ous_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
}.
-lemma segment_square_of_O_square:
- ∀O:ordered_set.∀u,v:O.∀x:O square.fst x ∈ [u,v] → snd x ∈ [u,v] → {[u,v]} square.
-intros; split; exists; [1: apply (fst x) |3: apply (snd x)] assumption;
+definition invert_os_relation ≝
+ λC:ordered_set.λU:C square → Prop.
+ λx:C square. U 〈\snd x,\fst x〉.
+
+interpretation "relation invertion" 'invert a = (invert_os_relation _ a).
+interpretation "relation invertion" 'invert_symbol = (invert_os_relation _).
+interpretation "relation invertion" 'invert_appl a x = (invert_os_relation _ a x).
+
+lemma segment_square_of_ordered_set_square:
+ ∀O:ordered_set.∀u,v:O.∀x:O square.
+ \fst x ∈ [u,v] → \snd x ∈ [u,v] → {[u,v]} square.
+intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
qed.
-coercion cic:/matita/dama/ordered_uniform/segment_square_of_O_square.con 0 2.
+coercion cic:/matita/dama/ordered_uniform/segment_square_of_ordered_set_square.con 0 2.
-alias symbol "pi1" (instance 4) = "sigma pi1".
-alias symbol "pi1" (instance 2) = "sigma pi1".
-lemma O_square_of_segment_square :
+alias symbol "pi1" (instance 4) = "exT \fst".
+alias symbol "pi1" (instance 2) = "exT \fst".
+lemma ordered_set_square_of_segment_square :
∀O:ordered_set.∀u,v:O.{[u,v]} square → O square ≝
- λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈fst(fst b),fst(snd b)〉.
+ λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈\fst(\fst b),\fst(\snd b)〉.
-coercion cic:/matita/dama/ordered_uniform/O_square_of_segment_square.con.
+coercion cic:/matita/dama/ordered_uniform/ordered_set_square_of_segment_square.con.
lemma restriction_agreement :
∀O:ordered_uniform_space.∀l,r:O.∀P:{[l,r]} square → Prop.∀OP:O square → Prop.Prop.
-apply(λO:ordered_uniform_space.λl,r:O.λP:{[l,r]} square → Prop.λOP:O square → Prop.
- ∀b:O square.∀H1,H2.
- (P b → OP b) ∧ (OP b → P b));
+apply(λO:ordered_uniform_space.λl,r:O.
+ λP:{[l,r]} square → Prop.λOP:O square → Prop.
+ ∀b:O square.∀H1,H2.(P b → OP b) ∧ (OP b → P b));
[5,7: apply H1|6,8:apply H2]skip;
qed.
lemma unrestrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} square.
restriction_agreement ? l r U u → U x → u x.
-intros 7; cases x (b b1); cases b; cases b1;
-cases (H 〈x1,x2〉 H1 H2) (L _); intros; apply L; assumption;
+intros 7; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b b1 x;
+cases (H 〈w1,w2〉 H1 H2) (L _); intro Uw; apply L; apply Uw;
qed.
lemma restrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} square.
restriction_agreement ? l r U u → u x → U x.
-intros 6; cases x (b b1); cases b; cases b1; intros (X);
-cases (X 〈x1,x2〉 H H1) (_ R); apply R; assumption;
+intros 6; cases x (b b1); cases b (w1 H1); cases b1 (w2 H2); clear b1 b x;
+intros (Ra uw); cases (Ra 〈w1,w2〉 H1 H2) (_ R); apply R; apply uw;
qed.
lemma invert_restriction_agreement:
- ∀O:ordered_uniform_space.∀l,r:O.∀U:{[l,r]} square → Prop.∀u.
+ ∀O:ordered_uniform_space.∀l,r:O.
+ ∀U:{[l,r]} square → Prop.∀u:O square → Prop.
restriction_agreement ? l r U u →
restriction_agreement ? l r (inv U) (inv u).
intros 9; split; intro;
-[1: apply (unrestrict ????? (segment_square_of_O_square ??? 〈snd b,fst b〉 H2 H1) H H3);
-|2: apply (restrict ????? (segment_square_of_O_square ??? 〈snd b,fst b〉 H2 H1) H H3);]
+[1: apply (unrestrict ????? (segment_square_of_ordered_set_square ??? 〈\snd b,\fst b〉 H2 H1) H H3);
+|2: apply (restrict ????? (segment_square_of_ordered_set_square ??? 〈\snd b,\fst b〉 H2 H1) H H3);]
qed.
+alias symbol "square" (instance 8) = "bishop set square".
lemma bs_of_ss:
∀O:ordered_set.∀u,v:O.{[u,v]} square → (bishop_set_of_ordered_set O) square ≝
- λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈fst(fst b),fst(snd b)〉.
+ λO:ordered_set.λu,v:O.λb:{[u,v]} square.〈\fst(\fst b),\fst(\snd b)〉.
-notation < "x \sub \neq" non associative with precedence 91 for @{'bsss $x}.
+notation < "x \sub \neq" with precedence 91 for @{'bsss $x}.
interpretation "bs_of_ss" 'bsss x = (bs_of_ss _ _ _ x).
+alias symbol "square" (instance 7) = "ordered set square".
+lemma ss_of_bs:
+ ∀O:ordered_set.∀u,v:O.
+ ∀b:O square.\fst b ∈ [u,v] → \snd b ∈ [u,v] → {[u,v]} square ≝
+ λO:ordered_set.λu,v:O.
+ λb:(O:bishop_set) square.λH1,H2.〈≪\fst b,H1≫,≪\snd b,H2≫〉.
+
+notation < "x \sub \nleq" with precedence 91 for @{'ssbs $x}.
+interpretation "ss_of_bs" 'ssbs x = (ss_of_bs _ _ _ x _ _).
+
lemma segment_ordered_uniform_space:
∀O:ordered_uniform_space.∀u,v:O.ordered_uniform_space.
intros (O l r); apply mk_ordered_uniform_space;
letin f ≝ (λP:{[l,r]} square → Prop. ∃OP:O square → Prop.
(us_unifbase O OP) ∧ restriction_agreement ??? P OP);
apply (mk_uniform_space (bishop_set_of_ordered_set {[l,r]}) f);
- [1: intros (U H); intro x; unfold mk_set; simplify;
+ [1: intros (U H); intro x; simplify;
cases H (w Hw); cases Hw (Gw Hwp); clear H Hw; intro Hm;
lapply (us_phi1 ?? Gw x Hm) as IH;
apply (restrict ?????? Hwp IH);
exists; [apply (λb:{[l,r]} square.w b)] split;
[1: unfold f; simplify; clearbody f;
exists; [apply w]; split; [assumption] intro b; simplify;
- unfold segment_square_of_O_square; (* ??? *)
+ unfold segment_square_of_ordered_set_square;
cases b; intros; split; intros; assumption;
- |2: intros 2 (x Hx); unfold mk_set; cases (Hw ? Hx); split;
+ |2: intros 2 (x Hx); cases (Hw ? Hx); split;
[apply (restrict ?????? HuU H)|apply (restrict ?????? HvV H1);]]
|3: intros (U Hu); cases Hu (u HU); cases HU (Gu HuU); clear Hu HU;
cases (us_phi3 ?? Gu) (w HW); cases HW (Gw Hwu); clear HW;
exists; [apply (λx:{[l,r]} square.w x)] split;
[1: exists;[apply w];split;[assumption] intros; simplify; intro;
- unfold segment_square_of_O_square; (* ??? *)
+ unfold segment_square_of_ordered_set_square;
cases b; intros; split; intro; assumption;
|2: intros 2 (x Hx); apply (restrict ?????? HuU); apply Hwu;
- cases Hx (m Hm); exists[apply (fst m)] apply Hm;]
+ cases Hx (m Hm); exists[apply (\fst m)] apply Hm;]
|4: intros (U HU x); cases HU (u Hu); cases Hu (Gu HuU); clear HU Hu;
cases (us_phi4 ?? Gu x) (Hul Hur);
split; intros;
- [1: apply (restrict ?????? (invert_restriction_agreement ????? HuU));
+ [1: lapply (invert_restriction_agreement ????? HuU) as Ra;
+ apply (restrict ????? x Ra);
apply Hul; apply (unrestrict ?????? HuU H);
|2: apply (restrict ?????? HuU); apply Hur;
apply (unrestrict ?????? (invert_restriction_agreement ????? HuU) H);]]
(segment_ordered_uniform_space _ a b).
(* Lemma 3.2 *)
-alias symbol "pi1" = "sigma pi1".
+alias symbol "pi1" = "exT \fst".
lemma restric_uniform_convergence:
∀O:ordered_uniform_space.∀l,u:O.
∀x:{[l,u]}.
∀a:sequence {[l,u]}.
- (λn.fst (a n)) uniform_converges (fst x) →
+ ⌊n,\fst (a n)⌋ uniform_converges (\fst x) →
a uniform_converges x.
intros 8; cases H1; cases H2; clear H2 H1;
cases (H ? H3) (m Hm); exists [apply m]; intros;
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