X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fdama%2Fsandwich.ma;h=aaea369f55ce82c67dfbf186e67a31408d23fd58;hb=cf07c50b03a49344eb4cbe2e1bc18fcef880b9e9;hp=c83136c770c85efc844c5529dc82b723b75215ce;hpb=11f667afbb87725dd5e243d4b3717d19f584a481;p=helm.git

diff --git a/helm/software/matita/dama/sandwich.ma b/helm/software/matita/dama/sandwich.ma
index c83136c77..aaea369f5 100644
--- a/helm/software/matita/dama/sandwich.ma
+++ b/helm/software/matita/dama/sandwich.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-set "baseuri" "cic:/matita/sandwich/".
+
 
 include "nat/plus.ma".
 include "nat/orders.ma".
@@ -26,98 +26,56 @@ lemma ltwr: ∀a,b,c:nat. a + b < c → a < c.
 intros 3 (x y z); rewrite > sym_plus; apply ltwl;
 qed. 
 
-include "sequence.ma".
+include "tend.ma".
 include "metric_lattice.ma".
 
-lemma sandwich_ineq: ∀G:todgroup.∀e:G.0<e → e/3 + e/2 + e/2 < e.
-intro G; cases G; unfold divide; intro e;
-alias num (instance 0) = "natural number".
-cases (dg_prop (mk_todgroup todg_order todg_division_ H) e 3) 0;
-cases (dg_prop (mk_todgroup todg_order todg_division_ H) e 2) 0; simplify;
-intro H3;
-cut (0<w1) as H4; [2: 
-  apply (ltpow_lt ??? 2); apply (lt_rewr ???? H2);
-  apply (lt_rewl ???? (powzero ? 3)); assumption]
-cut (0<w) as H5; [2:
-  apply (ltpow_lt ??? 3); apply (lt_rewr ???? H1);
-  apply (lt_rewl ???? (powzero ? 4)); assumption]
-cut (w<w1) as H6; [2: 
-  apply (poweqplus_lt ??? 2 O); try assumption; apply (Eq≈ ? H2 H1);]
-apply (plus_cancr_lt ??? w1);
-apply (lt_rewl ??? (w+e)); [
-  apply (Eq≈ (w+3*w1) ? H2);
-  apply (Eq≈ (w+w1+(w1+w1)) (plus_assoc ??w1 w1));
-  apply (Eq≈ (w+(w1+(w1+w1))) (plus_assoc ?w w1 ?));
-  simplify; repeat apply feq_plusl; apply eq_sym; 
-  apply (Eq≈ ? (plus_comm ???)); apply zero_neutral;]
-apply (lt_rewl ???? (plus_comm ???));
-apply flt_plusl; assumption;
-qed.
-
-definition d2s ≝ 
-  λR.λml:mlattice R.λs:sequence ml.λk.λn. δ (s n) k.
-
-notation "s ⇝ x" non associative with precedence 50 for @{'tends $s $x}.
-  
-interpretation "tends to" 'tends s x = 
-  (cic:/matita/sequence/tends0.con _ (cic:/matita/sandwich/d2s.con _ _ s x)).
-    
+alias symbol "leq" = "ordered sets less or equal than".
 alias symbol "and" = "constructive and".
 theorem sandwich:
+let ugo ≝ excess_OF_lattice1 in
   ∀R.∀ml:mlattice R.∀an,bn,xn:sequence ml.∀x:ml.
-    (∀n. (an n ≤ xn n) ∧ (xn n ≤ bn n)) → 
-    an ⇝ x → 
-    bn ⇝ x →
-    xn ⇝ x.
-intros (R ml an bn xn x H Ha Hb); unfold tends0 in Ha Hb ⊢ %. unfold d2s in Ha Hb ⊢ %.
-intros (e He);
-elim (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha; 
-elim (Hb (e/3) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
-constructor 1; [apply (n1 + n2);] intros (n3 Hn3);
-lapply (ltwr ??? Hn3) as Hn1n3; lapply (ltwl ??? Hn3) as Hn2n3;
-elim (H1 ? Hn1n3) (H3 H4); elim (H2 ? Hn2n3) (H5 H6); clear Hn1n3 Hn2n3;
-elim (H n3) (H7 H8); clear H H1 H2; 
-lapply (le_to_eqm ??? H7) as Dxm; lapply (le_to_eqj ??? H7) as Dym;
-(* the main step *)
-cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x); [2:
+    (∀n. (xn n ≤ an n) ∧ (bn n ≤ xn n)) →  
+    an ⇝ x → bn ⇝ x → xn ⇝ x.
+intros (R ml an bn xn x H Ha Hb); 
+unfold tends0 in Ha Hb ⊢ %; unfold d2s in Ha Hb ⊢ %; intros (e He);
+alias num (instance 0) = "natural number".
+cases (Ha (e/2) (divide_preserves_lt ??? He)) (n1 H1); clear Ha; 
+cases (Hb (e/2) (divide_preserves_lt ??? He)) (n2 H2); clear Hb;
+apply (ex_introT ?? (n1+n2)); intros (n3 Lt_n1n2_n3);
+lapply (ltwr ??? Lt_n1n2_n3) as Lt_n1n3; lapply (ltwl ??? Lt_n1n2_n3) as Lt_n2n3;
+cases (H1 ? Lt_n1n3) (c daxe); cases (H2 ? Lt_n2n3) (c dbxe); 
+cases (H n3) (H7 H8); clear Lt_n1n3 Lt_n2n3 Lt_n1n2_n3 c H1 H2 H n1 n2;     
+(* the main inequality *)
+cut (δ (xn n3) x ≤ δ (bn n3) x + δ (an n3) x + δ (an n3) x) as main_ineq; [2:
   apply (le_transitive ???? (mtineq ???? (an n3)));
-  lapply (le_mtri ????? H7 H8);
-  lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? Hletin);
-  cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))); [2:
-    apply (Eq≈  (0 + δ(an n3) (xn n3)) ? (zero_neutral ??));
-    apply (Eq≈  (δ(an n3) (xn n3) + 0) ? (plus_comm ???));
-    apply (Eq≈  (δ(an n3) (xn n3) +  (-δ(xn n3) (bn n3) +  δ(xn n3) (bn n3))) ? (opp_inverse ??));
-    apply (Eq≈  (δ(an n3) (xn n3) +  (δ(xn n3) (bn n3) + -δ(xn n3) (bn n3))) ? (plus_comm ?? (δ(xn n3) (bn n3))));
-    apply (Eq≈  ? ? (eq_sym ??? (plus_assoc ????))); assumption;] clear Hletin1;
-  apply (le_rewl ???  ( δ(an n3) (xn n3)+ δ(an n3) x));[
-    apply feq_plusr; apply msymmetric;]
-  apply (le_rewl ???  (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x));[
-    apply feq_plusr; assumption;]
-  clear Hcut Hletin Dym Dxm H8 H7 H6 H5 H4 H3;
-  apply (le_rewl ??? (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x));[
-    apply feq_plusr; apply plus_comm;]
-  apply (le_rewl ??? (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????));
-  apply (le_rewl ??? ((δ(an n3) (bn n3)+δ(an n3) x)+- δ(xn n3) (bn n3)) (plus_comm ???));
+  cut ( δ(an n3) (bn n3)+- δ(xn n3) (bn n3)≈( δ(an n3) (xn n3))) as H11; [2:
+    lapply (le_mtri ?? ??? H8 H7) as H9; clear H7 H8;
+    lapply (Eq≈ ? (msymmetric ????) H9) as H10; clear H9;
+    lapply (feq_plusr ? (- δ(xn n3) (bn n3)) ?? H10) as H9; clear H10;
+    apply (Eq≈ ? H9); clear H9;
+    apply (Eq≈ (δ(xn n3) (an n3)+ δ(bn n3) (xn n3)+- δ(xn n3) (bn n3)) (plus_comm ??( δ(xn n3) (an n3))));
+    apply (Eq≈ (δ(xn n3) (an n3)+ δ(bn n3) (xn n3)+- δ(bn n3) (xn n3)) (feq_opp ??? (msymmetric ????)));    
+    apply (Eq≈ ( δ(xn n3) (an n3)+ (δ(bn n3) (xn n3)+- δ(bn n3) (xn n3))) (plus_assoc ????));
+    apply (Eq≈ (δ(xn n3) (an n3) + (- δ(bn n3) (xn n3) + δ(bn n3) (xn n3))) (plus_comm ??(δ(bn n3) (xn n3))));
+    apply (Eq≈ (δ(xn n3) (an n3) + 0) (opp_inverse ??));
+    apply (Eq≈ ? (plus_comm ???));
+    apply (Eq≈ ? (zero_neutral ??));
+    apply (Eq≈ ? (msymmetric ????));
+    apply eq_reflexive;]
+  apply (Le≪ (δ(an n3) (xn n3)+ δ(an n3) x) (msymmetric ??(an n3)(xn n3)));    
+  apply (Le≪ (δ(an n3) (bn n3)+- δ(xn n3) (bn n3)+ δ(an n3) x) H11);
+  apply (Le≪ (- δ(xn n3) (bn n3)+ δ(an n3) (bn n3)+δ(an n3) x) (plus_comm ??(δ(an n3) (bn n3))));
+  apply (Le≪ (- δ(xn n3) (bn n3)+ (δ(an n3) (bn n3)+δ(an n3) x)) (plus_assoc ????));
+  apply (Le≪ ((δ(an n3) (bn n3)+δ(an n3) x)+- δ(xn n3) (bn n3)) (plus_comm ???));
   apply lew_opp; [apply mpositive] apply fle_plusr;
-  apply (le_rewr ???? (plus_comm ???));
-  apply (le_rewr ??? ( δ(an n3) x+ δx (bn n3)) (msymmetric ????));
+  apply (Le≫ ? (plus_comm ???));
+  apply (Le≫ ( δ(an n3) x+ δx (bn n3)) (msymmetric ????));
   apply mtineq;]
-split; [
+split; [ (* first the trivial case: -e< δ(xn n3) x *)
   apply (lt_le_transitive ????? (mpositive ????));
-  split; elim He; [apply  le_zero_x_to_le_opp_x_zero; assumption;]
-  cases t1; [ 
-    left; apply exc_zero_opp_x_to_exc_x_zero;
-    apply (Ex≫ ? (eq_opp_opp_x_x ??));assumption;]
-  right;  apply exc_opp_x_zero_to_exc_zero_x;
-  apply (Ex≪ ? (eq_opp_opp_x_x ??)); assumption;]
-clear Dxm Dym H7 H8 Hn3 H5 H3 n1 n2;
-apply (le_lt_transitive ???? ? (sandwich_ineq ?? He));
-apply (le_transitive ???? Hcut);
-apply (le_transitive ??  (e/3+ δ(an n3) x+ δ(an n3) x)); [
-  repeat apply fle_plusr; cases H6; assumption;]
-apply (le_transitive ??  (e/3+ e/2 + δ(an n3) x)); [
-  apply fle_plusr; apply fle_plusl; cases H4; assumption;]
-apply (le_transitive ??  (e/3+ e/2 + e/2)); [
-  apply fle_plusl; cases H4; assumption;]
-apply le_reflexive;
+  apply lt_zero_x_to_lt_opp_x_zero; assumption;]
+(* the main goal: δ(xn n3) x<e *)
+apply (le_lt_transitive ???? main_ineq);
+apply (Lt≫ (e/2+e/2+e/2)); [apply (divpow ?e 2)]
+repeat (apply ltplus; try assumption);
 qed.