(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 ⌊n,fst (a n)⌋.
-intros 5; change with (fst (a n) ≤ u); cases (a n); cases H; assumption;
+ ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound ⌊n,\fst (a n)⌋.
+intros 5; change with (\fst (a n) ≤ u); cases (a n); cases H; assumption;
qed.
lemma segment_lowerbound:
- ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound ⌊n,fst (a n)⌋.
-intros 5; change with (l ≤ fst (a n)); cases (a n); cases H; assumption;
+ ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound ⌊n,\fst (a n)⌋.
+intros 5; change with (l ≤ \fst (a n)); cases (a n); cases H; assumption;
qed.
lemma segment_preserves_uparrow:
∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- ⌊n,fst (a n)⌋ ↑ x → a ↑ 〈x,h〉.
+ ⌊n,\fst (a n)⌋ ↑ x → a ↑ ≪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;
+cases (H2 (\fst y)); [2: apply H3;] exists [apply w] assumption;
qed.
lemma segment_preserves_downarrow:
∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h.
- ⌊n,fst (a n)⌋ ↓ x → a ↓ 〈x,h〉.
+ ⌊n,\fst (a n)⌋ ↓ x → a ↓ ≪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;
+cases (H2 (\fst y));[2:apply H3]; exists [apply w] assumption;
qed.
(* Fact 2.18 *)
lemma segment_cauchy:
∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
- a is_cauchy → ⌊n,fst (a n)⌋ is_cauchy.
+ a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
intros 7;
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:{[l,u]} square.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;
lemma restrict_uniform_convergence_uparrow:
∀C:ordered_uniform_space.property_sigma C →
∀l,u:C.exhaustive {[l,u]} →
- ∀a:sequence {[l,u]}.∀x:C. ⌊n,fst (a n)⌋ ↑ x →
- xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↑ x →
+ xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
[1: split;
[1: apply (supremum_is_upper_bound C ?? Hx u);
|2: apply (le_transitive ? ??? ? (H2 O));
apply (segment_lowerbound ?l u);]
|2: intros;
- lapply (uparrow_upperlocated ? a â\8c©x,hâ\8cª) as Ha1;
+ lapply (uparrow_upperlocated ? a â\89ªx,hâ\89«) as Ha1;
[2: apply segment_preserves_uparrow;split; assumption;]
- lapply (segment_preserves_supremum ? l u a â\8c©?,hâ\8cª) as Ha2;
+ lapply (segment_preserves_supremum ? l u a â\89ª?,hâ\89«) 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;
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 →
- xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\8c©x,hâ\8cª.
+ ∀a:sequence {[l,u]}.∀x:C. ⌊n,\fst (a n)⌋ ↓ x →
+ xâ\88\88[l,u] â\88§ â\88\80h:x â\88\88 [l,u].a uniform_converges â\89ªx,hâ\89«.
intros; cases H2 (Ha Hx); clear H2; cases Hx; split;
[1: split;
[2: apply (infimum_is_lower_bound C ?? Hx l);
|1: apply (le_transitive ???? (H2 O));
apply (segment_upperbound ? l u);]
|2: intros;
- lapply (downarrow_lowerlocated ? a â\8c©x,hâ\8cª) as Ha1;
+ lapply (downarrow_lowerlocated ? a â\89ªx,hâ\89«) as Ha1;
[2: apply segment_preserves_downarrow;split; assumption;]
- lapply (segment_preserves_infimum ?l u a â\8c©?,hâ\8cª) as Ha2;
+ lapply (segment_preserves_infimum ?l u a â\89ª?,hâ\89«) 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;