(b is_decreasing → b is_lower_located → b is_cauchy).
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;
+ ∀O:half_ordered_set.∀s:segment O.∀x:O.
+ seg_l O s (λl.l ≤≤ x) → seg_u O s (λu.x ≤≤ u) → x ∈ s.
+intros; unfold; cases (wloss_prop 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 h_uparrow_to_in_segment:
+ ∀C:half_ordered_set.
+ ∀s:segment C.
+ ∀a:sequence C.
+ (∀i.a i ∈ s) →
+ ∀x:C. uparrow C a x →
+ in_segment C s x.
+intros (C H a H1 x H2); unfold in H2; cases H2; clear H2;unfold in H3 H4; cases H4; clear H4; unfold in H2;
+cases (wloss_prop C) (W W); apply prove_in_segment; unfold; rewrite <W;simplify;
+[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
+ cases K; unfold in H4 H6; rewrite <W in H6 H4; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+ cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H5 H6);
+| apply (hle_transitive ??? x ? (H2 O)); lapply (H1 0) as K; unfold in K; rewrite <W in K;
+ cases K; unfold in H4 H6; rewrite <W in H4 H6; simplify in H4 H6; assumption;
+| intro; cases (H5 ? H4); clear H5 H4;lapply(H1 w) as K; unfold in K; rewrite<W in K;
+ cases K; unfold in H5 H4; rewrite<W in H4 H5; simplify in H4 H5; apply (H4 H6);]
+qed.
+notation "'uparrow_to_in_segment'" non associative with precedence 90 for @{'uparrow_to_in_segment}.
+notation "'downarrow_to_in_segment'" non associative with precedence 90 for @{'downarrow_to_in_segment}.
+interpretation "uparrow_to_in_segment" 'uparrow_to_in_segment = (h_uparrow_to_in_segment (os_l _)).
+interpretation "downarrow_to_in_segment" 'downarrow_to_in_segment = (h_uparrow_to_in_segment (os_r _)).
+
(* Lemma 3.8 NON DUALIZZATO *)
lemma restrict_uniform_convergence_uparrow:
∀C:ordered_uniform_space.property_sigma C →
∀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: 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;
-
-
-
- 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);]
+[1: apply (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2);
+ simplify; intros; cases (a i); assumption;
|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;]
+ [2: apply (segment_preserves_uparrow s); assumption;]
+ lapply (segment_preserves_supremum s a ≪?,h≫ H2) as Ha2;
+ cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H5 H3 Ha1) as HaC;
+ lapply (segment_cauchy C s ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [left; assumption]
+ apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
qed.
lemma hint_mah1:
coercion hint_mah4 nocomposites.
+lemma hint_mah5:
+ ∀C. segment (hos_carr (os_r C)) → segment (hos_carr (os_l C)).
+ intros; assumption; qed.
+
+coercion hint_mah5 nocomposites.
+
+lemma hint_mah6:
+ ∀C. segment (hos_carr (os_l C)) → segment (hos_carr (os_r C)).
+ intros; assumption; qed.
+
+coercion hint_mah6 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);]
+ ∀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: apply (downarrow_to_in_segment s ⌊n,\fst (a n)⌋ ? x); [2: apply H2];
+ simplify; intros; cases (a i); assumption;
|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;]
+ [2: apply (segment_preserves_downarrow s a x h H2);]
+ lapply (segment_preserves_infimum s a ≪?,h≫ H2) as Ha2;
+ cases Ha2; clear Ha2;
+ cases (H1 a a); lapply (H6 H3 Ha1) as HaC;
+ lapply (segment_cauchy C s ? HaC) as Ha;
+ lapply (sigma_cauchy ? H ? x ? Ha); [right; assumption]
+ apply (restric_uniform_convergence C s ≪x,h≫ a Hletin)]
qed.
projects / helm.git / blobdiff