+++ /dev/null
-include ../../Makefile.defs
-
-DIR=$(shell basename $$PWD)
-
-$(DIR) all:
- $(BIN)../matitac
-$(DIR).opt opt all.opt:
- $(BIN)../matitac.opt
-clean:
- $(BIN)../matitaclean
-clean.opt:
- $(BIN)../matitaclean.opt
-depend:
- $(BIN)../matitadep -dot && rm depends.dot
-depend.opt:
- $(BIN)../matitadep.opt -dot && rm depends.dot
+++ /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 "nat_ordered_set.ma".
-include "models/q_support.ma".
-include "models/list_support.ma".
-include "logic/cprop_connectives.ma".
-
-definition bar ≝ ℚ × (ℚ × ℚ).
-
-notation < "\rationals \sup 2" non associative with precedence 90 for @{'q2}.
-interpretation "Q x Q" 'q2 = (Prod Q Q).
-
-definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉.
-notation "\rect" with precedence 90 for @{'empty_bar}.
-interpretation "q0" 'empty_bar = empty_bar.
-
-notation < "\ldots\rect\square\EmptySmallSquare\ldots" with precedence 90 for @{'lq2}.
-interpretation "lq2" 'lq2 = (list bar).
-
-definition q2_lt := mk_rel bar (λx,y:bar.\fst x < \fst y).
-
-interpretation "bar lt" 'lt x y = (rel_op _ q2_lt x y).
-
-lemma q2_trans : ∀a,b,c:bar. a < b → b < c → a < c.
-intros 3; cases a; cases b; cases c; unfold q2_lt; simplify; intros;
-apply (q_lt_trans ??? H H1);
-qed.
-
-definition q2_trel := mk_trans_rel bar q2_lt q2_trans.
-
-interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel q2_trel x y).
-
-definition canonical_q_lt : rel bar → trans_rel ≝ λx:rel bar.q2_trel.
-
-coercion canonical_q_lt with nocomposites.
-
-interpretation "bar lt" 'lt x y = (FunClass_2_OF_trans_rel (canonical_q_lt _) x y).
-
-definition nth_base ≝ λf,n. \fst (\nth f ▭ n).
-definition nth_height ≝ λf,n. \snd (\nth f ▭ n).
-
-record q_f : Type ≝ {
- bars: list bar;
- bars_sorted : sorted q2_lt bars;
- bars_begin_OQ : nth_base bars O = OQ;
- bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉
-}.
-
-lemma len_bases_gt_O: ∀f.O < \len (bars f).
-intros; generalize in match (bars_begin_OQ f); cases (bars f); intros;
-[2: simplify; apply le_S_S; apply le_O_n;
-|1: normalize in H; destruct H;]
-qed.
-
-lemma all_bases_positive : ∀f:q_f.∀i. OQ < nth_base (bars f) (S i).
-intro f; generalize in match (bars_begin_OQ f); generalize in match (bars_sorted f);
-cases (len_gt_non_empty ?? (len_bases_gt_O f)); intros;
-cases (cmp_nat (\len l) i);
-[2: lapply (sorted_tail_bigger q2_lt ?? ▭ H ? H2) as K;
- simplify in H1; rewrite < H1; apply K;
-|1: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)]
- cases n in H3; intros; [simplify in H3; cases (not_le_Sn_O ? H3)]
- apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);]
-qed.
-
-alias symbol "lt" (instance 9) = "Q less than".
-alias symbol "lt" (instance 7) = "natural 'less than'".
-alias symbol "lt" (instance 6) = "natural 'less than'".
-alias symbol "lt" (instance 5) = "Q less than".
-alias symbol "lt" (instance 4) = "natural 'less than'".
-alias symbol "lt" (instance 2) = "natural 'less than'".
-alias symbol "leq" = "Q less or equal than".
-coinductive value_spec (f : list bar) (i : ℚ) : ℚ × ℚ → CProp ≝
-| value_of : ∀j,q.
- nth_height f j = q → nth_base f j < i → j < \len f →
- (∀n.n<j → nth_base f n < i) →
- (∀n.j < n → n < \len f → i ≤ nth_base f n) → value_spec f i q.
-
-definition match_pred ≝
- λi.λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false].
-
-definition match_domain ≝
- λf: list bar.λi:ratio. pred (find ? (match_pred i) f ▭).
-
-definition value_simple ≝
- λf: list bar.λi:ratio. nth_height f (match_domain f i).
-
-alias symbol "lt" (instance 5) = "Q less than".
-alias symbol "lt" (instance 6) = "natural 'less than'".
-definition value_lemma :
- ∀f:list bar.sorted q2_lt f → O < length bar f →
- ∀i:ratio.nth_base f O < Qpos i → ∃p:ℚ×ℚ.value_spec f (Qpos i) p.
-intros (f bars_sorted_f len_bases_gt_O_f i bars_begin_OQ_f);
-exists [apply (value_simple f i);]
-apply (value_of ?? (match_domain f i));
-[1: reflexivity
-|2: unfold match_domain; cases (cases_find bar (match_pred i) f ▭);
- [1: cases i1 in H H1 H2 H3; simplify; intros;
- [1: generalize in match (bars_begin_OQ_f);
- cases (len_gt_non_empty ?? (len_bases_gt_O_f)); simplify; intros;
- assumption;
- |2: cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H3;
- intros; lapply (H3 n (le_n ?)) as K; unfold match_pred in K;
- cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ n))) in K;
- simplify; intros; [destruct H5] assumption]
- |2: destruct H; cases (len_gt_non_empty ?? (len_bases_gt_O_f)) in H2;
- simplify; intros; lapply (H (\len l) (le_n ?)) as K; clear H;
- unfold match_pred in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K;
- simplify; intros; [destruct H2] assumption;]
-|5: intro; unfold match_domain; cases (cases_find bar (match_pred i) f ▭); intros;
- [1: generalize in match (bars_sorted_f);
- cases (list_break ??? H) in H1; rewrite > H6;
- rewrite < H1; simplify; rewrite > nth_len; unfold match_pred;
- cases (q_cmp (Qpos i) (\fst x)); simplify;
- intros (X Hs); [2: destruct X] clear X;
- cases (sorted_pivot q2_lt ??? ▭ Hs);
- cut (\len l1 ≤ n) as Hn; [2:
- rewrite > H1; cases i1 in H4; simplify; intro X; [2: assumption]
- apply lt_to_le; assumption;]
- unfold nth_base; rewrite > (nth_append_ge_len ????? Hn);
- cut (n - \len l1 < \len (x::l2)) as K; [2:
- simplify; rewrite > H1; rewrite > (?:\len l2 = \len f - \len (l1 @ [x]));[2:
- rewrite > H6; repeat rewrite > len_append; simplify;
- repeat rewrite < plus_n_Sm; rewrite < plus_n_O; simplify;
- rewrite > sym_plus; rewrite < minus_plus_m_m; reflexivity;]
- rewrite > len_append; rewrite > H1; simplify; rewrite < plus_n_SO;
- apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H;
- elim (\len f) in i1 n H5; [cases (not_le_Sn_O ? H);]
- simplify; cases n2; [ repeat rewrite < minus_n_O; apply le_S_S_to_le; assumption]
- cases n1 in H1; [intros; rewrite > eq_minus_n_m_O; apply le_O_n]
- intros; simplify; apply H; apply le_S_S_to_le; assumption;]
- cases (n - \len l1) in K; simplify; intros; [ assumption]
- lapply (H9 ? (le_S_S_to_le ?? H10)) as W; apply (q_le_trans ??? H7);
- apply q_lt_to_le; apply W;
- |2: cases (not_le_Sn_n i1); rewrite > H in ⊢ (??%);
- apply (trans_le ??? ? H4); cases i1 in H3; intros; apply le_S_S;
- [ apply le_O_n; | assumption]]
-|3: unfold match_domain; cases (cases_find bar (match_pred i) f ▭); [
- cases i1 in H; intros; simplify; [assumption]
- apply lt_S_to_lt; assumption;]
- rewrite > H; cases (\len f) in len_bases_gt_O_f; intros; [cases (not_le_Sn_O ? H3)]
- simplify; apply le_n;
-|4: intros; unfold match_domain in H; cases (cases_find bar (match_pred i) f ▭) in H; simplify; intros;
- [1: lapply (H3 n); [2: cases i1 in H4; intros [assumption] apply le_S; assumption;]
- unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth f ▭ n))) in Hletin;
- simplify; intros; [destruct H6] assumption;
- |2: destruct H; cases f in len_bases_gt_O_f H2 H3; clear H1; simplify; intros;
- [cases (not_le_Sn_O ? H)] lapply (H1 n); [2: apply le_S; assumption]
- unfold match_pred in Hletin; cases (q_cmp (Qpos i) (\fst (\nth (b::l) ▭ n))) in Hletin;
- simplify; intros; [destruct H4] assumption;]]
-qed.
-
-lemma bars_begin_lt_Qpos : ∀q,r. nth_base (bars q) O<Qpos r.
-intros; rewrite > bars_begin_OQ; apply q_pos_OQ;
-qed.
-
-lemma value : q_f → ratio → ℚ × ℚ.
-intros; cases (value_lemma (bars q) ?? r);
-[ apply bars_sorted.
-| apply len_bases_gt_O;
-| apply w;
-| apply bars_begin_lt_Qpos;]
-qed.
-
-lemma cases_value : ∀f,i. value_spec (bars f) (Qpos i) (value f i).
-intros; unfold value;
-cases (value_lemma (bars f) (bars_sorted f) (len_bases_gt_O f) i (bars_begin_lt_Qpos f i));
-assumption;
-qed.
-
-definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input.
-
-definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i).
-
-lemma same_bases_cons: ∀a,b,l1,l2.
- same_bases l1 l2 → \fst a = \fst b → same_bases (a::l1) (b::l2).
-intros; intro; cases i; simplify; [assumption;] apply (H n);
-qed.
-
-alias symbol "lt" = "Q less than".
-lemma unpos: ∀x:ℚ.OQ < x → ∃r:ratio.Qpos r = x.
-intro; cases x; intros; [2:exists [apply r] reflexivity]
-cases (?:False);
-[ apply (q_lt_corefl ? H)| cases (q_lt_le_incompat ?? (q_neg_gt ?) (q_lt_to_le ?? H))]
-qed.
-
-notation < "x \blacksquare" non associative with precedence 50 for @{'unpos $x}.
-interpretation "hide unpos proof" 'unpos x = (unpos x _).
-
+++ /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".
-
-(* move in nat/minus *)
-lemma minus_lt : ∀i,j. i < j → j - i = S (j - S i).
-intros 2;
-apply (nat_elim2 ???? i j); simplify; intros;
-[1: cases n in H; intros; rewrite < minus_n_O; [cases (not_le_Sn_O ? H);]
- simplify; rewrite < minus_n_O; reflexivity;
-|2: cases (not_le_Sn_O ? H);
-|3: apply H; apply le_S_S_to_le; assumption;]
-qed.
-
-alias symbol "lt" = "bar lt".
-lemma inversion_sorted:
- ∀a,l. sorted q2_lt (a::l) → Or (a < \hd ▭ l) (l = []).
-intros 2; elim l; [right;reflexivity] left; inversion H1; intros;
-[1,2:destruct H2| destruct H5; assumption]
-qed.
-
-lemma inversion_sorted2:
- ∀a,b,l. sorted q2_lt (a::b::l) → a < b.
-intros; inversion H; intros; [1,2:destruct H1] destruct H4; assumption;
-qed.
-
-let rec copy (l : list bar) on l : list bar ≝
- match l with
- [ nil ⇒ []
- | cons x tl ⇒ 〈\fst x, 〈OQ,OQ〉〉 :: copy tl].
-
-lemma sorted_copy:
- ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l).
-intro l; elim l; [apply (sorted_nil q2_lt)] simplify;
-cases l1 in H H1; simplify; intros; [apply (sorted_one q2_lt)]
-apply (sorted_cons q2_lt); [2: apply H; apply (sorted_tail q2_lt ?? H1);]
-apply (inversion_sorted2 ??? H1);
-qed.
-
-lemma len_copy: ∀l. \len (copy l) = \len l.
-intro; elim l; [reflexivity] simplify; apply eq_f; assumption;
-qed.
-
-lemma copy_same_bases: ∀l. same_bases l (copy l).
-intros; elim l; [intro; reflexivity] intro; simplify; cases i; [reflexivity]
-simplify; apply (H n);
-qed.
-
-lemma copy_OQ : ∀l,n.nth_height (copy l) n = 〈OQ,OQ〉.
-intro; elim l; [elim n;[reflexivity] simplify; assumption]
-simplify; cases n; [reflexivity] simplify; apply (H n1);
-qed.
-
-lemma prepend_sorted_with_same_head:
- ∀r,x,l1,l2,d1,d2.
- sorted r (x::l1) → sorted r l2 →
- (r x (\nth l1 d1 O) → r x (\nth l2 d2 O)) → (l1 = [] → r x d1) →
- sorted r (x::l2).
-intros 8 (R x l1 l2 d1 d2 Sl1 Sl2); inversion Sl1; inversion Sl2;
-intros; destruct; try assumption; [3: apply (sorted_one R);]
-[1: apply sorted_cons;[2:assumption] apply H2; apply H3; reflexivity;
-|2: apply sorted_cons;[2: assumption] apply H5; apply H6; reflexivity;
-|3: apply sorted_cons;[2: assumption] apply H5; assumption;
-|4: apply sorted_cons;[2: assumption] apply H8; apply H4;]
-qed.
-
-lemma move_head_sorted: ∀x,l1,l2.
- sorted q2_lt (x::l1) → sorted q2_lt l2 → nth_base l2 O = nth_base l1 O →
- l1 ≠ [] → sorted q2_lt (x::l2).
-intros; apply (prepend_sorted_with_same_head q2_lt x l1 l2 ▭ ▭);
-try assumption; intros; unfold nth_base in H2; whd in H4;
-[1: rewrite < H2 in H4; assumption;
-|2: cases (H3 H4);]
-qed.
-
-
-lemma sort_q2lt_same_base:
- ∀b,h1,h2,l. sorted q2_lt (〈b,h1〉::l) → sorted q2_lt (〈b,h2〉::l).
-intros; cases (inversion_sorted ?? H); [2: rewrite > H1; apply (sorted_one q2_lt)]
-lapply (sorted_tail q2_lt ?? H) as K; clear H; cases l in H1 K; simplify; intros;
-[apply (sorted_one q2_lt);|apply (sorted_cons q2_lt);[2: assumption] apply H]
-qed.
-
-lemma value_head : ∀a,l,i.Qpos i ≤ \fst a → value_simple (a::l) i = \snd a.
-intros; unfold value_simple; unfold match_domain; cases (cases_find bar (match_pred i) (a::l) ▭);
-[1: cases i1 in H2 H3 H4; intros; [reflexivity] lapply (H4 O) as K; [2: apply le_S_S; apply le_O_n;]
- simplify in K; unfold match_pred in K; cases (q_cmp (Qpos i) (\fst a)) in K;
- simplify; intros; [destruct H6] lapply (q_le_lt_trans ??? H H5) as K; cases (q_lt_corefl ? K);
-|2: cases (?:False); lapply (H3 0); [2: simplify; apply le_S_S; apply le_O_n;]
- unfold match_pred in Hletin; simplify in Hletin; cases (q_cmp (Qpos i) (\fst a)) in Hletin;
- simplify; intros; [destruct H5] lapply (q_le_lt_trans ??? H H4); apply (q_lt_corefl ? Hletin);]
-qed.
-
-lemma value_unit : ∀x,i. value_simple [x] i = \snd x.
-intros; unfold value_simple; unfold match_domain;
-cases (cases_find bar (match_pred i) [x] ▭);
-[1: cases i1 in H; intros; [reflexivity] simplify in H;
- cases (not_le_Sn_O ? (le_S_S_to_le ?? H));
-|2: simplify in H; destruct H; reflexivity;]
-qed.
-
-lemma value_tail :
- ∀a,b,l,i.\fst a < Qpos i → \fst b ≤ Qpos i → value_simple (a::b::l) i = value_simple (b::l) i.
-intros; unfold value_simple; unfold match_domain;
-cases (cases_find bar (match_pred i) (a::b::l) ▭);
-[1: cases i1 in H3 H2 H4 H5; intros 1; simplify in ⊢ (? ? (? ? %) ?→?); unfold in ⊢ (? ? % ?→?); intros;
- [1: cases (?:False); cases (q_cmp (Qpos i) (\fst a)) in H3; simplify; intros;[2: destruct H6]
- apply (q_lt_corefl ? (q_lt_le_trans ??? H H3));
- |2:
-
-normalize in ⊢ (? ? % ?→?); simplify;
-[1: rewrite > (value_head);
-
-lemma value_copy :
- ∀l,i.rewrite > (value_u
- value_simple (copy l) i = 〈OQ,OQ〉.
-intros; elim l;
-[1: reflexivity;
-|2: cases l1 in H; intros; simplify in ⊢ (? ? (? % ?) ?);
- [1: rewrite > (value_unit); reflexivity;
- |2: cases (q_cmp (\fst b) (Qpos i));
-
- change with (\fst ▭ = \lamsimplify in ⊢ (? ? (? % ?) ?); unfold value_simple; unfold match_domain;
- cases (cases_find bar (match_pred i) [〈\fst x,〈OQ,OQ〉〉] ▭);
- [1: simplify in H1:(??%%);
-
- unfold match_pred;
- rewrite > (value_unit 〈\fst a,〈OQ,OQ〉〉); reflexivity;
-|2: intros; simplify in H2 H3 H4 ⊢ (? ? (? % ? ? ? ?) ?);
- cases (q_cmp (Qpos i) (\fst b));
- [2: rewrite > (value_tail ??? H2 H3 ? H4 H1); apply H;
- |1: rewrite > (value_head ??? H2 H3 ? H4 H1); reflexivity]]
-qed.
-
\ No newline at end of file
+++ /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 "russell_support.ma".
-include "models/q_copy.ma".
-(*
-definition rebase_spec ≝
- λl1,l2:q_f.λp:q_f × q_f.
- And3
- (same_bases (bars (\fst p)) (bars (\snd p)))
- (same_values l1 (\fst p))
- (same_values l2 (\snd p)).
-
-inductive rebase_cases : list bar → list bar → (list bar) × (list bar) → Prop ≝
-| rb_fst_full : ∀b,h1,xs.
- rebase_cases (〈b,h1〉::xs) [] 〈〈b,h1〉::xs,〈b,〈OQ,OQ〉〉::copy xs〉
-| rb_snd_full : ∀b,h1,ys.
- rebase_cases [] (〈b,h1〉::ys) 〈〈b,〈OQ,OQ〉〉::copy ys,〈b,h1〉::ys〉
-| rb_all_full : ∀b,h1,h2,h3,h4,xs,ys,r1,r2.
- \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h3〉::r1)) →
- \snd(\last ▭ (〈b,h2〉::ys)) = \snd(\last ▭ (〈b,h4〉::r2)) →
- rebase_cases (〈b,h1〉::xs) (〈b,h2〉::ys) 〈〈b,h3〉::r1,〈b,h4〉::r2〉
-| rb_all_full_l : ∀b1,b2,h1,h2,xs,ys,r1,r2.
- \snd(\last ▭ (〈b1,h1〉::xs)) = \snd(\last ▭ (〈b1,h1〉::r1)) →
- \snd(\last ▭ (〈b2,h2〉::ys)) = \snd(\last ▭ (〈b1,h2〉::r2)) →
- b1 < b2 →
- rebase_cases (〈b1,h1〉::xs) (〈b2,h2〉::ys) 〈〈b1,h1〉::r1,〈b1,〈OQ,OQ〉〉::r2〉
-| rb_all_full_r : ∀b1,b2,h1,h2,xs,ys,r1,r2.
- \snd(\last ▭ (〈b1,h1〉::xs)) = \snd(\last ▭ (〈b2,h1〉::r1)) →
- \snd(\last ▭ (〈b2,h2〉::ys)) = \snd(\last ▭ (〈b2,h2〉::r2)) →
- b2 < b1 →
- rebase_cases (〈b1,h1〉::xs) (〈b2,h2〉::ys) 〈〈b2,〈OQ,OQ〉〉::r1,〈b2,h2〉::r2〉
-| rb_all_empty : rebase_cases [] [] 〈[],[]〉.
-
-alias symbol "pi2" = "pair pi2".
-alias symbol "pi1" = "pair pi1".
-alias symbol "leq" = "natural 'less or equal to'".
-inductive rebase_spec_aux_p (l1, l2:list bar) (p:(list bar) × (list bar)) : Prop ≝
-| prove_rebase_spec_aux:
- rebase_cases l1 l2 p →
- (sorted q2_lt (\fst p)) →
- (sorted q2_lt (\snd p)) →
- (same_bases (\fst p) (\snd p)) →
- (same_values_simpl l1 (\fst p)) →
- (same_values_simpl l2 (\snd p)) →
- rebase_spec_aux_p l1 l2 p.
-
-lemma aux_preserves_sorting:
- ∀b,b3,l2,l3,w. rebase_cases l2 l3 w →
- sorted q2_lt (b::l2) → sorted q2_lt (b3::l3) → \fst b3 = \fst b →
- sorted q2_lt (\fst w) → sorted q2_lt (\snd w) →
- same_bases (\fst w) (\snd w) →
- sorted q2_lt (b :: \fst w).
-intros 6; cases H; simplify; intros; clear H;
-[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H1);
-| apply (sorted_cons q2_lt); [2:assumption]
- whd; rewrite < H3; apply (inversion_sorted2 ??? H2);
-| apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H3);
-| apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H4);
-| apply (sorted_cons q2_lt); [2:assumption]
- whd; rewrite < H6; apply (inversion_sorted2 ??? H5);
-| apply (sorted_one q2_lt);]
-qed.
-
-lemma aux_preserves_sorting2:
- ∀b,b3,l2,l3,w. rebase_cases l2 l3 w →
- sorted q2_lt (b::l2) → sorted q2_lt (b3::l3) → \fst b3 = \fst b →
- sorted q2_lt (\fst w) → sorted q2_lt (\snd w) → same_bases (\fst w) (\snd w) →
- sorted q2_lt (b :: \snd w).
-intros 6; cases H; simplify; intros; clear H;
-[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H1);
-| apply (sorted_cons q2_lt); [2:assumption]
- whd; rewrite < H3; apply (inversion_sorted2 ??? H2);
-| apply (sorted_cons q2_lt); [2: assumption] apply (inversion_sorted2 ??? H3);
-| apply (sorted_cons q2_lt); [2: assumption] apply (inversion_sorted2 ??? H4);
-| apply (sorted_cons q2_lt); [2: assumption]
- whd; rewrite < H6; apply (inversion_sorted2 ??? H5);
-| apply (sorted_one q2_lt);]
-qed.
-*)
-
-
-
-definition rebase_spec_aux ≝
- λl1,l2
- :list bar.λp:(list bar) × (list bar).
- sorted q2_lt l1 → (\snd (\last ▭ l1) = 〈OQ,OQ〉) →
- sorted q2_lt l2 → (\snd (\last ▭ l2) = 〈OQ,OQ〉) →
- rebase_spec_aux_p l1 l2 p.
-
-alias symbol "lt" = "Q less than".
-alias symbol "Q" = "Rationals".
-axiom q_unlimited: ∀x:ℚ.∃y:ratio.x<Qpos y.
-axiom q_halving: ∀x,y:ℚ.∃z:ℚ.x<z ∧ z<y.
-alias symbol "not" = "logical not".
-axiom q_not_OQ_lt_Qneg: ∀r. ¬ (OQ < Qneg r).
-lemma same_values_unit_OQ:
- ∀b1,b2,h1,l. OQ < b2 → b2 < b1 → sorted q2_lt (〈b1,h1〉::l) →
- sorted q2_lt [〈b2,〈OQ,OQ〉〉] →
- same_values_simpl (〈b1,h1〉::l) [〈b2,〈OQ,OQ〉〉] → h1 = 〈OQ,OQ〉.
-intros 5 (b1 b2 h1 l POS); cases l;
-[1: intros; cases (q_unlimited b1); cut (b2 < Qpos w); [2:apply (q_lt_trans ??? H H4);]
- lapply (H3 H1 ? H2 ? w H4 Hcut) as K; simplify; [1,2: autobatch]
- rewrite > (value_unit 〈b1,h1〉) in K;
- rewrite > (value_unit 〈b2,〈OQ,OQ〉〉) in K; assumption;
-|2: intros; unfold in H3; lapply depth=0 (H3 H1 ? H2 ?) as K; [1,2:simplify; autobatch]
- clear H3; cases (q_halving b1 (b1 + \fst p)) (w Pw); cases w in Pw; intros;
- [cases (q_lt_le_incompat ?? POS); apply q_lt_to_le; cases H3;
- apply (q_lt_trans ???? H4); assumption;
- |3: elim H3; lapply (q_lt_trans ??? H H4); lapply (q_lt_trans ??? POS Hletin);
- cases (q_not_OQ_lt_Qneg ? Hletin1);
- | cases H3; lapply (K r);
- [2: simplify; assumption
- |3: simplify; apply (q_lt_trans ???? H4); assumption;
- |rewrite > (value_head b1,h1 .. q) in Hletin;
-
-
-
- (* MANCA che le basi sono positive,
- poi con halving prendi tra b1 e \fst p e hai h1=OQ,OQ*)
-
-
-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: ∀l1,l2:q_f.∃p:q_f × q_f.rebase_spec l1 l2 p.
-intros 2 (f1 f2); cases f1 (b1 Hs1 Hb1 He1); cases f2 (b2 Hs2 Hb2 He2); clear f1 f2;
-alias symbol "leq" = "natural 'less or equal to'".
-alias symbol "minus" = "Q minus".
-letin aux ≝ (
-let rec aux (l1,l2:list bar) (n : nat) on n : (list bar) × (list bar) ≝
-match n with
-[ O ⇒ 〈[], []〉
-| S m ⇒
- match l1 with
- [ nil ⇒ 〈copy l2, l2〉
- | cons he1 tl1 ⇒
- match l2 with
- [ nil ⇒ 〈l1, copy l1〉
- | cons he2 tl2 ⇒
- let base1 ≝ \fst he1 in
- let base2 ≝ \fst he2 in
- let height1 ≝ \snd he1 in
- let height2 ≝ \snd he2 in
- match q_cmp base1 base2 with
- [ q_leq Hp1 ⇒
- match q_cmp base2 base1 with
- [ q_leq Hp2 ⇒
- let rc ≝ aux tl1 tl2 m in
- 〈he1 :: \fst rc,he2 :: \snd rc〉
- | q_gt 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 Hp ⇒
- let rest ≝ base1 - base2 in
- let rc ≝ aux (〈rest,height1〉 :: tl1) tl2 m in
- 〈〈base2,height1〉 :: \fst rc,〈base2,height2〉 :: \snd rc〉]]]]
-in aux : ∀l1,l2,m.∃z.\len l1 + \len l2 ≤ m → rebase_spec_aux l1 l2 z);
-[7: clearbody aux; cases (aux b1 b2 (\len b1 + \len b2)) (res Hres);
- exists; [split; constructor 1; [apply (\fst res)|5:apply (\snd res)]]
- [1,4: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); assumption;
- |2,5: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres aux;
- lapply (H3 O) as K; clear H1 H2 H3 H4 H5; unfold nth_base;
- cases H in K He1 He2 Hb1 Hb2; simplify; intros; assumption;
- |3,6: apply hide; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres aux;
- cases H in He1 He2; simplify; intros;
- [1,6,8,12: assumption;
- |2,7: rewrite > len_copy; generalize in match (\len ?); intro X;
- cases X; [1,3: reflexivity] simplify;
- [apply (copy_OQ ys n);|apply (copy_OQ xs n);]
- |3,4: rewrite < H6; assumption;
- |5: cases r1 in H6; simplify; intros; [reflexivity] rewrite < H6; assumption;
- |9,11: rewrite < H7; assumption;
- |10: cases r2 in H7; simplify; intros; [reflexivity] rewrite < H7; assumption]]
- split; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); clear Hres; unfold same_values; intros;
- [1: assumption
- |2,3: simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉);
- apply same_values_simpl_to_same_values; assumption]
-|3: cut (\fst b3 = \fst b) as E; [2: apply q_le_to_le_to_eq; assumption]
- clear H6 H5 H4 H3 He2 Hb2 Hs2 b2 He1 Hb1 Hs1 b1; cases (aux l2 l3 n1) (rc Hrc);
- clear aux; intro K; simplify in K; rewrite <plus_n_Sm in K;
- lapply le_S_S_to_le to K as W; lapply lt_to_le to W as R;
- simplify in match (? ≪rc,Hrc≫); intros (Hsbl2 Hendbl2 Hsb3l3 Hendb3l3);
- change in Hendbl2 with (\snd (\last ▭ (b::l2)) = 〈OQ,OQ〉);
- change in Hendb3l3 with (\snd (\last ▭ (b3::l3)) = 〈OQ,OQ〉);
- cases (Hrc R) (RC S1 S2 SB SV1 SV2); clear Hrc R W K;
- [2,4: apply (sorted_tail q2_lt);[apply b|3:apply b3]assumption;
- |3: cases l2 in Hendbl2; simplify; intros; [reflexivity] assumption;
- |5: cases l3 in Hendb3l3; simplify; intros; [reflexivity] assumption;]
- constructor 1; simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉);
- [1: cases b in E Hsbl2 Hendbl2; cases b3 in Hsb3l3 Hendb3l3; intros (Hsbl3 Hendbl2 E Hsb3l2 Hendb3l3);
- simplify in E; destruct E; constructor 3;
- [1: clear Hendbl2 Hsbl3 SV2 SB S2;
- cases RC in S1 SV1 Hsb3l2 Hendb3l3; intros;
- [1,6: reflexivity;
- |3,4: assumption;
- |5: simplify in H6:(??%) ⊢ %; rewrite > H3; cases r1 in H6; intros [2:reflexivity]
- use same_values_unit_OQ;
-
- |2: simplify in H3:(??%) ⊢ %; rewrite > H3; rewrite > len_copy; elim (\len ys); [reflexivity]
- symmetry; apply (copy_OQ ys n2);
- | cases H8 in H5 H7; simplify; intros; [2,6:reflexivity|3,4,5: assumption]
- simplify; rewrite > H5; rewrite > len_copy; elim (\len xs); [reflexivity]
- symmetry; apply (copy_OQ xs n2);]
- |2: apply (aux_preserves_sorting ? b3 ??? H8); assumption;
- |3: apply (aux_preserves_sorting2 ? b3 ??? H8); try assumption;
- try reflexivity; cases (inversion_sorted ?? H4);[2:rewrite >H3; apply (sorted_one q2_lt);]
- cases l2 in H3 H4; intros; [apply (sorted_one q2_lt)]
- apply (sorted_cons q2_lt);[2:apply (sorted_tail q2_lt ?? H3);] whd;
- rewrite > E; assumption;
- |4: apply I
- |5: apply I
- |6: intro; elim i; intros; simplify; solve [symmetry;assumption|apply H13]
- |7: unfold; intros; clear H9 H10 H11 H12 H13; simplify in Hi1 Hi2 H16 H18;
- cases H8 in H14 H15 H17 H3 H16 H18 H5 H6;
- simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉); intros;
- [1: reflexivity;
- |2: simplify in H3; rewrite > (value_unit b); rewrite > H3; symmetry;
- cases b in H3 H12 Hi2; intros 2; simplify in H12; rewrite > H12;
- intros; change in ⊢ (? ? (? % ? ? ? ?) ?) with (copy (〈q,〈OQ,OQ〉〉::〈b1,〈OQ,OQ〉〉::ys));
- apply (value_copy (〈q,〈OQ,OQ〉〉::〈b1,〈OQ,OQ〉〉::ys));
- |3: apply (same_value_tail b b1 h1 h3 xs r1 input); assumption;
- |4: apply (same_value_tail b b1 h1 h1 xs r1 input); assumption;
- |5: simplify in H9; STOP
-
- |6: reflexivity;]
-
- ]
- |8:
-
-
-include "Q/q/qtimes.ma".
-
-let rec area (l:list bar) on l ≝
- match l with
- [ nil ⇒ OQ
- | cons he tl ⇒ area tl + Qpos (\fst he) * ⅆ[OQ,\snd he]].
-
-alias symbol "pi1" = "exT \fst".
-alias symbol "minus" = "Q minus".
-alias symbol "exists" = "CProp exists".
-definition minus_spec_bar ≝
- λf,g,h:list bar.
- same_bases f g → len f = len g →
- ∀s,i:ℚ. \snd (\fst (value (mk_q_f s h) i)) =
- \snd (\fst (value (mk_q_f s f) i)) - \snd (\fst (value (mk_q_f s g) i)).
-
-definition minus_spec ≝
- λf,g:q_f.
- ∃h:q_f.
- ∀i:ℚ. \snd (\fst (value h i)) =
- \snd (\fst (value f i)) - \snd (\fst (value g i)).
-
-definition eject_bar : ∀P:list bar → CProp.(∃l:list bar.P l) → list bar ≝
- λP.λp.match p with [ex_introT x _ ⇒ x].
-definition inject_bar ≝ ex_introT (list bar).
-
-coercion inject_bar with 0 1 nocomposites.
-coercion eject_bar with 0 0 nocomposites.
-
-lemma minus_q_f : ∀f,g. minus_spec f g.
-intros;
-letin aux ≝ (
- let rec aux (l1, l2 : list bar) on l1 ≝
- match l1 with
- [ nil ⇒ []
- | cons he1 tl1 ⇒
- match l2 with
- [ nil ⇒ []
- | cons he2 tl2 ⇒ 〈\fst he1, \snd he1 - \snd he2〉 :: aux tl1 tl2]]
- in aux : ∀l1,l2 : list bar.∃h.minus_spec_bar l1 l2 h);
-[2: intros 4; simplify in H3; destruct H3;
-|3: intros 4; simplify in H3; cases l1 in H2; [2: intro X; simplify in X; destruct X]
- intros; rewrite > (value_OQ_e (mk_q_f s []) i); [2: reflexivity]
- rewrite > q_elim_minus; rewrite > q_plus_OQ; reflexivity;
-|1: cases (aux l2 l3); unfold in H2; intros 4;
- simplify in ⊢ (? ? (? ? ? (? ? ? (? % ?))) ?);
- cases (q_cmp i (s + Qpos (\fst b)));
-
-
-
-definition excess ≝
- λf,g.∃i.\snd (\fst (value f i)) < \snd (\fst (value g i)).
-
+++ /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 "Q/q/qtimes.ma".
-include "Q/q/qplus.ma".
-include "logic/cprop_connectives.ma".
-
-interpretation "Q" 'Q = Q.
-
-(* group over Q *)
-axiom qp : ℚ → ℚ → ℚ.
-
-interpretation "Q plus" 'plus x y = (qp x y).
-interpretation "Q minus" 'minus x y = (qp x (Qopp y)).
-
-axiom q_plus_OQ: ∀x:ℚ.x + OQ = x.
-axiom q_plus_sym: ∀x,y:ℚ.x + y = y + x.
-axiom q_plus_minus: ∀x.x - x = OQ.
-axiom q_plus_assoc: ∀x,y,z.x + (y + z) = x + y + z.
-axiom q_opp_plus: ∀x,y,z:Q. Qopp (y + z) = Qopp y + Qopp z.
-
-(* order over Q *)
-axiom qlt : ℚ → ℚ → Prop.
-axiom qle : ℚ → ℚ → Prop.
-interpretation "Q less than" 'lt x y = (qlt x y).
-interpretation "Q less or equal than" 'leq x y = (qle x y).
-
-inductive q_comparison (a,b:ℚ) : CProp ≝
- | q_leq : a ≤ b → q_comparison a b
- | q_gt : b < a → q_comparison a b.
-
-axiom q_cmp:∀a,b:ℚ.q_comparison a b.
-
-inductive q_le_elimination (a,b:ℚ) : CProp ≝
-| q_le_from_eq : a = b → q_le_elimination a b
-| q_le_from_lt : a < b → q_le_elimination a b.
-
-axiom q_le_cases : ∀x,y:ℚ.x ≤ y → q_le_elimination x y.
-
-axiom q_le_to_le_to_eq : ∀x,y. x ≤ y → y ≤ x → x = y.
-
-axiom q_le_plus_l: ∀a,b,c:ℚ. a ≤ c - b → a + b ≤ c.
-axiom q_le_plus_r: ∀a,b,c:ℚ. a - b ≤ c → a ≤ c + b.
-axiom q_lt_plus_l: ∀a,b,c:ℚ. a < c - b → a + b < c.
-axiom q_lt_plus_r: ∀a,b,c:ℚ. a - b < c → a < c + b.
-
-axiom q_lt_opp_opp: ∀a,b.b < a → Qopp a < Qopp b.
-
-axiom q_le_n: ∀x. x ≤ x.
-axiom q_lt_to_le: ∀a,b:ℚ.a < b → a ≤ b.
-
-axiom q_lt_corefl: ∀x:Q.x < x → False.
-axiom q_lt_le_incompat: ∀x,y:Q.x < y → y ≤ x → False.
-
-axiom q_neg_gt: ∀r:ratio.Qneg r < OQ.
-axiom q_pos_OQ: ∀x.OQ < Qpos x.
-
-axiom q_lt_trans: ∀x,y,z:Q. x < y → y < z → x < z.
-axiom q_lt_le_trans: ∀x,y,z:Q. x < y → y ≤ z → x < z.
-axiom q_le_lt_trans: ∀x,y,z:Q. x ≤ y → y < z → x < z.
-axiom q_le_trans: ∀x,y,z:Q. x ≤ y → y ≤ z → x ≤ z.
-
-axiom q_le_lt_OQ_plus_trans: ∀x,y:Q.OQ ≤ x → OQ < y → OQ < x + y.
-axiom q_lt_le_OQ_plus_trans: ∀x,y:Q.OQ < x → OQ ≤ y → OQ < x + y.
-axiom q_le_OQ_plus_trans: ∀x,y:Q.OQ ≤ x → OQ ≤ y → OQ ≤ x + y.
-
-axiom q_leWl: ∀x,y,z.OQ ≤ x → x + y ≤ z → y ≤ z.
-axiom q_ltWl: ∀x,y,z.OQ ≤ x → x + y < z → y < z.
-
-(* distance *)
-axiom q_dist : ℚ → ℚ → ℚ.
-
-notation "hbox(\dd [term 19 x, break term 19 y])" with precedence 90
-for @{'distance $x $y}.
-interpretation "ℚ distance" 'distance x y = (q_dist x y).
-
-axiom q_d_ge_OQ : ∀x,y:ℚ. OQ ≤ ⅆ[x,y].
-axiom q_d_OQ: ∀x:Q.ⅆ[x,x] = OQ.
-axiom q_d_noabs: ∀x,y. x ≤ y → ⅆ[y,x] = y - x.
-axiom q_d_sym: ∀x,y. ⅆ[x,y] = ⅆ[y,x].
-
-lemma q_2opp: ∀x:ℚ.Qopp (Qopp x) = x.
-intros; cases x; reflexivity; qed.
-
-(* derived *)
-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_plus_assoc; apply q_lt_plus_r; rewrite > q_2opp; 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_plus_minus;
-do 2 rewrite > q_plus_OQ; assumption;
-qed.
-
-lemma q_le_inj_plus_r:
- ∀x,y,z:Q.x ≤ y → x + z ≤ y + z.
-intros;cases (q_le_cases ?? H);
-[1: rewrite > H1; apply q_le_n;
-|2: apply q_lt_to_le; apply q_lt_inj_plus_r; assumption;]
-qed.
-
-lemma q_le_canc_plus_r:
- ∀x,y,z:Q.x + z ≤ y + z → x ≤ y.
-intros; lapply (q_le_inj_plus_r ?? (Qopp z) H) as H1;
-do 2 rewrite < q_plus_assoc in H1;
-rewrite > q_plus_minus in H1; do 2 rewrite > q_plus_OQ in H1; assumption;
-qed.
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