[ apply (λA,B.λr,r': binary_relation A B. ∀x,y. r x y ↔ r' x y)
| simplify; intros 3; split; intro; assumption
| simplify; intros 5; split; intro;
- [ apply (fi ?? (H ??)) | apply (if ?? (H ??))] assumption
+ [ apply (fi ?? (f ??)) | apply (if ?? (f ??))] assumption
| simplify; intros 7; split; intro;
- [ apply (if ?? (H1 ??)) | apply (fi ?? (H ??)) ]
- [ apply (if ?? (H ??)) | apply (fi ?? (H1 ??)) ]
+ [ apply (if ?? (f1 ??)) | apply (fi ?? (f ??)) ]
+ [ apply (if ?? (f ??)) | apply (fi ?? (f1 ??)) ]
assumption]]
qed.
(* carr to avoid universe inconsistency *)
apply (λs1:carr A.λs3:carr C.∃s2:carr B. s1 ♮R12 s2 ∧ s2 ♮R23 s3);
| intros;
- split; intro; cases H (w H3); clear H; exists; [1,3: apply w ]
+ split; intro; cases e2 (w H3); clear e2; exists; [1,3: apply w ]
[ apply (. (e‡#)‡(#‡e1)); assumption
| apply (. ((e \sup -1)‡#)‡(#‡(e1 \sup -1))); assumption]]
| intros 8; split; intro H2; simplify in H2 ⊢ %;
split; assumption
|6,7: intros 5; unfold composition; simplify; split; intro;
unfold setoid1_of_setoid in x y; simplify in x y;
- [1,3: cases H (w H1); clear H; cases H1; clear H1; unfold;
+ [1,3: cases e (w H1); clear e; cases H1; clear H1; unfold;
[ apply (. (e ^ -1 : eq1 ? w x)‡#); assumption
| apply (. #‡(e : eq1 ? w y)); assumption]
|2,4: exists; try assumption; split;
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.
+lemma Type_OF_setoid1_of_REL: ∀o1:Type_OF_category1 REL. Type_OF_objs1 o1 → Type_OF_setoid1 ?(*(setoid1_of_SET o1)*).
+ [ apply (setoid1_of_SET o1);
+ | intros; apply t;]
+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.
cases H7; clear H7; exists; [apply w2] split; [assumption] exists [apply w] split;
assumption]
qed.
+*)
(* the same as ⋄ for a basic pair *)
definition image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:U. x ♮r y ∧ x ∈ S});
- intros; simplify; split; intro; cases H1; exists [1,3: apply w]
- [ apply (. (#‡H)‡#); assumption
- | apply (. (#‡H \sup -1)‡#); assumption]
- | intros; split; simplify; intros; cases H2; exists [1,3: apply w]
- [ apply (. #‡(#‡H1)); cases x; split; try assumption;
- apply (if ?? (H ??)); assumption
- | apply (. #‡(#‡H1 \sup -1)); cases x; split; try assumption;
- apply (if ?? (H \sup -1 ??)); assumption]]
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∃x:carr U. x ♮r y ∧ x ∈ S });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ apply (. (#‡e)‡#); assumption
+ | apply (. (#‡e ^ -1)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); assumption]]
qed.
(* the same as □ for a basic pair *)
definition minus_star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup U) (Ω \sup V).
intros; constructor 1;
- [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:U. x ♮r y → x ∈ S});
- intros; simplify; split; intros; apply H1;
- [ apply (. #‡H \sup -1); assumption
- | apply (. #‡H); assumption]
- | intros; split; simplify; intros; [ apply (. #‡H1); | apply (. #‡H1 \sup -1)]
- apply H2; [ apply (if ?? (H \sup -1 ??)); | apply (if ?? (H ??)) ] assumption]
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup U. {y | ∀x:carr U. x ♮r y → x ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. #‡e ^ -1); assumption
+ | apply (. #‡e); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Rest for a basic pair *)
+definition star_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | ∀y:carr V. x ♮r y → y ∈ S});
+ intros; simplify; split; intros; apply f;
+ [ apply (. e ^ -1‡#); assumption
+ | apply (. e‡#); assumption]
+ | intros; split; simplify; intros; [ apply (. #‡e1); | apply (. #‡e1 ^ -1)]
+ apply f; [ apply (if ?? (e ^ -1 ??)); | apply (if ?? (e ??)) ] assumption]
+qed.
+
+(* the same as Ext for a basic pair *)
+definition minus_image: ∀U,V:REL. binary_morphism1 (arrows1 ? U V) (Ω \sup V) (Ω \sup U).
+ intros; constructor 1;
+ [ apply (λr: arrows1 ? U V.λS: Ω \sup V. {x | (*∃x:U. x ♮r y ∧ x ∈ S*)
+ exT ? (λy:carr V.x ♮r y ∧ y ∈ S) });
+ intros; simplify; split; intro; cases e1; exists [1,3: apply w]
+ [ apply (. (e‡#)‡#); assumption
+ | apply (. (e ^ -1‡#)‡#); assumption]
+ | intros; split; simplify; intros; cases e2; exists [1,3: apply w]
+ [ apply (. #‡(#‡e1)); cases x; split; try assumption;
+ apply (if ?? (e ??)); assumption
+ | apply (. #‡(#‡e1 ^ -1)); cases x; split; try assumption;
+ apply (if ?? (e ^ -1 ??)); assumption]]
qed.
+(*
(* minus_image is the same as ext *)
theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
[ cases H; cases x1; change in f2 with (eq1 ? x w); apply (. #‡f2 \sup -1);
assumption
| 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.
+
+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 *)
+lemma orelation_of_relation_preserves_composition:
+ ∀o1,o2,o3:REL.∀F: arrows1 ? o1 o2.∀G: arrows1 ? o2 o3.
+ orelation_of_relation ?? (G ∘ F) =
+ comp2 OA (SUBSETS o1) (SUBSETS o2) (SUBSETS o3)
+ ?? (*(orelation_of_relation ?? F) (orelation_of_relation ?? G)*).
+ [ apply (orelation_of_relation ?? F); | apply (orelation_of_relation ?? G); ]
+ intros; split; intro; split; whd; intro; whd in ⊢ (% → %); intros;
+ [ whd; intros; apply f; exists; [ apply x] split; assumption;
+ | cases f1; clear f1; cases x1; clear x1; apply (f w); assumption;
+ | 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 ]
+ split; [ exists; [apply w] split; assumption | assumption ]
+ | 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 ]
+ 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.
\ No newline at end of file
projects / helm.git / blobdiff