definition strictly_decreasing ≝
λC:ordered_set.λa:sequence C.∀n:nat.a n ≰ a (S n).
-notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 50
+notation < "s \nbsp 'is_strictly_increasing'" non associative with precedence 45
for @{'strictly_increasing $s}.
-notation > "s 'is_strictly_increasing'" non associative with precedence 50
+notation > "s 'is_strictly_increasing'" non associative with precedence 45
for @{'strictly_increasing $s}.
interpretation "Ordered set strict increasing" 'strictly_increasing s =
(strictly_increasing _ s).
-notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 50
+notation < "s \nbsp 'is_strictly_decreasing'" non associative with precedence 45
for @{'strictly_decreasing $s}.
-notation > "s 'is_strictly_decreasing'" non associative with precedence 50
+notation > "s 'is_strictly_decreasing'" non associative with precedence 45
for @{'strictly_decreasing $s}.
interpretation "Ordered set strict decreasing" 'strictly_decreasing s =
(strictly_decreasing _ s).
definition order_converge ≝
λO:ordered_set.λa:sequence O.λx:O.
exT23 (sequence O) (λl.l ↑ x) (λu.u ↓ x)
- (λl,u.∀i:nat. (l i) is_infimum ⌊w,a (w+i)⌋ ∧
+ (λl,u:sequence O.∀i:nat. (l i) is_infimum ⌊w,a (w+i)⌋ ∧
(u i) is_supremum ⌊w,a (w+i)⌋).
notation < "a \nbsp (\cir \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 45
interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).
(* Definition 2.8 *)
-alias symbol "and" = "constructive and".
definition segment ≝ λO:ordered_set.λa,b:O.λx:O.(x ≤ b) ∧ (a ≤ x).
-notation "[a,b]" left associative with precedence 70 for @{'segment $a $b}.
+notation "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}.
interpretation "Ordered set sergment" 'segment a b = (segment _ a b).
-notation "hvbox(x \in break [a,b])" non associative with precedence 45
+notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45
for @{'segment_in $a $b $x}.
interpretation "Ordered set sergment in" 'segment_in a b x= (segment _ a b x).
lemma segment_ordered_set:
∀O:ordered_set.∀u,v:O.ordered_set.
intros (O u v); apply (mk_ordered_set (∃x.x ∈ [u,v]));
-[1: intros (x y); apply (fst x ≰ fst y);
+[1: intros (x y); apply (\fst x ≰ \fst y);
|2: intro x; cases x; simplify; apply os_coreflexive;
|3: intros 3 (x y z); cases x; cases y ; cases z; simplify; apply os_cotransitive]
qed.
(* Lemma 2.9 *)
lemma segment_preserves_supremum:
∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
- ⌊n,fst (a n)⌋ is_increasing ∧
- (fst x) is_supremum ⌊n,fst (a n)⌋ → a ↑ x.
+ ⌊n,\fst (a n)⌋ is_increasing ∧
+ (\fst x) is_supremum ⌊n,\fst (a n)⌋ → a ↑ x.
intros; split; cases H; clear H;
[1: apply H1;
|2: cases H2; split; clear H2;
[1: apply H;
- |2: clear H; intro y0; apply (H3 (fst y0));]]
+ |2: clear H; intro y0; apply (H3 (\fst y0));]]
qed.
lemma segment_preserves_infimum:
∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
- ⌊n,fst (a n)⌋ is_decreasing ∧
- (fst x) is_infimum ⌊n,fst (a n)⌋ → a ↓ x.
+ ⌊n,\fst (a n)⌋ is_decreasing ∧
+ (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
intros; split; cases H; clear H;
[1: apply H1;
|2: cases H2; split; clear H2;
[1: apply H;
- |2: clear H; intro y0; apply (H3 (fst y0));]]
+ |2: clear H; intro y0; apply (H3 (\fst y0));]]
qed.
(* Definition 2.10 *)
alias symbol "pi1" = "pair pi1".
definition square_segment ≝
λO:ordered_set.λa,b:O.λx:O square.
- And4 (fst x ≤ b) (a ≤ fst x) (snd x ≤ b) (a ≤ snd x).
+ And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
definition convex ≝
λO:ordered_set.λU:O square → Prop.
- ∀p.U p → fst p ≤ snd p → ∀y. square_segment ? (fst p) (snd p) y → U y.
+ ∀p.U p → \fst p ≤ \snd p → ∀y. square_segment ? (\fst p) (\snd p) y → U y.
(* Definition 2.11 *)
definition upper_located ≝