From 8d367045e504f594c280d2c87f906695ef9671ee Mon Sep 17 00:00:00 2001 From: Enrico Tassi Date: Tue, 15 Jul 2008 09:18:23 +0000 Subject: [PATCH] new q_function representation --- .../matita/contribs/dama/dama/depends | 37 +++--- .../matita/contribs/dama/dama/depends.png | Bin 239491 -> 35260 bytes .../contribs/dama/dama/models/list_support.ma | 37 +++--- .../contribs/dama/dama/models/q_bars.ma | 113 +++--------------- .../contribs/dama/dama/models/q_function.ma | 3 +- .../contribs/dama/dama/models/q_shift.ma | 106 ---------------- .../contribs/dama/dama/models/q_value_skip.ma | 34 ------ 7 files changed, 56 insertions(+), 274 deletions(-) diff --git a/helm/software/matita/contribs/dama/dama/depends b/helm/software/matita/contribs/dama/dama/depends index b466b2e07..419a80369 100644 --- a/helm/software/matita/contribs/dama/dama/depends +++ b/helm/software/matita/contribs/dama/dama/depends @@ -1,29 +1,29 @@ -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 cprop_connectives.ma datatypes/constructors.ma logic/equality.ma -nat_ordered_set.ma bishop_set.ma nat/compare.ma +russell_support.ma cprop_connectives.ma nat/nat.ma lebesgue.ma property_exhaustivity.ma sandwich.ma ordered_set.ma cprop_connectives.ma -russell_support.ma cprop_connectives.ma nat/nat.ma -models/nat_lebesgue.ma lebesgue.ma models/nat_order_continuous.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 +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_support.ma Q/q/qplus.ma Q/q/qtimes.ma cprop_connectives.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 cprop_connectives.ma list/list.ma nat/le_arith.ma nat/minus.ma models/q_bars.ma cprop_connectives.ma models/list_support.ma models/q_support.ma nat_ordered_set.ma -models/q_function.ma Q/q/qtimes.ma models/q_shift.ma nat_ordered_set.ma -models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma +models/q_support.ma Q/q/qplus.ma Q/q/qtimes.ma cprop_connectives.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/q_value_skip.ma models/q_shift.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 +models/nat_uniform.ma models/discrete_uniformity.ma nat_ordered_set.ma +models/q_value_skip.ma Q/q/qplus.ma Q/q/qtimes.ma datatypes/constructors.ma @@ -31,5 +31,6 @@ list/list.ma logic/equality.ma nat/compare.ma nat/le_arith.ma +nat/minus.ma nat/nat.ma nat/plus.ma diff --git a/helm/software/matita/contribs/dama/dama/depends.png b/helm/software/matita/contribs/dama/dama/depends.png index 95829b4db95a05ae707ba728c11128866fbbf9ff..4fb8e3fd71e93076cfc0327fb6c91e1b028251ef 100644 GIT binary patch literal 35260 zcmZ7dby$?o_dkv=CEc+!NGbwLDhSf0fRfTK9g2c9xYE6JEg>zXA`MC`olAq#-QBWF zes7+y&-=Q5*YB@;?wNC+Idf+2na7-YOoY};RZ=2GA`l2f`dm#(2L!rj1_I%l;Nk!| 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\len (l1@l2) = \len l1 + \len l2. intros; elim l1; [reflexivity] simplify; rewrite < H; reflexivity; qed. -inductive non_empty_list (A:Type) : list A → Type := +coinductive non_empty_list (A:Type) : list A → Type := | show_head: ∀x,l. non_empty_list A (x::l). lemma len_gt_non_empty : @@ -97,21 +97,6 @@ cases i in H2; destruct H6; simplify; assumption;] 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 (bars_not_nil f); intros; -cases (cmp_nat i (len l)); -[1: lapply (sorted_tail_bigger ?? H ? H2) as K; simplify in H1; - rewrite > H1 in K; apply K; -|2: rewrite > H2; simplify; elim l; simplify; [apply (q_pos_OQ one)] - assumption; -|3: simplify; elim l in i H2;[simplify; rewrite > nth_nil; apply (q_pos_OQ one)] - cases n in H3; intros; [cases (not_le_Sn_O ? H3)] apply (H2 n1); - apply (le_S_S_to_le ?? H3);] -qed. -*) - (* move in nat/ *) lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m. intros; rewrite > sym_plus; apply (le_S_S n (m+n)); alias id "le_plus_n" = "cic:/matita/nat/le_arith/le_plus_n.con". @@ -162,7 +147,7 @@ intros 2 (r l); elim l; |2: apply H3; assumption]] qed. -inductive cases_bool (p:bool) : bool → CProp ≝ +coinductive cases_bool (p:bool) : bool → CProp ≝ | case_true : p = true → cases_bool p true | cases_false : p = false → cases_bool p false. @@ -170,6 +155,22 @@ lemma case_b : ∀A:Type.∀f:A → bool. ∀x.cases_bool (f x) (f x). intros; cases (f x);[left;|right] reflexivity; qed. +coinductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝ +| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l. + +lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l. +intros 2; elim n; +[1: elim l in H; [cases (not_le_Sn_O ? H)] + apply (break_to ?? ? [] a l1); reflexivity; +|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros; + [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1; + rewrite > len_append in H1; rewrite > plus_n_SO in H1; + cases (not_le_Sn_n ? H1); + |2: apply (break_to ?? ? (l1@[x]) t l3); + [2: simplify; rewrite > associative_append; assumption; + |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]] +qed. + include "cprop_connectives.ma". definition eject_N ≝ @@ -178,7 +179,7 @@ coercion eject_N. definition inject_N ≝ λP.λp:nat.λh:P p. ex_introT ? P p h. coercion inject_N with 0 1 nocomposites. -inductive find_spec (T:Type) (P:T→bool) (l:list T) (d:T) (res : nat) : nat → CProp ≝ +coinductive find_spec (T:Type) (P:T→bool) (l:list T) (d:T) (res : nat) : nat → CProp ≝ | found: ∀i. i < \len l → P (\nth l d i) = true → res = i → (∀j. j < i → P (\nth l d j) = false) → find_spec T P l d res i 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 65066590f..7279fe8c0 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_bars.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_bars.ma @@ -17,12 +17,12 @@ include "models/q_support.ma". include "models/list_support.ma". include "cprop_connectives.ma". -definition bar ≝ ℚ × ℚ. +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〉. +definition empty_bar : bar ≝ 〈Qpos one,〈OQ,OQ〉〉. notation "\rect" with precedence 90 for @{'empty_bar}. interpretation "q0" 'empty_bar = empty_bar. @@ -55,7 +55,7 @@ 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 + bars_end_OQ : nth_height bars (pred (\len bars)) = 〈OQ,OQ〉 }. lemma len_bases_gt_O: ∀f.O < \len (bars f). @@ -75,62 +75,17 @@ cases (cmp_nat (\len l) i); apply (H2 n1); simplify in H3; apply (le_S_S_to_le ?? H3);] qed. -(* -lemma lt_n_plus_n_Sm : ∀n,m:nat.n < n + S m. -intros; rewrite > sym_plus; apply (le_S_S n (m+n)); apply (le_plus_n m n); qed. -*) - -(* -lemma all_bigger_can_concat_bigger: - ∀l1,l2,start,b,x,n. - (∀i.i< len l1 → nth_base l1 i < \fst b) → - (∀i.i< len l2 → \fst b ≤ nth_base l2 i) → - (∀i.i< len l1 → start ≤ i → x ≤ nth_base l1 i) → - start ≤ n → n < len (l1@b::l2) → x ≤ \fst b → x ≤ nth_base (l1@b::l2) n. -intros; cases (cmp_nat n (len l1)); -[1: unfold nth_base; rewrite > (nth_concat_lt_len ????? H6); - apply (H2 n); assumption; -|2: rewrite > H6; unfold nth_base; rewrite > nth_len; assumption; -|3: unfold nth_base; rewrite > nth_concat_ge_len; [2: apply lt_to_le; assumption] - rewrite > len_concat in H4; simplify in H4; rewrite < plus_n_Sm in H4; - lapply linear le_S_S_to_le to H4 as K; rewrite > sym_plus in K; - lapply linear le_plus_to_minus to K as X; - generalize in match X; generalize in match (n - len l1); intro W; cases W; clear W X; - [intros; assumption] intros; - apply (q_le_trans ??? H5); apply (H1 n1); assumption;] -qed. -*) - - -inductive value_spec (f : q_f) (i : ℚ) : ℚ → nat → CProp ≝ - value_of : ∀q,j. - 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 j. - - -inductive break_spec (T : Type) (n : nat) (l : list T) : list T → CProp ≝ -| break_to: ∀l1,x,l2. \len l1 = n → l = l1 @ [x] @ l2 → break_spec T n l l. - -lemma list_break: ∀T,n,l. n < \len l → break_spec T n l l. -intros 2; elim n; -[1: elim l in H; [cases (not_le_Sn_O ? H)] - apply (break_to ?? ? [] a l1); reflexivity; -|2: cases (H l); [2: apply lt_S_to_lt; assumption;] cases l2 in H3; intros; - [1: rewrite < H2 in H1; rewrite > H3 in H1; rewrite > append_nil in H1; - rewrite > len_append in H1; rewrite > plus_n_SO in H1; - cases (not_le_Sn_n ? H1); - |2: apply (break_to ?? ? (l1@[x]) t l3); - [2: simplify; rewrite > associative_append; assumption; - |1: rewrite < H2; rewrite > len_append; rewrite > plus_n_SO; reflexivity]]] -qed. - -definition value : ∀f:q_f.∀i:ratio.∃p:ℚ.∃j.value_spec f (Qpos i) p j. +coinductive value_spec (f : q_f) (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. + +definition value_lemma : ∀f:q_f.∀i:ratio.∃p:ℚ×ℚ.value_spec f (Qpos i) p. intros; letin P ≝ (λx:bar.match q_cmp (Qpos i) (\fst x) with[ q_leq _ ⇒ true| q_gt _ ⇒ false]); exists [apply (nth_height (bars f) (pred (find ? P (bars f) ▭)));] -exists [apply (pred (find ? P (bars f) ▭))] apply value_of; +apply (value_of ?? (pred (find ? P (bars f) ▭))); [1: reflexivity |2: cases (cases_find bar P (bars f) ▭); [1: cases i1 in H H1 H2 H3; simplify; intros; @@ -175,55 +130,21 @@ exists [apply (pred (find ? P (bars f) ▭))] apply value_of; [ apply le_O_n; | assumption]]] qed. -lemma value_OQ_l: - ∀l,i.i < start l → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H H6); -qed. - -lemma value_OQ_r: - ∀l,i.start l + sum_bases (bars l) (len (bars l)) ≤ i → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (q_lt_le_incompat ?? H7 H); -qed. - -lemma value_OQ_e: - ∀l,i.bars l = [] → \snd (\fst (value l i)) = OQ. -intros; cases (value l i) (q Hq); cases Hq; clear Hq; simplify; cases H1; clear H1; -try assumption; cases H2; cases (?:False); apply (H1 H); -qed. +lemma value : q_f → ratio → ℚ × ℚ. +intros; cases (value_lemma q r); apply w; qed. -inductive value_ok_spec (f : q_f) (i : ℚ) : nat × ℚ → Type ≝ - | value_ok : ∀n,q. n ≤ (len (bars f)) → - q = \snd (nth (bars f) ▭ n) → - sum_bases (bars f) n ≤ ⅆ[i,start f] → - ⅆ[i, start f] < sum_bases (bars f) (S n) → value_ok_spec f i 〈n,q〉. - -lemma value_ok: - ∀f,i.bars f ≠ [] → start f ≤ i → i < start f + sum_bases (bars f) (len (bars f)) → - value_ok_spec f i (\fst (value f i)). -intros; cases (value f i); simplify; -cases H3; simplify; clear H3; cases H4; clear H4; -[1,2,3: cases (?:False); - [1: apply (q_lt_le_incompat ?? H3 H1); - |2: apply (q_lt_le_incompat ?? H2 H3); - |3: apply (H H3);] -|4: cases H7; clear H7; cases w in H3 H4 H5 H6 H8; simplify; intros; - constructor 1; 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. -definition same_values ≝ - λl1,l2:q_f. - ∀input.\snd (\fst (value l1 input)) = \snd (\fst (value l2 input)). +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)). +definition same_bases ≝ λl1,l2:list bar. ∀i.\fst (\nth l1 ▭ i) = \fst (\nth l2 ▭ i). 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)|apply (q_neg_gt ? H)] +[ 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}. 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 b1ff639e4..346ee5843 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_function.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_function.ma @@ -12,8 +12,7 @@ (* *) (**************************************************************************) -include "nat_ordered_set.ma". -include "models/q_shift.ma". +include "models/q_bars.ma". alias symbol "pi2" = "pair pi2". alias symbol "pi1" = "pair pi1". diff --git a/helm/software/matita/contribs/dama/dama/models/q_shift.ma b/helm/software/matita/contribs/dama/dama/models/q_shift.ma index a63066659..e69de29bb 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_shift.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_shift.ma @@ -1,106 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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. - diff --git a/helm/software/matita/contribs/dama/dama/models/q_value_skip.ma b/helm/software/matita/contribs/dama/dama/models/q_value_skip.ma index e639582e1..e69de29bb 100644 --- a/helm/software/matita/contribs/dama/dama/models/q_value_skip.ma +++ b/helm/software/matita/contribs/dama/dama/models/q_value_skip.ma @@ -1,34 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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_shift.ma". - -lemma q_cmp2: - ∀a,b:ℚ.a < b ∨ b ≤ a. -intros; cases (q_cmp a b); -[1: right; apply q_eq_to_le; symmetry; assumption; -|2: left; assumption; -|3: right; apply q_lt_to_le; assumption;] -qed. - - - -lemma value_skip: - ∀f,i,b,tl. bars f = b::tl → start f + Qpos (\fst b) < i → - \fst (value f i) = \fst (value (mk_q_f (start f + Qpos (\fst b)) tl) i). -intros; cases (value f i); cases H2; clear H2; -[1: cases (?:False); apply (q_lt_corefl (start f)); cases H3; clear H3 H4 H5; - apply (q_lt_trans ???? H2); -|2: -simplify in ⊢ (? ? % ?); -- 2.39.2