(**************************************************************************)
include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.ma".
-lemma key:
- ∀n,m,l.
- sum_bases l n < sum_bases l (S m) →
- sum_bases l m < sum_bases l (S n) →
- n = m.
-intros 2; apply (nat_elim2 ???? n m);
-[1: intro X; cases X; intros; [reflexivity] cases (?:False);
- cases l in H H1; simplify; intros;
- apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
- apply (q_lt_canc_plus_r ??? H1);
-|2: intros 2; cases l; simplify; intros; cases (?:False);
- apply (q_lt_le_incompat ??? (sum_bases_ge_OQ ? n1));
- apply (q_lt_canc_plus_r ??? H); (* magia ... *)
-|3: intros 4; cases l; simplify; intros;
- [1: rewrite > (H []); [reflexivity]
- apply (q_lt_canc_plus_r ??(Qpos one)); assumption;
- |2: rewrite > (H l1); [reflexivity]
- apply (q_lt_canc_plus_r ??(Qpos (\fst b))); assumption;]]
-qed.
-
-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 Hv1 (HV1 HV1 HV1 HV1); cases HV1 (Hi1 Hv11 Hv12); clear HV1 Hv1;
-[1: cut (input < start l1) as K;[2: apply (q_lt_trans ??? Hi1 H)]
- rewrite > (value_OQ_l ?? K); simplify; symmetry; assumption;
-|2: cut (start l1 + sum_bases (bars l1) (len (bars l1)) ≤ input) as K;[2:
- simplify in Hi1; apply (q_le_trans ???? Hi1); rewrite > H2;
- rewrite > q_plus_sym in ⊢ (? ? (? ? %));
- rewrite > q_plus_assoc; rewrite > q_elim_minus;
- rewrite > q_plus_sym in ⊢ (? ? (? (? ? %) ?));
- rewrite > q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_sym in ⊢ (? ? (? % ?));
- rewrite > q_plus_OQ; apply q_eq_to_le; reflexivity;]
- rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
-|3: simplify in Hi1; destruct Hi1;
-|4: cases (q_cmp input (start l1));
- [2: rewrite > (value_OQ_l ?? H4);
- change with (OQ = \snd v1); rewrite > Hv12;
- cases H3; clear H3; simplify in H5; cases (\fst v1) in H5;[intros;reflexivity]
- simplify; rewrite > q_d_sym; rewrite > q_d_noabs; [2:cases Hi1; apply H5]
- rewrite > H2; do 2 rewrite > q_elim_minus;rewrite > q_plus_assoc;
- intro X; lapply (q_le_canc_plus_r ??? X) as Y; clear X;
- (* OK *)
- |1,3: cases Hi1; clear Hi1; cases H3; clear H3;
- simplify in H5 H6 H8 H9 H7:(? ? (? % %)) ⊢ (? ? ? (? ? ? %));
- generalize in match (refl_eq ? (bars l1):bars l1 = bars l1);
- generalize in ⊢ (???% → ?); intro X; cases X; clear X; intro Hb;
- [1,3: rewrite > (value_OQ_e ?? Hb); rewrite > Hv12; rewrite > Hb in Hv11 ⊢ %;
- simplify in Hv11 ⊢ %; cases (\fst v1) in Hv11; [1,3:intros; reflexivity]
- cases n; [1,3: intros; reflexivity] intro X; cases (not_le_Sn_O ? (le_S_S_to_le ?? X));
- |2,4: cases (value_ok l1 input);
- [1,5: rewrite > Hv12; rewrite > Hb; clear Hv12; simplify;
- rewrite > H10; rewrite > Hb;
- cut (O < \fst v1);[2,4: cases (\fst v1) in H9; intros; [2,4: autobatch]
- cases (?:False); generalize in match H9;
- rewrite > q_d_sym; rewrite > q_d_noabs; [2,4: assumption]
- rewrite > H2; simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- repeat rewrite > q_elim_minus;
- intro X; lapply (q_lt_canc_plus_r ??? X) as Y;
- apply (q_lt_le_incompat ?? Y);
- [apply q_eq_to_le;symmetry|apply q_lt_to_le] assumption;]
- cases (\fst v1) in H8 H9 Hcut; [1,3:intros (_ _ X); cases (not_le_Sn_O ? X)]
- intros; clear H13; simplify;
- rewrite > (key n n1 (b::l)); [1,4: reflexivity] rewrite < Hb;
- [2,4: simplify in H8; apply (q_le_lt_trans ??? (q_le_plus_r ??? H8));
- apply (q_le_lt_trans ???? H12); rewrite > H2;
- rewrite > q_d_sym; rewrite > q_d_noabs; [2,4: assumption]
- rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
- rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
- rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? (? % ?) ?);
- rewrite > (q_plus_sym ? init); do 2 rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_OQ;
- rewrite > q_d_sym; rewrite > q_d_noabs;
- [2,4: [apply q_eq_to_le; symmetry|apply q_lt_to_le] assumption]
- apply q_eq_to_le; reflexivity;
- |*: apply (q_le_lt_trans ??? H11);
- rewrite > q_d_sym; rewrite > q_d_noabs;
- [2,4: [apply q_eq_to_le; symmetry|apply q_lt_to_le] assumption]
- generalize in match H9; rewrite > q_d_sym; rewrite > q_d_noabs;
- [2,4: assumption]
- rewrite > H2; intro X;
- lapply (q_lt_inj_plus_r ?? (Qopp (start l1-init)) X) as Y; clear X;
- rewrite < q_plus_assoc in Y; repeat rewrite < q_elim_minus in Y;
- rewrite > q_plus_minus in Y; rewrite > q_plus_OQ in Y;
- apply (q_le_lt_trans ???? Y);
- rewrite > (q_elim_minus (start l1) init); rewrite > q_minus_distrib;
- rewrite > q_elim_opp; repeat rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite > (q_plus_sym ? init);
- rewrite > q_plus_assoc;rewrite < q_plus_assoc in ⊢ (? ? (? % ?));
- rewrite > (q_plus_sym ? init); rewrite < (q_elim_minus init);
- rewrite > q_plus_minus; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity;]
- |2,6: rewrite > Hb; intro W; destruct W;
- |3,7: [apply q_eq_to_le;symmetry|apply q_lt_to_le] assumption;
- |4,8: apply (q_lt_le_trans ??? H7); rewrite > H2;
- rewrite > q_plus_sym; rewrite < q_plus_assoc;
- rewrite > q_plus_sym; apply q_le_inj_plus_r;
- apply q_le_minus; apply q_eq_to_le; reflexivity;]]]
-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)).
+
projects / helm.git / blobdiff