorem trans_eq : \forall A:Type.\forall x,y,z:A. x=y  \to y=z \to x=z
 theorem eq_elim_r:
  \forall A:Type.\forall x:A. \forall P: A \to Prop.
    P x \to \forall y:A. y=x \to P y.
-intros. elim sym_eq ? ? ? H1.assumption.
+intros. elim (sym_eq ? ? ? H1).assumption.
 qed.
 
 default "equality"
@@ -56,12 +68,147 @@ default "equality"
  cic:/matita/logic/equality/eq_elim_r.con. 
  
 theorem eq_f: \forall  A,B:Type.\forall f:A\to B.
-\forall x,y:A. x=y \to (f x)=(f y).
+\forall x,y:A. x=y \to f x = f y.
 intros.elim H.reflexivity.
 qed.
 
 theorem eq_f2: \forall  A,B,C:Type.\forall f:A\to B \to C.
 \forall x1,x2:A. \forall y1,y2:B.
-x1=x2 \to y1=y2 \to (f x1 y1)=(f x2 y2).
+x1=x2 \to y1=y2 \to f x1 y1 = f x2 y2.
 intros.elim H1.elim H.reflexivity.
 qed.
+
+definition comp \def
+ \lambda A.
+  \lambda x,y,y':A.
+   \lambda eq1:x=y.
+    \lambda eq2:x=y'.
+     eq_ind ? ? (\lambda a.a=y') eq2 ? eq1.
+     
+lemma trans_sym_eq:
+ \forall A.
+  \forall x,y:A.
+   \forall u:x=y.
+    comp ? ? ? ? u u = refl_eq ? y.
+ intros.
+ apply (eq_ind' ? ? ? ? ? u).
+ reflexivity.
+qed.
+
+definition nu \def
+ \lambda A.
+  \lambda H: \forall x,y:A. decidable (x=y).
+   \lambda x,y. \lambda p:x=y.
+     match H x y with
+      [ (or_introl p') \Rightarrow p'
+      | (or_intror K) \Rightarrow False_ind ? (K p) ].
+
+theorem nu_constant:
+ \forall A.
+  \forall H: \forall x,y:A. decidable (x=y).
+   \forall x,y:A.
+    \forall u,v:x=y.
+     nu ? H ? ? u = nu ? H ? ? v.
+ intros.
+ unfold nu.
+ unfold decidable in H.
+ apply (Or_ind' ? ? ? ? ? (H x y)); simplify.
+  intro; reflexivity.
+  intro; elim (q u).
+qed.
+
+definition nu_inv \def
+ \lambda A.
+  \lambda H: \forall x,y:A. decidable (x=y).
+   \lambda x,y:A.
+    \lambda v:x=y.
+     comp ? ? ? ? (nu ? H ? ? (refl_eq ? x)) v.
+
+theorem nu_left_inv:
+ \forall A.
+  \forall H: \forall x,y:A. decidable (x=y).
+   \forall x,y:A.
+    \forall u:x=y.
+     nu_inv ? H ? ? (nu ? H ? ? u) = u.
+ intros.
+ apply (eq_ind' ? ? ? ? ? u).
+ unfold nu_inv.
+ apply trans_sym_eq.
+qed.
+
+theorem eq_to_eq_to_eq_p_q:
+ \forall A. \forall x,y:A.
+  (\forall x,y:A. decidable (x=y)) \to
+   \forall p,q:x=y. p=q.
+ intros.
+ rewrite < (nu_left_inv ? H ? ? p).
+ rewrite < (nu_left_inv ? H ? ? q).
+ elim (nu_constant ? H ? ? q).
+ reflexivity.
+qed.
+
+(*CSC: alternative proof that does not pollute the environment with
+  technical lemmata. Unfortunately, it is a pain to do without proper
+  support for let-ins.
+theorem eq_to_eq_to_eq_p_q:
+ \forall A. \forall x,y:A.
+  (\forall x,y:A. decidable (x=y)) \to
+   \forall p,q:x=y. p=q.
+intros.
+letin nu \def
+ (\lambda x,y. \lambda p:x=y.
+   match H x y with
+    [ (or_introl p') \Rightarrow p'
+    | (or_intror K) \Rightarrow False_ind ? (K p) ]).
+cut
+ (\forall q:x=y.
+   eq_ind ? ? (\lambda z. z=y) (nu ? ? q) ? (nu ? ? (refl_eq ? x))
+   = q).
+focus 8.
+ clear q; clear p.
+ intro.
+ apply (eq_ind' ? ? ? ? ? q);
+ fold simplify (nu ? ? (refl_eq ? x)).
+ generalize in match (nu ? ? (refl_eq ? x)); intro.
+ apply
+  (eq_ind' A x
+   (\lambda y. \lambda u.
+    eq_ind A x (\lambda a.a=y) u y u = refl_eq ? y)
+   ? x H1).
+ reflexivity.
+unfocus.
+rewrite < (Hcut p); fold simplify (nu ? ? p).
+rewrite < (Hcut q); fold simplify (nu ? ? q).
+apply (Or_ind' (x=x) (x \neq x)
+ (\lambda p:decidable (x=x). eq_ind A x (\lambda z.z=y) (nu x y p) x
+   ([\lambda H1.eq A x x]
+    match p with
+    [(or_introl p') \Rightarrow p'
+    |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]) =
+   eq_ind A x (\lambda z.z=y) (nu x y q) x
+    ([\lambda H1.eq A x x]
+     match p with
+    [(or_introl p') \Rightarrow p'
+    |(or_intror K) \Rightarrow False_ind (x=x) (K (refl_eq A x))]))
+ ? ? (H x x)).
+intro; simplify; reflexivity.
+intro q; elim (q (refl_eq ? x)).
+qed.
+*)
+
+(*
+theorem a:\forall x.x=x\land True.
+[ 
+2:intros;
+  split;
+  [
+    exact (refl_eq Prop x);
+  |
+    exact I;
+  ]
+1:
+  skip
+]
+qed.
+*)
+