interpretation "basic pair relation for subsets" 'Vdash2 x y = (fun1 (concr _) __ (relS _) x y).
interpretation "basic pair relation for subsets (non applied)" 'Vdash = (fun1 ___ (relS _)).
*)
-
-include "o-basic_pairs.ma".
-(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
-definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
- intro;
- constructor 1;
- [ apply (SUBSETS (concr b));
- | apply (SUBSETS (form b));
- | apply (orelation_of_relation ?? (rel b)); ]
-qed.
-
-definition o_relation_pair_of_relation_pair:
- ∀BP1,BP2.cic:/matita/formal_topology/basic_pairs/relation_pair.ind#xpointer(1/1) BP1 BP2 →
- relation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
- intros;
- constructor 1;
- [ apply (orelation_of_relation ?? (r \sub \c));
- | apply (orelation_of_relation ?? (r \sub \f));
- | lapply (commute ?? r);
- lapply (orelation_of_relation_preserves_equality ???? Hletin);
- apply (.= (orelation_of_relation_preserves_composition (concr BP1) ??? (rel BP2)) ^ -1);
- apply (.= (orelation_of_relation_preserves_equality ???? (commute ?? r)));
- apply (orelation_of_relation_preserves_composition ?? (form BP2) (rel BP1) ?); ]
-qed.
\ No newline at end of file
--- /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 "basic_pairs.ma".
+include "o-basic_pairs.ma".
+include "relations_to_o-algebra.ma".
+
+(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
+definition o_basic_pair_of_basic_pair: cic:/matita/formal_topology/basic_pairs/basic_pair.ind#xpointer(1/1) → basic_pair.
+ intro;
+ constructor 1;
+ [ apply (SUBSETS (concr b));
+ | apply (SUBSETS (form b));
+ | apply (orelation_of_relation ?? (rel b)); ]
+qed.
+
+definition o_relation_pair_of_relation_pair:
+ ∀BP1,BP2.cic:/matita/formal_topology/basic_pairs/relation_pair.ind#xpointer(1/1) BP1 BP2 →
+ relation_pair (o_basic_pair_of_basic_pair BP1) (o_basic_pair_of_basic_pair BP2).
+ intros;
+ constructor 1;
+ [ apply (orelation_of_relation ?? (r \sub \c));
+ | apply (orelation_of_relation ?? (r \sub \f));
+ | lapply (commute ?? r);
+ lapply (orelation_of_relation_preserves_equality ???? Hletin);
+ apply (.= (orelation_of_relation_preserves_composition (concr BP1) ??? (rel BP2)) ^ -1);
+ apply (.= (orelation_of_relation_preserves_equality ???? (commute ?? r)));
+ apply (orelation_of_relation_preserves_composition ?? (form BP2) (rel BP1) ?); ]
+qed.
reduced: ∀U. U = J ? U → image ?? cont_rel U = J ? (image ?? cont_rel U);
saturated: ∀U. U = A ? U → minus_star_image ?? cont_rel U = A ? (minus_star_image ?? cont_rel U)
}.
-
+(*
definition continuous_relation_setoid: basic_topology → basic_topology → setoid1.
intros (S T); constructor 1;
[ apply (continuous_relation S T)
[2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;]
apply Hcut2; assumption.
qed.
+*)
*)
\ No newline at end of file
--- /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 "basic_topologies.ma".
+include "o-basic_topologies.ma".
+include "relations_to_o-algebra.ma".
+
+(* Qui, per fare le cose per bene, ci serve la nozione di funtore categorico *)
+definition o_basic_topology_of_basic_topology: cic:/matita/formal_topology/basic_topologies/basic_topology.ind#xpointer(1/1) → basic_topology.
+ intro;
+ constructor 1;
+ [ apply (SUBSETS (carrbt b));
+ | apply (A b);
+ | apply (J b);
+ | apply (A_is_saturation b);
+ | apply (J_is_reduction b);
+ | apply (compatibility b); ]
+qed.
+
+definition o_relation_pair_of_relation_pair:
+ ∀S,T.cic:/matita/formal_topology/basic_topologies/continuous_relation.ind#xpointer(1/1) S T →
+ continuous_relation (o_basic_topology_of_basic_topology S) (o_basic_topology_of_basic_topology T).
+ intros;
+ constructor 1;
+ [ apply (orelation_of_relation ?? (cont_rel ?? c));
+ | apply (reduced ?? c);
+ | apply (saturated ?? c); ]
+qed.
\ No newline at end of file
o-basic_pairs.ma o-algebra.ma
o-concrete_spaces.ma o-basic_pairs.ma o-saturations.ma
o-saturations.ma o-algebra.ma
-basic_pairs.ma o-basic_pairs.ma relations.ma
+basic_pairs.ma relations.ma
saturations.ma relations.ma
o-algebra.ma categories.ma
o-formal_topologies.ma o-basic_topologies.ma
categories.ma cprop_connectives.ma
-subsets.ma categories.ma o-algebra.ma
+saturations_to_o-saturations.ma o-saturations.ma relations_to_o-algebra.ma saturations.ma
+subsets.ma categories.ma
basic_topologies.ma relations.ma saturations.ma
-relations.ma o-algebra.ma subsets.ma
+relations.ma subsets.ma
o-basic_topologies.ma o-algebra.ma o-saturations.ma
+basic_pairs_to_o-basic_pairs.ma basic_pairs.ma o-basic_pairs.ma relations_to_o-algebra.ma
+basic_topologies_to_o-basic_topologies.ma basic_topologies.ma o-basic_topologies.ma relations_to_o-algebra.ma
cprop_connectives.ma logic/connectives.ma
+relations_to_o-algebra.ma o-algebra.ma relations.ma
logic/connectives.ma
| 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
--- /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.
--- /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 "saturations.ma".
+include "o-saturations.ma".
+include "relations_to_o-algebra.ma".
+
+(* These are only conversions :-) *)
+
+definition o_operator_of_operator:
+ ∀C:REL. (Ω \sup C => Ω \sup C) → (SUBSETS C ⇒ SUBSETS C).
+ intros;apply t;
+qed.
+
+definition is_o_saturation_of_is_saturation:
+ ∀C:REL.∀R: unary_morphism1 (Ω \sup C) (Ω \sup C).
+ is_saturation ? R → is_o_saturation ? (o_operator_of_operator ? R).
+ intros; apply i;
+qed.
+
+definition is_o_reduction_of_is_reduction:
+ ∀C:REL.∀R: unary_morphism1 (Ω \sup C) (Ω \sup C).
+ is_reduction ? R → is_o_reduction ? (o_operator_of_operator ? R).
+ intros; apply i;
+qed.
\ No newline at end of file
[ apply (. (#‡(e w))); apply x;
| apply (. (#‡(e w)\sup -1)); apply x]]
qed.
-
-(* incluso prima non funziona piu' nulla *)
-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.
projects / helm.git / commitdiff