(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).
+ (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.
-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}.
+
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
-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) 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 (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 _)).
+
+lemma hint_pippo:
+ ∀C,s.
+ sequence
+ (Type_of_ordered_set
+ (segment_ordered_set
+ (ordered_set_OF_ordered_uniform_space C) s))
+ →
+ sequence (Type_OF_uniform_space (segment_ordered_uniform_space C s)). intros; assumption;
+qed.
+
+coercion hint_pippo nocomposites.
+
(* 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".
-apply (H (λx:{[l,u]} square.U 〈fst (fst x),fst (snd x)〉));
+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.
-(* Lemma 3.8 *)
-lemma restrict_uniform_convergence:
+(* 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).
+ ∀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;
- [1: apply (supremum_is_upper_bound C ?? Hx u);
+ [1: apply (supremum_is_upper_bound ? x Hx u);
apply (segment_upperbound ? l);
- |2: apply (le_transitive ?? (fst (a 0))); [2: apply H2;]
- apply (segment_lowerbound ?l u);]
+ |2: apply (le_transitive l ? x ? (H2 O));
+ apply (segment_lowerbound ? l u a 0);]
|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;
+ 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); [split; assumption]
+ 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.
-
\ No newline at end of file
projects / helm.git / blobdiff