autobatch => [auto]

[helm.git] / matita / library / technicalities / setoids.ma

diff --git a/matita/library/technicalities/setoids.ma b/matita/library/technicalities/setoids.ma
index b2376f17df9ac9e7c1002e53e67eac0e0483ec93..b945931c14dab117679f714a0a3c3d7e3aa1cd72 100644 (file)
--- a/matita/library/technicalities/setoids.ma
+++ b/matita/library/technicalities/setoids.ma
@@ -19,6 +19,7 @@ set "baseuri" "cic:/matita/technicalities/setoids".
 
 include "datatypes/constructors.ma".
 include "logic/connectives2.ma".
+include "logic/coimplication.ma".
 
 (* DEFINITIONS OF Relation_Class AND n-ARY Morphism_Theory *)
 
@@ -51,42 +52,30 @@ inductive Areflexive_Relation_Class  : Type :=
  | AAsymmetric : ∀A.∀Aeq : relation A. Areflexive_Relation_Class.
 
 definition relation_class_of_argument_class : Argument_Class → Relation_Class.
- intro;
- unfold in a ⊢ %;
- elim a;
+ intros (a); cases a;
   [ apply (SymmetricReflexive ? ? ? H H1)
   | apply (AsymmetricReflexive ? something ? ? H)
   | apply (SymmetricAreflexive ? ? ? H)
   | apply (AsymmetricAreflexive ? something ? r)
-  | apply (Leibniz ? T1)
+  | apply (Leibniz ? T)
   ]
 qed.
 
 definition carrier_of_relation_class : ∀X. X_Relation_Class X → Type.
- intros;
- elim x;
- [1,2,3,4,5: exact T1]
+ intros (X x); cases x (A o o o o A o o A o o o A o A); exact A.
 qed.
 
-definition relation_of_relation_class :
+definition relation_of_relation_class:
  ∀X,R. carrier_of_relation_class X R → carrier_of_relation_class X R → Prop.
- intros 2;
- elim R 0;
- simplify;
-  [1,2: intros 4; apply r
-  |3,4: intros 3; apply r
-  | apply eq
- ]
+intros 2; cases R; simplify; [1,2,3,4: assumption | apply (eq T) ]
 qed.
 
 lemma about_carrier_of_relation_class_and_relation_class_of_argument_class :
  ∀R.
   carrier_of_relation_class ? (relation_class_of_argument_class R) =
    carrier_of_relation_class ? R.
- intro;
- elim R;
- reflexivity.
- qed.
+intro; cases R; reflexivity.
+qed.
 
 inductive nelistT (A : Type) : Type :=
    singl : A → nelistT A
@@ -96,11 +85,10 @@ definition Arguments := nelistT Argument_Class.
 
 definition function_type_of_morphism_signature :
  Arguments → Relation_Class → Type.
-  intros (In Out);
-  elim In
-   [ exact (carrier_of_relation_class ? t → carrier_of_relation_class ? Out)
-   | exact (carrier_of_relation_class ? t → T)
-   ]
+  intros (In Out); elim In; 
+  [ exact (carrier_of_relation_class ? t → carrier_of_relation_class ? Out)
+  | exact (carrier_of_relation_class ? t → T)
+  ]
 qed. 
 
 definition make_compatibility_goal_aux:
@@ -111,12 +99,12 @@ definition make_compatibility_goal_aux:
  generalize in match f; clear f;
   [ elim a; simplify in f f1;
      [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
-     | elim t;
+     | cases t;
         [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
         | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
         ]
      | exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
-     | elim t;
+     | cases t;
         [ exact (∀x1,x2. r x1 x2 → relation_of_relation_class ? Out (f x1) (f1 x2))
         | exact (∀x1,x2. r x2 x1 → relation_of_relation_class ? Out (f x1) (f1 x2))
         ]
@@ -127,15 +115,10 @@ definition make_compatibility_goal_aux:
       (carrier_of_relation_class ? t → function_type_of_morphism_signature n Out) →
       Prop).
     elim t; simplify in f f1;
-     [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
-     | elim t1;
-        [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
-        | exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
-        ]
-     | exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
-     | elim t1;
-        [ exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
-        | exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
+     [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
+     |2,4: cases t1;
+        [1,3: exact (∀x1,x2. r x1 x2 → R (f x1) (f1 x2))
+        |2,4: exact (∀x1,x2. r x2 x1 → R (f x1) (f1 x2))
         ]
      | exact (∀x. R (f x) (f1 x))
      ]
@@ -172,9 +155,7 @@ definition morphism_theory_of_function :
   generalize in match f; clear f;
   elim In;
    [ unfold make_compatibility_goal;
-     simplify;
-     intros;
-     whd;
+     whd; intros;
      reflexivity
    | simplify;
      intro;
@@ -228,7 +209,23 @@ definition equality_morphism_of_symmetric_areflexive_transitive_relation:
     unfold transitive in H;
     unfold symmetric in sym;
     intro;
-    auto new
+      [ apply (H x2 x1 x3 ? ?);
+          [apply (sym x1 x2 ?).
+           apply (H1).
+          |apply (H x1 x x3 ? ?);
+            [apply (H3).
+            |apply (H2).
+            ]
+          ]
+      | apply (H x1 x3 x ? ?);
+         [apply (H x1 x2 x3 ? ?);
+            [apply (H1).
+            |apply (H3).
+            ]
+         |apply (sym x x3 ?).
+          apply (H2).
+         ]
+      ]
   ].
 qed.
 
@@ -246,7 +243,26 @@ definition equality_morphism_of_symmetric_reflexive_transitive_relation:
     intro;
     unfold transitive in H;
     unfold symmetric in sym;
-    auto depth=4.
+    [ apply (H x2 x1 x3 ? ?);
+        [apply (sym x1 x2 ?).
+         apply (H1).
+        |apply (H x1 x x3 ? ?);
+          [apply (H3).
+          |apply (H2).
+          ]
+        ]
+    | apply (H x1 x2 x ? ?);
+      [apply (H1).
+      |apply (H x2 x3 x ? ?);
+        [apply (H3).
+        |apply (sym x x3 ?).
+         apply (H x x3 x3 ? ?);
+          [apply (H2).
+          |apply (refl x3).
+          ]
+        ]
+      ]
+    ]
   ]
 qed.
 
@@ -263,7 +279,13 @@ definition equality_morphism_of_asymmetric_areflexive_transitive_relation:
    intros;
    whd;
    intros;
-   auto
+   apply (H x2 x1 x3 ? ?);
+    [apply (H1).
+    |apply (H x1 x x3 ? ?);
+      [apply (H3).
+      |apply (H2).
+      ]
+    ]
  ].
 qed.
 
@@ -280,7 +302,13 @@ definition equality_morphism_of_asymmetric_reflexive_transitive_relation:
    intros;
    whd;
    intro;
-   auto
+   apply (H x2 x1 x3 ? ?);
+     [apply (H1).
+     |apply (H x1 x x3 ? ?);
+       [apply (H3).
+       |apply (H2).
+       ]
+     ]
  ].
 qed.
 
@@ -306,7 +334,6 @@ definition morphism_theory_of_predicate :
     elim In;
      [ reduce;
        intro;
-       alias id "iff_refl" = "cic:/matita/logic/coimplication/iff_refl.con".
        apply iff_refl
      | simplify;
        intro x;
@@ -321,7 +348,8 @@ theorem impl_trans: transitive ? impl.
  whd;
  unfold impl;
  intros;
- auto.
+ apply (H1 ?).apply (H ?).apply (H2).
+ autobatch.
 qed.
 
 (*DA PORTARE: Add Relation Prop impl
@@ -548,7 +576,7 @@ qed.
 
 definition get_rewrite_direction: rewrite_direction → Argument_Class → rewrite_direction.
  intros (dir R);
- cases (variance_of_argument_class R);
+ cases (variance_of_argument_class R) (a);
   [ exact dir
   | cases a;
      [ exact dir                      (* covariant *)
@@ -570,17 +598,12 @@ qed.
 definition directed_relation_of_argument_class:
  ∀dir:rewrite_direction.∀R: Argument_Class.
    carrier_of_relation_class ? R → carrier_of_relation_class ? R → Prop.
-  intros (dir R);
-  generalize in match
-   (about_carrier_of_relation_class_and_relation_class_of_argument_class R);
-  intro H;
-  apply (directed_relation_of_relation_class dir (relation_class_of_argument_class R));
-  apply (eq_rect ? ? (λX.X) ? ? (sym_eq ? ? ? H));
-   [ apply c
-   | apply c1
-   ]
+  intros (dir R c c1);
+  rewrite < (about_carrier_of_relation_class_and_relation_class_of_argument_class R) in c c1;
+  exact (directed_relation_of_relation_class dir (relation_class_of_argument_class R)  c c1).
 qed.
 
+
 definition relation_of_product_of_arguments:
  ∀dir:rewrite_direction.∀In.
   product_of_arguments In → product_of_arguments In → Prop.
@@ -592,12 +615,12 @@ definition relation_of_product_of_arguments:
   | intros;
     change in p with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
     change in p1 with (Prod (carrier_of_relation_class variance t) (product_of_arguments n));
-    cases p;
-    cases p1;
+    cases p (c p2);
+    cases p1 (c1 p3);
    apply And;
     [ exact
-      (directed_relation_of_argument_class (get_rewrite_direction r t) t a a1)
-    | exact (R b b1)
+      (directed_relation_of_argument_class (get_rewrite_direction r t) t c c1)
+    | exact (R p2 p3)
     ]
   ]
 qed. 
@@ -635,26 +658,21 @@ theorem apply_morphism_compatibility_Right2Left:
      generalize in match args1; clear args1;
      generalize in match m2; clear m2;
      generalize in match m1; clear m1;
-     elim t 0;
+     elim t 0; simplify;
       [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
-        simplify in H;
-        simplify in H1;
-        simplify;
         apply H;
         exact H1
       | intros 8 (v T1 r Hr m1 m2 args1 args2);
-        cases v;
+        cases v; 
+        simplify;
         intros (H H1);
-        simplify in H1;
-        apply H;
-        exact H1
+        apply (H ? ? H1);
       | intros;
         apply H1;
         exact H2
       | intros 7 (v);
-        cases v;
+        cases v; simplify;
         intros (H H1);
-        simplify in H1;
         apply H;
         exact H1
       | intros;
@@ -749,10 +767,9 @@ theorem apply_morphism_compatibility_Left2Right:
         directed_relation_of_relation_class Left2Right ?
          (apply_morphism ? ? m1 args1)
          (apply_morphism ? ? m2 args2).
-  intro;
-  elim In;
-   [ simplify in m1 m2 args1 args2 ⊢ %;
-     change in H1 with
+  intro; 
+  elim In 0; simplify; intros;
+   [ change in H1 with
       (directed_relation_of_argument_class
         (get_rewrite_direction Left2Right t) t args1 args2);
      generalize in match H1; clear H1;
@@ -761,11 +778,8 @@ theorem apply_morphism_compatibility_Left2Right:
      generalize in match args1; clear args1;
      generalize in match m2; clear m2;
      generalize in match m1; clear m1;
-     elim t 0;
+     elim t 0; simplify;
       [ intros (T1 r Hs Hr m1 m2 args1 args2 H H1);
-        simplify in H;
-        simplify in H1;
-        simplify;
         apply H;
         exact H1
       | intros 8 (v T1 r Hr m1 m2 args1 args2);
@@ -816,7 +830,6 @@ theorem apply_morphism_compatibility_Left2Right:
      generalize in match m1; clear m1;
      elim t 0;
       [ intros (T1 r Hs Hr m1 m2 H1 t1 t3 H3);
-        simplify in H3;
         change in H1 with
          (∀x1,x2:T1.r x1 x2 → make_compatibility_goal_aux n Out (m1 x1) (m2 x2));
      | intro v;
@@ -1005,39 +1018,39 @@ Qed.
 (* impl IS A MORPHISM *)
 
 Add Morphism impl with signature iff ==> iff ==> iff as Impl_Morphism.
-unfold impl; tauto.
+unfold impl; tautobatch.
 Qed.
 
 (* and IS A MORPHISM *)
 
 Add Morphism and with signature iff ==> iff ==> iff as And_Morphism.
- tauto.
+ tautobatch.
 Qed.
 
 (* or IS A MORPHISM *)
 
 Add Morphism or with signature iff ==> iff ==> iff as Or_Morphism.
- tauto.
+ tautobatch.
 Qed.
 
 (* not IS A MORPHISM *)
 
 Add Morphism not with signature iff ==> iff as Not_Morphism.
- tauto.
+ tautobatch.
 Qed.
 
 (* THE SAME EXAMPLES ON impl *)
 
 Add Morphism and with signature impl ++> impl ++> impl as And_Morphism2.
- unfold impl; tauto.
+ unfold impl; tautobatch.
 Qed.
 
 Add Morphism or with signature impl ++> impl ++> impl as Or_Morphism2.
- unfold impl; tauto.
+ unfold impl; tautobatch.
 Qed.
 
 Add Morphism not with signature impl -→ impl as Not_Morphism2.
- unfold impl; tauto.
+ unfold impl; tautobatch.
 Qed.
 
 *)