ous_convex: ∀U.us_unifbase ous_stuff U → convex ous_stuff U
}.
+definition Type_of_ordered_uniform_space : ordered_uniform_space → Type.
+intro; compose ordered_set_OF_ordered_uniform_space with os_l.
+apply (hos_carr (f o));
+qed.
+
+definition Type_of_ordered_uniform_space_dual : ordered_uniform_space → Type.
+intro; compose ordered_set_OF_ordered_uniform_space with os_r.
+apply (hos_carr (f o));
+qed.
+
+coercion Type_of_ordered_uniform_space.
+coercion Type_of_ordered_uniform_space_dual.
+
+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 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.∀u,v:O.∀x:O squareO.
+ \fst x ∈ [u,v] → \snd x ∈ [u,v] → {[u,v]} squareO.
intros; split; exists; [1: apply (\fst x) |3: apply (\snd x)] assumption;
qed.
-coercion segment_square_of_ordered_set_square with 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.∀u,v:O.{[u,v]} squareO → O squareO ≝
+ λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
-coercion ordered_set_square_of_segment_square.
+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.
+ ∀O:ordered_uniform_space.∀l,r:O.∀P:{[l,r]} squareO → Prop.∀OP:O squareO → 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));
+ λP:{[l,r]} 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.
+lemma unrestrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} squareO.
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;
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.
+lemma restrict: ∀O:ordered_uniform_space.∀l,r:O.∀U,u.∀x:{[l,r]} squareO.
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;
intros (Ra uw); cases (Ra 〈w1,w2〉 H1 H2) (_ R); apply R; apply uw;
lemma invert_restriction_agreement:
∀O:ordered_uniform_space.∀l,r:O.
- ∀U:{[l,r]} square → Prop.∀u:O square → Prop.
+ ∀U:{[l,r]} squareO → Prop.∀u:O squareO → 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);]
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)〉.
+
+lemma bs2_of_bss2:
+ ∀O:ordered_set.∀u,v:O.(bishop_set_of_ordered_set {[u,v]}) squareB → (bishop_set_of_ordered_set O) squareB ≝
+ λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\fst(\fst b),\fst(\snd b)〉.
+
+coercion bs2_of_bss2 nocomposites.
(*
notation < "x \sub \neq" with precedence 91 for @{'bsss $x}.
interpretation "bs_of_ss" 'bsss x = (bs_of_ss _ _ _ x).
*)
+(*
lemma ss_of_bs:
∀O:ordered_set.∀u,v:O.
- ∀b:O square.\fst b ∈ [u,v] → \snd b ∈ [u,v] → {[u,v]} square ≝
+ ∀b:O squareO.\fst b ∈ [u,v] → \snd b ∈ [u,v] → {[u,v]} squareO ≝
λO:ordered_set.λu,v:O.
- λb:(O:bishop_set) square.λH1,H2.〈≪\fst b,H1≫,≪\snd b,H2≫〉.
+ λb:O squareB.λH1,H2.〈≪\fst b,H1≫,≪\snd b,H2≫〉.
+*)
(*
notation < "x \sub \nleq" with precedence 91 for @{'ssbs $x}.
intros (O l r); apply mk_ordered_uniform_space;
[1: apply (mk_ordered_uniform_space_ {[l,r]});
[1: alias symbol "and" = "constructive and".
- letin f ≝ (λP:{[l,r]} square → Prop. ∃OP:O square → Prop.
+ letin f ≝ (λP:{[l,r]} squareO → Prop. ∃OP:O squareO → 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; 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 Hm) as IH;
+ apply (restrict ? l r ??? Hwp IH); STOP
|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;
constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity]
qed.
-notation "hvbox({[a, break b]})" non associative with precedence 90
- for @{'segment_set $a $b}.
-interpretation "Ordered set segment" 'segment_set a b =
+notation < "hvbox({[a, break b]/})" non associative with precedence 90
+ for @{'h_segment_set $a $b}.
+notation > "hvbox({[a, break b]/})" non associative with precedence 90
+ for @{'h_segment_set $a $b}.
+interpretation "Half ordered set segment" 'h_segment_set a b =
(half_segment_ordered_set _ a b).
+
+notation < "hvbox({[a, break b]})" non associative with precedence 90
+ for @{'segment_set $a $b}.
+notation > "hvbox({[a, break b]})" non associative with precedence 90
+ for @{'segment_set $a $b}.
interpretation "Ordered set segment" 'segment_set a b =
(segment_ordered_set _ a b).
(* Lemma 2.9 *)
lemma h_segment_preserves_supremum:
- ∀O:half_ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
+ ∀O:half_ordered_set.∀l,u:O.∀a:sequence {[l,u]/}.∀x:{[l,u]/}.
increasing ? ⌊n,\fst (a n)⌋ ∧
supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
intros; split; cases H; clear H;
interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
(* Definition 2.10 *)
-alias symbol "square" = "ordered set square".
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
definition square_segment ≝
- λO:ordered_set.λa,b:O.λx: O square.
+ λO:ordered_set.λa,b:O.λx: O squareO.
And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
definition convex ≝
- λO:ordered_set.λU:O square → Prop.
- ∀p.U p → \fst p ≤ \snd p → ∀y. square_segment ? (\fst p) (\snd p) y → U y.
+ λO:ordered_set.λU:O squareO → Prop.
+ ∀p.U p → \fst p ≤ \snd p → ∀y.
+ square_segment O (\fst p) (\snd p) y → U y.
(* Definition 2.11 *)
definition upper_located ≝
include "supremum.ma".
(* Definition 2.13 *)
-alias symbol "square" = "bishop set square".
alias symbol "pair" = "Pair construction".
alias symbol "exists" = "exists".
alias symbol "and" = "logical and".
definition compose_bs_relations ≝
- λC:bishop_set.λU,V:C square → Prop.
- λx:C square.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉.
+ λC:bishop_set.λU,V:C squareB → Prop.
+ λx:C squareB.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉.
definition compose_os_relations ≝
- λC:ordered_set.λU,V:C square → Prop.
- λx:C square.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉.
+ λC:ordered_set.λU,V:C squareB → Prop.
+ λx:C squareB.∃y:C. U 〈\fst x,y〉 ∧ V 〈y,\snd x〉.
interpretation "bishop set relations composition" 'compose a b = (compose_bs_relations _ a b).
interpretation "ordered set relations composition" 'compose a b = (compose_os_relations _ a b).
definition invert_bs_relation ≝
- λC:bishop_set.λU:C square → Prop.
- λx:C square. U 〈\snd x,\fst x〉.
+ λC:bishop_set.λU:C squareB → Prop.
+ λx:C squareB. U 〈\snd x,\fst x〉.
notation > "\inv" with precedence 60 for @{ 'invert_symbol }.
interpretation "relation invertion" 'invert a = (invert_bs_relation _ a).
alias symbol "and" (instance 9) = "constructive and".
record uniform_space : Type ≝ {
us_carr:> bishop_set;
- us_unifbase: (us_carr square → Prop) → CProp;
- us_phi1: ∀U:us_carr square → Prop. us_unifbase U →
- (λx:us_carr square.\fst x ≈ \snd x) ⊆ U;
- us_phi2: ∀U,V:us_carr square → Prop. us_unifbase U → us_unifbase V →
- ∃W:us_carr square → Prop.us_unifbase W ∧ (W ⊆ (λx.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 → ∀x.(U x → (\inv U) x) ∧ ((\inv U) x → U x)
+ us_unifbase: (us_carr squareB → Prop) → CProp;
+ us_phi1: ∀U:us_carr squareB → Prop. us_unifbase U →
+ (λx:us_carr squareB.\fst x ≈ \snd x) ⊆ U;
+ us_phi2: ∀U,V:us_carr squareB → Prop. us_unifbase U → us_unifbase V →
+ ∃W:us_carr squareB → Prop.us_unifbase W ∧ (W ⊆ (λx.U x ∧ V x));
+ us_phi3: ∀U:us_carr squareB → Prop. us_unifbase U →
+ ∃W:us_carr squareB → Prop.us_unifbase W ∧ (W ∘ W) ⊆ U;
+ us_phi4: ∀U:us_carr squareB → Prop. us_unifbase U → ∀x.(U x → (\inv U) x) ∧ ((\inv U) x → U x)
}.
(* Definition 2.14 *)
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