include "ordered_uniform.ma".
include "property_sigma.ma".
-(* 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 segment_upperbound:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound (λn.fst (a n)).
-intros 5; change with (fst (a n) ≤ u); cases (a n); cases H; assumption;
+lemma h_segment_upperbound:
+ ∀C:half_ordered_set.
+ ∀s:segment C.
+ ∀a:sequence (half_segment_ordered_set C s).
+ upper_bound ? ⌊n,\fst (a n)⌋ (seg_u C s).
+intros 4; simplify; cases (a n); simplify; unfold in H;
+cases (wloss_prop C); rewrite < H1 in H; simplify; cases H;
+assumption;
qed.
-lemma segment_lowerbound:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound (λn.fst (a n)).
-intros 5; change with (l ≤ fst (a n)); cases (a n); cases H; assumption;
-qed.
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
-lemma segment_preserves_uparrow:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- (λn.fst (a n)) ↑ x → a ↑ (sig_in ?? x h).
-intros; cases H (Ha Hx); split [apply Ha] cases Hx;
-split; [apply H1] intros;
-cases (H2 (fst y)); [2: apply H3;] exists [apply w] assumption;
-qed.
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
-lemma segment_preserves_downarrow:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- (λn.fst (a n)) ↓ x → a ↓ (sig_in ?? x h).
-intros; cases H (Ha Hx); split [apply Ha] cases Hx;
-split; [apply H1] intros;
-cases (H2 (fst y));[2:apply H3]; exists [apply w] assumption;
+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 rule H;
+| cases Hx; split;
+ [ intro n; intro H; apply (H1 n);apply rule H;
+ | intros; cases (H2 (\fst y)); [2: apply rule 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.∀l,u:C.∀a:sequence {[l,u]}.
- a is_cauchy → (λn:nat.fst (a n)) is_cauchy.
-intros 7;
+ ∀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:{[l,u]} square.U 〈fst (fst x),fst (snd x)〉));
+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.
-(* Lemma 3.8 *)
+(* 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 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;
+[ apply (hle_transitive ??? x ? (H2 O)); lapply (H1 O) as K; unfold in K; rewrite <W in K;
+ cases K; unfold in H4 H6; apply H4;
+| 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; apply H5; apply 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; apply H6;
+| 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; 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 →
- ∀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).
-intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
-[1: split;
- [1: apply (supremum_is_upper_bound C ?? Hx u);
- apply (segment_upperbound ? l);
- |2: apply (le_transitive ?? (fst (a 0))); [2: apply H2;]
- apply (segment_lowerbound ?l u);]
+ ∀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 (uparrow_to_in_segment s ⌊n,\fst (a \sub n)⌋ ? x H2);
+ simplify; intros; cases (a i); assumption;
|2: intros;
- lapply (uparrow_upperlocated ? a (sig_in ?? x h)) as Ha1;
- [2: apply segment_preserves_uparrow;split; assumption;]
- lapply (segment_preserves_supremum ?l u a (sig_in ??? 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;]
+ lapply (uparrow_upperlocated a ≪x,h≫) as Ha1;
+ [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:
+ ∀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 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. (λn.fst (a n)) ↓ x →
- x∈[l,u] ∧ ∀h:x ∈ [l,u].a uniform_converges (sig_in ?? x h).
-intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
-[1: split;
- [2: apply (infimum_is_lower_bound C ?? Hx l);
- apply (segment_lowerbound ? l u);
- |1: apply (le_transitive ?? (fst (a 0))); [apply H2;]
- apply (segment_upperbound ? l u);]
+ ∀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 (sig_in ?? x h)) as Ha1;
- [2: apply segment_preserves_downarrow;split; assumption;]
- lapply (segment_preserves_infimum ?l u a (sig_in ??? 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.
\ No newline at end of file
+ lapply (downarrow_lowerlocated a ≪x,h≫) as Ha1;
+ [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