--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "relations.ma".
+include "o-algebra.ma".
+
+definition SUBSETS: objs1 SET → OAlgebra.
+ intro A; constructor 1;
+ [ apply (Ω \sup A);
+ | apply subseteq;
+ | apply overlaps;
+ | apply big_intersects;
+ | apply big_union;
+ | apply ({x | True});
+ simplify; intros; apply (refl1 ? (eq1 CPROP));
+ | apply ({x | False});
+ simplify; intros; apply (refl1 ? (eq1 CPROP));
+ | intros; whd; intros; assumption
+ | 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 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]]
+ | 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 \sup -1‡#));
+ assumption]]
+qed.
+
+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.
projects / helm.git / blobdiff