(**************************************************************************)
include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.ma".
-lemma initial_shift_same_values:
- ∀l1:q_f.∀init.init < start l1 →
- same_values l1
- (mk_q_f init (〈\fst (unpos (start l1 - init) ?),OQ〉:: bars l1)).
-[apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
-intros; generalize in ⊢ (? ? (? ? (? ? (? ? ? (? ? ? (? ? %)) ?) ?))); intro;
-cases (unpos (start l1-init) H1); intro input;
-simplify in ⊢ (? ? ? (? ? ? (? ? ? (? (? ? (? ? (? ? ? % ?) ?)) ?))));
-cases (value (mk_q_f init (〈w,OQ〉::bars l1)) input) (v1 Hv1);
-(*cases (value l1 input) (v2 Hv2); *)
-cases Hv1 (HV1 HV1 HV1 HV1); (* cases Hv2 (HV2 HV2 HV2 HV2); clear Hv1 Hv2; *)
-cases HV1 (Hi1 Hv11 Hv12); (*cases HV2 (Hi2 Hv21 Hv22);*) clear HV1 (*HV2*);
-(* simplify; *)
-rewrite > Hv12; (*rewrite > Hv22;*) try reflexivity;
-[1: simplify in Hi1; cases (?:False);
- apply (q_lt_corefl (start l1)); cases (Hi2);
- autobatch by Hi2, Hi1, q_le_trans, H4, H, q_le_lt_trans, q_lt_le_trans.
-|2: simplify in Hi1; cases (?:False);
- apply (q_lt_corefl (start l1+sum_bases (bars l1) (len (bars l1))));
- cases Hi2; apply (q_le_lt_trans ???? H5);
- apply (q_le_trans ???? Hi1);
- rewrite > H2; rewrite > (q_plus_sym ? (start l1-init));
- rewrite > q_plus_assoc; apply q_le_inj_plus_r;
- apply q_eq_to_le;
- rewrite > q_elim_minus; rewrite > (q_plus_sym (start l1));
- rewrite > q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- reflexivity;
-|3: simplify in Hi1; destruct Hi1;
-|4: simplify in Hi1 H3 Hv12 Hv11 ⊢ %; cases H3; clear H3;
- cases (\fst v1) in H4; [intros;reflexivity] intros;
- simplify; simplify in H3;
-
-
-
-
-
-simplify in ⊢ (? ? ? (? ? ? %));
-cases (q_cmp input (start (mk_q_f init (〈w,OQ〉::bars l1)))) in H3;
-whd in ⊢ (% → ?); simplify in H3;
-[1: intro; cases H4; clear H4; rewrite > H3;
- cases (value l1 init); simplify; cases (q_cmp init (start l1)) in H4;
- [1: cases (?:False); apply (q_lt_corefl init); rewrite > H4 in ⊢ (?? %); apply H;
- |3: cases (?:False); apply (q_lt_antisym init (start l1)); assumption;
- |2: whd in ⊢ (% → ?); intro; rewrite > H8; clear H8 H4;
- rewrite > H7; clear H7; rewrite > (?:\fst w1 = O); [reflexivity]
- symmetry; apply le_n_O_to_eq;
- rewrite > (sum_bases_O (〈w,OQ〉::bars l1) (\fst w1)); [apply le_n]
- clear H6 w2; simplify in H5:(? ? (? ? %));
- destruct H3; rewrite > q_d_x_x in H5; assumption;]
-|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 (bars 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: intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
- simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
-
-axiom nth_nil: ∀T,n.∀d:T. nth [] d n = d.
-
-
-lemma case1 :
- ∀init,st,input,l.
- init<st → st<input →
- ⅆ[input,init] < sum_bases l O + (st-init) → False.
-intros 6; rewrite > q_d_sym; rewrite > q_d_noabs; [2:
- apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc;
-intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
-simplify in Y; cases (?:False);
-apply (q_lt_corefl st); apply (q_lt_trans ??? H1);
-apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
-apply q_eq_to_le; reflexivity;
-qed.
-
-lemma case2:
- ∀a,l1,init,st,input,n.
- init < st → st < input →
- sum_bases (a::l1) n + (st-init) ≤ ⅆ[input,init] →
- ⅆ[input,st] < sum_bases l1 O + Qpos (\fst a) →
- n = O.
-intros; cut (input - st < Qpos (\fst a)) as H6';[2:
- rewrite < q_d_noabs;[2:apply q_lt_to_le; assumption]
- rewrite > q_d_sym; apply (q_lt_le_trans ??? H3);
- rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity] clear H3;
-generalize in match H2; rewrite > q_d_sym; rewrite > q_d_noabs;
- [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-do 2 rewrite > q_elim_minus; rewrite > q_plus_assoc; intro X;
-lapply (q_le_canc_plus_r ??? X) as Y; clear X;
-lapply (q_le_inj_plus_r ?? (Qopp st) Y) as X; clear Y;
-cut (input + Qopp st < Qpos (\fst a)) as H6'';
- [2: rewrite < q_elim_minus; assumption;] clear H6';
-generalize in match (q_le_lt_trans ??? X H6''); clear X H6'';
-rewrite < q_plus_assoc; rewrite < q_elim_minus;
-rewrite > q_plus_minus; rewrite > q_plus_OQ; cases n; intro X; [reflexivity]
-cases (?:False);
-apply (q_lt_le_incompat (sum_bases l1 n1) OQ);[2: apply sum_bases_ge_OQ;]
-apply (q_lt_canc_plus_r ?? (Qpos (\fst a)));
-rewrite >(q_plus_sym OQ); rewrite > q_plus_OQ; apply X;
-qed.
-
-lemma case3:
- ∀init,st,input,l1,a,n.
- init<st → st<input →
- ⅆ[input,init]<OQ+Qpos a+(st-init) →
- sum_bases l1 n+Qpos a≤ⅆ[input,st] → False.
-intros;
-cut (sum_bases l1 n - ⅆ[input,st] < Qopp ⅆ[input,init] + (st - init)); [2:
- cut (sum_bases l1 n≤ⅆ[input,st]-Qpos a) as H7';[2:
- apply (q_le_canc_plus_r ?? (Qpos a));
- apply (q_le_trans ??? H3); rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity;] clear H3;
- rewrite > q_elim_minus; apply (q_lt_canc_plus_r ?? ⅆ[input,st]);
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply (q_le_lt_trans ??? H7'); clear H7'; rewrite > q_elim_minus;
- rewrite > q_plus_sym; apply q_lt_inj_plus_r;
- rewrite > q_plus_sym; apply q_lt_plus; rewrite > q_elim_opp;
- rewrite > q_plus_sym; apply (q_lt_canc_plus_r ?? (Qpos a));
- rewrite < q_plus_assoc; rewrite > (q_plus_sym (Qopp ?));
- rewrite < q_elim_minus; rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply (q_lt_le_trans ??? H2); rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
- rewrite > q_plus_sym; apply q_eq_to_le; reflexivity;]
-generalize in match Hcut; clear H2 H3 Hcut;
-rewrite > q_d_sym; rewrite > q_d_noabs; [2:apply q_lt_to_le; assumption]
-rewrite > q_d_sym; rewrite > q_d_noabs; [2: apply (q_le_trans ? st); apply q_lt_to_le; assumption]
-rewrite < q_plus_sym; rewrite < q_elim_minus;
-rewrite > (q_elim_minus input init);
-rewrite > q_minus_distrib; rewrite > q_elim_opp;
-rewrite > (q_elim_minus input st);
-rewrite > q_minus_distrib; rewrite > q_elim_opp;
-repeat rewrite > q_elim_minus;
-rewrite < q_plus_assoc in ⊢ (??% → ?);
-rewrite > (q_plus_sym (Qopp input) init);
-rewrite > q_plus_assoc;
-rewrite < q_plus_assoc in ⊢ (??(?%?) → ?);
-rewrite > (q_plus_sym (Qopp init) init);
-rewrite < (q_elim_minus init); rewrite >q_plus_minus;
-rewrite > q_plus_OQ; rewrite > (q_plus_sym st);
-rewrite < q_plus_assoc;
-rewrite < (q_plus_OQ (Qopp input + st)) in ⊢ (??% → ?);
-rewrite > (q_plus_sym ? OQ); intro X;
-lapply (q_lt_canc_plus_r ??? X) as Y; clear X;
-apply (q_lt_le_incompat ?? Y); apply sum_bases_ge_OQ;
-qed.
-
-lemma key:
- ∀init,input,l1,w1,w2,w.
- Qpos w = start l1 - init →
- init < start l1 →
- start l1 < input →
- sum_bases (〈w,OQ〉::bars l1) w1 ≤ ⅆ[input,init] →
- ⅆ[input,init] < sum_bases (bars l1) w1 + (start l1-init) →
- sum_bases (bars l1) w2 ≤ ⅆ[input,start l1] →
- ⅆ[input,start l1] < sum_bases (bars l1) (S w2) →
- \snd (nth (bars l1) ▭ w2) = \snd (nth (〈w,OQ〉::bars l1) ▭ w1).
-intros 3 (init input l); cases l (st l);
-change in match (start (mk_q_f st l)) with st;
-change in match (bars (mk_q_f st l)) with l;
-elim l; clear l;
-[1: rewrite > nth_nil; cases w1 in H4;
- [1: intro X; cases (case1 ?????? X); assumption;
- |2: intros; simplify; rewrite > nth_nil; reflexivity;]
-|2: cases w1 in H4 H5; clear w1;
- [1: intros (Y X); cases (case1 ?????? X); assumption;
- |2: intros; simplify in H4 H5 H7 ⊢ %;
- generalize in match H6; generalize in match H7;
- generalize in match H4; generalize in match H5; clear H4 H5 H6 H7;
- apply (nat_elim2 ???? w2 n); clear w2 n; intros;
- [1: rewrite > (case2 a l1 init st input n); [reflexivity]
- try rewrite < H1; assumption;
- |2: simplify in H4 H7; cases (case3 ???????? H4 H7); assumption;
- |3: (* dipende se vanno oltre la lunghezza di l1,
- forse dovevo gestire il caso prima dell'induzione *)
- simplify in ⊢ (? ? (? ? ? %) ?);
- rewrite > (H (S m) ? w); [reflexivity] try assumption;
-STOP
-
-qed.
-
-
-
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
definition rebase_spec ≝
∀l1,l2:q_f.∃p:q_f × q_f.
And4
- (*len (bars (\fst p)) = len (bars (\snd p))*)
(start (\fst p) = start (\snd p))
- (same_bases (\fst p) (\snd p))
+ (same_bases (bars (\fst p)) (bars (\snd p)))
(same_values l1 (\fst p))
(same_values l2 (\snd p)).
definition rebase_spec_simpl ≝
λstart.λl1,l2:list bar.λp:(list bar) × (list bar).
And3
- (same_bases (mk_q_f start (\fst p)) (mk_q_f start (\snd p)))
+ (same_bases (\fst p) (\snd p))
(same_values (mk_q_f start l1) (mk_q_f start (\fst p)))
(same_values (mk_q_f start l2) (mk_q_f start (\snd p))).
coercion eject.
definition inject ≝ λP.λp:(list bar) × (list bar).λh:P p. ex_introT ? P p h.
coercion inject with 0 1 nocomposites.
+
+axiom devil : False.
definition rebase: rebase_spec.
intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
[1,2: assumption;
|3: intro; apply (H3 input);
|4: intro; rewrite > H in H4;
- rewrite > (H4 input); reflexivity;]
+ rewrite > (H4 input) in ⊢ (? ? % ?); reflexivity;]
|2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
assumption]
[1: reflexivity
|2: assumption;
|3: assumption;
- |4: intro; rewrite < (H4 input); clear H3 H4 H2 w;
- cases (value (mk_q_f s1 l2') input);
- cases (q_cmp input (start (mk_q_f s1 l2'))) in H1;
- whd in ⊢ (% → ?);
- [1: intros; cases H2; clear H2; whd in ⊢ (??? %);
- cases (value (mk_q_f s2 l2) input);
- cases (q_cmp input (start (mk_q_f s2 l2))) in H2;
- whd in ⊢ (% → ?);
- [1: intros; cases H6; clear H6; change with (w1 = w);
-
- (* TODO *) ]]
+ |4: intro;
+ rewrite > (initial_shift_same_values (mk_q_f s2 l2) s1 H input) in ⊢ (? ? % ?);
+ rewrite < (H4 input)in ⊢ (? ? ? %); reflexivity;]
+ |3: letin l1' ≝ (〈\fst (unpos (s1-s2) ?),OQ〉::l1);[
+ apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption]
+ cases (aux l1' l2 (S (len l1' + len l2)));
+ cases (H1 s2 (le_n ?)); clear H1 aux;
+ exists [apply 〈mk_q_f s2 (\fst w), mk_q_f s2 (\snd w)〉] split;
+ [1: reflexivity
+ |2: assumption;
+ |4: assumption;
+ |3: intro; simplify in ⊢ (? ? ? (? ? ? (? ? ? (? % ?))));
+ rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input) in ⊢ (? ? % ?);
+ rewrite < (H3 input) in ⊢ (? ? ? %); reflexivity;]]
|1,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
assumption;
-|3:(* TODO *)
-|4:(* TODO *)
-|5:(* TODO *)
-|6:(* TODO *)
-|7:(* TODO *)
-|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);]
+|8: intros; cases (?:False); apply (not_le_Sn_O ? H1);
+|3: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ simplify in ⊢ (???? (??? (??? (??? (?? (? (?? (??? % ?) ?) ??)))) ?));
+ simplify in ⊢ (???? (???? (??? (??? (?? (? (?? (??? % ?) ?) ??))))));
+ clear H4; cases (aux (〈w,\snd b〉::l4) l5 n1); clear aux;
+ cut (len (〈w,\snd b〉::l4) + len l5 < n1) as K;[2:
+ simplify in H5; simplify; rewrite > sym_plus in H5; simplify in H5;
+ rewrite > sym_plus in H5; apply le_S_S_to_le; apply H5;]
+ split;
+ [1: simplify in ⊢ (? % ?); simplify in ⊢ (? ? %);
+ cases (H4 s K); clear K H4; intro input; cases input; [reflexivity]
+ simplify; apply H7;
+ |2: simplify in ⊢ (? ? %); cases (H4 s K); clear H4 K H5 spec;
+ intro;
+ (* input < s + b1 || input >= s + b1 *)
+ |3: simplify in ⊢ (? ? %);]
+|4: intros; generalize in match (unpos ??); intro X; cases X; clear X;
+ (* duale del 3 *)
+|5: intros; (* triviale, caso in cui non fa nulla *)
+|6,7: (* casi base in cui allunga la lista più corta *)
+]
+elim devil;
qed.
+
+include "Q/q/qtimes.ma".
+
+let rec area (l:list bar) on l ≝
+ match l with
+ [ nil ⇒ OQ
+ | cons he tl ⇒ area tl + Qpos (\fst he) * ⅆ[OQ,\snd he]].
+
+alias symbol "pi1" = "exT \fst".
+alias symbol "minus" = "Q minus".
+alias symbol "exists" = "CProp exists".
+definition minus_spec_bar ≝
+ λf,g,h:list bar.
+ same_bases f g → len f = len g →
+ ∀s,i:ℚ. \snd (\fst (value (mk_q_f s h) i)) =
+ \snd (\fst (value (mk_q_f s f) i)) - \snd (\fst (value (mk_q_f s g) i)).
+
+definition minus_spec ≝
+ λf,g:q_f.
+ ∃h:q_f.
+ ∀i:ℚ. \snd (\fst (value h i)) =
+ \snd (\fst (value f i)) - \snd (\fst (value g i)).
+
+definition eject_bar : ∀P:list bar → CProp.(∃l:list bar.P l) → list bar ≝
+ λP.λp.match p with [ex_introT x _ ⇒ x].
+definition inject_bar ≝ ex_introT (list bar).
+
+coercion inject_bar with 0 1 nocomposites.
+coercion eject_bar with 0 0 nocomposites.
+
+lemma minus_q_f : ∀f,g. minus_spec f g.
+intros;
+letin aux ≝ (
+ let rec aux (l1, l2 : list bar) on l1 ≝
+ match l1 with
+ [ nil ⇒ []
+ | cons he1 tl1 ⇒
+ match l2 with
+ [ nil ⇒ []
+ | cons he2 tl2 ⇒ 〈\fst he1, \snd he1 - \snd he2〉 :: aux tl1 tl2]]
+ in aux : ∀l1,l2 : list bar.∃h.minus_spec_bar l1 l2 h);
+[2: intros 4; simplify in H3; destruct H3;
+|3: intros 4; simplify in H3; cases l1 in H2; [2: intro X; simplify in X; destruct X]
+ intros; rewrite > (value_OQ_e (mk_q_f s []) i); [2: reflexivity]
+ rewrite > q_elim_minus; rewrite > q_plus_OQ; reflexivity;
+|1: cases (aux l2 l3); unfold in H2; intros 4;
+ simplify in ⊢ (? ? (? ? ? (? ? ? (? % ?))) ?);
+ cases (q_cmp i (s + Qpos (\fst b)));
+
+
+
+definition excess ≝
+ λf,g.∃i.\snd (\fst (value f i)) < \snd (\fst (value g i)).
+
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