notation made half decent

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / subsets.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
index 3c855236be7cbb1c1a6473b995e8cf16828c9bd5..8eedb552351134f7a9320fd6bdf88a4f2149e860 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/subsets.ma
@@ -14,10 +14,13 @@
 
 include "categories.ma".
 
-record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: unary_morphism1 A CPROP }.
+record powerset_carrier (A: objs1 SET) : Type1 ≝ { mem_operator: A ⇒_1 CPROP }.
+interpretation "powerset low" 'powerset A = (powerset_carrier A).
+notation "hvbox(a break ∈. b)" non associative with precedence 45 for @{ 'mem_low $a $b }.
+interpretation "memlow" 'mem_low a S = (mem_operator ? S a).
 
-definition subseteq_operator: ∀A: SET. powerset_carrier A → powerset_carrier A → CProp0 ≝
- λA:objs1 SET.λU,V.∀a:A. mem_operator ? U a → mem_operator ? V a.
+definition subseteq_operator: ∀A: objs1 SET. Ω^A → Ω^A → CProp0 ≝
+ λA:objs1 SET.λU,V.∀a:A. a ∈. U → a ∈. V.
 
 theorem transitive_subseteq_operator: ∀A. transitive2 ? (subseteq_operator A).
  intros 6; intros 2;
@@ -43,7 +46,7 @@ interpretation "powerset" 'powerset A = (powerset_setoid1 A).
 interpretation "subset construction" 'subset \eta.x =
  (mk_powerset_carrier ? (mk_unary_morphism1 ? CPROP x ?)).
 
-definition mem: ∀A. binary_morphism1 A (Ω \sup A) CPROP.
+definition mem: ∀A. A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
   [ apply (λx,S. mem_operator ? S x)
@@ -58,7 +61,7 @@ qed.
 
 interpretation "mem" 'mem a S = (fun21 ??? (mem ?) a S).
 
-definition subseteq: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition subseteq: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
   [ apply (λU,V. subseteq_operator ? U V)
@@ -73,21 +76,18 @@ qed.
 
 interpretation "subseteq" 'subseteq U V = (fun21 ??? (subseteq ?) U V).
 
-
-
-theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S.
+theorem subseteq_refl: ∀A.∀S:Ω^A.S ⊆ S.
  intros 4; assumption.
 qed.
 
-theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
+theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω^A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3.
  intros; apply transitive_subseteq_operator; [apply S2] assumption.
 qed.
 
-definition overlaps: ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) CPROP.
+definition overlaps: ∀A. Ω^A × Ω^A ⇒_1 CPROP.
  intros;
  constructor 1;
-  (* Se metto x al posto di ? ottengo una universe inconsistency *)
-  [ apply (λA:objs1 SET.λU,V:Ω \sup A.(exT2 ? (λx:A.?(*x*) ∈ U) (λx:A.?(*x*) ∈ V) : CProp0))
+  [ apply (λA:objs1 SET.λU,V:Ω^A.(exT2 ? (λx:A.x ∈ U) (λx:A.x ∈ V) : CProp0))
   | intros;
     constructor 1; intro; cases e2; exists; [1,4: apply w]
      [ apply (. #‡e^-1); assumption
@@ -98,8 +98,7 @@ qed.
 
 interpretation "overlaps" 'overlaps U V = (fun21 ??? (overlaps ?) U V).
 
-definition intersects:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition intersects: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
  intros;
  constructor 1;
   [ apply rule (λU,V. {x | x ∈ U ∧ x ∈ V });
@@ -112,8 +111,7 @@ qed.
 
 interpretation "intersects" 'intersects U V = (fun21 ??? (intersects ?) U V).
 
-definition union:
- ∀A. binary_morphism1 (Ω \sup A) (Ω \sup A) (Ω \sup A).
+definition union: ∀A. Ω^A × Ω^A ⇒_1 Ω^A.
  intros;
  constructor 1;
   [ apply (λU,V. {x | x ∈ U ∨ x ∈ V });
@@ -127,7 +125,7 @@ qed.
 interpretation "union" 'union U V = (fun21 ??? (union ?) U V).
 
 (* qua non riesco a mettere set *)
-definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
+definition singleton: ∀A:setoid. A ⇒_1 Ω^A.
  intros; constructor 1;
   [ apply (λa:A.{b | a =_0 b}); unfold setoid1_of_setoid; simplify;
     intros; simplify;
@@ -140,9 +138,9 @@ definition singleton: ∀A:setoid. unary_morphism1 A (Ω \sup A).
 qed.
 
 interpretation "singleton" 'singl a = (fun11 ?? (singleton ?) a).
+notation > "{ term 19 a : S }" with precedence 90 for @{fun11 ?? (singleton $S) $a}.
 
-definition big_intersects:
- ∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
+definition big_intersects: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
  intros; constructor 1;
   [ intro; whd; whd in I;
     apply ({x | ∀i:I. x ∈ c i});
@@ -153,8 +151,7 @@ definition big_intersects:
      | apply (. (#‡e i)); apply f]]
 qed.
 
-definition big_union:
- ∀A:SET.∀I:SET.unary_morphism2 (setoid1_of_setoid I ⇒ Ω \sup A) (setoid2_of_setoid1 (Ω \sup A)).
+definition big_union: ∀A:SET.∀I:SET.(I ⇒_2 Ω^A) ⇒_2 Ω^A.
  intros; constructor 1;
   [ intro; whd; whd in A; whd in I;
     apply ({x | ∃i:I. x ∈ c i });