(**************************************************************************)
include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.ma".
-lemma sum_bars_increasing:
- ∀l,x.sum_bases l x < sum_bases l (S x).
-intro; elim l;
-[1: elim x;
- [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_pos_lt_OQ;
- |2: simplify in H ⊢ %;
- 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: elim x;
- [1: simplify; rewrite > q_plus_sym; rewrite > q_plus_OQ;
- apply q_pos_lt_OQ;
- |2: simplify; change in ⊢ (? ? (? % ?)) with (sum_bases l1 (S n)) ;
- apply q_lt_plus; rewrite > q_elim_minus;
- rewrite < q_plus_assoc; rewrite < q_elim_minus;
- rewrite > q_plus_minus; rewrite > q_plus_OQ; apply H]]
-qed.
-
-lemma q_lt_canc_plus_r:
- ∀x,y,z:Q.x + z < y + z → x < y.
-intros; rewrite < (q_plus_OQ y); rewrite < (q_plus_minus z);
-rewrite > q_elim_minus; rewrite > q_plus_assoc;
-apply q_lt_plus; rewrite > q_elim_opp; assumption;
-qed.
-
-lemma q_lt_inj_plus_r:
- ∀x,y,z:Q.x < y → x + z < y + z.
-intros; apply (q_lt_canc_plus_r ?? (Qopp z));
-do 2 (rewrite < q_plus_assoc;rewrite < q_elim_minus);
-rewrite > q_plus_minus;
-do 2 rewrite > q_plus_OQ; assumption;
-qed.
-
-lemma sum_bases_lt_canc:
- ∀l,x,y.sum_bases l (S x) < sum_bases l (S y) → sum_bases l x < sum_bases l y.
-intro; elim l; [apply (q_lt_canc_plus_r ?? (Qpos one));apply H]
-generalize in match H1;apply (nat_elim2 (?:? → ? → CProp) ??? x y);
-intros 2;
-[3: intros 2; simplify; apply q_lt_inj_plus_r; apply H;
- apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H3;
-|2: cases (?:False); simplify in H2;
- apply (q_lt_le_incompat (sum_bases l1 (S n)) OQ);[2: apply sum_bases_ge_OQ;]
- apply (q_lt_canc_plus_r ?? (Qpos (\fst a))); apply H2;
-|1: cases n in H2; intro;
- [1: cases (?:False); apply (q_lt_corefl ? H2);
- |2: simplify; apply q_lt_plus_trans; [apply sum_bases_ge_OQ]
- apply q_pos_lt_OQ;]]
-qed.
-
-axiom q_minus_distrib:
- ∀x,y,z:Q.x - (y + z) = x - y - z.
-
-axiom q_le_OQ_Qpos: ∀x.OQ ≤ Qpos x.
-
-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]
-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 ⊢ (? ? ? (? ? ? (? ? % ? ?)));
- cut (\fst w1 = S (\fst w2)) as Key; [rewrite > Key; reflexivity;]
- simplify in H5 H6;
- cases (\fst w1) in H5 H6; intros;
- [1: cases (?:False); clear H5 H9 H10; simplify in H6;
- 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;
- lapply (q_lt_plus ??? H6) as X; clear H6 H2 H3 H1 H H4 w1 w2 w;
- rewrite > q_elim_minus in X; rewrite < q_plus_assoc in X;
- rewrite > (q_plus_sym (Qopp init)) in X;
- rewrite < q_elim_minus in X; rewrite > q_plus_minus in X;
- rewrite > q_plus_OQ in X; assumption;
- |2: simplify in H5; apply eq_f;
- cut (sum_bases (bars l1) (\fst w2) < sum_bases (bars l1) (S n)+Qpos w);[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]]
-qed.
-
-
-
alias symbol "pi2" = "pair pi2".
alias symbol "pi1" = "pair pi1".
definition rebase_spec ≝
[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);
+ rewrite < (H4 input); 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; rewrite > (initial_shift_same_values (mk_q_f s1 l1) s2 H input);
+ rewrite < (H3 input); 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 *)
+]
qed.
projects / helm.git / blobdiff