some notation for map_arrows2

[helm.git] / helm / software / matita / contribs / formal_topology / overlap / o-saturations.ma

diff --git a/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma b/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma
index 22c8fbdeaf7fe9e583a17a4d5230aa7a89b93545..bb193508e913086258728c46e74a34317b5f9dfe 100644 (file)
--- a/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma
+++ b/helm/software/matita/contribs/formal_topology/overlap/o-saturations.ma
@@ -14,30 +14,24 @@
 
 include "o-algebra.ma".
 
-alias symbol "eq" = "setoid1 eq".
-definition is_saturation ≝
- λC:OA.λA:unary_morphism (oa_P C) (oa_P C).
-  ∀U,V. (U ≤ A V) = (A U ≤ A V).
+definition is_o_saturation: ∀C:OA. C ⇒_1 C → CProp1 ≝
+ λC:OA.λA:C ⇒_1 C.∀U,V. (U ≤ A V) =_1 (A U ≤ A V).
 
-definition is_reduction ≝
- λC:OA.λJ:unary_morphism (oa_P C) (oa_P C).
-    ∀U,V. (J U ≤ V) = (J U ≤ J V).
+definition is_o_reduction: ∀C:OA. C ⇒_1 C → CProp1 ≝
+ λC:OA.λJ:C ⇒_1 C.∀U,V. (J U ≤ V) =_1 (J U ≤ J V).
 
-theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ≤ A U.
- intros; apply (fi ?? (H ??)); apply (oa_leq_refl C).
+theorem o_saturation_expansive: ∀C,A. is_o_saturation C A → ∀U. U ≤ A U.
+ intros; apply (fi ?? (i ??)); apply (oa_leq_refl C).
 qed.
 
-theorem saturation_monotone:
- ∀C,A. is_saturation C A →
-  ∀U,V. U ≤ V → A U ≤ A V.
- intros; apply (if ?? (H ??)); apply (oa_leq_trans C);
-  [apply V|3: apply saturation_expansive ]
+theorem o_saturation_monotone: ∀C:OA.∀A:C ⇒_1 C. is_o_saturation C A → ∀U,V. U ≤ V → A U ≤ A V.
+ intros; apply (if ?? (i ??)); apply (oa_leq_trans C);
+  [apply V|3: apply o_saturation_expansive ]
  assumption.
 qed.
 
-theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. 
- eq (oa_P C) (A (A U)) (A U).
+theorem o_saturation_idempotent: ∀C:OA.∀A:C ⇒_1 C. is_o_saturation C A → ∀U. A (A U) =_1 A U.
  intros; apply (oa_leq_antisym C);
-  [ apply (if ?? (H (A U) U)); apply (oa_leq_refl C).
-  | apply saturation_expansive; assumption]
+  [ apply (if ?? (i (A U) U)); apply (oa_leq_refl C).
+  | apply o_saturation_expansive; assumption]
 qed.