|5: cases (with_ X); simplify; apply (us_phi4 (ous_us_ X))]
qed.
-coercion cic:/matita/dama/ordered_uniform/ous_unifspace.con.
+coercion ous_unifspace.
record ordered_uniform_space : Type ≝ {
ous_stuff :> ordered_uniform_space_;
- ous_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
+ ous_convex: ∀U.us_unifbase ous_stuff U → convex (os_l ous_stuff) U
}.
+definition half_ordered_set_OF_ordered_uniform_space : ordered_uniform_space → half_ordered_set.
+intro; compose ordered_set_OF_ordered_uniform_space with os_l. apply (f o);
+qed.
+
definition invert_os_relation ≝
- λC:ordered_set.λU:C square → Prop.
- λx:C square. U 〈\snd x,\fst x〉.
+ λC:ordered_set.λU:C squareO → Prop.
+ λx:C squareO. 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 hint_segment: ∀O.
+ segment (Type_of_ordered_set O) →
+ segment (hos_carr (os_l O)).
+intros; assumption;
+qed.
+
+coercion hint_segment nocomposites.
+
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.
+ ∀O:ordered_set.∀s:‡O.∀x:O squareO.
+ \fst x ∈ s → \snd x ∈ s → {[s]} squareO.
intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
qed.
-coercion cic:/matita/dama/ordered_uniform/segment_square_of_ordered_set_square.con 0 2.
+coercion segment_square_of_ordered_set_square with 0 2 nocomposites.
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)〉.
+ ∀O:ordered_set.∀s:‡O.{[s]} squareO → O squareO ≝
+ λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
-coercion cic:/matita/dama/ordered_uniform/ordered_set_square_of_segment_square.con.
+coercion ordered_set_square_of_segment_square nocomposites.
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));
+ ∀O:ordered_uniform_space.∀s:‡O.∀P:{[s]} squareO → Prop.∀OP:O squareO → Prop.Prop.
+apply(λO:ordered_uniform_space.λs:‡O.
+ λP:{[s]} squareO → Prop. λOP:O squareO → Prop.
+ ∀b:O squareO.∀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 (w1 H1); cases b1 (w2 H2); clear b b1 x;
+lemma unrestrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
+ restriction_agreement ? s U u → U x → u x.
+intros 6; 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 (w1 H1); cases b1 (w2 H2); clear b1 b x;
+lemma restrict: ∀O:ordered_uniform_space.∀s:‡O.∀U,u.∀x:{[s]} squareO.
+ restriction_agreement ? s U u → u x → U x.
+intros 5; 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 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_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);]
+ ∀O:ordered_uniform_space.∀s:‡O.
+ ∀U:{[s]} squareO → Prop.∀u:O squareO → Prop.
+ restriction_agreement ? s U u →
+ restriction_agreement ? s (\inv U) (\inv u).
+intros 8; split; intro;
+[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)〉.
-
-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≫〉.
+lemma bs2_of_bss2:
+ ∀O:ordered_set.∀s:‡O.(bishop_set_of_ordered_set {[s]}) squareB → (bishop_set_of_ordered_set O) squareB ≝
+ λO:ordered_set.λs:‡O.λb:{[s]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion bs2_of_bss2 nocomposites.
+
+
+lemma a2sa :
+ ∀O:ordered_uniform_space.∀s:‡(ordered_set_OF_ordered_uniform_space O).
+ ∀x:
+ bs_carr
+ (square_bishop_set
+ (bishop_set_of_ordered_set
+ (segment_ordered_set
+ (ordered_set_OF_ordered_uniform_space O) s))).
+ (\fst x) ≈ (\snd x) →
+ (\fst (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x))
+ ≈
+ (\snd (bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x)).
+intros 3; cases x (a b); clear x; cases a (x Hx); cases b (y Hy); clear a b;
+simplify; intros 2 (K H); apply K; clear K; whd; whd in H; cases H;
+[left|right] apply x2sx; assumption;
+qed.
-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;
-[1: apply (mk_ordered_uniform_space_ {[l,r]});
+ ∀O:ordered_uniform_space.∀s:‡O.ordered_uniform_space.
+intros (O s); apply mk_ordered_uniform_space;
+[1: apply (mk_ordered_uniform_space_ {[s]});
[1: alias symbol "and" = "constructive and".
- 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);
+ letin f ≝ (λP:{[s]} squareO → Prop. ∃OP:O squareO → Prop.
+ (us_unifbase O OP) ∧ restriction_agreement ?? P OP);
+ apply (mk_uniform_space (bishop_set_of_ordered_set {[s]}) f);
[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);
+ lapply (us_phi1 O w Gw x (a2sa ??? Hm)) as IH;
+ apply (restrict ? s ??? Hwp IH);
|2: intros (U V HU HV); cases HU (u Hu); cases HV (v Hv); clear HU HV;
cases Hu (Gu HuU); cases Hv (Gv HvV); clear Hu Hv;
- cases (us_phi2 ??? Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
- exists; [apply (λb:{[l,r]} square.w b)] split;
+ cases (us_phi2 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
+ exists; [apply (λb:{[s]} squareB.w b)] split;
[1: unfold f; simplify; clearbody f;
exists; [apply w]; split; [assumption] intro b; simplify;
unfold segment_square_of_ordered_set_square;
cases b; intros; split; intros; assumption;
|2: intros 2 (x Hx); cases (Hw ? Hx); split;
- [apply (restrict ?????? HuU H)|apply (restrict ?????? HvV H1);]]
+ [apply (restrict O s ??? HuU H)|apply (restrict O s ??? 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;
+ cases (us_phi3 O u Gu) (w HW); cases HW (Gw Hwu); clear HW;
+ exists; [apply (λx:{[s]} squareB.w x)] split;
[1: exists;[apply w];split;[assumption] intros; simplify; intro;
unfold segment_square_of_ordered_set_square;
cases b; intros; split; intro; assumption;
- |2: intros 2 (x Hx); apply (restrict ?????? HuU); apply Hwu;
+ |2: intros 2 (x Hx); apply (restrict O s ??? HuU); apply Hwu;
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);
+ cases (us_phi4 O u Gu x) (Hul Hur);
split; intros;
- [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);]]
+ [1: lapply (invert_restriction_agreement O s ?? HuU) as Ra;
+ apply (restrict O s ?? x Ra);
+ apply Hul; apply (unrestrict O s ??? HuU H);
+ |2: apply (restrict O s ??? HuU); apply Hur;
+ apply (unrestrict O s ??? (invert_restriction_agreement O s ?? HuU) H);]]
|2: simplify; reflexivity;]
-|2: simplify; unfold convex; intros;
- cases H (u HU); cases HU (Gu HuU); clear HU H;
- lapply (ous_convex ?? Gu (bs_of_ss ? l r p) ? H2 (bs_of_ss ? l r y) H3) as Cu;
- [1: apply (unrestrict ?????? HuU); apply H1;
- |2: apply (restrict ?????? HuU Cu);]]
+|2: simplify; unfold convex; intros 3; cases s1; intros;
+ (* TODO: x2sx is for ≰, we need one for ≤ *)
+ cases H (u HU); cases HU (Gu HuU); clear HU H;
+ lapply depth=0 (ous_convex ?? Gu 〈\fst h,\fst h1〉 ???????) as K3;
+ [2: intro; apply H2; apply (x2sx (os_l O) s h h1 H);
+ |3: apply 〈\fst (\fst y),\fst (\snd y)〉;
+ |4: intro; change in H with (\fst (\fst y) ≰ \fst h1); apply H3;
+ apply (x2sx (os_l O) s (\fst y) h1 H);
+ |5: change with (\fst h ≤ \fst (\fst y)); intro; apply H4;
+ apply (x2sx (os_l O) s h (\fst y) H);
+ |6: change with (\fst (\snd y) ≤ \fst h1); intro; apply H5;
+ apply (x2sx (os_l O) s (\snd y) h1 H);
+ |7: change with (\fst (\fst y) ≤ \fst (\snd y)); intro; apply H6;
+ apply (x2sx (os_l O) s (\fst y) (\snd y) H);
+ |8: apply (restrict O s U u y HuU K3);
+ |1: apply (unrestrict O s ?? 〈h,h1〉 HuU H1);]]
qed.
interpretation "Ordered uniform space segment" 'segment_set a b =
(* Lemma 3.2 *)
alias symbol "pi1" = "exT \fst".
lemma restric_uniform_convergence:
- ∀O:ordered_uniform_space.∀l,u:O.
- ∀x:{[l,u]}.
- ∀a:sequence {[l,u]}.
- ⌊n,\fst (a n)⌋ uniform_converges (\fst x) →
+ ∀O:ordered_uniform_space.∀s:‡O.
+ ∀x:{[s]}.
+ ∀a:sequence {[s]}.
+ (⌊n, \fst (a n)⌋ : sequence O) 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;
definition order_continuity ≝
λC:ordered_uniform_space.∀a:sequence C.∀x:C.
(a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x).
+
+lemma hint_boh1: ∀O. Type_OF_ordered_uniform_space O → hos_carr (os_l O).
+intros; assumption;
+qed.
+
+coercion hint_boh1 nocomposites.
+
+lemma hint_boh2: ∀O:ordered_uniform_space. hos_carr (os_l O) → Type_OF_ordered_uniform_space O.
+intros; assumption;
+qed.
+
+coercion hint_boh2 nocomposites.
+
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