X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fsupremum.ma;h=0de61b292fcaca3e40f5b7bddfa7ba0b88d27398;hb=179574c117d34a39cebeaa66673cda83974e135a;hp=a3a341fea096d19bdc904aefd45a1dd7ca02b238;hpb=bf7f52019b3f65b6d635a8b49a63f0d95080f189;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/supremum.ma b/helm/software/matita/contribs/dama/dama/supremum.ma
index a3a341fea..0de61b292 100644
--- a/helm/software/matita/contribs/dama/dama/supremum.ma
+++ b/helm/software/matita/contribs/dama/dama/supremum.ma
@@ -63,16 +63,6 @@ interpretation "Ordered set decreasing"  'decreasing s = (increasing (os_r _) s)
 interpretation "Ordered set strong sup"  'supremum s x = (supremum (os_l _) s x).
 interpretation "Ordered set strong inf"  'infimum s x = (supremum (os_r _) s x).
   
-(* se non faccio il bs_of_hos perdo dualità qui *)  
-lemma uniq_supremum: 
-  ∀O:ordered_set.∀s:sequence O.∀t1,t2:O.
-    t1 is_supremum s → t2 is_supremum s → t1 ≈ t2.
-intros (O s t1 t2 Ht1 Ht2); cases Ht1 (U1 H1); cases Ht2 (U2 H2); 
-apply le_le_eq; intro X;
-[1: cases (H1 ? X); apply (U2 w); assumption
-|2: cases (H2 ? X); apply (U1 w); assumption]
-qed.
-
 (* Fact 2.5 *)
 lemma h_supremum_is_upper_bound: 
   ∀C:half_ordered_set.∀a:sequence C.∀u:C.
@@ -87,14 +77,6 @@ notation "'infimum_is_lower_bound'" non associative with precedence 90 for @{'in
 interpretation "supremum_is_upper_bound" 'supremum_is_upper_bound = (h_supremum_is_upper_bound (os_l _)).
 interpretation "infimum_is_lower_bound" 'infimum_is_lower_bound = (h_supremum_is_upper_bound (os_r _)).
 
-(* TEST DUALITY
-lemma test_infimum_is_lower_bound_duality: 
-  ∀C:ordered_set.∀a:sequence C.∀u:C.
-   u is_infimum a → ∀v.v is_lower_bound a → u ≥ v. 
-intros; lapply (infimum_is_lower_bound a u H v H1); assumption;
-qed.
-*)
-
 (* Lemma 2.6 *)
 definition strictly_increasing ≝ 
   λC:half_ordered_set.λa:sequence C.∀n:nat.a (S n) ≰≰ a n.
@@ -116,16 +98,8 @@ interpretation "Ordered set strict decreasing"  'strictly_decreasing s    =
 definition uparrow ≝
   λC:half_ordered_set.λs:sequence C.λu:C.
    increasing ? s ∧ supremum ? s u.
-(*   
-notation < "a \uparrow \nbsp u" non associative with precedence 45 for @{'sup_inc $a $u}.
-notation > "a \uparrow u" non associative with precedence 45 for @{'sup_inc $a $u}.
-*)
-interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
 
-(*
-notation < "a \downarrow \nbsp u" non associative with precedence 45 for @{'inf_dec $a $u}.
-notation > "a \downarrow u" non associative with precedence 45 for @{'inf_dec $a $u}.
-*)
+interpretation "Ordered set uparrow" 'funion s u = (uparrow (os_l _) s u).
 interpretation "Ordered set downarrow" 'fintersects s u = (uparrow (os_r _) s u).
 
 lemma h_trans_increasing: 
@@ -133,9 +107,9 @@ lemma h_trans_increasing:
    ∀n,m:nat_ordered_set. n ≤ m → a n ≤≤ a m.
 intros 5 (C a Hs n m); elim m; [
   rewrite > (le_n_O_to_eq n (not_lt_to_le O n H));
-  intro X; cases (hos_coreflexive ?? X);]
+  intro X; cases (hos_coreflexive ? (a n) X);]
 cases (le_to_or_lt_eq ?? (not_lt_to_le (S n1) n H1)); clear H1;
-[2: rewrite > H2; intro; cases (hos_coreflexive ?? H1);
+[2: rewrite > H2; intro; cases (hos_coreflexive ? (a (S n1)) H1);
 |1: apply (hle_transitive ???? (H ?) (Hs ?));
     intro; whd in H1; apply (not_le_Sn_n n); apply (transitive_le ??? H2 H1);]
 qed.
@@ -146,23 +120,24 @@ notation "'trans_decreasing'" non associative with precedence 90 for @{'trans_de
 interpretation "trans_increasing" 'trans_increasing = (h_trans_increasing (os_l _)).
 interpretation "trans_decreasing" 'trans_decreasing = (h_trans_increasing (os_r _)).
 
-(* TEST DUALITY
-lemma test_trans_decreasing_duality: 
-  ∀C:ordered_set.∀a:sequence C.a is_decreasing → 
-   ∀n,m:nat_ordered_set. n ≤ m → a m ≤ a n.
-intros; apply (trans_decreasing ? H ?? H1); qed.
-*)
+lemma hint_nat :
+ Type_of_ordered_set nat_ordered_set →
+   hos_carr (os_l (nat_ordered_set)).
+intros; assumption;
+qed.
+
+coercion hint_nat nocomposites.
 
 lemma h_trans_increasing_exc: 
   ∀C:half_ordered_set.∀a:sequence C.increasing ? a → 
-   ∀n,m:nat_ordered_set. m ≰ n → a n ≤≤ a m.
+   ∀n,m:nat_ordered_set. m ≰≰ n → a n ≤≤ a m.
 intros 5 (C a Hs n m); elim m; [cases (not_le_Sn_O n H);]
 intro; apply H; 
 [1: change in n1 with (hos_carr (os_l nat_ordered_set)); 
     change with (n<n1);
     cases (le_to_or_lt_eq ?? H1); [apply le_S_S_to_le;assumption]
     cases (Hs n); rewrite < H3 in H2; assumption;    
-|2: cases (hos_cotransitive ??? (a n1) H2); [assumption]
+|2: cases (hos_cotransitive ? (a n) (a (S n1)) (a n1) H2); [assumption]
     cases (Hs n1); assumption;]
 qed.
 
@@ -198,8 +173,8 @@ intros (C m Hm a u Ha); cases Ha (Ia Su); cases Su (Uu Hu); repeat split;
 |2: intro n; simplify; apply Uu;
 |3: intros (y Hy); simplify; cases (Hu ? Hy);
     cases (nat_strictly_increasing_reaches ? Hm w); 
-    exists [apply w1]; cases (hos_cotransitive ??? (a (m w1)) H); [2:assumption]  
-    cases (h_trans_increasing_exc ?? Ia ?? H1); assumption;]
+    exists [apply w1]; cases (hos_cotransitive ? (a w) y (a (m w1)) H); [2:assumption]  
+    cases (h_trans_increasing_exc ?? Ia w (m w1) H1); assumption;]
 qed.     
 
 notation "'selection_uparrow'" non associative with precedence 90 for @{'selection_uparrow}.
@@ -222,57 +197,180 @@ notation > "a 'order_converges' x" non associative with precedence 45
 interpretation "Order convergence" 'order_converge s u = (order_converge _ s u).   
     
 (* Definition 2.8 *)
-definition segment ≝ λO:half_ordered_set.λa,b:O.λx:O.(x ≤≤ b) ∧ (a ≤≤ x).
+record segment (O : Type) : Type ≝ {
+   seg_l_ : O;
+   seg_u_ : O
+}.
+
+notation > "𝕦_term 90 s" non associative with precedence 90 for @{'upp $s}.
+notation "𝕦 \sub term 90 s" non associative with precedence 90 for @{'upp $s}. 
+notation > "𝕝_term 90 s" non associative with precedence 90 for @{'low $s}.
+notation "𝕝 \sub term 90 s" non associative with precedence 90 for @{'low $s}. 
+ 
+definition seg_u ≝
+ 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.
+   wloss O ?? (λl,u.l) (seg_l_ ? s) (seg_u_ ? s). 
+ 
+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 ≤≤ x) (x ≤≤ seg_u ? s).
 
-notation "[term 19 a,term 19 b]" non associative with precedence 90 for @{'segment $a $b}.
-interpretation "Ordered set sergment" 'segment a b = (segment (os_l _) a b).
+notation "‡O" non associative with precedence 90 for @{'segment $O}.
+interpretation "Ordered set sergment" 'segment x = (segment x).
 
-notation "hvbox(x \in break [term 19 a, term 19 b])" non associative with precedence 45 
-  for @{'segment_in $a $b $x}.
-interpretation "Ordered set sergment in" 'segment_in a b x= (segment (os_l _) a b x).
+interpretation "Ordered set sergment in" 'mem x s = (in_segment _ s x).
 
 definition segment_ordered_set_carr ≝
-  λO:half_ordered_set.λu,v:O.∃x.segment ? u v x.
+  λO:half_ordered_set.λs:‡O.∃x.x ∈ s.
 definition segment_ordered_set_exc ≝ 
-  λO:half_ordered_set.λu,v:O.
-   λx,y:segment_ordered_set_carr ? u v.\fst x ≰≰ \fst y.
+  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,u,v. coreflexive ? (segment_ordered_set_exc O u v).
-intros 4; cases x; simplify; apply hos_coreflexive; qed.
+ ∀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;
+qed.
 lemma segment_ordered_set_cotrans : 
-  ∀O,u,v. cotransitive ? (segment_ordered_set_exc O u v).
-intros 6 (O u v x y z); cases x; cases y ; cases z; simplify; apply hos_cotransitive;
+  ∀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;
 qed.  
   
 lemma half_segment_ordered_set: 
-  ∀O:half_ordered_set.∀u,v:O.half_ordered_set.
-intros (O u v); apply (mk_half_ordered_set ?? (segment_ordered_set_corefl O u v) (segment_ordered_set_cotrans ???));
+  ∀O:half_ordered_set.∀s:segment O.half_ordered_set.
+intros (O a); constructor 1;
+[ apply (segment_ordered_set_carr O a);
+| apply (wloss O);
+| apply (wloss_prop O);
+| apply (segment_ordered_set_exc O a);
+| apply (segment_ordered_set_corefl O a);
+| apply (segment_ordered_set_cotrans ??);
+]
 qed.
 
 lemma segment_ordered_set: 
-  ∀O:ordered_set.∀u,v:O.ordered_set.
-intros (O u v); letin hos ≝ (half_segment_ordered_set (os_l O) u v);
-constructor 1; [apply hos; | apply (dual_hos hos); | reflexivity] 
+  ∀O:ordered_set.∀s:‡O.ordered_set.
+intros (O s); 
+apply half2full; apply (half_segment_ordered_set (os_l O) s); 
+qed.
+
+notation "{[ term 19 s ]}" non associative with precedence 90 for @{'segset $s}.
+interpretation "Ordered set segment" 'segset s = (segment_ordered_set _ s). 
+
+(* test :
+ ∀O:ordered_set.∀s: segment (os_l O).∀x:O.
+   in_segment (os_l O) s x
+   =
+   in_segment (os_r O) s x.
+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.
 
-notation "hvbox({[a, break b]})" non associative with precedence 90 
-  for @{'segment_set $a $b}.
-interpretation "Ordered set segment" 'segment_set a b = 
-  (half_segment_ordered_set _ a b).
-interpretation "Ordered set segment" 'segment_set a b = 
-  (segment_ordered_set _ a b).
+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).
+intros;assumption;
+qed.
+
+definition hint_sequence1: 
+  ∀C:ordered_set.
+    sequence (hos_carr (os_r C)) → sequence (Type_of_ordered_set_dual C).
+intros;assumption;
+qed.
+
+definition hint_sequence2: 
+  ∀C:ordered_set.
+    sequence (Type_of_ordered_set C) → sequence (hos_carr (os_l C)).
+intros;assumption;
+qed.
+
+definition hint_sequence3: 
+  ∀C:ordered_set.
+    sequence (Type_of_ordered_set_dual C) → sequence (hos_carr (os_r C)).
+intros;assumption;
+qed.
+
+coercion hint_sequence nocomposites.
+coercion hint_sequence1 nocomposites.
+coercion hint_sequence2 nocomposites.
+coercion hint_sequence3 nocomposites.
+
+(* Lemma 2.9 - non easily dualizable *)
+
+lemma x2sx_:
+  ∀O:half_ordered_set.
+   ∀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 ⊢ (?→%);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma sx2x_:
+  ∀O:half_ordered_set.
+   ∀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 ⊢ (% → ?);
+cases (wloss_prop O) (E E); do 2 rewrite < E; intros; assumption;
+qed.
+
+lemma l2sl_:
+  ∀C,s.∀x,y:half_segment_ordered_set C s. \fst x ≤≤ \fst y → x ≤≤ y.
+intros; intro; apply H; apply sx2x_; apply H1;
+qed.
+
+
+lemma sl2l_:
+  ∀C,s.∀x,y:half_segment_ordered_set C s. x ≤≤ y → \fst x ≤≤ \fst y.
+intros; intro; apply H; apply x2sx_; apply H1;
+qed.
+
+coercion x2sx_ nocomposites.
+coercion sx2x_ nocomposites.
+coercion l2sl_ nocomposites.
+coercion sl2l_ nocomposites.
 
-(* Lemma 2.9 *)
 lemma h_segment_preserves_supremum:
-  ∀O:half_ordered_set.∀l,u:O.∀a:sequence {[l,u]}.∀x:{[l,u]}. 
-    increasing ? ⌊n,\fst (a n)⌋ ∧ 
-    supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
+  ∀O:half_ordered_set.∀s:segment O.
+   ∀a:sequence (half_segment_ordered_set ? s).
+    ∀x:half_segment_ordered_set ? s. 
+      increasing ? ⌊n,\fst (a n)⌋ ∧ 
+      supremum ? ⌊n,\fst (a n)⌋ (\fst x) → uparrow ? a x.
 intros; split; cases H; clear H; 
-[1: apply H1;
+[1: intro n; lapply (H1 n) as K; clear H1 H2;
+    intro; apply K; clear K; apply rule H; 
 |2: cases H2; split; clear H2;
-    [1: apply H;
-    |2: clear H; intro y0; apply (H3 (\fst y0));]]
+    [1: intro n; lapply (H n) as K; intro W; apply K;
+        apply rule W;
+    |2: clear H1 H; intros (y0 Hy0); cases (H3 (\fst y0));[exists[apply w]]
+        [1: change in H with (\fst (a w) ≰≰ \fst y0); apply rule H;
+        |2: apply rule Hy0;]]]
 qed.
 
 notation "'segment_preserves_supremum'" non associative with precedence 90 for @{'segment_preserves_supremum}.
@@ -281,17 +379,33 @@ notation "'segment_preserves_infimum'" non associative with precedence 90 for @{
 interpretation "segment_preserves_supremum" 'segment_preserves_supremum = (h_segment_preserves_supremum (os_l _)).
 interpretation "segment_preserves_infimum" 'segment_preserves_infimum = (h_segment_preserves_supremum (os_r _)).
 
+(*
+test segment_preserves_infimum2:
+  ∀O:ordered_set.∀s:‡O.∀a:sequence {[s]}.∀x:{[s]}. 
+    ⌊n,\fst (a n)⌋ is_decreasing ∧ 
+    (\fst x) is_infimum ⌊n,\fst (a n)⌋ → a ↓ x.
+intros; apply (segment_preserves_infimum s a x H);
+qed.
+*)
+       
 (* Definition 2.10 *)
-alias symbol "square" = "ordered set square".
+
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
+(*
 definition square_segment ≝ 
-  λO:ordered_set.λa,b:O.λx: O square.
-    And4 (\fst x ≤ b) (a ≤ \fst x) (\snd x ≤ b) (a ≤ \snd x).
- 
+  λO:half_ordered_set.λs:segment O.λx: square_half_ordered_set O.
+    in_segment ? s (\fst x) ∧ in_segment ? s (\snd x).
+*) 
 definition convex ≝
-  λO:ordered_set.λU:O square → Prop.
-    ∀p.U p → \fst p ≤ \snd p → ∀y. square_segment ? (\fst p) (\snd p) y → U y.
+  λO:half_ordered_set.λU:square_half_ordered_set O → Prop.
+    ∀s.U s → le O (\fst s) (\snd s) → 
+     ∀y. 
+       le O (\fst y) (\snd s) → 
+       le O (\fst s) (\fst y) →
+       le O (\snd y) (\snd s) →
+       le O (\fst y) (\snd y) →
+       U y.
   
 (* Definition 2.11 *)  
 definition upper_located ≝
@@ -316,8 +430,8 @@ interpretation "Ordered set lower locatedness" 'lower_located s =
 lemma h_uparrow_upperlocated:
   ∀C:half_ordered_set.∀a:sequence C.∀u:C.uparrow ? a u → upper_located ? a.
 intros (C a u H); cases H (H2 H3); clear H; intros 3 (x y Hxy);
-cases H3 (H4 H5); clear H3; cases (hos_cotransitive ??? u Hxy) (W W);
-[2: cases (H5 ? W) (w Hw); left; exists [apply w] assumption;
+cases H3 (H4 H5); clear H3; cases (hos_cotransitive C y x u Hxy) (W W);
+[2: cases (H5 x W) (w Hw); left; exists [apply w] assumption;
 |1: right; exists [apply u]; split; [apply W|apply H4]]
 qed.