X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fnat_dedekind_sigma_complete.ma;h=73211e0233bfb2eb82df126cd5720d4ee0583f4f;hb=f116dde7daadaffc015e06b3c38a57398bfe0ef4;hp=b5610cbe0de1e82014050350e42855e77ee6eef7;hpb=a99ab6bf4e5bb993d363a9e62985371ba14cf71a;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/nat_dedekind_sigma_complete.ma b/helm/software/matita/contribs/dama/dama/models/nat_dedekind_sigma_complete.ma
index b5610cbe0..73211e023 100644
--- a/helm/software/matita/contribs/dama/dama/models/nat_dedekind_sigma_complete.ma
+++ b/helm/software/matita/contribs/dama/dama/models/nat_dedekind_sigma_complete.ma
@@ -17,18 +17,25 @@ include "supremum.ma".
 include "nat/le_arith.ma".
 include "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 "leq" = "natural 'less or equal to'".
+alias symbol "N" = "ordered set N".
 lemma nat_dedekind_sigma_complete:
-  ∀l,u:ℕ.∀a:sequence {[l,u]}.a is_increasing → 
-    ∀x.x is_supremum a → ∃i.∀j.i ≤ j → \fst x = \fst (a j).
-intros 5; cases x (s Hs); clear x; letin X ≝ (〈s,Hs〉); 
+  ∀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 "plus" = "natural plus".
-alias symbol "nat" = "Uniform space N".
-alias symbol "and" = "logical and".
-letin spec ≝ (λi,j:ℕ.(u≤i ∧ s = \fst (a j)) ∨ (i < u ∧ s+i ≤ u + \fst (a j))); 
-(* s - aj <= max 0 (u - i) *)  
+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
@@ -36,26 +43,29 @@ letin m ≝ (hide ? (
     | S m ⇒ 
         let pred ≝ aux m in
         let apred ≝ a pred in 
-        match cmp_nat s (\fst apred) with
+        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 u in H2 Hs a ⊢ %; intros (a Hs H);
+    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 Hs;
-        rewrite > (?:s = O); 
-        [2: cases Hs; lapply (os_le_to_nat_le ?? H1);
-            apply (symmetric_eq nat O s ?).apply (le_n_O_to_eq s ?).apply (Hletin).
-        |1: intros; lapply (os_le_to_nat_le (\fst (a O)) O (H2 O));
-            lapply (le_n_O_to_eq ? Hletin); assumption;]
-    |2: right; cases Hs; rewrite > (sym_plus s O); split; [apply le_S_S; apply le_O_n];
-        apply (trans_le ??? (os_le_to_nat_le ?? H1));
+        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;
+            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 (s = \fst (a (aux n1))); [2:
+|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;
@@ -66,25 +76,25 @@ letin m ≝ (hide ? (
     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 (S n1));
+    |*: cases (cmp_nat 𝕦_sg (S n1));
         [1,3: left; split; [1,3: assumption |2: assumption]
-            cut (u = S n1); [2: apply le_to_le_to_eq; 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 (s = S (\fst (a w)));
+            cut (x = S (\fst (a w)));
             [2: apply le_to_le_to_eq; [2: assumption]
-                change in H8 with (s + n1 ≤ S (n1 + \fst (a w)));
+                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 (s = \fst (a w1));
+            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)) s (H2 w1));
+            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 (s + S n1 ≤ u + \fst (a w1));rewrite < plus_n_Sm;
+                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;
@@ -92,31 +102,37 @@ 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))) s with
+  | 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 → s = \fst (a j));
-[1: cases (find (S n) n2); intro; change with (s = \fst (a w));
+  ∀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); cases H4; clear H4; cases H5; clear H5; [assumption]
+    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);
-exists [apply w]; intros; change with (s = \fst (a j));
+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 ?? H ? ? (nat_le_to_os_le ?? H5));
-|2: apply (trans_le ? s ?);[apply os_le_to_nat_le; apply (H2 j);]
+    apply (trans_increasing ? 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.
 
-alias symbol "pi1" = "exT \fst".
-alias symbol "leq" = "natural 'less or equal to'".
-alias symbol "nat" = "ordered set N".
+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 nat_dedekind_sigma_complete_r:
-  ∀l,u:ℕ.∀a:sequence {[l,u]}.a is_decreasing → 
+  ∀s:‡ℕ.∀a:sequence {[s]}.a is_decreasing → 
     ∀x.x is_infimum a → ∃i.∀j.i ≤ j → \fst x = \fst (a j).