include "nat_ordered_set.ma".
include "models/q_bars.ma".
-axiom le_le_eq: ∀x,y:Q. x ≤ y → y ≤ x → x = y.
+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 hide; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; assumption]
+[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);
-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 (mk_q_f init (〈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 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 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 4 (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;
-[1: rewrite > nth_nil; cases w1 in H4;
- [1: 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 ??? H2);
- apply (q_lt_le_trans ??? Y); rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_eq_to_le; reflexivity;
- |2: intros; simplify; rewrite > nth_nil; reflexivity;]
-|2: FACTORIZE w1>0
-
- (* interesting case: init < start < input *)
- intro; cases H8; clear H8; rewrite > H11; rewrite > H7; clear H11 H7;
- simplify in H5 H6 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
- elim (\fst w2) in H9 H10;
- [1: elim (\fst w1) in H5 H6;
- [1: cases (?:False); clear H5 H8 H7;
- apply (q_lt_antisym input (start l1)); [2: assumption]
- rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6;
- [2: apply q_lt_to_le; assumption]
- rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6;
- rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init));
- do 2 rewrite < q_elim_minus; assumption;
- |2:
-
- cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
- cases (\fst w1) in H5 H6; intros; [1:
- cases (?:False); clear H5 H9 H10;
- apply (q_lt_antisym input (start l1)); [2: assumption]
- rewrite > q_d_sym in H6; rewrite > q_d_noabs in H6;
- [2: apply q_lt_to_le; assumption]
- rewrite > q_plus_sym in H6; rewrite > q_plus_OQ in H6;
- rewrite > H2 in H6; apply (q_lt_canc_plus_r ?? (Qopp init));
- do 2 rewrite < q_elim_minus; assumption;]
- apply eq_f;
- cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n));[2:
- apply (q_le_lt_trans ??? H9);
- apply (q_lt_trans ??? ? H6);
- 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_lt_to_le;assumption]
- do 2 rewrite > q_elim_minus; rewrite > (q_plus_sym ? (Qopp init));
- apply q_lt_plus; rewrite > q_plus_sym;
- rewrite > q_elim_minus; rewrite < q_plus_assoc;
- rewrite < q_elim_minus; rewrite > q_plus_minus;
- rewrite > q_plus_OQ; apply q_lt_opp_opp; assumption]
- clear H9 H6;
- cut (ⅆ[input,init] - Qpos w = ⅆ[input,start l1]);[2:
- 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_lt_to_le;assumption]
- rewrite > H2; rewrite > (q_elim_minus (start ?));
- rewrite > q_minus_distrib; rewrite > q_elim_opp;
- do 2 rewrite > q_elim_minus;
- do 2 rewrite < q_plus_assoc;
- rewrite > (q_plus_sym ? init);
- rewrite > (q_plus_assoc ? init);
- rewrite > (q_plus_sym ? init);
- rewrite < (q_elim_minus init); rewrite > q_plus_minus;
- rewrite > (q_plus_sym OQ); rewrite > q_plus_OQ;
- rewrite < q_elim_minus; reflexivity;]
- cut (sum_bases (bars l1) n < sum_bases (bars l1) (S (\fst w2)));[2:
- apply (q_le_lt_trans ???? H10); rewrite < Hcut1;
- rewrite > q_elim_minus; apply q_le_minus_r; rewrite > q_elim_opp;
- assumption;] clear Hcut1 H5 H10;
- generalize in match Hcut;generalize in match Hcut2;clear Hcut Hcut2;
- apply (nat_elim2 ???? n (\fst w2));
- [3: intros (x y); apply eq_f; apply H5; clear H5;
- [1: clear H7; apply sum_bases_lt_canc; assumption;
- |2: clear H6; ]
- |2: intros; cases (?:False); clear H6;
- cases n1 in H5; intro;
- [1: apply (q_lt_corefl ? H5);
- |2: cases (bars l1) in H5; intro;
- [1: simplify in H5;
- apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
- apply q_le_plus_trans; [apply sum_bases_ge_OQ]
- apply q_le_OQ_Qpos;
- |2: simplify in H5:(??%);
- lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
- apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
- |1: intro; cases n1 [intros; reflexivity] intros; cases (?:False);
- elim n2 in H5 H6;
-
-
- elim (bars l1) 0;
- [1: intro; elim n1; [reflexivity] cases (?:False);
-
-
- intros; clear H5;
- elim n1 in H6; [reflexivity] cases (?:False);
- [1: apply (q_lt_corefl ? H5);
- |2: cases (bars l1) in H5; intro;
- [1: simplify in H5;
- apply (q_lt_le_incompat ?? (q_lt_canc_plus_r ??? H5));
- apply q_le_plus_trans; [apply sum_bases_ge_OQ]
- apply q_le_OQ_Qpos;
- |2: simplify in H5:(??%);
- lapply (q_lt_canc_plus_r (sum_bases l (S n2)) ?? H5) as X;
- apply (q_lt_le_incompat ?? X); apply sum_bases_ge_OQ]]
+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.
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