From 5bfea0766a39d05605b6e88508f5376a88351331 Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 25 Aug 2009 11:36:33 +0000 Subject: [PATCH] initial and incomplete port of the old demo about inductively generated formal topologies. we try to make slightly different choices... --- helm/software/matita/nlibrary/depends | 14 +- helm/software/matita/nlibrary/depends.dot | 51 ++++- helm/software/matita/nlibrary/depends.png | Bin 48586 -> 181674 bytes .../software/matita/nlibrary/topology/igft.ma | 174 ++++++++++++++++++ 4 files changed, 230 insertions(+), 9 deletions(-) create mode 100644 helm/software/matita/nlibrary/topology/igft.ma diff --git a/helm/software/matita/nlibrary/depends b/helm/software/matita/nlibrary/depends index f39678fb8..301939368 100644 --- a/helm/software/matita/nlibrary/depends +++ b/helm/software/matita/nlibrary/depends @@ -1,21 +1,23 @@ -sets/sets.ma logic/connectives.ma logic/cprop.ma properties/relations1.ma sets/setoids1.ma logic/cprop.ma sets/setoids1.ma +sets/sets.ma logic/connectives.ma logic/cprop.ma properties/relations1.ma sets/setoids1.ma +topology/igft.ma logic/connectives.ma datatypes/bool.ma logic/pts.ma sets/setoids1.ma properties/relations1.ma sets/setoids.ma -sets/setoids.ma logic/connectives.ma properties/relations.ma logic/equality.ma logic/connectives.ma properties/relations.ma +sets/setoids.ma logic/connectives.ma properties/relations.ma +.unnamed.ma logic/pts.ma logic/connectives.ma logic/pts.ma datatypes/pairs.ma logic/pts.ma algebra/abelian_magmas.ma algebra/magmas.ma nat/plus.ma algebra/abelian_magmas.ma algebra/unital_magmas.ma nat/big_ops.ma nat/minus.ma nat/order.ma algebra/magmas.ma sets/sets.ma -nat/nat.ma logic/equality.ma sets/setoids.ma -nat/big_ops.ma algebra/magmas.ma nat/order.ma properties/relations1.ma logic/pts.ma -nat/compare.ma datatypes/bool.ma nat/order.ma +nat/big_ops.ma algebra/magmas.ma nat/order.ma +nat/nat.ma logic/equality.ma sets/setoids.ma properties/relations.ma logic/pts.ma +nat/compare.ma datatypes/bool.ma nat/order.ma +logic/pts.ma nat/order.ma nat/nat.ma sets/sets.ma algebra/unital_magmas.ma algebra/magmas.ma -logic/pts.ma sets/partitions.ma datatypes/pairs.ma nat/compare.ma nat/minus.ma nat/plus.ma sets/sets.ma diff --git a/helm/software/matita/nlibrary/depends.dot b/helm/software/matita/nlibrary/depends.dot index 6a1e65939..ec3b85946 100644 --- a/helm/software/matita/nlibrary/depends.dot +++ b/helm/software/matita/nlibrary/depends.dot @@ -1,19 +1,64 @@ digraph g { + "logic/cprop.ma" []; + "logic/cprop.ma" -> "sets/setoids1.ma" []; "sets/sets.ma" []; - "sets/sets.ma" -> "logic/equality.ma" []; + "sets/sets.ma" -> "logic/connectives.ma" []; + "sets/sets.ma" -> "logic/cprop.ma" []; + "sets/sets.ma" -> "properties/relations1.ma" []; + "sets/sets.ma" -> "sets/setoids1.ma" []; + "topology/igft.ma" []; + "topology/igft.ma" -> "logic/connectives.ma" []; + "datatypes/bool.ma" []; + "datatypes/bool.ma" -> "logic/pts.ma" []; "sets/setoids1.ma" []; + "sets/setoids1.ma" -> "properties/relations1.ma" []; "sets/setoids1.ma" -> "sets/setoids.ma" []; + "logic/equality.ma" []; + "logic/equality.ma" -> "logic/connectives.ma" []; + "logic/equality.ma" -> "properties/relations.ma" []; "sets/setoids.ma" []; "sets/setoids.ma" -> "logic/connectives.ma" []; "sets/setoids.ma" -> "properties/relations.ma" []; - "logic/equality.ma" []; - "logic/equality.ma" -> "logic/connectives.ma" []; + ".unnamed.ma" []; + ".unnamed.ma" -> "logic/pts.ma" []; "logic/connectives.ma" []; "logic/connectives.ma" -> "logic/pts.ma" []; + "datatypes/pairs.ma" []; + "datatypes/pairs.ma" -> "logic/pts.ma" []; + "algebra/abelian_magmas.ma" []; + "algebra/abelian_magmas.ma" -> "algebra/magmas.ma" []; + "nat/plus.ma" []; + "nat/plus.ma" -> "algebra/abelian_magmas.ma" []; + "nat/plus.ma" -> "algebra/unital_magmas.ma" []; + "nat/plus.ma" -> "nat/big_ops.ma" []; + "nat/minus.ma" []; + "nat/minus.ma" -> "nat/order.ma" []; "algebra/magmas.ma" []; "algebra/magmas.ma" -> "sets/sets.ma" []; + "properties/relations1.ma" []; + "properties/relations1.ma" -> "logic/pts.ma" []; + "nat/big_ops.ma" []; + "nat/big_ops.ma" -> "algebra/magmas.ma" []; + "nat/big_ops.ma" -> "nat/order.ma" []; + "nat/nat.ma" []; + "nat/nat.ma" -> "logic/equality.ma" []; + "nat/nat.ma" -> "sets/setoids.ma" []; "properties/relations.ma" []; "properties/relations.ma" -> "logic/pts.ma" []; + "nat/compare.ma" []; + "nat/compare.ma" -> "datatypes/bool.ma" []; + "nat/compare.ma" -> "nat/order.ma" []; "logic/pts.ma" []; + "nat/order.ma" []; + "nat/order.ma" -> "nat/nat.ma" []; + "nat/order.ma" -> "sets/sets.ma" []; + "algebra/unital_magmas.ma" []; + "algebra/unital_magmas.ma" -> "algebra/magmas.ma" []; + "sets/partitions.ma" []; + "sets/partitions.ma" -> "datatypes/pairs.ma" []; + "sets/partitions.ma" -> "nat/compare.ma" []; + "sets/partitions.ma" -> "nat/minus.ma" []; + "sets/partitions.ma" -> "nat/plus.ma" []; + "sets/partitions.ma" -> "sets/sets.ma" []; } \ No newline at end of file diff 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b/helm/software/matita/nlibrary/topology/igft.ma new file mode 100644 index 000000000..29e1f1af3 --- /dev/null +++ b/helm/software/matita/nlibrary/topology/igft.ma @@ -0,0 +1,174 @@ +include "logic/connectives.ma". + +nrecord powerset (X : Type[0]) : Type[1] ≝ { char : X → CProp[0] }. + +interpretation "char" 'subset p = (mk_powerset ? p). + +interpretation "pwset" 'powerset a = (powerset a). + +interpretation "in" 'mem a X = (char ? X a). + +ndefinition subseteq ≝ λA.λU,V : Ω^A. + ∀x.x ∈ U → x ∈ V. + +interpretation "subseteq" 'subseteq u v = (subseteq ? u v). + +ndefinition overlaps ≝ λA.λU,V : Ω^A. + ∃x.x ∈ U ∧ x ∈ V. + +interpretation "overlaps" 'overlaps u v = (overlaps ? u v). +(* +ndefinition intersect ≝ λA.λu,v:Ω\sup A.{ y | y ∈ u ∧ y ∈ v }. + +interpretation "intersect" 'intersects u v = (intersect ? u v). +*) +nrecord axiom_set : Type[1] ≝ { + S:> Type[0]; + I: S → Type[0]; + C: ∀a:S. I a → Ω ^ S +}. + +ndefinition cover_set ≝ λc:∀A:axiom_set.Ω^A → A → CProp[0].λA,C,U. + ∀y.y ∈ C → c A U y. + +notation "hvbox(a break ◃ b)" non associative with precedence 45 +for @{ 'covers $a $b }. (* a \ltri b *) + +interpretation "covers set temp" 'covers C U = (cover_set ?? C U). + +ninductive cover (A : axiom_set) (U : Ω^A) : A → CProp[0] ≝ +| creflexivity : ∀a:A. a ∈ U → cover ? U a +| cinfinity : ∀a:A. ∀i:I ? a. (C ? a i ◃ U) → cover ? U a. +napply cover; +nqed. + +interpretation "covers" 'covers a U = (cover ? U a). +interpretation "covers set" 'covers a U = (cover_set cover ? a U). + +ndefinition fish_set ≝ λf:∀A:axiom_set.Ω^A → A → CProp[0]. + λA,U,V. + ∃a.a ∈ V ∧ f A U a. + +notation "hvbox(a break ⋉ b)" non associative with precedence 45 +for @{ 'fish $a $b }. (* a \ltimes b *) + +interpretation "fish set temp" 'fish A U = (fish_set ?? U A). + +ncoinductive fish (A : axiom_set) (F : Ω^A) : A → CProp[0] ≝ +| cfish : ∀a. a ∈ F → (∀i:I ? a.C A a i ⋉ F) → fish A F a. +napply fish; +nqed. + +interpretation "fish set" 'fish A U = (fish_set fish ? U A). +interpretation "fish" 'fish a U = (fish ? U a). + +nlet corec fish_rec (A:axiom_set) (U: Ω^A) + (P: Ω^A) (H1: P ⊆ U) + (H2: ∀a:A. a ∈ P → ∀j: I ? a. C ? a j ≬ P): + ∀a:A. ∀p: a ∈ P. a ⋉ U ≝ ?. +#a; #p; napply cfish; +##[ napply H1; napply p; +##| #i; ncases (H2 a p i); #x; *; #xC; #xP; napply ex_intro; ##[napply x] + napply conj; ##[ napply xC ] napply (fish_rec ? U P); nassumption; +##] +nqed. + +STOP + +theorem reflexivity: ∀A:axiom_set.∀a:A.∀V. a ∈ V → a ◃ V. + intros; + apply refl; + assumption. +qed. + +theorem transitivity: ∀A:axiom_set.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V. + intros; + apply (covers_elim ?? {a | a ◃ V} ??? H); simplify; intros; + [ cases H1 in H2; apply H2; + | apply infinity; + [ assumption + | constructor 1; + assumption]] +qed. + +theorem covers_elim2: + ∀A: axiom_set. ∀U:Ω \sup A.∀P: A → CProp. + (∀a:A. a ∈ U → P a) → + (∀a:A.∀V:Ω \sup A. a ◃ V → V ◃ U → (∀y. y ∈ V → P y) → P a) → + ∀a:A. a ◃ U → P a. + intros; + change with (a ∈ {a | P a}); + apply (covers_elim ?????? H2); + [ intros 2; simplify; apply H; assumption + | intros; + simplify in H4 ⊢ %; + apply H1; + [ apply (C ? a1 j); + | autobatch; + | assumption; + | assumption]] +qed. + +theorem coreflexivity: ∀A:axiom_set.∀a:A.∀V. a ⋉ V → a ∈ V. + intros; + cases H; + assumption. +qed. + +theorem cotransitivity: + ∀A:axiom_set.∀a:A.∀U,V. a ⋉ U → (∀b:A. b ⋉ U → b ∈ V) → a ⋉ V. + intros; + apply (fish_rec ?? {a|a ⋉ U} ??? H); simplify; intros; + [ apply H1; apply H2; + | cases H2 in j; clear H2; intro i; + cases (H4 i); clear H4; exists[apply a3] assumption] +qed. + +theorem compatibility: ∀A:axiom_set.∀a:A.∀U,V. a ⋉ V → a ◃ U → U ⋉ V. + intros; + generalize in match H; clear H; + apply (covers_elim ?? {a|a ⋉ V → U ⋉ V} ??? H1); + clear H1; simplify; intros; + [ exists [apply x] assumption + | cases H2 in j H H1; clear H2 a1; intros; + cases (H1 i); clear H1; apply (H3 a1); assumption] +qed. + +definition leq ≝ λA:axiom_set.λa,b:A. a ◃ {y|b=y}. + +interpretation "covered by one" 'leq a b = (leq ? a b). + +theorem leq_refl: ∀A:axiom_set.∀a:A. a ≤ a. + intros; + apply refl; + normalize; + reflexivity. +qed. + +theorem leq_trans: ∀A:axiom_set.∀a,b,c:A. a ≤ b → b ≤ c → a ≤ c. + intros; + unfold in H H1 ⊢ %; + apply (transitivity ???? H); + constructor 1; + intros; + normalize in H2; + rewrite < H2; + assumption. +qed. + +definition uparrow ≝ λA:axiom_set.λa:A.mk_powerset ? (λb:A. a ≤ b). + +interpretation "uparrow" 'uparrow a = (uparrow ? a). + +definition downarrow ≝ λA:axiom_set.λU:Ω \sup A.mk_powerset ? (λa:A. (↑a) ≬ U). + +interpretation "downarrow" 'downarrow a = (downarrow ? a). + +definition fintersects ≝ λA:axiom_set.λU,V:Ω \sup A.↓U ∩ ↓V. + +interpretation "fintersects" 'fintersects U V = (fintersects ? U V). + +record convergent_generated_topology : Type ≝ + { AA:> axiom_set; + convergence: ∀a:AA.∀U,V:Ω \sup AA. a ◃ U → a ◃ V → a ◃ (U ↓ V) + }. -- 2.39.2