X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Ftests%2Fcoercions_russell.ma;h=48639b3cb54afdad3c89def91c1b04be77f3e83b;hb=e880d6eab5e1700f4a625ddcd7d0fa8f0cce2dcc;hp=f504cef72e9e2a97576892e4d9d003c28df95722;hpb=5ec87fab28597f651b2792081da2264504f2dbf1;p=helm.git

diff --git a/helm/software/matita/tests/coercions_russell.ma b/helm/software/matita/tests/coercions_russell.ma
index f504cef72..48639b3cb 100644
--- a/helm/software/matita/tests/coercions_russell.ma
+++ b/helm/software/matita/tests/coercions_russell.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/test/russell/".
+
 
 include "nat/orders.ma".
 include "list/list.ma".
@@ -22,40 +22,33 @@ inductive sigma (A:Type) (P:A → Prop) : Type ≝
  sig_intro: ∀a:A. P a → sigma A P.
 
 interpretation "sigma" 'exists \eta.x =
-  (cic:/matita/test/russell/sigma.ind#xpointer(1/1) _ x).
+  (cic:/matita/tests/coercions_russell/sigma.ind#xpointer(1/1) _ x).
  
 definition inject ≝ λP.λa:list nat.λp:P a. sig_intro ? P ? p.
 
-coercion cic:/matita/test/russell/inject.con 0 1.
+coercion cic:/matita/tests/coercions_russell/inject.con 0 1.
 
 definition eject ≝ λP.λc: ∃n:list nat.P n. match c with [ sig_intro w _ ⇒ w].
 
-coercion cic:/matita/test/russell/eject.con.
+coercion cic:/matita/tests/coercions_russell/eject.con.
   
 alias symbol "exists" (instance 2) = "exists".
 lemma tl : ∀l:list nat. l ≠ [] → ∃l1.∃a.a :: l1 = l.
-letin program ≝ 
-  (λl:list nat. λH:l ≠ [].match l with [ nil ⇒ λH.[] | cons x l1 ⇒ λH.l1] H);
-letin program_spec ≝ (program : ∀l:list nat. l ≠ [] → ∃l1.∃a.a :: l1 = l);
- [ generalize in match H; cases l; [intros (h1); cases (h1 ?); reflexivity]
-   intros; apply (ex_intro ? ? n); apply eq_f; reflexivity; ]
-exact program_spec;
+ apply rule (λl:list nat. λ_.match l with [ nil ⇒ [] | cons x l1 ⇒ l1]);
+  [ exists; [2: reflexivity | skip]
+  | destruct; elim H; reflexivity]
 qed.
 
 alias symbol "exists" (instance 3) = "exists".
 lemma tl2 : ∀l:∃l:list nat. l ≠ []. ∃l1.∃a.a :: l1 = l.
-letin program ≝ 
-  (λl:list nat. match l with [ nil ⇒ [] | cons x l1 ⇒ l1]);
-letin program_spec ≝ 
-  (program : ∀l:∃l:list nat. l ≠ []. ∃l1.∃a.a :: l1 = l);
-  [ autobatch; | generalize in match H; clear H; cases s; simplify;
-    intros; cases (H H1); ]
-exact program_spec.
+apply rule (λl:list nat. match l with [ nil ⇒ [] | cons x l1 ⇒ l1]);
+[ autobatch; 
+| cases s in H; simplify; intros; cases (H H1); ]
 qed.
 
 definition nat_return := λn:nat. Some ? n.
 
-coercion cic:/matita/test/russell/nat_return.con.
+coercion cic:/matita/tests/coercions_russell/nat_return.con.
 
 definition raise_exn := None nat.
 
@@ -76,67 +69,63 @@ include "nat/compare.ma".
 
 definition inject_opt ≝ λP.λa:option nat.λp:P a. sig_intro ? P ? p.
 
-coercion cic:/matita/test/russell/inject_opt.con 0 1.
+coercion cic:/matita/tests/coercions_russell/inject_opt.con 0 1.
 
 definition eject_opt ≝ λP.λc: ∃n:option nat.P n. match c with [ sig_intro w _ ⇒ w].
 
-coercion cic:/matita/test/russell/eject_opt.con.
+coercion cic:/matita/tests/coercions_russell/eject_opt.con.
 
-definition find :
- ∀p:nat → bool.
-  ∀l:list nat. sigma ? (λres:option nat.
+(* we may define mem as in the following lemma and get rid of it *)
+definition find_spec ≝
+  λl,p.λres:option nat.
    match res with
     [ None ⇒ ∀y. mem ? eqb y l = true → p y = false
-    | Some x ⇒ mem ? eqb x l = true ∧ p x = true ]).
-letin program ≝
- (λp.
+    | Some x ⇒ mem ? eqb x l = true ∧ 
+               p x = true ∧ 
+               ∀y.mem ? eqb y l = true → p y = true → x ≠ y → 
+                    ∃l1,l2,l3.l = l1 @ [x] @ l2 @ [y] @ l3].
+
+lemma mem_x_to_ex_l1_l2 : ∀l,x.mem ? eqb x l = true → ∃l1,l2.l = l1 @ [x] @ l2.
+intros 2; elim l (H hd tl IH H); [simplify in H; destruct H]
+generalize in match H; clear H;
+simplify; apply (eqb_elim x hd); simplify; intros;
+[1:clear IH; rewrite < H; apply (ex_intro ? ? []); 
+|2:lapply(IH H1); clear H1 IH; decompose; rewrite > H2; clear H2]
+simplify; autobatch;
+qed.
+
+notation > "'If' b 'Then' t 'Else' f" non associative
+with precedence 90 for @{ match $b with [ true ⇒ $t | _ ⇒ $f ] }.
+
+alias symbol "exists" = "sigma".
+definition sigma_find_spec : ∀p,l. ∃res.find_spec l p res.
+apply rule (λp. 
   let rec aux l ≝
-   match l with
+    match l with
     [ nil ⇒ raise_exn
-    | cons x l ⇒ match p x with [ true ⇒ nat_return x | false ⇒ aux l ]
-    ]
-  in
-   aux);
-apply
- (program : ∀p:nat → bool.
-  ∀l:list nat. ∃res:option nat.
-   match res with
-    [ None ⇒ ∀y:nat. (mem nat eqb y l = true : Prop) → p y = false
-    | Some (x:nat) ⇒ mem nat eqb x l = true ∧ p x = true ]);
-clear program;
- [ cases (aux l1); clear aux;
-   simplify in ⊢ (match % in option return ? with [None⇒?|Some⇒?]);
-   generalize in match H2; clear H2;
-   cases a;
-    [ simplify;
-      intros 2;
-      apply (eqb_elim y n);
-       [ intros;
-         autobatch
-       | intros;
-         apply H2;
-         simplify in H4;
-         exact H4
-       ]
-    | simplify;
-      intros;
-      cases H2; clear H2;
-      split;
-       [ elim (eqb n1 n);
-         simplify;
-         autobatch
-       | assumption
-       ]
-    ]
- | unfold nat_return; simplify;
-   split;
-    [ rewrite > eqb_n_n;
-      reflexivity
-    | assumption
-    ]
- | unfold raise_exn; simplify;
-   intros;
-   change in H1 with (false = true);
-   destruct H1 
- ]
+    | cons x l ⇒ If p x Then nat_return x Else aux l]
+    in aux);
+(* l = x::tl ∧ p x = false *)
+[1: cases (aux l1); clear aux;
+    cases a in H2; simplify;
+    [1: intros 2; apply (eqb_elim y n); intros (Eyn); autobatch;
+    |2: intros; decompose; repeat split; [2: assumption]; intros;
+        [1: cases (eqb n1 n); simplify; autobatch;
+        |2: generalize in match (refl_eq ? (eqb y n));
+            generalize in ⊢ (? ? ? %→?); 
+            intro; cases b; clear b; intro Eyn; rewrite > Eyn in H3; simplify in H3;
+            [1: rewrite > (eqb_true_to_eq ? ? Eyn) in H6; rewrite > H1 in H6; destruct H6;
+            |2: lapply H4; try assumption; decompose; clear H4; rewrite > H8;
+                simplify; autobatch depth = 4;]]]
+(* l = x::tl ∧ p x = true *)
+|2: unfold find_spec; unfold nat_return; simplify; repeat split; [2: assumption]
+    [1: rewrite > eqb_n_n; reflexivity
+    |2: intro; generalize in match (refl_eq ? (eqb y n)); generalize in ⊢ (? ? ? %→?);
+        intro; cases b; clear b; intro Eyn; rewrite > Eyn; 
+        [1: rewrite > (eqb_true_to_eq ? ? Eyn);] clear Eyn; simplify; intros;
+          [1: cases H4; reflexivity
+          |2: lapply (mem_x_to_ex_l1_l2 ? ? H2); decompose; rewrite > H6;
+              apply (ex_intro ? ? []); simplify; autobatch;]]
+(* l = [] *)
+|3: unfold raise_exn; simplify; intros; destruct H1;]
 qed.
\ No newline at end of file