// used everywhere!

diff --git a/helm/software/matita/nlibrary/sets/sets.ma b/helm/software/matita/nlibrary/sets/sets.ma
index 113654ad330de847908ae68b3fcf65faaf9c3033..241282c1acde3865a377f1fea26436d4e0fa85ce 100644 (file)
--- a/helm/software/matita/nlibrary/sets/sets.ma
+++ b/helm/software/matita/nlibrary/sets/sets.ma
@@ -41,22 +41,19 @@ ndefinition big_intersection ≝ λA,B.λT:Ω^A.λf:A → Ω^B.{ x | ∀i. i ∈
 ndefinition full_set: ∀A. Ω^A ≝ λA.{ x | True }.
 
 nlemma subseteq_refl: ∀A.∀S: Ω^A. S ⊆ S.
- #A; #S; #x; #H; nassumption.
-nqed.
+//.nqed.
 
 nlemma subseteq_trans: ∀A.∀S,T,U: Ω^A. S ⊆ T → T ⊆ U → S ⊆ U.
- #A; #S; #T; #U; #H1; #H2; #x; #P; napply H2; napply H1; nassumption.
-nqed.
+/3/.nqed.
 
 include "properties/relations1.ma".
 
 ndefinition seteq: ∀A. equivalence_relation1 (Ω^A).
  #A; @
   [ napply (λS,S'. S ⊆ S' ∧ S' ⊆ S)
-  | #S; @; napply subseteq_refl
-  | #S; #S'; *; #H1; #H2; @; nassumption
-  | #S; #T; #U; *; #H1; #H2; *; #H3; #H4; @; napply subseteq_trans;
-     ##[##2,5: nassumption |##1,4: ##skip |##*: nassumption]##]
+  | /2/
+  | #S; #S'; *; /2/
+  | #S; #T; #U; *; #H1; #H2; *; /3/]
 nqed.
 
 include "sets/setoids1.ma".
@@ -65,7 +62,7 @@ include "sets/setoids1.ma".
 ncoercion full_set : ∀A:Type[0]. Ω^A ≝ full_set on A: Type[0] to (Ω^?).
 
 ndefinition powerclass_setoid: Type[0] → setoid1.
- #A; @[ napply (Ω^A)| napply seteq ]
+ #A; @(Ω^A);//.
 nqed.
 
 include "hints_declaration.ma". 
@@ -98,17 +95,11 @@ nqed.
 ncoercion Full_set: ∀A. ext_powerclass A ≝ Full_set on A: setoid to ext_powerclass ?.
 
 ndefinition ext_seteq: ∀A. equivalence_relation1 (𝛀^A).
- #A; @
-  [ napply (λS,S'. S = S')
-  | #S; napply (refl1 ? (seteq A))
-  | #S; #S'; napply (sym1 ? (seteq A))
-  | #S; #T; #U; napply (trans1 ? (seteq A))]
+ #A; @ [ napply (λS,S'. S = S') ] /2/.
 nqed.
 
 ndefinition ext_powerclass_setoid: setoid → setoid1.
- #A; @
-  [ napply (ext_powerclass A)
-  | napply (ext_seteq A) ]
+ #A; @ (ext_seteq A).
 nqed.
               
 unification hint 0 ≔ A;
@@ -126,7 +117,7 @@ on _x : (carr1 (ext_powerclass_setoid ?)) to (Ω^?).
 nlemma mem_ext_powerclass_setoid_is_morph: 
  ∀A. binary_morphism1 (setoid1_of_setoid A) (ext_powerclass_setoid A) CPROP.
  #A; @
-  [ napply (λx,S. x ∈ S) 
+  [ napply (λx,S. x ∈ S)
   | #a; #a'; #b; #b'; #Ha; *; #Hb1; #Hb2; @; #H;
      ##[ napply Hb1; napply (. (ext_prop … Ha^-1)); nassumption;
      ##| napply Hb2; napply (. (ext_prop … Ha)); nassumption;
@@ -147,11 +138,7 @@ unification hint 0 ≔  A:setoid, x, S;
 nlemma subseteq_is_morph: ∀A. binary_morphism1 (ext_powerclass_setoid A) (ext_powerclass_setoid A) CPROP.
  #A; @
   [ napply (λS,S'. S ⊆ S')
-  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #H
-     [ napply (subseteq_trans … a)
-        [ nassumption | napply (subseteq_trans … b); nassumption ]
-   ##| napply (subseteq_trans … a')
-        [ nassumption | napply (subseteq_trans … b'); nassumption ] ##]
+  | #a; #a'; #b; #b'; *; #Ha1; #Ha2; *;/4/]
 nqed.
 
 unification hint 0 ≔ A,a,a'
@@ -176,11 +163,7 @@ unification hint 0 ≔
 nlemma intersect_is_morph: 
  ∀A. binary_morphism1 (powerclass_setoid A) (powerclass_setoid A) (powerclass_setoid A).
  #A; @ (λS,S'. S ∩ S');
- #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *; #Ka; #Kb; @
-  [ napply Ha1; nassumption
-  | napply Hb1; nassumption
-  | napply Ha2; nassumption
-  | napply Hb2; nassumption]
+ #a; #a'; #b; #b'; *; #Ha1; #Ha2; *; #Hb1; #Hb2; @; #x; nnormalize; *;/3/.
 nqed.
 
 alias symbol "hint_decl" = "hint_decl_Type1".
@@ -298,30 +281,26 @@ nqed.
 ndefinition eqrel_of_morphism:
  ∀A,B. unary_morphism A B → compatible_equivalence_relation A.
  #A; #B; #f; @
-  [ @
-     [ napply (λx,y. f x = f y)
-     | #x; napply refl | #x; #y; napply sym | #x; #y; #z; napply trans]
+  [ @ [ napply (λx,y. f x = f y) ] /2/;
 ##| #x; #x'; #H; nwhd; alias symbol "prop1" = "prop1".
-napply (.= (†H)); napply refl ]
+napply (.= (†H)); // ]
 nqed.
 
 ndefinition canonical_proj: ∀A,R. unary_morphism A (quotient A R).
  #A; #R; @
-  [ napply (λx.x) | #a; #a'; #H; napply (compatibility … R … H) ]
+  [ napply (λx.x) |  #a; #a'; #H; napply (compatibility … R … H) ]
 nqed.
 
 ndefinition quotiented_mor:
  ∀A,B.∀f:unary_morphism A B.
   unary_morphism (quotient … (eqrel_of_morphism … f)) B.
- #A; #B; #f; @
-  [ napply f | #a; #a'; #H; nassumption]
+ #A; #B; #f; @ [ napply f ] //.
 nqed.
 
 nlemma first_omomorphism_theorem_functions1:
  ∀A,B.∀f: unary_morphism A B.
   ∀x. f x = quotiented_mor … (canonical_proj … (eqrel_of_morphism … f) x).
- #A; #B; #f; #x; napply refl;
-nqed.
+//. nqed.
 
 alias symbol "eq" = "setoid eq".
 ndefinition surjective ≝
@@ -335,9 +314,7 @@ ndefinition injective ≝
 nlemma first_omomorphism_theorem_functions2:
  ∀A,B.∀f: unary_morphism A B. 
    surjective … (Full_set ?) (Full_set ?) (canonical_proj ? (eqrel_of_morphism … f)).
- #A; #B; #f; nwhd; #y; #Hy; @ y; @ I ; napply refl; 
- (* bug, prova @ I refl *)
-nqed.
+/3/. nqed.
 
 nlemma first_omomorphism_theorem_functions3:
  ∀A,B.∀f: unary_morphism A B. 
@@ -353,16 +330,15 @@ nrecord isomorphism (A, B : setoid) (S: ext_powerclass A) (T: ext_powerclass B)
  }.
 
 nlemma subseteq_intersection_l: ∀A.∀U,V,W:Ω^A.U ⊆ W ∨ V ⊆ W → U ∩ V ⊆ W.
-#A; #U; #V; #W; *; #H; #x; *; #xU; #xV; napply H; nassumption;
+#A; #U; #V; #W; *; #H; #x; *; /2/.
 nqed.
 
 nlemma subseteq_union_l: ∀A.∀U,V,W:Ω^A.U ⊆ W → V ⊆ W → U ∪ V ⊆ W.
-#A; #U; #V; #W; #H; #H1; #x; *; #Hx; ##[ napply H; ##| napply H1; ##] nassumption;
-nqed. 
+#A; #U; #V; #W; #H; #H1; #x; *; /2/.
+nqed.
 
 nlemma subseteq_intersection_r: ∀A.∀U,V,W:Ω^A.W ⊆ U → W ⊆ V → W ⊆ U ∩ V.
-#A; #U; #V; #W; #H1; #H2; #x; #Hx; @; ##[ napply H1; ##| napply H2; ##] nassumption;
-nqed. 
+/3/. nqed.
 
 (*
 nrecord isomorphism (A, B : setoid) (S: qpowerclass A) (T: qpowerclass B) : CProp[0] ≝