Some notes for Enrico.

[helm.git] / matita / dama / reals.ma

diff --git a/matita/dama/reals.ma b/matita/dama/reals.ma
index f587435a3b38ff4f406fc125dd492c22b61e8513..d57e6cfba62f8cada56b0da615a5b07fa5df3d12 100644 (file)
--- 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  
  }.
 
@@ -27,20 +27,19 @@ record real: Type \def
    r_real_properties: is_real r_ordered_field_ch0
  }.
 
-(* serve l'esistenziale in CProp!
-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 H;
+ elim (r_complete ? (r_real_properties R) ? H);
+ exact a.
 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 
   ].
@@ -48,20 +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)
@@ -70,31 +71,35 @@ theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
       rewrite > (plus_comm ? x (-x));
       rewrite > opp_inverse;
       split;
-       [ elim daemon (* da finire *)
+       [ apply (le_zero_x_to_le_opp_x_zero R ?); 
+         apply lt_zero_to_le_inv_zero
        | apply lt_zero_to_le_inv_zero
        ]
     | simplify;
       split;
-       [ elim daemon (* da finire *)
+       [ 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)
  (not_eq_sum_field_zero R (S m) (le_S_S O m (le_O_n m)))));
      [ simplify;
        split;
-       [ elim daemon (* da finire *)
+       [ elim daemon
        | generalize in match (le_zero_x_to_le_opp_x_zero R ? t1);
          intro;
          unfold minus in H1;
          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
           ]
@@ -103,16 +108,65 @@ theorem cauchy_max_seq: ∀R:real.∀x,y. is_cauchy_seq ? (max_seq R x y).
        rewrite > (plus_comm ? y (-y));
        rewrite > opp_inverse;
        split;
-       [ elim daemon (* da finire *)
+       [ elim daemon
        | apply lt_zero_to_le_inv_zero
        ]
      ]
   ].
+  elim daemon.*)
 qed.
 
 definition max: ∀R:real.R → R → R.
  intros (R x y);
- elim (r_complete ? (r_real_properties R) ? ?);
-  [|| apply (cauchy_max_seq R x y) ]
+ apply (lim R (max_seq R x y));
+ apply cauchy_max_seq.
+qed.
+
+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.
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