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src/HOL/Decision_Procs/MIR.thy

author | wenzelm |

Mon, 06 Dec 2021 15:34:54 +0100 | |

changeset 74887 | 56247fdb8bbb |

parent 74406 | ed4149b3d7ab |

permissions | -rw-r--r-- |

discontinued old-style {* verbatim *} tokens;

(* Title: HOL/Decision_Procs/MIR.thy Author: Amine Chaieb *) theory MIR imports Complex_Main Dense_Linear_Order DP_Library "HOL-Library.Code_Target_Numeral" "HOL-Library.Old_Recdef" begin section \<open>Prelude\<close> abbreviation (input) UNION :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> 'b set" where "UNION A f \<equiv> \<Union> (f ` A)" \<comment> \<open>legacy\<close> section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, floor, <)\<close>\<close> declare of_int_floor_cancel [simp del] lemma myle: fixes a b :: "'a::{ordered_ab_group_add}" shows "(a \<le> b) = (0 \<le> b - a)" by (metis add_0_left add_le_cancel_right diff_add_cancel) lemma myless: fixes a b :: "'a::{ordered_ab_group_add}" shows "(a < b) = (0 < b - a)" by (metis le_iff_diff_le_0 less_le_not_le myle) (* Periodicity of dvd *) lemmas dvd_period = zdvd_period (* The Divisibility relation between reals *) definition rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50) where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real_of_int k)" lemma int_rdvd_real: "real_of_int (i::int) rdvd x = (i dvd \<lfloor>x\<rfloor> \<and> real_of_int \<lfloor>x\<rfloor> = x)" (is "?l = ?r") proof assume "?l" hence th: "\<exists> k. x=real_of_int (i*k)" by (simp add: rdvd_def) hence th': "real_of_int \<lfloor>x\<rfloor> = x" by (auto simp del: of_int_mult) with th have "\<exists> k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)" by simp hence "\<exists>k. \<lfloor>x\<rfloor> = i*k" by presburger thus ?r using th' by (simp add: dvd_def) next assume "?r" hence "(i::int) dvd \<lfloor>x::real\<rfloor>" .. hence "\<exists>k. real_of_int \<lfloor>x\<rfloor> = real_of_int (i*k)" by (metis (no_types) dvd_def) thus ?l using \<open>?r\<close> by (simp add: rdvd_def) qed lemma int_rdvd_iff: "(real_of_int (i::int) rdvd real_of_int t) = (i dvd t)" by (auto simp add: rdvd_def dvd_def) (rule_tac x="k" in exI, simp only: of_int_mult[symmetric]) lemma rdvd_abs1: "(\<bar>real_of_int d\<bar> rdvd t) = (real_of_int (d ::int) rdvd t)" proof assume d: "real_of_int d rdvd t" from d int_rdvd_real have d2: "d dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t" by auto from iffD2[OF abs_dvd_iff] d2 have "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" by blast with ti int_rdvd_real[symmetric] have "real_of_int \<bar>d\<bar> rdvd t" by blast thus "\<bar>real_of_int d\<bar> rdvd t" by simp next assume "\<bar>real_of_int d\<bar> rdvd t" hence "real_of_int \<bar>d\<bar> rdvd t" by simp with int_rdvd_real[where i="\<bar>d\<bar>" and x="t"] have d2: "\<bar>d\<bar> dvd \<lfloor>t\<rfloor>" and ti: "real_of_int \<lfloor>t\<rfloor> = t" by auto from iffD1[OF abs_dvd_iff] d2 have "d dvd \<lfloor>t\<rfloor>" by blast with ti int_rdvd_real[symmetric] show "real_of_int d rdvd t" by blast qed lemma rdvd_minus: "(real_of_int (d::int) rdvd t) = (real_of_int 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_of_int (n::int) * real_of_int (k::int) rdvd x * real_of_int k) = (real_of_int n rdvd x)" using knz by (simp add: rdvd_def) (*********************************************************************************) (**** SHADOW SYNTAX AND SEMANTICS ****) (*********************************************************************************) datatype (plugins del: size) 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 instantiation num :: size begin primrec size_num :: "num \<Rightarrow> nat" where "size_num (C c) = 1" | "size_num (Bound n) = 1" | "size_num (Neg a) = 1 + size_num a" | "size_num (Add a b) = 1 + size_num a + size_num b" | "size_num (Sub a b) = 3 + size_num a + size_num b" | "size_num (CN n c a) = 4 + size_num a " | "size_num (CF c a b) = 4 + size_num a + size_num b" | "size_num (Mul c a) = 1 + size_num a" | "size_num (Floor a) = 1 + size_num a" instance .. end (* Semantics of numeral terms (num) *) primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where "Inum bs (C c) = (real_of_int c)" | "Inum bs (Bound n) = bs!n" | "Inum bs (CN n c a) = (real_of_int 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_of_int c) * Inum bs a" | "Inum bs (Floor a) = real_of_int \<lfloor>Inum bs a\<rfloor>" | "Inum bs (CF c a b) = real_of_int c * real_of_int \<lfloor>Inum bs a\<rfloor> + Inum bs b" definition "isint t bs \<equiv> real_of_int \<lfloor>Inum bs t\<rfloor> = Inum bs t" lemma isint_iff: "isint n bs = (real_of_int \<lfloor>Inum bs n\<rfloor> = 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" assume be: "isint e bs" hence efe:"real_of_int \<lfloor>?e\<rfloor> = ?e" by (simp add: isint_iff) have "real_of_int \<lfloor>Inum bs (Mul c e)\<rfloor> = real_of_int \<lfloor>real_of_int (c * \<lfloor>?e\<rfloor>)\<rfloor>" using efe by simp also have "\<dots> = real_of_int (c* \<lfloor>?e\<rfloor>)" by (metis floor_of_int) also have "\<dots> = real_of_int 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_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def) have "real_of_int \<lfloor>?I (Neg e)\<rfloor> = real_of_int \<lfloor>- (real_of_int \<lfloor>?I e\<rfloor>)\<rfloor>" by (simp add: th) also have "\<dots> = - real_of_int \<lfloor>?I e\<rfloor>" by simp 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_of_int \<lfloor>?I e\<rfloor> = ?I e" by (simp add: isint_def) have "real_of_int \<lfloor>?I (Sub (C c) e)\<rfloor> = real_of_int \<lfloor>real_of_int (c - \<lfloor>?I e\<rfloor>)\<rfloor>" by (simp add: th) also have "\<dots> = real_of_int (c - \<lfloor>?I e\<rfloor>)" by simp 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_of_int \<lfloor>?a + ?b\<rfloor> = real_of_int \<lfloor>real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>\<rfloor>" by simp also have "\<dots> = real_of_int \<lfloor>?a\<rfloor> + real_of_int \<lfloor>?b\<rfloor>" 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 (plugins del: size) 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 instantiation fm :: size begin primrec size_fm :: "fm \<Rightarrow> nat" where "size_fm (Not p) = 1 + size_fm p" | "size_fm (And p q) = 1 + size_fm p + size_fm q" | "size_fm (Or p q) = 1 + size_fm p + size_fm q" | "size_fm (Imp p q) = 3 + size_fm p + size_fm q" | "size_fm (Iff p q) = 3 + 2 * (size_fm p + size_fm q)" | "size_fm (E p) = 1 + size_fm p" | "size_fm (A p) = 4 + size_fm p" | "size_fm (Dvd i t) = 2" | "size_fm (NDvd i t) = 2" | "size_fm T = 1" | "size_fm F = 1" | "size_fm (Lt _) = 1" | "size_fm (Le _) = 1" | "size_fm (Gt _) = 1" | "size_fm (Ge _) = 1" | "size_fm (Eq _) = 1" | "size_fm (NEq _) = 1" instance .. end lemma size_fm_pos [simp]: "size p > 0" for p :: fm by (induct p) simp_all (* Semantics of formulae (fm) *) primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where "Ifm bs T \<longleftrightarrow> True" | "Ifm bs F \<longleftrightarrow> False" | "Ifm bs (Lt a) \<longleftrightarrow> Inum bs a < 0" | "Ifm bs (Gt a) \<longleftrightarrow> Inum bs a > 0" | "Ifm bs (Le a) \<longleftrightarrow> Inum bs a \<le> 0" | "Ifm bs (Ge a) \<longleftrightarrow> Inum bs a \<ge> 0" | "Ifm bs (Eq a) \<longleftrightarrow> Inum bs a = 0" | "Ifm bs (NEq a) \<longleftrightarrow> Inum bs a \<noteq> 0" | "Ifm bs (Dvd i b) \<longleftrightarrow> real_of_int i rdvd Inum bs b" | "Ifm bs (NDvd i b) \<longleftrightarrow> \<not> (real_of_int i rdvd Inum bs b)" | "Ifm bs (Not p) \<longleftrightarrow> \<not> (Ifm bs p)" | "Ifm bs (And p q) \<longleftrightarrow> Ifm bs p \<and> Ifm bs q" | "Ifm bs (Or p q) \<longleftrightarrow> Ifm bs p \<or> Ifm bs q" | "Ifm bs (Imp p q) \<longleftrightarrow> (Ifm bs p \<longrightarrow> Ifm bs q)" | "Ifm bs (Iff p q) \<longleftrightarrow> (Ifm bs p \<longleftrightarrow> Ifm bs q)" | "Ifm bs (E p) \<longleftrightarrow> (\<exists>x. Ifm (x # bs) p)" | "Ifm bs (A p) \<longleftrightarrow> (\<forall>x. Ifm (x # bs) p)" fun prep :: "fm \<Rightarrow> fm" where "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" lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p" by (induct p rule: prep.induct) auto (* Quantifier freeness *) fun qfree:: "fm \<Rightarrow> bool" where "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 *) primrec numbound0 :: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where "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) auto 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:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where "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) auto primrec numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" (* substitute a num into a num for Bound 0 *) where "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 primrec subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *) where "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 fun decrnum:: "num \<Rightarrow> num" where "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" fun decr :: "fm \<Rightarrow> fm" where "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 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: decrnum) lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" by (induct p) simp_all fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where "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) 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) auto lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" by (induct p) simp_all definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where "djf f p q = (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))" definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where "evaldjf f ps = 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 apply (induct xs) apply (auto simp add: evaldjf_def djf_def Let_def) apply (case_tac "f a") apply auto done lemma evaldjf_qf: assumes nb: "\<forall> x\<in> set xs. qfree (f x)" shows "qfree (evaldjf f xs)" using nb apply (induct xs) apply (auto simp add: evaldjf_def djf_def Let_def) apply (case_tac "f a") apply auto done fun disjuncts :: "fm \<Rightarrow> fm list" where "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" | "disjuncts F = []" | "disjuncts p = [p]" fun conjuncts :: "fm \<Rightarrow> fm list" where "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)" | "conjuncts T = []" | "conjuncts p = [p]" lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm bs q) = Ifm bs p" by (induct p rule: conjuncts.induct) auto 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 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where "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 *) fun bnds:: "num \<Rightarrow> nat list" where "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 = []" fun lex_ns:: "nat list \<Rightarrow> nat list \<Rightarrow> bool" where "lex_ns [] ms = True" | "lex_ns ns [] = False" | "lex_ns (n#ns) (m#ms) = (n<m \<or> ((n = m) \<and> lex_ns ns ms)) " definition lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool" where "lex_bnd t s \<equiv> lex_ns (bnds t) (bnds s)" fun maxcoeff:: "num \<Rightarrow> int" where "maxcoeff (C i) = \<bar>i\<bar>" | "maxcoeff (CN n c t) = max \<bar>c\<bar> (maxcoeff t)" | "maxcoeff (CF c t s) = max \<bar>c\<bar> (maxcoeff s)" | "maxcoeff t = 1" lemma maxcoeff_pos: "maxcoeff t \<ge> 0" by (induct t rule: maxcoeff.induct) auto fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where "numgcdh (C i) = (\<lambda>g. gcd i g)" | "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))" | "numgcdh (CF c s t) = (\<lambda>g. gcd c (numgcdh t g))" | "numgcdh t = (\<lambda>g. 1)" definition numgcd :: "num \<Rightarrow> int" where "numgcd t = numgcdh t (maxcoeff t)" fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where "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)" definition reducecoeff :: "num \<Rightarrow> num" where "reducecoeff t = (let g = numgcd t in if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where "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 dvd_trans[OF gdg]) declare dvd_trans [trans add] 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) 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 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_of_int 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 assms 1 show ?case by (simp add: real_of_int_div[OF gd]) next case (2 n c t) hence gd: "g dvd c" by simp from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps) next case (3 c s t) hence gd: "g dvd c" by simp from assms 3 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps) qed (auto simp add: numgcd_def gp) fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where "ismaxcoeff (C i) = (\<lambda> x. \<bar>i\<bar> \<le> x)" | "ismaxcoeff (CN n c t) = (\<lambda>x. \<bar>c\<bar> \<le> x \<and> (ismaxcoeff t x))" | "ismaxcoeff (CF c s t) = (\<lambda>x. \<bar>c\<bar> \<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 \<bar>c\<bar> (maxcoeff t)" by simp from ismaxcoeff_mono[OF H thh] show ?case by simp 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 qed simp_all lemma zgcd_gt1: "\<bar>i\<bar> > 1 \<and> \<bar>j\<bar> > 1 \<or> \<bar>i\<bar> = 0 \<and> \<bar>j\<bar> > 1 \<or> \<bar>i\<bar> > 1 \<and> \<bar>j\<bar> = 0" if "gcd i j > 1" for i j :: int proof - have "\<bar>k\<bar> \<le> 1 \<longleftrightarrow> k = - 1 \<or> k = 0 \<or> k = 1" for k :: int by auto with that show ?thesis by (auto simp add: not_less) qed lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0" by (induct t rule: numgcdh.induct) auto lemma dvdnumcoeff_aux: assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1" shows "dvdnumcoeff t (numgcdh t m)" using assms proof(induct t rule: numgcdh.induct) case (2 n c t) let ?g = "numgcdh t m" from 2 have th:"gcd c ?g > 1" by simp from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 2 have th: "dvdnumcoeff t ?g" by simp have th': "gcd c ?g dvd ?g" by simp from dvdnumcoeff_trans[OF th' th] have ?case by simp } moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1" with 2 have th: "dvdnumcoeff t ?g" by simp have th': "gcd c ?g dvd ?g" by simp from dvdnumcoeff_trans[OF th' th] have ?case by simp hence ?case by simp } moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0" from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp } ultimately show ?case by blast next case (3 c s t) let ?g = "numgcdh t m" from 3 have th:"gcd c ?g > 1" by simp from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] have "(\<bar>c\<bar> > 1 \<and> ?g > 1) \<or> (\<bar>c\<bar> = 0 \<and> ?g > 1) \<or> (\<bar>c\<bar> > 1 \<and> ?g = 0)" by simp moreover {assume "\<bar>c\<bar> > 1" and gp: "?g > 1" with 3 have th: "dvdnumcoeff t ?g" by simp have th': "gcd c ?g dvd ?g" by simp from dvdnumcoeff_trans[OF th' th] have ?case by simp } moreover {assume "\<bar>c\<bar> = 0 \<and> ?g > 1" with 3 have th: "dvdnumcoeff t ?g" by simp have th': "gcd c ?g dvd ?g" by simp from dvdnumcoeff_trans[OF th' th] have ?case by simp hence ?case by simp } moreover {assume "\<bar>c\<bar> > 1" and g0:"?g = 0" from numgcdh0[OF g0] have "m=0". with 3 g0 have ?case by simp } ultimately show ?case by blast qed auto lemma dvdnumcoeff_aux2: assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0" using assms 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_of_int (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 numadd:: "num \<times> 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)" by (induct t s rule: numadd.induct) (simp_all add: Let_def algebra_simps add_eq_0_iff) lemma numadd_nb [simp]: "numbound0 t \<Longrightarrow> numbound0 s \<Longrightarrow> numbound0 (numadd (t, s))" by (induct t s rule: numadd.induct) (simp_all add: Let_def) fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where "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: algebra_simps) lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)" by (induct t rule: nummul.induct) auto definition numneg :: "num \<Rightarrow> num" where "numneg t \<equiv> nummul t (- 1)" definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where "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 fun split_int:: "num \<Rightarrow> num \<times> num" where "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 2(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ from 2(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def) from 2(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 3(1) have b:"Inum bs (Add ?bv ?bi) = Inum bs b" and bii: "isint ?bi bs" by blast+ from 3(2) have tibi: "ti = CF c a ?bi" by (simp add: Let_def split_def) from 3(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 algebra_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) definition numfloor:: "num \<Rightarrow> num" where "numfloor t = (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) 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_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)" 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_of_int \<lfloor>?N (Add ?tv ?ti)\<rfloor>" by simp also have "\<dots> = real_of_int (\<lfloor>?N ?tv\<rfloor> + \<lfloor>?N ?ti\<rfloor>)" 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) } 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) fun simpnum:: "num \<Rightarrow> num" where "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 fun nozerocoeff:: "num \<Rightarrow> bool" where "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 \<bar>c\<bar> (maxcoeff t) = 0" by simp+ have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith with 2 show ?case by simp next case (3 c s t) hence cnz: "c \<noteq>0" and mx: "max \<bar>c\<bar> (maxcoeff t) = 0" by simp+ have "max \<bar>c\<bar> (maxcoeff t) \<ge> \<bar>c\<bar>" by simp with cnz have "max \<bar>c\<bar> (maxcoeff t) > 0" by arith with 3 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 definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where "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' = gcd 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_of_int n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real_of_int n) (t,n))" (is "?lhs = ?rhs") proof- let ?t' = "simpnum t" let ?g = "numgcd ?t'" let ?g' = "gcd 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 g1 nnz have gp0: "?g' \<noteq> 0" by simp hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="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 have gpdd: "?g' dvd n" by simp have gpdgp: "?g' dvd ?g'" by simp from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] have th2:"real_of_int ?g' * ?t = Inum bs ?t'" by simp from nnz g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')" by (simp add: simp_num_pair_def Let_def) also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))" by simp also have "\<dots> = (Inum bs ?t' / real_of_int n)" using real_of_int_div[OF gpdd] th2 gp0 by simp finally have "?lhs = Inum bs t / real_of_int n" by simp then have ?thesis using nnz g1 g'1 by (simp add: simp_num_pair_def) } ultimately have ?thesis by auto } 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' = "gcd n ?g" { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) } moreover { assume nnz: "n \<noteq> 0" {assume "\<not> ?g > 1" hence ?thesis using assms by (auto simp add: Let_def simp_num_pair_def) } moreover {assume g1:"?g>1" hence g0: "?g > 0" by simp from g1 nnz have gp0: "?g' \<noteq> 0" by simp hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith hence "?g'= 1 \<or> ?g' > 1" by arith moreover {assume "?g'=1" hence ?thesis using assms g1 g0 by (auto simp add: Let_def simp_num_pair_def) } moreover {assume g'1:"?g'>1" have gpdg: "?g' dvd ?g" by simp have gpdd: "?g' dvd n" by simp 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 div_self[OF gp0]] have "n div ?g' >0" by simp hence ?thesis using assms g1 g'1 by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)} ultimately have ?thesis by auto } ultimately have ?thesis by blast } ultimately show ?thesis by blast qed fun not:: "fm \<Rightarrow> fm" where "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 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where "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 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where "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 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where "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) definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where "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 iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)" by (unfold iff_def,cases "p=q", auto) fun check_int:: "num \<Rightarrow> bool" where "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_of_int \<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_of_int (d::int) rdvd real_of_int (c::int)*t = (real_of_int (d div g) rdvd real_of_int (c div g)*t)" proof assume d: "real_of_int d rdvd real_of_int c * t" from d rdvd_def obtain k where k_def: "real_of_int c * t = real_of_int d* real_of_int (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_of_int g * real_of_int kc * t = real_of_int g * real_of_int kd * real_of_int k" by simp hence "real_of_int kc * t = real_of_int kd * real_of_int k" using gp by simp hence th:"real_of_int kd rdvd real_of_int 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_of_int (d div g) rdvd real_of_int (c div g) * t" by simp next assume d: "real_of_int (d div g) rdvd real_of_int (c div g) * t" from gp have gnz: "g \<noteq> 0" by simp thus "real_of_int d rdvd real_of_int c * t" using d rdvd_mult[OF gnz, where n="d div g" and x="real_of_int (c div g) * t"] real_of_int_div[OF gd] real_of_int_div[OF gc] by simp qed definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where "simpdvd d t \<equiv> (let g = numgcd t in if g > 1 then (let g' = gcd 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' = "gcd 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 g1 dnz have gp0: "?g' \<noteq> 0" by simp hence g'p: "?g' > 0" using gcd_ge_0_int[where x="d" and y="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 have gpdd: "?g' dvd d" by simp have gpdgp: "?g' dvd ?g'" by simp from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] have th2:"real_of_int ?g' * ?t = Inum bs t" by simp from assms g1 g0 g'1 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_of_int d rdvd (Inum bs t))" using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]] th2[symmetric] by simp finally have ?thesis by simp } ultimately have ?thesis by auto } ultimately show ?thesis by blast qed fun simpfm :: "fm \<Rightarrow> fm" where "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 (\<bar>i\<bar> = 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 (\<bar>i\<bar> = 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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a < 0 = (real_of_int ?g * ?r < real_of_int ?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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a \<le> 0 = (real_of_int ?g * ?r \<le> real_of_int ?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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a > 0 = (real_of_int ?g * ?r > real_of_int ?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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a \<ge> 0 = (real_of_int ?g * ?r \<ge> real_of_int ?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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a = 0 = (real_of_int ?g * ?r = 0)" by simp also have "\<dots> = (?r = 0)" using gp by simp 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_of_int ?g > 0" by simp have "Inum bs ?sa = real_of_int ?g* ?r" by (simp add: reducecoeff) with sa have "Inum bs a \<noteq> 0 = (real_of_int ?g * ?r \<noteq> 0)" by simp also have "\<dots> = (?r \<noteq> 0)" using gp by simp 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> (\<bar>i\<bar> = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((\<bar>i\<bar> \<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: "\<bar>i\<bar> = 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: "(\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)" {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond by (cases "\<bar>i\<bar> = 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> (\<bar>i\<bar> = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((\<bar>i\<bar> \<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: "\<bar>i\<bar> = 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: "(\<bar>i\<bar> \<noteq> 1) \<or> (\<not> check_int a)" {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond by (cases "\<bar>i\<bar> = 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 *) definition list_conj :: "fm list \<Rightarrow> fm" where "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) definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = List.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 (List.partition bound0 ?cjs)" let ?no = "snd (List.partition bound0 ?cjs)" let ?cno = "list_conj ?no" let ?cyes = "list_conj ?yes" have part: "List.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 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 fix y 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] simp del: partition_filter_conv) 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 fun qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where "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) text \<open>The \<open>\<int>\<close> Part\<close> text\<open>Linearity for fm where Bound 0 ranges over \<open>\<int>\<close>\<close> fun zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*) where "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'))" lemma zsplit0_I: shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real_of_int (x::int)) #bs) (CN 0 n a) = Inum (real_of_int 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 5 by (simp add: Let_def split_def) from abj 5 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 6 by (simp add: Let_def split_def) from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*) with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast from abjs 6 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: distrib_right) 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 7 by (simp add: Let_def split_def) from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*) with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast from abjs 7 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 8 by (simp add: Let_def split_def) from abj 8 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast hence "?I x (Mul i t) = (real_of_int i) * ?I x (CN 0 ?nt ?at)" by simp also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: distrib_left) 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 9 by (simp add: Let_def split_def) from abj 9 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_of_int ?nt)*(real_of_int x) = real_of_int (?nt * x)" by simp have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp also have "\<dots> = real_of_int \<lfloor>real_of_int ?nt * real_of_int x + ?I x ?at\<rfloor>" by simp also have "\<dots> = real_of_int \<lfloor>?I x ?at + real_of_int (?nt * x)\<rfloor>" by (simp add: ac_simps) also have "\<dots> = real_of_int (\<lfloor>?I x ?at\<rfloor> + (?nt * x))" by (simp add: of_int_mult[symmetric] del: of_int_mult) also have "\<dots> = real_of_int (?nt)*(real_of_int x) + real_of_int \<lfloor>?I x ?at\<rfloor>" by (simp add: ac_simps) finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp with na show ?case by simp qed fun iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool" (* Linearity test for fm *) where "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) fun zlfm :: "fm \<Rightarrow> fm" (* Linearity transformation for fm *) where "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 \<bar>i\<bar> r else if c>0 then And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (CN 0 c (Floor r))) else And (Eq (Sub (Floor r) r)) (Dvd \<bar>i\<bar> (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 \<bar>i\<bar> r else if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (CN 0 c (Floor r))) else Or (NEq (Sub (Floor r) r)) (NDvd \<bar>i\<bar> (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" lemma split_int_less_real: "(real_of_int (a::int) < b) = (a < \<lfloor>b\<rfloor> \<or> (a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b))" proof( auto) assume alb: "real_of_int a < b" and agb: "\<not> a < \<lfloor>b\<rfloor>" from agb have "\<lfloor>b\<rfloor> \<le> a" by simp hence th: "b < real_of_int a + 1" by (simp only: floor_le_iff) from floor_eq[OF alb th] show "a = \<lfloor>b\<rfloor>" by simp next assume alb: "a < \<lfloor>b\<rfloor>" hence "real_of_int a < real_of_int \<lfloor>b\<rfloor>" by simp moreover have "real_of_int \<lfloor>b\<rfloor> \<le> b" by simp ultimately show "real_of_int a < b" by arith qed lemma split_int_less_real': "(real_of_int (a::int) + b < 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> < 0 \<or> (real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0))" proof- have "(real_of_int a + b <0) = (real_of_int 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_of_int (a::int) + b > 0) = (real_of_int a + real_of_int \<lfloor>b\<rfloor> > 0 \<or> (real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0))" proof- have th: "(real_of_int a + b >0) = (real_of_int (-a) + (-b)< 0)" by arith show ?thesis by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) (auto simp add: algebra_simps) qed lemma split_int_le_real: "(real_of_int (a::int) \<le> b) = (a \<le> \<lfloor>b\<rfloor> \<or> (a = \<lfloor>b\<rfloor> \<and> real_of_int \<lfloor>b\<rfloor> < b))" proof( auto) assume alb: "real_of_int a \<le> b" and agb: "\<not> a \<le> \<lfloor>b\<rfloor>" from alb have "\<lfloor>real_of_int a\<rfloor> \<le> \<lfloor>b\<rfloor>" by (simp only: floor_mono) hence "a \<le> \<lfloor>b\<rfloor>" by simp with agb show "False" by simp next assume alb: "a \<le> \<lfloor>b\<rfloor>" hence "real_of_int a \<le> real_of_int \<lfloor>b\<rfloor>" by (simp only: floor_mono) also have "\<dots>\<le> b" by simp finally show "real_of_int a \<le> b" . qed lemma split_int_le_real': "(real_of_int (a::int) + b \<le> 0) = (real_of_int a - real_of_int \<lfloor>- b\<rfloor> \<le> 0 \<or> (real_of_int a - real_of_int \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b < 0))" proof- have "(real_of_int a + b \<le>0) = (real_of_int 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_of_int (a::int) + b \<ge> 0) = (real_of_int a + real_of_int \<lfloor>b\<rfloor> \<ge> 0 \<or> (real_of_int a + real_of_int \<lfloor>b\<rfloor> = 0 \<and> real_of_int \<lfloor>b\<rfloor> - b < 0))" proof- have th: "(real_of_int a + b \<ge>0) = (real_of_int (-a) + (-b) \<le> 0)" by arith show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"]) (simp add: algebra_simps ,arith) qed lemma split_int_eq_real: "(real_of_int (a::int) = b) = ( a = \<lfloor>b\<rfloor> \<and> b = real_of_int \<lfloor>b\<rfloor>)" (is "?l = ?r") by auto lemma split_int_eq_real': "(real_of_int (a::int) + b = 0) = ( a - \<lfloor>- b\<rfloor> = 0 \<and> real_of_int \<lfloor>- b\<rfloor> + b = 0)" (is "?l = ?r") proof- have "?l = (real_of_int 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_of_int i #bs) (zlfm p) = Ifm (real_of_int i# bs) p) \<and> iszlfm (zlfm p) (real_of_int (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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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,rename_tac nat a b,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_of_int (?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) 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_of_int (?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 ac_simps, 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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, rename_tac nat a b, 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_of_int (?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) 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_of_int (?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 ac_simps, 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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, rename_tac nat a b, 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_of_int (?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) 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_of_int (?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 ac_simps, 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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, rename_tac nat a b, 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_of_int (?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) 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_of_int (?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 ac_simps, 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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, rename_tac nat a b, 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_of_int (?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 of_int_mult[symmetric] del: 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_of_int (?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 of_int_mult[symmetric] del: 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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, rename_tac nat a b, 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_of_int (?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 of_int_mult[symmetric] del: 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_of_int (?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 of_int_mult[symmetric] del: 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) hence ?case using 11 j 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, rename_tac nat a b, 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_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" using Ia by (simp add: Let_def split_def) also have "\<dots> = (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r))" by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp also have "\<dots> = (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> (real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r))))" by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz by (simp add: Let_def split_def int_rdvd_iff[symmetric] del: of_int_mult) (auto simp add: ac_simps) 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_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" using Ia by (simp add: Let_def split_def) also have "\<dots> = (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r))" by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp also have "\<dots> = (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> (real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r))))" by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz using rdvd_minus [where d="\<bar>j\<bar>" and t="real_of_int (?c*i + \<lfloor>?N ?r\<rfloor>)", simplified, symmetric] by (simp add: Let_def split_def int_rdvd_iff[symmetric] del: of_int_mult) (auto simp add: ac_simps) 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_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto let ?N = "\<lambda> t. Inum (real_of_int 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: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) hence ?case using 12 j 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, rename_tac nat a b, 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_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" using Ia by (simp add: Let_def split_def) also have "\<dots> = (\<not> (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r)))" by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp also have "\<dots> = (\<not> (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> (real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r)))))" by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz by (simp add: Let_def split_def int_rdvd_iff[symmetric] del: of_int_mult) (auto simp add: ac_simps) 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_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" using Ia by (simp add: Let_def split_def) also have "\<dots> = (\<not> (real_of_int \<bar>j\<bar> rdvd real_of_int (?c*i) + (?N ?r)))" by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp also have "\<dots> = (\<not> (\<bar>j\<bar> dvd \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> \<and> (real_of_int \<lfloor>(?N ?r) + real_of_int (?c*i)\<rfloor> = (real_of_int (?c*i) + (?N ?r)))))" by(simp only: int_rdvd_real[where i="\<bar>j\<bar>" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps) also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz using rdvd_minus [where d="\<bar>j\<bar>" and t="real_of_int (?c*i + \<lfloor>?N ?r\<rfloor>)", simplified, symmetric] by (simp add: Let_def split_def int_rdvd_iff[symmetric] del: of_int_mult) (auto simp add: ac_simps) finally have ?case using l jnz by blast } ultimately show ?case by blast qed auto text\<open>plusinf : Virtual substitution of \<open>+\<infinity>\<close> minusinf: Virtual substitution of \<open>-\<infinity>\<close> \<open>\<delta>\<close> Compute lcm \<open>d| Dvd d c*x+t \<in> p\<close> \<open>d_\<delta>\<close> checks if a given l divides all the ds above\<close> fun minusinf:: "fm \<Rightarrow> fm" where "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) fun plusinf:: "fm \<Rightarrow> fm" where "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" fun \<delta> :: "fm \<Rightarrow> int" where "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)" | "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)" | "\<delta> (Dvd i (CN 0 c e)) = i" | "\<delta> (NDvd i (CN 0 c e)) = i" | "\<delta> p = 1" fun d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" where "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: dvd_trans[of "i" "d" "d'"]) next case (10 i c e) thus ?case using d by (simp add: dvd_trans[of "i" "d" "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 1 lcm_pos_int have dp: "?d >0" by simp have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp hence th: "d_\<delta> p ?d" using delta_mono 1 by (simp only: iszlfm.simps) blast have "\<delta> q dvd \<delta> (And p q)" using 1 by simp hence th': "d_\<delta> q ?d" using delta_mono 1 by (simp only: iszlfm.simps) blast from th th' dp show ?case by simp next case (2 p q) let ?d = "\<delta> (And p q)" from 2 lcm_pos_int have dp: "?d >0" by simp have "\<delta> p dvd \<delta> (And p q)" using 2 by simp hence th: "d_\<delta> p ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast have "\<delta> q dvd \<delta> (And p q)" using 2 by simp hence th': "d_\<delta> q ?d" using delta_mono 2 by (simp only: iszlfm.simps) blast 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_of_int x)#bs) (minusinf p) = Ifm ((real_of_int 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) then have "?P f" by simp then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast with 1 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) then have "?P f" by simp then obtain z1 where z1_def: "\<forall> x < z1. ?I x (?M f) = ?I x f" by blast with 2 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) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 3 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos by simp thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] by simp qed thus ?case by blast next case (4 c e) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 4 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos by simp thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] by simp qed thus ?case by blast next case (5 c e) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 5 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp qed thus ?case by blast next case (6 c e) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 6 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp qed thus ?case by blast next case (7 c e) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 7 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) thus "\<not> (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e>0)" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp qed thus ?case by blast next case (8 c e) then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp from 8 have nbe: "numbound0 e" by simp fix y have "\<forall> x < \<lfloor>- (Inum (y#bs) e) / (real_of_int c)\<rfloor>. ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))" proof (simp add: less_floor_iff , rule allI, rule impI) fix x :: int assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)" hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp with rcpos have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))" by (simp only: mult_strict_left_mono [OF th1 rcpos]) thus "\<not> real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0" using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int 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_of_int(x - k*d))#bs) (minusinf p) = Ifm (real_of_int 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_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI) assume "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int 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_of_int l)" by (simp add: rdvd_def) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" by (simp add: algebra_simps di_def) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))" by (simp add: algebra_simps) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp next assume "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e") hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" by (simp add: rdvd_def) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" by simp hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)" by (simp add: di_def) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))" by (simp add: algebra_simps) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)" by blast thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int 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_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI) assume "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int 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_of_int l)" by (simp add: rdvd_def) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" by (simp add: algebra_simps di_def) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))" by (simp add: algebra_simps) hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp next assume "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e") hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" by (simp add: rdvd_def) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" by simp hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int i * real_of_int di)" by (simp add: di_def) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int (l - c*k*di))" by (simp add: algebra_simps) hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l)" by blast thus "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e" using rdvd_def by simp qed qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int(x - k*d)" and b'="real_of_int x"] simp del: of_int_mult of_int_diff) lemma minusinf_ex: assumes lin: "iszlfm p (real_of_int (a::int) #bs)" and exmi: "\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)" (is "\<exists> x. ?P1 x") shows "\<exists> (x::int). Ifm (real_of_int 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 lemma minusinf_bex: assumes lin: "iszlfm p (real_of_int (a::int) #bs)" shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)) = (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real_of_int 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 periodic_finite_ex[OF dpos th1] show ?thesis by blast qed lemma dvd1_eq1: "x > 0 \<Longrightarrow> is_unit x \<longleftrightarrow> x = 1" for x :: int by simp fun a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coefficients of a formula *) where "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)" fun d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*) where "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)" fun \<zeta> :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*) where "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)" | "\<zeta> (Or p q) = lcm (\<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" fun \<beta> :: "fm \<Rightarrow> num list" where "\<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 = []" fun \<alpha> :: "fm \<Rightarrow> num list" where "\<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 = []" fun mirror :: "fm \<Rightarrow> fm" where "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_of_int (i::int)#bs)) ` set (\<alpha> p) = (Inum (real_of_int 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_of_int (x::int)#bs) (mirror p) = Ifm (real_of_int (- x)#bs) p" using lp proof(induct p rule: iszlfm.induct) case (9 j c e) have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) = (real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))" by (simp only: rdvd_minus[symmetric]) from 9 th show ?case by (simp add: algebra_simps numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"]) next case (10 j c e) have th: "(real_of_int j rdvd real_of_int c * real_of_int x - Inum (real_of_int x # bs) e) = (real_of_int j rdvd - (real_of_int c * real_of_int x - Inum (real_of_int x # bs) e))" by (simp only: rdvd_minus[symmetric]) from 10 th show ?case by (simp add: algebra_simps numbound0_I[where bs="bs" and b'="real_of_int x" and b="- real_of_int x"]) qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int x" and b'="- real_of_int x"]) 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_of_int (i::int)#bs)" shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real_of_int 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 dvd_trans[of _ "l" "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) then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] dl1 dl2 show ?case by (auto simp add: lcm_pos_int) next case (2 p q) then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] dl1 dl2 show ?case by (auto simp add: lcm_pos_int) qed (auto simp add: lcm_pos_int) 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_of_int (l * x) #bs) (a_\<beta> p l) = Ifm ((real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) < (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0)" using mult_less_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<le> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0)" using mult_le_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) > (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e > 0)" using zero_less_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<ge> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0)" using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) = (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = 0)" using mult_eq_0_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> (0::real)) = (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> 0)" by simp also have "\<dots> = (real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e) \<noteq> (real_of_int (l div c)) * 0)" by (simp add: algebra_simps) also have "\<dots> = (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0)" using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e"] ldcp by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)" by simp also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps) also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)" using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int 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: div_self[OF cnz]) have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp hence cl:"c * (l div c) =l" using mult_div_mod_eq [where a="l" and b="c"] by simp hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)" by simp also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps) also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)" using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def be isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp qed (simp_all add: numbound0_I[where bs="bs" and b="real_of_int (l * x)" and b'="real_of_int x"] isint_Mul del: 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_of_int x #bs) (a_\<beta> p l)) = (\<exists> (x::int). Ifm (real_of_int 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_of_int x = b + real_of_int j)" and p: "Ifm (real_of_int 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_all with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 5 show ?case by (simp del: of_int_minus) next case (6 c e) hence c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp_all with dp p c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] 6 show ?case by (simp del: of_int_minus) next case (7 c e) hence p: "Ifm (real_of_int 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_all let ?e = "Inum (real_of_int x # bs) e" from ie1 have ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="a#bs"] numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"] by (simp add: isint_iff) {assume "real_of_int (x-d) +?e > 0" hence ?case using c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] by (simp del: of_int_minus)} moreover {assume H: "\<not> real_of_int (x-d) + ?e > 0" let ?v="Neg e" have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x = - ?e + real_of_int j)" by auto from H p have "real_of_int x + ?e > 0 \<and> real_of_int x + ?e \<le> real_of_int d" by (simp add: c1) hence "real_of_int (x + \<lfloor>?e\<rfloor>) > real_of_int (0::int) \<and> real_of_int (x + \<lfloor>?e\<rfloor>) \<le> real_of_int d" using ie by simp hence "x + \<lfloor>?e\<rfloor> \<ge> 1 \<and> x + \<lfloor>?e\<rfloor> \<le> d" by simp hence "\<exists> (j::int) \<in> {1 .. d}. j = x + \<lfloor>?e\<rfloor>" by simp hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = real_of_int (- \<lfloor>?e\<rfloor> + j)" by force hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = - ?e + real_of_int 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_of_int 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_of_int x # bs) e" from ie1 have ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"] by (simp add: isint_iff) {assume "real_of_int (x-d) +?e \<ge> 0" hence ?case using c1 numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] by (simp del: of_int_minus)} moreover {assume H: "\<not> real_of_int (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 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x = - ?e - 1 + real_of_int j)" by auto from H p have "real_of_int x + ?e \<ge> 0 \<and> real_of_int x + ?e < real_of_int d" by (simp add: c1) hence "real_of_int (x + \<lfloor>?e\<rfloor>) \<ge> real_of_int (0::int) \<and> real_of_int (x + \<lfloor>?e\<rfloor>) < real_of_int d" using ie by simp hence "x + \<lfloor>?e\<rfloor> + 1 \<ge> 1 \<and> x + \<lfloor>?e\<rfloor> + 1 \<le> d" by simp hence "\<exists> (j::int) \<in> {1 .. d}. j = x + \<lfloor>?e\<rfloor> + 1" by simp hence "\<exists> (j::int) \<in> {1 .. d}. x= - \<lfloor>?e\<rfloor> - 1 + j" by (simp add: algebra_simps) hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= real_of_int (- \<lfloor>?e\<rfloor> - 1 + j)" by presburger hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= - ?e - 1 + real_of_int 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_of_int 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_of_int 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_of_int x= - ?e" by (simp add: c1) with 3(5) show ?case using dp by simp (erule ballE[where x="1"], simp_all add:algebra_simps numbound0_I[OF bn,where b="real_of_int x"and b'="a"and bs="bs"]) next case (4 c e)hence p: "Ifm (real_of_int 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_of_int x # bs) e" let ?v="Neg e" have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp {assume "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e \<noteq> 0" hence ?case by (simp add: c1)} moreover {assume H: "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e = 0" hence "real_of_int x = - Inum ((real_of_int (x -d)) # bs) e + real_of_int d" by simp hence "real_of_int x = - Inum (a # bs) e + real_of_int d" by (simp add: numbound0_I[OF bn,where b="real_of_int x - real_of_int d"and b'="a"and bs="bs"]) with 4(5) have ?case using dp by simp} ultimately show ?case by blast next case (9 j c e) hence p: "Ifm (real_of_int 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_of_int x # bs) e" from 9 have "isint e (a #bs)" by simp hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="(real_of_int x)#bs"] numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] by (simp add: isint_iff) from 9 have id: "j dvd d" by simp from c1 ie[symmetric] have "?p x = (real_of_int j rdvd real_of_int (x + \<lfloor>?e\<rfloor>))" by simp also have "\<dots> = (j dvd x + \<lfloor>?e\<rfloor>)" using int_rdvd_real[where i="j" and x="real_of_int (x + \<lfloor>?e\<rfloor>)"] by simp also have "\<dots> = (j dvd x - d + \<lfloor>?e\<rfloor>)" using dvd_period[OF id, where x="x" and c="-1" and t="\<lfloor>?e\<rfloor>"] by simp also have "\<dots> = (real_of_int j rdvd real_of_int (x - d + \<lfloor>?e\<rfloor>))" using int_rdvd_real[where i="j" and x="real_of_int (x - d + \<lfloor>?e\<rfloor>)",symmetric, simplified] ie by simp also have "\<dots> = (real_of_int j rdvd real_of_int x - real_of_int d + ?e)" using ie by simp finally show ?case using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp next case (10 j c e) hence p: "Ifm (real_of_int 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_of_int x # bs) e" from 10 have "isint e (a#bs)" by simp hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"] by (simp add: isint_iff) from 10 have id: "j dvd d" by simp from c1 ie[symmetric] have "?p x = (\<not> real_of_int j rdvd real_of_int (x + \<lfloor>?e\<rfloor>))" by simp also have "\<dots> = (\<not> j dvd x + \<lfloor>?e\<rfloor>)" using int_rdvd_real[where i="j" and x="real_of_int (x + \<lfloor>?e\<rfloor>)"] by simp also have "\<dots> = (\<not> j dvd x - d + \<lfloor>?e\<rfloor>)" using dvd_period[OF id, where x="x" and c="-1" and t="\<lfloor>?e\<rfloor>"] by simp also have "\<dots> = (\<not> real_of_int j rdvd real_of_int (x - d + \<lfloor>?e\<rfloor>))" using int_rdvd_real[where i="j" and x="real_of_int (x - d + \<lfloor>?e\<rfloor>)",symmetric, simplified] ie by simp also have "\<dots> = (\<not> real_of_int j rdvd real_of_int x - real_of_int d + ?e)" using ie by simp finally show ?case using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int (x - d)" and b'="real_of_int x"] simp del: 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_of_int j) #bs) p) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (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_of_int x = b + real_of_int 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) lemma cpmi_eq: "0 < D \<Longrightarrow> (\<exists>z::int. \<forall>x. x < z \<longrightarrow> (P x = P1 x)) \<Longrightarrow> \<forall>x. \<not>(\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. P(b+j)) \<longrightarrow> P (x) \<longrightarrow> P (x - D) \<Longrightarrow> (\<forall>(x::int). \<forall>(k::int). ((P1 x)= (P1 (x-k*D)))) \<Longrightarrow> (\<exists>(x::int). P(x)) = ((\<exists>(j::int) \<in> {1..D} . (P1(j))) | (\<exists>(j::int) \<in> {1..D}. \<exists>(b::int) \<in> B. P (b+j)))" apply(rule iffI) prefer 2 apply(drule minusinfinity) apply assumption+ apply(fastforce) apply clarsimp apply(subgoal_tac "\<And>k. 0<=k \<Longrightarrow> \<forall>x. P x \<longrightarrow> P (x - k*D)") apply(frule_tac x = x and z=z in decr_lemma) apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)") prefer 2 apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)") prefer 2 apply arith apply fastforce apply(drule (1) periodic_finite_ex) apply blast apply(blast dest:decr_mult_lemma) done 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_of_int x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real_of_int j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p))" (is "(\<exists> (x::int). ?P (real_of_int x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real_of_int j)))") proof- from minusinf_inf[OF lp] have th: "\<exists>(z::int). \<forall>x<z. ?P (real_of_int x) = ?M x" by blast let ?B' = "{\<lfloor>?I b\<rfloor> | b. b\<in> ?B}" from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real_of_int \<lfloor>?I b\<rfloor> = ?I b" by simp from B[rule_format] have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int \<lfloor>?I b\<rfloor> + real_of_int j))" by simp also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (\<lfloor>?I b\<rfloor> + j)))" by simp also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" by blast finally have BB': "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" by blast hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j))) \<longrightarrow> ?P (real_of_int x) \<longrightarrow> ?P (real_of_int (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 *) fun \<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy and Loveland Bset*) where "\<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 = []" fun \<sigma>_\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy and Loveland*) where "\<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)" fun \<alpha>_\<rho> :: "fm \<Rightarrow> (num \<times> int) list" where "\<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 *) definition \<sigma> :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where "\<sigma> p k t \<equiv> And (Dvd k t) (\<sigma>_\<rho> p (t,k))" lemma \<sigma>_\<rho>: assumes linp: "iszlfm p (real_of_int (x::int)#bs)" and kpos: "real_of_int k > 0" and tnb: "numbound0 t" and tint: "isint t (real_of_int x#bs)" and kdt: "k dvd \<lfloor>Inum (b'#bs) t\<rfloor>" shows "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (t,k)) = (Ifm ((real_of_int (\<lfloor>Inum (b'#bs) t\<rfloor> div k))#bs) p)" (is "?I (real_of_int x) (?s p) = (?I (real_of_int (\<lfloor>?N b' t\<rfloor> div k)) p)" is "_ = (?I ?tk p)") using linp kpos tnb proof(induct p rule: \<sigma>_\<rho>.induct) case (3 c e) from 3 have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from kpos have knz': "real_of_int k \<noteq> 0" by simp from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Eq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k = 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (Eq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (4 c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from kpos have knz': "real_of_int k \<noteq> 0" by simp from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (NEq (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<noteq> 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (NEq (CN 0 c e)))" using nonzero_eq_divide_eq[OF knz', where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (5 c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Lt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k < 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (Lt (CN 0 c e)))" using pos_less_divide_eq[OF kpos, where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (6 c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Le (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<le> 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (Le (CN 0 c e)))" using pos_le_divide_eq[OF kpos, where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (7 c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Gt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k > 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (Gt (CN 0 c e)))" using pos_divide_less_eq[OF kpos, where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (8 c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Ge (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<ge> 0)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_simps) also have "\<dots> = (?I ?tk (Ge (CN 0 c e)))" using pos_divide_le_eq[OF kpos, where a="real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e" and b="0", symmetric] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (9 i c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (Dvd i (CN 0 c e))) = (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k)" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_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 kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast next case (10 i c e) then have cp: "c > 0" and nb: "numbound0 e" by auto { assume kdc: "k dvd c" from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } moreover { assume *: "\<not> k dvd c" from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp from tint have ti: "real_of_int \<lfloor>?N (real_of_int x) t\<rfloor> = ?N (real_of_int x) t" using isint_def by simp from assms * have "?I (real_of_int x) (?s (NDvd i (CN 0 c e))) = (\<not> (real_of_int i * real_of_int k rdvd (real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k))" using real_of_int_div[OF kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti algebra_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 kdt] numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"] numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) finally have ?case . } ultimately show ?case by blast qed (simp_all add: bound0_I[where bs="bs" and b="real_of_int (\<lfloor>?N b' t\<rfloor> div k)" and b'="real_of_int x"] numbound0_I[where bs="bs" and b="real_of_int (\<lfloor>?N b' t\<rfloor> div k)" and b'="real_of_int x"]) 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_of_int (i::int)#bs)" shows "\<forall> (b,k) \<in> set (\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int 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_of_int (i::int)#bs)" shows "\<forall> (b,k) \<in> set (\<alpha>_\<rho> p). k >0 \<and> numbound0 b \<and> isint b (real_of_int i#bs)" using lp isint_add [OF isint_c[where j="- 1"],where bs="real_of_int i#bs"] by (induct p rule: \<alpha>_\<rho>.induct, auto) lemma \<rho>: assumes lp: "iszlfm p (real_of_int (i::int) #bs)" and pi: "Ifm (real_of_int 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_of_int (c*i) \<noteq> Inum (real_of_int i#bs) e + real_of_int j" (is "\<forall>(e,c) \<in> set (\<rho> p). \<forall> j\<in> {1 .. c*d}. _ \<noteq> ?N i e + _") shows "Ifm (real_of_int(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_of_int i#bs)" and pi: "real_of_int (c*i) = - 1 - ?N i e + real_of_int (1::int)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> -1 - ?N i e + real_of_int 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_of_int i#bs)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j" by simp+ {assume "real_of_int (c*i) \<noteq> - ?N i e + real_of_int (c*d)" with numbound0_I[OF nb, where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] have ?case by (simp add: algebra_simps)} moreover {assume pi: "real_of_int (c*i) = - ?N i e + real_of_int (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 5 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] of_int_mult] show ?case using 5 dp apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] algebra_simps del: mult_pos_pos) by (metis add.right_neutral of_int_0_less_iff of_int_mult pos_add_strict) next case (6 c e) hence cp: "c > 0" by simp from 6 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] of_int_mult] show ?case using 6 dp apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] algebra_simps del: mult_pos_pos) using order_trans by fastforce next case (7 c e) hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j" and pi: "real_of_int (c*i) + ?N i e > 0" and cp': "real_of_int c >0" by simp+ let ?fe = "\<lfloor>?N i e\<rfloor>" from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c > 0" by (simp add: algebra_simps) from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) > real_of_int (0::int)" by simp hence pi': "c*i + ?fe > 0" by (simp only: of_int_less_iff[symmetric]) have "real_of_int (c*i) + ?N i e > real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" by auto moreover {assume "real_of_int (c*i) + ?N i e > real_of_int (c*d)" hence ?case by (simp add: algebra_simps numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} moreover {assume H:"real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<le> real_of_int (c*d)" by simp hence pid: "c*i + ?fe \<le> c*d" by (simp only: 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_of_int (c*i) = - ?N i e + real_of_int j1" unfolding Bex_def using ei[simplified isint_iff] by fastforce 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_of_int i#bs)" and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - 1 - ?N i e + real_of_int j" and pi: "real_of_int (c*i) + ?N i e \<ge> 0" and cp': "real_of_int c >0" by simp+ let ?fe = "\<lfloor>?N i e\<rfloor>" from pi cp have th:"(real_of_int i +?N i e / real_of_int c)*real_of_int c \<ge> 0" by (simp add: algebra_simps) from pi ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<ge> real_of_int (0::int)" by simp hence pi': "c*i + 1 + ?fe \<ge> 1" by (simp only: of_int_le_iff[symmetric]) have "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e < real_of_int (c*d)" by auto moreover {assume "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d)" hence ?case by (simp add: algebra_simps numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} moreover {assume H:"real_of_int (c*i) + ?N i e < real_of_int (c*d)" with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) < real_of_int (c*d)" by simp hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: 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_of_int (c*i) + 1= - ?N i e + real_of_int j1" unfolding Bex_def using ei[simplified isint_iff] by fastforce hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = (- ?N i e + real_of_int j1) - 1" by (simp only: algebra_simps) hence "\<exists> j1\<in> {1 .. c*d}. real_of_int (c*i) = - 1 - ?N i e + real_of_int j1" by (simp add: algebra_simps) with nob have ?case by blast } ultimately show ?case by blast next case (9 j c e) hence p: "real_of_int j rdvd real_of_int (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ let ?e = "Inum (real_of_int i # bs) e" from 9 have "isint e (real_of_int i #bs)" by simp hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"] by (simp add: isint_iff) from 9 have id: "j dvd d" by simp from ie[symmetric] have "?p i = (real_of_int j rdvd real_of_int (c*i + \<lfloor>?e\<rfloor>))" by simp also have "\<dots> = (j dvd c*i + \<lfloor>?e\<rfloor>)" using int_rdvd_iff [where i="j" and t="c*i + \<lfloor>?e\<rfloor>"] by simp also have "\<dots> = (j dvd c*i - c*d + \<lfloor>?e\<rfloor>)" using dvd_period[OF id, where x="c*i" and c="-c" and t="\<lfloor>?e\<rfloor>"] by simp also have "\<dots> = (real_of_int j rdvd real_of_int (c*i - c*d + \<lfloor>?e\<rfloor>))" using int_rdvd_iff[where i="j" and t="(c*i - c*d + \<lfloor>?e\<rfloor>)",symmetric, simplified] ie by simp also have "\<dots> = (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" using ie by (simp add:algebra_simps) finally show ?case using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p by (simp add: algebra_simps) next case (10 j c e) hence p: "\<not> (real_of_int j rdvd real_of_int (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e" by simp+ let ?e = "Inum (real_of_int i # bs) e" from 10 have "isint e (real_of_int i #bs)" by simp hence ie: "real_of_int \<lfloor>?e\<rfloor> = ?e" using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"] by (simp add: isint_iff) from 10 have id: "j dvd d" by simp from ie[symmetric] have "?p i = (\<not> (real_of_int j rdvd real_of_int (c*i + \<lfloor>?e\<rfloor>)))" by simp also have "\<dots> \<longleftrightarrow> \<not> (j dvd c*i + \<lfloor>?e\<rfloor>)" using int_rdvd_iff [where i="j" and t="c*i + \<lfloor>?e\<rfloor>"] by simp also have "\<dots> \<longleftrightarrow> \<not> (j dvd c*i - c*d + \<lfloor>?e\<rfloor>)" using dvd_period[OF id, where x="c*i" and c="-c" and t="\<lfloor>?e\<rfloor>"] by simp also have "\<dots> \<longleftrightarrow> \<not> (real_of_int j rdvd real_of_int (c*i - c*d + \<lfloor>?e\<rfloor>))" using int_rdvd_iff[where i="j" and t="(c*i - c*d + \<lfloor>?e\<rfloor>)",symmetric, simplified] ie by simp also have "\<dots> \<longleftrightarrow> \<not> (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" using ie by (simp add:algebra_simps) finally show ?case using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p by (simp add: algebra_simps) qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"]) 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_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (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_of_int x"] have lp': "iszlfm p (real_of_int x#bs)". have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * x) \<noteq> Inum (real_of_int x # bs) e + real_of_int 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_of_int (c*x) = Inum (real_of_int x#bs) e + real_of_int j" let ?e = "Inum (real_of_int x#bs) e" from \<rho>_l[OF lp'] ecR have ei:"isint e (real_of_int 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_of_int (c*x) = real_of_int (\<lfloor>?e\<rfloor> + j)" by simp hence cx: "c*x = \<lfloor>?e\<rfloor> + j" by (simp only: of_int_eq_iff) hence cdej:"c dvd \<lfloor>?e\<rfloor> + j" by (simp add: dvd_def) (rule_tac x="x" in exI, simp) hence "real_of_int c rdvd real_of_int (\<lfloor>?e\<rfloor> + j)" by (simp only: int_rdvd_iff) hence rcdej: "real_of_int c rdvd ?e + real_of_int j" by (simp add: ei[simplified isint_iff]) from cx have "(c*x) div c = (\<lfloor>?e\<rfloor> + j) div c" by simp with cp have "x = (\<lfloor>?e\<rfloor> + j) div c" by simp with px have th: "?P ((\<lfloor>?e\<rfloor> + j) div c)" by auto from cp have cp': "real_of_int c > 0" by simp from cdej have cdej': "c dvd \<lfloor>Inum (real_of_int x#bs) (Add e (C j))\<rfloor>" by simp from nb have nb': "numbound0 (Add e (C j))" by simp have ji: "isint (C j) (real_of_int x#bs)" by (simp add: isint_def) from isint_add[OF ei ji] have ei':"isint (Add e (C j)) (real_of_int x#bs)" . from th \<sigma>_\<rho>[where b'="real_of_int x", OF lp' cp' nb' ei' cdej',symmetric] have "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (Add e (C j), c))" by simp with rcdej have th: "Ifm (real_of_int 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_of_int 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_of_int (i::int)#bs)" shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int 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_of_int i#bs)" and cp: "c > 0" by auto have "isint (C j) (real_of_int i#bs)" by (simp add: isint_iff) with isint_add[OF numbound0_gen[OF nb ei,rule_format, where y="real_of_int i"]] have eji:"isint (Add e (C j)) (real_of_int 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_of_int i"] have spx': "Ifm (real_of_int i # bs) (\<sigma> p c (Add e (C j)))" by blast from spx' have rcdej:"real_of_int c rdvd (Inum (real_of_int i#bs) (Add e (C j)))" and sr:"Ifm (real_of_int i#bs) (\<sigma>_\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+ from rcdej eji[simplified isint_iff] have "real_of_int c rdvd real_of_int \<lfloor>Inum (real_of_int i#bs) (Add e (C j))\<rfloor>" by simp hence cdej:"c dvd \<lfloor>Inum (real_of_int i#bs) (Add e (C j))\<rfloor>" by (simp only: int_rdvd_iff) from cp have cp': "real_of_int c > 0" by simp from \<sigma>_\<rho>[OF lp cp' nb' eji cdej] spx' have "?P (\<lfloor>Inum (real_of_int 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: "\<bar>x - z\<bar> + 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) text \<open>The \<open>\<real>\<close> part\<close> text\<open>Linearity for fm where Bound 0 ranges over \<open>\<real>\<close>\<close> fun isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *) where "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))" definition fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm" where "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*) fun rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list" where "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)) [(a,b). a\<leftarrow>acs,b\<leftarrow>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) = concat (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 [0 .. n] else [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 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 [0..n] else [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)) [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)) [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>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. 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)) [n..0]" by auto hence U3: "(\<Union> ((\<lambda>(p,n,s). set (?ff (p,n,s))) ` {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0})) = (\<Union> ((\<lambda>(p,n,s). set (map (?f(p,n,s)) [n..0])) ` {(p,n,s). (p,n,s)\<in> ?SS a\<and>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>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" by (auto simp add: split_def) qed have "?SS (Floor a) = \<Union> ((\<lambda>x. set (?ff x)) ` ?SS a)" by auto also have "\<dots> = \<Union> ((\<lambda> (p,n,s). set (?ff (p,n,s))) ` ?SS a)" by blast 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)))))" by (auto split: if_splits) 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)) [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)) [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 set_upto list.set) 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 5(1) 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 5 pns have H:"(Ifm ((x::real) # (bs::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 let ?nxs = "CN 0 n' s'" let ?l = "\<lfloor>?N s'\<rfloor> + j" from H have "?I (?p (p',n',s') j) \<longrightarrow> (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" by (simp add: fp_def np algebra_simps) also have "\<dots> \<longrightarrow> \<lfloor>?N ?nxs\<rfloor> = ?l \<and> ?N a = ?N ?nxs" using floor_eq_iff[where x="?N ?nxs" and a="?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 5 pns have H:"(Ifm ((x::real) # (bs::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 let ?nxs = "CN 0 n' s'" let ?l = "\<lfloor>?N s'\<rfloor> + j" from H have "?I (?p (p',n',s') j) \<longrightarrow> (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" by (simp add: np fp_def algebra_simps) also have "\<dots> \<longrightarrow> \<lfloor>?N ?nxs\<rfloor> = ?l \<and> ?N a = ?N ?nxs" using floor_eq_iff[where x="?N ?nxs" and a="?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 fastforce qed next case (3 a b) then show ?case by auto qed (auto simp add: Let_def split_def algebra_simps) lemma real_in_int_intervals: assumes xb: "real_of_int m \<le> x \<and> x < real_of_int ((n::int) + 1)" shows "\<exists> j\<in> {m.. n}. real_of_int j \<le> x \<and> x < real_of_int (j+1)" (is "\<exists> j\<in> ?N. ?P j") by (rule bexI[where P="?P" and x="\<lfloor>x\<rfloor>" and A="?N"]) (auto simp add: floor_less_iff[where x="x" and z="n+1", simplified] xb[simplified] floor_mono[where x="real_of_int m" and y="x", OF conjunct1[OF xb], simplified floor_of_int[where z="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) then have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast with 2 have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by blast 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[(x,y). x\<leftarrow>rsplit0 a, y\<leftarrow>rsplit0 b]" by (auto) 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 [0..n] else [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)) [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)) [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>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. 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)) [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)) [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>(a, b, c)\<in>M. set (f (a, b, c))) = (\<Union>(a, b, c)\<in>M. set (g a b c))" by (auto simp add: split_def) qed have "?SS (Floor a) = \<Union> ((\<lambda>x. set (?ff x)) ` ?SS a)" by auto also have "\<dots> = \<Union> ((\<lambda> (p,n,s). set (?ff (p,n,s))) ` ?SS a)" by blast 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)))))" by (auto split: if_splits) 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)) [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)) [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 set_upto list.set) 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 5 have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto 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 of_int_floor_le[of "?N s"] have "?N (Floor s) \<le> ?N s" by simp also from mult_left_mono[OF xp] np have "?N s \<le> real_of_int n * x + ?N s" by simp finally have "?N (Floor s) \<le> real_of_int n * x + ?N s" . moreover {from x1 np have "real_of_int n *x + ?N s < real_of_int n + ?N s" by simp also from real_of_int_floor_add_one_gt[where r="?N s"] have "\<dots> < real_of_int n + ?N (Floor s) + 1" by simp finally have "real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp} ultimately have "?N (Floor s) \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp hence th: "0 \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (n+1)" by simp from real_in_int_intervals th have "\<exists> j\<in> {0 .. n}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp hence "\<exists> j\<in> {0 .. n}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0" by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int 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 algebra_simps) 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 of_int_floor_le[of "?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_of_int n * x + ?N s" by simp ultimately have "?N (Floor s) + 1 > real_of_int n * x + ?N s" by arith moreover {from x1 nn have "real_of_int n *x + ?N s \<ge> real_of_int n + ?N s" by simp moreover from of_int_floor_le[of "?N s"] have "real_of_int n + ?N s \<ge> real_of_int n + ?N (Floor s)" by simp ultimately have "real_of_int n *x + ?N s \<ge> ?N (Floor s) + real_of_int n" by (simp only: algebra_simps)} ultimately have "?N (Floor s) + real_of_int n \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (1::int)" by simp hence th: "real_of_int n \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (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_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp hence "\<exists> j\<in> {n .. 0}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0" by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j) \<and> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (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 algebra_simps 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: nn) } ultimately show ?case by blast qed (auto simp add: Let_def split_def) (* Linearize a formula where Bound 0 ranges over [0,1) *) definition rsplit :: "(int \<Rightarrow> num \<Rightarrow> fm) \<Rightarrow> num \<Rightarrow> fm" where "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 definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) else (Gt (CN 0 (-c) (Neg t))))" definition le :: "int \<Rightarrow> num \<Rightarrow> fm" where le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) else (Ge (CN 0 (-c) (Neg t))))" definition gt :: "int \<Rightarrow> num \<Rightarrow> fm" where gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) else (Lt (CN 0 (-c) (Neg t))))" definition ge :: "int \<Rightarrow> num \<Rightarrow> fm" where ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) else (Le (CN 0 (-c) (Neg t))))" definition eq :: "int \<Rightarrow> num \<Rightarrow> fm" where eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) else (Eq (CN 0 (-c) (Neg t))))" definition neq :: "int \<Rightarrow> num \<Rightarrow> fm" where neq_def: "neq c t = (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 algebra_simps myless[of _ "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, rename_tac nat a b, 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 algebra_simps myle[of _ "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, rename_tac nat a b, 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 algebra_simps myless[of _ "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, rename_tac nat a b, 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 algebra_simps myle[of _ "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, rename_tac nat a b, 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 algebra_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, rename_tac nat a b, 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 algebra_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, rename_tac nat a b, case_tac"nat", simp_all) lemma small_le: assumes u0:"0 \<le> u" and u1: "u < 1" shows "(-u \<le> real_of_int (n::int)) = (0 \<le> n)" using u0 u1 by auto lemma small_lt: assumes u0:"0 \<le> u" and u1: "u < 1" shows "(real_of_int (n::int) < real_of_int (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_of_int n > 0" shows "(real_of_int (i::int) rdvd real_of_int (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real_of_int n * u = s - real_of_int \<lfloor>s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>s\<rfloor>))" (is "?lhs = ?rhs") proof- let ?ss = "s - real_of_int \<lfloor>s\<rfloor>" from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]] of_int_floor_le have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" by (auto simp: floor_less_cancel) from np have n0: "real_of_int n \<ge> 0" by simp from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto from int_rdvd_real[where i="i" and x="real_of_int (n::int) * u - s"] have "real_of_int i rdvd real_of_int n * u - s = (i dvd \<lfloor>real_of_int n * u - s\<rfloor> \<and> (real_of_int \<lfloor>real_of_int n * u - s\<rfloor> = real_of_int n * u - s ))" (is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp also have "\<dots> = (?DE \<and> real_of_int (\<lfloor>real_of_int n * u - s\<rfloor> + \<lfloor>s\<rfloor>) \<ge> -?ss \<and> real_of_int (\<lfloor>real_of_int n * u - s\<rfloor> + \<lfloor>s\<rfloor>) < real_of_int n - ?ss)" (is "_=(?DE \<and>real_of_int ?a \<ge> _ \<and> real_of_int ?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_of_int (\<lfloor>real_of_int n * u - s\<rfloor>) = real_of_int j - real_of_int \<lfloor>s\<rfloor> ))" by (simp only: algebra_simps of_int_diff[symmetric] of_int_eq_iff) also have "\<dots> = ((\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * u - s = real_of_int j - real_of_int \<lfloor>s\<rfloor> \<and> real_of_int i rdvd real_of_int n * u - s))" using int_rdvd_iff[where i="i" and t="\<lfloor>real_of_int n * u - s\<rfloor>"] by (auto cong: conj_cong) also have "\<dots> = ?rhs" by(simp cong: conj_cong) (simp add: algebra_simps ) finally show ?thesis . qed definition DVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where DVDJ_def: "DVDJ i n s = (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))))) [0..n - 1]) F)" definition NDVDJ:: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where NDVDJ_def: "NDVDJ i n s = (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))))) [0..n - 1]) T)" lemma DVDJ_DVD: assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int 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="[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: np DVDJ_def) also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int \<lfloor>- ?s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>- ?s\<rfloor>))" by (simp add: algebra_simps) also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] have "\<dots> = (real_of_int i rdvd real_of_int 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_of_int 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="[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: np NDVDJ_def) also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int \<lfloor>- ?s\<rfloor> + real_of_int j \<and> real_of_int i rdvd real_of_int (j - \<lfloor>- ?s\<rfloor>)))" by (simp add: algebra_simps) also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] have "\<dots> = (\<not> (real_of_int i rdvd real_of_int 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 definition DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where DVD_def: "DVD i c t = (if i=0 then eq c t else if c = 0 then (Dvd i t) else if c >0 then DVDJ \<bar>i\<bar> c t else DVDJ \<bar>i\<bar> (-c) (Neg t))" definition NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where "NDVD i c t = (if i=0 then neq c t else if c = 0 then (NDvd i t) else if c >0 then NDVDJ \<bar>i\<bar> c t else NDVDJ \<bar>i\<bar> (-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_of_int 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_of_int 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, rename_tac nat a b, 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, rename_tac nat a b, case_tac "nat", simp_all) fun rlfm :: "fm \<Rightarrow> fm" where "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" lemma bound0at_l : "\<lbrakk>isatom p ; bound0 p\<rbrakk> \<Longrightarrow> isrlfm p" by (induct p rule: isrlfm.induct, auto) 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, rename_tac nat a b, 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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with Lt a 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, rename_tac nat a b, 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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with Le a have ?case by (simp add: Let_def reducecoeff_def 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, rename_tac nat a b,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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with Gt a have ?case by (simp add: Let_def reducecoeff_def 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, rename_tac nat a b, 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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with Ge a have ?case by (simp add: Let_def reducecoeff_def 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, rename_tac nat a b, 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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with Eq a have ?case by (simp add: Let_def reducecoeff_def 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, rename_tac nat a b, 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: "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 \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" by (simp add: numgcd_def) from \<open>c > 0\<close> have th': "c\<noteq>0" by auto from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp from zdiv_mono2[OF cp th1 th, simplified div_self[OF th']] have "0 < c div numgcd (CN 0 c (simpnum e))" by simp } with NEq a have ?case by (simp add: Let_def reducecoeff_def 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) 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 3 have nb: "numbound0 e" by simp from 3 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int c * x + ?e < 0" by arith hence "real_of_int 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 4 have nb: "numbound0 e" by simp from 4 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int c * x + ?e < 0" by arith hence "real_of_int 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 5 have nb: "numbound0 e" by simp from 5 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int 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 6 have nb: "numbound0 e" by simp from 6 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int 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 7 have nb: "numbound0 e" by simp from 7 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int 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 8 have nb: "numbound0 e" by simp from 8 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x < ?z" hence "(real_of_int c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) hence "real_of_int 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 3 have nb: "numbound0 e" by simp from 3 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int c * x + ?e > 0" by arith hence "real_of_int 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 4 have nb: "numbound0 e" by simp from 4 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int c * x + ?e > 0" by arith hence "real_of_int 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 5 have nb: "numbound0 e" by simp from 5 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int 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 6 have nb: "numbound0 e" by simp from 6 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int 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 7 have nb: "numbound0 e" by simp from 7 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int 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 8 have nb: "numbound0 e" by simp from 8 have cp: "real_of_int c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real_of_int c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real_of_int c * x > - ?e)" by (simp add: ac_simps) hence "real_of_int 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 fun \<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list" where "\<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 = []" fun \<upsilon> :: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" where "\<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_of_int n > 0" and nbt: "numbound0 t" shows "(Ifm (x#bs) (\<upsilon> p (t,n)) = Ifm (((Inum (x#bs) t)/(real_of_int 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 5 have cp: "c >0" and nb: "numbound0 e" by simp_all have "?I ?u (Lt (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) < 0)" by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) < 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (6 c e) from 6 have cp: "c >0" and nb: "numbound0 e" by simp_all have "?I ?u (Le (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) \<le> 0)" by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<le> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (7 c e) from 7 have cp: "c >0" and nb: "numbound0 e" by simp_all have "?I ?u (Gt (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) > 0)" by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) > 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (8 c e) from 8 have cp: "c >0" and nb: "numbound0 e" by simp_all have "?I ?u (Ge (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) \<ge> 0)" by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<ge> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (3 c e) from 3 have cp: "c >0" and nb: "numbound0 e" by simp_all from np have np: "real_of_int n \<noteq> 0" by simp have "?I ?u (Eq (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) = 0)" by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) = 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (4 c e) from 4 have cp: "c >0" and nb: "numbound0 e" by simp_all from np have np: "real_of_int n \<noteq> 0" by simp have "?I ?u (NEq (CN 0 c e)) = (real_of_int 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_of_int c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)" by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" and b="0", simplified div_0]) (simp only: algebra_simps) also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<noteq> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real_of_int n" and b'="x"]) 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_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real_of_int m") proof- have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int 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"]) then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<ge> ?N a s" by blast from \<Upsilon>_l[OF lp] smU have mp: "real_of_int 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_of_int 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_of_int m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real_of_int m") proof- have "\<exists> (s,m) \<in> set (\<Upsilon> p). real_of_int m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real_of_int 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"]) then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<le> ?N a s" by blast from \<Upsilon>_l[OF lp] smU have mp: "real_of_int 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_of_int 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_of_int n) ` set (\<Upsilon> p)" (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real_of_int 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_of_int c > 0" and nb: "numbound0 e" by simp_all from 5 have "x * real_of_int c + ?N x e < 0" by (simp add: algebra_simps) hence pxc: "x < (- ?N x e) / real_of_int c" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto moreover {assume y: "y < (-?N x e)/ real_of_int c" hence "y * real_of_int c < - ?N x e" by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_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_of_int c" with yu have eu: "u > (- ?N x e) / real_of_int c" by auto with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int 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_of_int c > 0" and nb: "numbound0 e" by simp_all from 6 have "x * real_of_int c + ?N x e \<le> 0" by (simp add: algebra_simps) hence pxc: "x \<le> (- ?N x e) / real_of_int c" by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto moreover {assume y: "y < (-?N x e)/ real_of_int c" hence "y * real_of_int c < - ?N x e" by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_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_of_int c" with yu have eu: "u > (- ?N x e) / real_of_int c" by auto with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int 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_of_int c > 0" and nb: "numbound0 e" by simp_all from 7 have "x * real_of_int c + ?N x e > 0" by (simp add: algebra_simps) hence pxc: "x > (- ?N x e) / real_of_int c" by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto moreover {assume y: "y > (-?N x e)/ real_of_int c" hence "y * real_of_int c > - ?N x e" by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_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_of_int c" with ly have eu: "l < (- ?N x e) / real_of_int c" by auto with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int 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_of_int c > 0" and nb: "numbound0 e" by simp_all from 8 have "x * real_of_int c + ?N x e \<ge> 0" by (simp add: algebra_simps) hence pxc: "x \<ge> (- ?N x e) / real_of_int c" by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto hence "y < (- ?N x e) / real_of_int c \<or> y > (-?N x e) / real_of_int c" by auto moreover {assume y: "y > (-?N x e)/ real_of_int c" hence "y * real_of_int c > - ?N x e" by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_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_of_int c" with ly have eu: "l < (- ?N x e) / real_of_int c" by auto with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int 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_of_int c > 0" and nb: "numbound0 e" by simp_all from cp have cnz: "real_of_int c \<noteq> 0" by simp from 3 have "x * real_of_int c + ?N x e = 0" by (simp add: algebra_simps) hence pxc: "x = (- ?N x e) / real_of_int c" by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with lx xu have yne: "x \<noteq> - ?N x e / real_of_int c" by auto with pxc show ?case by simp next case (4 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all from cp have cnz: "real_of_int c \<noteq> 0" by simp from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto hence "y* real_of_int 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_of_int c + ?N x e \<noteq> 0" by (simp add: algebra_simps) thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by (simp add: algebra_simps) qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"]) 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_of_int n + Inum (x#bs) s / real_of_int 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_of_int n + ?N a s /real_of_int m) / 2) p" proof- let ?M = "(\<lambda> (t,c). ?N a t / real_of_int 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_of_int n \<and> a \<ge> ?N x s / real_of_int 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_of_int m" and tx1: "a \<ge> ?N x t / real_of_int 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_of_int m" and tx: "a \<ge> ?N a t / real_of_int 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_of_int n \<in> ?M" using tnU by auto have smM: "?N a s / real_of_int 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_of_int n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp have "?N a s / real_of_int 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_of_int nu" by auto then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real_of_int nu" by blast have "(u + u) / 2 = u" by auto with pu tuu have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int 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_of_int t1n" by auto then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real_of_int t1n" by blast from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real_of_int t2n" by auto then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real_of_int 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_of_int n + ?N a s / real_of_int 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_of_int n + ?N x s / real_of_int 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_of_int n + (Inum (x#bs) s) / real_of_int 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_of_int 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_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0" by auto let ?st = "Add (Mul m t) (Mul n s)" from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute) from tnb snb have st_nb: "numbound0 ?st" by simp have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" using mnp mp np by (simp add: algebra_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_of_int n + ?N s / real_of_int 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_of_int k > 0" and snb: "numbound0 s" and mp:"real_of_int l > 0" by auto let ?st = "Add (Mul l t) (Mul k s)" from np mp have mnp: "real_of_int (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 text\<open>The overall Part\<close> 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_of_int i + u))" proof(auto) fix x assume Px: "P x" let ?i = "\<lfloor>x\<rfloor>" let ?u = "x - real_of_int ?i" have "x = real_of_int ?i + ?u" by simp hence "P (real_of_int ?i + ?u)" using Px by simp moreover have "real_of_int ?i \<le> x" using 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_of_int i + u))" by blast qed fun exsplitnum :: "num \<Rightarrow> num" where "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)" fun exsplit :: "fm \<Rightarrow> fm" where "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: algebra_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_of_int 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_of_int i)#bs) (exsplit p))" by auto also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real_of_int i + x) #bs) p)" by (simp only: exsplit[OF qf] ac_simps) 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. *) definition ferrack01 :: "fm \<Rightarrow> fm" where "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 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 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_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int 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 np mp have mnp: "real_of_int (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_of_int 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_of_int n + Inum (x#bs) s /real_of_int 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_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" using mnz nnz by (simp add: algebra_simps add_divide_distrib) thus "(real_of_int m * Inum (x # bs) t + real_of_int n * Inum (x # bs) s) / (2 * real_of_int n * real_of_int m) \<in> (\<lambda>((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set U \<times> set U)"using mnz nnz th apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def) by (rule_tac x="(s,m)" in bexI,simp_all) (rule_tac x="(t,n)" in bexI,simp_all add: mult.commute) 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_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" using mnz nnz by (simp add: algebra_simps add_divide_distrib) let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int 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_of_int n' + ?N s' / real_of_int m')/2 = ?N ?st' / real_of_int (2*n'*m')" using mnz' nnz' by (simp add: algebra_simps add_divide_distrib) from Pts' have "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2" by simp also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real_of_int 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_of_int n + Inum (x # bs) s / real_of_int m) / 2 \<in> (\<lambda>(t, n). Inum (x # bs) t / real_of_int 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_of_int n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real_of_int n + Inum (x#bs) s /real_of_int 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 np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult) from tnb snb have stnb: "numbound0 ?st" by simp have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" using mp np by (simp add: algebra_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_of_int 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_of_int (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 np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult) from tnb snb have stnb: "numbound0 ?st" by simp have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)" using mp np by (simp add: algebra_simps add_divide_distrib) from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int 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_of_int (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" fix x 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_of_int n)" let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real_of_int n + ?N s/ real_of_int 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) 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 "\<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) 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 simp del: map_map) (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_comp comp_assoc) 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_comp) 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_comp] 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_of_int 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 prod.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_of_int (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_of_int x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real_of_int i#bs)) ` set (\<beta> p). Ifm ((b+real_of_int j)#bs) p))" using cp_thm[OF lp up dd dp] by auto definition unit :: "fm \<Rightarrow> fm \<times> num list \<times> int" where "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_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> (Inum (real_of_int i#bs)) ` set B = (Inum (real_of_int i#bs)) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q (real_of_int (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_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> Inum (real_of_int i#bs) ` set B = Inum (real_of_int i#bs) ` set (\<beta> q) \<and> d_\<beta> q 1 \<and> d_\<delta> q d \<and> 0 < d \<and> iszlfm q (real_of_int i # bs) \<and> (\<forall> b\<in> set B. numbound0 b)" let ?I = "\<lambda> (x::int) p. Ifm (real_of_int 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_of_int i#bs)" by simp hence lp'': "iszlfm ?p' (real_of_int (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_of_int 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_of_int (i::int)#bs) t" have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_comp) also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real_of_int i #bs"] by auto finally have BB': "?N ` set ?B' = ?N ` ?B" . have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_comp) also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int 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 from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b" by simp { 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_of_int i"] have lq': "iszlfm q (real_of_int 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 *) definition cooper :: "fm \<Rightarrow> fm" where "cooper p \<equiv> (let (q,B,d) = unit p; js = [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))) [(b,j). b\<leftarrow>B,j\<leftarrow>js])) in decr (disj md qd)))" lemma cooper: assumes qf: "qfree p" shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" (is "(?lhs = ?rhs) \<and> _") proof- let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" let ?q = "fst (unit p)" let ?B = "fst (snd(unit p))" let ?d = "snd (snd (unit p))" let ?js = "[1..?d]" let ?mq = "minusinf ?q" let ?smq = "simpfm ?mq" let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" fix i let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t" let ?bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]" let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs" 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_of_int 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 from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp from Bn jsnb have "\<forall> (b,j) \<in> set ?bjs. numbound0 (Add b (C j))" by simp hence "\<forall> (b,j) \<in> set ?bjs. 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_of_int 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_of_int j)#bs) ?q))" 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 ?bjs. Ifm (t #bs) ?q))" by 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 ?bjs. 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] 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 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 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_of_int 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_of_int x#bs) q)" using cooper disjuncts_qf[OF qf] by blast also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int 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' show ?rhs apply (auto simp: cc') 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' show ?lhs apply (auto simp: cc') 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_of_int (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_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int 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 definition chooset :: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int" where "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_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (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_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" let ?I = "\<lambda> (x::int) p. Ifm (real_of_int 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_of_int i"] have lq: "iszlfm ?q (real_of_int (i::int)#bs)" . from \<delta>[OF lq] have dp:"?d >0" by blast let ?N = "\<lambda> (t,c). (Inum (real_of_int (i::int)#bs) t,c)" have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_comp) also have "\<dots> = ?N ` ?B" by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int 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_comp) also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"] by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int 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: 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: 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_of_int i"] have lq': "iszlfm q (real_of_int 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 definition stage :: "fm \<Rightarrow> int \<Rightarrow> (num \<times> int) \<Rightarrow> fm" where "stage p d \<equiv> (\<lambda> (e,c). evaldjf (\<lambda> j. simpfm (\<sigma> p c (Add e (C j)))) [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 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 [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 definition redlove :: "fm \<Rightarrow> fm" where "redlove p \<equiv> (let (q,B,d) = chooset p; mq = simpfm (minusinf q); md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) [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_of_int x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" (is "(?lhs = ?rhs) \<and> _") proof- let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p" let ?q = "fst (chooset p)" let ?B = "fst (snd(chooset p))" let ?d = "snd (snd (chooset p))" let ?js = "[1..?d]" let ?mq = "minusinf ?q" let ?smq = "simpfm ?mq" let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js" fix i let ?N = "\<lambda> (t,k). (Inum (real_of_int (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_of_int 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 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_of_int 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: 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 set_upto simpfm) also have "\<dots> = (?I i (disj ?md ?qd))" by simp 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_of_int 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_of_int x#bs) q)" using redlove disjuncts_qf[OF qf] by blast also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int 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) definition mircfr :: "fm \<Rightarrow> fm" where "mircfr = DJ cooper o ferrack01 o simpfm o exsplit" definition mirlfr :: "fm \<Rightarrow> fm" where "mirlfr = 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_of_int 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_of_int 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_of_int 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_of_int 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 definition mircfrqe:: "fm \<Rightarrow> fm" where "mircfrqe p = qelim (prep p) mircfr" definition mirlfrqe:: "fm \<Rightarrow> fm" where "mirlfrqe 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) definition "problem1 = A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0))))))" definition "problem2 = A (Iff (Eq (Add (Floor (Bound 0)) (Floor (Neg (Bound 0))))) (Eq (Sub (Floor (Bound 0)) (Bound 0))))" definition "problem3 = A (And (Le (Sub (Floor (Bound 0)) (Bound 0))) (Le (Add (Bound 0) (Floor (Neg (Bound 0))))))" definition "problem4 = E (And (Ge (Sub (Bound 1) (Bound 0))) (Eq (Add (Floor (Bound 1)) (Floor (Neg (Bound 0))))))" ML_val \<open>@{code mircfrqe} @{code problem1}\<close> ML_val \<open>@{code mirlfrqe} @{code problem1}\<close> ML_val \<open>@{code mircfrqe} @{code problem2}\<close> ML_val \<open>@{code mirlfrqe} @{code problem2}\<close> ML_val \<open>@{code mircfrqe} @{code problem3}\<close> ML_val \<open>@{code mirlfrqe} @{code problem3}\<close> ML_val \<open>@{code mircfrqe} @{code problem4}\<close> ML_val \<open>@{code mirlfrqe} @{code problem4}\<close> (*code_reflect Mir functions mircfrqe mirlfrqe file "mir.ML"*) oracle mirfr_oracle = \<open> let val mk_C = @{code C} o @{code int_of_integer}; val mk_Dvd = @{code Dvd} o apfst @{code int_of_integer}; val mk_Bound = @{code Bound} o @{code nat_of_integer}; fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (=) vs t of NONE => error "Variable not found in the list!" | SOME n => mk_Bound n) | num_of_term vs \<^term>\<open>of_int (0::int)\<close> = mk_C 0 | num_of_term vs \<^term>\<open>of_int (1::int)\<close> = mk_C 1 | num_of_term vs \<^term>\<open>0::real\<close> = mk_C 0 | num_of_term vs \<^term>\<open>1::real\<close> = mk_C 1 | num_of_term vs \<^term>\<open>- 1::real\<close> = mk_C (~ 1) | num_of_term vs (Bound i) = mk_Bound i | num_of_term vs \<^Const_>\<open>uminus \<^Type>\<open>real\<close> for t'\<close> = @{code Neg} (num_of_term vs t') | num_of_term vs \<^Const_>\<open>plus \<^Type>\<open>real\<close> for t1 t2\<close> = @{code Add} (num_of_term vs t1, num_of_term vs t2) | num_of_term vs \<^Const_>\<open>minus \<^Type>\<open>real\<close> for t1 t2\<close> = @{code Sub} (num_of_term vs t1, num_of_term vs t2) | num_of_term vs \<^Const_>\<open>times \<^Type>\<open>real\<close> for t1 t2\<close> = (case num_of_term vs t1 of @{code C} i => @{code Mul} (i, num_of_term vs t2) | _ => error "num_of_term: unsupported Multiplication") | num_of_term vs \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>numeral \<^Type>\<open>int\<close> for t'\<close>\<close> = mk_C (HOLogic.dest_numeral t') | num_of_term vs \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>uminus \<^Type>\<open>int\<close> for \<^Const_>\<open>numeral \<^Type>\<open>int\<close> for t'\<close>\<close>\<close> = mk_C (~ (HOLogic.dest_numeral t')) | num_of_term vs \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>floor \<^Type>\<open>real\<close> for t'\<close>\<close> = @{code Floor} (num_of_term vs t') | num_of_term vs \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>ceiling \<^Type>\<open>real\<close> for t'\<close>\<close> = @{code Neg} (@{code Floor} (@{code Neg} (num_of_term vs t'))) | num_of_term vs \<^Const_>\<open>numeral \<^Type>\<open>real\<close> for t'\<close> = mk_C (HOLogic.dest_numeral t') | num_of_term vs \<^Const_>\<open>uminus \<^Type>\<open>real\<close> for \<^Const_>\<open>numeral \<^Type>\<open>real\<close> for t'\<close>\<close> = mk_C (~ (HOLogic.dest_numeral t')) | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term \<^context> t); fun fm_of_term vs \<^Const_>\<open>True\<close> = @{code T} | fm_of_term vs \<^Const_>\<open>False\<close> = @{code F} | fm_of_term vs \<^Const_>\<open>less \<^Type>\<open>real\<close> for t1 t2\<close> = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs \<^Const_>\<open>less_eq \<^Type>\<open>real\<close> for t1 t2\<close> = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs \<^Const_>\<open>HOL.eq \<^Type>\<open>real\<close> for t1 t2\<close> = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs \<^Const_>\<open>rdvd for \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>numeral \<^Type>\<open>int\<close> for t1\<close>\<close> t2\<close> = mk_Dvd (HOLogic.dest_numeral t1, num_of_term vs t2) | fm_of_term vs \<^Const_>\<open>rdvd for \<^Const_>\<open>of_int \<^Type>\<open>real\<close> for \<^Const_>\<open>uminus \<^Type>\<open>int\<close> for \<^Const_>\<open>numeral \<^Type>\<open>int\<close> for t1\<close>\<close>\<close> t2\<close> = mk_Dvd (~ (HOLogic.dest_numeral t1), num_of_term vs t2) | fm_of_term vs \<^Const_>\<open>HOL.eq \<^Type>\<open>bool\<close> for t1 t2\<close> = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs \<^Const_>\<open>HOL.conj for t1 t2\<close> = @{code And} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs \<^Const_>\<open>HOL.disj for t1 t2\<close> = @{code Or} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs \<^Const_>\<open>HOL.implies for t1 t2\<close> = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs \<^Const_>\<open>HOL.Not for t'\<close> = @{code Not} (fm_of_term vs t') | fm_of_term vs \<^Const_>\<open>Ex _ for \<open>Abs (xn, xT, p)\<close>\<close> = @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) | fm_of_term vs \<^Const_>\<open>All _ for \<open>Abs (xn, xT, p)\<close>\<close> = @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term \<^context> t); fun term_of_num vs (@{code C} i) = \<^Const>\<open>of_int \<^Type>\<open>real\<close> for \<open>HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)\<close>\<close> | term_of_num vs (@{code Bound} n) = let val m = @{code integer_of_nat} n; in fst (the (find_first (fn (_, q) => m = q) vs)) end | term_of_num vs (@{code Neg} (@{code Floor} (@{code Neg} t'))) = \<^Const>\<open>of_int \<^Type>\<open>real\<close> for \<^Const>\<open>ceiling \<^Type>\<open>real\<close> for \<open>term_of_num vs t'\<close>\<close>\<close> | term_of_num vs (@{code Neg} t') = \<^Const>\<open>uminus \<^Type>\<open>real\<close> for \<open>term_of_num vs t'\<close>\<close> | term_of_num vs (@{code Add} (t1, t2)) = \<^Const>\<open>plus \<^Type>\<open>real\<close> for \<open>term_of_num vs t1\<close> \<open>term_of_num vs t2\<close>\<close> | term_of_num vs (@{code Sub} (t1, t2)) = \<^Const>\<open>minus \<^Type>\<open>real\<close> for \<open>term_of_num vs t1\<close> \<open>term_of_num vs t2\<close>\<close> | term_of_num vs (@{code Mul} (i, t2)) = \<^Const>\<open>times \<^Type>\<open>real\<close> for \<open>term_of_num vs (@{code C} i)\<close> \<open>term_of_num vs t2\<close>\<close> | term_of_num vs (@{code Floor} t) = \<^Const>\<open>of_int \<^Type>\<open>real\<close> for \<^Const>\<open>floor \<^Type>\<open>real\<close> for \<open>term_of_num vs t\<close>\<close>\<close> | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)) | term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s)); fun term_of_fm vs @{code T} = \<^Const>\<open>True\<close> | term_of_fm vs @{code F} = \<^Const>\<open>False\<close> | term_of_fm vs (@{code Lt} t) = \<^Const>\<open>less \<^Type>\<open>real\<close> for \<open>term_of_num vs t\<close> \<^term>\<open>0::real\<close>\<close> | term_of_fm vs (@{code Le} t) = \<^Const>\<open>less_eq \<^Type>\<open>real\<close> for \<open>term_of_num vs t\<close> \<^term>\<open>0::real\<close>\<close> | term_of_fm vs (@{code Gt} t) = \<^Const>\<open>less \<^Type>\<open>real\<close> for \<^term>\<open>0::real\<close> \<open>term_of_num vs t\<close>\<close> | term_of_fm vs (@{code Ge} t) = \<^Const>\<open>less_eq \<^Type>\<open>real\<close> for \<^term>\<open>0::real\<close> \<open>term_of_num vs t\<close>\<close> | term_of_fm vs (@{code Eq} t) = \<^Const>\<open>HOL.eq \<^Type>\<open>real\<close> for \<open>term_of_num vs t\<close> \<^term>\<open>0::real\<close>\<close> | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code Not} (@{code Eq} t)) | term_of_fm vs (@{code Dvd} (i, t)) = \<^Const>\<open>rdvd for \<open>term_of_num vs (@{code C} i)\<close> \<open>term_of_num vs t\<close>\<close> | term_of_fm vs (@{code NDvd} (i, t)) = term_of_fm vs (@{code Not} (@{code Dvd} (i, t))) | term_of_fm vs (@{code Not} t') = HOLogic.Not $ term_of_fm vs t' | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2 | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2 | term_of_fm vs (@{code Imp} (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2 | term_of_fm vs (@{code Iff} (t1, t2)) = \<^Const>\<open>HOL.eq \<^Type>\<open>bool\<close> for \<open>term_of_fm vs t1\<close> \<open>term_of_fm vs t2\<close>\<close>; in fn (ctxt, t) => let val fs = Misc_Legacy.term_frees t; val vs = map_index swap fs; (*If quick_and_dirty then run without proof generation as oracle*) val qe = if Config.get ctxt quick_and_dirty then @{code mircfrqe} else @{code mirlfrqe}; val t' = term_of_fm vs (qe (fm_of_term vs t)); in Thm.cterm_of ctxt (HOLogic.mk_Trueprop (HOLogic.mk_eq (t, t'))) end end \<close> lemmas iff_real_of_int = of_int_eq_iff [where 'a = real, symmetric] of_int_less_iff [where 'a = real, symmetric] of_int_le_iff [where 'a = real, symmetric] ML_file \<open>mir_tac.ML\<close> method_setup mir = \<open> Scan.lift (Args.mode "no_quantify") >> (fn q => fn ctxt => SIMPLE_METHOD' (Mir_Tac.mir_tac ctxt (not q))) \<close> "decision procedure for MIR arithmetic" lemma "\<forall>x::real. (\<lfloor>x\<rfloor> = \<lceil>x\<rceil> \<longleftrightarrow> (x = real_of_int \<lfloor>x\<rfloor>))" by mir lemma "\<forall>x::real. real_of_int (2::int)*x - (real_of_int (1::int)) < real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil> \<and> real_of_int \<lfloor>x\<rfloor> + real_of_int \<lceil>x\<rceil> \<le> real_of_int (2::int)*x + (real_of_int (1::int))" by mir lemma "\<forall>x::real. 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>" by mir lemma "\<forall>x::real. \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)" by mir lemma "\<forall>(x::real) (y::real). \<lfloor>x\<rfloor> = \<lfloor>y\<rfloor> \<longrightarrow> 0 \<le> \<bar>y - x\<bar> \<and> \<bar>y - x\<bar> \<le> 1" by mir end