-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "models/q_bars.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 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)]
- simplify in ⊢ (? ? ? (? ? ? %));
- 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;]
- simplify in ⊢ (? ? ? (? ? ? %));
- rewrite > (value_OQ_r ?? K); simplify; symmetry; assumption;
-|3: simplify in Hi1; destruct Hi1;
-|4: cut (start l1 ≤ input → \snd (\fst (value l1 input))=\snd v1) as solution;[2:
- intro H4; 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: rewrite > (value_OQ_e ?? Hb); rewrite > Hv12; rewrite > Hb in Hv11 ⊢ %;
- simplify in Hv11 ⊢ %; cases (\fst v1) in Hv11; [intros; reflexivity]
- cases n; [intros; reflexivity] intro X; cases (not_le_Sn_O ? (le_S_S_to_le ?? X));
- |2: cases (value_ok l1 input);
- [2: rewrite > Hb; intro W; destruct W;
- |3: assumption;
- |4: 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;
- |1: rewrite > Hv12; rewrite > Hb; clear Hv12; simplify;
- rewrite > H10; rewrite > Hb;
- cut (O < \fst v1);[2: cases (\fst v1) in H9; intros; [2: 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); assumption;]
- cases (\fst v1) in H8 H9 Hcut; [1:intros (_ _ X); cases (not_le_Sn_O ? X)]
- intros; clear H13; simplify;
- rewrite > (sum_bases_n_m n n1 (b::l)); [1,4: reflexivity] rewrite < Hb;
- [2: 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: 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:assumption]
- apply q_eq_to_le; reflexivity;
- |*: apply (q_le_lt_trans ??? H11);
- rewrite > q_d_sym; rewrite > q_d_noabs; [2:assumption;]
- generalize in match H9; rewrite > q_d_sym; rewrite > q_d_noabs; [2: 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;]]]]]
- cases (q_cmp input (start l1));
- [2: simplify in ⊢ (? ? ? (? ? ? %)); 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;
- cases (?:False); apply (q_lt_le_incompat input (start l1)); try assumption;
- apply (q_le_S ???? Y); try assumption; apply sum_bases_ge_OQ;
- |1: apply solution; apply (q_eq_to_le ?? (sym_eq ??? H4));
- |3: apply solution; apply (q_lt_to_le ?? H4);]
-qed.
-
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