(* *)
(**************************************************************************)
-include "models/q_function.ma".
-
+include "dama/russell_support.ma".
+include "models/q_copy.ma".
+(*
+definition rebase_spec ≝
+ λl1,l2:q_f.λp:q_f × q_f.
+ And3
+ (same_bases (bars (\fst p)) (bars (\snd p)))
+ (same_values l1 (\fst p))
+ (same_values l2 (\snd p)).
+
+inductive rebase_cases : list bar → list bar → (list bar) × (list bar) → Prop ≝
+| rb_fst_full : ∀b,h1,xs.
+ rebase_cases (〈b,h1〉::xs) [] 〈〈b,h1〉::xs,〈b,〈OQ,OQ〉〉::copy xs〉
+| rb_snd_full : ∀b,h1,ys.
+ rebase_cases [] (〈b,h1〉::ys) 〈〈b,〈OQ,OQ〉〉::copy ys,〈b,h1〉::ys〉
+| rb_all_full : ∀b,h1,h2,h3,h4,xs,ys,r1,r2.
+ \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h3〉::r1)) →
+ \snd(\last ▭ (〈b,h2〉::ys)) = \snd(\last ▭ (〈b,h4〉::r2)) →
+ rebase_cases (〈b,h1〉::xs) (〈b,h2〉::ys) 〈〈b,h3〉::r1,〈b,h4〉::r2〉
+| rb_all_full_l : ∀b1,b2,h1,h2,xs,ys,r1,r2.
+ \snd(\last ▭ (〈b1,h1〉::xs)) = \snd(\last ▭ (〈b1,h1〉::r1)) →
+ \snd(\last ▭ (〈b2,h2〉::ys)) = \snd(\last ▭ (〈b1,h2〉::r2)) →
+ b1 < b2 →
+ rebase_cases (〈b1,h1〉::xs) (〈b2,h2〉::ys) 〈〈b1,h1〉::r1,〈b1,〈OQ,OQ〉〉::r2〉
+| rb_all_full_r : ∀b1,b2,h1,h2,xs,ys,r1,r2.
+ \snd(\last ▭ (〈b1,h1〉::xs)) = \snd(\last ▭ (〈b2,h1〉::r1)) →
+ \snd(\last ▭ (〈b2,h2〉::ys)) = \snd(\last ▭ (〈b2,h2〉::r2)) →
+ b2 < b1 →
+ rebase_cases (〈b1,h1〉::xs) (〈b2,h2〉::ys) 〈〈b2,〈OQ,OQ〉〉::r1,〈b2,h2〉::r2〉
+| rb_all_empty : rebase_cases [] [] 〈[],[]〉.
+
+alias symbol "pi2" = "pair pi2".
+alias symbol "pi1" = "pair pi1".
+alias symbol "leq" = "natural 'less or equal to'".
+inductive rebase_spec_aux_p (l1, l2:list bar) (p:(list bar) × (list bar)) : Prop ≝
+| prove_rebase_spec_aux:
+ rebase_cases l1 l2 p →
+ (sorted q2_lt (\fst p)) →
+ (sorted q2_lt (\snd p)) →
+ (same_bases (\fst p) (\snd p)) →
+ (same_values_simpl l1 (\fst p)) →
+ (same_values_simpl l2 (\snd p)) →
+ rebase_spec_aux_p l1 l2 p.
+
+lemma aux_preserves_sorting:
+ ∀b,b3,l2,l3,w. rebase_cases l2 l3 w →
+ sorted q2_lt (b::l2) → sorted q2_lt (b3::l3) → \fst b3 = \fst b →
+ sorted q2_lt (\fst w) → sorted q2_lt (\snd w) →
+ same_bases (\fst w) (\snd w) →
+ sorted q2_lt (b :: \fst w).
+intros 6; cases H; simplify; intros; clear H;
+[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H1);
+| apply (sorted_cons q2_lt); [2:assumption]
+ whd; rewrite < H3; apply (inversion_sorted2 ??? H2);
+| apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H3);
+| apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H4);
+| apply (sorted_cons q2_lt); [2:assumption]
+ whd; rewrite < H6; apply (inversion_sorted2 ??? H5);
+| apply (sorted_one q2_lt);]
+qed.
+
+lemma aux_preserves_sorting2:
+ ∀b,b3,l2,l3,w. rebase_cases l2 l3 w →
+ sorted q2_lt (b::l2) → sorted q2_lt (b3::l3) → \fst b3 = \fst b →
+ sorted q2_lt (\fst w) → sorted q2_lt (\snd w) → same_bases (\fst w) (\snd w) →
+ sorted q2_lt (b :: \snd w).
+intros 6; cases H; simplify; intros; clear H;
+[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H1);
+| apply (sorted_cons q2_lt); [2:assumption]
+ whd; rewrite < H3; apply (inversion_sorted2 ??? H2);
+| apply (sorted_cons q2_lt); [2: assumption] apply (inversion_sorted2 ??? H3);
+| apply (sorted_cons q2_lt); [2: assumption] apply (inversion_sorted2 ??? H4);
+| apply (sorted_cons q2_lt); [2: assumption]
+ whd; rewrite < H6; apply (inversion_sorted2 ??? H5);
+| apply (sorted_one q2_lt);]
+qed.
+*)
+
+
+
+definition rebase_spec_aux ≝
+ λl1,l2
+ :list bar.λp:(list bar) × (list bar).
+ sorted q2_lt l1 → (\snd (\last ▭ l1) = 〈OQ,OQ〉) →
+ sorted q2_lt l2 → (\snd (\last ▭ l2) = 〈OQ,OQ〉) →
+ rebase_spec_aux_p l1 l2 p.
+
+alias symbol "lt" = "Q less than".
+alias symbol "Q" = "Rationals".
+axiom q_unlimited: ∀x:ℚ.∃y:ratio.x<Qpos y.
+axiom q_halving: ∀x,y:ℚ.∃z:ℚ.x<z ∧ z<y.
+alias symbol "not" = "logical not".
+axiom q_not_OQ_lt_Qneg: ∀r. ¬ (OQ < Qneg r).
+lemma same_values_unit_OQ:
+ ∀b1,b2,h1,l. OQ < b2 → b2 < b1 → sorted q2_lt (〈b1,h1〉::l) →
+ sorted q2_lt [〈b2,〈OQ,OQ〉〉] →
+ same_values_simpl (〈b1,h1〉::l) [〈b2,〈OQ,OQ〉〉] → h1 = 〈OQ,OQ〉.
+intros 5 (b1 b2 h1 l POS); cases l;
+[1: intros; cases (q_unlimited b1); cut (b2 < Qpos w); [2:apply (q_lt_trans ??? H H4);]
+ lapply (H3 H1 ? H2 ? w H4 Hcut) as K; simplify; [1,2: autobatch]
+ rewrite > (value_unit 〈b1,h1〉) in K;
+ rewrite > (value_unit 〈b2,〈OQ,OQ〉〉) in K; assumption;
+|2: intros; unfold in H3; lapply depth=0 (H3 H1 ? H2 ?) as K; [1,2:simplify; autobatch]
+ clear H3; cases (q_halving b1 (b1 + \fst p)) (w Pw); cases w in Pw; intros;
+ [cases (q_lt_le_incompat ?? POS); apply q_lt_to_le; cases H3;
+ apply (q_lt_trans ???? H4); assumption;
+ |3: elim H3; lapply (q_lt_trans ??? H H4); lapply (q_lt_trans ??? POS Hletin);
+ cases (q_not_OQ_lt_Qneg ? Hletin1);
+ | cases H3; lapply (K r);
+ [2: simplify; assumption
+ |3: simplify; apply (q_lt_trans ???? H4); assumption;
+ |rewrite > (value_head b1,h1 .. q) in Hletin;
+
+
+
+ (* MANCA che le basi sono positive,
+ poi con halving prendi tra b1 e \fst p e hai h1=OQ,OQ*)
+
+
definition eject ≝
λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ p].
coercion eject.
[7: clearbody aux; cases (aux b1 b2 (\len b1 + \len b2)) (res Hres);
exists; [split; constructor 1; [apply (\fst res)|5:apply (\snd res)]]
[1,4: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); assumption;
- |2,5: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); lapply (H5 O) as K;
- clear H1 H2 H3 H4 H5 H6 H7 Hres aux; unfold nth_base;
+ |2,5: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres aux;
+ lapply (H3 O) as K; clear H1 H2 H3 H4 H5; unfold nth_base;
cases H in K He1 He2 Hb1 Hb2; simplify; intros; assumption;
- |3,6: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2);
+ |3,6: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres aux;
cases H in He1 He2; simplify; intros;
[1,6,8,12: assumption;
|2,7: rewrite > len_copy; generalize in match (\len ?); intro X;
cases X; [1,3: reflexivity] simplify;
[apply (copy_OQ ys n);|apply (copy_OQ xs n);]
- |3,4,5: rewrite < H8; assumption;
- |9,10,11: rewrite < H9; assumption]]
- split; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); unfold same_values; intros;
+ |3,4: rewrite < H6; assumption;
+ |5: cases r1 in H6; simplify; intros; [reflexivity] rewrite < H6; assumption;
+ |9,11: rewrite < H7; assumption;
+ |10: cases r2 in H7; simplify; intros; [reflexivity] rewrite < H7; assumption]]
+ split; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres; unfold same_values; intros;
[1: assumption
|2,3: simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉);
apply same_values_simpl_to_same_values; assumption]
|3: cut (\fst b3 = \fst b) as E; [2: apply q_le_to_le_to_eq; assumption]
- clear H6 H5 H4 H3 He2 Hb2 Hs2 b2 He1 Hb1 Hs1 b1; cases (aux l2 l3 n1);
+ clear H6 H5 H4 H3 He2 Hb2 Hs2 b2 He1 Hb1 Hs1 b1; cases (aux l2 l3 n1) (rc Hrc);
clear aux; intro K; simplify in K; rewrite <plus_n_Sm in K;
lapply le_S_S_to_le to K as W; lapply lt_to_le to W as R;
- simplify in match (? ≪w,H3≫); intros; cases (H3 R); clear H3 R W K;
+ simplify in match (? ≪rc,Hrc≫); intros (Hsbl2 Hendbl2 Hsb3l3 Hendb3l3);
+ change in Hendbl2 with (\snd (\last ▭ (b::l2)) = 〈OQ,OQ〉);
+ change in Hendb3l3 with (\snd (\last ▭ (b3::l3)) = 〈OQ,OQ〉);
+ cases (Hrc R) (RC S1 S2 SB SV1 SV2); clear Hrc R W K;
[2,4: apply (sorted_tail q2_lt);[apply b|3:apply b3]assumption;
- |3: cases l2 in H5; simplify; intros; try reflexivity; assumption;
- |5: cases l3 in H7; simplify; intros; try reflexivity; assumption;]
+ |3: cases l2 in Hendbl2; simplify; intros; [reflexivity] assumption;
+ |5: cases l3 in Hendb3l3; simplify; intros; [reflexivity] assumption;]
constructor 1; simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉);
- [1: cases b in E H5 H7 H11 H14; cases b3; intros (E H5 H7 H11 H14); simplify in E;
- destruct E; constructor 3;
- [ cases H8 in H5 H7; intros; [1,6:reflexivity|3,4,5: assumption;]
- simplify; rewrite > H3; rewrite > len_copy; elim (\len ys); [reflexivity]
+ [1: cases b in E Hsbl2 Hendbl2; cases b3 in Hsb3l3 Hendb3l3; intros (Hsbl3 Hendbl2 E Hsb3l2 Hendb3l3);
+ simplify in E; destruct E; constructor 3;
+ [1: clear Hendbl2 Hsbl3 SV2 SB S2;
+ cases RC in S1 SV1 Hsb3l2 Hendb3l3; intros;
+ [1,6: reflexivity;
+ |3,4: assumption;
+ |5: simplify in H6:(??%) ⊢ %; rewrite > H3; cases r1 in H6; intros [2:reflexivity]
+ use same_values_unit_OQ;
+
+ |2: simplify in H3:(??%) ⊢ %; rewrite > H3; rewrite > len_copy; elim (\len ys); [reflexivity]
symmetry; apply (copy_OQ ys n2);
| cases H8 in H5 H7; simplify; intros; [2,6:reflexivity|3,4,5: assumption]
simplify; rewrite > H5; rewrite > len_copy; elim (\len xs); [reflexivity]
|4: apply I
|5: apply I
|6: intro; elim i; intros; simplify; solve [symmetry;assumption|apply H13]
- |7: unfold; intros; cases H8 in H13 H14 H15 Hi1 Hi2 H17 H18 H3 H16; intros;
- [1: simplify in match (\fst 〈?,?〉) in H16 H17 H20 H21 H22 H23 ⊢ %;
- cases (q_cmp (Qpos input) b1);
- [1: do 2 (rewrite > value_head; [id|assumption]); reflexivity;
- |2: do 2 (rewrite > value_tail;[id|assumption]); apply H16;]
- |2: simplify in match (\fst 〈?,?〉) in H16 H17 H20 H21 H22 H23 ⊢ %;
- cases (q_cmp (Qpos input) b1);
- [1: rewrite > value_head; [2:assumption]
- |2: rewrite > value_tail;[2:assumption]
- simplify in H15;
+ |7: unfold; intros; clear H9 H10 H11 H12 H13; simplify in Hi1 Hi2 H16 H18;
+ cases H8 in H14 H15 H17 H3 H16 H18 H5 H6;
+ simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉); intros;
+ [1: reflexivity;
+ |2: simplify in H3; rewrite > (value_unit b); rewrite > H3; symmetry;
+ cases b in H3 H12 Hi2; intros 2; simplify in H12; rewrite > H12;
+ intros; change in ⊢ (? ? (? % ? ? ? ?) ?) with (copy (〈q,〈OQ,OQ〉〉::〈b1,〈OQ,OQ〉〉::ys));
+ apply (value_copy (〈q,〈OQ,OQ〉〉::〈b1,〈OQ,OQ〉〉::ys));
+ |3: apply (same_value_tail b b1 h1 h3 xs r1 input); assumption;
+ |4: apply (same_value_tail b b1 h1 h1 xs r1 input); assumption;
+ |5: simplify in H9; STOP
+
+ |6: reflexivity;]
+
]
|8:
projects / helm.git / blobdiff