X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fdama%2Freals.ma;h=d57e6cfba62f8cada56b0da615a5b07fa5df3d12;hb=f2e7cb9757a151382ed3a70d1a610e8f729a6597;hp=97ca4151fed9225137df2fd185aaeb219f06aa87;hpb=8bf2443a45a06c53153344201213afab724bf142;p=helm.git

diff --git a/matita/dama/reals.ma b/matita/dama/reals.ma
index 97ca4151f..d57e6cfba 100644
--- a/matita/dama/reals.ma
+++ b/matita/dama/reals.ma
@@ -18,7 +18,7 @@ include "ordered_fields_ch0.ma".
 
 record is_real (F:ordered_field_ch0) : Type
 ≝
- { r_archimedean: ∀x:F. ∃n:nat. x ≤ (sum_field F n);
+ { r_archimedean: ∀x:F. ∃n:nat. x ≤ (sum_field ? n);
    r_complete: is_complete F  
  }.
 
@@ -27,7 +27,7 @@ record real: Type \def
    r_real_properties: is_real r_ordered_field_ch0
  }.
 
-definition lim: ∀R:real.∀f:nat→R.is_cauchy_seq R f → R.
+definition lim: ∀R:real.∀f:nat→R.is_cauchy_seq ? f → R.
  intros;
  elim (r_complete ? (r_real_properties R) ? H);
  exact a.
@@ -35,11 +35,11 @@ qed.
 
 definition max_seq: ∀R:real.∀x,y:R. nat → R.
  intros (R x y);
- elim (to_cotransitive R (of_le R) R 0 (inv ? (sum_field R (S n)) ?) (x-y));
+ elim (cos_cotransitive R 0 (inv ? (sum_field ? (S n)) ?) (x-y));
   [ apply x
   | apply not_eq_sum_field_zero ;
     unfold;
-    auto new
+    autobatch
   | apply y
   | apply lt_zero_to_le_inv_zero 
   ].
@@ -47,20 +47,22 @@ qed.
 
 axiom daemon: False.
 
-theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
+theorem cauchy_max_seq: ∀R:real.∀x,y:R. is_cauchy_seq ? (max_seq ? x y).
+elim daemon.
+(*
  intros;
  unfold;
  intros;
  exists; [ exact m | ]; (* apply (ex_intro ? ? m); *)
  intros;
  unfold max_seq;
- elim (to_cotransitive R (of_le R) R 0
+ elim (of_cotransitive R 0
 (inv R (sum_field R (S N))
  (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))) (x-y)
 (lt_zero_to_le_inv_zero R (S N)
  (not_eq_sum_field_zero R (S N) (le_S_S O N (le_O_n N)))));
   [ simplify;
-    elim (to_cotransitive R (of_le R) R 0
+    elim (of_cotransitive R  0
 (inv R (1+sum R (plus R) 0 1 m)
  (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
 (lt_zero_to_le_inv_zero R (S m)
@@ -75,16 +77,15 @@ theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
        ]
     | simplify;
       split;
-       [  apply (to_transitive ? ? 
-       (of_total_order_relation ? ? R) ? 0 ?);
-         [ apply (le_zero_x_to_le_opp_x_zero R ?)
-         | assumption
-         ]        
+       [ apply (or_transitive ? ? R ? 0);
+          [ apply (le_zero_x_to_le_opp_x_zero R ?)
+          | assumption
+          ]
        | assumption
        ]
     ]
   | simplify;
-    elim (to_cotransitive R (of_le R) R 0
+    elim (of_cotransitive R 0
 (inv R (1+sum R (plus R) 0 1 m)
  (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))) (x-y)
 (lt_zero_to_le_inv_zero R (S m)
@@ -98,7 +99,7 @@ theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
          rewrite > eq_opp_plus_plus_opp_opp in H1;
          rewrite > eq_opp_opp_x_x in H1;
          rewrite > plus_comm in H1;
-         apply (to_transitive ? ? (of_total_order_relation ? ? R) ? 0 ?);
+         apply (or_transitive ? ? R ? 0);
           [ assumption
           | apply lt_zero_to_le_inv_zero
           ]
@@ -112,7 +113,7 @@ theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
        ]
      ]
   ].
-  elim daemon.
+  elim daemon.*)
 qed.
 
 definition max: ∀R:real.R → R → R.
@@ -121,4 +122,51 @@ definition max: ∀R:real.R → R → R.
  apply cauchy_max_seq.
 qed.
 
-definition abs \def λR:real.λx:R. max R x (-x).
\ No newline at end of file
+definition abs \def λR:real.λx:R. max R x (-x).
+
+lemma comparison:
+ ∀R:real.∀f,g:nat→R. is_cauchy_seq ? f → is_cauchy_seq ? g →
+  (∀n:nat.f n ≤ g n) → lim ? f ? ≤ lim ? g ?.
+ [ assumption
+ | assumption
+ | intros;
+   elim daemon
+ ].
+qed.
+
+definition to_zero ≝
+ λR:real.λn.
+  -(inv R (sum_field R (S n))
+   (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))).
+  
+axiom is_cauchy_seq_to_zero: ∀R:real. is_cauchy_seq ? (to_zero R).
+
+lemma technical1: ∀R:real.lim R (to_zero R) (is_cauchy_seq_to_zero R) = 0.
+ intros;
+ unfold lim;
+ elim daemon.
+qed.
+ 
+lemma abs_x_ge_O: ∀R:real.∀x:R. 0 ≤ abs ? x.
+ intros;
+ unfold abs;
+ unfold max;
+ rewrite < technical1;
+ apply comparison;
+ intros;
+ unfold to_zero;
+ unfold max_seq;
+ elim
+     (cos_cotransitive R 0
+(inv R (sum_field R (S n))
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))) (x--x)
+(lt_zero_to_le_inv_zero R (S n)
+ (not_eq_sum_field_zero R (S n) (le_S_S O n (le_O_n n)))));
+ [ simplify;
+   (* facile *)
+   elim daemon
+ | simplify;
+   (* facile *)
+   elim daemon
+ ].
+qed.
\ No newline at end of file