X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Flibrary%2Fdama%2Fmodels%2Fincreasing_supremum_stabilizes.ma;h=fc4424e5ec7cde8616c68e0d96d237cfc37baa4e;hb=a88be1ca42c0969dbab9a5c76240f5931df876d9;hp=4f7a2f847469050911a9861814792068f59e4cfe;hpb=c5a4db6c1020488d0792cee00dcf395a0ce54735;p=helm.git

diff --git a/helm/software/matita/library/dama/models/increasing_supremum_stabilizes.ma b/helm/software/matita/library/dama/models/increasing_supremum_stabilizes.ma
index 4f7a2f847..fc4424e5e 100644
--- a/helm/software/matita/library/dama/models/increasing_supremum_stabilizes.ma
+++ b/helm/software/matita/library/dama/models/increasing_supremum_stabilizes.ma
@@ -27,6 +27,7 @@ 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 → 
@@ -34,7 +35,7 @@ lemma increasing_supremum_stabilizes:
 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))); 
+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 ≝
@@ -51,14 +52,14 @@ letin m ≝ (hide ? (
    ∀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);
+    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;
+        |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);]
@@ -76,9 +77,9 @@ 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_sg (S n1));
+    |*: 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 ]
+            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]
@@ -94,10 +95,11 @@ letin m ≝ (hide ? (
             [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;
+                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
@@ -107,19 +109,19 @@ letin find ≝ (
           | false ⇒ find (S i) w]]
  in find
  :
-  ∀i,bound.∃j.i + bound = 𝕦_sg → x = \fst (a j));
+  ∀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 (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);
+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));
+    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.