(* *)
(**************************************************************************)
-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.
-
-record q_f : Type ≝ {
- start : ℚ;
- bars: list (ℚ × ℚ) (* base, height *)
-}.
-
-axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-
-axiom qm : ℚ → ℚ → ℚ.
-
-interpretation "Q minus" 'minus x y = (qm x y).
-
-axiom qlt : ℚ → ℚ → CProp.
-
-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" 'le x y = (qle x y).
-
-notation "'nth'" with precedence 90 for @{'nth}.
-notation < "'nth' \nbsp l \nbsp d \nbsp i" with precedence 71
-for @{'nth_appl $l $d $i}.
-interpretation "list nth" 'nth = (cic:/matita/list/list/nth.con _).
-interpretation "list nth" 'nth_appl l d i = (cic:/matita/list/list/nth.con _ l d i).
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-let rec make_list (A:Type) (def:nat→A) (n:nat) on n ≝
- match n with
- [ O ⇒ nil ?
- | S m ⇒ def m :: make_list A def m].
-
-notation "'mk_list'" with precedence 90 for @{'mk_list}.
-interpretation "'mk_list'" 'mk_list = (make_list _).
-notation < "'mk_list' \nbsp f \nbsp n"
-with precedence 71 for @{'mk_list_appl $f $n}.
-interpretation "'mk_list' appl" 'mk_list_appl f n = (make_list _ f n).
-
-definition q0 : ℚ × ℚ ≝ 〈OQ,OQ〉.
-notation < "0 \sub \rationals" with precedence 90 for @{'q0}.
-interpretation "q0" 'q0 = q0.
-
-notation < "[ \rationals \sup 2]" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list (Prod Q Q)).
-notation < "[ \rationals \sup 2] \sup 2" with precedence 90 for @{'lq22}.
-interpretation "lq22" 'lq22 = (Prod (list (Prod Q Q)) (list (Prod Q Q))).
-
-
-notation "'len'" with precedence 90 for @{'len}.
-interpretation "len" 'len = length.
-notation < "'len' \nbsp l" with precedence 70 for @{'len_appl $l}.
-interpretation "len appl" 'len_appl l = (length _ l).
-
-definition eject ≝
- λP.λp:∃x:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).P x.match p with [ex_introT p _ ⇒ p].
-coercion cic:/matita/dama/models/q_function/eject.con.
-definition inject ≝
- λP.λp:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).λh:P p. ex_introT ? P p h.
-(*coercion inject with 0 1 nocomposites.*)
-coercion cic:/matita/dama/models/q_function/inject.con 0 1 nocomposites.
-
-definition cb0h ≝ (λl.mk_list (λi.〈\fst (nth l q0 i),OQ〉) (length ? l)).
-
+include "nat_ordered_set.ma".
+include "models/q_bars.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:
- q_f → q_f →
- ∃p:q_f × q_f.∀i.
- \fst (nth (bars (\fst p)) q0 i) =
- \fst (nth (bars (\snd p)) q0 i).
-intros (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
-letin spec ≝ (λl1,l2:list (ℚ × ℚ).λm:nat.λz:(list (ℚ × ℚ)) × (list (ℚ × ℚ)).True);
+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_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_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))).
+
+(* a local letin makes russell fail *)
+definition cb0h : list bar → list bar ≝
+ λl.mk_list (λi.〈\fst (nth l ▭ i),OQ〉) (len l).
+
+definition eject ≝
+ λP.λp:∃x:(list bar) × (list bar).P x.match p with [ex_introT p _ ⇒ 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.
+
+definition rebase: rebase_spec.
+intros 2 (f1 f2); cases f1 (s1 l1); cases f2 (s2 l2); clear f1 f2;
+letin spec ≝ (
+ λs.λl1,l2.λm.λz.len l1 + len l2 < m → rebase_spec_simpl s l1 l2 z);
+alias symbol "pi1" (instance 34) = "exT \fst".
+alias symbol "pi1" (instance 21) = "exT \fst".
letin aux ≝ (
-let rec aux (l1,l2:list (ℚ × ℚ)) (n:nat) on n : (list (ℚ × ℚ)) × (list (ℚ × ℚ)) ≝
+let rec aux (l1,l2:list bar) (n:nat) on n : (list bar) × (list bar) ≝
match n with
[ O ⇒ 〈 nil ? , nil ? 〉
| S m ⇒
match l2 with
[ nil ⇒ 〈l1, cb0h l1〉
| cons he2 tl2 ⇒
- let base1 ≝ (\fst he1) in
- let base2 ≝ (\fst he2) in
+ let base1 ≝ Qpos (\fst he1) in
+ let base2 ≝ Qpos (\fst he2) in
let height1 ≝ (\snd he1) in
let height2 ≝ (\snd he2) in
match q_cmp base1 base2 with
- [ q_eq _ ⇒
+ [ q_eq _ ⇒
let rc ≝ aux tl1 tl2 m in
- 〈he1 :: \fst rc,he2 :: \snd rc〉
- | q_lt _ ⇒
+ 〈he1 :: \fst rc,he2 :: \snd rc〉
+ | q_lt Hp ⇒
let rest ≝ base2 - base1 in
- let rc ≝ aux tl1 (〈rest,height2〉 :: tl2) m in
- 〈〈base1,height1〉 :: \fst rc,〈base1,height2〉 :: \snd rc〉
- | q_gt _ ⇒
+ let rc ≝ aux tl1 (〈\fst (unpos rest ?),height2〉 :: tl2) m in
+ 〈〈\fst he1,height1〉 :: \fst rc,〈\fst he1,height2〉 :: \snd rc〉
+ | q_gt Hp ⇒
let rest ≝ base1 - base2 in
- let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
- 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉
+ let rc ≝ aux (〈\fst (unpos rest ?),height1〉 :: tl1) tl2 m in
+ 〈〈\fst he2,height1〉 :: \fst rc,〈\fst he2,height2〉 :: \snd rc〉
]]]]
-in aux : ∀l1,l2,m.∃z.spec l1 l2 m z);
-qed.
\ No newline at end of file
+in aux : ∀l1,l2,m.∃z.∀s.spec s l1 l2 m z); unfold spec;
+[9: clearbody aux; unfold spec in aux; clear spec;
+ cases (q_cmp s1 s2);
+ [1: cases (aux l1 l2 (S (len l1 + len l2)));
+ cases (H1 s1 (le_n ?)); clear H1;
+ exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s2 (\snd w)〉] split;
+ [1,2: assumption;
+ |3: intro; apply (H3 input);
+ |4: intro; rewrite > H in H4;
+ rewrite > (H4 input); reflexivity;]
+ |2: letin l2' ≝ (〈\fst (unpos (s2-s1) ?),OQ〉::l2);[
+ apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ;
+ assumption]
+ cases (aux l1 l2' (S (len l1 + len l2')));
+ cases (H1 s1 (le_n ?)); clear H1 aux;
+ exists [apply 〈mk_q_f s1 (\fst w), mk_q_f s1 (\snd w)〉] split;
+ [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 *) ]]
+|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);]
+qed.
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