X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2Fdama%2Fdama%2Fmodels%2Fq_function.ma;h=32122371004982cad24e4deccf33ec5f71dad0e1;hb=6b61a9e6698a7c1936adf217b599e34e65a5e4c9;hp=21e51808a10189754f4e0e7e80736a562c9168df;hpb=b82d421fb24bd0eb8848dc7b978ce1165fb526ab;p=helm.git

diff --git a/helm/software/matita/contribs/dama/dama/models/q_function.ma b/helm/software/matita/contribs/dama/dama/models/q_function.ma
index 21e51808a..321223710 100644
--- a/helm/software/matita/contribs/dama/dama/models/q_function.ma
+++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma
@@ -12,220 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Q/q/q.ma".
-include "list/list.ma".
-include "cprop_connectives.ma". 
-
-
-notation "\rationals" non associative with precedence 99 for @{'q}.
-interpretation "Q" 'q = Q. 
-
-definition bar ≝ ratio × ℚ. (* base (Qpos) , height *)
-record q_f : Type ≝ { start : ℚ; bars: list bar }.
-
-axiom qp : ℚ → ℚ → ℚ.
-axiom qm : ℚ → ℚ → ℚ.
-axiom qlt : ℚ → ℚ → CProp.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-interpretation "Q minus" 'minus x y = (qm x y).
-interpretation "Q less than" 'lt x y = (qlt x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_eq : a = b → q_comparison a b 
- | q_lt : a < b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-definition qle ≝ λa,b:ℚ.a = b ∨ a < b.
-
-interpretation "Q less or equal than" 'leq x y = (qle x y).
-
-axiom q_le_minus: ∀a,b,c:ℚ. a ≤ c - b → a + b ≤ c.
-axiom q_lt_plus: ∀a,b,c:ℚ. a - b < c → a < c + b.
-axiom q_lt_minus: ∀a,b,c:ℚ. a + b < c → a < c - b.
-
-axiom q_dist : ℚ → ℚ → ℚ.
-
-notation "hbox(\dd [term 19 x, break term 19 y])" with precedence 90
-for @{'distance $x $y}.
-interpretation "ℚ distance" 'distance x y = (q_dist x y).
-
-axiom q_dist_ge_OQ : ∀x,y:ℚ. OQ ≤ ⅆ[x,y].
-
-axiom q_lt_to_le: ∀a,b:ℚ.a < b → a ≤ b.
-axiom q_le_to_diff_ge_OQ : ∀a,b.a ≤ b → OQ ≤ b-a.
-axiom q_plus_OQ: ∀x:ℚ.x + OQ = x.
-axiom q_plus_sym: ∀x,y:ℚ.x + y = y + x.
-axiom nat_of_q: ℚ → nat. 
-
-interpretation "list nth" 'nth = (nth _).
-interpretation "list nth" 'nth_appl l d i = (nth _ l d i).
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp term 90 l \nbsp term 90 d \nbsp term 90 i" 
-with precedence 69 for @{'nth_appl $l $d $i}.
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-definition make_list ≝
-  λA:Type.λdef:nat→A.
-    let rec make_list (n:nat) on n ≝
-      match n with [ O ⇒ nil ? | S m ⇒ def m :: make_list m]
-    in make_list.
-
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-interpretation "'mk_list'" 'mk_list = (make_list _).   
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-notation < "'mk_list' \nbsp term 90 f \nbsp term 90 n" 
-with precedence 69 for @{'mk_list_appl $f $n}.
-
-
-definition empty_bar : bar ≝ 〈one,OQ〉.
-notation "\rect" with precedence 90 for @{'empty_bar}.
-interpretation "q0" 'empty_bar = empty_bar.
-
-notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list bar).
-
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = (length _).
-notation < "'len' \nbsp term 90 l" with precedence 69 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-lemma len_mk_list : ∀T:Type.∀f:nat→T.∀n.len (mk_list f n) = n.
-intros; elim n; [reflexivity] simplify; rewrite > H; reflexivity;
-qed.
-
-let rec sum_bases (l:list bar) (i:nat)on i ≝
-    match i with
-    [ O ⇒ OQ
-    | S m ⇒ 
-         match l with
-         [ nil ⇒ sum_bases l m + Qpos one
-         | cons x tl ⇒ sum_bases tl m + Qpos (\fst x)]].
-         
-axiom sum_bases_empty_nat_of_q_ge_OQ:
-  ∀q:ℚ.OQ ≤ sum_bases [] (nat_of_q q). 
-axiom sum_bases_empty_nat_of_q_le_q:
-  ∀q:ℚ.sum_bases [] (nat_of_q q) ≤ q.
-axiom sum_bases_empty_nat_of_q_le_q_one:
-  ∀q:ℚ.q < sum_bases [] (nat_of_q q) + Qpos one.
-
-definition eject1 ≝
-  λP.λp:∃x:nat × ℚ.P x.match p with [ex_introT p _ ⇒ p].
-coercion eject1.
-definition inject1 ≝ λP.λp:nat × ℚ.λh:P p. ex_introT ? P p h.
-coercion inject1 with 0 1 nocomposites.
-
-definition value : 
-  ∀f:q_f.∀i:ℚ.∃p:nat × ℚ. 
-   match q_cmp i (start f) with
-   [ q_lt _ ⇒ \snd p = OQ
-   | _ ⇒ 
-        And3
-         (sum_bases (bars f) (\fst p) ≤ ⅆ[i,start f]) 
-         (ⅆ[i, start f] < sum_bases (bars f) (S (\fst p))) 
-         (\snd p = \snd (nth (bars f) ▭ (\fst p)))].
-intros;
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-letin value ≝ (
-  let rec value (p: ℚ) (l : list bar) on l ≝
-    match l with
-    [ nil ⇒ 〈nat_of_q p,OQ〉
-    | cons x tl ⇒
-        match q_cmp p (Qpos (\fst x)) with
-        [ q_lt _ ⇒ 〈O, \snd x〉
-        | _ ⇒
-           let rc ≝ value (p - Qpos (\fst x)) tl in
-           〈S (\fst rc),\snd rc〉]]
-  in value :
-  ∀acc,l.∃p:nat × ℚ. OQ ≤ acc →
-     And3
-       (sum_bases l (\fst p) ≤ acc) 
-       (acc < sum_bases l (S (\fst p))) 
-       (\snd p = \snd (nth l ▭ (\fst p))));
-[5: clearbody value; 
-    cases (q_cmp i (start f));
-    [2: exists [apply 〈O,OQ〉] simplify; reflexivity;
-    |*: cases (value ⅆ[i,start f] (bars f)) (p Hp);
-        cases (Hp (q_dist_ge_OQ ? ?)); clear Hp value;
-        exists[1,3:apply p]; simplify; split; assumption;]
-|1,3: intros; split;
-    [1,4: clear H2; cases (value (q-Qpos (\fst b)) l1);
-           cases (H2 (q_le_to_diff_ge_OQ ?? (? H1)));
-          [1,3: intros; [right|left;symmetry] assumption]
-          simplify; apply q_le_minus; assumption;
-    |2,5: cases (value (q-Qpos (\fst b)) l1); 
-          cases (H4 (q_le_to_diff_ge_OQ ?? (? H1)));
-          [1,3: intros; [right|left;symmetry] assumption]
-          clear H3 H2 value;
-          change with (q < sum_bases l1 (S (\fst w)) + Qpos (\fst b));
-          apply q_lt_plus; assumption;
-    |*: cases (value (q-Qpos (\fst b)) l1); simplify; 
-        cases (H4 (q_le_to_diff_ge_OQ ?? (? H1))); 
-        [1,3: intros; [right|left;symmetry] assumption]
-        assumption;]
-|2: clear value H2; simplify; intros; split; [assumption|3:reflexivity]
-    rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption;
-|4: simplify; intros; split; 
-    [1: apply sum_bases_empty_nat_of_q_le_q;
-    |2: apply sum_bases_empty_nat_of_q_le_q_one;
-    |3: elim (nat_of_q q); [reflexivity] simplify; assumption]] 
-qed.
-    
-          
-definition same_values ≝
-  λl1,l2:q_f.
-   ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). 
-
-definition same_bases ≝ 
-  λl1,l2:q_f.
-    (∀i.\fst (nth (bars l1) ▭ i) = \fst (nth (bars l2) ▭ i)).
-
-axiom q_lt_corefl: ∀x:Q.x < x → False.
-axiom q_lt_antisym: ∀x,y:Q.x < y → y < x → False.
-axiom q_neg_gt: ∀r:ratio.OQ < Qneg r → False.
-axiom q_d_x_x: ∀x:Q.ⅆ[x,x] = OQ.
-axiom q_pos_OQ: ∀x.Qpos x ≤ OQ → False.
-axiom q_lt_plus_trans:
-  ∀x,y:Q.OQ ≤ x → OQ < y → OQ < x + y.
-axiom q_pos_lt_OQ: ∀x.OQ < Qpos x.
-axiom q_le_plus_trans:
-  ∀x,y:Q. OQ ≤ x → OQ ≤ y → OQ ≤ x + y. 
-
-lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
-intro; cases x; intros; [2:exists [apply r] reflexivity]
-cases (?:False);
-[ apply (q_lt_corefl ? H)|apply (q_neg_gt ? H)]
-qed.
-
-notation < "\blacksquare" non associative with precedence 90 for @{'hide}.
-definition hide ≝ λT:Type.λx:T.x.
-interpretation "hide" 'hide = (hide _ _).
-
-lemma sum_bases_ge_OQ:
-  ∀l,n. OQ ≤ sum_bases (bars l) n.
-intro; elim (bars l); simplify; intros;
-[1: elim n; [left;reflexivity] simplify;
-    apply q_le_plus_trans; try assumption; apply q_lt_to_le; apply q_pos_lt_OQ;
-|2: cases n; [left;reflexivity] simplify; 
-    apply q_le_plus_trans; [apply H| apply q_lt_to_le; apply q_pos_lt_OQ;]]
-qed.
-
-lemma sum_bases_O:
-  ∀l:q_f.∀x.sum_bases (bars l) x ≤ OQ → x = O.
-intros; cases x in H; [intros; reflexivity] intro; cases (?:False);
-cases H; 
-[1: apply (q_lt_corefl OQ); rewrite < H1 in ⊢ (?? %); 
-|2: apply (q_lt_antisym ??? H1);] clear H H1; cases (bars l);
-simplify; apply q_lt_plus_trans;
-try apply q_pos_lt_OQ; 
-try apply (sum_bases_ge_OQ (mk_q_f OQ []));
-apply (sum_bases_ge_OQ (mk_q_f OQ l1));
-qed.
+include "models/q_bars.ma".
 
 lemma initial_shift_same_values:
   ∀l1:q_f.∀init.init < start l1 →
@@ -247,11 +34,60 @@ whd in ⊢ (% → ?); simplify in H3;
         rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
         symmetry; apply le_n_O_to_eq;
         rewrite > (sum_bases_O (mk_q_f init (〈w,OQ〉::bars l1)) (\fst w1)); [apply le_n]   
-        clear H6 w2;
-        simplify in H5:(? ? (? ? %));  
+        clear H6 w2; simplify in H5:(? ? (? ? %));  
         destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|2: intros;
-  
+|2: intros; cases (value l1 input); simplify in ⊢ (? ? (? ? ? %) ?);
+    cases (q_cmp input (start l1)) in H5; whd in ⊢ (% → ?);
+    [1: cases (?:False); clear w2 H4 w1 H2 w H1; 
+        apply (q_lt_antisym init (start l1)); [assumption] rewrite < H5; assumption
+    |2: intros; rewrite > H6; clear H6; rewrite > H4; reflexivity;
+    |3: cases (?:False); apply (q_lt_antisym input (start l1)); [2: assumption]
+        apply (q_lt_trans ??? H3 H);]
+|3: intro; cases H4; clear H4;   
+    cases (value l1 input); simplify; cases (q_cmp input (start l1)) in H4; whd in ⊢ (% → ?);
+    [1: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
+        simplify in ⊢ (? ? ? (? ? ? (? ? % ? ?)));
+        cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
+        cut (\fst w2 = O); [2: clear H10;
+          symmetry; apply le_n_O_to_eq; rewrite > (sum_bases_O l1 (\fst w2)); [apply le_n]
+          apply (q_le_trans ??? H9); rewrite < H4; rewrite > q_d_x_x; 
+          apply q_eq_to_le; reflexivity;]
+        rewrite > Hcut; clear Hcut H10 H9; simplify in H5 H6;
+        cut (ⅆ[input,init] = Qpos w) as E; [2:
+          rewrite > H2; rewrite < H4; rewrite > q_d_sym; 
+          rewrite > q_d_noabs; [reflexivity] apply q_lt_to_le; assumption;]
+        cases (\fst w1) in H5 H6; intros;
+        [1: cases (?:False); clear H5; simplify in H6;
+            apply (q_lt_corefl ⅆ[input,init]);
+            rewrite > E in ⊢ (??%); rewrite < q_plus_OQ in ⊢ (??%);
+            rewrite > q_plus_sym; assumption;
+        |2: cases n in H5 H6; [intros; reflexivity] intros;
+            cases (?:False); clear H6; cases (bars l1) in H5; simplify; intros;
+            [apply (q_pos_OQ one);|apply (q_pos_OQ (\fst b));] 
+            apply (q_le_S ??? (sum_bases_ge_OQ ? n1));[apply []|3:apply l]
+            simplify in ⊢ (? (? (? % ?) ?) ?); rewrite < (q_plus_minus (Qpos w));
+            rewrite > q_elim_minus; apply q_le_minus_r; 
+            rewrite > q_elim_opp; rewrite < E in ⊢ (??%); assumption;]
+    |2: intros; rewrite > H8; rewrite > H7; clear H8 H7;
+        simplify in H5 H6 ⊢ %; 
+        cases (\fst w1) in H5 H6; [intros; reflexivity]
+        cases (bars l1);
+        [1: intros; simplify; elim n [reflexivity] simplify; assumption;
+        |2: simplify; intros; cases (?:False); clear H6;
+            apply (q_lt_le_incompat (input - init) (Qpos w) );
+            [1: rewrite > H2; do 2 rewrite > q_elim_minus;
+                apply q_lt_plus; rewrite > q_elim_minus;
+                rewrite < q_plus_assoc; rewrite < q_elim_minus;
+                rewrite > q_plus_minus;
+                rewrite > q_plus_OQ; assumption;
+            |2: rewrite < q_d_noabs; [2: apply q_lt_to_le; assumption]
+                rewrite > q_d_sym; apply (q_le_S ???? H5);
+                apply sum_bases_ge_OQ;]]
+    |3:
+    
+        
+STOP            
+        
 alias symbol "pi2" = "pair pi2".
 alias symbol "pi1" = "pair pi1".
 definition rebase_spec ≝ 
@@ -352,4 +188,4 @@ in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
 |6:(* TODO *)
 |7:(* TODO *)
 |8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
-qed.
\ No newline at end of file
+qed.