From 31126ffda75cba3fbc6572d86e910c10940da46c Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Fri, 19 Sep 2008 07:55:24 +0000 Subject: [PATCH] new reorganization --- .../matita/contribs/dama/dama/depends | 41 +- .../matita/contribs/dama/dama/depends.png | Bin 34866 -> 251072 bytes .../contribs/dama/dama/models/list_support.ma | 16 + .../contribs/dama/dama/models/q_bars.ma | 158 +++++- .../contribs/dama/dama/models/q_copy.ma | 139 +++++ .../contribs/dama/dama/models/q_function.ma | 514 ++---------------- .../contribs/dama/dama/models/q_rebase.ma | 167 ++++++ 7 files changed, 516 insertions(+), 519 deletions(-) create mode 100644 helm/software/matita/contribs/dama/dama/models/q_copy.ma create mode 100644 helm/software/matita/contribs/dama/dama/models/q_rebase.ma diff --git a/helm/software/matita/contribs/dama/dama/depends b/helm/software/matita/contribs/dama/dama/depends index 124d9017c..393b547d8 100644 --- a/helm/software/matita/contribs/dama/dama/depends +++ b/helm/software/matita/contribs/dama/dama/depends @@ -1,28 +1,31 @@ -ordered_uniform.ma uniform.ma -russell_support.ma logic/cprop_connectives.ma nat/nat.ma -lebesgue.ma property_exhaustivity.ma sandwich.ma -ordered_set.ma logic/cprop_connectives.ma bishop_set.ma ordered_set.ma -bishop_set_rewrite.ma bishop_set.ma -sandwich.ma ordered_uniform.ma nat_ordered_set.ma bishop_set.ma nat/compare.ma -property_sigma.ma ordered_uniform.ma russell_support.ma -sequence.ma nat/nat.ma +models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma +models/q_rebase.ma Q/q/qtimes.ma models/q_function.ma property_exhaustivity.ma ordered_uniform.ma property_sigma.ma -supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma -uniform.ma supremum.ma -models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma -models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma -models/q_function.ma Q/q/qtimes.ma models/q_bars.ma -models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma -models/list_support.ma list/list.ma logic/cprop_connectives.ma nat/le_arith.ma nat/minus.ma -models/q_bars.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma -models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma logic/cprop_connectives.ma +ordered_uniform.ma uniform.ma models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.ma -models/q_shift.ma models/nat_dedekind_sigma_complete.ma models/nat_uniform.ma nat/le_arith.ma russell_support.ma supremum.ma -models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma +ordered_set.ma datatypes/constructors.ma logic/cprop_connectives.ma +models/list_support.ma list/list.ma logic/cprop_connectives.ma nat/minus.ma +uniform.ma supremum.ma +sandwich.ma ordered_uniform.ma models/q_value_skip.ma +models/q_shift.ma +models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma logic/cprop_connectives.ma +models/q_bars.ma logic/cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma russell_support.ma +models/q_copy.ma models/q_bars.ma +models/nat_order_continuous.ma models/nat_dedekind_sigma_complete.ma models/nat_ordered_uniform.ma +property_sigma.ma ordered_uniform.ma russell_support.ma +lebesgue.ma property_exhaustivity.ma sandwich.ma +models/discrete_uniformity.ma bishop_set_rewrite.ma uniform.ma +models/nat_ordered_uniform.ma bishop_set_rewrite.ma models/nat_uniform.ma ordered_uniform.ma +bishop_set_rewrite.ma bishop_set.ma +russell_support.ma logic/cprop_connectives.ma nat/nat.ma +.unnamed.ma models/q_bars.ma +supremum.ma datatypes/constructors.ma nat/plus.ma nat_ordered_set.ma sequence.ma +models/q_function.ma models/q_copy.ma russell_support.ma +sequence.ma nat/nat.ma Q/q/qplus.ma Q/q/qtimes.ma datatypes/constructors.ma diff --git a/helm/software/matita/contribs/dama/dama/depends.png b/helm/software/matita/contribs/dama/dama/depends.png index 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last" 'last = (last _). +interpretation "list last applied" 'last2 d l = (last _ d l). + +definition head ≝ + λT:Type.λd.λl:list T.\nth l d O. + +notation > "\hd" non associative with precedence 90 for @{'hd}. +notation < "\hd d l" non associative with precedence 70 for @{'hd2 $d $l}. +interpretation "list head" 'hd = (head _). +interpretation "list head applied" 'hd2 d l = (head _ d l). + diff --git a/helm/software/matita/contribs/dama/dama/models/q_bars.ma b/helm/software/matita/contribs/dama/dama/models/q_bars.ma index 5098fa724..baf0bcfda 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -75,38 +75,45 @@ cases (cmp_nat (\len l) i); 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". -alias symbol "Q" = "Rationals". -coinductive value_spec (f : q_f) (i : ℚ) : ℚ × ℚ → CProp ≝ +coinductive value_spec (f : list bar) (i : ℚ) : ℚ × ℚ → CProp ≝ | value_of : ∀j,q. - nth_height (bars f) j = q → nth_base (bars f) j < i → - (∀n.j < n → n < \len (bars f) → i ≤ nth_base (bars f) n) → value_spec f i q. + nth_height f j = q → nth_base f j < i → j < \len f → + (∀n.n H4; apply q_pos_OQ; - |2: cases (len_gt_non_empty ?? (len_bases_gt_O f)) in H3; + [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 P 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; + |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 P in K; cases (q_cmp (Qpos i) (\fst (\nth (x::l) ▭ (\len l)))) in K; simplify; intros; [destruct H2] assumption;] -|3: intro; cases (cases_find bar P (bars f) ▭); intros; - [1: generalize in match (bars_sorted f); +|5: intro; cases (cases_find bar P f ▭); intros; + [1: generalize in match (bars_sorted_f); cases (list_break ??? H) in H1; rewrite > H6; rewrite < H1; simplify; rewrite > nth_len; unfold P; cases (q_cmp (Qpos i) (\fst x)); simplify; @@ -117,13 +124,13 @@ apply (value_of ?? (pred (find ? P (bars f) ▭))); 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 (bars f) - \len (l1 @ [x]));[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 P i; - elim (\len (bars f)) in i1 n H5; [cases (not_le_Sn_O ? H);] + apply le_S_S; clear H1 H6 H7 Hs H8 H9 Hn x l2 l1 H4 H3 H2 H P; + 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;] @@ -132,19 +139,124 @@ apply (value_of ?? (pred (find ? P (bars f) ▭))); 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]]] + [ apply le_O_n; | assumption]] +|3: cases (cases_find bar P 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; cases (cases_find bar P f ▭) in H; simplify; intros; + [1: lapply (H3 n); [2: cases i1 in H4; intros [assumption] apply le_S; assumption;] + unfold P 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 P 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 bars_begin_OQ; apply q_pos_OQ; +qed. + lemma value : q_f → ratio → ℚ × ℚ. -intros; cases (value_lemma q r); apply w; qed. +intros; cases (value_lemma (bars q) ?? r); +[ apply bars_sorted. +| apply len_bases_gt_O; +| apply w; +| apply bars_begin_lt_Qpos;] +qed. + +alias symbol "lt" (instance 5) = "natural 'less than'". +alias symbol "lt" (instance 4) = "Q less than". +lemma value_simpl: + ∀f:list bar.sorted q2_lt f → O < (length bar f) → + ∀i:ratio.nth_base f O < Qpos i → ℚ × ℚ. +intros; cases (value_lemma f H H1 i H2); assumption; +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. -lemma cases_value : ∀f,i. value_spec f (Qpos i) (value f i). -intros; unfold value; cases (value_lemma f i); assumption; qed. +lemma cases_value_simpl : + ∀f,H1,H2,i,Hi.value_spec f (Qpos i) (value_simpl f H1 H2 i Hi). +intros; unfold value_simpl; cases (value_lemma f H1 H2 i Hi); +assumption; +qed. definition same_values ≝ λl1,l2:q_f.∀input. value l1 input = value l2 input. +definition same_values_simpl ≝ + λl1,l2:list bar.∀H1,H2,H3,H4,input,Hi1,Hi2. + value_simpl l1 H1 H2 input Hi1 = value_simpl l2 H3 H4 input Hi2. + +lemma value_head : + ∀x,y,l,H1,H2,i,H3. + Qpos i ≤ \fst x → value_simpl (y::x::l) H1 H2 i H3 = \snd y. +intros; cases (cases_value_simpl ? H1 H2 i H3); +cases j in H4 H5 H6 H7 H8 (j); simplify; intros; +[1: symmetry; assumption; +|2: cases (?:False); cases j in H4 H5 H6 H7 H8; intros; + [1: lapply (q_le_lt_trans ??? H H5) as K;cases (q_lt_corefl ? K); + |2: lapply (H7 1); [2: do 2 apply le_S_S; apply le_O_n;] + simplify in Hletin; + lapply (q_le_lt_trans ??? H Hletin) as K;cases (q_lt_corefl ? K);]] +qed. + +lemma same_values_simpl_to_same_values: + ∀b1,b2,Hs1,Hs2,Hb1,Hb2,He1,He2,input. + same_values_simpl b1 b2 → + value (mk_q_f b1 Hs1 Hb1 He1) input = + value (mk_q_f b2 Hs2 Hb2 He2) input. +intros; +lapply (len_bases_gt_O (mk_q_f b1 Hs1 Hb1 He1)); +lapply (len_bases_gt_O (mk_q_f b2 Hs2 Hb2 He2)); +lapply (H ???? input) as K; try assumption; +[2: rewrite > Hb1; apply q_pos_OQ; +|3: rewrite > Hb2; apply q_pos_OQ; +|1: apply K;] +qed. +include "russell_support.ma". + +lemma value_tail : + ∀x,y,l,H1,H2,i,H3. + \fst x < Qpos i → + value_simpl (y::x::l) H1 H2 i H3 = + value_simpl (x::l) ?? i ?. +[1: apply hide; apply (sorted_tail q2_lt); [apply y| assumption] +|2: apply hide; simplify; apply le_S_S; apply le_O_n; +|3: apply hide; assumption;] +intros;cases (cases_value_simpl ? H1 H2 i H3); +generalize in ⊢ (? ? ? (? ? % ? ? ?)); intro; +generalize in ⊢ (? ? ? (? ? ? % ? ?)); intro; +generalize in ⊢ (? ? ? (? ? ? ? ? %)); intro; +cases (cases_value_simpl (x::l) H9 H10 i H11); +cut (j = S j1) as E; [ destruct E; destruct H12; reflexivity;] +clear H12 H4; cases j in H8 H5 H6 H7; +[1: intros;cases (?:False); lapply (H7 1 (le_n ?)); [2: simplify; do 2 apply le_S_S; apply le_O_n] + simplify in Hletin; apply (q_lt_corefl (\fst x)); + apply (q_lt_le_trans ??? H Hletin); +|2: simplify; intros; clear q q1 j H11 H10 H1 H2; simplify in H3 H14; apply eq_f; + cases (cmp_nat n j1); [cases (cmp_nat j1 n);[apply le_to_le_to_eq; assumption]] + [1: clear H1; cases (?:False); + lapply (H7 (S j1)); [2: cases j1 in H2; intros[cases (not_le_Sn_O ? H1)] apply le_S_S; assumption] + [2: apply le_S_S; assumption;] simplify in Hletin; + apply (q_lt_corefl ? (q_le_lt_trans ??? Hletin H13)); + |2: cases (?:False); + lapply (H16 n); [2: assumption|3:simplify; apply le_S_S_to_le; assumption] + apply (q_lt_corefl ? (q_le_lt_trans ??? Hletin H4));]] +qed. + 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] diff --git a/helm/software/matita/contribs/dama/dama/models/q_copy.ma b/helm/software/matita/contribs/dama/dama/models/q_copy.ma new file mode 100644 index 000000000..78e4ea821 --- /dev/null +++ b/helm/software/matita/contribs/dama/dama/models/q_copy.ma @@ -0,0 +1,139 @@ +(**************************************************************************) +(* ___ *) +(* ||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. + +definition copy ≝ + λl:list bar.make_list ? (λn.〈nth_base l (\len l - S n),〈OQ,OQ〉〉) (\len l). + +lemma sorted_copy: + ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l). +intros 2; unfold copy; generalize in match (le_n (\len l)); +elim (\len l) in ⊢ (?%?→? ? (? ? ? %)); +simplify; [apply (sorted_nil q2_lt);] cases n in H1 H2; +simplify; intros; [apply (sorted_one q2_lt);] +apply (sorted_cons q2_lt); +[2: apply H1; apply lt_to_le; apply H2; +|1: elim l in H2 H; simplify; [simplify in H2; cases (not_le_Sn_O ? H2)] + simplify in H3; unfold nth_base; + unfold canonical_q_lt; unfold q2_trel; unfold q2_lt; simplify; + change with (q2_lt (\nth (a::l1) ▭ (\len l1-S n1)) (\nth (a::l1) ▭ (\len l1-n1))); + cut (∃w.w = \len l1 - S n1); [2: exists[apply (\len l1 - S n1)] reflexivity] + cases Hcut; rewrite < H4; rewrite < (?:S w = \len l1 - n1); + [1: apply (sorted_near q2_lt (a::l1) H2); rewrite > H4; + simplify; apply le_S_S; elim (\len l1) in H3; simplify; + [ cases (not_le_Sn_O ? (le_S_S_to_le ?? H3)); + | lapply le_S_S_to_le to H5 as H6; + lapply le_S_S_to_le to H6 as H7; clear H5 H6; + cases H7 in H3; intros; [rewrite < minus_n_n; apply le_S_S; apply le_O_n] + simplify in H5; apply le_S_S; apply (trans_le ???? (H5 ?)); + [2: apply le_S_S; apply le_S_S; assumption; + |1: apply (lt_minus_S_n_to_le_minus_n n1 (S m) (S (minus m n1)) ?). + apply (not_le_to_lt (S (minus m n1)) (minus (S m) (S n1)) ?). + apply (not_le_Sn_n (minus (S m) (S n1))).]] + |2: rewrite > H4; lapply le_S_S_to_le to H3 as K; + clear H4 Hcut H3 H H1 H2; generalize in match K; clear K; + apply (nat_elim2 ???? n1 (\len l1)); simplify; intros; + [1: rewrite < minus_n_O; cases n2 in H; [intro; cases (not_le_Sn_O ? H)] + intros; cases n3; simplify; reflexivity; + |2: cases (not_le_Sn_O ? H); + |3: apply H; apply le_S_S_to_le; apply H1;]]] +qed. + +lemma len_copy: ∀l. \len (copy l) = \len l. +intro; unfold copy; apply len_mk_list; +qed. + +lemma copy_same_bases: ∀l. same_bases l (copy l). +intro; unfold copy; elim l using list_elim_with_len; [1: intro;reflexivity] +simplify; rewrite < minus_n_n; +simplify in ⊢ (? ? (? ? (? ? ? % ?) ?)); +apply same_bases_cons; [2: reflexivity] +cases l1 in H; [intros 2; reflexivity] +simplify in ⊢ (? ? (? ? (λ_:?.? ? ? (? ? %) ?) ?)→?); +simplify in ⊢ (?→? ? (? ? (λ_:?.? ? ? (? ? (? % ?)) ?) ?)); +intro; rewrite > (mk_list_ext ?? (λn:nat.〈nth_base (b::l2) (\len l2-n),〈OQ,OQ〉〉));[assumption] +intro; elim x; [simplify; rewrite < minus_n_O; reflexivity] +simplify in ⊢ (? ? (? ? ? (? ? %) ?) ?); +simplify in H2:(? ? %); rewrite > minus_lt; [reflexivity] apply le_S_S_to_le; +assumption; +qed. + +lemma rewrite_ext_nth_height : ∀n,m,f,f1,g,g1. (∀t.g t = g1 t) → +nth_height (\mk_list (λn.〈f n,g n〉) m) n = nth_height (\mk_list (λn.〈f1 n,g1 n〉) m) n. +intros; generalize in match n; clear n; elim m; [reflexivity] simplify; +cases n1; [simplify;apply H;] simplify; apply (H1 n2); +qed. + +lemma copy_OQ : ∀l,n.nth_height (copy l) n = 〈OQ,OQ〉. +intro; elim l; [cases n; [reflexivity] simplify; rewrite > nth_nil; reflexivity] +cases n; simplify [reflexivity] +change with (nth_height (\mk_list (λn:ℕ.〈nth_base (a::l1) (\len l1-n),〈OQ,OQ〉〉) (\len l1)) n1 = 〈OQ,OQ〉); +rewrite > (rewrite_ext_nth_height ??? (λ_.OQ) ? (λ_.〈OQ,OQ〉)); [2: intros; reflexivity] +generalize in match n1; +elim (\len l1); simplify; unfold nth_height; [rewrite > nth_nil; reflexivity] +cases n3; simplify; [reflexivity] apply (H1 n4); +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. + +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. + +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. + diff --git a/helm/software/matita/contribs/dama/dama/models/q_function.ma b/helm/software/matita/contribs/dama/dama/models/q_function.ma index 33898007a..e16f75116 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_function.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma @@ -13,101 +13,8 @@ (**************************************************************************) include "russell_support.ma". -include "models/q_bars.ma". +include "models/q_copy.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. - -definition copy ≝ - λl:list bar.make_list ? (λn.〈nth_base l (\len l - S n),〈OQ,OQ〉〉) (\len l). - -lemma sorted_copy: - ∀l:list bar.sorted q2_lt l → sorted q2_lt (copy l). -intros 2; unfold copy; generalize in match (le_n (\len l)); -elim (\len l) in ⊢ (?%?→? ? (? ? ? %)); -simplify; [apply (sorted_nil q2_lt);] cases n in H1 H2; -simplify; intros; [apply (sorted_one q2_lt);] -apply (sorted_cons q2_lt); -[2: apply H1; apply lt_to_le; apply H2; -|1: elim l in H2 H; simplify; [simplify in H2; cases (not_le_Sn_O ? H2)] - simplify in H3; unfold nth_base; - unfold canonical_q_lt; unfold q2_trel; unfold q2_lt; simplify; - change with (q2_lt (\nth (a::l1) ▭ (\len l1-S n1)) (\nth (a::l1) ▭ (\len l1-n1))); - cut (∃w.w = \len l1 - S n1); [2: exists[apply (\len l1 - S n1)] reflexivity] - cases Hcut; rewrite < H4; rewrite < (?:S w = \len l1 - n1); - [1: apply (sorted_near q2_lt (a::l1) H2); rewrite > H4; - simplify; apply le_S_S; elim (\len l1) in H3; simplify; - [ cases (not_le_Sn_O ? (le_S_S_to_le ?? H3)); - | lapply le_S_S_to_le to H5 as H6; - lapply le_S_S_to_le to H6 as H7; clear H5 H6; - cases H7 in H3; intros; [rewrite < minus_n_n; apply le_S_S; apply le_O_n] - simplify in H5; apply le_S_S; apply (trans_le ???? (H5 ?)); - [2: apply le_S_S; apply le_S_S; assumption; - |1: apply (lt_minus_S_n_to_le_minus_n n1 (S m) (S (minus m n1)) ?). - apply (not_le_to_lt (S (minus m n1)) (minus (S m) (S n1)) ?). - apply (not_le_Sn_n (minus (S m) (S n1))).]] - |2: rewrite > H4; lapply le_S_S_to_le to H3 as K; - clear H4 Hcut H3 H H1 H2; generalize in match K; clear K; - apply (nat_elim2 ???? n1 (\len l1)); simplify; intros; - [1: rewrite < minus_n_O; cases n2 in H; [intro; cases (not_le_Sn_O ? H)] - intros; cases n3; simplify; reflexivity; - |2: cases (not_le_Sn_O ? H); - |3: apply H; apply le_S_S_to_le; apply H1;]]] -qed. - -lemma len_copy: ∀l. \len (copy l) = \len l. -intro; unfold copy; apply len_mk_list; -qed. - -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. - -lemma copy_same_bases: ∀l. same_bases l (copy l). -intro; unfold copy; elim l using list_elim_with_len; [1: intro;reflexivity] -simplify; rewrite < minus_n_n; -simplify in ⊢ (? ? (? ? (? ? ? % ?) ?)); -apply same_bases_cons; [2: reflexivity] -cases l1 in H; [intros 2; reflexivity] -simplify in ⊢ (? ? (? ? (λ_:?.? ? ? (? ? %) ?) ?)→?); -simplify in ⊢ (?→? ? (? ? (λ_:?.? ? ? (? ? (? % ?)) ?) ?)); -intro; rewrite > (mk_list_ext ?? (λn:nat.〈nth_base (b::l2) (\len l2-n),〈OQ,OQ〉〉));[assumption] -intro; elim x; [simplify; rewrite < minus_n_O; reflexivity] -simplify in ⊢ (? ? (? ? ? (? ? %) ?) ?); -simplify in H2:(? ? %); rewrite > minus_lt; [reflexivity] apply le_S_S_to_le; -assumption; -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. - definition rebase_spec ≝ λl1,l2:q_f.λp:q_f × q_f. And3 @@ -115,51 +22,29 @@ definition rebase_spec ≝ (same_values l1 (\fst p)) (same_values l2 (\snd p)). -definition last ≝ - λT:Type.λd.λl:list T. \nth l d (pred (\len l)). - -notation > "\last" non associative with precedence 90 for @{'last}. -notation < "\last d l" non associative with precedence 70 for @{'last2 $d $l}. -interpretation "list last" 'last = (last _). -interpretation "list last applied" 'last2 d l = (last _ d l). - -definition head ≝ - λT:Type.λd.λl:list T.\nth l d O. - -notation > "\hd" non associative with precedence 90 for @{'hd}. -notation < "\hd d l" non associative with precedence 70 for @{'hd2 $d $l}. -interpretation "list head" 'hd = (head _). -interpretation "list head applied" 'hd2 d l = (head _ d l). - -alias symbol "lt" = "bar lt". -lemma inversion_sorted: - ∀a,l. sorted q2_lt (a::l) → 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. - -definition same_values_simpl ≝ - λl1,l2.∀p1,p2,p3,p4,p5,p6. - same_values (mk_q_f l1 p1 p2 p3) (mk_q_f l2 p4 p5 p6). - inductive rebase_cases : list bar → list bar → (list bar) × (list bar) → Prop ≝ -| rb_fst_full : ∀b,h1,h2,xs,ys,r1,r2. 〈b,h1〉 < \hd ▭ ys → - \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h1〉::r1)) → - \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h2〉::r2)) → - rebase_cases (〈b,h1〉::xs) ys 〈〈b,h1〉::r1,〈b,h2〉::r2〉 -| rb_snd_full : ∀b,h1,h2,xs,ys,r1,r2. 〈b,h1〉 < \hd ▭ xs → - \snd(\last ▭ (〈b,h1〉::ys)) = \snd(\last ▭ (〈b,h1〉::r2)) → - \snd(\last ▭ (〈b,h1〉::ys)) = \snd(\last ▭ (〈b,h2〉::r1)) → - rebase_cases xs (〈b,h1〉::ys) 〈〈b,h2〉::r1,〈b,h1〉::r2〉 +| rb_fst_full : ∀b,h1,xs. (*〈b,h1〉 < \hd ▭ ys → *) +(* \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h1〉::r1)) → + \snd(\last ▭ (〈b,h1〉::xs)) = \snd(\last ▭ (〈b,h2〉::r2)) →*) + rebase_cases (〈b,h1〉::xs) [] 〈〈b,h1〉::xs,〈b,〈OQ,OQ〉〉::copy xs〉 +| rb_snd_full : ∀b,h1,ys. (*〈b,h1〉 < \hd ▭ xs →*) + (* \snd(\last ▭ (〈b,h1〉::ys)) = \snd(\last ▭ (〈b,h1〉::r2)) → + \snd(\last ▭ (〈b,h1〉::ys)) = \snd(\last ▭ (〈b,h2〉::r1)) →*) + 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〉 + 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,h2〉::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,h1〉::r1,〈b2,h2〉::r2〉 | rb_all_empty : rebase_cases [] [] 〈[],[]〉. alias symbol "pi2" = "pair pi2". @@ -181,23 +66,20 @@ inductive rebase_spec_aux_p (l1, l2:list bar) (p:(list bar) × (list bar)) : Pro (*\len (\fst p) = \len (\snd p) → *) rebase_spec_aux_p l1 l2 p. -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 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; -[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H4); -| apply (sorted_cons q2_lt); [2:apply (sort_q2lt_same_base ???? H7);] + 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_cons q2_lt); [2: assumption] apply (inversion_sorted2 ??? H3); | apply (sorted_one q2_lt);] qed. @@ -206,341 +88,19 @@ lemma aux_preserves_sorting2: 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; -[ apply (sorted_cons q2_lt); [2:assumption] apply (inversion_sorted2 ??? H4); +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 < H6; apply (inversion_sorted2 ??? H5); + 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. +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. - -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); lapply (H5 O) as K; - clear H1 H2 H3 H4 H5 H6 H7 Hres aux; 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); - cases H in He1 He2; simplify; intros; - [4,8: assumption; - |1,6,7: rewrite < H9; assumption; - |2,3,5: rewrite < H10; [2: symmetry] assumption;]] - split; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); unfold same_values; intros; - [1: assumption - |2: simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉); apply H6; - |3: simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉); apply H7] -|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); - clear aux; intro K; simplify in K; rewrite H16; rewrite < H7; symmetry; apply H17; | reflexivity] - | cases H8 in H5 H7; simplify; intros; [2,3: assumption] - [ rewrite < H7; rewrite > H16; apply H17; | reflexivity]] - |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; apply H14; - |8: - - clear H15 H14 H11 H12 H7 H5; cases H8; clear H8; cases H3; clear H3; - [1: apply (move_head ?? H4 H5 ?? H9); symmetry; assumption; - |2: apply (move_head ??? H5 ?? H9); [apply (soted_q2lt_rewrite_hd ??? E H6)] - symmetry; rewrite > (H13 O); assumption; - |3: apply (soted_q2lt_rewrite_hd ??? E); apply (move_head ?? H6 H7); [symmetry;] assumption; - |4: rewrite > H8; apply (sorted_one q2_lt);] - - - cases l2 in H4 H8 H16; cases l3 in H6; - intros; cases H5; clear H5; cases H7; clear H7; - [1,2: cases (q_lt_corefl ? H5); - |3: rewrite >(?:\fst w = []); [apply (sorted_one q2_lt)] - simplify in H6; lapply (le_n_O_to_eq ? H6) as K; - cases (\fst w) in K; simplify; intro X [reflexivity|destruct X] - |4: - - rewrite >H8 in H5; cases (\fst w) in H9 H5; intros [apply (sorted_one q2_lt)] - - - - - - - - - - -definition same_values_simpl ≝ - λl1,l2.∀p1,p2,p3,p4,p5,p6.same_values (mk_q_f l1 p1 p2 p3) (mk_q_f l2 p4 p5 p6). - -alias symbol "pi2" = "pair pi2". -alias symbol "pi1" = "pair pi1". -definition rebase_spec_aux ≝ - λl1,l2:list bar.λp:(list bar) × (list bar). - sorted q2_lt l1 → sorted q2_lt l2 → - (l1 ≠ [] → \snd (\nth l1 ▭ (pred (\len l1))) = 〈OQ,OQ〉) → - (l2 ≠ [] → \snd (\nth l2 ▭ (pred (\len l2))) = 〈OQ,OQ〉) → - And4 - (nth_base l1 O = nth_base (\fst p) O ∨ - nth_base l2 O = nth_base (\fst p) O) - (sorted q2_lt (\fst p) ∧ sorted q2_lt (\snd p)) - ((l1 ≠ [] → \snd (\nth (\fst p) ▭ (pred (\len (\fst p)))) = 〈OQ,OQ〉) ∧ - (l2 ≠ [] → \snd (\nth (\snd p) ▭ (pred (\len (\snd p)))) = 〈OQ,OQ〉)) - (And3 - (same_bases (\fst p) (\snd p)) - (same_values_simpl l1 (\fst p)) - (same_values_simpl l2 (\snd p))). - -lemma copy_rebases: - ∀l1.rebase_spec_aux l1 [] 〈l1, copy l1〉. -intros; elim l1; intros 4; -[1: split; [left; reflexivity]; split; try assumption; unfold; intros; - unfold same_values; intros; reflexivity; -|2: rewrite > H3; [2: intro X; destruct X] - split; [left; reflexivity] split; - unfold same_values_simpl; unfold same_values; intros; try reflexivity; - try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;] - [1: apply (sorted_copy ? H1); - |2: apply (copy_same_bases (a::l));]] -qed. - -lemma copy_rebases_r: - ∀l1.rebase_spec_aux [] l1 〈copy l1, l1〉. -intros; elim l1; intros 4; -[1: split; [left; reflexivity]; split; try assumption; unfold; intros; - unfold same_values; intros; reflexivity; -|2: rewrite > H4; [2: intro X; destruct X] - split; [right; simplify; rewrite < minus_n_n; reflexivity] split; - unfold same_values_simpl; unfold same_values; intros; try reflexivity; - try assumption; [4: normalize in p2; destruct p2|2: cases H5; reflexivity;] - [1: apply (sorted_copy ? H2); - |2: intro; symmetry; apply (copy_same_bases (a::l));]] -qed. - - -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)) (w Hw); clear aux; - cases (Hw (le_n ?) Hs1 Hs2 (λ_.He1) (λ_.He2)); clear Hw; cases H1; cases H2; cases H3; clear H3 H1 H2; - exists [constructor 1;constructor 1;[apply (\fst w)|5:apply (\snd w)]] try assumption; - [1,3: apply hide; cases H (X X); try rewrite < (H8 O); try rewrite < X; assumption - |2,4: apply hide;[apply H6|apply H7]intro X;[rewrite > X in Hb1|rewrite > X in Hb2] - normalize in Hb1 Hb2; [destruct Hb1|destruct Hb2]] - unfold; unfold same_values; simplify in ⊢ (? (? % %) ? ?); - simplify in match (\snd 〈?,?〉); simplify in match (\fst 〈?,?〉); - split; [assumption; |apply H9;|apply H10] -|6: intro ABS; unfold; intros 4; clear H1 H2; - cases l in ABS H3; intros 1; [2: simplify in H1; cases (not_le_Sn_O ? H1)] - cases l1 in H4 H1; intros; [2: simplify in H2; cases (not_le_Sn_O ? H2)] - split; [ left; reflexivity|split; apply (sorted_nil q2_lt);|split; assumption;] - split; unfold; intros; unfold same_values; intros; reflexivity; -|5: intros; apply copy_rebases_r; -|4: intros; rewrite < H1; apply copy_rebases; -|3: cut (\fst b = \fst b3) as K; [2: apply q_le_to_le_to_eq; assumption] clear H6 H5 H4 H3; - intros; cases (aux l2 l3 n1); cases w in H4 (w1 w2); clear w; - intros 5; - simplify in match (\fst 〈?,?〉); simplify in match (\snd 〈?,?〉); - cases H5; - [2: apply le_S_S_to_le; apply (trans_le ???? H3); simplify; - rewrite < plus_n_Sm; apply le_S; apply le_n; - |3,4: apply (sorted_tail q2_lt); [2: apply H4|4:apply H6] - |5: intro; cases l2 in H7 H9; intros; [cases H9; reflexivity] - simplify in H7 ⊢ %; apply H7; intro; destruct H10; - |6: intro; cases l3 in H8 H9; intros; [cases H9; reflexivity] - simplify in H8 ⊢ %; apply H8; intro; destruct H10;] - clear aux H5; - simplify in match (\fst 〈?,?〉) in H9 H10 H11 H12; - simplify in match (\snd 〈?,?〉) in H9 H10 H11 H12; - split; - [1: left; reflexivity; - |2: cases H10; cases H12; clear H15 H16 H12 H7 H8 H11 H10; - cases H9; clear H9; - [1: lapply (H14 O) as K1; clear H14; change in K1 with (nth_base w1 O = nth_base w2 O); - split; - [1: apply (move_head_sorted ??? H4 H5 H7); STOP - - - - unfold rebase_spec_aux; intros; cases l1 in H2 H4 H6; intros; [ simplify in H2; destruct H2;] - lapply H6 as H7; [2: intro X; destruct X] clear H6 H5; - rewrite > H7; split; [right; simplify; - - split; [left;reflexivity] - split; - -,2: unfold rest; apply q_lt_minus; rewrite > q_plus_sym; rewrite > q_plus_OQ; - assumption; -|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 *) -] -elim devil; -qed. - -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)). - diff --git a/helm/software/matita/contribs/dama/dama/models/q_rebase.ma b/helm/software/matita/contribs/dama/dama/models/q_rebase.ma new file mode 100644 index 000000000..f2423954f --- /dev/null +++ b/helm/software/matita/contribs/dama/dama/models/q_rebase.ma @@ -0,0 +1,167 @@ +(**************************************************************************) +(* ___ *) +(* ||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_function.ma". + +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); lapply (H5 O) as K; + clear H1 H2 H3 H4 H5 H6 H7 Hres aux; 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); + 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);] + |3,4,5: rewrite < H8; assumption; + |9,10,11: rewrite < H9; assumption]] + split; cases (Hres (le_n ?) Hs1 He1 Hs2 He2); 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); + clear aux; intro K; simplify in K; 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; cases H8 in H13 H14 H15 Hi1 Hi2 H17 H18 H3 H16; intros; + [1: simplify in match (\fst 〈?,?〉) in H16 H17 H20 H21 H22 H23 ⊢ %; + cases (q_cmp (Qpos input) b1); + [1: do 2 (rewrite > value_head; [id|assumption]); reflexivity; + |2: do 2 (rewrite > value_tail;[id|assumption]); apply H16;] + |2: simplify in match (\fst 〈?,?〉) in H16 H17 H20 H21 H22 H23 ⊢ %; + cases (q_cmp (Qpos input) b1); + [1: rewrite > value_head; [2:assumption] + |2: rewrite > value_tail;[2:assumption] + simplify in H15; + ] + |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)). + -- 2.39.2