include "o-algebra.ma".
-definition hint1: OA → Type ≝ λc:OA.carr (oa_P c).
-coercion hint1.
-
-definition hint2: ∀C.hint1 C → carr1 ((λx.x) (setoid1_of_setoid (oa_P C))).
-intros; assumption;
-qed.
-coercion hint2.
-
alias symbol "eq" = "setoid1 eq".
-definition is_saturation ≝
- λC:OA.λA:C → C.
+definition is_o_saturation: ∀C:OA. unary_morphism1 C C → CProp1 ≝
+ λC:OA.λA:unary_morphism1 C C.
∀U,V. (U ≤ A V) = (A U ≤ A V).
-definition is_reduction ≝
- λC:OA.λJ:C → C.
+definition is_o_reduction: ∀C:OA. unary_morphism1 C C → CProp1 ≝
+ λC:OA.λJ:unary_morphism1 C C.
∀U,V. (J U ≤ V) = (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:C. 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,A. 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,A. is_o_saturation C A → ∀U.
+ eq1 C (A (A U)) (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.