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author | chaieb |

Tue, 05 Jun 2007 20:44:12 +0200 | |

changeset 23264 | 324622260d29 |

parent 23263 | 0c227412b285 |

child 23265 | a6992b91fdde |

Added twe Examples for Quantifier elimination ofer linear real arithmetic and over the mixed theory of linear real artihmetic with integers

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Complex/ex/MIR.thy Tue Jun 05 20:44:12 2007 +0200 @@ -0,0 +1,5784 @@ +(* Title: Complex/ex/MIR.thy + Author: Amine Chaieb +*) + +header {* Quatifier elimination for R(0,1,+,floor,<) *} + +theory MIR + imports Real GCD + uses ("mireif.ML") ("mirtac.ML") + begin + +declare real_of_int_floor_cancel [simp del] + + (* All pairs from two lists *) + +lemma allpairs_set: "set (allpairs Pair xs ys) = {(x,y). x\<in> set xs \<and> y \<in> set ys}" +by (induct xs) auto + +fun alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where + "alluopairs [] = []" +| "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)" + +lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}" +by (induct xs, auto) + +lemma alluopairs_set: + "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) " +by (induct xs, auto) + +lemma alluopairs_ex: + assumes Pc: "\<forall> x y. P x y = P y x" + shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)" +proof + assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" + then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y" by blast + from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" + by auto +next + assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y" + then obtain "x" and "y" where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+ + from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast + with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast +qed + + (* generate a list from i to j*) +consts iupt :: "int \<times> int \<Rightarrow> int list" +recdef iupt "measure (\<lambda> (i,j). nat (j-i +1))" + "iupt (i,j) = (if j <i then [] else (i# iupt(i+1, j)))" + +lemma iupt_set: "set (iupt(i,j)) = {i .. j}" +proof(induct rule: iupt.induct) + case (1 a b) + show ?case + using prems by (simp add: simp_from_to) +qed + +lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)" +using Nat.gr0_conv_Suc +by clarsimp + + +lemma myl: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a \<le> b) = (0 \<le> b - a)" +proof(clarify) + fix x y ::"'a" + have "(x \<le> y) = (x - y \<le> 0)" by (simp only: le_iff_diff_le_0[where a="x" and b="y"]) + also have "\<dots> = (- (y - x) \<le> 0)" by simp + also have "\<dots> = (0 \<le> y - x)" by (simp only: neg_le_0_iff_le[where a="y-x"]) + finally show "(x \<le> y) = (0 \<le> y - x)" . +qed + +lemma myless: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a < b) = (0 < b - a)" +proof(clarify) + fix x y ::"'a" + have "(x < y) = (x - y < 0)" by (simp only: less_iff_diff_less_0[where a="x" and b="y"]) + also have "\<dots> = (- (y - x) < 0)" by simp + also have "\<dots> = (0 < y - x)" by (simp only: neg_less_0_iff_less[where a="y-x"]) + finally show "(x < y) = (0 < y - x)" . +qed + +lemma myeq: "\<forall> (a::'a::{pordered_ab_group_add}) (b::'a). (a = b) = (0 = b - a)" + by auto + + (* Maybe should be added to the library \<dots> *) +lemma floor_int_eq: "(real n\<le> x \<and> x < real (n+1)) = (floor x = n)" +proof( auto) + assume lb: "real n \<le> x" + and ub: "x < real n + 1" + have "real (floor x) \<le> x" by simp + hence "real (floor x) < real (n + 1) " using ub by arith + hence "floor x < n+1" by simp + moreover from lb have "n \<le> floor x" using floor_mono2[where x="real n" and y="x"] + by simp ultimately show "floor x = n" by simp +qed + +(* Periodicity of dvd *) +lemma dvd_period: + assumes advdd: "(a::int) dvd d" + shows "(a dvd (x + t)) = (a dvd ((x+ c*d) + t))" + using advdd +proof- + from advdd have "\<forall>x.\<forall>k. (((a::int) dvd (x + t)) = (a dvd (x+k*d + t)))" + by (rule dvd_modd_pinf) + then show ?thesis by simp +qed + + (* The Divisibility relation between reals *) +consts rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl 50) +defs rdvd_def: "x rdvd y \<equiv> \<exists> (k::int). y=x*(real k)" + +lemma int_rdvd_real: + shows "real (i::int) rdvd x = (i dvd (floor x) \<and> real (floor x) = x)" (is "?l = ?r") +proof + assume "?l" + hence th: "\<exists> k. x=real (i*k)" by (simp add: rdvd_def) + hence th': "real (floor x) = x" by (auto simp del: real_of_int_mult) + with th have "\<exists> k. real (floor x) = real (i*k)" by simp + hence "\<exists> k. floor x = i*k" by (simp only: real_of_int_inject) + thus ?r using th' by (simp add: dvd_def) +next + assume "?r" hence "(i\<Colon>int) dvd \<lfloor>x\<Colon>real\<rfloor>" .. + hence "\<exists> k. real (floor x) = real (i*k)" + by (simp only: real_of_int_inject) (simp add: dvd_def) + thus ?l using prems by (simp add: rdvd_def) +qed + +lemma int_rdvd_iff: "(real (i::int) rdvd real t) = (i dvd t)" +by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only :real_of_int_mult[symmetric]) + + +lemma rdvd_abs1: + "(abs (real d) rdvd t) = (real (d ::int) rdvd t)" +proof + assume d: "real d rdvd t" + from d int_rdvd_real have d2: "d dvd (floor t)" and ti: "real (floor t) = t" by auto + + from iffD1[OF zdvd_abs1] d2 have "(abs d) dvd (floor t)" by blast + with ti int_rdvd_real[symmetric] have "real (abs d) rdvd t" by blast + thus "abs (real d) rdvd t" by simp +next + assume "abs (real d) rdvd t" hence "real (abs d) rdvd t" by simp + with int_rdvd_real[where i="abs d" and x="t"] have d2: "abs d dvd floor t" and ti: "real (floor t) =t" by auto + from iffD2[OF zdvd_abs1] d2 have "d dvd floor t" by blast + with ti int_rdvd_real[symmetric] show "real d rdvd t" by blast +qed + +lemma rdvd_minus: "(real (d::int) rdvd t) = (real d rdvd -t)" + apply (auto simp add: rdvd_def) + apply (rule_tac x="-k" in exI, simp) + apply (rule_tac x="-k" in exI, simp) +done + +lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)" +by (auto simp add: rdvd_def) + +lemma rdvd_mult: + assumes knz: "k\<noteq>0" + shows "(real (n::int) * real (k::int) rdvd x * real k) = (real n rdvd x)" +using knz by (simp add:rdvd_def) + +lemma rdvd_trans: assumes mn:"m rdvd n" and nk:"n rdvd k" + shows "m rdvd k" +proof- + from rdvd_def mn obtain c where nmc:"n = m * real (c::int)" by auto + from rdvd_def nk obtain c' where nkc:"k = n * real (c'::int)" by auto + hence "k = m * real (c * c')" using nmc by simp + thus ?thesis using rdvd_def by blast +qed + + (*********************************************************************************) + (**** SHADOW SYNTAX AND SEMANTICS ****) + (*********************************************************************************) + +datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num + | Mul int num | Floor num| CF int num num + + (* A size for num to make inductive proofs simpler*) +fun num_size :: "num \<Rightarrow> nat" where + "num_size (C c) = 1" +| "num_size (Bound n) = 1" +| "num_size (Neg a) = 1 + num_size a" +| "num_size (Add a b) = 1 + num_size a + num_size b" +| "num_size (Sub a b) = 3 + num_size a + num_size b" +| "num_size (CN n c a) = 4 + num_size a " +| "num_size (CF c a b) = 4 + num_size a + num_size b" +| "num_size (Mul c a) = 1 + num_size a" +| "num_size (Floor a) = 1 + num_size a" + + (* Semantics of numeral terms (num) *) +fun Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where + "Inum bs (C c) = (real c)" +| "Inum bs (Bound n) = bs!n" +| "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)" +| "Inum bs (Neg a) = -(Inum bs a)" +| "Inum bs (Add a b) = Inum bs a + Inum bs b" +| "Inum bs (Sub a b) = Inum bs a - Inum bs b" +| "Inum bs (Mul c a) = (real c) * Inum bs a" +| "Inum bs (Floor a) = real (floor (Inum bs a))" +| "Inum bs (CF c a b) = real c * real (floor (Inum bs a)) + Inum bs b" +definition "isint t bs \<equiv> real (floor (Inum bs t)) = Inum bs t" + +lemma isint_iff: "isint n bs = (real (floor (Inum bs n)) = Inum bs n)" +by (simp add: isint_def) + +lemma isint_Floor: "isint (Floor n) bs" + by (simp add: isint_iff) + +lemma isint_Mul: "isint e bs \<Longrightarrow> isint (Mul c e) bs" +proof- + let ?e = "Inum bs e" + let ?fe = "floor ?e" + assume be: "isint e bs" hence efe:"real ?fe = ?e" by (simp add: isint_iff) + have "real ((floor (Inum bs (Mul c e)))) = real (floor (real (c * ?fe)))" using efe by simp + also have "\<dots> = real (c* ?fe)" by (simp only: floor_real_of_int) + also have "\<dots> = real c * ?e" using efe by simp + finally show ?thesis using isint_iff by simp +qed + +lemma isint_neg: "isint e bs \<Longrightarrow> isint (Neg e) bs" +proof- + let ?I = "\<lambda> t. Inum bs t" + assume ie: "isint e bs" + hence th: "real (floor (?I e)) = ?I e" by (simp add: isint_def) + have "real (floor (?I (Neg e))) = real (floor (- (real (floor (?I e)))))" by (simp add: th) + also have "\<dots> = - real (floor (?I e))" by(simp add: floor_minus_real_of_int) + finally show "isint (Neg e) bs" by (simp add: isint_def th) +qed + +lemma isint_sub: + assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs" +proof- + let ?I = "\<lambda> t. Inum bs t" + from ie have th: "real (floor (?I e)) = ?I e" by (simp add: isint_def) + have "real (floor (?I (Sub (C c) e))) = real (floor ((real (c -floor (?I e)))))" by (simp add: th) + also have "\<dots> = real (c- floor (?I e))" by(simp add: floor_minus_real_of_int) + finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th) +qed + +lemma isint_add: assumes + ai:"isint a bs" and bi: "isint b bs" shows "isint (Add a b) bs" +proof- + let ?a = "Inum bs a" + let ?b = "Inum bs b" + from ai bi isint_iff have "real (floor (?a + ?b)) = real (floor (real (floor ?a) + real (floor ?b)))" by simp + also have "\<dots> = real (floor ?a) + real (floor ?b)" by simp + also have "\<dots> = ?a + ?b" using ai bi isint_iff by simp + finally show "isint (Add a b) bs" by (simp add: isint_iff) +qed + +lemma isint_c: "isint (C j) bs" + by (simp add: isint_iff) + + + (* FORMULAE *) +datatype fm = + T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num| + NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm + + + (* A size for fm *) +fun fmsize :: "fm \<Rightarrow> nat" where + "fmsize (NOT p) = 1 + fmsize p" +| "fmsize (And p q) = 1 + fmsize p + fmsize q" +| "fmsize (Or p q) = 1 + fmsize p + fmsize q" +| "fmsize (Imp p q) = 3 + fmsize p + fmsize q" +| "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" +| "fmsize (E p) = 1 + fmsize p" +| "fmsize (A p) = 4+ fmsize p" +| "fmsize (Dvd i t) = 2" +| "fmsize (NDvd i t) = 2" +| "fmsize p = 1" + (* several lemmas about fmsize *) +lemma fmsize_pos: "fmsize p > 0" +by (induct p rule: fmsize.induct) simp_all + + (* Semantics of formulae (fm) *) +fun Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where + "Ifm bs T = True" +| "Ifm bs F = False" +| "Ifm bs (Lt a) = (Inum bs a < 0)" +| "Ifm bs (Gt a) = (Inum bs a > 0)" +| "Ifm bs (Le a) = (Inum bs a \<le> 0)" +| "Ifm bs (Ge a) = (Inum bs a \<ge> 0)" +| "Ifm bs (Eq a) = (Inum bs a = 0)" +| "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)" +| "Ifm bs (Dvd i b) = (real i rdvd Inum bs b)" +| "Ifm bs (NDvd i b) = (\<not>(real i rdvd Inum bs b))" +| "Ifm bs (NOT p) = (\<not> (Ifm bs p))" +| "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)" +| "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)" +| "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))" +| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)" +| "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)" +| "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)" + +consts prep :: "fm \<Rightarrow> fm" +recdef prep "measure fmsize" + "prep (E T) = T" + "prep (E F) = F" + "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))" + "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))" + "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" + "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))" + "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" + "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" + "prep (E p) = E (prep p)" + "prep (A (And p q)) = And (prep (A p)) (prep (A q))" + "prep (A p) = prep (NOT (E (NOT p)))" + "prep (NOT (NOT p)) = prep p" + "prep (NOT (And p q)) = Or (prep (NOT p)) (prep (NOT q))" + "prep (NOT (A p)) = prep (E (NOT p))" + "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))" + "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))" + "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))" + "prep (NOT p) = NOT (prep p)" + "prep (Or p q) = Or (prep p) (prep q)" + "prep (And p q) = And (prep p) (prep q)" + "prep (Imp p q) = prep (Or (NOT p) q)" + "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))" + "prep p = p" +(hints simp add: fmsize_pos) +lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p" +by (induct p rule: prep.induct, auto) + + + (* Quantifier freeness *) +consts qfree:: "fm \<Rightarrow> bool" +recdef qfree "measure size" + "qfree (E p) = False" + "qfree (A p) = False" + "qfree (NOT p) = qfree p" + "qfree (And p q) = (qfree p \<and> qfree q)" + "qfree (Or p q) = (qfree p \<and> qfree q)" + "qfree (Imp p q) = (qfree p \<and> qfree q)" + "qfree (Iff p q) = (qfree p \<and> qfree q)" + "qfree p = True" + + (* Boundedness and substitution *) +consts + numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) + bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) + numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) + subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) +primrec + "numbound0 (C c) = True" + "numbound0 (Bound n) = (n>0)" + "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)" + "numbound0 (Neg a) = numbound0 a" + "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)" + "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" + "numbound0 (Mul i a) = numbound0 a" + "numbound0 (Floor a) = numbound0 a" + "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)" +lemma numbound0_I: + assumes nb: "numbound0 a" + shows "Inum (b#bs) a = Inum (b'#bs) a" +using nb +by (induct a rule: numbound0.induct) (auto simp add: nth_pos2) + + +lemma numbound0_gen: + assumes nb: "numbound0 t" and ti: "isint t (x#bs)" + shows "\<forall> y. isint t (y#bs)" +using nb ti +proof(clarify) + fix y + from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def] + show "isint t (y#bs)" + by (simp add: isint_def) +qed + +primrec + "bound0 T = True" + "bound0 F = True" + "bound0 (Lt a) = numbound0 a" + "bound0 (Le a) = numbound0 a" + "bound0 (Gt a) = numbound0 a" + "bound0 (Ge a) = numbound0 a" + "bound0 (Eq a) = numbound0 a" + "bound0 (NEq a) = numbound0 a" + "bound0 (Dvd i a) = numbound0 a" + "bound0 (NDvd i a) = numbound0 a" + "bound0 (NOT p) = bound0 p" + "bound0 (And p q) = (bound0 p \<and> bound0 q)" + "bound0 (Or p q) = (bound0 p \<and> bound0 q)" + "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))" + "bound0 (Iff p q) = (bound0 p \<and> bound0 q)" + "bound0 (E p) = False" + "bound0 (A p) = False" + +lemma bound0_I: + assumes bp: "bound0 p" + shows "Ifm (b#bs) p = Ifm (b'#bs) p" +using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] +by (induct p rule: bound0.induct) (auto simp add: nth_pos2) + +primrec + "numsubst0 t (C c) = (C c)" + "numsubst0 t (Bound n) = (if n=0 then t else Bound n)" + "numsubst0 t (CN n i a) = (if n=0 then Add (Mul i t) (numsubst0 t a) else CN n i (numsubst0 t a))" + "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)" + "numsubst0 t (Neg a) = Neg (numsubst0 t a)" + "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)" + "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" + "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)" + "numsubst0 t (Floor a) = Floor (numsubst0 t a)" + +lemma numsubst0_I: + shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t" + by (induct t) (simp_all add: nth_pos2) + +lemma numsubst0_I': + assumes nb: "numbound0 a" + shows "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t" + by (induct t) (simp_all add: nth_pos2 numbound0_I[OF nb, where b="b" and b'="b'"]) + + +primrec + "subst0 t T = T" + "subst0 t F = F" + "subst0 t (Lt a) = Lt (numsubst0 t a)" + "subst0 t (Le a) = Le (numsubst0 t a)" + "subst0 t (Gt a) = Gt (numsubst0 t a)" + "subst0 t (Ge a) = Ge (numsubst0 t a)" + "subst0 t (Eq a) = Eq (numsubst0 t a)" + "subst0 t (NEq a) = NEq (numsubst0 t a)" + "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)" + "subst0 t (NDvd i a) = NDvd i (numsubst0 t a)" + "subst0 t (NOT p) = NOT (subst0 t p)" + "subst0 t (And p q) = And (subst0 t p) (subst0 t q)" + "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)" + "subst0 t (Imp p q) = Imp (subst0 t p) (subst0 t q)" + "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)" + +lemma subst0_I: assumes qfp: "qfree p" + shows "Ifm (b#bs) (subst0 a p) = Ifm ((Inum (b#bs) a)#bs) p" + using qfp numsubst0_I[where b="b" and bs="bs" and a="a"] + by (induct p) (simp_all add: nth_pos2 ) + +consts + decrnum:: "num \<Rightarrow> num" + decr :: "fm \<Rightarrow> fm" + +recdef decrnum "measure size" + "decrnum (Bound n) = Bound (n - 1)" + "decrnum (Neg a) = Neg (decrnum a)" + "decrnum (Add a b) = Add (decrnum a) (decrnum b)" + "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" + "decrnum (Mul c a) = Mul c (decrnum a)" + "decrnum (Floor a) = Floor (decrnum a)" + "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" + "decrnum (CF c a b) = CF c (decrnum a) (decrnum b)" + "decrnum a = a" + +recdef decr "measure size" + "decr (Lt a) = Lt (decrnum a)" + "decr (Le a) = Le (decrnum a)" + "decr (Gt a) = Gt (decrnum a)" + "decr (Ge a) = Ge (decrnum a)" + "decr (Eq a) = Eq (decrnum a)" + "decr (NEq a) = NEq (decrnum a)" + "decr (Dvd i a) = Dvd i (decrnum a)" + "decr (NDvd i a) = NDvd i (decrnum a)" + "decr (NOT p) = NOT (decr p)" + "decr (And p q) = And (decr p) (decr q)" + "decr (Or p q) = Or (decr p) (decr q)" + "decr (Imp p q) = Imp (decr p) (decr q)" + "decr (Iff p q) = Iff (decr p) (decr q)" + "decr p = p" + +lemma decrnum: assumes nb: "numbound0 t" + shows "Inum (x#bs) t = Inum bs (decrnum t)" + using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) + +lemma decr: assumes nb: "bound0 p" + shows "Ifm (x#bs) p = Ifm bs (decr p)" + using nb + by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) + +lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" +by (induct p, simp_all) + +consts + isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) +recdef isatom "measure size" + "isatom T = True" + "isatom F = True" + "isatom (Lt a) = True" + "isatom (Le a) = True" + "isatom (Gt a) = True" + "isatom (Ge a) = True" + "isatom (Eq a) = True" + "isatom (NEq a) = True" + "isatom (Dvd i b) = True" + "isatom (NDvd i b) = True" + "isatom p = False" + +lemma numsubst0_numbound0: assumes nb: "numbound0 t" + shows "numbound0 (numsubst0 t a)" +using nb by (induct a rule: numsubst0.induct, auto) + +lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t" + shows "bound0 (subst0 t p)" +using qf numsubst0_numbound0[OF nb] by (induct p rule: subst0.induct, auto) + +lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" +by (induct p, simp_all) + + +constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" + "djf f p q \<equiv> (if q=T then T else if q=F then f p else + (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))" +constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" + "evaldjf f ps \<equiv> foldr (djf f) ps F" + +lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" +by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) +(cases "f p", simp_all add: Let_def djf_def) + +lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))" + by(induct ps, simp_all add: evaldjf_def djf_Or) + +lemma evaldjf_bound0: + assumes nb: "\<forall> x\<in> set xs. bound0 (f x)" + shows "bound0 (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +lemma evaldjf_qf: + assumes nb: "\<forall> x\<in> set xs. qfree (f x)" + shows "qfree (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +consts + disjuncts :: "fm \<Rightarrow> fm list" + conjuncts :: "fm \<Rightarrow> fm list" +recdef disjuncts "measure size" + "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" + "disjuncts F = []" + "disjuncts p = [p]" + +recdef conjuncts "measure size" + "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)" + "conjuncts T = []" + "conjuncts p = [p]" +lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: disjuncts.induct, auto) +lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: conjuncts.induct, auto) + +lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed +lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed + +lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (disjuncts p)" + by (induct p rule: disjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed +lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (conjuncts p)" + by (induct p rule: conjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed + +constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" + "DJ f p \<equiv> evaldjf f (disjuncts p)" + +lemma DJ: assumes fdj: "\<forall> p q. f (Or p q) = Or (f p) (f q)" + and fF: "f F = F" + shows "Ifm bs (DJ f p) = Ifm bs (f p)" +proof- + have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) + finally show ?thesis . +qed + +lemma DJ_qf: assumes + fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)" + shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) " +proof(clarify) + fix p assume qf: "qfree p" + have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) + from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" . + with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast + + from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp +qed + +lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" + shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))" +proof(clarify) + fix p::fm and bs + assume qf: "qfree p" + from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast + from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto + have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto + also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) + finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast +qed + (* Simplification *) + + (* Algebraic simplifications for nums *) +consts bnds:: "num \<Rightarrow> nat list" + lex_ns:: "nat list \<times> nat list \<Rightarrow> bool" +recdef bnds "measure size" + "bnds (Bound n) = [n]" + "bnds (CN n c a) = n#(bnds a)" + "bnds (Neg a) = bnds a" + "bnds (Add a b) = (bnds a)@(bnds b)" + "bnds (Sub a b) = (bnds a)@(bnds b)" + "bnds (Mul i a) = bnds a" + "bnds (Floor a) = bnds a" + "bnds (CF c a b) = (bnds a)@(bnds b)" + "bnds a = []" +recdef lex_ns "measure (\<lambda> (xs,ys). length xs + length ys)" + "lex_ns ([], ms) = True" + "lex_ns (ns, []) = False" + "lex_ns (n#ns, m#ms) = (n<m \<or> ((n = m) \<and> lex_ns (ns,ms))) " +constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" + "lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)" + +consts + numgcd :: "num \<Rightarrow> int" + numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" + reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" + reducecoeff :: "num \<Rightarrow> num" + dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" +consts maxcoeff:: "num \<Rightarrow> int" +recdef maxcoeff "measure size" + "maxcoeff (C i) = abs i" + "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)" + "maxcoeff (CF c t s) = max (abs c) (maxcoeff s)" + "maxcoeff t = 1" + +lemma maxcoeff_pos: "maxcoeff t \<ge> 0" + apply (induct t rule: maxcoeff.induct, auto) + done + +recdef numgcdh "measure size" + "numgcdh (C i) = (\<lambda>g. igcd i g)" + "numgcdh (CN n c t) = (\<lambda>g. igcd c (numgcdh t g))" + "numgcdh (CF c s t) = (\<lambda>g. igcd c (numgcdh t g))" + "numgcdh t = (\<lambda>g. 1)" +defs numgcd_def: "numgcd t \<equiv> numgcdh t (maxcoeff t)" + +recdef reducecoeffh "measure size" + "reducecoeffh (C i) = (\<lambda> g. C (i div g))" + "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))" + "reducecoeffh (CF c s t) = (\<lambda> g. CF (c div g) s (reducecoeffh t g))" + "reducecoeffh t = (\<lambda>g. t)" + +defs reducecoeff_def: "reducecoeff t \<equiv> + (let g = numgcd t in + if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" + +recdef dvdnumcoeff "measure size" + "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)" + "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" + "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" + "dvdnumcoeff t = (\<lambda>g. False)" + +lemma dvdnumcoeff_trans: + assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" + shows "dvdnumcoeff t g" + using dgt' gdg + by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg]) + +declare zdvd_trans [trans add] + +lemma natabs0: "(nat (abs x) = 0) = (x = 0)" +by arith + +lemma numgcd0: + assumes g0: "numgcd t = 0" + shows "Inum bs t = 0" +proof- + have "\<And>x. numgcdh t x= 0 \<Longrightarrow> Inum bs t = 0" + by (induct t rule: numgcdh.induct, auto simp add: igcd_def gcd_zero natabs0 max_def maxcoeff_pos) + thus ?thesis using g0[simplified numgcd_def] by blast +qed + +lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0" + using gp + by (induct t rule: numgcdh.induct, auto simp add: igcd_def) + +lemma numgcd_pos: "numgcd t \<ge>0" + by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) + +lemma reducecoeffh: + assumes gt: "dvdnumcoeff t g" and gp: "g > 0" + shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t" + using gt +proof(induct t rule: reducecoeffh.induct) + case (1 i) hence gd: "g dvd i" by simp + from gp have gnz: "g \<noteq> 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd]) +next + case (2 n c t) hence gd: "g dvd c" by simp + from gp have gnz: "g \<noteq> 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_eq_simps) +next + case (3 c s t) hence gd: "g dvd c" by simp + from gp have gnz: "g \<noteq> 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_eq_simps) +qed (auto simp add: numgcd_def gp) +consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" +recdef ismaxcoeff "measure size" + "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)" + "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))" + "ismaxcoeff (CF c s t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))" + "ismaxcoeff t = (\<lambda>x. True)" + +lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'" +by (induct t rule: ismaxcoeff.induct, auto) + +lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence H:"ismaxcoeff t (maxcoeff t)" . + have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2) + from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1) +next + case (3 c t s) + hence H1:"ismaxcoeff s (maxcoeff s)" by auto + have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def) + from ismaxcoeff_mono[OF H1 thh1] show ?case by (simp add: le_maxI1) +qed simp_all + +lemma igcd_gt1: "igcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))" + apply (unfold igcd_def) + apply (cases "i = 0", simp_all) + apply (cases "j = 0", simp_all) + apply (cases "abs i = 1", simp_all) + apply (cases "abs j = 1", simp_all) + apply auto + done +lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0" + by (induct t rule: numgcdh.induct, auto simp add:igcd0) + +lemma dvdnumcoeff_aux: + assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1" + shows "dvdnumcoeff t (numgcdh t m)" +using prems +proof(induct t rule: numgcdh.induct) + case (2 n c t) + let ?g = "numgcdh t m" + from prems have th:"igcd c ?g > 1" by simp + from igcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)} + moreover {assume "abs c = 0 \<and> ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +next + case (3 c s t) + let ?g = "numgcdh t m" + from prems have th:"igcd c ?g > 1" by simp + from igcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)} + moreover {assume "abs c = 0 \<and> ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +qed(auto simp add: igcd_dvd1) + +lemma dvdnumcoeff_aux2: + assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0" + using prems +proof (simp add: numgcd_def) + let ?mc = "maxcoeff t" + let ?g = "numgcdh t ?mc" + have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) + have th2: "?mc \<ge> 0" by (rule maxcoeff_pos) + assume H: "numgcdh t ?mc > 1" + from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . +qed + +lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" +proof- + let ?g = "numgcd t" + have "?g \<ge> 0" by (simp add: numgcd_pos) + hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto + moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} + moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} + moreover { assume g1:"?g > 1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ + from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis + by (simp add: reducecoeff_def Let_def)} + ultimately show ?thesis by blast +qed + +lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)" +by (induct t rule: reducecoeffh.induct, auto) + +lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)" +using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) + +consts + simpnum:: "num \<Rightarrow> num" + numadd:: "num \<times> num \<Rightarrow> num" + nummul:: "num \<Rightarrow> int \<Rightarrow> num" + numfloor:: "num \<Rightarrow> num" + +recdef numadd "measure (\<lambda> (t,s). size t + size s)" + "numadd (CN n1 c1 r1,CN n2 c2 r2) = + (if n1=n2 then + (let c = c1 + c2 + in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) + else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2)) + else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" + "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" + "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" + "numadd (CF c1 t1 r1,CF c2 t2 r2) = + (if t1 = t2 then + (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s)) + else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2)) + else CF c2 t2 (numadd(CF c1 t1 r1,r2)))" + "numadd (CF c1 t1 r1,C c) = CF c1 t1 (numadd (r1, C c))" + "numadd (C c,CF c1 t1 r1) = CF c1 t1 (numadd (r1, C c))" + "numadd (C b1, C b2) = C (b1+b2)" + "numadd (a,b) = Add a b" + +lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" +apply (induct t s rule: numadd.induct, simp_all add: Let_def) +apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all) +apply (case_tac "n1 = n2", simp_all add: ring_eq_simps) +apply (simp only: ring_eq_simps(1)[symmetric]) +apply simp +apply (case_tac "lex_bnd t1 t2", simp_all) +apply (case_tac "c1+c2 = 0") +by (case_tac "t1 = t2", simp_all add: ring_eq_simps left_distrib[symmetric] real_of_int_mult[symmetric] real_of_int_add[symmetric]del: real_of_int_mult real_of_int_add left_distrib) + +lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))" +by (induct t s rule: numadd.induct, auto simp add: Let_def) + +recdef nummul "measure size" + "nummul (C j) = (\<lambda> i. C (i*j))" + "nummul (CN n c t) = (\<lambda> i. CN n (c*i) (nummul t i))" + "nummul (CF c t s) = (\<lambda> i. CF (c*i) t (nummul s i))" + "nummul (Mul c t) = (\<lambda> i. nummul t (i*c))" + "nummul t = (\<lambda> i. Mul i t)" + +lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)" +by (induct t rule: nummul.induct, auto simp add: ring_eq_simps) + +lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)" +by (induct t rule: nummul.induct, auto) + +constdefs numneg :: "num \<Rightarrow> num" + "numneg t \<equiv> nummul t (- 1)" + +constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num" + "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))" + +lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" +using numneg_def nummul by simp + +lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)" +using numneg_def by simp + +lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" +using numsub_def by simp + +lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)" +using numsub_def by simp + +lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs" +proof- + have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor) + + have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def) + also have "\<dots>" by (simp add: isint_add cti si) + finally show ?thesis . +qed + +consts split_int:: "num \<Rightarrow> num\<times>num" +recdef split_int "measure num_size" + "split_int (C c) = (C 0, C c)" + "split_int (CN n c b) = + (let (bv,bi) = split_int b + in (CN n c bv, bi))" + "split_int (CF c a b) = + (let (bv,bi) = split_int b + in (bv, CF c a bi))" + "split_int a = (a,C 0)" + +lemma split_int:"\<And> tv ti. split_int t = (tv,ti) \<Longrightarrow> (Inum bs (Add tv ti) = Inum bs t) \<and> isint ti bs" +proof (induct t rule: split_int.induct) + case (2 c n b tv ti) + let ?bv = "fst (split_int b)" + let ?bi = "snd (split_int b)" + have "split_int b = (?bv,?bi)" by simp + with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ + from prems(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def) + from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def) +next + case (3 c a b tv ti) + let ?bv = "fst (split_int b)" + let ?bi = "snd (split_int b)" + have "split_int b = (?bv,?bi)" by simp + with prems(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ + from prems(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def) + from prems(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def isint_Floor isint_add isint_Mul isint_CF) +qed (auto simp add: Let_def isint_iff isint_Floor isint_add isint_Mul split_def ring_eq_simps) + +lemma split_int_nb: "numbound0 t \<Longrightarrow> numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t)) " +by (induct t rule: split_int.induct, auto simp add: Let_def split_def) + +defs numfloor_def: "numfloor t \<equiv> (let (tv,ti) = split_int t in + (case tv of C i \<Rightarrow> numadd (tv,ti) + | _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))" + +lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)") +proof- + let ?tv = "fst (split_int t)" + let ?ti = "snd (split_int t)" + have tvti:"split_int t = (?tv,?ti)" by simp + {assume H: "\<forall> v. ?tv \<noteq> C v" + hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" + by (cases ?tv, auto simp add: numfloor_def Let_def split_def numadd) + from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ + hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp + also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))" + by (simp,subst tii[simplified isint_iff, symmetric]) simp + also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) + finally have ?thesis using th1 by simp} + moreover {fix v assume H:"?tv = C v" + from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+ + hence "?N (Floor t) = real (floor (?N (Add ?tv ?ti)))" by simp + also have "\<dots> = real (floor (?N ?tv) + (floor (?N ?ti)))" + by (simp,subst tii[simplified isint_iff, symmetric]) simp + also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff]) + finally have ?thesis by (simp add: H numfloor_def Let_def split_def numadd) } + ultimately show ?thesis by auto +qed + +lemma numfloor_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numfloor t)" + using split_int_nb[where t="t"] + by (cases "fst(split_int t)" , auto simp add: numfloor_def Let_def split_def numadd_nb) + +recdef simpnum "measure num_size" + "simpnum (C j) = C j" + "simpnum (Bound n) = CN n 1 (C 0)" + "simpnum (Neg t) = numneg (simpnum t)" + "simpnum (Add t s) = numadd (simpnum t,simpnum s)" + "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" + "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" + "simpnum (Floor t) = numfloor (simpnum t)" + "simpnum (CN n c t) = (if c=0 then simpnum t else CN n c (simpnum t))" + "simpnum (CF c t s) = simpnum(Add (Mul c (Floor t)) s)" + +lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" +by (induct t rule: simpnum.induct, auto) + +lemma simpnum_numbound0[simp]: + "numbound0 t \<Longrightarrow> numbound0 (simpnum t)" +by (induct t rule: simpnum.induct, auto) + +consts nozerocoeff:: "num \<Rightarrow> bool" +recdef nozerocoeff "measure size" + "nozerocoeff (C c) = True" + "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)" + "nozerocoeff (CF c s t) = (c \<noteq> 0 \<and> nozerocoeff t)" + "nozerocoeff (Mul c t) = (c\<noteq>0 \<and> nozerocoeff t)" + "nozerocoeff t = True" + +lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))" +by (induct a b rule: numadd.induct,auto simp add: Let_def) + +lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)" + by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz) + +lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)" +by (simp add: numneg_def nummul_nz) + +lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)" +by (simp add: numsub_def numneg_nz numadd_nz) + +lemma split_int_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (fst (split_int t)) \<and> nozerocoeff (snd (split_int t))" +by (induct t rule: split_int.induct,auto simp add: Let_def split_def) + +lemma numfloor_nz: "nozerocoeff t \<Longrightarrow> nozerocoeff (numfloor t)" +by (simp add: numfloor_def Let_def split_def) +(cases "fst (split_int t)", simp_all add: split_int_nz numadd_nz) + +lemma simpnum_nz: "nozerocoeff (simpnum t)" +by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz numfloor_nz) + +lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +next + case (3 c s t) + hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +qed auto + +lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" +proof- + from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) + from numgcdh0[OF th] have th:"maxcoeff t = 0" . + from maxcoeff_nz[OF nz th] show ?thesis . +qed + +constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int" + "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else + (let t' = simpnum t ; g = numgcd t' in + if g > 1 then (let g' = igcd n g in + if g' = 1 then (t',n) + else (reducecoeffh t' g', n div g')) + else (t',n))))" + +lemma simp_num_pair_ci: + shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))" + (is "?lhs = ?rhs") +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "igcd n ?g" + {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \<noteq> 0" + {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp + hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \<or> ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. + let ?tt = "reducecoeffh ?t' ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2) + have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs ?t'" by simp + from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def) + also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp + also have "\<dots> = (Inum bs ?t' / real n)" + using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp + finally have "?lhs = Inum bs t / real n" by simp + then have ?thesis using prems by (simp add: simp_num_pair_def)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" + shows "numbound0 t' \<and> n' >0" +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "igcd n ?g" + {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \<noteq> 0" + {assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp + hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \<or> ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis using prems + by (auto simp add: Let_def simp_num_pair_def)} + moreover {assume g'1:"?g'>1" + have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2) + have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) + have gpdgp: "?g' dvd ?g'" by simp + from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" . + from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]] + have "n div ?g' >0" by simp + hence ?thesis using prems + by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +consts not:: "fm \<Rightarrow> fm" +recdef not "measure size" + "not (NOT p) = p" + "not T = F" + "not F = T" + "not (Lt t) = Ge t" + "not (Le t) = Gt t" + "not (Gt t) = Le t" + "not (Ge t) = Lt t" + "not (Eq t) = NEq t" + "not (NEq t) = Eq t" + "not (Dvd i t) = NDvd i t" + "not (NDvd i t) = Dvd i t" + "not (And p q) = Or (not p) (not q)" + "not (Or p q) = And (not p) (not q)" + "not p = NOT p" +lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" +by (induct p) auto +lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)" +by (induct p, auto) +lemma not_nb[simp]: "bound0 p \<Longrightarrow> bound0 (not p)" +by (induct p, auto) + +constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else + if p = q then p else And p q)" +lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" +by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all) + +lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)" +using conj_def by auto +lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)" +using conj_def by auto + +constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p + else if p=q then p else Or p q)" + +lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" +by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all) +lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)" +using disj_def by auto +lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)" +using disj_def by auto + +constdefs imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p + else Imp p q)" +lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" +by (cases "p=F \<or> q=T",simp_all add: imp_def) +lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)" +using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def) +lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)" +using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def) + +constdefs iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else + if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else + Iff p q)" +lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" + by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) +(cases "not p= q", auto simp add:not) +lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)" + by (unfold iff_def,cases "p=q", auto) +lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)" +using iff_def by (unfold iff_def,cases "p=q", auto) + +consts check_int:: "num \<Rightarrow> bool" +recdef check_int "measure size" + "check_int (C i) = True" + "check_int (Floor t) = True" + "check_int (Mul i t) = check_int t" + "check_int (Add t s) = (check_int t \<and> check_int s)" + "check_int (Neg t) = check_int t" + "check_int (CF c t s) = check_int s" + "check_int t = False" +lemma check_int: "check_int t \<Longrightarrow> isint t bs" +by (induct t, auto simp add: isint_add isint_Floor isint_Mul isint_neg isint_c isint_CF) + +lemma rdvd_left1_int: "real \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t" + by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp + +lemma rdvd_reduce: + assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0" + shows "real (d::int) rdvd real (c::int)*t = (real (d div g) rdvd real (c div g)*t)" +proof + assume d: "real d rdvd real c * t" + from d rdvd_def obtain k where k_def: "real c * t = real d* real (k::int)" by auto + from gd dvd_def obtain kd where kd_def: "d = g * kd" by auto + from gc dvd_def obtain kc where kc_def: "c = g * kc" by auto + from k_def kd_def kc_def have "real g * real kc * t = real g * real kd * real k" by simp + hence "real kc * t = real kd * real k" using gp by simp + hence th:"real kd rdvd real kc * t" using rdvd_def by blast + from kd_def gp have th':"kd = d div g" by simp + from kc_def gp have "kc = c div g" by simp + with th th' show "real (d div g) rdvd real (c div g) * t" by simp +next + assume d: "real (d div g) rdvd real (c div g) * t" + from gp have gnz: "g \<noteq> 0" by simp + thus "real d rdvd real c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real (c div g) * t"] real_of_int_div[OF gnz gd] real_of_int_div[OF gnz gc] by simp +qed + +constdefs simpdvd:: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" + "simpdvd d t \<equiv> + (let g = numgcd t in + if g > 1 then (let g' = igcd d g in + if g' = 1 then (d, t) + else (d div g',reducecoeffh t g')) + else (d, t))" +lemma simpdvd: + assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0" + shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)" +proof- + let ?g = "numgcd t" + let ?g' = "igcd d ?g" + {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simpdvd_def)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from igcd0 g1 dnz have gp0: "?g' \<noteq> 0" by simp + hence g'p: "?g' > 0" using igcd_pos[where i="d" and j="numgcd t"] by arith + hence "?g'= 1 \<or> ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simpdvd_def)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" .. + let ?tt = "reducecoeffh t ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2) + have gpdd: "?g' dvd d" by (simp add: igcd_dvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs t" by simp + from prems have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)" + by (simp add: simpdvd_def Let_def) + also have "\<dots> = (real d rdvd (Inum bs t))" + using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified zdiv_self[OF gp0]] + th2[symmetric] by simp + finally have ?thesis by simp } + ultimately have ?thesis by blast + } + ultimately show ?thesis by blast +qed + +consts simpfm :: "fm \<Rightarrow> fm" +recdef simpfm "measure fmsize" + "simpfm (And p q) = conj (simpfm p) (simpfm q)" + "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + "simpfm (NOT p) = not (simpfm p)" + "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F + | _ \<Rightarrow> Lt (reducecoeff a'))" + "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0) then T else F | _ \<Rightarrow> Le (reducecoeff a'))" + "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt (reducecoeff a'))" + "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0) then T else F | _ \<Rightarrow> Ge (reducecoeff a'))" + "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq (reducecoeff a'))" + "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0) then T else F | _ \<Rightarrow> NEq (reducecoeff a'))" + "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a) + else if (abs i = 1) \<and> check_int a then T + else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))" + "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) + else if (abs i = 1) \<and> check_int a then F + else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))" + "simpfm p = p" + +lemma simpfm[simp]: "Ifm bs (simpfm p) = Ifm bs p" +proof(induct p rule: simpfm.induct) + case (6 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a < 0 = (real ?g * ?r < real ?g * 0)" by simp + also have "\<dots> = (?r < 0)" using gp + by (simp only: mult_less_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (7 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \<le> 0 = (real ?g * ?r \<le> real ?g * 0)" by simp + also have "\<dots> = (?r \<le> 0)" using gp + by (simp only: mult_le_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (8 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a > 0 = (real ?g * ?r > real ?g * 0)" by simp + also have "\<dots> = (?r > 0)" using gp + by (simp only: mult_less_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (9 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \<ge> 0 = (real ?g * ?r \<ge> real ?g * 0)" by simp + also have "\<dots> = (?r \<ge> 0)" using gp + by (simp only: mult_le_cancel_left) simp + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (10 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a = 0 = (real ?g * ?r = 0)" by simp + also have "\<dots> = (?r = 0)" using gp + by (simp add: mult_eq_0_iff) + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (11 a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + let ?g = "numgcd ?sa" + let ?rsa = "reducecoeff ?sa" + let ?r = "Inum bs ?rsa" + have sa_nz: "nozerocoeff ?sa" by (rule simpnum_nz) + {assume gz: "?g=0" from numgcd_nz[OF sa_nz gz] H have "False" by auto} + with numgcd_pos[where t="?sa"] have "?g > 0" by (cases "?g=0", auto) + hence gp: "real ?g > 0" by simp + have "Inum bs ?sa = real ?g* ?r" by (simp add: reducecoeff) + with sa have "Inum bs a \<noteq> 0 = (real ?g * ?r \<noteq> 0)" by simp + also have "\<dots> = (?r \<noteq> 0)" using gp + by (simp add: mult_eq_0_iff) + finally have ?case using H by (cases "?sa" , simp_all add: Let_def)} + ultimately show ?case by blast +next + case (12 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto + {assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)} + moreover + {assume ai1: "abs i = 1" and ai: "check_int a" + hence "i=1 \<or> i= - 1" by arith + moreover {assume i1: "i = 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + have ?case using i1 ai by simp } + moreover {assume i1: "i = - 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + rdvd_abs1[where d="- 1" and t="Inum bs a"] + have ?case using i1 ai by simp } + ultimately have ?case by blast} + moreover + {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto simp add: int_rdvd_iff) } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def) + from simpnum_nz have nz:"nozerocoeff ?sa" by simp + from simpdvd [OF nz inz] th have ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +next + case (13 i a) let ?sa = "simpnum a" have sa: "Inum bs ?sa = Inum bs a" by simp + have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto + {assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)} + moreover + {assume ai1: "abs i = 1" and ai: "check_int a" + hence "i=1 \<or> i= - 1" by arith + moreover {assume i1: "i = 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + have ?case using i1 ai by simp } + moreover {assume i1: "i = - 1" + from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] + rdvd_abs1[where d="- 1" and t="Inum bs a"] + have ?case using i1 ai by simp } + ultimately have ?case by blast} + moreover + {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)" + {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond + by (cases "abs i = 1", auto simp add: int_rdvd_iff) } + moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" + hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond + by (cases ?sa, auto simp add: Let_def split_def) + from simpnum_nz have nz:"nozerocoeff ?sa" by simp + from simpdvd [OF nz inz] th have ?case using sa by simp} + ultimately have ?case by blast} + ultimately show ?case by blast +qed (induct p rule: simpfm.induct, simp_all) + +lemma simpdvd_numbound0: "numbound0 t \<Longrightarrow> numbound0 (snd (simpdvd d t))" + by (simp add: simpdvd_def Let_def split_def reducecoeffh_numbound0) + +lemma simpfm_bound0[simp]: "bound0 p \<Longrightarrow> bound0 (simpfm p)" +proof(induct p rule: simpfm.induct) + case (6 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (7 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (8 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (9 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (10 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (11 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0) +next + case (12 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) +next + case (13 i a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def reducecoeff_numbound0 simpdvd_numbound0 split_def) +qed(auto simp add: disj_def imp_def iff_def conj_def) + +lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)" +by (induct p rule: simpfm.induct, auto simp add: Let_def) +(case_tac "simpnum a",auto simp add: split_def Let_def)+ + + + (* Generic quantifier elimination *) + +constdefs list_conj :: "fm list \<Rightarrow> fm" + "list_conj ps \<equiv> foldr conj ps T" +lemma list_conj: "Ifm bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm bs p)" + by (induct ps, auto simp add: list_conj_def) +lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)" + by (induct ps, auto simp add: list_conj_def) +lemma list_conj_nb: " \<forall>p\<in> set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)" + by (induct ps, auto simp add: list_conj_def) +constdefs CJNB:: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" + "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs + in conj (decr (list_conj yes)) (f (list_conj no)))" + +lemma CJNB_qe: + assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" + shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm bs ((CJNB qe p)) = Ifm bs (E p))" +proof(clarify) + fix bs p + assume qfp: "qfree p" + let ?cjs = "conjuncts p" + let ?yes = "fst (partition bound0 ?cjs)" + let ?no = "snd (partition bound0 ?cjs)" + let ?cno = "list_conj ?no" + let ?cyes = "list_conj ?yes" + have part: "partition bound0 ?cjs = (?yes,?no)" by simp + from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast + hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb) + hence yes_qf: "qfree (decr ?cyes )" by (simp add: decr_qf) + from conjuncts_qf[OF qfp] partition_set[OF part] + have " \<forall>q\<in> set ?no. qfree q" by auto + hence no_qf: "qfree ?cno"by (simp add: list_conj_qf) + with qe have cno_qf:"qfree (qe ?cno )" + and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+ + from cno_qf yes_qf have qf: "qfree (CJNB qe p)" + by (simp add: CJNB_def Let_def conj_qf split_def) + {fix bs + from conjuncts have "Ifm bs p = (\<forall>q\<in> set ?cjs. Ifm bs q)" by blast + also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm bs q) \<and> (\<forall>q\<in> set ?no. Ifm bs q))" + using partition_set[OF part] by auto + finally have "Ifm bs p = ((Ifm bs ?cyes) \<and> (Ifm bs ?cno))" using list_conj by simp} + hence "Ifm bs (E p) = (\<exists>x. (Ifm (x#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" by simp + also have "\<dots> = (\<exists>x. (Ifm (y#bs) ?cyes) \<and> (Ifm (x#bs) ?cno))" + using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast + also have "\<dots> = (Ifm bs (decr ?cyes) \<and> Ifm bs (E ?cno))" + by (auto simp add: decr[OF yes_nb]) + also have "\<dots> = (Ifm bs (conj (decr ?cyes) (qe ?cno)))" + using qe[rule_format, OF no_qf] by auto + finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)" + by (simp add: Let_def CJNB_def split_def) + with qf show "qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast +qed + +consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" +recdef qelim "measure fmsize" + "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))" + "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))" + "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))" + "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" + "qelim (Or p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" + "qelim (Imp p q) = (\<lambda> qe. disj (qelim (NOT p) qe) (qelim q qe))" + "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))" + "qelim p = (\<lambda> y. simpfm p)" + +lemma qelim_ci: + assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" + shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)" +using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] +by(induct p rule: qelim.induct) +(auto simp del: simpfm.simps) + + + + (**********************************************************************************) + (******* THE \<int>-PART ***) + (**********************************************************************************) + (* Linearity for fm where Bound 0 ranges over \<int> *) +consts + zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*) +recdef zsplit0 "measure num_size" + "zsplit0 (C c) = (0,C c)" + "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))" + "zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)" + "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)" + "zsplit0 (Neg a) = (let (i',a') = zsplit0 a in (-i', Neg a'))" + "zsplit0 (Add a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia+ib, Add a' b'))" + "zsplit0 (Sub a b) = (let (ia,a') = zsplit0 a ; + (ib,b') = zsplit0 b + in (ia-ib, Sub a' b'))" + "zsplit0 (Mul i a) = (let (i',a') = zsplit0 a in (i*i', Mul i a'))" + "zsplit0 (Floor a) = (let (i',a') = zsplit0 a in (i',Floor a'))" +(hints simp add: Let_def) + +lemma zsplit0_I: + shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real (x::int)) #bs) (CN 0 n a) = Inum (real x #bs) t) \<and> numbound0 a" + (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a") +proof(induct t rule: zsplit0.induct) + case (1 c n a) thus ?case by auto +next + case (2 m n a) thus ?case by (cases "m=0") auto +next + case (3 n i a n a') thus ?case by auto +next + case (4 c a b n a') thus ?case by auto +next + case (5 t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast + from th2[simplified] th[simplified] show ?case by simp +next + case (6 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast + from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_distrib) +next + case (7 s t n a) + let ?ns = "fst (zsplit0 s)" + let ?as = "snd (zsplit0 s)" + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abjs: "zsplit0 s = (?ns,?as)" by simp + moreover have abjt: "zsplit0 t = (?nt,?at)" by simp + ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems + by (simp add: Let_def split_def) + from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast + from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real x # bs) (CN 0 xa xb) = Inum (real x # bs) t \<and> numbound0 xb)" by simp + with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast + from abjs prems have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast + from th3[simplified] th2[simplified] th[simplified] show ?case + by (simp add: left_diff_distrib) +next + case (8 i t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast + hence " ?I x (Mul i t) = (real i) * ?I x (CN 0 ?nt ?at)" by simp + also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib) + finally show ?case using th th2 by simp +next + case (9 t n a) + let ?nt = "fst (zsplit0 t)" + let ?at = "snd (zsplit0 t)" + have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a= Floor ?at \<and> n=?nt" using prems + by (simp add: Let_def split_def) + from abj prems have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast + hence na: "?N a" using th by simp + have th': "(real ?nt)*(real x) = real (?nt * x)" by simp + have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp + also have "\<dots> = real (floor ((real ?nt)* real(x) + ?I x ?at))" by simp + also have "\<dots> = real (floor (?I x ?at + real (?nt* x)))" by (simp add: add_ac) + also have "\<dots> = real (floor (?I x ?at) + (?nt* x))" + using floor_add[where x="?I x ?at" and a="?nt* x"] by simp + also have "\<dots> = real (?nt)*(real x) + real (floor (?I x ?at))" by (simp add: add_ac) + finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp + with na show ?case by simp +qed + +consts + iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool" (* Linearity test for fm *) + zlfm :: "fm \<Rightarrow> fm" (* Linearity transformation for fm *) +recdef iszlfm "measure size" + "iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" + "iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" + "iszlfm (Eq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (NEq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (Lt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (Le (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (Gt (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (Ge (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (Dvd i (CN 0 c e)) = + (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm (NDvd i (CN 0 c e))= + (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)" + "iszlfm p = (\<lambda> bs. isatom p \<and> (bound0 p))" + +lemma zlin_qfree: "iszlfm p bs \<Longrightarrow> qfree p" + by (induct p rule: iszlfm.induct) auto + +lemma iszlfm_gen: + assumes lp: "iszlfm p (x#bs)" + shows "\<forall> y. iszlfm p (y#bs)" +proof + fix y + show "iszlfm p (y#bs)" + using lp + by(induct p rule: iszlfm.induct, simp_all add: numbound0_gen[rule_format, where x="x" and y="y"]) +qed + +lemma conj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (conj p q) bs" + using conj_def by (cases p,auto) +lemma disj_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm q bs \<Longrightarrow> iszlfm (disj p q) bs" + using disj_def by (cases p,auto) +lemma not_zl[simp]: "iszlfm p bs \<Longrightarrow> iszlfm (not p) bs" + by (induct p rule:iszlfm.induct ,auto) + +recdef zlfm "measure fmsize" + "zlfm (And p q) = conj (zlfm p) (zlfm q)" + "zlfm (Or p q) = disj (zlfm p) (zlfm q)" + "zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)" + "zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))" + "zlfm (Lt a) = (let (c,r) = zsplit0 a in + if c=0 then Lt r else + if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) + else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" + "zlfm (Le a) = (let (c,r) = zsplit0 a in + if c=0 then Le r else + if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) + else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))" + "zlfm (Gt a) = (let (c,r) = zsplit0 a in + if c=0 then Gt r else + if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) + else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" + "zlfm (Ge a) = (let (c,r) = zsplit0 a in + if c=0 then Ge r else + if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) + else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))" + "zlfm (Eq a) = (let (c,r) = zsplit0 a in + if c=0 then Eq r else + if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r))) + else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))" + "zlfm (NEq a) = (let (c,r) = zsplit0 a in + if c=0 then NEq r else + if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r))) + else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))" + "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) + else (let (c,r) = zsplit0 a in + if c=0 then Dvd (abs i) r else + if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r))) + else And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 (-c) (Neg (Floor r))))))" + "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) + else (let (c,r) = zsplit0 a in + if c=0 then NDvd (abs i) r else + if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r))) + else Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 (-c) (Neg (Floor r))))))" + "zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))" + "zlfm (NOT (Imp p q)) = conj (zlfm p) (zlfm (NOT q))" + "zlfm (NOT (Iff p q)) = disj (conj(zlfm p) (zlfm(NOT q))) (conj (zlfm(NOT p)) (zlfm q))" + "zlfm (NOT (NOT p)) = zlfm p" + "zlfm (NOT T) = F" + "zlfm (NOT F) = T" + "zlfm (NOT (Lt a)) = zlfm (Ge a)" + "zlfm (NOT (Le a)) = zlfm (Gt a)" + "zlfm (NOT (Gt a)) = zlfm (Le a)" + "zlfm (NOT (Ge a)) = zlfm (Lt a)" + "zlfm (NOT (Eq a)) = zlfm (NEq a)" + "zlfm (NOT (NEq a)) = zlfm (Eq a)" + "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)" + "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)" + "zlfm p = p" (hints simp add: fmsize_pos) + +lemma split_int_less_real: + "(real (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real (floor b) < b))" +proof( auto) + assume alb: "real a < b" and agb: "\<not> a < floor b" + from agb have "floor b \<le> a" by simp hence th: "b < real a + 1" by (simp only: floor_le_eq) + from floor_eq[OF alb th] show "a= floor b" by simp +next + assume alb: "a < floor b" + hence "real a < real (floor b)" by simp + moreover have "real (floor b) \<le> b" by simp ultimately show "real a < b" by arith +qed + +lemma split_int_less_real': + "(real (a::int) + b < 0) = (real a - real (floor(-b)) < 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))" +proof- + have "(real a + b <0) = (real a < -b)" by arith + with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith +qed + +lemma split_int_gt_real': + "(real (a::int) + b > 0) = (real a + real (floor b) > 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))" +proof- + have th: "(real a + b >0) = (real (-a) + (-b)< 0)" by arith + show ?thesis using myless[rule_format, where b="real (floor b)"] + by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) + (simp add: ring_eq_simps diff_def[symmetric],arith) +qed + +lemma split_int_le_real: + "(real (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real (floor b) < b))" +proof( auto) + assume alb: "real a \<le> b" and agb: "\<not> a \<le> floor b" + from alb have "floor (real a) \<le> floor b " by (simp only: floor_mono2) + hence "a \<le> floor b" by simp with agb show "False" by simp +next + assume alb: "a \<le> floor b" + hence "real a \<le> real (floor b)" by (simp only: floor_mono2) + also have "\<dots>\<le> b" by simp finally show "real a \<le> b" . +qed + +lemma split_int_le_real': + "(real (a::int) + b \<le> 0) = (real a - real (floor(-b)) \<le> 0 \<or> (real a - real (floor (-b)) = 0 \<and> real (floor (-b)) + b < 0))" +proof- + have "(real a + b \<le>0) = (real a \<le> -b)" by arith + with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith +qed + +lemma split_int_ge_real': + "(real (a::int) + b \<ge> 0) = (real a + real (floor b) \<ge> 0 \<or> (real a + real (floor b) = 0 \<and> real (floor b) - b < 0))" +proof- + have th: "(real a + b \<ge>0) = (real (-a) + (-b) \<le> 0)" by arith + show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"]) + (simp add: ring_eq_simps diff_def[symmetric],arith) +qed + +lemma split_int_eq_real: "(real (a::int) = b) = ( a = floor b \<and> b = real (floor b))" (is "?l = ?r") +by auto + +lemma split_int_eq_real': "(real (a::int) + b = 0) = ( a - floor (-b) = 0 \<and> real (floor (-b)) + b = 0)" (is "?l = ?r") +proof- + have "?l = (real a = -b)" by arith + with split_int_eq_real[where a="a" and b="-b"] show ?thesis by simp arith +qed + +lemma zlfm_I: + assumes qfp: "qfree p" + shows "(Ifm (real i #bs) (zlfm p) = Ifm (real i# bs) p) \<and> iszlfm (zlfm p) (real (i::int) #bs)" + (is "(?I (?l p) = ?I p) \<and> ?L (?l p)") +using qfp +proof(induct p rule: zlfm.induct) + case (5 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def,case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Lt a) = (real (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (6 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat",simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Le a) = (real (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac ,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (7 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Gt a) = (real (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (8 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def diff_def) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Ge a) = (real (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def diff_def[symmetric] add_ac, arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (9 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (Eq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Eq a) = (real (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (10 a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def) + also have "\<dots> = (?I (?l (NEq a)))" using cp cnz by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult) + finally have ?case using l by simp} + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NEq a) = (real (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def) + also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia real_of_int_mult[symmetric] del: real_of_int_mult,arith) + finally have ?case using l by simp} + ultimately show ?case by blast +next + case (11 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)} + moreover + {assume "?c=0" and "j\<noteq>0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" + using Ia by (simp add: Let_def split_def) + also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by simp } + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (Dvd j a) = (real j rdvd (real (?c * i) + (?N ?r)))" + using Ia by (simp add: Let_def split_def) + also have "\<dots> = (real (abs j) rdvd real (?c*i) + (?N ?r))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz + using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric] + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by blast } + ultimately show ?case by blast +next + case (12 j a) + let ?c = "fst (zsplit0 a)" + let ?r = "snd (zsplit0 a)" + have spl: "zsplit0 a = (?c,?r)" by simp + from zsplit0_I[OF spl, where x="i" and bs="bs"] + have Ia:"Inum (real i # bs) a = Inum (real i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto + let ?N = "\<lambda> t. Inum (real i#bs) t" + have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith + moreover + {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) + hence ?case using prems by (simp del: zlfm.simps add: rdvd_left_0_eq)} + moreover + {assume "?c=0" and "j\<noteq>0" hence ?case + using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"] + by (cases "?r", simp_all add: Let_def split_def, case_tac "nat", simp_all)} + moreover + {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" + using Ia by (simp add: Let_def split_def) + also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by simp } + moreover + {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" + by (simp add: nb Let_def split_def isint_Floor isint_neg) + have "?I (NDvd j a) = (\<not> (real j rdvd (real (?c * i) + (?N ?r))))" + using Ia by (simp add: Let_def split_def) + also have "\<dots> = (\<not> (real (abs j) rdvd real (?c*i) + (?N ?r)))" + by (simp only: rdvd_abs1[where d="j" and t="real (?c*i) + ?N ?r", symmetric]) simp + also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real (?c*i))) \<and> + (real (floor ((?N ?r) + real (?c*i))) = (real (?c*i) + (?N ?r)))))" + by(simp only: int_rdvd_real[where i="abs j" and x="real (?c*i) + (?N ?r)"]) (simp only: add_ac) + also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz + using rdvd_minus [where d="abs j" and t="real (?c*i + floor (?N ?r))", simplified, symmetric] + by (simp add: Let_def split_def int_rdvd_iff[symmetric] + del: real_of_int_mult) (auto simp add: add_ac) + finally have ?case using l jnz by blast } + ultimately show ?case by blast +qed auto + +consts + plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) + minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) + \<delta> :: "fm \<Rightarrow> int" (* Compute lcm {d| N\<^isup>?\<^isup> Dvd c*x+t \<in> p}*) + d\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* checks if a given l divides all the ds above*) + +recdef minusinf "measure size" + "minusinf (And p q) = conj (minusinf p) (minusinf q)" + "minusinf (Or p q) = disj (minusinf p) (minusinf q)" + "minusinf (Eq (CN 0 c e)) = F" + "minusinf (NEq (CN 0 c e)) = T" + "minusinf (Lt (CN 0 c e)) = T" + "minusinf (Le (CN 0 c e)) = T" + "minusinf (Gt (CN 0 c e)) = F" + "minusinf (Ge (CN 0 c e)) = F" + "minusinf p = p" + +lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)" + by (induct p rule: minusinf.induct, auto) + +recdef plusinf "measure size" + "plusinf (And p q) = conj (plusinf p) (plusinf q)" + "plusinf (Or p q) = disj (plusinf p) (plusinf q)" + "plusinf (Eq (CN 0 c e)) = F" + "plusinf (NEq (CN 0 c e)) = T" + "plusinf (Lt (CN 0 c e)) = F" + "plusinf (Le (CN 0 c e)) = F" + "plusinf (Gt (CN 0 c e)) = T" + "plusinf (Ge (CN 0 c e)) = T" + "plusinf p = p" + +recdef \<delta> "measure size" + "\<delta> (And p q) = ilcm (\<delta> p) (\<delta> q)" + "\<delta> (Or p q) = ilcm (\<delta> p) (\<delta> q)" + "\<delta> (Dvd i (CN 0 c e)) = i" + "\<delta> (NDvd i (CN 0 c e)) = i" + "\<delta> p = 1" + +recdef d\<delta> "measure size" + "d\<delta> (And p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" + "d\<delta> (Or p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" + "d\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)" + "d\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)" + "d\<delta> p = (\<lambda> d. True)" + +lemma delta_mono: + assumes lin: "iszlfm p bs" + and d: "d dvd d'" + and ad: "d\<delta> p d" + shows "d\<delta> p d'" + using lin ad d +proof(induct p rule: iszlfm.induct) + case (9 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +next + case (10 i c e) thus ?case using d + by (simp add: zdvd_trans[where m="i" and n="d" and k="d'"]) +qed simp_all + +lemma \<delta> : assumes lin:"iszlfm p bs" + shows "d\<delta> p (\<delta> p) \<and> \<delta> p >0" +using lin +proof (induct p rule: iszlfm.induct) + case (1 p q) + let ?d = "\<delta> (And p q)" + from prems ilcm_pos have dp: "?d >0" by simp + have d1: "\<delta> p dvd \<delta> (And p q)" using prems ilcm_dvd1 by simp + hence th: "d\<delta> p ?d" using delta_mono prems by auto + have "\<delta> q dvd \<delta> (And p q)" using prems ilcm_dvd2 by simp + hence th': "d\<delta> q ?d" using delta_mono prems by auto + from th th' dp show ?case by simp +next + case (2 p q) + let ?d = "\<delta> (And p q)" + from prems ilcm_pos have dp: "?d >0" by simp + have "\<delta> p dvd \<delta> (And p q)" using prems ilcm_dvd1 by simp hence th: "d\<delta> p ?d" using delta_mono prems by auto + have "\<delta> q dvd \<delta> (And p q)" using prems ilcm_dvd2 by simp hence th': "d\<delta> q ?d" using delta_mono prems by auto + from th th' dp show ?case by simp +qed simp_all + + +lemma minusinf_inf: + assumes linp: "iszlfm p (a # bs)" + shows "\<exists> (z::int). \<forall> x < z. Ifm ((real x)#bs) (minusinf p) = Ifm ((real x)#bs) p" + (is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p") +using linp +proof (induct p rule: minusinf.induct) + case (1 f g) + from prems have "?P f" by simp + then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast + from prems have "?P g" by simp + then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast + let ?z = "min z1 z2" + from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (And f g)) = ?I x (And f g)" by simp + thus ?case by blast +next + case (2 f g) from prems have "?P f" by simp + then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast + from prems have "?P g" by simp + then obtain z2 where z2_def: "\<forall> x < z2. ?I x (?M g) = ?I x g" by blast + let ?z = "min z1 z2" + from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp + thus ?case by blast +next + case (3 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos by simp + thus "real c * real x + Inum (real x # bs) e \<noteq> 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp + qed + thus ?case by blast +next + case (4 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + hence "real c * real x + Inum (y # bs) e \<noteq> 0"using rcpos by simp + thus "real c * real x + Inum (real x # bs) e \<noteq> 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] by simp + qed + thus ?case by blast +next + case (5 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "real c * real x + Inum (real x # bs) e < 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (6 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "real c * real x + Inum (real x # bs) e \<le> 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (7 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "\<not> (real c * real x + Inum (real x # bs) e>0)" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +next + case (8 c e) + from prems have "c > 0" by simp hence rcpos: "real c > 0" by simp + from prems have nbe: "numbound0 e" by simp + have "\<forall> x < (floor (- (Inum (y#bs) e) / (real c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))" + proof (simp add: less_floor_eq , rule allI, rule impI) + fix x + assume A: "real x + (1\<Colon>real) \<le> - (Inum (y # bs) e / real c)" + hence th1:"real x < - (Inum (y # bs) e / real c)" by simp + with rcpos have "(real c)*(real x) < (real c)*(- (Inum (y # bs) e / real c))" + by (simp only: real_mult_less_mono2[OF rcpos th1]) + thus "\<not> real c * real x + Inum (real x # bs) e \<ge> 0" + using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real x"] rcpos by simp + qed + thus ?case by blast +qed simp_all + +lemma minusinf_repeats: + assumes d: "d\<delta> p d" and linp: "iszlfm p (a # bs)" + shows "Ifm ((real(x - k*d))#bs) (minusinf p) = Ifm (real x #bs) (minusinf p)" +using linp d +proof(induct p rule: iszlfm.induct) + case (9 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\<exists> k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI) + assume + "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" + (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") + hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" + by (simp add: ring_eq_simps di_def) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))" + by (simp add: ring_eq_simps) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast + thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp + next + assume + "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e") + hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: ring_eq_simps) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)" + by blast + thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp + qed +next + case (10 i c e) hence nbe: "numbound0 e" and id: "i dvd d" by simp+ + hence "\<exists> k. d=i*k" by (simp add: dvd_def) + then obtain "di" where di_def: "d=i*di" by blast + show ?case + proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real x - real k * real d" and b'="real x"] right_diff_distrib, rule iffI) + assume + "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" + (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt") + hence "\<exists> (l::int). ?rt = ?ri * (real l)" by (simp add: rdvd_def) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real l)+?rc*(?rk * (real i) * (real di))" + by (simp add: ring_eq_simps di_def) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real (l + c*k*di))" + by (simp add: ring_eq_simps) + hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real l)" by blast + thus "real i rdvd real c * real x + Inum (real x # bs) e" using rdvd_def by simp + next + assume + "real i rdvd real c * real x + Inum (real x # bs) e" (is "?ri rdvd ?rc*?rx+?e") + hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real l)" by (simp add: rdvd_def) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real d)" by simp + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l) - real c * (real k * real i * real di)" by (simp add: di_def) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real (l - c*k*di))" by (simp add: ring_eq_simps) + hence "\<exists> (l::int). ?rc*?rx - real c * (real k * real d) +?e = ?ri * (real l)" + by blast + thus "real i rdvd real c * real x - real c * (real k * real d) + Inum (real x # bs) e" using rdvd_def by simp + qed +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="real(x - k*d)" and b'="real x"] simp del: real_of_int_mult real_of_int_diff) + + (* Is'nt this beautiful?*) +lemma minusinf_ex: + assumes lin: "iszlfm p (real (a::int) #bs)" + and exmi: "\<exists> (x::int). Ifm (real x#bs) (minusinf p)" (is "\<exists> x. ?P1 x") + shows "\<exists> (x::int). Ifm (real x#bs) p" (is "\<exists> x. ?P x") +proof- + let ?d = "\<delta> p" + from \<delta> [OF lin] have dpos: "?d >0" by simp + from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp + from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P1 x = ?P1 (x - (k * ?d))" by simp + from minusinf_inf[OF lin] have th2:"\<exists> z. \<forall> x. x<z \<longrightarrow> (?P x = ?P1 x)" by blast + from minusinfinity [OF dpos th1 th2] exmi show ?thesis by blast +qed + + (* And This ???*) +lemma minusinf_bex: + assumes lin: "iszlfm p (real (a::int) #bs)" + shows "(\<exists> (x::int). Ifm (real x#bs) (minusinf p)) = + (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real x#bs) (minusinf p))" + (is "(\<exists> x. ?P x) = _") +proof- + let ?d = "\<delta> p" + from \<delta> [OF lin] have dpos: "?d >0" by simp + from \<delta> [OF lin] have alld: "d\<delta> p ?d" by simp + from minusinf_repeats[OF alld lin] have th1:"\<forall> x k. ?P x = ?P (x - (k * ?d))" by simp + from minf_vee[OF dpos th1] show ?thesis by blast +qed + + (* Lemmas for the correctness of \<sigma>\<rho> *) +lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)" by auto + +consts + a\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *) + d\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*) + \<zeta> :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*) + \<beta> :: "fm \<Rightarrow> num list" + \<alpha> :: "fm \<Rightarrow> num list" + +recdef a\<beta> "measure size" + "a\<beta> (And p q) = (\<lambda> k. And (a\<beta> p k) (a\<beta> q k))" + "a\<beta> (Or p q) = (\<lambda> k. Or (a\<beta> p k) (a\<beta> q k))" + "a\<beta> (Eq (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (Lt (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (Le (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (Gt (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (Ge (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))" + "a\<beta> p = (\<lambda> k. p)" + +recdef d\<beta> "measure size" + "d\<beta> (And p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" + "d\<beta> (Or p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" + "d\<beta> (Eq (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (Lt (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (Le (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (Gt (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (Ge (CN 0 c e)) = (\<lambda> k. c dvd k)" + "d\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)" + "d\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)" + "d\<beta> p = (\<lambda> k. True)" + +recdef \<zeta> "measure size" + "\<zeta> (And p q) = ilcm (\<zeta> p) (\<zeta> q)" + "\<zeta> (Or p q) = ilcm (\<zeta> p) (\<zeta> q)" + "\<zeta> (Eq (CN 0 c e)) = c" + "\<zeta> (NEq (CN 0 c e)) = c" + "\<zeta> (Lt (CN 0 c e)) = c" + "\<zeta> (Le (CN 0 c e)) = c" + "\<zeta> (Gt (CN 0 c e)) = c" + "\<zeta> (Ge (CN 0 c e)) = c" + "\<zeta> (Dvd i (CN 0 c e)) = c" + "\<zeta> (NDvd i (CN 0 c e))= c" + "\<zeta> p = 1" + +recdef \<beta> "measure size" + "\<beta> (And p q) = (\<beta> p @ \<beta> q)" + "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" + "\<beta> (Eq (CN 0 c e)) = [Sub (C -1) e]" + "\<beta> (NEq (CN 0 c e)) = [Neg e]" + "\<beta> (Lt (CN 0 c e)) = []" + "\<beta> (Le (CN 0 c e)) = []" + "\<beta> (Gt (CN 0 c e)) = [Neg e]" + "\<beta> (Ge (CN 0 c e)) = [Sub (C -1) e]" + "\<beta> p = []" + +recdef \<alpha> "measure size" + "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" + "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" + "\<alpha> (Eq (CN 0 c e)) = [Add (C -1) e]" + "\<alpha> (NEq (CN 0 c e)) = [e]" + "\<alpha> (Lt (CN 0 c e)) = [e]" + "\<alpha> (Le (CN 0 c e)) = [Add (C -1) e]" + "\<alpha> (Gt (CN 0 c e)) = []" + "\<alpha> (Ge (CN 0 c e)) = []" + "\<alpha> p = []" +consts mirror :: "fm \<Rightarrow> fm" +recdef mirror "measure size" + "mirror (And p q) = And (mirror p) (mirror q)" + "mirror (Or p q) = Or (mirror p) (mirror q)" + "mirror (Eq (CN 0 c e)) = Eq (CN 0 c (Neg e))" + "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))" + "mirror (Lt (CN 0 c e)) = Gt (CN 0 c (Neg e))" + "mirror (Le (CN 0 c e)) = Ge (CN 0 c (Neg e))" + "mirror (Gt (CN 0 c e)) = Lt (CN 0 c (Neg e))" + "mirror (Ge (CN 0 c e)) = Le (CN 0 c (Neg e))" + "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))" + "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))" + "mirror p = p" + +lemma mirror\<alpha>\<beta>: + assumes lp: "iszlfm p (a#bs)" + shows "(Inum (real (i::int)#bs)) ` set (\<alpha> p) = (Inum (real i#bs)) ` set (\<beta> (mirror p))" +using lp +by (induct p rule: mirror.induct, auto) + +lemma mirror: + assumes lp: "iszlfm p (a#bs)" + shows "Ifm (real (x::int)#bs) (mirror p) = Ifm (real (- x)#bs) p" +using lp +proof(induct p rule: iszlfm.induct) + case (9 j c e) + have th: "(real j rdvd real c * real x - Inum (real x # bs) e) = + (real j rdvd - (real c * real x - Inum (real x # bs) e))" + by (simp only: rdvd_minus[symmetric]) + from prems show ?case + by (simp add: ring_eq_simps th[simplified ring_eq_simps diff_def] + numbound0_I[where bs="bs" and b'="real x" and b="- real x"]) +next + case (10 j c e) + have th: "(real j rdvd real c * real x - Inum (real x # bs) e) = + (real j rdvd - (real c * real x - Inum (real x # bs) e))" + by (simp only: rdvd_minus[symmetric]) + from prems show ?case + by (simp add: ring_eq_simps th[simplified ring_eq_simps diff_def] + numbound0_I[where bs="bs" and b'="real x" and b="- real x"]) +qed (auto simp add: numbound0_I[where bs="bs" and b="real x" and b'="- real x"] nth_pos2) + +lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)" +by (induct p rule: mirror.induct, auto simp add: isint_neg) + +lemma mirror_d\<beta>: "iszlfm p (a#bs) \<and> d\<beta> p 1 + \<Longrightarrow> iszlfm (mirror p) (a#bs) \<and> d\<beta> (mirror p) 1" +by (induct p rule: mirror.induct, auto simp add: isint_neg) + +lemma mirror_\<delta>: "iszlfm p (a#bs) \<Longrightarrow> \<delta> (mirror p) = \<delta> p" +by (induct p rule: mirror.induct,auto) + + +lemma mirror_ex: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "(\<exists> (x::int). Ifm (real x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real x#bs) p)" + (is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)") +proof(auto) + fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast + thus "\<exists> x. ?I x p" by blast +next + fix x assume "?I x p" hence "?I (- x) ?mp" + using mirror[OF lp, where x="- x", symmetric] by auto + thus "\<exists> x. ?I x ?mp" by blast +qed + +lemma \<beta>_numbound0: assumes lp: "iszlfm p bs" + shows "\<forall> b\<in> set (\<beta> p). numbound0 b" + using lp by (induct p rule: \<beta>.induct,auto) + +lemma d\<beta>_mono: + assumes linp: "iszlfm p (a #bs)" + and dr: "d\<beta> p l" + and d: "l dvd l'" + shows "d\<beta> p l'" +using dr linp zdvd_trans[where n="l" and k="l'", simplified d] +by (induct p rule: iszlfm.induct) simp_all + +lemma \<alpha>_l: assumes lp: "iszlfm p (a#bs)" + shows "\<forall> b\<in> set (\<alpha> p). numbound0 b \<and> isint b (a#bs)" +using lp +by(induct p rule: \<alpha>.induct, auto simp add: isint_add isint_c) + +lemma \<zeta>: + assumes linp: "iszlfm p (a #bs)" + shows "\<zeta> p > 0 \<and> d\<beta> p (\<zeta> p)" +using linp +proof(induct p rule: iszlfm.induct) + case (1 p q) + from prems have dl1: "\<zeta> p dvd ilcm (\<zeta> p) (\<zeta> q)" + by (simp add: ilcm_dvd1[where a="\<zeta> p" and b="\<zeta> q"]) + from prems have dl2: "\<zeta> q dvd ilcm (\<zeta> p) (\<zeta> q)" + by (simp add: ilcm_dvd2[where a="\<zeta> p" and b="\<zeta> q"]) + from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="ilcm (\<zeta> p) (\<zeta> q)"] + d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="ilcm (\<zeta> p) (\<zeta> q)"] + dl1 dl2 show ?case by (auto simp add: ilcm_pos) +next + case (2 p q) + from prems have dl1: "\<zeta> p dvd ilcm (\<zeta> p) (\<zeta> q)" + by (simp add: ilcm_dvd1[where a="\<zeta> p" and b="\<zeta> q"]) + from prems have dl2: "\<zeta> q dvd ilcm (\<zeta> p) (\<zeta> q)" + by (simp add: ilcm_dvd2[where a="\<zeta> p" and b="\<zeta> q"]) + from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="ilcm (\<zeta> p) (\<zeta> q)"] + d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="ilcm (\<zeta> p) (\<zeta> q)"] + dl1 dl2 show ?case by (auto simp add: ilcm_pos) +qed (auto simp add: ilcm_pos) + +lemma a\<beta>: assumes linp: "iszlfm p (a #bs)" and d: "d\<beta> p l" and lp: "l > 0" + shows "iszlfm (a\<beta> p l) (a #bs) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm (real (l * x) #bs) (a\<beta> p l) = Ifm ((real x)#bs) p)" +using linp d +proof (induct p rule: iszlfm.induct) + case (5 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e < (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e < 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) < (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e < 0)" + using mult_less_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (6 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<le> (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<le> 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<le> (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e \<le> 0)" + using mult_le_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (7 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e > (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e > 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) > (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e > 0)" + using zero_less_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (8 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<ge> (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<ge> 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<ge> (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e \<ge> 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (3 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e = (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) = (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e = 0)" + using mult_eq_0_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (4 c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(real l * real x + real (l div c) * Inum (real x # bs) e \<noteq> (0\<Colon>real)) = + (real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e \<noteq> 0)" + by simp + also have "\<dots> = (real (l div c) * (real c * real x + Inum (real x # bs) e) \<noteq> (real (l div c)) * 0)" by (simp add: ring_eq_simps) + also have "\<dots> = (real c * real x + Inum (real x # bs) e \<noteq> 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e"] ldcp by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] be isint_Mul[OF ei] by simp +next + case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp + also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: ring_eq_simps) + also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp + also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp +next + case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+ + from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp]) + from cp have cnz: "c \<noteq> 0" by simp + have "c div c\<le> l div c" + by (simp add: zdiv_mono1[OF clel cp]) + then have ldcp:"0 < l div c" + by (simp add: zdiv_self[OF cnz]) + have "c * (l div c) = c* (l div c) + l mod c" using d' zdvd_iff_zmod_eq_0[where m="c" and n="l"] by simp + hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] + by simp + hence "(\<exists> (k::int). real l * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k) = (\<exists> (k::int). real (c * (l div c)) * real x + real (l div c) * Inum (real x # bs) e = (real (l div c) * real j) * real k)" by simp + also have "\<dots> = (\<exists> (k::int). real (l div c) * (real c * real x + Inum (real x # bs) e - real j * real k) = real (l div c)*0)" by (simp add: ring_eq_simps) + also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e - real j * real k = 0)" + using zero_le_mult_iff [where a="real (l div c)" and b="real c * real x + Inum (real x # bs) e - real j * real k"] ldcp by simp + also have "\<dots> = (\<exists> (k::int). real c * real x + Inum (real x # bs) e = real j * real k)" by simp + finally show ?case using numbound0_I[OF be,where b="real (l * x)" and b'="real x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp +qed (simp_all add: nth_pos2 numbound0_I[where bs="bs" and b="real (l * x)" and b'="real x"] isint_Mul del: real_of_int_mult) + +lemma a\<beta>_ex: assumes linp: "iszlfm p (a#bs)" and d: "d\<beta> p l" and lp: "l>0" + shows "(\<exists> x. l dvd x \<and> Ifm (real x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm (real x#bs) p)" + (is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)") +proof- + have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))" + using unity_coeff_ex[where l="l" and P="?P", simplified] by simp + also have "\<dots> = (\<exists> (x::int). ?P' x)" using a\<beta>[OF linp d lp] by simp + finally show ?thesis . +qed + +lemma \<beta>: + assumes lp: "iszlfm p (a#bs)" + and u: "d\<beta> p 1" + and d: "d\<delta> p d" + and dp: "d > 0" + and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)" + and p: "Ifm (real x#bs) p" (is "?P x") + shows "?P (x - d)" +using lp u d dp nob p +proof(induct p rule: iszlfm.induct) + case (5 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems + show ?case by (simp del: real_of_int_minus) +next + case (6 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + with dp p c1 numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] prems + show ?case by (simp del: real_of_int_minus) +next + case (7 c e) hence p: "Ifm (real x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" and ie1:"isint e (a#bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from ie1 have ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"] + numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"] + by (simp add: isint_iff) + {assume "real (x-d) +?e > 0" hence ?case using c1 + numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] + by (simp del: real_of_int_minus)} + moreover + {assume H: "\<not> real (x-d) + ?e > 0" + let ?v="Neg e" + have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] + have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x = - ?e + real j)" by auto + from H p have "real x + ?e > 0 \<and> real x + ?e \<le> real d" by (simp add: c1) + hence "real (x + floor ?e) > real (0::int) \<and> real (x + floor ?e) \<le> real d" + using ie by simp + hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d" by simp + hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp + hence "\<exists> (j::int) \<in> {1 .. d}. real x = real (- floor ?e + j)" + by (simp only: real_of_int_inject) (simp add: ring_eq_simps) + hence "\<exists> (j::int) \<in> {1 .. d}. real x = - ?e + real j" + by (simp add: ie[simplified isint_iff]) + with nob have ?case by auto} + ultimately show ?case by blast +next + case (8 c e) hence p: "Ifm (real x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" + and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from ie1 have ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"] + by (simp add: isint_iff) + {assume "real (x-d) +?e \<ge> 0" hence ?case using c1 + numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] + by (simp del: real_of_int_minus)} + moreover + {assume H: "\<not> real (x-d) + ?e \<ge> 0" + let ?v="Sub (C -1) e" + have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp + from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="real x" and bs="bs"]] + have nob: "\<not> (\<exists> j\<in> {1 ..d}. real x = - ?e - 1 + real j)" by auto + from H p have "real x + ?e \<ge> 0 \<and> real x + ?e < real d" by (simp add: c1) + hence "real (x + floor ?e) \<ge> real (0::int) \<and> real (x + floor ?e) < real d" + using ie by simp + hence "x + floor ?e +1 \<ge> 1 \<and> x + floor ?e + 1 \<le> d" by simp + hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp + hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: ring_eq_simps) + hence "\<exists> (j::int) \<in> {1 .. d}. real x= real (- floor ?e - 1 + j)" + by (simp only: real_of_int_inject) + hence "\<exists> (j::int) \<in> {1 .. d}. real x= - ?e - 1 + real j" + by (simp add: ie[simplified isint_iff]) + with nob have ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence p: "Ifm (real x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + let ?v="(Sub (C -1) e)" + have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp + from p have "real x= - ?e" by (simp add: c1) with prems(11) show ?case using dp + by simp (erule ballE[where x="1"], + simp_all add:ring_eq_simps numbound0_I[OF bn,where b="real x"and b'="a"and bs="bs"]) +next + case (4 c e)hence p: "Ifm (real x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + let ?v="Neg e" + have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp + {assume "real x - real d + Inum ((real (x -d)) # bs) e \<noteq> 0" + hence ?case by (simp add: c1)} + moreover + {assume H: "real x - real d + Inum ((real (x -d)) # bs) e = 0" + hence "real x = - Inum ((real (x -d)) # bs) e + real d" by simp + hence "real x = - Inum (a # bs) e + real d" + by (simp add: numbound0_I[OF bn,where b="real x - real d"and b'="a"and bs="bs"]) + with prems(11) have ?case using dp by simp} + ultimately show ?case by blast +next + case (9 j c e) hence p: "Ifm (real x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" + and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from prems have "isint e (a #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real x)#bs"] numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from c1 ie[symmetric] have "?p x = (real j rdvd real (x+ floor ?e))" by simp + also have "\<dots> = (j dvd x + floor ?e)" + using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp + also have "\<dots> = (j dvd x - d + floor ?e)" + using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp + also have "\<dots> = (real j rdvd real (x - d + floor ?e))" + using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified] + ie by simp + also have "\<dots> = (real j rdvd real x - real d + ?e)" + using ie by simp + finally show ?case + using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp +next + case (10 j c e) hence p: "Ifm (real x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+ + let ?e = "Inum (real x # bs) e" + from prems have "isint e (a#bs)" by simp + hence ie: "real (floor ?e) = ?e" using numbound0_I[OF bn,where b="real x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real x)#bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from c1 ie[symmetric] have "?p x = (\<not> real j rdvd real (x+ floor ?e))" by simp + also have "\<dots> = (\<not> j dvd x + floor ?e)" + using int_rdvd_real[where i="j" and x="real (x+ floor ?e)"] by simp + also have "\<dots> = (\<not> j dvd x - d + floor ?e)" + using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp + also have "\<dots> = (\<not> real j rdvd real (x - d + floor ?e))" + using int_rdvd_real[where i="j" and x="real (x-d + floor ?e)",symmetric, simplified] + ie by simp + also have "\<dots> = (\<not> real j rdvd real x - real d + ?e)" + using ie by simp + finally show ?case using numbound0_I[OF bn,where b="real (x-d)" and b'="real x" and bs="bs"] c1 p by simp +qed (auto simp add: numbound0_I[where bs="bs" and b="real (x - d)" and b'="real x"] nth_pos2 simp del: real_of_int_diff) + +lemma \<beta>': + assumes lp: "iszlfm p (a #bs)" + and u: "d\<beta> p 1" + and d: "d\<delta> p d" + and dp: "d > 0" + shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)") +proof(clarify) + fix x + assume nb:"?b" and px: "?P x" + hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real x = b + real j)" + by auto + from \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" . +qed + +lemma \<beta>_int: assumes lp: "iszlfm p bs" + shows "\<forall> b\<in> set (\<beta> p). isint b bs" +using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub) + +theorem cp_thm: + assumes lp: "iszlfm p (a #bs)" + and u: "d\<beta> p 1" + and d: "d\<delta> p d" + and dp: "d > 0" + shows "(\<exists> (x::int). Ifm (real x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real j) #bs) p))" + (is "(\<exists> (x::int). ?P (real x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real j)))") +proof- + from minusinf_inf[OF lp] + have th: "\<exists>(z::int). \<forall>x<z. ?P (real x) = ?M x" by blast + let ?B' = "{floor (?I b) | b. b\<in> ?B}" + from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real (floor (?I b)) = ?I b" by simp + from B[rule_format] + have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b)) + real j))" + by simp + also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real (floor (?I b) + j)))" by simp + also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))" by blast + finally have BB': + "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j)))" + by blast + hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real (b + j))) \<longrightarrow> ?P (real x) \<longrightarrow> ?P (real (x - d))" using \<beta>'[OF lp u d dp] by blast + from minusinf_repeats[OF d lp] + have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp + from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast +qed + + (* Reddy and Loveland *) + + +consts + \<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy/Loveland Bset*) + \<sigma>\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy/Loveland*) + \<alpha>\<rho> :: "fm \<Rightarrow> (num\<times>int) list" + a\<rho> :: "fm \<Rightarrow> int \<Rightarrow> fm" +recdef \<rho> "measure size" + "\<rho> (And p q) = (\<rho> p @ \<rho> q)" + "\<rho> (Or p q) = (\<rho> p @ \<rho> q)" + "\<rho> (Eq (CN 0 c e)) = [(Sub (C -1) e,c)]" + "\<rho> (NEq (CN 0 c e)) = [(Neg e,c)]" + "\<rho> (Lt (CN 0 c e)) = []" + "\<rho> (Le (CN 0 c e)) = []" + "\<rho> (Gt (CN 0 c e)) = [(Neg e, c)]" + "\<rho> (Ge (CN 0 c e)) = [(Sub (C (-1)) e, c)]" + "\<rho> p = []" + +recdef \<sigma>\<rho> "measure size" + "\<sigma>\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))" + "\<sigma>\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>\<rho> p (t,k)) (\<sigma>\<rho> q (t,k)))" + "\<sigma>\<rho> (Eq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e)) + else (Eq (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e)) + else (NEq (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (Lt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e)) + else (Lt (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (Le (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e)) + else (Le (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (Gt (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e)) + else (Gt (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (Ge (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e)) + else (Ge (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e)) + else (Dvd (i*k) (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e)) + else (NDvd (i*k) (Add (Mul c t) (Mul k e))))" + "\<sigma>\<rho> p = (\<lambda> (t,k). p)" + +recdef \<alpha>\<rho> "measure size" + "\<alpha>\<rho> (And p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)" + "\<alpha>\<rho> (Or p q) = (\<alpha>\<rho> p @ \<alpha>\<rho> q)" + "\<alpha>\<rho> (Eq (CN 0 c e)) = [(Add (C -1) e,c)]" + "\<alpha>\<rho> (NEq (CN 0 c e)) = [(e,c)]" + "\<alpha>\<rho> (Lt (CN 0 c e)) = [(e,c)]" + "\<alpha>\<rho> (Le (CN 0 c e)) = [(Add (C -1) e,c)]" + "\<alpha>\<rho> p = []" + + (* Simulates normal substituion by modifying the formula see correctness theorem *) + +recdef a\<rho> "measure size" + "a\<rho> (And p q) = (\<lambda> k. And (a\<rho> p k) (a\<rho> q k))" + "a\<rho> (Or p q) = (\<lambda> k. Or (a\<rho> p k) (a\<rho> q k))" + "a\<rho> (Eq (CN 0 c e)) = (\<lambda> k. if k dvd c then (Eq (CN 0 (c div k) e)) + else (Eq (CN 0 c (Mul k e))))" + "a\<rho> (NEq (CN 0 c e)) = (\<lambda> k. if k dvd c then (NEq (CN 0 (c div k) e)) + else (NEq (CN 0 c (Mul k e))))" + "a\<rho> (Lt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Lt (CN 0 (c div k) e)) + else (Lt (CN 0 c (Mul k e))))" + "a\<rho> (Le (CN 0 c e)) = (\<lambda> k. if k dvd c then (Le (CN 0 (c div k) e)) + else (Le (CN 0 c (Mul k e))))" + "a\<rho> (Gt (CN 0 c e)) = (\<lambda> k. if k dvd c then (Gt (CN 0 (c div k) e)) + else (Gt (CN 0 c (Mul k e))))" + "a\<rho> (Ge (CN 0 c e)) = (\<lambda> k. if k dvd c then (Ge (CN 0 (c div k) e)) + else (Ge (CN 0 c (Mul k e))))" + "a\<rho> (Dvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (Dvd i (CN 0 (c div k) e)) + else (Dvd (i*k) (CN 0 c (Mul k e))))" + "a\<rho> (NDvd i (CN 0 c e)) = (\<lambda> k. if k dvd c then (NDvd i (CN 0 (c div k) e)) + else (NDvd (i*k) (CN 0 c (Mul k e))))" + "a\<rho> p = (\<lambda> k. p)" + +constdefs \<sigma> :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" + "\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>\<rho> p (t,k))" + +lemma \<sigma>\<rho>: + assumes linp: "iszlfm p (real (x::int)#bs)" + and kpos: "real k > 0" + and tnb: "numbound0 t" + and tint: "isint t (real x#bs)" + and kdt: "k dvd floor (Inum (b'#bs) t)" + shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) = + (Ifm ((real ((floor (Inum (b'#bs) t)) div k))#bs) p)" + (is "?I (real x) (?s p) = (?I (real ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)") +using linp kpos tnb +proof(induct p rule: \<sigma>\<rho>.induct) + case (3 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Eq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k = 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (4 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (NEq (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<noteq> 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (5 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Lt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k < 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (6 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Le (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<le> 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (7 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Gt (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k > 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (8 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Ge (CN 0 c e))) = ((real c * (?N (real x) t / real k) + ?N (real x) e)* real k \<ge> 0)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real c * (?N (real x) t / real k) + ?N (real x) e" and b="0", symmetric] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (9 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (Dvd i (CN 0 c e))) = (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k)" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (Dvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +next + case (10 i c e) from prems have cp: "c > 0" and nb: "numbound0 e" by auto + {assume kdc: "k dvd c" + from kpos have knz: "k\<noteq>0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have ?case using real_of_int_div[OF knz kdc] real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti) } + moreover + {assume "\<not> k dvd c" + from kpos have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + from tint have ti: "real (floor (?N (real x) t)) = ?N (real x) t" using isint_def by simp + from prems have "?I (real x) (?s (NDvd i (CN 0 c e))) = (\<not> (real i * real k rdvd (real c * (?N (real x) t / real k) + ?N (real x) e)* real k))" + using real_of_int_div[OF knz kdt] + numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] by (simp add: ti ring_eq_simps) + also have "\<dots> = (?I ?tk (NDvd i (CN 0 c e)))" using rdvd_mult[OF knz, where n="i"] real_of_int_div[OF knz kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real x"] + numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real x"] + by (simp add: ti) + finally have ?case . } + ultimately show ?case by blast +qed (simp_all add: nth_pos2 bound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"] numbound0_I[where bs="bs" and b="real ((floor (?N b' t)) div k)" and b'="real x"]) + + +lemma a\<rho>: + assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "real k > 0" + shows "Ifm (real (x*k)#bs) (a\<rho> p k) = Ifm (real x#bs) p" (is "?I (x*k) (?f p k) = ?I x p") +using lp bound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] numbound0_I[where bs="bs" and b="real (x*k)" and b'="real x"] +proof(induct p rule: a\<rho>.induct) + case (3 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (4 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] nonzero_eq_divide_eq[OF knz', where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (5 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_less_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (6 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_le_divide_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (7 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_less_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (8 c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume nkdc: "\<not> k dvd c" hence ?case using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] pos_divide_le_eq[OF kp, where b="0" and a="real c * real x + Inum (real x # bs) e", symmetric] by (simp add: ring_eq_simps)} + ultimately show ?case by blast +next + case (9 i c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume "\<not> k dvd c" + hence "Ifm (real (x*k)#bs) (a\<rho> (Dvd i (CN 0 c e)) k) = + (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k)" + using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] + by (simp add: ring_eq_simps) + also have "\<dots> = (Ifm (real x#bs) (Dvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"]) + finally have ?case . } + ultimately show ?case by blast +next + case (10 i c e) + from prems have cp: "c > 0" and nb: "numbound0 e" by auto + from kp have knz: "k\<noteq>0" by simp hence knz': "real k \<noteq> 0" by simp + {assume kdc: "k dvd c" from prems have ?case using real_of_int_div[OF knz kdc] by simp } + moreover + {assume "\<not> k dvd c" + hence "Ifm (real (x*k)#bs) (a\<rho> (NDvd i (CN 0 c e)) k) = + (\<not> (real i * real k rdvd (real c * real x + Inum (real x#bs) e) * real k))" + using numbound0_I[OF nb, where bs="bs" and b="real (x*k)" and b'="real x"] + by (simp add: ring_eq_simps) + also have "\<dots> = (Ifm (real x#bs) (NDvd i (CN 0 c e)))" by (simp add: rdvd_mult[OF knz, where n="i"]) + finally have ?case . } + ultimately show ?case by blast +qed (simp_all add: nth_pos2) + +lemma a\<rho>_ex: + assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" + shows "(\<exists> (x::int). real k rdvd real x \<and> Ifm (real x#bs) (a\<rho> p k)) = + (\<exists> (x::int). Ifm (real x#bs) p)" (is "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. ?P x)") +proof- + have "(\<exists> x. ?D x \<and> ?P' x) = (\<exists> x. k dvd x \<and> ?P' x)" using int_rdvd_iff by simp + also have "\<dots> = (\<exists>x. ?P' (x*k))" using unity_coeff_ex[where P="?P'" and l="k", simplified] + by (simp add: ring_eq_simps) + also have "\<dots> = (\<exists> x. ?P x)" using a\<rho> iszlfm_gen[OF lp] kp by auto + finally show ?thesis . +qed + +lemma \<sigma>\<rho>': assumes lp: "iszlfm p (real (x::int)#bs)" and kp: "k > 0" and nb: "numbound0 t" + shows "Ifm (real x#bs) (\<sigma>\<rho> p (t,k)) = Ifm ((Inum (real x#bs) t)#bs) (a\<rho> p k)" +using lp +by(induct p rule: \<sigma>\<rho>.induct, simp_all add: + numbound0_I[OF nb, where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] + numbound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] + bound0_I[where bs="bs" and b="Inum (real x#bs) t" and b'="real x"] nth_pos2 cong: imp_cong) + +lemma \<sigma>\<rho>_nb: assumes lp:"iszlfm p (a#bs)" and nb: "numbound0 t" + shows "bound0 (\<sigma>\<rho> p (t,k))" + using lp + by (induct p rule: iszlfm.induct, auto simp add: nb) + +lemma \<rho>_l: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)" +using lp by (induct p rule: \<rho>.induct, auto simp add: isint_sub isint_neg) + +lemma \<alpha>\<rho>_l: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "\<forall> (b,k) \<in> set (\<alpha>\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real i#bs)" +using lp isint_add [OF isint_c[where j="- 1"],where bs="real i#bs"] + by (induct p rule: \<alpha>\<rho>.induct, auto) + +lemma zminusinf_\<rho>: + assumes lp: "iszlfm p (real (i::int)#bs)" + and nmi: "\<not> (Ifm (real i#bs) (minusinf p))" (is "\<not> (Ifm (real i#bs) (?M p))") + and ex: "Ifm (real i#bs) p" (is "?I i p") + shows "\<exists> (e,c) \<in> set (\<rho> p). real (c*i) > Inum (real i#bs) e" (is "\<exists> (e,c) \<in> ?R p. real (c*i) > ?N i e") + using lp nmi ex +by (induct p rule: minusinf.induct, auto) + + +lemma \<sigma>_And: "Ifm bs (\<sigma> (And p q) k t) = Ifm bs (And (\<sigma> p k t) (\<sigma> q k t))" +using \<sigma>_def by auto +lemma \<sigma>_Or: "Ifm bs (\<sigma> (Or p q) k t) = Ifm bs (Or (\<sigma> p k t) (\<sigma> q k t))" +using \<sigma>_def by auto + +lemma \<rho>: assumes lp: "iszlfm p (real (i::int) #bs)" + and pi: "Ifm (real i#bs) p" + and d: "d\<delta> p d" + and dp: "d > 0" + and nob: "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> Inum (real i#bs) e + real j" + (is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _") + shows "Ifm (real(i - d)#bs) p" + using lp pi d nob +proof(induct p rule: iszlfm.induct) + case (3 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and pi: "real (c*i) = - 1 - ?N i e + real (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> -1 - ?N i e + real j" + by simp+ + from mult_strict_left_mono[OF dp cp] have one:"1 \<in> {1 .. c*d}" by auto + from nob[rule_format, where j="1", OF one] pi show ?case by simp +next + case (4 c e) + hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j" + by simp+ + {assume "real (c*i) \<noteq> - ?N i e + real (c*d)" + with numbound0_I[OF nb, where bs="bs" and b="real i - real d" and b'="real i"] + have ?case by (simp add: ring_eq_simps)} + moreover + {assume pi: "real (c*i) = - ?N i e + real (c*d)" + from mult_strict_left_mono[OF dp cp] have d: "(c*d) \<in> {1 .. c*d}" by simp + from nob[rule_format, where j="c*d", OF d] pi have ?case by simp } + ultimately show ?case by blast +next + case (5 c e) hence cp: "c > 0" by simp + from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] + real_of_int_mult] + show ?case using prems dp + by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] + ring_eq_simps) +next + case (6 c e) hence cp: "c > 0" by simp + from prems mult_strict_left_mono[OF dp cp, simplified real_of_int_less_iff[symmetric] + real_of_int_mult] + show ?case using prems dp + by (simp add: add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] + ring_eq_simps) +next + case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - ?N i e + real j" + and pi: "real (c*i) + ?N i e > 0" and cp': "real c >0" + by simp+ + let ?fe = "floor (?N i e)" + from pi cp have th:"(real i +?N i e / real c)*real c > 0" by (simp add: ring_eq_simps) + from pi ei[simplified isint_iff] have "real (c*i + ?fe) > real (0::int)" by simp + hence pi': "c*i + ?fe > 0" by (simp only: real_of_int_less_iff[symmetric]) + have "real (c*i) + ?N i e > real (c*d) \<or> real (c*i) + ?N i e \<le> real (c*d)" by auto + moreover + {assume "real (c*i) + ?N i e > real (c*d)" hence ?case + by (simp add: ring_eq_simps + numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} + moreover + {assume H:"real (c*i) + ?N i e \<le> real (c*d)" + with ei[simplified isint_iff] have "real (c*i + ?fe) \<le> real (c*d)" by simp + hence pid: "c*i + ?fe \<le> c*d" by (simp only: real_of_int_le_iff) + with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + ?fe = j1" by auto + hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - ?N i e + real j1" + by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] ring_eq_simps) + with nob have ?case by blast } + ultimately show ?case by blast +next + case (8 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real i#bs)" + and nob: "\<forall> j\<in> {1 .. c*d}. real (c*i) \<noteq> - 1 - ?N i e + real j" + and pi: "real (c*i) + ?N i e \<ge> 0" and cp': "real c >0" + by simp+ + let ?fe = "floor (?N i e)" + from pi cp have th:"(real i +?N i e / real c)*real c \<ge> 0" by (simp add: ring_eq_simps) + from pi ei[simplified isint_iff] have "real (c*i + ?fe) \<ge> real (0::int)" by simp + hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: real_of_int_le_iff[symmetric]) + have "real (c*i) + ?N i e \<ge> real (c*d) \<or> real (c*i) + ?N i e < real (c*d)" by auto + moreover + {assume "real (c*i) + ?N i e \<ge> real (c*d)" hence ?case + by (simp add: ring_eq_simps + numbound0_I[OF nb,where bs="bs" and b="real i - real d" and b'="real i"])} + moreover + {assume H:"real (c*i) + ?N i e < real (c*d)" + with ei[simplified isint_iff] have "real (c*i + ?fe) < real (c*d)" by simp + hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: real_of_int_le_iff) + with pi' have "\<exists> j1\<in> {1 .. c*d}. c*i + 1+ ?fe = j1" by auto + hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) + 1= - ?N i e + real j1" + by (simp only: diff_def[symmetric] real_of_int_mult real_of_int_add real_of_int_inject[symmetric] ei[simplified isint_iff] ring_eq_simps real_of_one) + hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = (- ?N i e + real j1) - 1" + by (simp only: ring_eq_simps diff_def[symmetric]) + hence "\<exists> j1\<in> {1 .. c*d}. real (c*i) = - 1 - ?N i e + real j1" + by (simp only: add_ac diff_def) + with nob have ?case by blast } + ultimately show ?case by blast +next + case (9 j c e) hence p: "real j rdvd real (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ + let ?e = "Inum (real i # bs) e" + from prems have "isint e (real i #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from ie[symmetric] have "?p i = (real j rdvd real (c*i+ floor ?e))" by simp + also have "\<dots> = (j dvd c*i + floor ?e)" + using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp + also have "\<dots> = (j dvd c*i - c*d + floor ?e)" + using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp + also have "\<dots> = (real j rdvd real (c*i - c*d + floor ?e))" + using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified] + ie by simp + also have "\<dots> = (real j rdvd real (c*(i - d)) + ?e)" + using ie by (simp add:ring_eq_simps) + finally show ?case + using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p + by (simp add: ring_eq_simps) +next + case (10 j c e) hence p: "\<not> (real j rdvd real (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ + let ?e = "Inum (real i # bs) e" + from prems have "isint e (real i #bs)" by simp + hence ie: "real (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real i)#bs"] numbound0_I[OF bn,where b="real i" and b'="real i" and bs="bs"] + by (simp add: isint_iff) + from prems have id: "j dvd d" by simp + from ie[symmetric] have "?p i = (\<not> (real j rdvd real (c*i+ floor ?e)))" by simp + also have "\<dots> = Not (j dvd c*i + floor ?e)" + using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp + also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)" + using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp + also have "\<dots> = Not (real j rdvd real (c*i - c*d + floor ?e))" + using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified] + ie by simp + also have "\<dots> = Not (real j rdvd real (c*(i - d)) + ?e)" + using ie by (simp add:ring_eq_simps) + finally show ?case + using numbound0_I[OF bn,where b="real i - real d" and b'="real i" and bs="bs"] p + by (simp add: ring_eq_simps) +qed(auto simp add: numbound0_I[where bs="bs" and b="real i - real d" and b'="real i"] nth_pos2) + +lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t" + shows "bound0 (\<sigma> p k t)" + using \<sigma>\<rho>_nb[OF lp nb] nb by (simp add: \<sigma>_def) + +lemma \<rho>': assumes lp: "iszlfm p (a #bs)" + and d: "d\<delta> p d" + and dp: "d > 0" + shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real x#bs) p \<longrightarrow> Ifm (real (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)") +proof(clarify) + fix x + assume nob1:"?b x" and px: "?P x" + from iszlfm_gen[OF lp, rule_format, where y="real x"] have lp': "iszlfm p (real x#bs)". + have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real (c * x) \<noteq> Inum (real x # bs) e + real j" + proof(clarify) + fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}" + and cx: "real (c*x) = Inum (real x#bs) e + real j" + let ?e = "Inum (real x#bs) e" + let ?fe = "floor ?e" + from \<rho>_l[OF lp'] ecR have ei:"isint e (real x#bs)" and cp:"c>0" and nb:"numbound0 e" + by auto + from numbound0_gen [OF nb ei, rule_format,where y="a"] have "isint e (a#bs)" . + from cx ei[simplified isint_iff] have "real (c*x) = real (?fe + j)" by simp + hence cx: "c*x = ?fe + j" by (simp only: real_of_int_inject) + hence cdej:"c dvd ?fe + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp) + hence "real c rdvd real (?fe + j)" by (simp only: int_rdvd_iff) + hence rcdej: "real c rdvd ?e + real j" by (simp add: ei[simplified isint_iff]) + from cx have "(c*x) div c = (?fe + j) div c" by simp + with cp have "x = (?fe + j) div c" by simp + with px have th: "?P ((?fe + j) div c)" by auto + from cp have cp': "real c > 0" by simp + from cdej have cdej': "c dvd floor (Inum (real x#bs) (Add e (C j)))" by simp + from nb have nb': "numbound0 (Add e (C j))" by simp + have ji: "isint (C j) (real x#bs)" by (simp add: isint_def) + from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real x#bs)" . + from th \<sigma>\<rho>[where b'="real x", OF lp' cp' nb' ei' cdej',symmetric] + have "Ifm (real x#bs) (\<sigma>\<rho> p (Add e (C j), c))" by simp + with rcdej have th: "Ifm (real x#bs) (\<sigma> p c (Add e (C j)))" by (simp add: \<sigma>_def) + from th bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"],where bs="bs" and b="real x" and b'="a"] + have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast + with ecR jD nob1 show "False" by blast + qed + from \<rho>[OF lp' px d dp nob] show "?P (x -d )" . +qed + + +lemma rl_thm: + assumes lp: "iszlfm p (real (i::int)#bs)" + shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))" + (is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))" + is "?lhs = (?MD \<or> ?RD)" is "?lhs = ?rhs") +proof- + let ?d= "\<delta> p" + from \<delta>[OF lp] have d:"d\<delta> p ?d" and dp: "?d > 0" by auto + { assume H:"?MD" hence th:"\<exists> (x::int). ?MP x" by blast + from H minusinf_ex[OF lp th] have ?thesis by blast} + moreover + { fix e c j assume exR:"(e,c) \<in> ?R" and jD:"j\<in> {1 .. c*?d}" and spx:"?SP c e j" + from exR \<rho>_l[OF lp] have nb: "numbound0 e" and ei:"isint e (real i#bs)" and cp: "c > 0" + by auto + have "isint (C j) (real i#bs)" by (simp add: isint_iff) + with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real i"]] + have eji:"isint (Add e (C j)) (real i#bs)" by simp + from nb have nb': "numbound0 (Add e (C j))" by simp + from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real i"] + have spx': "Ifm (real i # bs) (\<sigma> p c (Add e (C j)))" by blast + from spx' have rcdej:"real c rdvd (Inum (real i#bs) (Add e (C j)))" + and sr:"Ifm (real i#bs) (\<sigma>\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+ + from rcdej eji[simplified isint_iff] + have "real c rdvd real (floor (Inum (real i#bs) (Add e (C j))))" by simp + hence cdej:"c dvd floor (Inum (real i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff) + from cp have cp': "real c > 0" by simp + from \<sigma>\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real i # bs) (Add e (C j))\<rfloor> div c)" + by (simp add: \<sigma>_def) + hence ?lhs by blast + with exR jD spx have ?thesis by blast} + moreover + { fix x assume px: "?P x" and nob: "\<not> ?RD" + from iszlfm_gen [OF lp,rule_format, where y="a"] have lp':"iszlfm p (a#bs)" . + from \<rho>'[OF lp' d dp, rule_format, OF nob] have th:"\<forall> x. ?P x \<longrightarrow> ?P (x - ?d)" by blast + from minusinf_inf[OF lp] obtain z where z:"\<forall> x<z. ?MP x = ?P x" by blast + have zp: "abs (x - z) + 1 \<ge> 0" by arith + from decr_lemma[OF dp,where x="x" and z="z"] + decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\<exists> x. ?MP x" by auto + with minusinf_bex[OF lp] px nob have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma mirror_\<alpha>\<rho>: assumes lp: "iszlfm p (a#bs)" + shows "(\<lambda> (t,k). (Inum (a#bs) t, k)) ` set (\<alpha>\<rho> p) = (\<lambda> (t,k). (Inum (a#bs) t,k)) ` set (\<rho> (mirror p))" +using lp +by (induct p rule: mirror.induct, simp_all add: split_def image_Un ) + + + + (********************************************************************) + (*** THE \<real>-PART ***) + (********************************************************************) + + + (* Linearity for fm where Bound 0 ranges over \<real> *) +consts + isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *) +recdef isrlfm "measure size" + "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" + "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" + "isrlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Ge (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm p = (isatom p \<and> (bound0 p))" + +constdefs fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm" + "fp p n s j \<equiv> (if n > 0 then + (And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j))))) + (Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1)))))))) + else + (And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j))))) + (Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))" + + (* splits the bounded from the unbounded part*) + (* FIXME: Abscence of simplification of formulae and numeral-terms + here is also a problem!!!!! Redundancy!!!!!*) +consts rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list" +recdef rsplit0 "measure num_size" + "rsplit0 (Bound 0) = [(T,1,C 0)]" + "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b + in map (\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) (allpairs Pair acs bcs))" + "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" + "rsplit0 (Neg a) = map (\<lambda> (p,n,s). (p,-n,Neg s)) (rsplit0 a)" + "rsplit0 (Floor a) = foldl (op @) [] (map + (\<lambda> (p,n,s). if n=0 then [(p,0,Floor s)] + else (map (\<lambda> j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then iupt (0,n) else iupt(n,0)))) + (rsplit0 a))" + "rsplit0 (CN 0 c a) = map (\<lambda> (p,n,s). (p,n+c,s)) (rsplit0 a)" + "rsplit0 (CN m c a) = map (\<lambda> (p,n,s). (p,n,CN m c s)) (rsplit0 a)" + "rsplit0 (CF c t s) = rsplit0 (Add (Mul c (Floor t)) s)" + "rsplit0 (Mul c a) = map (\<lambda> (p,n,s). (p,c*n,Mul c s)) (rsplit0 a)" + "rsplit0 t = [(T,0,t)]" + +lemma not_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm (not p)" + by (induct p rule: isrlfm.induct, auto) +lemma conj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)" + using conj_def by (cases p, auto) +lemma disj_rl[simp]: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)" + using disj_def by (cases p, auto) + + +lemma rsplit0_cs: + shows "\<forall> (p,n,s) \<in> set (rsplit0 t). + (Ifm (x#bs) p \<longrightarrow> (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" + (is "\<forall> (p,n,s) \<in> ?SS t. (?I p \<longrightarrow> ?N t = ?N (CN 0 n s)) \<and> _ \<and> _ ") +proof(induct t rule: rsplit0.induct) + case (5 a) + let ?p = "\<lambda> (p,n,s) j. fp p n s j" + let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))" + let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)" + let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" + have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith + have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto + have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. + ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" by auto + hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). + set (map (?f(p,n,s)) (iupt(0,n)))))" + proof- + fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g + assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" + thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))" + by auto + hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = + (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n<0} (\<lambda>(p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))" + proof- + fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g + assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" + thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + from foldl_append_map_Nil_set[where xs="rsplit0 a" and f="?ff"] + have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" by auto + also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" + using int_cases[rule_format] by blast + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un + (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) (iupt(0,n))))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). + set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3) + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))" + by (simp only: set_map iupt_set set.simps) + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast + finally + have FS: "?SS (Floor a) = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast + show ?case + proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -) + fix p n s + let ?ths = "(?I p \<longrightarrow> (?N (Floor a) = ?N (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" + assume "(\<exists>ba. (p, 0, ba) \<in> set (rsplit0 a) \<and> n = 0 \<and> s = Floor ba) \<or> + (\<exists>ab ac ba. + (ab, ac, ba) \<in> set (rsplit0 a) \<and> + 0 < ac \<and> + (\<exists>j. p = fp ab ac ba j \<and> + n = 0 \<and> s = Add (Floor ba) (C j) \<and> 0 \<le> j \<and> j \<le> ac)) \<or> + (\<exists>ab ac ba. + (ab, ac, ba) \<in> set (rsplit0 a) \<and> + ac < 0 \<and> + (\<exists>j. p = fp ab ac ba j \<and> + n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))" + moreover + {fix s' + assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'" + hence ?ths using prems by auto} + moreover + { fix p' n' s' j + assume pns: "(p', n', s') \<in> ?SS a" + and np: "0 < n'" + and p_def: "p = ?p (p',n',s') j" + and n0: "n = 0" + and s_def: "s = (Add (Floor s') (C j))" + and jp: "0 \<le> j" and jn: "j \<le> n'" + from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow> + Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and> + numbound0 s' \<and> isrlfm p'" by blast + hence nb: "numbound0 s'" by simp + from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numsub_nb) + let ?nxs = "CN 0 n' s'" + let ?l = "floor (?N s') + j" + from H + have "?I (?p (p',n',s') j) \<longrightarrow> + (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" + by (simp add: fp_def np ring_eq_simps numsub numadd numfloor) + also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))" + using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp + moreover + have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp + ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" + by blast + with s_def n0 p_def nb nf have ?ths by auto} + moreover + {fix p' n' s' j + assume pns: "(p', n', s') \<in> ?SS a" + and np: "n' < 0" + and p_def: "p = ?p (p',n',s') j" + and n0: "n = 0" + and s_def: "s = (Add (Floor s') (C j))" + and jp: "n' \<le> j" and jn: "j \<le> 0" + from prems pns have H:"(Ifm ((x\<Colon>real) # (bs\<Colon>real list)) p' \<longrightarrow> + Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and> + numbound0 s' \<and> isrlfm p'" by blast + hence nb: "numbound0 s'" by simp + from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by (simp add: numneg_nb) + let ?nxs = "CN 0 n' s'" + let ?l = "floor (?N s') + j" + from H + have "?I (?p (p',n',s') j) \<longrightarrow> + (((?N ?nxs \<ge> real ?l) \<and> (?N ?nxs < real (?l + 1))) \<and> (?N a = ?N ?nxs ))" + by (simp add: np fp_def ring_eq_simps numneg numfloor numadd numsub) + also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))" + using floor_int_eq[where x="?N ?nxs" and n="?l"] by simp + moreover + have "\<dots> \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" by simp + ultimately have "?I (?p (p',n',s') j) \<longrightarrow> (?N (Floor a) = ?N ((Add (Floor s') (C j))))" + by blast + with s_def n0 p_def nb nf have ?ths by auto} + ultimately show ?ths by auto + qed +next + case (3 a b) thus ?case by auto +qed (auto simp add: Let_def allpairs_set split_def ring_eq_simps conj_rl) + +lemma real_in_int_intervals: + assumes xb: "real m \<le> x \<and> x < real ((n::int) + 1)" + shows "\<exists> j\<in> {m.. n}. real j \<le> x \<and> x < real (j+1)" (is "\<exists> j\<in> ?N. ?P j") +by (rule bexI[where P="?P" and x="floor x" and A="?N"]) +(auto simp add: floor_less_eq[where x="x" and a="n+1", simplified] xb[simplified] floor_mono2[where x="real m" and y="x", OF conjunct1[OF xb], simplified floor_real_of_int[where n="m"]]) + +lemma rsplit0_complete: + assumes xp:"0 \<le> x" and x1:"x < 1" + shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p") +proof(induct t rule: rsplit0.induct) + case (2 a b) + from prems have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by simp + then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast + from prems have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by simp + then obtain "pb" "nb" "sb" where pb: "(pb,nb,sb)\<in> ?SS b \<and> ?I pb" by blast + from pa pb have th: "((pa,na,sa),(pb,nb,sb)) \<in> set (allpairs Pair (rsplit0 a) (rsplit0 b))" + by (auto simp add: allpairs_set) + let ?f="(\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s))" + from imageI[OF th, where f="?f"] have "?f ((pa,na,sa),(pb,nb,sb)) \<in> ?SS (Add a b)" + by (simp add: Let_def) + hence "(And pa pb, na +nb, Add sa sb) \<in> ?SS (Add a b)" by simp + moreover from pa pb have "?I (And pa pb)" by simp + ultimately show ?case by blast +next + case (5 a) + let ?p = "\<lambda> (p,n,s) j. fp p n s j" + let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))" + let ?J = "\<lambda> n. if n>0 then iupt (0,n) else iupt (n,0)" + let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))" + have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith + have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto + have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(0,n))" + by auto + hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n)))))" + proof- + fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g + assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" + thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) (iupt(n,0))" + by auto + hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0)))))" + proof- + fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g + assume "\<forall> (a,b,c) \<in> M. f (a,b,c) = g a b c" + thus "(UNION M (\<lambda> (a,b,c). set (f (a,b,c)))) = (UNION M (\<lambda> (a,b,c). set (g a b c)))" + by (auto simp add: split_def) + qed + from foldl_append_map_Nil_set[where xs="rsplit0 a" and f="?ff"] + have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" by auto + also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by auto + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" + using int_cases[rule_format] by blast + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(0,n))))) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) (iupt(n,0))))))" by (simp only: U1 U2 U3) + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))" + by (simp only: set_map iupt_set set.simps) + also have "\<dots> = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast + finally + have FS: "?SS (Floor a) = + ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un + (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast + from prems have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by simp + then obtain "p" "n" "s" where pns: "(p,n,s) \<in> ?SS a \<and> ?I p" by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p" + by auto + + have "n=0 \<or> n >0 \<or> n <0" by arith + moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto } + moreover + { + assume np: "n > 0" + from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) \<le> ?N s" by simp + also from mult_left_mono[OF xp] np have "?N s \<le> real n * x + ?N s" by simp + finally have "?N (Floor s) \<le> real n * x + ?N s" . + moreover + {from mult_strict_left_mono[OF x1] np + have "real n *x + ?N s < real n + ?N s" by simp + also from real_of_int_floor_add_one_gt[where r="?N s"] + have "\<dots> < real n + ?N (Floor s) + 1" by simp + finally have "real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp} + ultimately have "?N (Floor s) \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (n+1)" by simp + hence th: "0 \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (n+1)" by simp + from real_in_int_intervals th have "\<exists> j\<in> {0 .. n}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp + + hence "\<exists> j\<in> {0 .. n}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0" + by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) + hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)" + using pns by (simp add: fp_def np ring_eq_simps numsub numadd) + then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast + hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto + hence ?case using pns + by (simp only: FS,simp add: bex_Un) + (rule disjI2, rule disjI1,rule exI [where x="p"], + rule exI [where x="n"],rule exI [where x="s"],simp_all add: np) + } + moreover + { assume nn: "n < 0" hence np: "-n >0" by simp + from real_of_int_floor_le[where r="?N s"] have "?N (Floor s) + 1 > ?N s" by simp + moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real n * x + ?N s" by simp + ultimately have "?N (Floor s) + 1 > real n * x + ?N s" by arith + moreover + {from mult_strict_left_mono_neg[OF x1, where c="real n"] nn + have "real n *x + ?N s \<ge> real n + ?N s" by simp + moreover from real_of_int_floor_le[where r="?N s"] have "real n + ?N s \<ge> real n + ?N (Floor s)" by simp + ultimately have "real n *x + ?N s \<ge> ?N (Floor s) + real n" + by (simp only: ring_eq_simps)} + ultimately have "?N (Floor s) + real n \<le> real n *x + ?N s\<and> real n *x + ?N s < ?N (Floor s) + real (1::int)" by simp + hence th: "real n \<le> real n *x + ?N s - ?N (Floor s) \<and> real n *x + ?N s - ?N (Floor s) < real (1::int)" by simp + have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto + have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto + from real_in_int_intervals th have "\<exists> j\<in> {n .. 0}. real j \<le> real n *x + ?N s - ?N (Floor s)\<and> real n *x + ?N s - ?N (Floor s) < real (j+1)" by simp + + hence "\<exists> j\<in> {n .. 0}. 0 \<le> real n *x + ?N s - ?N (Floor s) - real j \<and> real n *x + ?N s - ?N (Floor s) - real (j+1) < 0" + by(simp only: myl[rule_format, where b="real n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real n *x + ?N s - ?N (Floor s)"]) + hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real n *x + ?N s - ?N (Floor s) - real j) \<and> - (real n *x + ?N s - ?N (Floor s) - real (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format]) + hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)" + using pns by (simp add: fp_def nn diff_def add_ac mult_ac numfloor numadd numneg + del: diff_less_0_iff_less diff_le_0_iff_le) + then obtain "j" where j_def: "j\<in> {n .. 0} \<and> ?I (?p (p,n,s) j)" by blast + hence "\<exists>x \<in> {?p (p,n,s) j |j. n\<le> j \<and> j \<le> 0 }. ?I x" by auto + hence ?case using pns + by (simp only: FS,simp add: bex_Un) + (rule disjI2, rule disjI2,rule exI [where x="p"], + rule exI [where x="n"],rule exI [where x="s"],simp_all add: np) + } + ultimately show ?case by blast +qed (auto simp add: Let_def split_def) + + (* Linearize a formula where Bound 0 ranges over [0,1) *) + +constdefs rsplit :: "(int \<Rightarrow> num \<Rightarrow> fm) \<Rightarrow> num \<Rightarrow> fm" + "rsplit f a \<equiv> foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) (rsplit0 a)) F" + +lemma foldr_disj_map: "Ifm bs (foldr disj (map f xs) F) = (\<exists> x \<in> set xs. Ifm bs (f x))" +by(induct xs, simp_all) + +lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\<forall> x \<in> set xs. Ifm bs (f x))" +by(induct xs, simp_all) + +lemma foldr_disj_map_rlfm: + assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)" + and \<phi>: "\<forall> (\<phi>,n,s) \<in> set xs. numbound0 s \<and> isrlfm \<phi>" + shows "isrlfm (foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) xs) F)" +using lf \<phi> by (induct xs, auto) + +lemma rsplit_ex: "Ifm bs (rsplit f a) = (\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). Ifm bs (conj \<phi> (f n s)))" +using foldr_disj_map[where xs="rsplit0 a"] rsplit_def by (simp add: split_def) + +lemma rsplit_l: assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)" + shows "isrlfm (rsplit f a)" +proof- + from rsplit0_cs[where t="a"] have th: "\<forall> (\<phi>,n,s) \<in> set (rsplit0 a). numbound0 s \<and> isrlfm \<phi>" by blast + from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp +qed + +lemma rsplit: + assumes xp: "x \<ge> 0" and x1: "x < 1" + and f: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))" + shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)" +proof(auto) + let ?I = "\<lambda>x p. Ifm (x#bs) p" + let ?N = "\<lambda> x t. Inum (x#bs) t" + assume "?I x (rsplit f a)" + hence "\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). ?I x (And \<phi> (f n s))" using rsplit_ex by simp + then obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and "?I x (And \<phi> (f n s))" by blast + hence \<phi>: "?I x \<phi>" and fns: "?I x (f n s)" by auto + from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi> + have th: "(?N x a = ?N x (CN 0 n s)) \<and> numbound0 s" by auto + from f[rule_format, OF th] fns show "?I x (g a)" by simp +next + let ?I = "\<lambda>x p. Ifm (x#bs) p" + let ?N = "\<lambda> x t. Inum (x#bs) t" + assume ga: "?I x (g a)" + from rsplit0_complete[OF xp x1, where bs="bs" and t="a"] + obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and fx: "?I x \<phi>" by blast + from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx + have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto + with ga f have "?I x (f n s)" by auto + with rsplit_ex fnsS fx show "?I x (rsplit f a)" by auto +qed + +consts + lt :: "int \<Rightarrow> num \<Rightarrow> fm" + le :: "int \<Rightarrow> num \<Rightarrow> fm" + gt :: "int \<Rightarrow> num \<Rightarrow> fm" + ge :: "int \<Rightarrow> num \<Rightarrow> fm" + eq :: "int \<Rightarrow> num \<Rightarrow> fm" + neq :: "int \<Rightarrow> num \<Rightarrow> fm" + +defs lt_def: "lt c t \<equiv> (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) + else (Gt (CN 0 (-c) (Neg t))))" +defs le_def: "le c t \<equiv> (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) + else (Ge (CN 0 (-c) (Neg t))))" +defs gt_def: "gt c t \<equiv> (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) + else (Lt (CN 0 (-c) (Neg t))))" +defs ge_def: "ge c t \<equiv> (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) + else (Le (CN 0 (-c) (Neg t))))" +defs eq_def: "eq c t \<equiv> (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) + else (Eq (CN 0 (-c) (Neg t))))" +defs neq_def: "neq c t \<equiv> (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) + else (NEq (CN 0 (-c) (Neg t))))" + +lemma lt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)" + (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _\<longrightarrow> ?I (lt n s) = ?I (Lt a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (lt n s) = ?I (Lt a)" using H by (cases "n=0", (simp add: lt_def)) + (cases "n > 0", simp_all add: lt_def ring_eq_simps myless[rule_format, where b="0"]) +qed + +lemma lt_l: "isrlfm (rsplit lt a)" + by (rule rsplit_l[where f="lt" and a="a"], auto simp add: lt_def, + case_tac s, simp_all, case_tac "nat", simp_all) + +lemma le_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (le n s) = Ifm (x#bs) (Le a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (le n s) = ?I (Le a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (le n s) = ?I (Le a)" using H by (cases "n=0", (simp add: le_def)) + (cases "n > 0", simp_all add: le_def ring_eq_simps myl[rule_format, where b="0"]) +qed + +lemma le_l: "isrlfm (rsplit le a)" + by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def) +(case_tac s, simp_all, case_tac "nat",simp_all) + +lemma gt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (gt n s) = ?I (Gt a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (gt n s) = ?I (Gt a)" using H by (cases "n=0", (simp add: gt_def)) + (cases "n > 0", simp_all add: gt_def ring_eq_simps myless[rule_format, where b="0"]) +qed +lemma gt_l: "isrlfm (rsplit gt a)" + by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma ge_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (ge n s) = ?I (Ge a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def)) + (cases "n > 0", simp_all add: ge_def ring_eq_simps myl[rule_format, where b="0"]) +qed +lemma ge_l: "isrlfm (rsplit ge a)" + by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma eq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (eq n s) = ?I (Eq a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" + show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def ring_eq_simps) +qed +lemma eq_l: "isrlfm (rsplit eq a)" + by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def) +(case_tac s, simp_all, case_tac"nat", simp_all) + +lemma neq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (neq n s) = ?I (NEq a)") +proof(clarify) + fix a n s bs + assume H: "?N a = ?N (CN 0 n s)" + show "?I (neq n s) = ?I (NEq a)" using H by (auto simp add: neq_def ring_eq_simps) +qed + +lemma neq_l: "isrlfm (rsplit neq a)" + by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def) +(case_tac s, simp_all, case_tac"nat", simp_all) + +consts + DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" + DVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" + NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" + NDVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" + +lemma small_le: + assumes u0:"0 \<le> u" and u1: "u < 1" + shows "(-u \<le> real (n::int)) = (0 \<le> n)" +using u0 u1 by auto + +lemma small_lt: + assumes u0:"0 \<le> u" and u1: "u < 1" + shows "(real (n::int) < real (m::int) - u) = (n < m)" +using u0 u1 by auto + +lemma rdvd01_cs: + assumes up: "u \<ge> 0" and u1: "u<1" and np: "real n > 0" + shows "(real (i::int) rdvd real (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real n * u = s - real (floor s) + real j \<and> real i rdvd real (j - floor s))" (is "?lhs = ?rhs") +proof- + let ?ss = "s - real (floor s)" + from real_of_int_floor_add_one_gt[where r="s", simplified myless[rule_format,where a="s"]] + real_of_int_floor_le[where r="s"] have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" + by (auto simp add: myl[rule_format, where b="s", symmetric] myless[rule_format, where a="?ss"]) + from np have n0: "real n \<ge> 0" by simp + from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] + have nu0:"real n * u - s \<ge> -s" and nun:"real n * u -s < real n - s" by auto + from int_rdvd_real[where i="i" and x="real (n::int) * u - s"] + have "real i rdvd real n * u - s = + (i dvd floor (real n * u -s) \<and> (real (floor (real n * u - s)) = real n * u - s ))" + (is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp + also have "\<dots> = (?DE \<and> real(floor (real n * u - s) + floor s)\<ge> -?ss + \<and> real(floor (real n * u - s) + floor s)< real n - ?ss)" (is "_=(?DE \<and>real ?a \<ge> _ \<and> real ?a < _)") + using nu0 nun by auto + also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1]) + also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. ?a = j))" by simp + also have "\<dots> = (?DE \<and> (\<exists> j\<in> {0 .. (n - 1)}. real (\<lfloor>real n * u - s\<rfloor>) = real j - real \<lfloor>s\<rfloor> ))" + by (simp only: ring_eq_simps real_of_int_diff[symmetric] real_of_int_inject del: real_of_int_diff) + also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real n * u - s = real j - real \<lfloor>s\<rfloor> \<and> real i rdvd real n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real n * u - s\<rfloor>"] + by (auto cong: conj_cong) + also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: ring_eq_simps ) + finally show ?thesis . +qed + +defs DVDJ_def: "DVDJ i n s \<equiv> (foldr disj (map (\<lambda> j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) F)" +defs NDVDJ_def: "NDVDJ i n s \<equiv> (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) (iupt(0,n - 1))) T)" + +lemma DVDJ_DVD: + assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0" + shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))" +proof- + let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))" + let ?s= "Inum (x#bs) s" + from foldr_disj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"] + have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" + by (simp add: iupt_set np DVDJ_def del: iupt.simps) + also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s)))" by (simp add: ring_eq_simps diff_def[symmetric]) + also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] + have "\<dots> = (real i rdvd real n * x - (-?s))" by simp + finally show ?thesis by simp +qed + +lemma NDVDJ_NDVD: + assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real n > 0" + shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))" +proof- + let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))" + let ?s= "Inum (x#bs) s" + from foldr_conj_map[where xs="iupt(0,n - 1)" and bs="x#bs" and f="?f"] + have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" + by (simp add: iupt_set np NDVDJ_def del: iupt.simps) + also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real n * x = (- ?s) - real (floor (- ?s)) + real j \<and> real i rdvd real (j - floor (- ?s))))" by (simp add: ring_eq_simps diff_def[symmetric]) + also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] + have "\<dots> = (\<not> (real i rdvd real n * x - (-?s)))" by simp + finally show ?thesis by simp +qed + +lemma foldr_disj_map_rlfm2: + assumes lf: "\<forall> n . isrlfm (f n)" + shows "isrlfm (foldr disj (map f xs) F)" +using lf by (induct xs, auto) +lemma foldr_And_map_rlfm2: + assumes lf: "\<forall> n . isrlfm (f n)" + shows "isrlfm (foldr conj (map f xs) T)" +using lf by (induct xs, auto) + +lemma DVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" + shows "isrlfm (DVDJ i n s)" +proof- + let ?f="\<lambda>j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) + (Dvd i (Sub (C j) (Floor (Neg s))))" + have th: "\<forall> j. isrlfm (?f j)" using nb np by auto + from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp +qed + +lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s" + shows "isrlfm (NDVDJ i n s)" +proof- + let ?f="\<lambda>j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) + (NDvd i (Sub (C j) (Floor (Neg s))))" + have th: "\<forall> j. isrlfm (?f j)" using nb np by auto + from NDVDJ_def foldr_And_map_rlfm2[OF th] show ?thesis by auto +qed + +defs DVD_def: "DVD i c t \<equiv> + (if i=0 then eq c t else + if c = 0 then (Dvd i t) else if c >0 then DVDJ (abs i) c t else DVDJ (abs i) (-c) (Neg t))" +defs NDVD_def: "NDVD i c t \<equiv> + (if i=0 then neq c t else + if c = 0 then (NDvd i t) else if c >0 then NDVDJ (abs i) c t else NDVDJ (abs i) (-c) (Neg t))" + +lemma DVD_mono: + assumes xp: "0\<le> x" and x1: "x < 1" + shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)" + (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (DVD i n s) = ?I (Dvd i a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" + let ?th = "?I (DVD i n s) = ?I (Dvd i a)" + have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith + moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]] + by (simp add: DVD_def rdvd_left_0_eq)} + moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) } + moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th + by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 + rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } + moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)} + ultimately show ?th by blast +qed + +lemma NDVD_mono: assumes xp: "0\<le> x" and x1: "x < 1" + shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)" + (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (NDVD i n s) = ?I (NDvd i a)") +proof(clarify) + fix a n s + assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s" + let ?th = "?I (NDVD i n s) = ?I (NDvd i a)" + have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith + moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]] + by (simp add: NDVD_def rdvd_left_0_eq)} + moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) } + moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th + by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 + rdvd_minus[where d="i" and t="real n * x + Inum (x # bs) s"]) } + moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th + by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)} + ultimately show ?th by blast +qed + +lemma DVD_l: "isrlfm (rsplit (DVD i) a)" + by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l) +(case_tac s, simp_all, case_tac "nat", simp_all) + +lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)" + by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l) +(case_tac s, simp_all, case_tac "nat", simp_all) + +consts rlfm :: "fm \<Rightarrow> fm" +recdef rlfm "measure fmsize" + "rlfm (And p q) = conj (rlfm p) (rlfm q)" + "rlfm (Or p q) = disj (rlfm p) (rlfm q)" + "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" + "rlfm (Iff p q) = disj (conj(rlfm p) (rlfm q)) (conj(rlfm (NOT p)) (rlfm (NOT q)))" + "rlfm (Lt a) = rsplit lt a" + "rlfm (Le a) = rsplit le a" + "rlfm (Gt a) = rsplit gt a" + "rlfm (Ge a) = rsplit ge a" + "rlfm (Eq a) = rsplit eq a" + "rlfm (NEq a) = rsplit neq a" + "rlfm (Dvd i a) = rsplit (\<lambda> t. DVD i t) a" + "rlfm (NDvd i a) = rsplit (\<lambda> t. NDVD i t) a" + "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" + "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" + "rlfm (NOT (NOT p)) = rlfm p" + "rlfm (NOT T) = F" + "rlfm (NOT F) = T" + "rlfm (NOT (Lt a)) = simpfm (rlfm (Ge a))" + "rlfm (NOT (Le a)) = simpfm (rlfm (Gt a))" + "rlfm (NOT (Gt a)) = simpfm (rlfm (Le a))" + "rlfm (NOT (Ge a)) = simpfm (rlfm (Lt a))" + "rlfm (NOT (Eq a)) = simpfm (rlfm (NEq a))" + "rlfm (NOT (NEq a)) = simpfm (rlfm (Eq a))" + "rlfm (NOT (Dvd i a)) = simpfm (rlfm (NDvd i a))" + "rlfm (NOT (NDvd i a)) = simpfm (rlfm (Dvd i a))" + "rlfm p = p" (hints simp add: fmsize_pos) + +lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p" + by (induct p rule: isrlfm.induct, auto) +lemma igcd_le1: assumes ip: "0 < i" shows "igcd i j \<le> i" +proof- + from igcd_dvd1 have th: "igcd i j dvd i" by blast + from zdvd_imp_le[OF th ip] show ?thesis . +qed + + +lemma simpfm_rl: "isrlfm p \<Longrightarrow> isrlfm (simpfm p)" +proof (induct p) + case (Lt a) + hence "bound0 (Lt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Le a) + hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Gt a) + hence "bound0 (Gt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Ge a) + hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Eq a) + hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (NEq a) + hence "bound0 (NEq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)" + by (cases a,simp_all, case_tac "nat", simp_all) + moreover + {assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))" + using simpfm_bound0 by blast + have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def) + with bn bound0at_l have ?case by blast} + moreover + {fix c e assume "a = CN 0 c e" and "c>0" and "numbound0 e" + { + assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0" + with numgcd_pos[where t="CN 0 c (simpnum e)"] + have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp + from prems have th:"numgcd (CN 0 c (simpnum e)) \<le> c" + by (simp add: numgcd_def igcd_le1) + from prems have th': "c\<noteq>0" by auto + from prems have cp: "c \<ge> 0" by simp + from zdiv_mono2[OF cp th1 th, simplified zdiv_self[OF th']] + have "0 < c div numgcd (CN 0 c (simpnum e))" by simp + } + with prems have ?case + by (simp add: Let_def reducecoeff_def simpnum_numbound0 reducecoeffh_numbound0)} + ultimately show ?case by blast +next + case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))" + using simpfm_bound0 by blast + have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) + with bn bound0at_l show ?case by blast +next + case (NDvd i a) hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))" + using simpfm_bound0 by blast + have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def) + with bn bound0at_l show ?case by blast +qed(auto simp add: conj_def imp_def disj_def iff_def Let_def simpfm_bound0 numadd_nb numneg_nb) + + + +lemma rlfm_I: + assumes qfp: "qfree p" + and xp: "0 \<le> x" and x1: "x < 1" + shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \<and> isrlfm (rlfm p)" + using qfp +by (induct p rule: rlfm.induct) +(auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l + rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l + rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl) +lemma rlfm_l: + assumes qfp: "qfree p" + shows "isrlfm (rlfm p)" + using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l +by (induct p rule: rlfm.induct,auto simp add: simpfm_rl) + + (* Operations needed for Ferrante and Rackoff *) +lemma rminusinf_inf: + assumes lp: "isrlfm p" + shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") +using lp +proof (induct p rule: minusinf.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rplusinf_inf: + assumes lp: "isrlfm p" + shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") +using lp +proof (induct p rule: isrlfm.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rminusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (minusinf p)" + using lp + by (induct p rule: minusinf.induct) simp_all + +lemma rplusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (plusinf p)" + using lp + by (induct p rule: plusinf.induct) simp_all + +lemma rminusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (minusinf p)" + shows "\<exists> x. Ifm (x#bs) p" +proof- + from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto + from rminusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp + moreover have "z - 1 < z" by simp + ultimately show ?thesis using z_def by auto +qed + +lemma rplusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (plusinf p)" + shows "\<exists> x. Ifm (x#bs) p" +proof- + from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto + from rplusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp + moreover have "z + 1 > z" by simp + ultimately show ?thesis using z_def by auto +qed + +consts + \<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list" + \<upsilon> :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm " +recdef \<Upsilon> "measure size" + "\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)" + "\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)" + "\<Upsilon> (Eq (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> (NEq (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> (Lt (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> (Le (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> (Gt (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> (Ge (CN 0 c e)) = [(Neg e,c)]" + "\<Upsilon> p = []" + +recdef \<upsilon> "measure size" + "\<upsilon> (And p q) = (\<lambda> (t,n). And (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))" + "\<upsilon> (Or p q) = (\<lambda> (t,n). Or (\<upsilon> p (t,n)) (\<upsilon> q (t,n)))" + "\<upsilon> (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))" + "\<upsilon> (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))" + "\<upsilon> (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))" + "\<upsilon> (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))" + "\<upsilon> (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))" + "\<upsilon> (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))" + "\<upsilon> p = (\<lambda> (t,n). p)" + +lemma \<upsilon>_I: assumes lp: "isrlfm p" + and np: "real n > 0" and nbt: "numbound0 t" + shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (\<upsilon> p (t,n))" (is "(?I x (\<upsilon> p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _") + using lp +proof(induct p rule: \<upsilon>.induct) + case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)" + by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)" + by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)" + by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)" + by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \<noteq> 0" by simp + have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \<noteq> 0" by simp + have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2) + +lemma \<Upsilon>_l: + assumes lp: "isrlfm p" + shows "\<forall> (t,k) \<in> set (\<Upsilon> p). numbound0 t \<and> k >0" +using lp +by(induct p rule: \<Upsilon>.induct) auto + +lemma rminusinf_\<Upsilon>: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m") +proof- + have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<ge> ?N a s" by blast + from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto + from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma rplusinf_\<Upsilon>: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\<exists> (s,m) \<in> set (\<Upsilon> p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m") +proof- + have "\<exists> (s,m) \<in> set (\<Upsilon> p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real m * x \<le> ?N a s" by blast + from \<Upsilon>_l[OF lp] smU have mp: "real m > 0" by auto + from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma lin_dense: + assumes lp: "isrlfm p" + and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (\<Upsilon> p)" + (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)") + and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" + and ly: "l < y" and yu: "y < u" + shows "Ifm (y#bs) p" +using lp px noS +proof (induct p rule: isrlfm.induct) + case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e < 0" by (simp add: ring_eq_simps) + hence pxc: "x < (- ?N x e) / real c" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp + + from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_eq_simps) + hence pxc: "x \<le> (- ?N x e) / real c" + by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e > 0" by (simp add: ring_eq_simps) + hence pxc: "x > (- ?N x e) / real c" + by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_eq_simps) + hence pxc: "x \<ge> (- ?N x e) / real c" + by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \<noteq> 0" by simp + from prems have "x * real c + ?N x e = 0" by (simp add: ring_eq_simps) + hence pxc: "x = (- ?N x e) / real c" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto + with pxc show ?case by simp +next + case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \<noteq> 0" by simp + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y* real c \<noteq> -?N x e" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp + hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_eq_simps) + thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] + by (simp add: ring_eq_simps) +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"]) + +lemma finite_set_intervals: + assumes px: "P (x::real)" + and lx: "l \<le> x" and xu: "x \<le> u" + and linS: "l\<in> S" and uinS: "u \<in> S" + and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" + shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x" +proof- + let ?Mx = "{y. y\<in> S \<and> y \<le> x}" + let ?xM = "{y. y\<in> S \<and> x \<le> y}" + let ?a = "Max ?Mx" + let ?b = "Min ?xM" + have MxS: "?Mx \<subseteq> S" by blast + hence fMx: "finite ?Mx" using fS finite_subset by auto + from lx linS have linMx: "l \<in> ?Mx" by blast + hence Mxne: "?Mx \<noteq> {}" by blast + have xMS: "?xM \<subseteq> S" by blast + hence fxM: "finite ?xM" using fS finite_subset by auto + from xu uinS have linxM: "u \<in> ?xM" by blast + hence xMne: "?xM \<noteq> {}" by blast + have ax:"?a \<le> x" using Mxne fMx by auto + have xb:"x \<le> ?b" using xMne fxM by auto + have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast + have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast + have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S" + proof(clarsimp) + fix y + assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" + from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto + moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp} + moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp} + ultimately show "False" by blast + qed + from ainS binS noy ax xb px show ?thesis by blast +qed + +lemma finite_set_intervals2: + assumes px: "P (x::real)" + and lx: "l \<le> x" and xu: "x \<le> u" + and linS: "l\<in> S" and uinS: "u \<in> S" + and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" + shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)" +proof- + from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] + obtain a and b where + as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto + from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto + thus ?thesis using px as bs noS by blast +qed + +lemma rinf_\<Upsilon>: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))") + and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))") + and ex: "\<exists> x. Ifm (x#bs) p" (is "\<exists> x. ?I x p") + shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" +proof- + let ?N = "\<lambda> x t. Inum (x#bs) t" + let ?U = "set (\<Upsilon> p)" + from ex obtain a where pa: "?I a p" by blast + from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi + have nmi': "\<not> (?I a (?M p))" by simp + from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi + have npi': "\<not> (?I a (?P p))" by simp + have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). ?I ((?N a l/real n + ?N a s /real m) / 2) p" + proof- + let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U" + have fM: "finite ?M" by auto + from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa] + have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast + then obtain "t" "n" "s" "m" where + tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" + and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast + from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto + from tnU have Mne: "?M \<noteq> {}" by auto + hence Une: "?U \<noteq> {}" by simp + let ?l = "Min ?M" + let ?u = "Max ?M" + have linM: "?l \<in> ?M" using fM Mne by simp + have uinM: "?u \<in> ?M" using fM Mne by simp + have tnM: "?N a t / real n \<in> ?M" using tnU by auto + have smM: "?N a s / real m \<in> ?M" using smU by auto + have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto + have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto + have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp + have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp + from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu] + have "(\<exists> s\<in> ?M. ?I s p) \<or> + (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" . + moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p" + hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto + then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast + have "(u + u) / 2 = u" by auto with pu tuu + have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp + with tuU have ?thesis by blast} + moreover{ + assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p" + then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" + and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" + by blast + from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto + then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast + from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto + then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast + from t1x xt2 have t1t2: "t1 < t2" by simp + let ?u = "(t1 + t2) / 2" + from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto + from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . + with t1uU t2uU t1u t2u have ?thesis by blast} + ultimately show ?thesis by blast + qed + then obtain "l" "n" "s" "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" + and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast + from lnU smU \<Upsilon>_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto + from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] + numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu + have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp + with lnU smU + show ?thesis by auto +qed + (* The Ferrante - Rackoff Theorem *) + +theorem fr_eq: + assumes lp: "isrlfm p" + shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/ real n + (Inum (x#bs) s) / real m) /2)#bs) p))" + (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") +proof + assume px: "\<exists> x. ?I x p" + have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast + moreover {assume "?M \<or> ?P" hence "?D" by blast} + moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" + from rinf_\<Upsilon>[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {assume f:"?F" hence "?E" by blast} + ultimately show "?E" by blast +qed + + +lemma fr_eq\<upsilon>: + assumes lp: "isrlfm p" + shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (\<Upsilon> p). \<exists> (s,l) \<in> set (\<Upsilon> p). Ifm (x#bs) (\<upsilon> p (Add(Mul l t) (Mul k s) , 2*k*l))))" + (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") +proof + assume px: "\<exists> x. ?I x p" + have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast + moreover {assume "?M \<or> ?P" hence "?D" by blast} + moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" + let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n" + let ?N = "\<lambda> t. Inum (x#bs) t" + {fix t n s m assume "(t,n)\<in> set (\<Upsilon> p)" and "(s,m) \<in> set (\<Upsilon> p)" + with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" + by auto + let ?st = "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnp mp np by (simp add: ring_eq_simps add_divide_distrib) + from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"] + have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])} + with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)" + and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))" + with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto + let ?st = "Add (Mul l t) (Mul k s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + from \<upsilon>_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} + ultimately show "?E" by blast +qed + + (********************************************************************) + (*** THE OVERALL-PART ***) + (********************************************************************) + +lemma real_ex_int_real01: + shows "(\<exists> (x::real). P x) = (\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))" +proof(auto) + fix x + assume Px: "P x" + let ?i = "floor x" + let ?u = "x - real ?i" + have "x = real ?i + ?u" by simp + hence "P (real ?i + ?u)" using Px by simp + moreover have "real ?i \<le> x" using real_of_int_floor_le by simp hence "0 \<le> ?u" by arith + moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith + ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real i + u))" by blast +qed + +consts exsplitnum :: "num \<Rightarrow> num" + exsplit :: "fm \<Rightarrow> fm" +recdef exsplitnum "measure size" + "exsplitnum (C c) = (C c)" + "exsplitnum (Bound 0) = Add (Bound 0) (Bound 1)" + "exsplitnum (Bound n) = Bound (n+1)" + "exsplitnum (Neg a) = Neg (exsplitnum a)" + "exsplitnum (Add a b) = Add (exsplitnum a) (exsplitnum b) " + "exsplitnum (Sub a b) = Sub (exsplitnum a) (exsplitnum b) " + "exsplitnum (Mul c a) = Mul c (exsplitnum a)" + "exsplitnum (Floor a) = Floor (exsplitnum a)" + "exsplitnum (CN 0 c a) = CN 0 c (Add (Mul c (Bound 1)) (exsplitnum a))" + "exsplitnum (CN n c a) = CN (n+1) c (exsplitnum a)" + "exsplitnum (CF c s t) = CF c (exsplitnum s) (exsplitnum t)" + +recdef exsplit "measure size" + "exsplit (Lt a) = Lt (exsplitnum a)" + "exsplit (Le a) = Le (exsplitnum a)" + "exsplit (Gt a) = Gt (exsplitnum a)" + "exsplit (Ge a) = Ge (exsplitnum a)" + "exsplit (Eq a) = Eq (exsplitnum a)" + "exsplit (NEq a) = NEq (exsplitnum a)" + "exsplit (Dvd i a) = Dvd i (exsplitnum a)" + "exsplit (NDvd i a) = NDvd i (exsplitnum a)" + "exsplit (And p q) = And (exsplit p) (exsplit q)" + "exsplit (Or p q) = Or (exsplit p) (exsplit q)" + "exsplit (Imp p q) = Imp (exsplit p) (exsplit q)" + "exsplit (Iff p q) = Iff (exsplit p) (exsplit q)" + "exsplit (NOT p) = NOT (exsplit p)" + "exsplit p = p" + +lemma exsplitnum: + "Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t" + by(induct t rule: exsplitnum.induct) (simp_all add: ring_eq_simps) + +lemma exsplit: + assumes qfp: "qfree p" + shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p" +using qfp exsplitnum[where x="x" and y="y" and bs="bs"] +by(induct p rule: exsplit.induct) simp_all + +lemma splitex: + assumes qf: "qfree p" + shows "(Ifm bs (E p)) = (\<exists> (i::int). Ifm (real i#bs) (E (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (exsplit p))))" (is "?lhs = ?rhs") +proof- + have "?rhs = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm (x#(real i)#bs) (exsplit p))" + by (simp add: myless[rule_format, where b="1"] myless[rule_format, where b="0"] add_ac diff_def) + also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real i + x) #bs) p)" + by (simp only: exsplit[OF qf] add_ac) + also have "\<dots> = (\<exists> x. Ifm (x#bs) p)" + by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"]) + finally show ?thesis by simp +qed + + (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) + (* NOTE THAT THIS ONLY HOLDS IN THE CONTEXT OF THE MIXED THEORY!!! MAY BE SHOULD ALSO IMPLEMENT FERRANTE AND RACKOFF TO MAKE IT AVAILABLE AS SAND ALONE!!!! *) + (* SINCE x is constrained to be between 0 and 1, plusinf and minusinf will always evaluate to False !!!!! *) + +constdefs ferrack01:: "fm \<Rightarrow> fm" + "ferrack01 p \<equiv> (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p); + U = remdups(map simp_num_pair + (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) + (alluopairs (\<Upsilon> p')))) + in decr (evaldjf (\<upsilon> p') U ))" + +lemma fr_eq_01: + assumes qf: "qfree p" + shows "(\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\<exists> (t,n) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \<exists> (s,m) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\<upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))" + (is "(\<exists> x. ?I x ?q) = ?F") +proof- + let ?rq = "rlfm ?q" + let ?M = "?I x (minusinf ?rq)" + let ?P = "?I x (plusinf ?rq)" + have MF: "?M = False" + apply (simp add: Let_def reducecoeff_def numgcd_def igcd_def rsplit_def ge_def lt_def conj_def disj_def) + by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all) + have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def igcd_def rsplit_def ge_def lt_def conj_def disj_def) + by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C -1)))", simp_all) + have "(\<exists> x. ?I x ?q ) = + ((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))" + (is "(\<exists> x. ?I x ?q) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") + proof + assume "\<exists> x. ?I x ?q" + then obtain x where qx: "?I x ?q" by blast + hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p" + by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf]) + from qx have "?I x ?rq " + by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) + hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto + from qf have qfq:"isrlfm ?rq" + by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1]) + with lqx fr_eq\<upsilon>[OF qfq] show "?M \<or> ?P \<or> ?F" by blast + next + assume D: "?D" + let ?U = "set (\<Upsilon> ?rq )" + from MF PF D have "?F" by auto + then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast + from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] + by (auto simp add: rsplit_def lt_def ge_def) + from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by (auto simp add: split_def) + let ?st = "Add (Mul m t) (Mul n s)" + from tnb snb have stnb: "numbound0 ?st" by simp + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx + have "\<exists> x. ?I x ?rq" by auto + thus "?E" + using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def) + qed + with MF PF show ?thesis by blast +qed + +lemma \<Upsilon>_cong_aux: + assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0" + shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))" + (is "?lhs = ?rhs") +proof(auto) + fix t n s m + assume "((t,n),(s,m)) \<in> set (alluopairs U)" + hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)" + using alluopairs_set1[where xs="U"] by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul th have mnz: "m \<noteq> 0" by auto + from Ul th have nnz: "n \<noteq> 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: ring_eq_simps add_divide_distrib) + + thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) / + (2 * real n * real m) + \<in> (\<lambda>((t, n), s, m). + (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` + (set U \<times> set U)"using mnz nnz th + apply (auto simp add: th add_divide_distrib ring_eq_simps split_def image_def) + by (rule_tac x="(s,m)" in bexI,simp_all) + (rule_tac x="(t,n)" in bexI,simp_all) +next + fix t n s m + assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul smU have mnz: "m \<noteq> 0" by auto + from Ul tnU have nnz: "n \<noteq> 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: ring_eq_simps add_divide_distrib) + let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" + have Pc:"\<forall> a b. ?P a b = ?P b a" + by auto + from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast + from alluopairs_ex[OF Pc, where xs="U"] tnU smU + have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')" + by blast + then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" + and Pts': "?P (t',n') (s',m')" by blast + from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto + let ?st' = "Add (Mul m' t') (Mul n' s')" + have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')" + using mnz' nnz' by (simp add: ring_eq_simps add_divide_distrib) + from Pts' have + "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp + also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') + finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 + \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) ` + (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` + set (alluopairs U)" + using ts'_U by blast +qed + +lemma \<Upsilon>_cong: + assumes lp: "isrlfm p" + and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)") + and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0" + and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0" + shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (\<upsilon> p (t,n)))" + (is "?lhs = ?rhs") +proof + assume ?lhs + then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and + Pst: "Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))" by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: ring_eq_simps add_divide_distrib) + from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast + hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + from conjunct1[OF \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] + have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st) + then show ?rhs using tnU' by auto +next + assume ?rhs + then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (\<upsilon> p (t', n'))" + by blast + from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast + hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and + th: "?f (t',n') = ?g((t,n),(s,m)) "by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: ring_eq_simps add_divide_distrib) + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + with \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast +qed + +lemma ferrack01: + assumes qf: "qfree p" + shows "((\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \<and> qfree (ferrack01 p)" (is "(?lhs = ?rhs) \<and> _") +proof- + let ?I = "\<lambda> x p. Ifm (x#bs) p" + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?q = "rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)" + let ?U = "\<Upsilon> ?q" + let ?Up = "alluopairs ?U" + let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" + let ?S = "map ?g ?Up" + let ?SS = "map simp_num_pair ?S" + let ?Y = "remdups ?SS" + let ?f= "(\<lambda> (t,n). ?N t / real n)" + let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2" + let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))" + let ?ep = "evaldjf (\<upsilon> ?q) ?Y" + from rlfm_l[OF qf] have lq: "isrlfm ?q" + by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def igcd_def) + from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp + from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" . + from U_l UpU + have Up_: "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto + hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 " + by (auto simp add: mult_pos_pos) + have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" + proof- + { fix t n assume tnY: "(t,n) \<in> set ?Y" + hence "(t,n) \<in> set ?SS" by simp + hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)" + by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) + then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast + from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto + from simp_num_pair_l[OF tnb np tns] + have "numbound0 t \<and> n > 0" . } + thus ?thesis by blast + qed + + have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))" + proof- + from simp_num_pair_ci[where bs="x#bs"] have + "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto + hence th: "?f o simp_num_pair = ?f" using ext by blast + have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose) + also have "\<dots> = (?f ` set ?S)" by (simp add: th) + also have "\<dots> = ((?f o ?g) ` set ?Up)" + by (simp only: set_map o_def image_compose[symmetric]) + also have "\<dots> = (?h ` (set ?U \<times> set ?U))" + using \<Upsilon>_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast + finally show ?thesis . + qed + have "\<forall> (t,n) \<in> set ?Y. bound0 (\<upsilon> ?q (t,n))" + proof- + { fix t n assume tnY: "(t,n) \<in> set ?Y" + with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto + from \<upsilon>_I[OF lq np tnb] + have "bound0 (\<upsilon> ?q (t,n))" by simp} + thus ?thesis by blast + qed + hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="\<upsilon> ?q"] + by auto + + from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q" + by (simp only: split_def fst_conv snd_conv) + also have "\<dots> = (\<exists> (t,n) \<in> set ?Y. ?I x (\<upsilon> ?q (t,n)))" using \<Upsilon>_cong[OF lq YU U_l Y_l] + by (simp only: split_def fst_conv snd_conv) + also have "\<dots> = (Ifm (x#bs) ?ep)" + using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\<upsilon> ?q",symmetric] + by (simp only: split_def pair_collapse) + also have "\<dots> = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast + finally have lr: "?lhs = ?rhs" by (simp only: ferrack01_def Let_def) + from decr_qf[OF ep_nb] have "qfree (ferrack01 p)" by (simp only: Let_def ferrack01_def) + with lr show ?thesis by blast +qed + +lemma cp_thm': + assumes lp: "iszlfm p (real (i::int)#bs)" + and up: "d\<beta> p 1" and dd: "d\<delta> p d" and dp: "d > 0" + shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real i#bs)) ` set (\<beta> p). Ifm ((b+real j)#bs) p))" + using cp_thm[OF lp up dd dp] by auto + +constdefs unit:: "fm \<Rightarrow> fm \<times> num list \<times> int" + "unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\<beta> p' l); d = \<delta> q; + B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q)) + in if length B \<le> length a then (q,B,d) else (mirror q, a,d))" + +lemma unit: assumes qf: "qfree p" + shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> (Inum (real i#bs)) ` set B = (Inum (real i#bs)) ` set (\<beta> q) \<and> d\<beta> q 1 \<and> d\<delta> q d \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)" +proof- + fix q B d + assume qBd: "unit p = (q,B,d)" + let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> + Inum (real i#bs) ` set B = Inum (real i#bs) ` set (\<beta> q) \<and> + d\<beta> q 1 \<and> d\<delta> q d \<and> 0 < d \<and> iszlfm q (real i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)" + let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p" + let ?p' = "zlfm p" + let ?l = "\<zeta> ?p'" + let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\<beta> ?p' ?l)" + let ?d = "\<delta> ?q" + let ?B = "set (\<beta> ?q)" + let ?B'= "remdups (map simpnum (\<beta> ?q))" + let ?A = "set (\<alpha> ?q)" + let ?A'= "remdups (map simpnum (\<alpha> ?q))" + from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] + have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto + from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]] + have lp': "\<forall> (i::int). iszlfm ?p' (real i#bs)" by simp + hence lp'': "iszlfm ?p' (real (i::int)#bs)" by simp + from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d\<beta> ?p' ?l" by auto + from a\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp' + have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff) + from lp'' lp a\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real i#bs)" and uq: "d\<beta> ?q 1" + by (auto simp add: isint_def) + from \<delta>[OF lq] have dp:"?d >0" and dd: "d\<delta> ?q ?d" by blast+ + let ?N = "\<lambda> t. Inum (real (i::int)#bs) t" + have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_compose) + also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real i #bs"] by auto + finally have BB': "?N ` set ?B' = ?N ` ?B" . + have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_compose) + also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] by auto + finally have AA': "?N ` set ?A' = ?N ` ?A" . + from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b" + by (simp add: simpnum_numbound0) + from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b" + by (simp add: simpnum_numbound0) + {assume "length ?B' \<le> length ?A'" + hence q:"q=?q" and "B = ?B'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)" + and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ + with pq_ex dp uq dd lq q d have ?thes by simp} + moreover + {assume "\<not> (length ?B' \<le> length ?A')" + hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" + using qBd by (auto simp add: Let_def unit_def) + with AA' mirror\<alpha>\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" + and bn: "\<forall>b\<in> set B. numbound0 b" by simp+ + from mirror_ex[OF lq] pq_ex q + have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp + from lq uq q mirror_d\<beta> [where p="?q" and bs="bs" and a="real i"] + have lq': "iszlfm q (real i#bs)" and uq: "d\<beta> q 1" by auto + from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d\<delta> q d " by auto + from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp + } + ultimately show ?thes by blast +qed + (* Cooper's Algorithm *) + +constdefs cooper :: "fm \<Rightarrow> fm" + "cooper p \<equiv> + (let (q,B,d) = unit p; js = iupt (1,d); + mq = simpfm (minusinf q); + md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js + in if md = T then T else + (let qd = evaldjf (\<lambda> t. simpfm (subst0 t q)) + (remdups (map (\<lambda> (b,j). simpnum (Add b (C j))) + (allpairs Pair B js))) + in decr (disj md qd)))" +lemma cooper: assumes qf: "qfree p" + shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" + (is "(?lhs = ?rhs) \<and> _") +proof- + + let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p" + let ?q = "fst (unit p)" + let ?B = "fst (snd(unit p))" + let ?d = "snd (snd (unit p))" + let ?js = "iupt (1,?d)" + let ?mq = "minusinf ?q" + let ?smq = "simpfm ?mq" + let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" + let ?N = "\<lambda> t. Inum (real (i::int)#bs) t" + let ?bjs = "allpairs Pair ?B ?js" + let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) (allpairs Pair ?B ?js)" + let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)" + have qbf:"unit p = (?q,?B,?d)" by simp + from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and + B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and + uq:"d\<beta> ?q 1" and dd: "d\<delta> ?q ?d" and dp: "?d > 0" and + lq: "iszlfm ?q (real i#bs)" and + Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto + from zlin_qfree[OF lq] have qfq: "qfree ?q" . + from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". + have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp + hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" + by (auto simp only: subst0_bound0[OF qfmq]) + hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))" + by (auto simp add: simpfm_bound0) + from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp + from Bn jsnb have "\<forall> (b,j) \<in> set (allpairs Pair ?B ?js). numbound0 (Add b (C j))" + by (simp add: allpairs_set) + hence "\<forall> (b,j) \<in> set (allpairs Pair ?B ?js). numbound0 (simpnum (Add b (C j)))" + using simpnum_numbound0 by blast + hence "\<forall> t \<in> set ?sbjs. numbound0 t" by simp + hence "\<forall> t \<in> set (remdups ?sbjs). bound0 (subst0 t ?q)" + using subst0_bound0[OF qfq] by auto + hence th': "\<forall> t \<in> set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))" + using simpfm_bound0 by blast + from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp + from mdb qdb + have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all) + from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B + have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real j)#bs) ?q))" by auto + also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real j)#bs) ?q))" apply (simp only: iupt_set simpfm) by auto + also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set (allpairs Pair ?B ?js). Ifm (t #bs) ?q))" by (simp only: allpairs_set) simp + also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set (allpairs Pair ?B ?js). Ifm (t #bs) ?q))" by (simp only: simpnum_ci) + also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))" + by (auto simp add: split_def) + also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> t \<in> set (remdups ?sbjs). (\<lambda> t. ?I i (simpfm (subst0 t ?q))) t))" by (simp only: simpfm subst0_I[OF qfq] simpfm Inum.simps subst0_I[OF qfmq] set_remdups) + also have "\<dots> = ((?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js)) \<or> (?I i (evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex) + finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by (simp add: disj) + hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp + {assume mdT: "?md = T" + hence cT:"cooper p = T" + by (simp only: cooper_def unit_def split_def Let_def if_True) simp + from mdT mdqd have lhs:"?lhs" by (auto simp add: disj) + from mdT have "?rhs" by (simp add: cooper_def unit_def split_def) + with lhs cT have ?thesis by simp } + moreover + {assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)" + by (simp only: cooper_def unit_def split_def Let_def if_False) + with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } + ultimately show ?thesis by blast +qed + +lemma DJcooper: + assumes qf: "qfree p" + shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)" +proof- + from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by blast + from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast + have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs) q)" + using cooper disjuncts_qf[OF qf] by blast + also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto) + finally show ?thesis using thqf by blast +qed + + (* Redy and Loveland *) + +lemma \<sigma>\<rho>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" + shows "Ifm (a#bs) (\<sigma>\<rho> p (t,c)) = Ifm (a#bs) (\<sigma>\<rho> p (t',c))" + using lp + by (induct p rule: iszlfm.induct, auto simp add: tt') + +lemma \<sigma>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'" + shows "Ifm (a#bs) (\<sigma> p c t) = Ifm (a#bs) (\<sigma> p c t')" + by (simp add: \<sigma>_def tt' \<sigma>\<rho>_cong[OF lp tt']) + +lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)" + and RR: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" + shows "(\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))) = (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j))))" + (is "?lhs = ?rhs") +proof + let ?d = "\<delta> p" + assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}" + and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast + from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" by auto + hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" using RR by simp + hence "\<exists> (e',c') \<in> set (\<rho> p). Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto + then obtain e' c' where ecRo:"(e',c') \<in> set (\<rho> p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'" + and cc':"c = c'" by blast + from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp + + from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp + from ecRo jD px' cc' show ?rhs apply auto + by (rule_tac x="(e', c')" in bexI,simp_all) + (rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) +next + let ?d = "\<delta> p" + assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}" + and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast + from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" by auto + hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp + hence "\<exists> (e',c') \<in> R. Inum (a#bs) e = Inum (a#bs) e' \<and> c = c'" by auto + then obtain e' c' where ecRo:"(e',c') \<in> R" and ee':"Inum (a#bs) e = Inum (a#bs) e'" + and cc':"c = c'" by blast + from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp + from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp + from ecRo jD px' cc' show ?lhs apply auto + by (rule_tac x="(e', c')" in bexI,simp_all) + (rule_tac x="j" in bexI, simp_all add: cc'[symmetric]) +qed + +lemma rl_thm': + assumes lp: "iszlfm p (real (i::int)#bs)" + and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R = (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" + shows "(\<exists> (x::int). Ifm (real x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))" + using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp + +constdefs chooset:: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int" + "chooset p \<equiv> (let q = zlfm p ; d = \<delta> q; + B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ; + a = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<alpha>\<rho> q)) + in if length B \<le> length a then (q,B,d) else (mirror q, a,d))" + +lemma chooset: assumes qf: "qfree p" + shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> ((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" +proof- + fix q B d + assume qBd: "chooset p = (q,B,d)" + let ?thes = "((\<exists> (x::int). Ifm (real x#bs) p) = (\<exists> (x::int). Ifm (real x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" + let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p" + let ?q = "zlfm p" + let ?d = "\<delta> ?q" + let ?B = "set (\<rho> ?q)" + let ?f = "\<lambda> (t,k). (simpnum t,k)" + let ?B'= "remdups (map ?f (\<rho> ?q))" + let ?A = "set (\<alpha>\<rho> ?q)" + let ?A'= "remdups (map ?f (\<alpha>\<rho> ?q))" + from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] + have pp': "\<forall> i. ?I i ?q = ?I i p" by auto + hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp + from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real i"] + have lq: "iszlfm ?q (real (i::int)#bs)" . + from \<delta>[OF lq] have dp:"?d >0" by blast + let ?N = "\<lambda> (t,c). (Inum (real (i::int)#bs) t,c)" + have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_compose) + also have "\<dots> = ?N ` ?B" + by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def) + finally have BB': "?N ` set ?B' = ?N ` ?B" . + have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_compose) + also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real i #bs"] + by(simp add: split_def image_compose simpnum_ci[where bs="real i #bs"] image_def) + finally have AA': "?N ` set ?A' = ?N ` ?A" . + from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0" + by (simp add: simpnum_numbound0 split_def) + from \<alpha>\<rho>_l[OF lq] have A_nb: "\<forall> (e,c)\<in> set ?A'. numbound0 e \<and> c > 0" + by (simp add: simpnum_numbound0 split_def) + {assume "length ?B' \<le> length ?A'" + hence q:"q=?q" and "B = ?B'" and d:"d = ?d" + using qBd by (auto simp add: Let_def chooset_def) + with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)" + and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto + with pq_ex dp lq q d have ?thes by simp} + moreover + {assume "\<not> (length ?B' \<le> length ?A')" + hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d" + using qBd by (auto simp add: Let_def chooset_def) + with AA' mirror_\<alpha>\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)" + and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto + from mirror_ex[OF lq] pq_ex q + have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp + from lq q mirror_l [where p="?q" and bs="bs" and a="real i"] + have lq': "iszlfm q (real i#bs)" by auto + from mirror_\<delta>[OF lq] pqm_eq b bn lq' dp q dp d have ?thes by simp + } + ultimately show ?thes by blast +qed + +constdefs stage:: "fm \<Rightarrow> int \<Rightarrow> (num \<times> int) \<Rightarrow> fm" + "stage p d \<equiv> (\<lambda> (e,c). evaldjf (\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))) (iupt (1,c*d)))" +lemma stage: + shows "Ifm bs (stage p d (e,c)) = (\<exists> j\<in>{1 .. c*d}. Ifm bs (\<sigma> p c (Add e (C j))))" + by (unfold stage_def split_def ,simp only: evaldjf_ex iupt_set simpfm) simp + +lemma stage_nb: assumes lp: "iszlfm p (a#bs)" and cp: "c >0" and nb:"numbound0 e" + shows "bound0 (stage p d (e,c))" +proof- + let ?f = "\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))" + have th: "\<forall> j\<in> set (iupt(1,c*d)). bound0 (?f j)" + proof + fix j + from nb have nb':"numbound0 (Add e (C j))" by simp + from simpfm_bound0[OF \<sigma>_nb[OF lp nb', where k="c"]] + show "bound0 (simpfm (\<sigma> p c (Add e (C j))))" . + qed + from evaldjf_bound0[OF th] show ?thesis by (unfold stage_def split_def) simp +qed + +constdefs redlove:: "fm \<Rightarrow> fm" + "redlove p \<equiv> + (let (q,B,d) = chooset p; + mq = simpfm (minusinf q); + md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) (iupt (1,d)) + in if md = T then T else + (let qd = evaldjf (stage q d) B + in decr (disj md qd)))" + +lemma redlove: assumes qf: "qfree p" + shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" + (is "(?lhs = ?rhs) \<and> _") +proof- + + let ?I = "\<lambda> (x::int) p. Ifm (real x#bs) p" + let ?q = "fst (chooset p)" + let ?B = "fst (snd(chooset p))" + let ?d = "snd (snd (chooset p))" + let ?js = "iupt (1,?d)" + let ?mq = "minusinf ?q" + let ?smq = "simpfm ?mq" + let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" + let ?N = "\<lambda> (t,k). (Inum (real (i::int)#bs) t,k)" + let ?qd = "evaldjf (stage ?q ?d) ?B" + have qbf:"chooset p = (?q,?B,?d)" by simp + from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and + B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and + lq: "iszlfm ?q (real i#bs)" and + Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto + from zlin_qfree[OF lq] have qfq: "qfree ?q" . + from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq". + have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp + hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" + by (auto simp only: subst0_bound0[OF qfmq]) + hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))" + by (auto simp add: simpfm_bound0) + from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp + from Bn stage_nb[OF lq] have th:"\<forall> x \<in> set ?B. bound0 (stage ?q ?d x)" by auto + from evaldjf_bound0[OF th] have qdb: "bound0 ?qd" . + from mdb qdb + have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all) + from trans [OF pq_ex rl_thm'[OF lq B]] dd + have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto + also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))" + by (simp add: simpfm stage split_def) + also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq)) \<or> ?I i ?qd)" + by (simp add: evaldjf_ex subst0_I[OF qfmq]) + finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex iupt_set simpfm) + also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj) + also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) + finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" . + {assume mdT: "?md = T" + hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def) + from mdT have lhs:"?lhs" using mdqd by simp + from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def) + with lhs cT have ?thesis by simp } + moreover + {assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)" + by (simp add: redlove_def chooset_def split_def Let_def) + with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp } + ultimately show ?thesis by blast +qed + +lemma DJredlove: + assumes qf: "qfree p" + shows "((\<exists> (x::int). Ifm (real x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)" +proof- + from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by blast + from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast + have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real x#bs) q)" + using redlove disjuncts_qf[OF qf] by blast + also have "\<dots> = (\<exists> (x::int). Ifm (real x#bs) p)" by (induct p rule: disjuncts.induct, auto) + finally show ?thesis using thqf by blast +qed + + +lemma exsplit_qf: assumes qf: "qfree p" + shows "qfree (exsplit p)" +using qf by (induct p rule: exsplit.induct, auto) + +constdefs mircfr :: "fm \<Rightarrow> fm" +"mircfr \<equiv> (DJ cooper) o ferrack01 o simpfm o exsplit" + +constdefs mirlfr :: "fm \<Rightarrow> fm" +"mirlfr \<equiv> (DJ redlove) o ferrack01 o simpfm o exsplit" + + +lemma mircfr: "\<forall> bs p. qfree p \<longrightarrow> qfree (mircfr p) \<and> Ifm bs (mircfr p) = Ifm bs (E p)" +proof(clarsimp simp del: Ifm.simps) + fix bs p + assume qf: "qfree p" + show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)") + proof- + let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" + have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" + using splitex[OF qf] by simp + with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ + with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def) + qed +qed + +lemma mirlfr: "\<forall> bs p. qfree p \<longrightarrow> qfree(mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)" +proof(clarsimp simp del: Ifm.simps) + fix bs p + assume qf: "qfree p" + show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)") + proof- + let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))" + have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real i#bs) ?es)" + using splitex[OF qf] by simp + with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+ + with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def) + qed +qed + +constdefs mircfrqe:: "fm \<Rightarrow> fm" + "mircfrqe \<equiv> (\<lambda> p. qelim (prep p) mircfr)" + +constdefs mirlfrqe:: "fm \<Rightarrow> fm" + "mirlfrqe \<equiv> (\<lambda> p. qelim (prep p) mirlfr)" + +theorem mircfrqe: "(Ifm bs (mircfrqe p) = Ifm bs p) \<and> qfree (mircfrqe p)" + using qelim_ci[OF mircfr] prep by (auto simp add: mircfrqe_def) + +theorem mirlfrqe: "(Ifm bs (mirlfrqe p) = Ifm bs p) \<and> qfree (mirlfrqe p)" + using qelim_ci[OF mirlfr] prep by (auto simp add: mirlfrqe_def) + +declare zdvd_iff_zmod_eq_0 [code] +declare max_def [code unfold] + +code_module Mir +file "mir.ML" +contains + mircfrqe = "mircfrqe" + mirlfrqe = "mirlfrqe" + test = "%x . mircfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))" + test2 = "%x . mircfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))" + test' = "%x . mirlfrqe (A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0)))))))" + test2' = "%x . mirlfrqe (A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0)))))" +test3 = "%x .mircfrqe (A(E(And (Ge(Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg(Bound 0))))))))" + +ML {* use "mir.ML" *} +ML "set Toplevel.timing" +ML "Mir.test ()" +ML "Mir.test2 ()" +ML "Mir.test' ()" +ML "Mir.test2' ()" +ML "Mir.test3 ()" + +use "mireif.ML" +oracle mircfr_oracle ("term") = ReflectedMir.mircfr_oracle +oracle mirlfr_oracle ("term") = ReflectedMir.mirlfr_oracle +use"mirtac.ML" +setup "MirTac.setup" + +ML "set Toplevel.timing" +lemma "ALL (x::real). (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> = (x = real \<lfloor>x\<rfloor>))" +apply mir +done + +lemma "ALL (x::real). real (2::int)*x - (real (1::int)) < real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<and> real \<lfloor>x\<rfloor> + real \<lceil>x\<rceil> \<le> real (2::int)*x + (real (1::int))" +apply mir +done + +lemma "ALL (x::real). 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>" +apply mir +done + + +lemma "ALL (x::real). \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)" +apply mir +done +ML "reset Toplevel.timing" + +end

--- a/src/HOL/Complex/ex/ROOT.ML Tue Jun 05 19:23:09 2007 +0200 +++ b/src/HOL/Complex/ex/ROOT.ML Tue Jun 05 20:44:12 2007 +0200 @@ -22,3 +22,5 @@ use_thy "DenumRat"; use_thy "Ferrante_Rackoff_Ex"; +use_thy "MIR"; +use_thy "ReflectedFerrack"; \ No newline at end of file

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Complex/ex/ReflectedFerrack.thy Tue Jun 05 20:44:12 2007 +0200 @@ -0,0 +1,1991 @@ +(* Title: Complex/ex/ReflectedFerrack.thy + Author: Amine Chaieb +*) + +header {* Quatifier elimination for R(0,1,+,<) *} + +theory ReflectedFerrack + imports GCD Real + uses ("linreif.ML") ("linrtac.ML") +begin + + + (*********************************************************************************) + (* SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB *) + (*********************************************************************************) + +consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" +primrec + "alluopairs [] = []" + "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)" + +lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}" +by (induct xs, auto) + +lemma alluopairs_set: + "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) " +by (induct xs, auto) + +lemma alluopairs_ex: + assumes Pc: "\<forall> x y. P x y = P y x" + shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)" +proof + assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" + then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y" by blast + from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" + by auto +next + assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y" + then obtain "x" and "y" where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+ + from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast + with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast +qed + +lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)" +using Nat.gr0_conv_Suc +by clarsimp + +lemma filter_length: "length (List.filter P xs) < Suc (length xs)" + apply (induct xs, auto) done + +consts remdps:: "'a list \<Rightarrow> 'a list" + +recdef remdps "measure size" + "remdps [] = []" + "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))" +(hints simp add: filter_length[rule_format]) + +lemma remdps_set[simp]: "set (remdps xs) = set xs" + by (induct xs rule: remdps.induct, auto) + + + + (*********************************************************************************) + (**** SHADOW SYNTAX AND SEMANTICS ****) + (*********************************************************************************) + +datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num + | Mul int num + + (* A size for num to make inductive proofs simpler*) +consts num_size :: "num \<Rightarrow> nat" +primrec + "num_size (C c) = 1" + "num_size (Bound n) = 1" + "num_size (Neg a) = 1 + num_size a" + "num_size (Add a b) = 1 + num_size a + num_size b" + "num_size (Sub a b) = 3 + num_size a + num_size b" + "num_size (Mul c a) = 1 + num_size a" + "num_size (CN n c a) = 3 + num_size a " + + (* Semantics of numeral terms (num) *) +consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real" +primrec + "Inum bs (C c) = (real c)" + "Inum bs (Bound n) = bs!n" + "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)" + "Inum bs (Neg a) = -(Inum bs a)" + "Inum bs (Add a b) = Inum bs a + Inum bs b" + "Inum bs (Sub a b) = Inum bs a - Inum bs b" + "Inum bs (Mul c a) = (real c) * Inum bs a" + (* FORMULAE *) +datatype fm = + T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| + NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm + + + (* A size for fm *) +consts fmsize :: "fm \<Rightarrow> nat" +recdef fmsize "measure size" + "fmsize (NOT p) = 1 + fmsize p" + "fmsize (And p q) = 1 + fmsize p + fmsize q" + "fmsize (Or p q) = 1 + fmsize p + fmsize q" + "fmsize (Imp p q) = 3 + fmsize p + fmsize q" + "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" + "fmsize (E p) = 1 + fmsize p" + "fmsize (A p) = 4+ fmsize p" + "fmsize p = 1" + (* several lemmas about fmsize *) +lemma fmsize_pos: "fmsize p > 0" +by (induct p rule: fmsize.induct) simp_all + + (* Semantics of formulae (fm) *) +consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" +primrec + "Ifm bs T = True" + "Ifm bs F = False" + "Ifm bs (Lt a) = (Inum bs a < 0)" + "Ifm bs (Gt a) = (Inum bs a > 0)" + "Ifm bs (Le a) = (Inum bs a \<le> 0)" + "Ifm bs (Ge a) = (Inum bs a \<ge> 0)" + "Ifm bs (Eq a) = (Inum bs a = 0)" + "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)" + "Ifm bs (NOT p) = (\<not> (Ifm bs p))" + "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)" + "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)" + "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))" + "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)" + "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)" + "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)" + +lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')" +apply simp +done + +lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')" +apply simp +done +lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')" +apply simp +done +lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))" +apply simp +done +lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))" +apply simp +done +lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))" +apply simp +done +lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))" +apply simp +done +lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))" +apply simp +done + +lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))" +apply simp +done +lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))" +apply simp +done + +consts not:: "fm \<Rightarrow> fm" +recdef not "measure size" + "not (NOT p) = p" + "not T = F" + "not F = T" + "not p = NOT p" +lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" +by (cases p) auto + +constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else + if p = q then p else And p q)" +lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" +by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all) + +constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p + else if p=q then p else Or p q)" + +lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" +by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all) + +constdefs imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p + else Imp p q)" +lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" +by (cases "p=F \<or> q=T",simp_all add: imp_def) + +constdefs iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" + "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else + if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else + Iff p q)" +lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" + by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto) + +lemma conj_simps: + "conj F Q = F" + "conj P F = F" + "conj T Q = Q" + "conj P T = P" + "conj P P = P" + "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q" + by (simp_all add: conj_def) + +lemma disj_simps: + "disj T Q = T" + "disj P T = T" + "disj F Q = Q" + "disj P F = P" + "disj P P = P" + "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q" + by (simp_all add: disj_def) +lemma imp_simps: + "imp F Q = T" + "imp P T = T" + "imp T Q = Q" + "imp P F = not P" + "imp P P = T" + "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q" + by (simp_all add: imp_def) +lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p" +apply (induct p, auto) +done + +lemma iff_simps: + "iff p p = T" + "iff p (NOT p) = F" + "iff (NOT p) p = F" + "iff p F = not p" + "iff F p = not p" + "p \<noteq> NOT T \<Longrightarrow> iff T p = p" + "p\<noteq> NOT T \<Longrightarrow> iff p T = p" + "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q" + using trivNOT + by (simp_all add: iff_def, cases p, auto) + (* Quantifier freeness *) +consts qfree:: "fm \<Rightarrow> bool" +recdef qfree "measure size" + "qfree (E p) = False" + "qfree (A p) = False" + "qfree (NOT p) = qfree p" + "qfree (And p q) = (qfree p \<and> qfree q)" + "qfree (Or p q) = (qfree p \<and> qfree q)" + "qfree (Imp p q) = (qfree p \<and> qfree q)" + "qfree (Iff p q) = (qfree p \<and> qfree q)" + "qfree p = True" + + (* Boundedness and substitution *) +consts + numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) + bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) +primrec + "numbound0 (C c) = True" + "numbound0 (Bound n) = (n>0)" + "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)" + "numbound0 (Neg a) = numbound0 a" + "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)" + "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" + "numbound0 (Mul i a) = numbound0 a" +lemma numbound0_I: + assumes nb: "numbound0 a" + shows "Inum (b#bs) a = Inum (b'#bs) a" +using nb +by (induct a rule: numbound0.induct,auto simp add: nth_pos2) + +primrec + "bound0 T = True" + "bound0 F = True" + "bound0 (Lt a) = numbound0 a" + "bound0 (Le a) = numbound0 a" + "bound0 (Gt a) = numbound0 a" + "bound0 (Ge a) = numbound0 a" + "bound0 (Eq a) = numbound0 a" + "bound0 (NEq a) = numbound0 a" + "bound0 (NOT p) = bound0 p" + "bound0 (And p q) = (bound0 p \<and> bound0 q)" + "bound0 (Or p q) = (bound0 p \<and> bound0 q)" + "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))" + "bound0 (Iff p q) = (bound0 p \<and> bound0 q)" + "bound0 (E p) = False" + "bound0 (A p) = False" + +lemma bound0_I: + assumes bp: "bound0 p" + shows "Ifm (b#bs) p = Ifm (b'#bs) p" +using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] +by (induct p rule: bound0.induct) (auto simp add: nth_pos2) + +lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)" +by (cases p, auto) +lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)" +by (cases p, auto) + + +lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)" +using conj_def by auto +lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)" +using conj_def by auto + +lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)" +using disj_def by auto +lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)" +using disj_def by auto + +lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)" +using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def) +lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)" +using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def) + +lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)" + by (unfold iff_def,cases "p=q", auto) +lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)" +using iff_def by (unfold iff_def,cases "p=q", auto) + +consts + decrnum:: "num \<Rightarrow> num" + decr :: "fm \<Rightarrow> fm" + +recdef decrnum "measure size" + "decrnum (Bound n) = Bound (n - 1)" + "decrnum (Neg a) = Neg (decrnum a)" + "decrnum (Add a b) = Add (decrnum a) (decrnum b)" + "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" + "decrnum (Mul c a) = Mul c (decrnum a)" + "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" + "decrnum a = a" + +recdef decr "measure size" + "decr (Lt a) = Lt (decrnum a)" + "decr (Le a) = Le (decrnum a)" + "decr (Gt a) = Gt (decrnum a)" + "decr (Ge a) = Ge (decrnum a)" + "decr (Eq a) = Eq (decrnum a)" + "decr (NEq a) = NEq (decrnum a)" + "decr (NOT p) = NOT (decr p)" + "decr (And p q) = conj (decr p) (decr q)" + "decr (Or p q) = disj (decr p) (decr q)" + "decr (Imp p q) = imp (decr p) (decr q)" + "decr (Iff p q) = iff (decr p) (decr q)" + "decr p = p" + +lemma decrnum: assumes nb: "numbound0 t" + shows "Inum (x#bs) t = Inum bs (decrnum t)" + using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) + +lemma decr: assumes nb: "bound0 p" + shows "Ifm (x#bs) p = Ifm bs (decr p)" + using nb + by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) + +lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" +by (induct p, simp_all) + +consts + isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) +recdef isatom "measure size" + "isatom T = True" + "isatom F = True" + "isatom (Lt a) = True" + "isatom (Le a) = True" + "isatom (Gt a) = True" + "isatom (Ge a) = True" + "isatom (Eq a) = True" + "isatom (NEq a) = True" + "isatom p = False" + +lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" +by (induct p, simp_all) + +constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" + "djf f p q \<equiv> (if q=T then T else if q=F then f p else + (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))" +constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" + "evaldjf f ps \<equiv> foldr (djf f) ps F" + +lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" +by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) +(cases "f p", simp_all add: Let_def djf_def) + + +lemma djf_simps: + "djf f p T = T" + "djf f p F = f p" + "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)" + by (simp_all add: djf_def) + +lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))" + by(induct ps, simp_all add: evaldjf_def djf_Or) + +lemma evaldjf_bound0: + assumes nb: "\<forall> x\<in> set xs. bound0 (f x)" + shows "bound0 (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +lemma evaldjf_qf: + assumes nb: "\<forall> x\<in> set xs. qfree (f x)" + shows "qfree (evaldjf f xs)" + using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) + +consts disjuncts :: "fm \<Rightarrow> fm list" +recdef disjuncts "measure size" + "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" + "disjuncts F = []" + "disjuncts p = [p]" + +lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p" +by(induct p rule: disjuncts.induct, auto) + +lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q" +proof- + assume nb: "bound0 p" + hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) + thus ?thesis by (simp only: list_all_iff) +qed + +lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q" +proof- + assume qf: "qfree p" + hence "list_all qfree (disjuncts p)" + by (induct p rule: disjuncts.induct, auto) + thus ?thesis by (simp only: list_all_iff) +qed + +constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" + "DJ f p \<equiv> evaldjf f (disjuncts p)" + +lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))" + and fF: "f F = F" + shows "Ifm bs (DJ f p) = Ifm bs (f p)" +proof- + have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) + finally show ?thesis . +qed + +lemma DJ_qf: assumes + fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)" + shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) " +proof(clarify) + fix p assume qf: "qfree p" + have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) + from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" . + with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast + + from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp +qed + +lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" + shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))" +proof(clarify) + fix p::fm and bs + assume qf: "qfree p" + from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast + from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto + have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))" + by (simp add: DJ_def evaldjf_ex) + also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto + also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) + finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast +qed + (* Simplification *) +consts + numgcd :: "num \<Rightarrow> int" + numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" + reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" + reducecoeff :: "num \<Rightarrow> num" + dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" +consts maxcoeff:: "num \<Rightarrow> int" +recdef maxcoeff "measure size" + "maxcoeff (C i) = abs i" + "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)" + "maxcoeff t = 1" + +lemma maxcoeff_pos: "maxcoeff t \<ge> 0" + by (induct t rule: maxcoeff.induct, auto) + +recdef numgcdh "measure size" + "numgcdh (C i) = (\<lambda>g. igcd i g)" + "numgcdh (CN n c t) = (\<lambda>g. igcd c (numgcdh t g))" + "numgcdh t = (\<lambda>g. 1)" +defs numgcd_def: "numgcd t \<equiv> numgcdh t (maxcoeff t)" + +recdef reducecoeffh "measure size" + "reducecoeffh (C i) = (\<lambda> g. C (i div g))" + "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))" + "reducecoeffh t = (\<lambda>g. t)" + +defs reducecoeff_def: "reducecoeff t \<equiv> + (let g = numgcd t in + if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" + +recdef dvdnumcoeff "measure size" + "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)" + "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" + "dvdnumcoeff t = (\<lambda>g. False)" + +lemma dvdnumcoeff_trans: + assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" + shows "dvdnumcoeff t g" + using dgt' gdg + by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg]) + +declare zdvd_trans [trans add] + +lemma natabs0: "(nat (abs x) = 0) = (x = 0)" +by arith + +lemma numgcd0: + assumes g0: "numgcd t = 0" + shows "Inum bs t = 0" + using g0[simplified numgcd_def] + by (induct t rule: numgcdh.induct, auto simp add: igcd_def gcd_zero natabs0 max_def maxcoeff_pos) + +lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0" + using gp + by (induct t rule: numgcdh.induct, auto simp add: igcd_def) + +lemma numgcd_pos: "numgcd t \<ge>0" + by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) + +lemma reducecoeffh: + assumes gt: "dvdnumcoeff t g" and gp: "g > 0" + shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t" + using gt +proof(induct t rule: reducecoeffh.induct) + case (1 i) hence gd: "g dvd i" by simp + from gp have gnz: "g \<noteq> 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd]) +next + case (2 n c t) hence gd: "g dvd c" by simp + from gp have gnz: "g \<noteq> 0" by simp + from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_eq_simps) +qed (auto simp add: numgcd_def gp) +consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" +recdef ismaxcoeff "measure size" + "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)" + "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))" + "ismaxcoeff t = (\<lambda>x. True)" + +lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'" +by (induct t rule: ismaxcoeff.induct, auto) + +lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence H:"ismaxcoeff t (maxcoeff t)" . + have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2) + from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1) +qed simp_all + +lemma igcd_gt1: "igcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))" + apply (cases "abs i = 0", simp_all add: igcd_def) + apply (cases "abs j = 0", simp_all) + apply (cases "abs i = 1", simp_all) + apply (cases "abs j = 1", simp_all) + apply auto + done +lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0" + by (induct t rule: numgcdh.induct, auto simp add:igcd0) + +lemma dvdnumcoeff_aux: + assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1" + shows "dvdnumcoeff t (numgcdh t m)" +using prems +proof(induct t rule: numgcdh.induct) + case (2 n c t) + let ?g = "numgcdh t m" + from prems have th:"igcd c ?g > 1" by simp + from igcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] + have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp + moreover {assume "abs c > 1" and gp: "?g > 1" with prems + have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)} + moreover {assume "abs c = 0 \<and> ?g > 1" + with prems have th: "dvdnumcoeff t ?g" by simp + have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2) + from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1) + hence ?case by simp } + moreover {assume "abs c > 1" and g0:"?g = 0" + from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } + ultimately show ?case by blast +qed(auto simp add: igcd_dvd1) + +lemma dvdnumcoeff_aux2: + assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0" + using prems +proof (simp add: numgcd_def) + let ?mc = "maxcoeff t" + let ?g = "numgcdh t ?mc" + have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) + have th2: "?mc \<ge> 0" by (rule maxcoeff_pos) + assume H: "numgcdh t ?mc > 1" + from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . +qed + +lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" +proof- + let ?g = "numgcd t" + have "?g \<ge> 0" by (simp add: numgcd_pos) + hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto + moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} + moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} + moreover { assume g1:"?g > 1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ + from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis + by (simp add: reducecoeff_def Let_def)} + ultimately show ?thesis by blast +qed + +lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)" +by (induct t rule: reducecoeffh.induct, auto) + +lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)" +using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) + +consts + simpnum:: "num \<Rightarrow> num" + numadd:: "num \<times> num \<Rightarrow> num" + nummul:: "num \<Rightarrow> int \<Rightarrow> num" +recdef numadd "measure (\<lambda> (t,s). size t + size s)" + "numadd (CN n1 c1 r1,CN n2 c2 r2) = + (if n1=n2 then + (let c = c1 + c2 + in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) + else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) + else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" + "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" + "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" + "numadd (C b1, C b2) = C (b1+b2)" + "numadd (a,b) = Add a b" + +lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" +apply (induct t s rule: numadd.induct, simp_all add: Let_def) +apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all) +apply (case_tac "n1 = n2", simp_all add: ring_eq_simps) +by (simp only: ring_eq_simps(1)[symmetric],simp) + +lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))" +by (induct t s rule: numadd.induct, auto simp add: Let_def) + +recdef nummul "measure size" + "nummul (C j) = (\<lambda> i. C (i*j))" + "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))" + "nummul t = (\<lambda> i. Mul i t)" + +lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)" +by (induct t rule: nummul.induct, auto simp add: ring_eq_simps) + +lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)" +by (induct t rule: nummul.induct, auto ) + +constdefs numneg :: "num \<Rightarrow> num" + "numneg t \<equiv> nummul t (- 1)" + +constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num" + "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))" + +lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" +using numneg_def by simp + +lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)" +using numneg_def by simp + +lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" +using numsub_def by simp + +lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)" +using numsub_def by simp + +recdef simpnum "measure size" + "simpnum (C j) = C j" + "simpnum (Bound n) = CN n 1 (C 0)" + "simpnum (Neg t) = numneg (simpnum t)" + "simpnum (Add t s) = numadd (simpnum t,simpnum s)" + "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" + "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" + "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))" + +lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" +by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul) + +lemma simpnum_numbound0[simp]: + "numbound0 t \<Longrightarrow> numbound0 (simpnum t)" +by (induct t rule: simpnum.induct, auto) + +consts nozerocoeff:: "num \<Rightarrow> bool" +recdef nozerocoeff "measure size" + "nozerocoeff (C c) = True" + "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)" + "nozerocoeff t = True" + +lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))" +by (induct a b rule: numadd.induct,auto simp add: Let_def) + +lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)" +by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz) + +lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)" +by (simp add: numneg_def nummul_nz) + +lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)" +by (simp add: numsub_def numneg_nz numadd_nz) + +lemma simpnum_nz: "nozerocoeff (simpnum t)" +by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz) + +lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0" +proof (induct t rule: maxcoeff.induct) + case (2 n c t) + hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ + have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1) + with cnz have "max (abs c) (maxcoeff t) > 0" by arith + with prems show ?case by simp +qed auto + +lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" +proof- + from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) + from numgcdh0[OF th] have th:"maxcoeff t = 0" . + from maxcoeff_nz[OF nz th] show ?thesis . +qed + +constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int" + "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else + (let t' = simpnum t ; g = numgcd t' in + if g > 1 then (let g' = igcd n g in + if g' = 1 then (t',n) + else (reducecoeffh t' g', n div g')) + else (t',n))))" + +lemma simp_num_pair_ci: + shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))" + (is "?lhs = ?rhs") +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "igcd n ?g" + {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \<noteq> 0" + {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp + hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \<or> ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} + moreover {assume g'1:"?g'>1" + from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. + let ?tt = "reducecoeffh ?t' ?g'" + let ?t = "Inum bs ?tt" + have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2) + have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) + have gpdgp: "?g' dvd ?g'" by simp + from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] + have th2:"real ?g' * ?t = Inum bs ?t'" by simp + from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def) + also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp + also have "\<dots> = (Inum bs ?t' / real n)" + using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp + finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci) + then have ?thesis using prems by (simp add: simp_num_pair_def)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" + shows "numbound0 t' \<and> n' >0" +proof- + let ?t' = "simpnum t" + let ?g = "numgcd ?t'" + let ?g' = "igcd n ?g" + {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)} + moreover + { assume nnz: "n \<noteq> 0" + {assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} + moreover + {assume g1:"?g>1" hence g0: "?g > 0" by simp + from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp + hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith + hence "?g'= 1 \<or> ?g' > 1" by arith + moreover {assume "?g'=1" hence ?thesis using prems + by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} + moreover {assume g'1:"?g'>1" + have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2) + have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) + have gpdgp: "?g' dvd ?g'" by simp + from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" . + from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]] + have "n div ?g' >0" by simp + hence ?thesis using prems + by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)} + ultimately have ?thesis by blast} + ultimately have ?thesis by blast} + ultimately show ?thesis by blast +qed + +consts simpfm :: "fm \<Rightarrow> fm" +recdef simpfm "measure fmsize" + "simpfm (And p q) = conj (simpfm p) (simpfm q)" + "simpfm (Or p q) = disj (simpfm p) (simpfm q)" + "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" + "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" + "simpfm (NOT p) = not (simpfm p)" + "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F + | _ \<Rightarrow> Lt a')" + "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0) then T else F | _ \<Rightarrow> Le a')" + "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt a')" + "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0) then T else F | _ \<Rightarrow> Ge a')" + "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq a')" + "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0) then T else F | _ \<Rightarrow> NEq a')" + "simpfm p = p" +lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p" +proof(induct p rule: simpfm.induct) + case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (7 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (8 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (9 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (10 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +next + case (11 a) let ?sa = "simpnum a" + from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp + {fix v assume "?sa = C v" hence ?case using sa by simp } + moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa + by (cases ?sa, simp_all add: Let_def)} + ultimately show ?case by blast +qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not) + + +lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)" +proof(induct p rule: simpfm.induct) + case (6 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (7 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (8 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (9 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (10 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +next + case (11 a) hence nb: "numbound0 a" by simp + hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) + thus ?case by (cases "simpnum a", auto simp add: Let_def) +qed(auto simp add: disj_def imp_def iff_def conj_def not_bn) + +lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)" +by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def) + (case_tac "simpnum a",auto)+ + +consts prep :: "fm \<Rightarrow> fm" +recdef prep "measure fmsize" + "prep (E T) = T" + "prep (E F) = F" + "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))" + "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))" + "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" + "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))" + "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" + "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" + "prep (E p) = E (prep p)" + "prep (A (And p q)) = conj (prep (A p)) (prep (A q))" + "prep (A p) = prep (NOT (E (NOT p)))" + "prep (NOT (NOT p)) = prep p" + "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))" + "prep (NOT (A p)) = prep (E (NOT p))" + "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))" + "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))" + "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))" + "prep (NOT p) = not (prep p)" + "prep (Or p q) = disj (prep p) (prep q)" + "prep (And p q) = conj (prep p) (prep q)" + "prep (Imp p q) = prep (Or (NOT p) q)" + "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))" + "prep p = p" +(hints simp add: fmsize_pos) +lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p" +by (induct p rule: prep.induct, auto) + + (* Generic quantifier elimination *) +consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" +recdef qelim "measure fmsize" + "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))" + "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))" + "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))" + "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" + "qelim (Or p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" + "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))" + "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))" + "qelim p = (\<lambda> y. simpfm p)" + +lemma qelim_ci: + assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" + shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)" +using qe_inv DJ_qe[OF qe_inv] +by(induct p rule: qelim.induct) +(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf + simpfm simpfm_qf simp del: simpfm.simps) + +consts + plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) + minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) +recdef minusinf "measure size" + "minusinf (And p q) = conj (minusinf p) (minusinf q)" + "minusinf (Or p q) = disj (minusinf p) (minusinf q)" + "minusinf (Eq (CN 0 c e)) = F" + "minusinf (NEq (CN 0 c e)) = T" + "minusinf (Lt (CN 0 c e)) = T" + "minusinf (Le (CN 0 c e)) = T" + "minusinf (Gt (CN 0 c e)) = F" + "minusinf (Ge (CN 0 c e)) = F" + "minusinf p = p" + +recdef plusinf "measure size" + "plusinf (And p q) = conj (plusinf p) (plusinf q)" + "plusinf (Or p q) = disj (plusinf p) (plusinf q)" + "plusinf (Eq (CN 0 c e)) = F" + "plusinf (NEq (CN 0 c e)) = T" + "plusinf (Lt (CN 0 c e)) = F" + "plusinf (Le (CN 0 c e)) = F" + "plusinf (Gt (CN 0 c e)) = T" + "plusinf (Ge (CN 0 c e)) = T" + "plusinf p = p" + +consts + isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *) +recdef isrlfm "measure size" + "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" + "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" + "isrlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm (Ge (CN 0 c e)) = (c>0 \<and> numbound0 e)" + "isrlfm p = (isatom p \<and> (bound0 p))" + + (* splits the bounded from the unbounded part*) +consts rsplit0 :: "num \<Rightarrow> int \<times> num" +recdef rsplit0 "measure num_size" + "rsplit0 (Bound 0) = (1,C 0)" + "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b + in (ca+cb, Add ta tb))" + "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" + "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))" + "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))" + "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))" + "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))" + "rsplit0 t = (0,t)" +lemma rsplit0: + shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))" +proof (induct t rule: rsplit0.induct) + case (2 a b) + let ?sa = "rsplit0 a" let ?sb = "rsplit0 b" + let ?ca = "fst ?sa" let ?cb = "fst ?sb" + let ?ta = "snd ?sa" let ?tb = "snd ?sb" + from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" + by(cases "rsplit0 a",auto simp add: Let_def split_def) + have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = + Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)" + by (simp add: Let_def split_def ring_eq_simps) + also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all) + finally show ?case using nb by simp +qed(auto simp add: Let_def split_def ring_eq_simps , simp add: ring_eq_simps(2)[symmetric]) + + (* Linearize a formula*) +consts + lt :: "int \<Rightarrow> num \<Rightarrow> fm" + le :: "int \<Rightarrow> num \<Rightarrow> fm" + gt :: "int \<Rightarrow> num \<Rightarrow> fm" + ge :: "int \<Rightarrow> num \<Rightarrow> fm" + eq :: "int \<Rightarrow> num \<Rightarrow> fm" + neq :: "int \<Rightarrow> num \<Rightarrow> fm" + +defs lt_def: "lt c t \<equiv> (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) + else (Gt (CN 0 (-c) (Neg t))))" +defs le_def: "le c t \<equiv> (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) + else (Ge (CN 0 (-c) (Neg t))))" +defs gt_def: "gt c t \<equiv> (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) + else (Lt (CN 0 (-c) (Neg t))))" +defs ge_def: "ge c t \<equiv> (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) + else (Le (CN 0 (-c) (Neg t))))" +defs eq_def: "eq c t \<equiv> (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) + else (Eq (CN 0 (-c) (Neg t))))" +defs neq_def: "neq c t \<equiv> (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) + else (NEq (CN 0 (-c) (Neg t))))" + +lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))" +using rsplit0[where bs = "bs" and t="t"] +by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) + +lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)" +by (auto simp add: conj_def) +lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)" +by (auto simp add: disj_def) + +consts rlfm :: "fm \<Rightarrow> fm" +recdef rlfm "measure fmsize" + "rlfm (And p q) = conj (rlfm p) (rlfm q)" + "rlfm (Or p q) = disj (rlfm p) (rlfm q)" + "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" + "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))" + "rlfm (Lt a) = split lt (rsplit0 a)" + "rlfm (Le a) = split le (rsplit0 a)" + "rlfm (Gt a) = split gt (rsplit0 a)" + "rlfm (Ge a) = split ge (rsplit0 a)" + "rlfm (Eq a) = split eq (rsplit0 a)" + "rlfm (NEq a) = split neq (rsplit0 a)" + "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" + "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" + "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" + "rlfm (NOT (NOT p)) = rlfm p" + "rlfm (NOT T) = F" + "rlfm (NOT F) = T" + "rlfm (NOT (Lt a)) = rlfm (Ge a)" + "rlfm (NOT (Le a)) = rlfm (Gt a)" + "rlfm (NOT (Gt a)) = rlfm (Le a)" + "rlfm (NOT (Ge a)) = rlfm (Lt a)" + "rlfm (NOT (Eq a)) = rlfm (NEq a)" + "rlfm (NOT (NEq a)) = rlfm (Eq a)" + "rlfm p = p" (hints simp add: fmsize_pos) + +lemma rlfm_I: + assumes qfp: "qfree p" + shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)" + using qfp +by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin) + + (* Operations needed for Ferrante and Rackoff *) +lemma rminusinf_inf: + assumes lp: "isrlfm p" + shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") +using lp +proof (induct p rule: minusinf.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x < ?z" + hence "(real c * x < - ?e)" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) + hence "real c * x + ?e < 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rplusinf_inf: + assumes lp: "isrlfm p" + shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") +using lp +proof (induct p rule: isrlfm.induct) + case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto +next + case (3 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (Eq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp + thus ?case by blast +next + case (4 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + hence "real c * x + ?e \<noteq> 0" by simp + with xz have "?P ?z x (NEq (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp + thus ?case by blast +next + case (5 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Lt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp + thus ?case by blast +next + case (6 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Le (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp + thus ?case by blast +next + case (7 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Gt (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp + thus ?case by blast +next + case (8 c e) + from prems have nb: "numbound0 e" by simp + from prems have cp: "real c > 0" by simp + let ?e="Inum (a#bs) e" + let ?z = "(- ?e) / real c" + {fix x + assume xz: "x > ?z" + with mult_strict_right_mono [OF xz cp] cp + have "(real c * x > - ?e)" by (simp add: mult_ac) + hence "real c * x + ?e > 0" by arith + with xz have "?P ?z x (Ge (CN 0 c e))" + using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } + hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp + thus ?case by blast +qed simp_all + +lemma rminusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (minusinf p)" + using lp + by (induct p rule: minusinf.induct) simp_all + +lemma rplusinf_bound0: + assumes lp: "isrlfm p" + shows "bound0 (plusinf p)" + using lp + by (induct p rule: plusinf.induct) simp_all + +lemma rminusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (minusinf p)" + shows "\<exists> x. Ifm (x#bs) p" +proof- + from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto + from rminusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp + moreover have "z - 1 < z" by simp + ultimately show ?thesis using z_def by auto +qed + +lemma rplusinf_ex: + assumes lp: "isrlfm p" + and ex: "Ifm (a#bs) (plusinf p)" + shows "\<exists> x. Ifm (x#bs) p" +proof- + from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex + have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto + from rplusinf_inf[OF lp, where bs="bs"] + obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast + from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp + moreover have "z + 1 > z" by simp + ultimately show ?thesis using z_def by auto +qed + +consts + uset:: "fm \<Rightarrow> (num \<times> int) list" + usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm " +recdef uset "measure size" + "uset (And p q) = (uset p @ uset q)" + "uset (Or p q) = (uset p @ uset q)" + "uset (Eq (CN 0 c e)) = [(Neg e,c)]" + "uset (NEq (CN 0 c e)) = [(Neg e,c)]" + "uset (Lt (CN 0 c e)) = [(Neg e,c)]" + "uset (Le (CN 0 c e)) = [(Neg e,c)]" + "uset (Gt (CN 0 c e)) = [(Neg e,c)]" + "uset (Ge (CN 0 c e)) = [(Neg e,c)]" + "uset p = []" +recdef usubst "measure size" + "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))" + "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))" + "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))" + "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))" + "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))" + "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))" + "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))" + "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))" + "usubst p = (\<lambda> (t,n). p)" + +lemma usubst_I: assumes lp: "isrlfm p" + and np: "real n > 0" and nbt: "numbound0 t" + shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _") + using lp +proof(induct p rule: usubst.induct) + case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)" + by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)" + by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)" + by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)" + by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \<noteq> 0" by simp + have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +next + case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ + from np have np: "real n \<noteq> 0" by simp + have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)" + using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp + also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)" + by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" + and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps) + also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)" + using np by simp + finally show ?case using nbt nb by (simp add: ring_eq_simps) +qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2) + +lemma uset_l: + assumes lp: "isrlfm p" + shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0" +using lp +by(induct p rule: uset.induct,auto) + +lemma rminusinf_uset: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m") +proof- + have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast + from uset_l[OF lp] smU have mp: "real m > 0" by auto + from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma rplusinf_uset: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") + and ex: "Ifm (x#bs) p" (is "?I x p") + shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m") +proof- + have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s") + using lp nmi ex + by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) + then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast + from uset_l[OF lp] smU have mp: "real m > 0" by auto + from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" + by (auto simp add: mult_commute) + thus ?thesis using smU by auto +qed + +lemma lin_dense: + assumes lp: "isrlfm p" + and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" + (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)") + and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" + and ly: "l < y" and yu: "y < u" + shows "Ifm (y#bs) p" +using lp px noS +proof (induct p rule: isrlfm.induct) + case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e < 0" by (simp add: ring_eq_simps) + hence pxc: "x < (- ?N x e) / real c" + by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp + + from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_eq_simps) + hence pxc: "x \<le> (- ?N x e) / real c" + by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y < (-?N x e)/ real c" + hence "y * real c < - ?N x e" + by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y > (- ?N x e) / real c" + with yu have eu: "u > (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) + with lx pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e > 0" by (simp add: ring_eq_simps) + hence pxc: "x > (- ?N x e) / real c" + by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_eq_simps) + hence pxc: "x \<ge> (- ?N x e) / real c" + by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto + moreover {assume y: "y > (-?N x e)/ real c" + hence "y * real c > - ?N x e" + by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) + hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps) + hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp} + moreover {assume y: "y < (- ?N x e) / real c" + with ly have eu: "l < (- ?N x e) / real c" by auto + with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) + with xu pxc have "False" by auto + hence ?case by simp } + ultimately show ?case by blast +next + case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \<noteq> 0" by simp + from prems have "x * real c + ?N x e = 0" by (simp add: ring_eq_simps) + hence pxc: "x = (- ?N x e) / real c" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto + with pxc show ?case by simp +next + case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ + from cp have cnz: "real c \<noteq> 0" by simp + from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto + with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto + hence "y* real c \<noteq> -?N x e" + by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp + hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_eq_simps) + thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] + by (simp add: ring_eq_simps) +qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"]) + +lemma finite_set_intervals: + assumes px: "P (x::real)" + and lx: "l \<le> x" and xu: "x \<le> u" + and linS: "l\<in> S" and uinS: "u \<in> S" + and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" + shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x" +proof- + let ?Mx = "{y. y\<in> S \<and> y \<le> x}" + let ?xM = "{y. y\<in> S \<and> x \<le> y}" + let ?a = "Max ?Mx" + let ?b = "Min ?xM" + have MxS: "?Mx \<subseteq> S" by blast + hence fMx: "finite ?Mx" using fS finite_subset by auto + from lx linS have linMx: "l \<in> ?Mx" by blast + hence Mxne: "?Mx \<noteq> {}" by blast + have xMS: "?xM \<subseteq> S" by blast + hence fxM: "finite ?xM" using fS finite_subset by auto + from xu uinS have linxM: "u \<in> ?xM" by blast + hence xMne: "?xM \<noteq> {}" by blast + have ax:"?a \<le> x" using Mxne fMx by auto + have xb:"x \<le> ?b" using xMne fxM by auto + have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast + have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast + have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S" + proof(clarsimp) + fix y + assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" + from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto + moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp} + moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp} + ultimately show "False" by blast + qed + from ainS binS noy ax xb px show ?thesis by blast +qed + +lemma finite_set_intervals2: + assumes px: "P (x::real)" + and lx: "l \<le> x" and xu: "x \<le> u" + and linS: "l\<in> S" and uinS: "u \<in> S" + and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" + shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)" +proof- + from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] + obtain a and b where + as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto + from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto + thus ?thesis using px as bs noS by blast +qed + +lemma rinf_uset: + assumes lp: "isrlfm p" + and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))") + and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))") + and ex: "\<exists> x. Ifm (x#bs) p" (is "\<exists> x. ?I x p") + shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" +proof- + let ?N = "\<lambda> x t. Inum (x#bs) t" + let ?U = "set (uset p)" + from ex obtain a where pa: "?I a p" by blast + from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi + have nmi': "\<not> (?I a (?M p))" by simp + from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi + have npi': "\<not> (?I a (?P p))" by simp + have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p" + proof- + let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U" + have fM: "finite ?M" by auto + from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] + have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast + then obtain "t" "n" "s" "m" where + tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" + and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast + from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto + from tnU have Mne: "?M \<noteq> {}" by auto + hence Une: "?U \<noteq> {}" by simp + let ?l = "Min ?M" + let ?u = "Max ?M" + have linM: "?l \<in> ?M" using fM Mne by simp + have uinM: "?u \<in> ?M" using fM Mne by simp + have tnM: "?N a t / real n \<in> ?M" using tnU by auto + have smM: "?N a s / real m \<in> ?M" using smU by auto + have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto + have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto + have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp + have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp + from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu] + have "(\<exists> s\<in> ?M. ?I s p) \<or> + (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" . + moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p" + hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto + then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast + have "(u + u) / 2 = u" by auto with pu tuu + have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp + with tuU have ?thesis by blast} + moreover{ + assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p" + then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" + and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" + by blast + from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto + then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast + from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto + then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast + from t1x xt2 have t1t2: "t1 < t2" by simp + let ?u = "(t1 + t2) / 2" + from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto + from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . + with t1uU t2uU t1u t2u have ?thesis by blast} + ultimately show ?thesis by blast + qed + then obtain "l" "n" "s" "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" + and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast + from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto + from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] + numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu + have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp + with lnU smU + show ?thesis by auto +qed + (* The Ferrante - Rackoff Theorem *) + +theorem fr_eq: + assumes lp: "isrlfm p" + shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/ real n + (Inum (x#bs) s) / real m) /2)#bs) p))" + (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") +proof + assume px: "\<exists> x. ?I x p" + have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast + moreover {assume "?M \<or> ?P" hence "?D" by blast} + moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" + from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {assume f:"?F" hence "?E" by blast} + ultimately show "?E" by blast +qed + + +lemma fr_equsubst: + assumes lp: "isrlfm p" + shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))" + (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") +proof + assume px: "\<exists> x. ?I x p" + have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast + moreover {assume "?M \<or> ?P" hence "?D" by blast} + moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" + let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n" + let ?N = "\<lambda> t. Inum (x#bs) t" + {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)" + with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" + by auto + let ?st = "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnp mp np by (simp add: ring_eq_simps add_divide_distrib) + from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] + have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])} + with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} + ultimately show "?D" by blast +next + assume "?D" + moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} + moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } + moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" + and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))" + with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto + let ?st = "Add (Mul l t) (Mul k s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" + by (simp add: mult_commute) + from tnb snb have st_nb: "numbound0 ?st" by simp + from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} + ultimately show "?E" by blast +qed + +consts allpairs:: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list" +primrec + "allpairs [] ys = []" + "allpairs (x#xs) ys = (map (Pair x) ys)@(allpairs xs ys)" + +lemma allpairs_set: "set (allpairs xs ys) = {(x,y). x\<in> set xs \<and> y \<in> set ys}" +by (induct xs) auto + + (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *) +constdefs ferrack:: "fm \<Rightarrow> fm" + "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p' + in if (mp = T \<or> pp = T) then T else + (let U = remdps(map simp_num_pair + (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) + (alluopairs (uset p')))) + in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))" + +lemma uset_cong_aux: + assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0" + shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))" + (is "?lhs = ?rhs") +proof(auto) + fix t n s m + assume "((t,n),(s,m)) \<in> set (alluopairs U)" + hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)" + using alluopairs_set1[where xs="U"] by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul th have mnz: "m \<noteq> 0" by auto + from Ul th have nnz: "n \<noteq> 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: ring_eq_simps add_divide_distrib) + + thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) / + (2 * real n * real m) + \<in> (\<lambda>((t, n), s, m). + (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` + (set U \<times> set U)"using mnz nnz th + apply (auto simp add: th add_divide_distrib ring_eq_simps split_def image_def) + by (rule_tac x="(s,m)" in bexI,simp_all) + (rule_tac x="(t,n)" in bexI,simp_all) +next + fix t n s m + assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?st= "Add (Mul m t) (Mul n s)" + from Ul smU have mnz: "m \<noteq> 0" by auto + from Ul tnU have nnz: "n \<noteq> 0" by auto + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mnz nnz by (simp add: ring_eq_simps add_divide_distrib) + let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" + have Pc:"\<forall> a b. ?P a b = ?P b a" + by auto + from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast + from alluopairs_ex[OF Pc, where xs="U"] tnU smU + have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')" + by blast + then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" + and Pts': "?P (t',n') (s',m')" by blast + from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto + let ?st' = "Add (Mul m' t') (Mul n' s')" + have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')" + using mnz' nnz' by (simp add: ring_eq_simps add_divide_distrib) + from Pts' have + "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp + also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') + finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 + \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) ` + (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` + set (alluopairs U)" + using ts'_U by blast +qed + +lemma uset_cong: + assumes lp: "isrlfm p" + and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)") + and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0" + and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0" + shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))" + (is "?lhs = ?rhs") +proof + assume ?lhs + then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and + Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: ring_eq_simps add_divide_distrib) + from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast + hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] + have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) + then show ?rhs using tnU' by auto +next + assume ?rhs + then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" + by blast + from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast + hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" + by auto (rule_tac x="(a,b)" in bexI, auto) + then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and + th: "?f (t',n') = ?g((t,n),(s,m)) "by blast + let ?N = "\<lambda> t. Inum (x#bs) t" + from tnU smU U have tnb: "numbound0 t" and np: "n > 0" + and snb: "numbound0 s" and mp:"m > 0" by auto + let ?st= "Add (Mul m t) (Mul n s)" + from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" + by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) + from tnb snb have stnb: "numbound0 ?st" by simp + have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" + using mp np by (simp add: ring_eq_simps add_divide_distrib) + from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto + from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' + have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp + with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast +qed + +lemma ferrack: + assumes qf: "qfree p" + shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))" + (is "_ \<and> (?rhs = ?lhs)") +proof- + let ?I = "\<lambda> x p. Ifm (x#bs) p" + let ?N = "\<lambda> t. Inum (x#bs) t" + let ?q = "rlfm (simpfm p)" + let ?U = "uset ?q" + let ?Up = "alluopairs ?U" + let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" + let ?S = "map ?g ?Up" + let ?SS = "map simp_num_pair ?S" + let ?Y = "remdps ?SS" + let ?f= "(\<lambda> (t,n). ?N t / real n)" + let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2" + let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))" + let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y" + from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast + from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp + from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" . + from U_l UpU + have Up_: "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto + hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 " + by (auto simp add: mult_pos_pos) + have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" + proof- + { fix t n assume tnY: "(t,n) \<in> set ?Y" + hence "(t,n) \<in> set ?SS" by simp + hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)" + by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) + then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast + from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto + from simp_num_pair_l[OF tnb np tns] + have "numbound0 t \<and> n > 0" . } + thus ?thesis by blast + qed + + have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))" + proof- + from simp_num_pair_ci[where bs="x#bs"] have + "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto + hence th: "?f o simp_num_pair = ?f" using ext by blast + have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose) + also have "\<dots> = (?f ` set ?S)" by (simp add: th) + also have "\<dots> = ((?f o ?g) ` set ?Up)" + by (simp only: set_map o_def image_compose[symmetric]) + also have "\<dots> = (?h ` (set ?U \<times> set ?U))" + using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast + finally show ?thesis . + qed + have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))" + proof- + { fix t n assume tnY: "(t,n) \<in> set ?Y" + with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto + from usubst_I[OF lq np tnb] + have "bound0 (usubst ?q (t,n))" by simp hence "bound0 (simpfm (usubst ?q (t,n)))" + using simpfm_bound0 by simp} + thus ?thesis by blast + qed + hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto + let ?mp = "minusinf ?q" + let ?pp = "plusinf ?q" + let ?M = "?I x ?mp" + let ?P = "?I x ?pp" + let ?res = "disj ?mp (disj ?pp ?ep)" + from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb + have nbth: "bound0 ?res" by auto + thm rlfm_I[OF simpfm_qf[OF qf]] + + from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm + + have th: "?lhs = (\<exists> x. ?I x ?q)" by auto + from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)" + by (simp only: split_def fst_conv snd_conv) + also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" + using uset_cong[OF lq YU U_l Y_l] by (simp only: split_def fst_conv snd_conv simpfm) + also have "\<dots> = (Ifm (x#bs) ?res)" + using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric] + by (simp add: split_def pair_collapse) + finally have lheq: "?lhs = (Ifm bs (decr ?res))" using decr[OF nbth] by blast + hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def) + by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+ + from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def) + with lr show ?thesis by blast +qed + +constdefs linrqe:: "fm \<Rightarrow> fm" + "linrqe \<equiv> (\<lambda> p. qelim (prep p) ferrack)" + +theorem linrqe: "(Ifm bs (linrqe p) = Ifm bs p) \<and> qfree (linrqe p)" +using ferrack qelim_ci prep +unfolding linrqe_def by auto + +declare max_def [code unfold] + +code_module Ferrack +file "generated_ferrack.ML" +contains linrqe = "linrqe" +test = "%x . linrqe (A(A(Imp (Lt(Sub (Bound 1) (Bound 0))) (E(Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))" + +ML{* use "generated_ferrack.ML"*} +ML "Ferrack.test ()" + +use "linreif.ML" +oracle linr_oracle ("term") = ReflectedFerrack.linrqe_oracle + +use"linrtac.ML" +setup "LinrTac.setup" + +end

--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/HOL/Complex/ex/linreif.ML Tue Jun 05 20:44:12 2007 +0200 @@ -0,0 +1,123 @@ +(* + The oracle for Mixed Real-Integer auantifier elimination + based on the verified Code in ~/work/MIR/MIR.thy +*) + +structure ReflectedFerrack = +struct + +open Ferrack; + +exception LINR; + +(* pseudo reification : term -> intterm *) +val iT = HOLogic.intT; +val rT = Type ("RealDef.real",[]); +val bT = HOLogic.boolT; +val realC = Const("RealDef.real",iT --> rT); +val rzero = Const("0",rT); + +fun num_of_term vs t = + case t of + Free(xn,xT) => (case AList.lookup (op =) vs t of + NONE => error "Variable not found in the list!!" + | SOME n => Bound n) + | Const("RealDef.real",_)$ @{term "0::int"} => C 0 + | Const("RealDef.real",_)$ @{term "1::int"} => C 1 + | @{term "0::real"} => C 0 + | @{term "0::real"} => C 1 + | Term.Bound i => Bound (nat (IntInf.fromInt i)) + | Const(@{const_name "HOL.uminus"},_)$t' => Neg (num_of_term vs t') + | Const (@{const_name "HOL.plus"},_)$t1$t2 => Add (num_of_term vs t1,num_of_term vs t2) + | Const (@{const_name "HOL.minus"},_)$t1$t2 => Sub (num_of_term vs t1,num_of_term vs t2) + | Const (@{const_name "HOL.times"},_)$t1$t2 => + (case (num_of_term vs t1) of C i => + Mul (i,num_of_term vs t2) + | _ => error "num_of_term: unsupported Multiplication") + | Const("RealDef.real",_) $ Const (@{const_name "Numeral.number_of"},_)$t' => C (HOLogic.dest_numeral t') + | Const (@{const_name "Numeral.number_of"},_)$t' => C (HOLogic.dest_numeral t') + | _ => error ("num_of_term: unknown term " ^ (Disp