+sandwich.ma ordered_uniform.ma
property_sigma.ma ordered_uniform.ma russell_support.ma
+uniform.ma supremum.ma
bishop_set.ma ordered_set.ma
+sequence.ma nat/nat.ma
ordered_uniform.ma uniform.ma
+supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
+property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
bishop_set_rewrite.ma bishop_set.ma
-sequence.ma nat/nat.ma
+cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
nat_ordered_set.ma bishop_set.ma nat/compare.ma
lebesgue.ma property_exhaustivity.ma sandwich.ma
-property_exhaustivity.ma ordered_uniform.ma property_sigma.ma
-cprop_connectives.ma datatypes/constructors.ma logic/equality.ma
ordered_set.ma cprop_connectives.ma
-sandwich.ma ordered_uniform.ma
russell_support.ma cprop_connectives.ma nat/nat.ma
-uniform.ma supremum.ma
-supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma
+models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma
-models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
models/q_support.ma Q/q/q.ma cprop_connectives.ma
-models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
-models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma
-models/list_support.ma list/list.ma
-models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma
-models/q_function.ma models/q_bars.ma nat_ordered_set.ma
models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma
+models/q_function.ma models/q_shift.ma nat_ordered_set.ma
+models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma
+models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma
+models/q_shift.ma models/q_bars.ma
+models/list_support.ma list/list.ma
+models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma
Q/q/q.ma
datatypes/constructors.ma
list/list.ma
(**************************************************************************)
include "nat_ordered_set.ma".
-include "models/q_bars.ma".
+include "models/q_shift.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_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.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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.
projects / helm.git / commitdiff
