comp1: ∀o1,o2,o3. binary_morphism1 (arrows1 o1 o2) (arrows1 o2 o3) (arrows1 o1 o3);
comp_assoc1: ∀o1,o2,o3,o4. ∀a12,a23,a34.
comp1 o1 o3 o4 (comp1 o1 o2 o3 a12 a23) a34 = comp1 o1 o2 o4 a12 (comp1 o2 o3 o4 a23 a34);
- id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
- id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
+ id_neutral_right1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? (id1 o1) a = a;
+ id_neutral_left1: ∀o1,o2. ∀a: arrows1 o1 o2. comp1 ??? a (id1 o2) = a
}.
notation "'ASSOC'" with precedence 90 for @{'assoc}.
notation "'ASSOC1'" with precedence 90 for @{'assoc1}.
-interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) x y).
+interpretation "category1 composition" 'compose x y = (fun1 ___ (comp1 ____) y x).
interpretation "category1 assoc" 'assoc1 = (comp_assoc1 ________).
-interpretation "category composition" 'compose x y = (fun ___ (comp ____) x y).
+interpretation "category composition" 'compose x y = (fun ___ (comp ____) y x).
interpretation "category assoc" 'assoc = (comp_assoc ________).
record relation_pair (BP1,BP2: basic_pair): Type ≝
{ concr_rel: arrows1 ? (concr BP1) (concr BP2);
form_rel: arrows1 ? (form BP1) (form BP2);
- commute: concr_rel ∘ ⊩ = ⊩ ∘ form_rel
+ commute: ⊩ ∘ concr_rel = form_rel ∘ ⊩
}.
notation "hvbox (r \sub \c)" with precedence 90 for @{'concr_rel $r}.
∀o1,o2. equivalence_relation1 (relation_pair o1 o2).
intros;
constructor 1;
- [ apply (λr,r'. r \sub\c ∘ ⊩ = r' \sub\c ∘ ⊩);
+ [ apply (λr,r'. ⊩ ∘ r \sub\c = ⊩ ∘ r' \sub\c);
| simplify;
intros;
apply refl1;
]
qed.
-lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → ⊩ \circ r \sub\f = ⊩ \circ r'\sub\f.
+lemma eq_to_eq': ∀o1,o2.∀r,r': relation_pair_setoid o1 o2. r=r' → r \sub\f ∘ ⊩ = r'\sub\f ∘ ⊩.
intros 7 (o1 o2 r r' H c1 f2);
split; intro H1;
[ lapply (fi ?? (commute ?? r c1 f2) H1) as H2;
intro;
constructor 1;
[1,2: apply id1;
- | lapply (id_neutral_left1 ? (concr o) ? (⊩)) as H;
- lapply (id_neutral_right1 ?? (form o) (⊩)) as H1;
+ | lapply (id_neutral_right1 ? (concr o) ? (⊩)) as H;
+ lapply (id_neutral_left1 ?? (form o) (⊩)) as H1;
apply (.= H);
apply (H1 \sup -1);]
qed.
constructor 1;
[ intros (r r1);
constructor 1;
- [ apply (r \sub\c ∘ r1 \sub\c)
- | apply (r \sub\f ∘ r1 \sub\f)
+ [ apply (r1 \sub\c ∘ r \sub\c)
+ | apply (r1 \sub\f ∘ r \sub\f)
| lapply (commute ?? r) as H;
lapply (commute ?? r1) as H1;
apply (.= ASSOC1);
apply (.= H‡#);
apply ASSOC1]
| intros;
- change with (a\sub\c ∘ b\sub\c ∘ ⊩ = a'\sub\c ∘ b'\sub\c ∘ ⊩);
- change in H with (a \sub\c ∘ ⊩ = a' \sub\c ∘ ⊩);
- change in H1 with (b \sub\c ∘ ⊩ = b' \sub\c ∘ ⊩);
+ change with (⊩ ∘ (b\sub\c ∘ a\sub\c) = ⊩ ∘ (b'\sub\c ∘ a'\sub\c));
+ change in H with (⊩ ∘ a \sub\c = ⊩ ∘ a' \sub\c);
+ change in H1 with (⊩ ∘ b \sub\c = ⊩ ∘ b' \sub\c);
apply (.= ASSOC1);
apply (.= #‡H1);
apply (.= #‡(commute ?? b'));
| apply id_relation_pair
| apply relation_pair_composition
| intros;
- change with (a12\sub\c ∘ a23\sub\c ∘ a34\sub\c ∘ ⊩ =
- (a12\sub\c ∘ (a23\sub\c ∘ a34\sub\c) ∘ ⊩));
+ change with (⊩ ∘ (a34\sub\c ∘ (a23\sub\c ∘ a12\sub\c)) =
+ ⊩ ∘ ((a34\sub\c ∘ a23\sub\c) ∘ a12\sub\c));
apply (ASSOC1‡#);
| intros;
- change with ((id_relation_pair o1)\sub\c ∘ a\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
- apply ((id_neutral_left1 ????)‡#);
- | intros;
- change with (a\sub\c ∘ (id_relation_pair o2)\sub\c ∘ ⊩ = a\sub\c ∘ ⊩);
+ change with (⊩ ∘ (a\sub\c ∘ (id_relation_pair o1)\sub\c) = ⊩ ∘ a\sub\c);
apply ((id_neutral_right1 ????)‡#);
- ]
+ | intros;
+ change with (⊩ ∘ ((id_relation_pair o2)\sub\c ∘ a\sub\c) = ⊩ ∘ a\sub\c);
+ apply ((id_neutral_left1 ????)‡#);]
qed.
definition BPext: ∀o: BP. form o ⇒ Ω \sup (concr o) ≝ λo.ext ? ? (rel o).
--- /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 "formal_topology/relations.ma".
+include "datatypes/categories.ma".
+
+definition is_saturation ≝
+ λC:REL.λA:unary_morphism (Ω \sup C) (Ω \sup C).
+ ∀U,V. (U ⊆ A V) = (A U ⊆ A V).
+
+definition is_reduction ≝
+ λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C).
+ ∀U,V. (J U ⊆ V) = (J U ⊆ J V).
+
+record basic_topology: Type ≝
+ { carrbt:> REL;
+ A: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt);
+ J: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt);
+ A_is_saturation: is_saturation ? A;
+ J_is_reduction: is_reduction ? J;
+ compatibility: ∀U,V. (A U ≬ J V) = (U ≬ J V)
+ }.
+
+(* 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]]
+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]
+qed.
+
+(* minus_image is the same as ext *)
+
+theorem image_id: ∀o,U. image o o (id1 REL o) U = U.
+ intros; unfold image; simplify; split; simplify; intros;
+ [ change with (a ∈ U);
+ cases H; cases x; change in f with (eq1 ? w a); apply (. f‡#); assumption
+ | change in f with (a ∈ U);
+ exists; [apply a] split; [ change with (a = a); apply refl | assumption]]
+qed.
+
+theorem minus_star_image_id: ∀o,U. minus_star_image o o (id1 REL o) U = U.
+ intros; unfold minus_star_image; simplify; split; simplify; intros;
+ [ change with (a ∈ U); apply H; change with (a=a); apply refl
+ | change in f1 with (eq1 ? x a); apply (. f1 \sup -1‡#); apply f]
+qed.
+
+theorem image_comp: ∀A,B,C,r,s,X. image A C (r ∘ s) X = image B C r (image A B s X).
+ intros; unfold image; simplify; split; simplify; intros; cases H; clear H; cases x;
+ clear x; [ cases f; clear f; | cases f1; clear f1 ]
+ exists; try assumption; cases x; clear x; split; try assumption;
+ exists; try assumption; split; assumption.
+qed.
+
+theorem minus_star_image_comp:
+ ∀A,B,C,r,s,X.
+ minus_star_image A C (r ∘ s) X = minus_star_image B C r (minus_star_image A B s X).
+ intros; unfold minus_star_image; simplify; split; simplify; intros; whd; intros;
+ [ apply H; exists; try assumption; split; assumption
+ | change with (x ∈ X); cases f; cases x1; apply H; assumption]
+qed.
+
+record continuous_relation (S,T: basic_topology) : Type ≝
+ { cont_rel:> arrows1 ? S T;
+ 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)
+ | constructor 1;
+ [ apply (λr,s:continuous_relation S T.∀b. A ? (ext ?? r b) = A ? (ext ?? s b));
+ | simplify; intros; apply refl1;
+ | simplify; intros; apply sym1; apply H
+ | simplify; intros; apply trans1; [2: apply H |3: apply H1; |1: skip]]]
+qed.
+
+definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → arrows1 ? S T ≝ cont_rel.
+
+coercion cont_rel'.
+(*
+definition BTop: category1.
+ constructor 1;
+ [ apply basic_topology
+ | apply continuous_relation_setoid
+ | intro; constructor 1;
+ [ apply id1
+ | intros;
+ apply (.= (image_id ??));
+ apply sym1;
+ apply (.= †(image_id ??));
+ apply sym1;
+ assumption
+ | intros;
+ apply (.= (minus_star_image_id ??));
+ apply sym1;
+ apply (.= †(minus_star_image_id ??));
+ apply sym1;
+ assumption]
+ | intros; constructor 1;
+ [ intros (r s); constructor 1;
+ [ apply (s ∘ r)
+ | intros;
+ apply sym1;
+ apply (.= †(image_comp ??????));
+ apply (.= (reduced ?????)\sup -1);
+ [ apply (.= (reduced ?????)); [ assumption | apply refl1 ]
+ | apply (.= (image_comp ??????)\sup -1);
+ apply refl1]
+ | intros;
+ apply sym1;
+ apply (.= †(minus_star_image_comp ??????));
+ apply (.= (saturated ?????)\sup -1);
+ [ apply (.= (saturated ?????)); [ assumption | apply refl1 ]
+ | apply (.= (minus_star_image_comp ??????)\sup -1);
+ apply refl1]]]
+ | intros; simplify; intro; simplify;
+ | intros; simplify; intro; simplify;
+ | intros; simplify; intro; simplify;
+ ]
+qed.
+*)
\ No newline at end of file
constructor 1;
[ apply (fun1 ??? (comp1 BP ???)); [apply (bp o2) |*: apply rp; assumption]
| intros;
- change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c \sub \c ∘ c1 \sub \c);
+ change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? (? ? ? (? ? ? %) ?) ?)))
- with (c \sub \f ∘ c1 \sub \f);
+ with (c1 \sub \f ∘ c \sub \f);
change in ⊢ (? ? ? ? (? ? ? ? (? ? ? ? ? ? (? ? ? (? ? ? %) ?))))
- with (c \sub \f ∘ c1 \sub \f);
+ with (c1 \sub \f ∘ c \sub \f);
apply (.= (extS_com ??????));
apply (.= (†(respects_converges ?????)));
apply (.= (respects_converges ?????));
apply (.= (†(((extS_com ??????) \sup -1)‡(extS_com ??????)\sup -1)));
apply refl1;
- | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c \sub \c ∘ c1 \sub \c);
+ | change in ⊢ (? ? ? (? ? ? (? ? ? %) ?) ?) with (c1 \sub \c ∘ c \sub \c);
apply (.= (extS_com ??????));
apply (.= (†(respects_all_covered ???)));
apply (.= respects_all_covered ???);
apply refl1]
| intros;
- change with (a ∘ b = a' ∘ b');
+ change with (b ∘ a = b' ∘ a');
change in H with (rp'' ?? a = rp'' ?? a');
change in H1 with (rp'' ?? b = rp ?? b');
apply (.= (H‡H1));
apply refl1]
| apply convergent_relation_space_composition
| intros; simplify;
- change with ((a12 ∘ a23) ∘ a34 = a12 ∘ (a23 ∘ a34));
+ change with (a34 ∘ (a23 ∘ a12) = (a34 ∘ a23) ∘ a12);
apply (.= ASSOC1);
apply refl1
| intros; simplify;
- change with (id1 ? o1 ∘ a = a);
- apply (.= id_neutral_left1 ????);
+ change with (a ∘ id1 ? o1 = a);
+ apply (.= id_neutral_right1 ????);
apply refl1
| intros; simplify;
- change with (a ∘ id1 ? o2 = a);
- apply (.= id_neutral_right1 ????);
+ change with (id1 ? o2 ∘ a = a);
+ apply (.= id_neutral_left1 ????);
apply refl1]
qed.
| change with (a = a); apply refl]]]
qed.
-lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c1 ∘ c2) S = extS o1 o2 c1 (extS o2 o3 c2 S).
+lemma extS_com: ∀o1,o2,o3,c1,c2,S. extS o1 o3 (c2 ∘ c1) S = extS o1 o2 c1 (extS o2 o3 c2 S).
intros; unfold extS; simplify; split; intros; simplify; intros;
[ cases f (H1 H2); cases H2 (w H3); clear f H2; split; [assumption]
cases H3 (H4 H5); cases H5 (w1 H6); clear H3 H5; cases H6 (H7 H8); clear H6;
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