--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "dama/models/nat_uniform.ma".
+include "dama/supremum.ma".
+include "nat/le_arith.ma".
+include "dama/russell_support.ma".
+
+lemma hint1:
+ ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s))
+ → sequence (hos_carr (os_l (segment_ordered_set nat_ordered_set s))).
+intros; assumption;
+qed.
+
+coercion hint1 nocomposites.
+
+alias symbol "pi1" = "exT \fst".
+alias symbol "N" = "ordered set N".
+alias symbol "dependent_pair" = "dependent pair".
+lemma increasing_supremum_stabilizes:
+ ∀sg:‡ℕ.∀a:sequence {[sg]}.
+ a is_increasing →
+ ∀X.X is_supremum a → ∃i.∀j.i ≤ j → \fst X = \fst (a j).
+intros 4; cases X (x Hx); clear X; letin X ≝ ≪x,Hx≫;
+fold normalize X; intros; cases H1;
+alias symbol "N" = "Natural numbers".
+letin spec ≝ (λi,j:ℕ.(𝕦_ sg ≤ i ∧ x = \fst (a j)) ∨ (i < 𝕦_ sg ∧ x + i ≤ 𝕦_ sg + \fst (a j)));
+(* x - aj <= max 0 (u - i) *)
+letin m ≝ (hide ? (
+ let rec aux i ≝
+ match i with
+ [ O ⇒ O
+ | S m ⇒
+ let pred ≝ aux m in
+ let apred ≝ a pred in
+ match cmp_nat x (\fst apred) with
+ [ cmp_le _ ⇒ pred
+ | cmp_gt nP ⇒ \fst (H3 apred ?)]]
+ in aux
+ :
+ ∀i:nat.∃j:nat.spec i j));[whd; apply nP;] unfold spec in aux ⊢ %;
+[3: unfold X in H2; clear H4 n aux spec H3 H1 H X;
+ cases (cases_in_segment ??? Hx);
+ elim 𝕦_ sg in H1 ⊢ %; intros (a Hs H);
+ [1: left; split; [apply le_n]
+ generalize in match H;
+ generalize in match Hx;
+ rewrite > (?:x = O);
+ [2: cases Hx; lapply (os_le_to_nat_le ?? H1);
+ apply (symmetric_eq nat O x ?).apply (le_n_O_to_eq x ?).apply (Hletin).
+ |1: intros; unfold Type_OF_ordered_set in sg a; whd in a:(? %);
+ lapply (H2 O) as K; lapply (sl2l_ ?? (a O) ≪x,Hx≫ K) as P;
+ simplify in P:(???%); lapply (le_transitive ??? P H1) as W;
+ lapply (os_le_to_nat_le ?? W) as R; apply (le_n_O_to_eq (\fst (a O)) R);]
+ |2: right; cases Hx; rewrite > (sym_plus x O); split; [apply le_S_S; apply le_O_n];
+ apply (trans_le ??? (os_le_to_nat_le ?? H3));
+ apply le_plus_n_r;]
+|2: clear H6; cut (x = \fst (a (aux n1))); [2:
+ cases (le_to_or_lt_eq ?? H5); [2: assumption]
+ cases (?:False); apply (H2 (aux n1) H6);] clear H5;
+ generalize in match Hcut; clear Hcut; intro H5;
+|1: clear H6]
+[2,1:
+ cases (aux n1) in H5 ⊢ %; intros;
+ change in match (a ≪w,H5≫) in H6 ⊢ % with (a w);
+ cases H5; clear H5; cases H7; clear H7;
+ [1: left; split; [ apply (le_S ?? H5); | assumption]
+ |3: cases (?:False); rewrite < H8 in H6; apply (not_le_Sn_n ? H6);
+ |*: cases (cmp_nat 𝕦_ sg (S n1));
+ [1,3: left; split; [1,3: assumption |2: assumption]
+ cut (𝕦_ sg = S n1); [2: apply le_to_le_to_eq; assumption ]
+ clear H7 H5 H4;rewrite > Hcut in H8:(? ? (? % ?)); clear Hcut;
+ cut (x = S (\fst (a w)));
+ [2: apply le_to_le_to_eq; [2: assumption]
+ change in H8 with (x + n1 ≤ S (n1 + \fst (a w)));
+ rewrite > plus_n_Sm in H8; rewrite > sym_plus in H8;
+ apply (le_plus_to_le ??? H8);]
+ cases (H3 (a w) H6);
+ change with (x = \fst (a w1));
+ change in H4 with (\fst (a w) < \fst (a w1));
+ apply le_to_le_to_eq; [ rewrite > Hcut; assumption ]
+ apply (os_le_to_nat_le (\fst (a w1)) x (H2 w1));
+ |*: right; split; try assumption;
+ [1: rewrite > sym_plus in ⊢ (? ? %);
+ rewrite < H6; apply le_plus_r; assumption;
+ |2: cases (H3 (a w) H6);
+ change with (x + S n1 ≤ 𝕦_ sg + \fst (a w1));rewrite < plus_n_Sm;
+ apply (trans_le ??? (le_S_S ?? H8)); rewrite > plus_n_Sm;
+ apply (le_plus ???? (le_n ?) H9);]]]]
+clearbody m; unfold spec in m; clear spec;
+alias symbol "exists" = "CProp exists".
+letin find ≝ (
+ let rec find i u on u : nat ≝
+ match u with
+ [ O ⇒ (m i:nat)
+ | S w ⇒ match eqb (\fst (a (m i))) x with
+ [ true ⇒ (m i:nat)
+ | false ⇒ find (S i) w]]
+ in find
+ :
+ ∀i,bound.∃j.i + bound = 𝕦_ sg → x = \fst (a j));
+[1: cases (find (S n) n2); intro; change with (x = \fst (a w));
+ apply H6; rewrite < H7; simplify; apply plus_n_Sm;
+|2: intros; rewrite > (eqb_true_to_eq ?? H5); reflexivity
+|3: intros; rewrite > sym_plus in H5; rewrite > H5; clear H5 H4 n n1;
+ cases (m 𝕦_ sg); cases H4; clear H4; cases H5; clear H5; [assumption]
+ cases (not_le_Sn_n ? H4)]
+clearbody find; cases (find O 𝕦_ sg);
+exists [apply w]; intros; change with (x = \fst (a j));
+rewrite > (H4 ?); [2: reflexivity]
+apply le_to_le_to_eq;
+[1: apply os_le_to_nat_le;
+ apply (trans_increasing a H ? ? (nat_le_to_os_le ?? H5));
+|2: apply (trans_le ? x ?);[apply os_le_to_nat_le; apply (H2 j);]
+ rewrite < (H4 ?); [2: reflexivity] apply le_n;]
+qed.
+
+lemma hint2:
+ ∀s.sequence (Type_of_ordered_set (segment_ordered_set nat_ordered_set s))
+ → sequence (hos_carr (os_r (segment_ordered_set nat_ordered_set s))).
+intros; assumption;
+qed.
+
+coercion hint2 nocomposites.
+
+alias symbol "N" = "ordered set N".
+axiom increasing_supremum_stabilizes_r:
+ ∀s:‡ℕ.∀a:sequence {[s]}.a is_decreasing →
+ ∀x.x is_infimum a → ∃i.∀j.i ≤ j → \fst x = \fst (a j).
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