From 7c9b20db66af78579a5312e4a6a5a42471d6312b Mon Sep 17 00:00:00 2001 From: Claudio Sacerdoti Coen Date: Mon, 15 Sep 2008 20:26:34 +0000 Subject: [PATCH] BTop is a category. --- .../formal_topology/basic_topologies.ma | 196 ++++++++++++++++-- 1 file changed, 174 insertions(+), 22 deletions(-) diff --git a/helm/software/matita/library/formal_topology/basic_topologies.ma b/helm/software/matita/library/formal_topology/basic_topologies.ma index 749089a89..92f4cdf46 100644 --- a/helm/software/matita/library/formal_topology/basic_topologies.ma +++ b/helm/software/matita/library/formal_topology/basic_topologies.ma @@ -23,6 +23,31 @@ definition is_reduction ≝ λC:REL.λJ:unary_morphism (Ω \sup C) (Ω \sup C). ∀U,V. (J U ⊆ V) = (J U ⊆ J V). +theorem subseteq_refl: ∀A.∀S:Ω \sup A.S ⊆ S. + intros 4; assumption. +qed. + +theorem subseteq_trans: ∀A.∀S1,S2,S3: Ω \sup A. S1 ⊆ S2 → S2 ⊆ S3 → S1 ⊆ S3. + intros; apply transitive_subseteq_operator; [apply S2] assumption. +qed. + +theorem saturation_expansive: ∀C,A. is_saturation C A → ∀U. U ⊆ A U. + intros; apply (fi ?? (H ??)); apply subseteq_refl. +qed. + +theorem saturation_monotone: + ∀C,A. is_saturation C A → + ∀U,V. U ⊆ V → A U ⊆ A V. + intros; apply (if ?? (H ??)); apply subseteq_trans; [apply V|3: apply saturation_expansive ] + assumption. +qed. + +theorem saturation_idempotent: ∀C,A. is_saturation C A → ∀U. A (A U) = A U. + intros; split; + [ apply (if ?? (H ??)); apply subseteq_refl + | apply saturation_expansive; assumption] +qed. + record basic_topology: Type ≝ { carrbt:> REL; A: unary_morphism (Ω \sup carrbt) (Ω \sup carrbt); @@ -108,7 +133,129 @@ definition cont_rel': ∀S,T: basic_topology. continuous_relation_setoid S T → coercion cont_rel'. +definition cont_rel'': ∀S,T: basic_topology. continuous_relation_setoid S T → binary_relation S T ≝ cont_rel. + +coercion cont_rel''. + +theorem ext_comp: + ∀o1,o2,o3: REL. + ∀a: arrows1 ? o1 o2. + ∀b: arrows1 ? o2 o3. + ∀x. ext ?? (b∘a) x = extS ?? a (ext ?? b x). + intros; + unfold ext; unfold extS; simplify; split; intro; simplify; intros; + cases f; clear f; split; try assumption; + [ cases f2; clear f2; cases x1; clear x1; exists; [apply w] split; + [1: split] assumption; + | cases H; clear H; cases x1; clear x1; exists [apply w]; split; + [2: cases f] assumption] +qed. + (* +(* this proof is more logic-oriented than set/lattice oriented *) +theorem continuous_relation_eqS: + ∀o1,o2:basic_topology.∀a,a': continuous_relation_setoid o1 o2. + a = a' → ∀X. A ? (extS ?? a X) = A ? (extS ?? a' X). + intros; + cut (∀a: arrows1 ? o1 ?.∀x. x ∈ extS ?? a X → ∃y:o2.y ∈ X ∧ x ∈ ext ?? a y); + [2: intros; cases f; clear f; cases H1; exists [apply w] cases x1; split; + try assumption; split; assumption] + cut (∀a,a':continuous_relation_setoid o1 o2.eq1 ? a a' → ∀x. x ∈ extS ?? a X → ∃y:o2. y ∈ X ∧ x ∈ A ? (ext ?? a' y)); + [2: intros; cases (Hcut ?? f); exists; [apply w] cases x1; split; try assumption; + apply (. #‡(H1 ?)); + apply (saturation_expansive ?? (A_is_saturation o1) (ext ?? a1 w) x); + assumption;] clear Hcut; + split; apply (if ?? (A_is_saturation ???)); intros 2; + [lapply (Hcut1 a a' H a1 f) | lapply (Hcut1 a' a (H \sup -1) a1 f)] + cases Hletin; clear Hletin; cases x; clear x; + cut (∀a: arrows1 ? o1 ?. ext ?? a w ⊆ extS ?? a X); + [2,4: intros 3; cases f3; clear f3; simplify in f5; split; try assumption; + exists [1,3: apply w] split; assumption;] + cut (∀a. A ? (ext o1 o2 a w) ⊆ A ? (extS o1 o2 a X)); + [2,4: intros; apply saturation_monotone; try (apply A_is_saturation); apply Hcut;] + apply Hcut2; assumption. +qed. +*) + +theorem continuous_relation_eq': + ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2. + a = a' → ∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X). + intros; split; intro; unfold minus_star_image; simplify; intros; + [ cut (ext ?? a a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;] + lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut; + cut (A ? (ext ?? a' a1) ⊆ A ? X); [2: apply (. (H ?)‡#); assumption] + lapply (fi ?? (A_is_saturation ???) Hcut); + apply (Hletin1 x); change with (x ∈ ext ?? a' a1); split; simplify; + [ apply I | assumption ] + | cut (ext ?? a' a1 ⊆ A ? X); [2: intros 2; apply (H1 a2); cases f1; assumption;] + lapply (if ?? (A_is_saturation ???) Hcut); clear Hcut; + cut (A ? (ext ?? a a1) ⊆ A ? X); [2: apply (. (H ?)\sup -1‡#); assumption] + lapply (fi ?? (A_is_saturation ???) Hcut); + apply (Hletin1 x); change with (x ∈ ext ?? a a1); split; simplify; + [ apply I | assumption ]] +qed. + +theorem extS_singleton: + ∀o1,o2.∀a:arrows1 ? o1 o2.∀x.extS o1 o2 a (singleton o2 x) = ext o1 o2 a x. + intros; unfold extS; unfold ext; unfold singleton; simplify; + split; intros 2; simplify; cases f; split; try assumption; + [ 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. + +theorem continuous_relation_eq_inv': + ∀o1,o2.∀a,a': continuous_relation_setoid o1 o2. + (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → a=a'. + intros 6; + cut (∀a,a': continuous_relation_setoid o1 o2. + (∀X.minus_star_image ?? a (A o1 X) = minus_star_image ?? a' (A o1 X)) → + ∀V:o2. A ? (ext ?? a' V) ⊆ A ? (ext ?? a V)); + [2: clear b H a' a; intros; + lapply depth=0 (λV.saturation_expansive ??? (extS ?? a V)); [2: apply A_is_saturation;|skip] + (* fundamental adjunction here! to be taken out *) + cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a (A ? (extS ?? a V))); + [2: intro; intros 2; unfold minus_star_image; simplify; intros; + apply (Hletin V1 x); whd; split; [ exact I | exists; [apply a1] split; assumption]] + clear Hletin; + cut (∀V:Ω \sup o2.V ⊆ minus_star_image ?? a' (A ? (extS ?? a V))); + [2: intro; apply (. #‡(H ?)); apply Hcut] clear H Hcut; + (* second half of the fundamental adjunction here! to be taken out too *) + intro; lapply (Hcut1 (singleton ? V)); clear Hcut1; + unfold minus_star_image in Hletin; unfold singleton in Hletin; simplify in Hletin; + whd in Hletin; whd in Hletin:(?→?→%); simplify in Hletin; + apply (if ?? (A_is_saturation ???)); + intros 2 (x H); lapply (Hletin V ? x ?); + [ apply refl | cases H; assumption; ] + change with (x ∈ A ? (ext ?? a V)); + apply (. #‡(†(extS_singleton ????))); + assumption;] + split; apply Hcut; [2: assumption | intro; apply sym1; apply H] +qed. + +definition continuous_relation_comp: + ∀o1,o2,o3. + continuous_relation_setoid o1 o2 → + continuous_relation_setoid o2 o3 → + continuous_relation_setoid o1 o3. + intros (o1 o2 o3 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]] +qed. + definition BTop: category1. constructor 1; [ apply basic_topology @@ -128,32 +275,37 @@ definition BTop: category1. 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; whd in H H1; - apply (.= †(ext_comp ???)); - ] - | intros; simplify; intro; simplify; + [ apply continuous_relation_comp; + | intros; simplify; intro x; simplify; + lapply depth=0 (continuous_relation_eq' ???? H) as H'; + lapply depth=0 (continuous_relation_eq' ???? H1) as H1'; + letin K ≝ (λX.H1' (minus_star_image ?? a (A ? X))); clearbody K; + cut (∀X:Ω \sup o1. + minus_star_image o2 o3 b (A o2 (minus_star_image o1 o2 a (A o1 X))) + = minus_star_image o2 o3 b' (A o2 (minus_star_image o1 o2 a' (A o1 X)))); + [2: intro; apply sym1; apply (.= #‡(†((H' ?)\sup -1))); apply sym1; apply (K X);] + clear K H' H1'; + cut (∀X:Ω \sup o1. + minus_star_image o1 o3 (b ∘ a) (A o1 X) = minus_star_image o1 o3 (b'∘a') (A o1 X)); + [2: intro; + apply (.= (minus_star_image_comp ??????)); + apply (.= #‡(saturated ?????)); + [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ] + apply sym1; + apply (.= (minus_star_image_comp ??????)); + apply (.= #‡(saturated ?????)); + [ apply ((saturation_idempotent ????) \sup -1); apply A_is_saturation ] + apply ((Hcut X) \sup -1)] + clear Hcut; generalize in match x; clear x; + apply (continuous_relation_eq_inv'); + apply Hcut1;] + | intros; simplify; intro; do 2 (unfold continuous_relation_comp); simplify; apply (.= †(ASSOC1‡#)); apply refl1 - | intros; simplify; intro; simplify; + | intros; simplify; intro; unfold continuous_relation_comp; simplify; apply (.= †((id_neutral_right1 ????)‡#)); apply refl1 | intros; simplify; intro; simplify; apply (.= †((id_neutral_left1 ????)‡#)); apply refl1] -qed.*) \ No newline at end of file +qed. \ No newline at end of file -- 2.39.2