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 squareO.
- \fst x ∈ [u,v] → \snd x ∈ [u,v] → {[u,v]} squareO.
+ ∀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.
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]} squareO → O squareO ≝
- λO:ordered_set.λu,v:O.λb:{[u,v]} squareO.〈\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 ordered_set_square_of_segment_square nocomposites.
lemma restriction_agreement :
- ∀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]} squareO → Prop. λOP:O squareO → Prop.
+ ∀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]} 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;
+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]} 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;
+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]} 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);]
+ ∀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.
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)〉.
+ ∀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 xxx :
+ ∀O,s,x.bs2_of_bss2 (ordered_set_OF_ordered_uniform_space O) s x
+ =
+ x.
+intros; reflexivity;
+*)
+
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]} 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);
+ 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 O w Gw x Hm) as IH;
- apply (restrict ? l r ??? Hwp IH);
+ lapply (us_phi1 O w Gw x) as IH;[2:intro;apply Hm;cases H; clear H;
+ [left;apply (x2sx ? s (\fst x) (\snd x) H1);
+ |right;apply (x2sx ? s ?? H1);]
+
+ 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 O u v Gu Gv) (w HW); cases HW (Gw Hw); clear HW;
apply (restrict ? l u ??? H4); apply (Hm ? H1);
qed.
-definition hint_sequence:
- ∀C:ordered_set.
- sequence (hos_carr (os_l C)) → sequence (Type_of_ordered_set C).
-intros;assumption;
-qed.
-
-definition hint_sequence1:
- ∀C:ordered_set.
- sequence (hos_carr (os_r C)) → sequence (Type_of_ordered_set_dual C).
-intros;assumption;
-qed.
-
-definition hint_sequence2:
- ∀C:ordered_set.
- sequence (Type_of_ordered_set C) → sequence (hos_carr (os_l C)).
-intros;assumption;
-qed.
-
-definition hint_sequence3:
- ∀C:ordered_set.
- sequence (Type_of_ordered_set_dual C) → sequence (hos_carr (os_r C)).
-intros;assumption;
-qed.
-
-coercion hint_sequence nocomposites.
-coercion hint_sequence1 nocomposites.
-coercion hint_sequence2 nocomposites.
-coercion hint_sequence3 nocomposites.
-
definition order_continuity ≝
λC:ordered_uniform_space.∀a:sequence C.∀x:C.
(a ↑ x → a uniform_converges x) ∧ (a ↓ x → a uniform_converges x).
projects / helm.git / blobdiff