| intros; whd; split; assumption
| intros; apply transitive_subseteq_operator; [2: apply f; | skip | assumption]
| intros; cases f; exists [apply w] assumption
- | intros; intros 2; apply (f ? f1 i);
- | intros; intros 2; apply f;
- (* senza questa change, universe inconsistency *)
- whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists; [apply i] assumption;
+ | intros; split; [ intros 4; apply (f ? f1 i); | intros 3; intro; apply (f i ? f1); ]
+ | intros; split;
+ [ intros 4; apply f; exists; [apply i] assumption;
+ | intros 3; intros; cases f1; apply (f w a x); ]
| intros 3; cases f;
| intros 3; constructor 1;
| intros; cases f; exists; [apply w]
[ assumption
| whd; intros; cases i; simplify; assumption]
| intros; split; intro;
- [ cases f; cases x1;
- (* senza questa change, universe inconsistency *)
- change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists [apply w1] exists [apply w] assumption;
- | cases e; cases x; exists; [apply w1]
- [ assumption
- | (* senza questa change, universe inconsistency *)
- whd; change in ⊢ (? ? (λ_:%.?)) with (carr I);
- exists; [apply w] assumption]]
+ [ cases f; cases x1; exists [apply w1] exists [apply w] assumption;
+ | cases e; cases x; exists; [apply w1] [ assumption | exists; [apply w] assumption]]
| intros; intros 2; cases (f (singleton ? a) ?);
[ exists; [apply a] [assumption | change with (a = a); apply refl1;]
| change in x1 with (a = w); change with (mem A a q); apply (. (x1‡#));
intros;
constructor 1;
[ constructor 1;
- [ apply (λU.image ?? t U);
+ [ apply (λU.image ?? c U);
| intros; apply (#‡e); ]
| constructor 1;
- [ apply (λU.minus_star_image ?? t U);
+ [ apply (λU.minus_star_image ?? c U);
| intros; apply (#‡e); ]
| constructor 1;
- [ apply (λU.star_image ?? t U);
+ [ apply (λU.star_image ?? c U);
| intros; apply (#‡e); ]
| constructor 1;
- [ apply (λU.minus_image ?? t U);
+ [ apply (λU.minus_image ?? c 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);
+ [ change in f with (∀a. a ∈ image ?? c p → a ∈ q);
+ change with (∀a:o1. a ∈ p → a ∈ star_image ?? c 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);
+ | change in f with (∀a:o1. a ∈ p → a ∈ star_image ?? c q);
+ change with (∀a. a ∈ image ?? c 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);
+ [ change in f with (∀a. a ∈ minus_image ?? c p → a ∈ q);
+ change with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c 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);
+ | change in f with (∀a:o2. a ∈ p → a ∈ minus_star_image ?? c q);
+ change with (∀a. a ∈ minus_image ?? c 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]
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'.
+ ∀o1,o2:REL.∀t,t': arrows1 ? o1 o2. t = t' → eq2 ? (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‡#)); ]
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).
+ ∀o1:REL. eq2 ? (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‡#); apply f;
| alias symbol "and" = "and_morphism".
- change with ((∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a) → a1 ∈ a);
+ change with ((∃y: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‡#); apply f1;
- | change with (a1 ∈ a → ∃y: carr o1.a1 ♮(id1 REL o1) y ∧ y∈a);
+ | change with (a1 ∈ a → ∃y: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);
+ | change with ((∃x: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^-1‡#); apply f1;
- | change with (a1 ∈ a → ∃x: carr o1.x ♮(id1 REL o1) a1∧x∈a);
+ | change with (a1 ∈ a → ∃x: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 in f1 with (a1 = y); apply (. f1^-1‡#); apply f;]
qed.
-lemma hint2: ∀S,T. carr2 (arrows2 OA S T) → Type_OF_setoid2 (arrows2 OA S T).
- intros; apply c;
-qed.
-coercion hint2.
-
(* CSC: ???? forse un uncertain mancato *)
+alias symbol "eq" = "setoid2 eq".
+alias symbol "compose" = "category1 composition".
lemma orelation_of_relation_preserves_composition:
∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
orelation_of_relation ?? (G ∘ F) =
split; [ assumption | exists; [apply w] split; assumption ]
| cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
split; [ exists; [apply w] split; assumption | assumption ]
- | cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
+ | unfold arrows1_of_ORelation_setoid;
+ cases e; cases x; cases f; cases x1; clear e x f x1; exists; [ apply w1 ]
split; [ assumption | exists; [apply w] split; assumption ]
- | cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
+ | unfold arrows1_of_ORelation_setoid in e;
+ cases e; cases x; cases f1; cases x1; clear e x f1 x1; exists; [apply w1 ]
split; [ exists; [apply w] split; assumption | assumption ]
| whd; intros; apply f; exists; [ apply y] split; assumption;
| cases f1; clear f1; cases x; clear x; apply (f w); assumption;]
qed.
+
+definition SUBSETS': carr3 (arrows3 CAT2 (category2_of_category1 REL) OA).
+ constructor 1;
+ [ apply SUBSETS;
+ | intros; constructor 1;
+ [ apply (orelation_of_relation S T);
+ | intros; apply (orelation_of_relation_preserves_equality S T a a' e); ]
+ | apply orelation_of_relation_preserves_identity;
+ | simplify; intros;
+ apply (.= (orelation_of_relation_preserves_composition o1 o2 o4 f1 (f3∘f2)));
+ apply (#‡(orelation_of_relation_preserves_composition o2 o3 o4 f2 f3)); ]
+qed.
+
+theorem SUBSETS_faithful:
+ ∀S,T.∀f,g:arrows2 (category2_of_category1 REL) S T.
+ map_arrows2 ?? SUBSETS' ?? f = map_arrows2 ?? SUBSETS' ?? g → f=g.
+ intros; unfold SUBSETS' in e; simplify in e; cases e;
+ unfold orelation_of_relation in e3; simplify in e3; clear e e1 e2 e4;
+ intros 2; lapply (e3 (singleton ? x)); cases Hletin;
+ split; intro; [ lapply (s y); | lapply (s1 y); ]
+ [2,4: exists; [1,3:apply x] split; [1,3: assumption |*: change with (x=x); apply rule #]
+ |*: cases Hletin1; cases x1; change in f3 with (eq1 ? x w); apply (. f3‡#); assumption;]
+qed.
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projects / helm.git / blobdiff