first [apply refl | assumption]]]
qed.
-(*
definition full_subset: ∀s:REL. Ω \sup s.
apply (λs.{x | True});
intros; simplify; split; intro; assumption.
qed.
coercion full_subset.
-*)
definition setoid1_of_REL: REL → setoid ≝ λS. S.
coercion setoid1_of_REL.
qed.
coercion Type_OF_setoid1_of_REL.
-(*
-definition comprehension: ∀b:REL. (b ⇒ CPROP) → Ω \sup b.
- apply (λb:REL. λP: b ⇒ CPROP. {x | x ∈ b ∧ P x});
- intros; simplify; apply (.= (H‡#)‡(†H)); apply refl1.
+definition comprehension: ∀b:REL. (unary_morphism1 b CPROP) → Ω \sup b.
+ apply (λb:REL. λP: b ⇒ CPROP. {x | P x});
+ intros; simplify;
+ alias symbol "trans" = "trans1".
+ alias symbol "prop1" = "prop11".
+ apply (.= †e); apply refl1.
qed.
interpretation "subset comprehension" 'comprehension s p =
- (comprehension s (mk_unary_morphism __ p _)).
+ (comprehension s (mk_unary_morphism1 __ p _)).
definition ext: ∀X,S:REL. binary_morphism1 (arrows1 ? X S) S (Ω \sup X).
- apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 ? X S.λf:S.{x ∈ X | x ♮r f}) ?);
- [ intros; simplify; apply (.= (H‡#)); apply refl1
- | intros; simplify; split; intros; simplify; intros; cases f; split; try assumption;
- [ apply (. (#‡H1)); whd in H; apply (if ?? (H ??)); assumption
- | apply (. (#‡H1\sup -1)); whd in H; apply (fi ?? (H ??));assumption]]
+ apply (λX,S.mk_binary_morphism1 ??? (λr:arrows1 REL X S.λf:S.{x ∈ X | x ♮r f}) ?);
+ [ intros; simplify; apply (.= (e‡#)); apply refl1
+ | intros; simplify; split; intros; simplify;
+ [ change with (∀x. x ♮a b → x ♮a' b'); intros;
+ apply (. (#‡e1)); whd in e; apply (if ?? (e ??)); assumption
+ | change with (∀x. x ♮a' b' → x ♮a b); intros;
+ apply (. (#‡e1\sup -1)); whd in e; apply (fi ?? (e ??));assumption]]
qed.
-
+(*
definition extS: ∀X,S:REL. ∀r: arrows1 ? X S. Ω \sup S ⇒ Ω \sup X.
(* ∃ is not yet a morphism apply (λX,S,r,F.{x ∈ X | ∃a. a ∈ F ∧ x ♮r a});*)
intros (X S r); constructor 1;
| exists; try assumption; split; try assumption; change with (x = x); apply refl]
qed.
*)
-
-include "o-algebra.ma".
-
-definition orelation_of_relation: ∀o1,o2:REL. arrows1 ? o1 o2 → arrows2 OA (SUBSETS o1) (SUBSETS o2).
- intros;
- constructor 1;
- [ constructor 1;
- [ apply (λU.image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_star_image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.star_image ?? t U);
- | intros; apply (#‡e); ]
- | constructor 1;
- [ apply (λU.minus_image ?? t U);
- | intros; apply (#‡e); ]
- | intros; split; intro;
- [ change in f with (∀a. a ∈ image ?? t p → a ∈ q);
- change with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? t q);
- change with (∀a. a ∈ image ?? t p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
- | intros; split; intro;
- [ change in f with (∀a. a ∈ minus_image ?? t p → a ∈ q);
- change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
- intros 4; apply f; exists; [apply a] split; assumption;
- | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? t q);
- change with (∀a. a ∈ minus_image ?? t p → a ∈ q);
- intros; cases f1; cases x; clear f1 x; apply (f ? f3); assumption; ]
- | intros; split; intro; cases f; clear f;
- [ cases x; cases x2; clear x x2; exists; [apply w1]
- [ assumption;
- | exists; [apply w] split; assumption]
- | cases x1; cases x2; clear x1 x2; exists; [apply w1]
- [ exists; [apply w] split; assumption;
- | assumption; ]]]
-qed.
-
-lemma orelation_of_relation_preserves_equality:
- ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. eq1 ? t t' → orelation_of_relation ?? t = orelation_of_relation ?? t'.
- intros; split; unfold orelation_of_relation; simplify; intro; split; intro;
- simplify; whd in o1 o2;
- [ change with (a1 ∈ minus_star_image ?? t a → a1 ∈ minus_star_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ minus_star_image ?? t' a → a1 ∈ minus_star_image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ minus_image ?? t a → a1 ∈ minus_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ minus_image ?? t' a → a1 ∈ minus_image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ image ?? t a → a1 ∈ image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ image ?? t' a → a1 ∈ image ?? t a);
- apply (. #‡(e ^ -1‡#));
- | change with (a1 ∈ star_image ?? t a → a1 ∈ star_image ?? t' a);
- apply (. #‡(e‡#));
- | change with (a1 ∈ star_image ?? t' a → a1 ∈ star_image ?? t a);
- apply (. #‡(e ^ -1‡#)); ]
-qed.
-
-lemma hint: ∀o1,o2:OA. Type_OF_setoid2 (arrows2 ? o1 o2) → carr2 (arrows2 OA o1 o2).
- intros; apply t;
-qed.
-coercion hint.
-
-lemma orelation_of_relation_preserves_identity:
- ∀o1:REL. orelation_of_relation ?? (id1 ? o1) = id2 OA (SUBSETS o1).
- intros; split; intro; split; whd; intro;
- [ change with ((∀x. x ♮(id1 REL o1) a1→x∈a) → a1 ∈ a); intros;
- apply (f a1); change with (a1 = a1); apply refl1;
- | change with (a1 ∈ a → ∀x. x ♮(id1 REL o1) a1→x∈a); intros;
- change in f1 with (x = a1); apply (. f1 ^ -1‡#); apply f;
- | alias symbol "and" = "and_morphism".
- change with ((∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
- intro; cases e; clear e; cases x; clear x; change in f with (a1=w);
- apply (. f^-1‡#); apply f1;
- | change with (a1 ∈ a → ∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a);
- intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
- | change with ((∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a) → a1 ∈ a);
- intro; cases e; clear e; cases x; clear x; change in f with (w=a1);
- apply (. f‡#); apply f1;
- | change with (a1 ∈ a → ∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a);
- intro; exists; [apply a1]; split; [ change with (a1=a1); apply refl1; | apply f]
- | change with ((∀y.a1 ♮(id1 REL o1) y→y∈a) → a1 ∈ a); intros;
- apply (f a1); change with (a1 = a1); apply refl1;
- | change with (a1 ∈ a → ∀y.a1 ♮(id1 REL o1) y→y∈a); intros;
- change in f1 with (a1 = y); apply (. f1‡#); apply f;]
-qed.
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projects / helm.git / blobdiff