interpretation "Bishop subset" 'subset a b =
(cic:/matita/dama/uniform/subset.con _ a b).
-notation "{ ident x : t | p }" non associative with precedence 50
+notation "hvbox({ ident x : t | break p })" non associative with precedence 50
for @{ 'explicitset (\lambda ${ident x} : $t . $p) }.
definition mk_set ≝ λT:bishop_set.λx:T→Prop.x.
interpretation "explicit set" 'explicitset t =
(cic:/matita/dama/uniform/mk_set.con _ t).
-notation < "s \sup 2" non associative with precedence 90
- for @{ 'square $s }.
+notation < "s 2 \atop \neq" non associative with precedence 90
+ for @{ 'square2 $s }.
notation > "s 'square'" non associative with precedence 90
for @{ 'square $s }.
-interpretation "bishop suqare set" 'square x =
+interpretation "bishop set square" 'square x =
+ (cic:/matita/dama/uniform/square_bishop_set.con x).
+interpretation "bishop set square" 'square2 x =
(cic:/matita/dama/uniform/square_bishop_set.con x).
+
alias symbol "exists" = "exists".
alias symbol "and" = "logical and".
definition compose_relations ≝
λC:bishop_set.λU,V:C square → Prop.
- λx:C square.∃y:C. U (prod ?? (fst x) y) ∧ V (prod ?? y (snd x)).
+ λx:C square.∃y:C. U 〈fst x,y〉 ∧ V 〈y,snd x〉.
notation "a \circ b" left associative with precedence 60
for @{'compose $a $b}.
interpretation "relations composition" 'compose a b =
(cic:/matita/dama/uniform/compose_relations.con _ a b).
+notation "hvbox(x \in break a \circ break b)" non associative with precedence 50
+ for @{'compose2 $a $b $x}.
+interpretation "relations composition" 'compose2 a b x =
+ (cic:/matita/dama/uniform/compose_relations.con _ a b x).
definition invert_relation ≝
λC:bishop_set.λU:C square → Prop.
- λx:C square. U (prod ?? (snd x) (fst x)).
+ λx:C square. U 〈snd x,fst x〉.
notation < "s \sup (-1)" non associative with precedence 90
for @{ 'invert $s }.
+notation < "s \sup (-1) x" non associative with precedence 90
+ for @{ 'invert2 $s $x}.
notation > "'inv' s" non associative with precedence 90
for @{ 'invert $s }.
interpretation "relation invertion" 'invert a =
(cic:/matita/dama/uniform/invert_relation.con _ a).
+interpretation "relation invertion" 'invert2 a x =
+ (cic:/matita/dama/uniform/invert_relation.con _ a x).
+alias symbol "exists" = "CProp exists".
+alias symbol "and" (instance 18) = "constructive and".
+alias symbol "and" (instance 10) = "constructive and".
record uniform_space : Type ≝ {
us_carr:> bishop_set;
us_unifbase: (us_carr square → Prop) → CProp;
∃W:us_carr square → Prop.us_unifbase W ∧ (W ⊆ {x:?|U x ∧ V x});
us_phi3: ∀U:us_carr square → Prop. us_unifbase U →
∃W:us_carr square → Prop.us_unifbase W ∧ (W ∘ W) ⊆ U;
- us_phi4: ∀U:us_carr square → Prop. us_unifbase U → U = inv U (* I should use double inclusion ... *)
+ us_phi4: ∀U:us_carr square → Prop. us_unifbase U → ∀x.(U x → (inv U) x) ∧ ((inv U) x → U x)
}.
(* Definition 2.14 *)
alias symbol "leq" = "natural 'less or equal to'".
definition cauchy ≝
λC:uniform_space.λa:sequence C.∀U.us_unifbase C U →
- ∃n. ∀i,j. n ≤ i → n ≤ j → U (prod ?? (a i) (a j)).
+ ∃n. ∀i,j. n ≤ i → n ≤ j → U 〈a i,a j〉.
notation < "a \nbsp 'is_cauchy'" non associative with precedence 50
for @{'cauchy $a}.
(* Definition 2.15 *)
definition uniform_converge ≝
λC:uniform_space.λa:sequence C.λu:C.
- ∀U.us_unifbase C U → ∃n. ∀i. n ≤ i → U (prod ?? u (a i)).
+ ∀U.us_unifbase C U → ∃n. ∀i. n ≤ i → U 〈u,a i〉.
notation < "a \nbsp (\u \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 50
for @{'uniform_converge $a $x}.
cases (us_phi3 ?? Hu) (v Hv0); cases Hv0 (Hv H); clear Hv0;
cases (Ha ? Hv) (n Hn); exists [apply n]; intros;
apply H; unfold; exists [apply x]; split [2: apply (Hn ? H2)]
-rewrite > (us_phi4 ?? Hv); simplify; apply (Hn ? H1);
+cases (us_phi4 ?? Hv 〈a i,x〉) (P1 P2); apply P2;
+apply (Hn ? H1);
qed.
(* Definition 2.17 *)
definition restrict_uniformity ≝
λC:uniform_space.λX:C→Prop.
{U:C square → Prop| (U ⊆ {x:C square|X (fst x) ∧ X(snd x)}) ∧ us_unifbase C U}.
-
-