even more polymorphic dualizer

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

diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index bad35177b12bd3d2c4601f6eb588f9af999a70d1..e3a83a09a4f5d2d72646ee2a360b65ef82c03f90 100644 (file)
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -202,26 +202,26 @@ record segment (O : Type) : Type ≝ {
    seg_u_ : O
 }.
 
-notation > "𝕦_term 90 s p" non associative with precedence 45 for @{'upp $s $p}.
-notation "𝕦 \sub term 90 s p" non associative with precedence 45 for @{'upp $s $p}. 
-notation > "𝕝_term 90 s p" non associative with precedence 45 for @{'low $s $p}.
-notation "𝕝 \sub term 90 s p" non associative with precedence 45 for @{'low $s $p}. 
+notation > "𝕦_term 90 s" non associative with precedence 45 for @{'upp $s}.
+notation "𝕦 \sub term 90 s" non associative with precedence 45 for @{'upp $s}. 
+notation > "𝕝_term 90 s" non associative with precedence 45 for @{'low $s}.
+notation "𝕝 \sub term 90 s" non associative with precedence 45 for @{'low $s}. 
  
 definition seg_u ≝
- λO:half_ordered_set.λs:segment O.λP: O → CProp.
-   wloss O ? (λl,u.P l) (seg_u_ ? s) (seg_l_ ? s).
+ λO:half_ordered_set.λs:segment O.
+   wloss O ?? (λl,u.l) (seg_u_ ? s) (seg_l_ ? s).
 definition seg_l ≝
- λO:half_ordered_set.λs:segment O.λP: O → CProp.
-   wloss O ? (λl,u.P l) (seg_l_ ? s) (seg_u_ ? s). 
+ λO:half_ordered_set.λs:segment O.
+   wloss O ?? (λl,u.l) (seg_l_ ? s) (seg_u_ ? s). 
  
-interpretation "uppper" 'upp s P = (seg_u (os_l _) s P).
-interpretation "lower" 'low s P = (seg_l (os_l _) s P).
-interpretation "uppper dual" 'upp s P = (seg_l (os_r _) s P).
-interpretation "lower dual" 'low s P = (seg_u (os_r _) s P).
+interpretation "uppper" 'upp s = (seg_u (os_l _) s).
+interpretation "lower" 'low s = (seg_l (os_l _) s).
+interpretation "uppper dual" 'upp s = (seg_l (os_r _) s).
+interpretation "lower dual" 'low s = (seg_u (os_r _) s).
  
 definition in_segment ≝ 
   λO:half_ordered_set.λs:segment O.λx:O.
-    wloss O ? (λp1,p2.p1 ∧ p2) (seg_l ? s (λl.l ≤≤ x)) (seg_u ? s (λu.x ≤≤ u)).
+    wloss O ?? (λp1,p2.p1 ∧ p2) (seg_l ? s ≤≤ x) (x ≤≤ seg_u ? s).
 
 notation "‡O" non associative with precedence 90 for @{'segment $O}.
 interpretation "Ordered set sergment" 'segment x = (segment x).
@@ -234,19 +234,19 @@ definition segment_ordered_set_exc ≝
   λO:half_ordered_set.λs:‡O.
    λx,y:segment_ordered_set_carr O s.hos_excess_ O (\fst x) (\fst y).
 lemma segment_ordered_set_corefl:
- ∀O,s. coreflexive ? (wloss O ? (segment_ordered_set_exc O s)).
+ ∀O,s. coreflexive ? (wloss O ?? (segment_ordered_set_exc O s)).
 intros 3; cases x; cases (wloss_prop O);
 generalize in match (hos_coreflexive O w);
-rewrite < (H1 ? (segment_ordered_set_exc O s));
-rewrite < (H1 ? (hos_excess_ O)); intros; assumption;
+rewrite < (H1 ?? (segment_ordered_set_exc O s));
+rewrite < (H1 ?? (hos_excess_ O)); intros; assumption;
 qed.
 lemma segment_ordered_set_cotrans : 
-  ∀O,s. cotransitive ? (wloss O ? (segment_ordered_set_exc O s)).
+  ∀O,s. cotransitive ? (wloss O ?? (segment_ordered_set_exc O s)).
 intros 5 (O s x y z); cases x; cases y ; cases z; clear x y z;
 generalize in match (hos_cotransitive O w w1 w2);
 cases (wloss_prop O); 
-do 3 rewrite < (H3 ? (segment_ordered_set_exc O s));
-do 3 rewrite < (H3 ? (hos_excess_ O)); intros; apply H4; assumption;
+do 3 rewrite < (H3 ?? (segment_ordered_set_exc O s));
+do 3 rewrite < (H3 ?? (hos_excess_ O)); intros; apply H4; assumption;
 qed.  
   
 lemma half_segment_ordered_set: 
@@ -278,6 +278,19 @@ interpretation "Ordered set segment" 'segset s = (segment_ordered_set _ s).
 intros; try reflexivity;
 *)
 
+lemma prove_in_segment: 
+ ∀O:half_ordered_set.∀s:segment O.∀x:O.
+   (seg_l O s) ≤≤ x → x ≤≤ (seg_u O s) → x ∈ s.
+intros; unfold; cases (wloss_prop O); rewrite < H2;
+split; assumption;
+qed.
+
+lemma cases_in_segment: 
+  ∀C:half_ordered_set.∀s:segment C.∀x. x ∈ s → (seg_l C s) ≤≤ x ∧ x ≤≤ (seg_u C s).
+intros; unfold in H; cases (wloss_prop C) (W W); rewrite<W in H; [cases H; split;assumption]
+cases H; split; assumption;
+qed. 
+
 definition hint_sequence: 
   ∀C:ordered_set.
     sequence (hos_carr (os_l C)) → sequence (Type_of_ordered_set C).
@@ -314,7 +327,7 @@ lemma x2sx:
    ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
     \fst x ≰≰ \fst y → x ≰≰ y.
 intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
-whd in ⊢ (?→? (% ? ?) ? ? ? ?); simplify in ⊢ (?→%);
+whd in ⊢ (?→? (% ? ?)? ? ? ? ?); simplify in ⊢ (?→%);
 cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
 qed.
 
@@ -323,7 +336,7 @@ lemma sx2x:
    ∀s:segment O.∀x,y:half_segment_ordered_set ? s.
     x ≰≰ y → \fst x ≰≰ \fst y.
 intros 4; cases x; cases y; clear x y; simplify; unfold hos_excess;
-whd in ⊢ (? (% ? ?) ? ? ? ? → ?); simplify in ⊢ (% → ?);
+whd in ⊢ (? (% ? ?) ?? ? ? ? → ?); simplify in ⊢ (% → ?);
 cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
 qed.