include "ordered_uniform.ma".
include "property_sigma.ma".
+lemma h_segment_upperbound:
+ ∀C:half_ordered_set.
+ ∀s:segment C.
+ ∀a:sequence (half_segment_ordered_set C s).
+ (seg_u C s) (upper_bound ? ⌊n,\fst (a n)⌋).
+intros; cases (wloss_prop C); unfold; rewrite < H; simplify; intro n;
+cases (a n); simplify; unfold in H1; rewrite < H in H1; cases H1;
+simplify in H2 H3; rewrite < H in H2 H3; assumption;
+qed.
+
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
+
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
+
+lemma h_segment_preserves_uparrow:
+ ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
+ ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
+intros; cases H (Ha Hx); split;
+[ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
+| cases Hx; split;
+ [ intro n; intro H; apply (H1 n);apply (sx2x ???? H);
+ | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);]
+ exists [apply w] apply (x2sx ?? (a w) y H4);]]
+qed.
+
+notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
+notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
+
+interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
+interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
+
+(* Fact 2.18 *)
+lemma segment_cauchy:
+ ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
+ a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
+intros 6;
+alias symbol "pi1" (instance 3) = "pair pi1".
+alias symbol "pi2" = "pair pi2".
+apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
+(unfold segment_ordered_uniform_space; simplify);
+exists [apply U] split; [assumption;]
+intro; cases b; intros; simplify; split; intros; assumption;
+qed.
+
(* Definition 3.7 *)
definition exhaustive ≝
λC:ordered_uniform_space.
∀a,b:sequence C.
(a is_increasing → a is_upper_located → a is_cauchy) ∧
(b is_decreasing → b is_lower_located → b is_cauchy).
-
-(* Lemma 3.8 *)
-lemma xxx:
+
+lemma prove_in_segment:
+ ∀O:ordered_set.∀s:segment (os_l O).∀x:O.
+ 𝕝_s (λl.l ≤ x) → 𝕦_s (λu.x ≤ u) → x ∈ s.
+intros; unfold; cases (wloss_prop (os_l O)); rewrite < H2;
+split; assumption;
+qed.
+
+lemma under_wloss_upperbound:
+ ∀C:half_ordered_set.∀s:segment C.∀a:sequence C.
+ seg_u C s (upper_bound C a) →
+ ∀i.seg_u C s (λu.a i ≤≤ u).
+intros; unfold in H; unfold;
+cases (wloss_prop C); rewrite <H1 in H ⊢ %;
+apply (H i);
+qed.
+
+
+(* Lemma 3.8 NON DUALIZZATO *)
+lemma restrict_uniform_convergence_uparrow:
∀C:ordered_uniform_space.property_sigma C →
- ∀l,u:C.exhaustive {[l,u]} →
- ∀a:sequence {[l,u]}.∀x:C. (λn.fst (a n)) ↑ x →
- x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges (sig_in ?? x h).
+ ∀s:segment (os_l C).exhaustive (segment_ordered_uniform_space C s) →
+ ∀a:sequence (segment_ordered_uniform_space C s).
+ ∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
+ in_segment (os_l C) s x ∧ ∀h:x ∈ s.a uniform_converges ≪x,h≫.
intros; split;
-[1: (* manca fact 2.5 *)
+[1: unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
+ cases (wloss_prop (os_l C)) (W W); apply prove_in_segment; unfold; rewrite <W;
+ simplify;
+ [ apply (le_transitive ?? x ? (H2 O));
+ lapply (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K; apply K;
+ | intro; cases (H5 ? H4); clear H5 H4;
+ lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
+ apply K; apply H6;
+ | intro; unfold in H4; rewrite <W in H4;
+ lapply depth=0 (H5 (seg_u_ (os_l C) s)) as k; unfold in k:(%???→?);
+ simplify in k; rewrite <W in k; lapply (k
+ simplify;intro; cases (H5 ? H4); clear H5 H4;
+ lapply (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) w) as K;
+ unfold in K; whd in K:(?%????); simplify in K; rewrite <W in K;
+ apply K; apply H6;
+
-
\ No newline at end of file
+
+ cases H2 (Ha Hx); clear H2; cases Hx; split;
+ lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a) O) as W1;
+ lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a) O) as W2;
+ lapply (H2 O); simplify in Hletin; simplify in W2 W1;
+ cases a in Hletin W2 W1; simplify; cases (f O); simplify; intros;
+ whd in H6:(? % ? ? ? ?);
+ simplify in H6:(%);
+ cases (wloss_prop (os_l C)); rewrite <H8 in H5 H6 ⊢ %;
+ [ change in H6 with (le (os_l C) (seg_l_ (os_l C) s) w);
+ apply (le_transitive ??? H6 H7);
+ | apply (le_transitive (seg_u_ (os_l C) s) w x H6 H7);
+ |
+ lapply depth=0 (supremum_is_upper_bound ? x Hx (seg_u_ (os_l C) s)) as K;
+ lapply depth=0 (under_wloss_upperbound (os_l C) ?? (segment_upperbound s a));
+ apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+ rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+ | lapply depth=0 (h_supremum_is_upper_bound (os_r C) ⌊n,\fst (a n)⌋ x Hx (seg_l_ (os_r C) s)) as K;
+ lapply depth=0 (under_wloss_upperbound (os_r C) ?? (h_segment_upperbound (os_r C) s a));
+ apply K; intro; lapply (Hletin n); unfold; unfold in Hletin1;
+whd in Hletin1:(? % ? ? ? ?);
+simplify in Hletin1:(%);
+ rewrite < H8 in Hletin1; intro; apply Hletin1; clear Hletin1; apply H9;
+
+
+ apply (segment_upperbound ? l);
+ generalize in match (H2 O); generalize in match Hx; unfold supremum;
+ unfold upper_bound; whd in ⊢ (?→%→?); rewrite < H4;
+ split; unfold; rewrite < H4; simplify;
+ [1: lapply (infimum_is_lower_bound ? ? Hx u);
+
+
+
+split;
+ [1: apply (supremum_is_upper_bound ? x Hx u);
+ apply (segment_upperbound ? l);
+ |2: apply (le_transitive l ? x ? (H2 O));
+ apply (segment_lowerbound ? l u a 0);]
+|2: intros;
+ lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_uparrow C l u);split; assumption;]
+ lapply (segment_preserves_supremum C l u a ≪?,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H6 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
+
+lemma hint_mah1:
+ ∀C. Type_OF_ordered_uniform_space1 C → hos_carr (os_r C).
+ intros; assumption; qed.
+
+coercion hint_mah1 nocomposites.
+
+lemma hint_mah2:
+ ∀C. sequence (hos_carr (os_l C)) → sequence (hos_carr (os_r C)).
+ intros; assumption; qed.
+
+coercion hint_mah2 nocomposites.
+
+lemma hint_mah3:
+ ∀C. Type_OF_ordered_uniform_space C → hos_carr (os_r C).
+ intros; assumption; qed.
+
+coercion hint_mah3 nocomposites.
+
+lemma hint_mah4:
+ ∀C. sequence (hos_carr (os_r C)) → sequence (hos_carr (os_l C)).
+ intros; assumption; qed.
+
+coercion hint_mah4 nocomposites.
+
+lemma restrict_uniform_convergence_downarrow:
+ ∀C:ordered_uniform_space.property_sigma C →
+ ∀l,u:C.exhaustive {[l,u]} →
+ ∀a:sequence {[l,u]}.∀x: C. ⌊n,\fst (a n)⌋ ↓ x →
+ x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges ≪x,h≫.
+intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
+[1: split;
+ [2: apply (infimum_is_lower_bound ? x Hx l);
+ apply (segment_lowerbound ? l u);
+ |1: lapply (ge_transitive ? ? x ? (H2 O)); [apply u||assumption]
+ apply (segment_upperbound ? l u a 0);]
+|2: intros;
+ lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+ [2: apply (segment_preserves_downarrow ? l u);split; assumption;]
+ lapply (segment_preserves_infimum C l u a ≪x,h≫) as Ha2;
+ [2:split; assumption]; cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H7 H4 Ha1) as HaC;
+ lapply (segment_cauchy ? l u ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; split; assumption]
+ apply restric_uniform_convergence; assumption;]
+qed.
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