interpretation "constructive and" 'and x y = (And x y).
-inductive exT (A:Type) (P:A→CProp) : CProp ≝
- ex_introT: ∀w:A. P w → exT A P.
+inductive Or (A,B:CProp) : CProp ≝
+ | or_intro_l: A → Or A B
+ | or_intro_r: B → Or A B.
+
+interpretation "constructive or" 'or x y = (Or x y).
-interpretation "CProp exists" 'exists \eta.x = (exT _ x).
+inductive exT2 (A:Type) (P,Q:A→CProp) : CProp ≝
+ ex_introT2: ∀w:A. P w → Q w → exT2 A P Q.
record powerset (A: Type) : Type ≝ { char: A → CProp }.
interpretation "powerset" 'powerset A = (powerset A).
+notation < "hvbox({ ident i | term 19 p })" with precedence 90
+for @{ 'subset (\lambda ${ident i} : $nonexistent . $p)}.
+
+notation > "hvbox({ ident i | term 19 p })" with precedence 90
+for @{ 'subset (\lambda ${ident i}. $p)}.
+
+interpretation "subset construction" 'subset \eta.x = (mk_powerset _ x).
+
definition mem ≝ λA.λS:2 \sup A.λx:A. match S with [mk_powerset c ⇒ c x].
notation "hvbox(a break ∈ b)" non associative with precedence 45
interpretation "mem" 'mem a S = (mem _ S a).
-record axiom_set : Type ≝
- { A:> Type;
- i: A → Type;
- C: ∀a:A. i a → 2 \sup A
- }.
+definition overlaps ≝ λA:Type.λU,V:2 \sup A.exT2 ? (λa:A. a ∈ U) (λa.a ∈ V).
+
+notation "hvbox(a break ≬ b)" non associative with precedence 45
+for @{ 'overlaps $a $b }. (* \between *)
+
+interpretation "overlaps" 'overlaps U V = (overlaps _ U V).
+
+definition subseteq ≝ λA:Type.λU,V:2 \sup A.∀a:A. a ∈ U → a ∈ V.
+
+notation "hvbox(a break ⊆ b)" non associative with precedence 45
+for @{ 'subseteq $a $b }. (* \subseteq *)
+
+interpretation "subseteq" 'subseteq U V = (subseteq _ U V).
+
+definition intersects ≝ λA:Type.λU,V:2 \sup A.{a | a ∈ U ∧ a ∈ V}.
+
+notation "hvbox(a break ∩ b)" non associative with precedence 55
+for @{ 'intersects $a $b }. (* \cap *)
+
+interpretation "intersects" 'intersects U V = (intersects _ U V).
+
+definition union ≝ λA:Type.λU,V:2 \sup A.{a | a ∈ U ∨ a ∈ V}.
+
+notation "hvbox(a break ∪ b)" non associative with precedence 55
+for @{ 'union $a $b }. (* \cup *)
+
+interpretation "union" 'union U V = (union _ U V).
+
+record axiom_set : Type ≝ {
+ A:> Type;
+ i: A → Type;
+ C: ∀a:A. i a → 2 \sup A
+}.
+
+inductive for_all (A: axiom_set) (U,V: 2 \sup A) (covers: A → CProp) : CProp ≝
+ iter: (∀a:A.a ∈ V → covers a) → for_all A U V covers.
inductive covers (A: axiom_set) (U: 2 \sup A) : A → CProp ≝
refl: ∀a:A. a ∈ U → covers A U a
- | infinity: ∀a:A. ∀j: i ? a. coversl A U (C ? a j) → covers A U a
-with coversl : (2 \sup A) → CProp ≝
- iter: ∀V:2 \sup A.(∀a:A.a ∈ V → covers A U a) → coversl A U V.
+ | infinity: ∀a:A. ∀j: i ? a. for_all A U (C ? a j) (covers A U) → covers A U a.
notation "hvbox(a break ◃ b)" non associative with precedence 45
-for @{ 'covers $a $b }.
+for @{ 'covers $a $b }. (* a \ltri b *)
+interpretation "coversl" 'covers A U = (for_all _ U A (covers _ U)).
interpretation "covers" 'covers a U = (covers _ U a).
-interpretation "coversl" 'covers A U = (coversl _ U A).
definition covers_elim ≝
λA:axiom_set.λU: 2 \sup A.λP:2 \sup A.
λH1:∀a:A. a ∈ U → a ∈ P.
- λH2:∀a:A.∀j:i ? a. C ? a j ◃ U → (∀b. b ∈ C ? a j → b ∈ P) → a ∈ P.
+ λH2:∀a:A.∀j:i ? a. C ? a j ◃ U → C ? a j ⊆ P → a ∈ P.
let rec aux (a:A) (p:a ◃ U) on p : a ∈ P ≝
match p return λaa.λ_:aa ◃ U.aa ∈ P with
[ refl a q ⇒ H1 a q
- | infinity a j q ⇒ H2 a j q (auxl (C ? a j) q)
- ]
- and auxl (V: 2 \sup A) (q: V ◃ U) on q : ∀b. b ∈ V → b ∈ P ≝
- match q return λVV.λ_:VV ◃ U.∀b. b ∈ VV → b ∈ P with
- [ iter VV f ⇒ λb.λr. aux b (f b r) ]
+ | infinity a j q ⇒
+ H2 a j q
+ match q return λ_:(C ? a j) ◃ U. C ? a j ⊆ P with
+ [ iter f ⇒ λb.λr. aux b (f b r) ]]
in
aux.
+inductive ex_such (A : axiom_set) (U,V : 2 \sup A) (fish: A → CProp) : CProp ≝
+ found : ∀a. a ∈ V → fish a → ex_such A U V fish.
+
coinductive fish (A:axiom_set) (U: 2 \sup A) : A → CProp ≝
- mk_fish: ∀a:A. (a ∈ U ∧ ∀j: i ? a. ∃y: A. y ∈ C ? a j ∧ fish A U y) → fish A U a.
+ mk_fish: ∀a:A. a ∈ U → (∀j: i ? a. ex_such A U (C ? a j) (fish A U)) → fish A U a.
notation "hvbox(a break ⋉ b)" non associative with precedence 45
-for @{ 'fish $a $b }.
+for @{ 'fish $a $b }. (* a \ltimes b *)
+interpretation "fishl" 'fish A U = (ex_such _ U A (fish _ U)).
interpretation "fish" 'fish a U = (fish _ U a).
let corec fish_rec (A:axiom_set) (U: 2 \sup A)
(P: 2 \sup A) (H1: ∀a:A. a ∈ P → a ∈ U)
- (H2: ∀a:A. a ∈ P → ∀j: i ? a. ∃y: A. y ∈ C ? a j ∧ y ∈ P) :
+ (H2: ∀a:A. a ∈ P → ∀j: i ? a. C ? a j ≬ P):
∀a:A. ∀p: a ∈ P. a ⋉ U ≝
λa,p.
mk_fish A U a
- (conj ? ? (H1 ? p)
+ (H1 ? p)
(λj: i ? a.
match H2 a p j with
- [ ex_introT (y: A) (Ha: y ∈ C ? a j ∧ y ∈ P) ⇒
- match Ha with
- [ conj (fHa: y ∈ C ? a j) (sHa: y ∈ P) ⇒
- ex_introT A (λy.y ∈ C ? a j ∧ fish A U y) y
- (conj ? ? fHa (fish_rec A U P H1 H2 y sHa))
- ]
- ])).
+ [ ex_introT2 (y: A) (HyC : y ∈ C ? a j) (HyP : y ∈ P) ⇒
+ found ???? y HyC (fish_rec A U P H1 H2 y HyP)
+ ]).
theorem reflexivity: ∀A:axiom_set.∀a:A.∀V. a ∈ V → a ◃ V.
intros;
theorem transitivity: ∀A:axiom_set.∀a:A.∀U,V. a ◃ U → U ◃ V → a ◃ V.
intros;
- apply (covers_elim ?? (mk_powerset A (λa.a ◃ V)) ??? H); intros;
- [ cases H1 in H2;
- intro;
- apply H2;
- assumption
+ apply (covers_elim ?? {a | a ◃ V} ??? H); simplify; intros;
+ [ cases H1 in H2; apply H2;
| apply infinity;
[ assumption
| constructor 1;
theorem coreflexivity: ∀A:axiom_set.∀a:A.∀V. a ⋉ V → a ∈ V.
intros;
cases H;
- cases H1;
assumption.
qed.
theorem cotransitivity:
- ∀A:axiom_set.∀a:A.∀U,V. a ⋉ U → (∀b. b ⋉ U → b ∈ V) → a ⋉ V.
+ ∀A:axiom_set.∀a:A.∀U,V. a ⋉ U → (∀b:A. b ⋉ U → b ∈ V) → a ⋉ V.
intros;
- apply (fish_rec ?? (mk_powerset A (λa.a ⋉ U)) ??? H); simplify; intros;
- [ apply H1;
- assumption
- | cases H2 in j; clear H2; cases H3; clear H3;
- assumption]
-qed.
\ No newline at end of file
+ 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 a1] assumption
+ | cases H2 in j H H1; clear H2 a1; intros;
+ cases (H1 i); clear H1; apply (H3 a1); assumption]
+qed.
+
+definition singleton ≝ λA:axiom_set.λa:A.{b | a=b}.
+
+notation "hvbox({ term 19 a })" with precedence 90 for @{ 'singl $a}.
+
+interpretation "singleton" 'singl a = (singleton _ a).
+
+definition leq ≝ λA:axiom_set.λa,b:A. a ◃ {b}.
+
+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).
+
+notation "↑a" with precedence 80 for @{ 'uparrow $a }.
+
+interpretation "uparrow" 'uparrow a = (uparrow _ a).
+
+definition downarrow ≝ λA:axiom_set.λU:2 \sup A.mk_powerset ? (λa:A. ↑a ≬ U).
+
+notation "↓a" with precedence 80 for @{ 'downarrow $a }.
+
+interpretation "downarrow" 'downarrow a = (downarrow _ a).
+
+definition fintersects ≝ λA:axiom_set.λU,V:2 \sup A.↓U ∩ ↓V.
+
+notation "hvbox(U break ↓ V)" non associative with precedence 80 for @{ 'fintersects $U $V }.
+
+interpretation "fintersects" 'fintersects U V = (fintersects _ U V).
+
+record convergent_generated_topology : Type ≝
+ { AA:> axiom_set;
+ convergence: ∀a:AA.∀U,V:2 \sup AA. a ◃ U → a ◃ V → a ◃ U ↓ V
+ }.
+
projects / helm.git / blobdiff