seg_u/l were inverted, more work

[helm.git] / helm / software / matita / contribs / dama / dama / property_exhaustivity.ma

diff --git a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
index 41674ec004dd3ccdff0700922112746089fa72e2..713588e7066f0feebcccdbe62f2cedc7179804bd 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
+++ b/helm/software/matita/contribs/dama/dama/property_exhaustivity.ma
@@ -22,40 +22,59 @@ definition exhaustive ≝
      (a is_increasing → a is_upper_located → a is_cauchy) ∧
      (b is_decreasing → b is_lower_located → b is_cauchy).
 
-lemma segment_upperbound:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.u is_upper_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (\fst (a n) ≤ u); cases (a n); cases H; assumption;
+lemma h_segment_upperbound:
+  ∀C:half_ordered_set.
+   ∀s:segment C.
+    ∀a:sequence (half_segment_ordered_set C s).
+     (seg_u C s) (upper_bound ? ⌊n,\fst (a n)⌋). 
+intros; cases (wloss_prop C); unfold; rewrite < H; simplify; intro n;
+cases (a n); simplify; unfold in H1; rewrite < H in H1; cases H1; 
+simplify in H2 H3; rewrite < H in H2 H3; assumption;
 qed.
 
-lemma segment_lowerbound:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.l is_lower_bound ⌊n,\fst (a n)⌋.
-intros 5; change with (l ≤ \fst (a n)); cases (a n); cases H; assumption;
-qed.
+notation "'segment_upperbound'" non associative with precedence 90 for @{'segment_upperbound}.
+notation "'segment_lowerbound'" non associative with precedence 90 for @{'segment_lowerbound}.
 
-lemma segment_preserves_uparrow:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    ⌊n,\fst (a n)⌋ ↑ x → a ↑ ≪x,h≫.
-intros; cases H (Ha Hx); split [apply Ha] cases Hx; 
-split; [apply H1] intros;
-cases (H2 (\fst y)); [2: apply H3;] exists [apply w] assumption;
-qed.
+interpretation "segment_upperbound" 'segment_upperbound = (h_segment_upperbound (os_l _)).
+interpretation "segment_lowerbound" 'segment_lowerbound = (h_segment_upperbound (os_r _)).
 
-lemma segment_preserves_downarrow:
-  ∀C:ordered_set.∀l,u:C.∀a:sequence {[l,u]}.∀x,h. 
-    ⌊n,\fst (a n)⌋ ↓ x → a ↓ ≪x,h≫.
-intros; cases H (Ha Hx); split [apply Ha] cases Hx; 
-split; [apply H1] intros;
-cases (H2 (\fst y));[2:apply H3]; exists [apply w] assumption;
+lemma h_segment_preserves_uparrow:
+  ∀C:half_ordered_set.∀s:segment C.∀a:sequence (half_segment_ordered_set C s).
+   ∀x,h. uparrow C ⌊n,\fst (a n)⌋ x → uparrow (half_segment_ordered_set C s) a ≪x,h≫.
+intros; cases H (Ha Hx); split;
+[ intro n; intro H; apply (Ha n); apply (sx2x ???? H);
+| cases Hx; split; 
+  [ intro n; intro H; apply (H1 n);apply (sx2x ???? H); 
+  | intros; cases (H2 (\fst y)); [2: apply (sx2x ???? H3);] 
+    exists [apply w] apply (x2sx ?? (a w) y H4);]]
 qed.
-    
+
+notation "'segment_preserves_uparrow'" non associative with precedence 90 for @{'segment_preserves_uparrow}.
+notation "'segment_preserves_downarrow'" non associative with precedence 90 for @{'segment_preserves_downarrow}.
+
+interpretation "segment_preserves_uparrow" 'segment_preserves_uparrow = (h_segment_preserves_uparrow (os_l _)).
+interpretation "segment_preserves_downarrow" 'segment_preserves_downarrow = (h_segment_preserves_uparrow (os_r _)).
+
+lemma hint_pippo:
+ ∀C,s.
+  sequence
+   (Type_of_ordered_set
+    (segment_ordered_set
+     (ordered_set_OF_ordered_uniform_space C) s))
+ →  
+ sequence (Type_OF_uniform_space (segment_ordered_uniform_space C s)). intros; assumption;
+qed. 
+
+coercion hint_pippo nocomposites.
+ 
 (* Fact 2.18 *)
 lemma segment_cauchy:
-  ∀C:ordered_uniform_space.∀l,u:C.∀a:sequence {[l,u]}.
+  ∀C:ordered_uniform_space.∀s:‡C.∀a:sequence {[s]}.
     a is_cauchy → ⌊n,\fst (a n)⌋ is_cauchy.
-intros 7; 
+intros 6; 
 alias symbol "pi1" (instance 3) = "pair pi1".
 alias symbol "pi2" = "pair pi2".
-apply (H (λx:{[l,u]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
+apply (H (λx:{[s]} squareB.U 〈\fst (\fst x),\fst (\snd x)〉));
 (unfold segment_ordered_uniform_space; simplify);
 exists [apply U] split; [assumption;]
 intro; cases b; intros; simplify; split; intros; assumption;