+[1: intro n; simplify; apply trans_increasing_exc; [assumption] apply (Hm n);
+|2: intro n; simplify; apply Uu;
+|3: intros (y Hy); simplify; cases (Hu ? Hy);
+ cases (strictly_increasing_reaches C ? Hm w);
+ exists [apply w1]; cases (os_cotransitive ??? (a (m w1)) H); [2:assumption]
+ cases (trans_increasing_exc C ? Ia ?? H1); assumption;]
+qed.
+
+(* Definition 2.7 *)
+alias symbol "exists" = "CProp exists".
+alias symbol "and" = "constructive and".
+definition order_converge ≝
+ λO:ordered_set.λa:sequence O.λx:O.
+ ∃l:sequence O.∃u:sequence O.
+ l is_increasing ∧ u is_decreasing ∧ l ↑ x ∧ u ↓ x ∧
+ ∀i:nat. (l i) is_infimum (λw.a (w+i)) ∧ (u i) is_supremum (λw.a (w+i)).
+
+notation < "a \nbsp (\circ \atop (\horbar\triangleright)) \nbsp x" non associative with precedence 50
+ for @{'order_converge $a $x}.
+notation > "a 'order_converges' x" non associative with precedence 50
+ for @{'order_converge $a $x}.
+interpretation "Order convergence" 'order_converge s u =
+ (cic:/matita/dama/supremum/order_converge.con _ s u).
+
+(* Definition 2.8 *)
+
+definition segment ≝ λO:ordered_set.λa,b:O.λx:O.
+ (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (x ≤ b) (a ≤ x)).
+
+notation "[a,b]" non associative with precedence 50
+ for @{'segment $a $b}.
+interpretation "Ordered set sergment" 'segment a b =
+ (cic:/matita/dama/supremum/segment.con _ a b).
+
+notation "hvbox(x \in break [a,b])" non associative with precedence 50
+ for @{'segment2 $a $b $x}.
+interpretation "Ordered set sergment in" 'segment2 a b x=
+ (cic:/matita/dama/supremum/segment.con _ a b x).
+
+coinductive sigma (A:Type) (P:A→Prop) : Type ≝ sig_in : ∀x.P x → sigma A P.
+
+definition pi1 : ∀A.∀P.sigma A P → A ≝ λA,P,s.match s with [sig_in x _ ⇒ x].
+
+notation < "'fst' \nbsp x" non associative with precedence 50 for @{'pi1 $x}.
+notation < "'snd' \nbsp x" non associative with precedence 50 for @{'pi2 $x}.
+notation > "'fst' x" non associative with precedence 50 for @{'pi1 $x}.
+notation > "'snd' x" non associative with precedence 50 for @{'pi2 $x}.
+interpretation "sigma pi1" 'pi1 x =
+ (cic:/matita/dama/supremum/pi1.con _ _ x).
+
+interpretation "Type exists" 'exists \eta.x =
+ (cic:/matita/dama/supremum/sigma.ind#xpointer(1/1) _ 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);
+|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.
+
+notation "hvbox({[a, break b]})" non associative with precedence 90
+ for @{'segment_set $a $b}.
+interpretation "Ordered set segment" 'segment_set a b =
+ (cic:/matita/dama/supremum/segment_ordered_set.con _ a b).
+
+(* Lemma 2.9 *)
+lemma segment_preserves_supremum:
+ ∀O:ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}.
+ (λn.fst (a n)) is_increasing ∧
+ (fst x) is_supremum (λn.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));]]
+qed.
+
+(* Definition 2.10 *)
+coinductive pair (A,B:Type) : Type ≝ prod : ∀a:A.∀b:B.pair A B.
+definition first : ∀A.∀P.pair A P → A ≝ λA,P,s.match s with [prod x _ ⇒ x].
+definition second : ∀A.∀P.pair A P → P ≝ λA,P,s.match s with [prod _ y ⇒ y].
+
+interpretation "pair pi1" 'pi1 x =
+ (cic:/matita/dama/supremum/first.con _ _ x).
+interpretation "pair pi2" 'pi2 x =
+ (cic:/matita/dama/supremum/second.con _ _ x).
+
+notation "hvbox(\langle a, break b\rangle)" non associative with precedence 91 for @{ 'pair $a $b}.
+interpretation "pair" 'pair a b =
+ (cic:/matita/dama/supremum/pair.ind#xpointer(1/1/1) _ _ a b).
+
+notation "a \times b" left associative with precedence 60 for @{'prod $a $b}.
+interpretation "prod" 'prod a b =
+ (cic:/matita/dama/supremum/pair.ind#xpointer(1/1) a b).
+
+lemma square_ordered_set: ordered_set → ordered_set.
+intro O; apply (mk_ordered_set (O × O));
+[1: intros (x y); apply (fst x ≰ fst y ∨ snd x ≰ snd y);
+|2: intro x0; cases x0 (x y); clear x0; simplify; intro H;
+ cases H (X X); apply (os_coreflexive ?? X);
+|3: intros 3 (x0 y0 z0); cases x0 (x1 x2); cases y0 (y1 y2) ; cases z0 (z1 z2);
+ clear x0 y0 z0; simplify; intro H; cases H (H1 H1); clear H;
+ [1: cases (os_cotransitive ??? z1 H1); [left; left|right;left]assumption;
+ |2: cases (os_cotransitive ??? z2 H1); [left;right|right;right]assumption]]
+qed.
+
+notation < "s 2 \atop \nleq" non associative with precedence 90
+ for @{ 'square $s }.
+notation > "s 'square'" non associative with precedence 90
+ for @{ 'square $s }.
+interpretation "ordered set square" 'square s =
+ (cic:/matita/dama/supremum/square_ordered_set.con s).
+
+definition square_segment ≝
+ λO:ordered_set.λa,b:O.λx:square_ordered_set O.
+ (cic:/matita/logic/connectives/And.ind#xpointer(1/1)
+ (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (fst x ≤ b) (a ≤ fst x))
+ (cic:/matita/logic/connectives/And.ind#xpointer(1/1) (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.
+
+(* Definition 2.11 *)
+definition upper_located ≝
+ λO:ordered_set.λa:sequence O.∀x,y:O. y ≰ x →
+ (∃i:nat.a i ≰ x) ∨ (∃b:O.y≰b ∧ ∀i:nat.a i ≤ b).
+
+definition lower_located ≝
+ λO:ordered_set.λa:sequence O.∀x,y:O. x ≰ y →
+ (∃i:nat.x ≰ a i) ∨ (∃b:O.b≰y ∧ ∀i:nat.b ≤ a i).
+
+notation < "s \nbsp 'is_upper_located'" non associative with precedence 50
+ for @{'upper_located $s}.
+notation > "s 'is_upper_located'" non associative with precedence 50
+ for @{'upper_located $s}.
+interpretation "Ordered set upper locatedness" 'upper_located s =
+ (cic:/matita/dama/supremum/upper_located.con _ s).
+
+notation < "s \nbsp 'is_lower_located'" non associative with precedence 50
+ for @{'lower_located $s}.
+notation > "s 'is_lower_located'" non associative with precedence 50
+ for @{'lower_located $s}.
+interpretation "Ordered set lower locatedness" 'lower_located s =
+ (cic:/matita/dama/supremum/lower_located.con _ s).
+
+(* Lemma 2.12 *)
+lemma uparrow_upperlocated:
+ ∀C:ordered_set.∀a:sequence C.∀u:C.a ↑ u → a is_upper_located.
+intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
+cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
+[2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+|1: right; exists [apply u]; split; [apply W|apply H4]]
+qed.
+
+lemma downarrow_lowerlocated:
+ ∀C:ordered_set.∀a:sequence C.∀u:C.a ↓ u → a is_lower_located.
+intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
+cases H3 (H4 H5); clear H3; cases (os_cotransitive ??? u Hxy) (W W);
+[1: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+|2: right; exists [apply u]; split; [apply W|apply H4]]
+qed.
+