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src/HOL/Decision_Procs/Ferrack.thy

author | hoelzl |

Wed, 11 Mar 2009 10:58:18 +0100 | |

changeset 30439 | 57c68b3af2ea |

parent 30042 | 31039ee583fa |

child 30684 | c98a64746c69 |

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

Updated paths in Decision_Procs comments and NEWS

(* Title: HOL/Decision_Procs/Ferrack.thy Author: Amine Chaieb *) theory Ferrack imports Complex_Main Dense_Linear_Order Efficient_Nat uses ("ferrack_tac.ML") begin section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *} (*********************************************************************************) (* SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB *) (*********************************************************************************) consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" primrec "alluopairs [] = []" "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)" lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}" by (induct xs, auto) lemma alluopairs_set: "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) " by (induct xs, auto) lemma alluopairs_ex: assumes Pc: "\<forall> x y. P x y = P y x" shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)" proof assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y" by blast from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" by auto next assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y" then obtain "x" and "y" where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+ from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast qed lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)" using Nat.gr0_conv_Suc by clarsimp lemma filter_length: "length (List.filter P xs) < Suc (length xs)" apply (induct xs, auto) done consts remdps:: "'a list \<Rightarrow> 'a list" recdef remdps "measure size" "remdps [] = []" "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))" (hints simp add: filter_length[rule_format]) lemma remdps_set[simp]: "set (remdps xs) = set xs" by (induct xs rule: remdps.induct, auto) (*********************************************************************************) (**** SHADOW SYNTAX AND SEMANTICS ****) (*********************************************************************************) datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num | Mul int num (* A size for num to make inductive proofs simpler*) consts num_size :: "num \<Rightarrow> nat" primrec "num_size (C c) = 1" "num_size (Bound n) = 1" "num_size (Neg a) = 1 + num_size a" "num_size (Add a b) = 1 + num_size a + num_size b" "num_size (Sub a b) = 3 + num_size a + num_size b" "num_size (Mul c a) = 1 + num_size a" "num_size (CN n c a) = 3 + num_size a " (* Semantics of numeral terms (num) *) consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real" primrec "Inum bs (C c) = (real c)" "Inum bs (Bound n) = bs!n" "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)" "Inum bs (Neg a) = -(Inum bs a)" "Inum bs (Add a b) = Inum bs a + Inum bs b" "Inum bs (Sub a b) = Inum bs a - Inum bs b" "Inum bs (Mul c a) = (real c) * Inum bs a" (* FORMULAE *) datatype fm = T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| NOT fm| And fm fm| Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm (* A size for fm *) consts fmsize :: "fm \<Rightarrow> nat" recdef fmsize "measure size" "fmsize (NOT p) = 1 + fmsize p" "fmsize (And p q) = 1 + fmsize p + fmsize q" "fmsize (Or p q) = 1 + fmsize p + fmsize q" "fmsize (Imp p q) = 3 + fmsize p + fmsize q" "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)" "fmsize (E p) = 1 + fmsize p" "fmsize (A p) = 4+ fmsize p" "fmsize p = 1" (* several lemmas about fmsize *) lemma fmsize_pos: "fmsize p > 0" by (induct p rule: fmsize.induct) simp_all (* Semantics of formulae (fm) *) consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" primrec "Ifm bs T = True" "Ifm bs F = False" "Ifm bs (Lt a) = (Inum bs a < 0)" "Ifm bs (Gt a) = (Inum bs a > 0)" "Ifm bs (Le a) = (Inum bs a \<le> 0)" "Ifm bs (Ge a) = (Inum bs a \<ge> 0)" "Ifm bs (Eq a) = (Inum bs a = 0)" "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)" "Ifm bs (NOT p) = (\<not> (Ifm bs p))" "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)" "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)" "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))" "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)" "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)" "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)" lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')" apply simp done lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')" apply simp done lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')" apply simp done lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))" apply simp done lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))" apply simp done lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))" apply simp done lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))" apply simp done lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))" apply simp done lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))" apply simp done lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))" apply simp done consts not:: "fm \<Rightarrow> fm" recdef not "measure size" "not (NOT p) = p" "not T = F" "not F = T" "not p = NOT p" lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)" by (cases p) auto constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else if p = q then p else And p q)" lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)" by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all) constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p else if p=q then p else Or p q)" lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)" by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all) constdefs imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p else Imp p q)" lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)" by (cases "p=F \<or> q=T",simp_all add: imp_def) constdefs iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else Iff p q)" lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)" by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto) lemma conj_simps: "conj F Q = F" "conj P F = F" "conj T Q = Q" "conj P T = P" "conj P P = P" "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q" by (simp_all add: conj_def) lemma disj_simps: "disj T Q = T" "disj P T = T" "disj F Q = Q" "disj P F = P" "disj P P = P" "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q" by (simp_all add: disj_def) lemma imp_simps: "imp F Q = T" "imp P T = T" "imp T Q = Q" "imp P F = not P" "imp P P = T" "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q" by (simp_all add: imp_def) lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p" apply (induct p, auto) done lemma iff_simps: "iff p p = T" "iff p (NOT p) = F" "iff (NOT p) p = F" "iff p F = not p" "iff F p = not p" "p \<noteq> NOT T \<Longrightarrow> iff T p = p" "p\<noteq> NOT T \<Longrightarrow> iff p T = p" "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q" using trivNOT by (simp_all add: iff_def, cases p, auto) (* Quantifier freeness *) consts qfree:: "fm \<Rightarrow> bool" recdef qfree "measure size" "qfree (E p) = False" "qfree (A p) = False" "qfree (NOT p) = qfree p" "qfree (And p q) = (qfree p \<and> qfree q)" "qfree (Or p q) = (qfree p \<and> qfree q)" "qfree (Imp p q) = (qfree p \<and> qfree q)" "qfree (Iff p q) = (qfree p \<and> qfree q)" "qfree p = True" (* Boundedness and substitution *) consts numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) primrec "numbound0 (C c) = True" "numbound0 (Bound n) = (n>0)" "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)" "numbound0 (Neg a) = numbound0 a" "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)" "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" "numbound0 (Mul i a) = numbound0 a" lemma numbound0_I: assumes nb: "numbound0 a" shows "Inum (b#bs) a = Inum (b'#bs) a" using nb by (induct a rule: numbound0.induct,auto simp add: nth_pos2) primrec "bound0 T = True" "bound0 F = True" "bound0 (Lt a) = numbound0 a" "bound0 (Le a) = numbound0 a" "bound0 (Gt a) = numbound0 a" "bound0 (Ge a) = numbound0 a" "bound0 (Eq a) = numbound0 a" "bound0 (NEq a) = numbound0 a" "bound0 (NOT p) = bound0 p" "bound0 (And p q) = (bound0 p \<and> bound0 q)" "bound0 (Or p q) = (bound0 p \<and> bound0 q)" "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))" "bound0 (Iff p q) = (bound0 p \<and> bound0 q)" "bound0 (E p) = False" "bound0 (A p) = False" lemma bound0_I: assumes bp: "bound0 p" shows "Ifm (b#bs) p = Ifm (b'#bs) p" using bp numbound0_I[where b="b" and bs="bs" and b'="b'"] by (induct p rule: bound0.induct) (auto simp add: nth_pos2) lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)" by (cases p, auto) lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)" by (cases p, auto) lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)" using conj_def by auto lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)" using conj_def by auto lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)" using disj_def by auto lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)" using disj_def by auto lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)" using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def) lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)" using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def) lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)" by (unfold iff_def,cases "p=q", auto) lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)" using iff_def by (unfold iff_def,cases "p=q", auto) consts decrnum:: "num \<Rightarrow> num" decr :: "fm \<Rightarrow> fm" recdef decrnum "measure size" "decrnum (Bound n) = Bound (n - 1)" "decrnum (Neg a) = Neg (decrnum a)" "decrnum (Add a b) = Add (decrnum a) (decrnum b)" "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)" "decrnum (Mul c a) = Mul c (decrnum a)" "decrnum (CN n c a) = CN (n - 1) c (decrnum a)" "decrnum a = a" recdef decr "measure size" "decr (Lt a) = Lt (decrnum a)" "decr (Le a) = Le (decrnum a)" "decr (Gt a) = Gt (decrnum a)" "decr (Ge a) = Ge (decrnum a)" "decr (Eq a) = Eq (decrnum a)" "decr (NEq a) = NEq (decrnum a)" "decr (NOT p) = NOT (decr p)" "decr (And p q) = conj (decr p) (decr q)" "decr (Or p q) = disj (decr p) (decr q)" "decr (Imp p q) = imp (decr p) (decr q)" "decr (Iff p q) = iff (decr p) (decr q)" "decr p = p" lemma decrnum: assumes nb: "numbound0 t" shows "Inum (x#bs) t = Inum bs (decrnum t)" using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2) lemma decr: assumes nb: "bound0 p" shows "Ifm (x#bs) p = Ifm bs (decr p)" using nb by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum) lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)" by (induct p, simp_all) consts isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) recdef isatom "measure size" "isatom T = True" "isatom F = True" "isatom (Lt a) = True" "isatom (Le a) = True" "isatom (Gt a) = True" "isatom (Ge a) = True" "isatom (Eq a) = True" "isatom (NEq a) = True" "isatom p = False" lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p" by (induct p, simp_all) constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" "djf f p q \<equiv> (if q=T then T else if q=F then f p else (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))" constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" "evaldjf f ps \<equiv> foldr (djf f) ps F" lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)" by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) (cases "f p", simp_all add: Let_def djf_def) lemma djf_simps: "djf f p T = T" "djf f p F = f p" "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)" by (simp_all add: djf_def) lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))" by(induct ps, simp_all add: evaldjf_def djf_Or) lemma evaldjf_bound0: assumes nb: "\<forall> x\<in> set xs. bound0 (f x)" shows "bound0 (evaldjf f xs)" using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) lemma evaldjf_qf: assumes nb: "\<forall> x\<in> set xs. qfree (f x)" shows "qfree (evaldjf f xs)" using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) consts disjuncts :: "fm \<Rightarrow> fm list" recdef disjuncts "measure size" "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)" "disjuncts F = []" "disjuncts p = [p]" lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p" by(induct p rule: disjuncts.induct, auto) lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q" proof- assume nb: "bound0 p" hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto) thus ?thesis by (simp only: list_all_iff) qed lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q" proof- assume qf: "qfree p" hence "list_all qfree (disjuncts p)" by (induct p rule: disjuncts.induct, auto) thus ?thesis by (simp only: list_all_iff) qed constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" "DJ f p \<equiv> evaldjf f (disjuncts p)" lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))" and fF: "f F = F" shows "Ifm bs (DJ f p) = Ifm bs (f p)" proof- have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))" by (simp add: DJ_def evaldjf_ex) also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto) finally show ?thesis . qed lemma DJ_qf: assumes fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)" shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) " proof(clarify) fix p assume qf: "qfree p" have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def) from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" . with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp qed lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))" proof(clarify) fix p::fm and bs assume qf: "qfree p" from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))" by (simp add: DJ_def evaldjf_ex) also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto) finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast qed (* Simplification *) consts numgcd :: "num \<Rightarrow> int" numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" reducecoeff :: "num \<Rightarrow> num" dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" consts maxcoeff:: "num \<Rightarrow> int" recdef maxcoeff "measure size" "maxcoeff (C i) = abs i" "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)" "maxcoeff t = 1" lemma maxcoeff_pos: "maxcoeff t \<ge> 0" by (induct t rule: maxcoeff.induct, auto) recdef numgcdh "measure size" "numgcdh (C i) = (\<lambda>g. zgcd i g)" "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))" "numgcdh t = (\<lambda>g. 1)" defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)" recdef reducecoeffh "measure size" "reducecoeffh (C i) = (\<lambda> g. C (i div g))" "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))" "reducecoeffh t = (\<lambda>g. t)" defs reducecoeff_def: "reducecoeff t \<equiv> (let g = numgcd t in if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)" recdef dvdnumcoeff "measure size" "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)" "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))" "dvdnumcoeff t = (\<lambda>g. False)" lemma dvdnumcoeff_trans: assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'" shows "dvdnumcoeff t g" using dgt' gdg by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg]) declare dvd_trans [trans add] lemma natabs0: "(nat (abs x) = 0) = (x = 0)" by arith lemma numgcd0: assumes g0: "numgcd t = 0" shows "Inum bs t = 0" using g0[simplified numgcd_def] by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos) lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0" using gp by (induct t rule: numgcdh.induct, auto simp add: zgcd_def) lemma numgcd_pos: "numgcd t \<ge>0" by (simp add: numgcd_def numgcdh_pos maxcoeff_pos) lemma reducecoeffh: assumes gt: "dvdnumcoeff t g" and gp: "g > 0" shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t" using gt proof(induct t rule: reducecoeffh.induct) case (1 i) hence gd: "g dvd i" by simp from gp have gnz: "g \<noteq> 0" by simp from prems show ?case by (simp add: real_of_int_div[OF gnz gd]) next case (2 n c t) hence gd: "g dvd c" by simp from gp have gnz: "g \<noteq> 0" by simp from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps) qed (auto simp add: numgcd_def gp) consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" recdef ismaxcoeff "measure size" "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)" "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))" "ismaxcoeff t = (\<lambda>x. True)" lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'" by (induct t rule: ismaxcoeff.induct, auto) lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)" proof (induct t rule: maxcoeff.induct) case (2 n c t) hence H:"ismaxcoeff t (maxcoeff t)" . have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2) from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1) qed simp_all lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))" apply (cases "abs i = 0", simp_all add: zgcd_def) apply (cases "abs j = 0", simp_all) apply (cases "abs i = 1", simp_all) apply (cases "abs j = 1", simp_all) apply auto done lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow> m =0" by (induct t rule: numgcdh.induct, auto simp add:zgcd0) lemma dvdnumcoeff_aux: assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1" shows "dvdnumcoeff t (numgcdh t m)" using prems proof(induct t rule: numgcdh.induct) case (2 n c t) let ?g = "numgcdh t m" from prems have th:"zgcd c ?g > 1" by simp from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"] have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp moreover {assume "abs c > 1" and gp: "?g > 1" with prems have th: "dvdnumcoeff t ?g" by simp have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)} moreover {assume "abs c = 0 \<and> ?g > 1" with prems have th: "dvdnumcoeff t ?g" by simp have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2) from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1) hence ?case by simp } moreover {assume "abs c > 1" and g0:"?g = 0" from numgcdh0[OF g0] have "m=0". with prems have ?case by simp } ultimately show ?case by blast qed(auto simp add: zgcd_zdvd1) lemma dvdnumcoeff_aux2: assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0" using prems proof (simp add: numgcd_def) let ?mc = "maxcoeff t" let ?g = "numgcdh t ?mc" have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff) have th2: "?mc \<ge> 0" by (rule maxcoeff_pos) assume H: "numgcdh t ?mc > 1" from dvdnumcoeff_aux[OF th1 th2 H] show "dvdnumcoeff t ?g" . qed lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t" proof- let ?g = "numgcd t" have "?g \<ge> 0" by (simp add: numgcd_pos) hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} moreover { assume g1:"?g > 1" from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+ from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis by (simp add: reducecoeff_def Let_def)} ultimately show ?thesis by blast qed lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)" by (induct t rule: reducecoeffh.induct, auto) lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)" using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def) consts simpnum:: "num \<Rightarrow> num" numadd:: "num \<times> num \<Rightarrow> num" nummul:: "num \<Rightarrow> int \<Rightarrow> num" recdef numadd "measure (\<lambda> (t,s). size t + size s)" "numadd (CN n1 c1 r1,CN n2 c2 r2) = (if n1=n2 then (let c = c1 + c2 in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2)))) else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))" "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))" "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" "numadd (C b1, C b2) = C (b1+b2)" "numadd (a,b) = Add a b" lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)" apply (induct t s rule: numadd.induct, simp_all add: Let_def) apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all) apply (case_tac "n1 = n2", simp_all add: algebra_simps) by (simp only: left_distrib[symmetric],simp) lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))" by (induct t s rule: numadd.induct, auto simp add: Let_def) recdef nummul "measure size" "nummul (C j) = (\<lambda> i. C (i*j))" "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))" "nummul t = (\<lambda> i. Mul i t)" lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)" by (induct t rule: nummul.induct, auto simp add: algebra_simps) lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)" by (induct t rule: nummul.induct, auto ) constdefs numneg :: "num \<Rightarrow> num" "numneg t \<equiv> nummul t (- 1)" constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num" "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))" lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)" using numneg_def by simp lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)" using numneg_def by simp lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)" using numsub_def by simp lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)" using numsub_def by simp recdef simpnum "measure size" "simpnum (C j) = C j" "simpnum (Bound n) = CN n 1 (C 0)" "simpnum (Neg t) = numneg (simpnum t)" "simpnum (Add t s) = numadd (simpnum t,simpnum s)" "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)" "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)" "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))" lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t" by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul) lemma simpnum_numbound0[simp]: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)" by (induct t rule: simpnum.induct, auto) consts nozerocoeff:: "num \<Rightarrow> bool" recdef nozerocoeff "measure size" "nozerocoeff (C c) = True" "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)" "nozerocoeff t = True" lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))" by (induct a b rule: numadd.induct,auto simp add: Let_def) lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)" by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz) lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)" by (simp add: numneg_def nummul_nz) lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)" by (simp add: numsub_def numneg_nz numadd_nz) lemma simpnum_nz: "nozerocoeff (simpnum t)" by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz) lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0" proof (induct t rule: maxcoeff.induct) case (2 n c t) hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+ have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1) with cnz have "max (abs c) (maxcoeff t) > 0" by arith with prems show ?case by simp qed auto lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" proof- from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) from numgcdh0[OF th] have th:"maxcoeff t = 0" . from maxcoeff_nz[OF nz th] show ?thesis . qed constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int" "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else (let t' = simpnum t ; g = numgcd t' in if g > 1 then (let g' = zgcd n g in if g' = 1 then (t',n) else (reducecoeffh t' g', n div g')) else (t',n))))" lemma simp_num_pair_ci: shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))" (is "?lhs = ?rhs") proof- let ?t' = "simpnum t" let ?g = "numgcd ?t'" let ?g' = "zgcd n ?g" {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)} moreover { assume nnz: "n \<noteq> 0" {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} moreover {assume g1:"?g>1" hence g0: "?g > 0" by simp from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith hence "?g'= 1 \<or> ?g' > 1" by arith moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)} moreover {assume g'1:"?g'>1" from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" .. let ?tt = "reducecoeffh ?t' ?g'" let ?t = "Inum bs ?tt" have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) have gpdgp: "?g' dvd ?g'" by simp from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] have th2:"real ?g' * ?t = Inum bs ?t'" by simp from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def) also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp also have "\<dots> = (Inum bs ?t' / real n)" using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci) then have ?thesis using prems by (simp add: simp_num_pair_def)} ultimately have ?thesis by blast} ultimately have ?thesis by blast} ultimately show ?thesis by blast qed lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')" shows "numbound0 t' \<and> n' >0" proof- let ?t' = "simpnum t" let ?g = "numgcd ?t'" let ?g' = "zgcd n ?g" {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)} moreover { assume nnz: "n \<noteq> 0" {assume "\<not> ?g > 1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} moreover {assume g1:"?g>1" hence g0: "?g > 0" by simp from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith hence "?g'= 1 \<or> ?g' > 1" by arith moreover {assume "?g'=1" hence ?thesis using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)} moreover {assume g'1:"?g'>1" have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2) have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1) have gpdgp: "?g' dvd ?g'" by simp from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" . from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]] have "n div ?g' >0" by simp hence ?thesis using prems by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)} ultimately have ?thesis by blast} ultimately have ?thesis by blast} ultimately show ?thesis by blast qed consts simpfm :: "fm \<Rightarrow> fm" recdef simpfm "measure fmsize" "simpfm (And p q) = conj (simpfm p) (simpfm q)" "simpfm (Or p q) = disj (simpfm p) (simpfm q)" "simpfm (Imp p q) = imp (simpfm p) (simpfm q)" "simpfm (Iff p q) = iff (simpfm p) (simpfm q)" "simpfm (NOT p) = not (simpfm p)" "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F | _ \<Rightarrow> Lt a')" "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0) then T else F | _ \<Rightarrow> Le a')" "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0) then T else F | _ \<Rightarrow> Gt a')" "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0) then T else F | _ \<Rightarrow> Ge a')" "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0) then T else F | _ \<Rightarrow> Eq a')" "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0) then T else F | _ \<Rightarrow> NEq a')" "simpfm p = p" lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p" proof(induct p rule: simpfm.induct) case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast next case (7 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast next case (8 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast next case (9 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast next case (10 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast next case (11 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp {fix v assume "?sa = C v" hence ?case using sa by simp } moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa by (cases ?sa, simp_all add: Let_def)} ultimately show ?case by blast qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not) lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)" proof(induct p rule: simpfm.induct) case (6 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) next case (7 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) next case (8 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) next case (9 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) next case (10 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) next case (11 a) hence nb: "numbound0 a" by simp hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb]) thus ?case by (cases "simpnum a", auto simp add: Let_def) qed(auto simp add: disj_def imp_def iff_def conj_def not_bn) lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)" by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def) (case_tac "simpnum a",auto)+ consts prep :: "fm \<Rightarrow> fm" recdef prep "measure fmsize" "prep (E T) = T" "prep (E F) = F" "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))" "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))" "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))" "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))" "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))" "prep (E p) = E (prep p)" "prep (A (And p q)) = conj (prep (A p)) (prep (A q))" "prep (A p) = prep (NOT (E (NOT p)))" "prep (NOT (NOT p)) = prep p" "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))" "prep (NOT (A p)) = prep (E (NOT p))" "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))" "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))" "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))" "prep (NOT p) = not (prep p)" "prep (Or p q) = disj (prep p) (prep q)" "prep (And p q) = conj (prep p) (prep q)" "prep (Imp p q) = prep (Or (NOT p) q)" "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))" "prep p = p" (hints simp add: fmsize_pos) lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p" by (induct p rule: prep.induct, auto) (* Generic quantifier elimination *) consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" recdef qelim "measure fmsize" "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))" "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))" "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))" "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" "qelim (Or p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))" "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))" "qelim p = (\<lambda> y. simpfm p)" lemma qelim_ci: assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))" shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)" using qe_inv DJ_qe[OF qe_inv] by(induct p rule: qelim.induct) (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf simpfm simpfm_qf simp del: simpfm.simps) consts plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) recdef minusinf "measure size" "minusinf (And p q) = conj (minusinf p) (minusinf q)" "minusinf (Or p q) = disj (minusinf p) (minusinf q)" "minusinf (Eq (CN 0 c e)) = F" "minusinf (NEq (CN 0 c e)) = T" "minusinf (Lt (CN 0 c e)) = T" "minusinf (Le (CN 0 c e)) = T" "minusinf (Gt (CN 0 c e)) = F" "minusinf (Ge (CN 0 c e)) = F" "minusinf p = p" recdef plusinf "measure size" "plusinf (And p q) = conj (plusinf p) (plusinf q)" "plusinf (Or p q) = disj (plusinf p) (plusinf q)" "plusinf (Eq (CN 0 c e)) = F" "plusinf (NEq (CN 0 c e)) = T" "plusinf (Lt (CN 0 c e)) = F" "plusinf (Le (CN 0 c e)) = F" "plusinf (Gt (CN 0 c e)) = T" "plusinf (Ge (CN 0 c e)) = T" "plusinf p = p" consts isrlfm :: "fm \<Rightarrow> bool" (* Linearity test for fm *) recdef isrlfm "measure size" "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" "isrlfm (Eq (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm (Lt (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm (Le (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm (Gt (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm (Ge (CN 0 c e)) = (c>0 \<and> numbound0 e)" "isrlfm p = (isatom p \<and> (bound0 p))" (* splits the bounded from the unbounded part*) consts rsplit0 :: "num \<Rightarrow> int \<times> num" recdef rsplit0 "measure num_size" "rsplit0 (Bound 0) = (1,C 0)" "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b in (ca+cb, Add ta tb))" "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))" "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))" "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))" "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))" "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))" "rsplit0 t = (0,t)" lemma rsplit0: shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))" proof (induct t rule: rsplit0.induct) case (2 a b) let ?sa = "rsplit0 a" let ?sb = "rsplit0 b" let ?ca = "fst ?sa" let ?cb = "fst ?sb" let ?ta = "snd ?sa" let ?tb = "snd ?sb" from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" by(cases "rsplit0 a",auto simp add: Let_def split_def) have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)" by (simp add: Let_def split_def algebra_simps) also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all) finally show ?case using nb by simp qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric]) (* Linearize a formula*) definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where "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 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 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 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 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 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: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))" using rsplit0[where bs = "bs" and t="t"] by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto) lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)" by (auto simp add: conj_def) lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)" by (auto simp add: disj_def) consts rlfm :: "fm \<Rightarrow> fm" recdef rlfm "measure fmsize" "rlfm (And p q) = conj (rlfm p) (rlfm q)" "rlfm (Or p q) = disj (rlfm p) (rlfm q)" "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)" "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))" "rlfm (Lt a) = split lt (rsplit0 a)" "rlfm (Le a) = split le (rsplit0 a)" "rlfm (Gt a) = split gt (rsplit0 a)" "rlfm (Ge a) = split ge (rsplit0 a)" "rlfm (Eq a) = split eq (rsplit0 a)" "rlfm (NEq a) = split neq (rsplit0 a)" "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))" "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))" "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))" "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))" "rlfm (NOT (NOT p)) = rlfm p" "rlfm (NOT T) = F" "rlfm (NOT F) = T" "rlfm (NOT (Lt a)) = rlfm (Ge a)" "rlfm (NOT (Le a)) = rlfm (Gt a)" "rlfm (NOT (Gt a)) = rlfm (Le a)" "rlfm (NOT (Ge a)) = rlfm (Lt a)" "rlfm (NOT (Eq a)) = rlfm (NEq a)" "rlfm (NOT (NEq a)) = rlfm (Eq a)" "rlfm p = p" (hints simp add: fmsize_pos) lemma rlfm_I: assumes qfp: "qfree p" shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)" using qfp by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin) (* Operations needed for Ferrante and Rackoff *) lemma rminusinf_inf: assumes lp: "isrlfm p" shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") using lp proof (induct p rule: minusinf.induct) case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto next case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto next case (3 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith hence "real c * x + ?e \<noteq> 0" by simp with xz have "?P ?z x (Eq (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp thus ?case by blast next case (4 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith hence "real c * x + ?e \<noteq> 0" by simp with xz have "?P ?z x (NEq (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp thus ?case by blast next case (5 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith with xz have "?P ?z x (Lt (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp thus ?case by blast next case (6 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith with xz have "?P ?z x (Le (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp thus ?case by blast next case (7 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith with xz have "?P ?z x (Gt (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp thus ?case by blast next case (8 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x < ?z" hence "(real c * x < - ?e)" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) hence "real c * x + ?e < 0" by arith with xz have "?P ?z x (Ge (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp thus ?case by blast qed simp_all lemma rplusinf_inf: assumes lp: "isrlfm p" shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p") using lp proof (induct p rule: isrlfm.induct) case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto next case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto next case (3 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith hence "real c * x + ?e \<noteq> 0" by simp with xz have "?P ?z x (Eq (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp thus ?case by blast next case (4 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith hence "real c * x + ?e \<noteq> 0" by simp with xz have "?P ?z x (NEq (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp thus ?case by blast next case (5 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith with xz have "?P ?z x (Lt (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp thus ?case by blast next case (6 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith with xz have "?P ?z x (Le (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp thus ?case by blast next case (7 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith with xz have "?P ?z x (Gt (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp thus ?case by blast next case (8 c e) from prems have nb: "numbound0 e" by simp from prems have cp: "real c > 0" by simp fix a let ?e="Inum (a#bs) e" let ?z = "(- ?e) / real c" {fix x assume xz: "x > ?z" with mult_strict_right_mono [OF xz cp] cp have "(real c * x > - ?e)" by (simp add: mult_ac) hence "real c * x + ?e > 0" by arith with xz have "?P ?z x (Ge (CN 0 c e))" using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp } hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp thus ?case by blast qed simp_all lemma rminusinf_bound0: assumes lp: "isrlfm p" shows "bound0 (minusinf p)" using lp by (induct p rule: minusinf.induct) simp_all lemma rplusinf_bound0: assumes lp: "isrlfm p" shows "bound0 (plusinf p)" using lp by (induct p rule: plusinf.induct) simp_all lemma rminusinf_ex: assumes lp: "isrlfm p" and ex: "Ifm (a#bs) (minusinf p)" shows "\<exists> x. Ifm (x#bs) p" proof- from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto from rminusinf_inf[OF lp, where bs="bs"] obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp moreover have "z - 1 < z" by simp ultimately show ?thesis using z_def by auto qed lemma rplusinf_ex: assumes lp: "isrlfm p" and ex: "Ifm (a#bs) (plusinf p)" shows "\<exists> x. Ifm (x#bs) p" proof- from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto from rplusinf_inf[OF lp, where bs="bs"] obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp moreover have "z + 1 > z" by simp ultimately show ?thesis using z_def by auto qed consts uset:: "fm \<Rightarrow> (num \<times> int) list" usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm " recdef uset "measure size" "uset (And p q) = (uset p @ uset q)" "uset (Or p q) = (uset p @ uset q)" "uset (Eq (CN 0 c e)) = [(Neg e,c)]" "uset (NEq (CN 0 c e)) = [(Neg e,c)]" "uset (Lt (CN 0 c e)) = [(Neg e,c)]" "uset (Le (CN 0 c e)) = [(Neg e,c)]" "uset (Gt (CN 0 c e)) = [(Neg e,c)]" "uset (Ge (CN 0 c e)) = [(Neg e,c)]" "uset p = []" recdef usubst "measure size" "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))" "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))" "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))" "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))" "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))" "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))" "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))" "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))" "usubst p = (\<lambda> (t,n). p)" lemma usubst_I: assumes lp: "isrlfm p" and np: "real n > 0" and nbt: "numbound0 t" shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _") using lp proof(induct p rule: usubst.induct) case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)" by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)" by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)" by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)" by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ from np have np: "real n \<noteq> 0" by simp have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)" by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) next case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+ from np have np: "real n \<noteq> 0" by simp have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)" using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)" by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" and b="0", simplified divide_zero_left]) (simp only: algebra_simps) also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)" using np by simp finally show ?case using nbt nb by (simp add: algebra_simps) qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2) lemma uset_l: assumes lp: "isrlfm p" shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0" using lp by(induct p rule: uset.induct,auto) lemma rminusinf_uset: assumes lp: "isrlfm p" and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") and ex: "Ifm (x#bs) p" (is "?I x p") shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m") proof- have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s") using lp nmi ex by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast from uset_l[OF lp] smU have mp: "real m > 0" by auto from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" by (auto simp add: mult_commute) thus ?thesis using smU by auto qed lemma rplusinf_uset: assumes lp: "isrlfm p" and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))") and ex: "Ifm (x#bs) p" (is "?I x p") shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m") proof- have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s") using lp nmi ex by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2) then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast from uset_l[OF lp] smU have mp: "real m > 0" by auto from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" by (auto simp add: mult_commute) thus ?thesis using smU by auto qed lemma lin_dense: assumes lp: "isrlfm p" and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)") and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p" and ly: "l < y" and yu: "y < u" shows "Ifm (y#bs) p" using lp px noS proof (induct p rule: isrlfm.induct) case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps) hence pxc: "x < (- ?N x e) / real c" by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"]) from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto moreover {assume y: "y < (-?N x e)/ real c" hence "y * real c < - ?N x e" by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real c * y + ?N x e < 0" by (simp add: 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 c" with yu have eu: "u > (- ?N x e) / real c" by auto with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) with lx pxc have "False" by auto hence ?case by simp } ultimately show ?case by blast next case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp + from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps) hence pxc: "x \<le> (- ?N x e) / real c" by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"]) from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto moreover {assume y: "y < (-?N x e)/ real c" hence "y * real c < - ?N x e" by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real c * y + ?N x e < 0" by (simp add: 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 c" with yu have eu: "u > (- ?N x e) / real c" by auto with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto) with lx pxc have "False" by auto hence ?case by simp } ultimately show ?case by blast next case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps) hence pxc: "x > (- ?N x e) / real c" by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"]) from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto moreover {assume y: "y > (-?N x e)/ real c" hence "y * real c > - ?N x e" by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real c * y + ?N x e > 0" by (simp add: 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 c" with ly have eu: "l < (- ?N x e) / real c" by auto with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) with xu pxc have "False" by auto hence ?case by simp } ultimately show ?case by blast next case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps) hence pxc: "x \<ge> (- ?N x e) / real c" by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"]) from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto moreover {assume y: "y > (-?N x e)/ real c" hence "y * real c > - ?N x e" by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric]) hence "real c * y + ?N x e > 0" by (simp add: 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 c" with ly have eu: "l < (- ?N x e) / real c" by auto with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto) with xu pxc have "False" by auto hence ?case by simp } ultimately show ?case by blast next case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ from cp have cnz: "real c \<noteq> 0" by simp from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps) hence pxc: "x = (- ?N x e) / real c" by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"]) from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto with pxc show ?case by simp next case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+ from cp have cnz: "real c \<noteq> 0" by simp from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto hence "y* real c \<noteq> -?N x e" by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp hence "y* real c + ?N x e \<noteq> 0" by (simp add: 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: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"]) lemma finite_set_intervals: assumes px: "P (x::real)" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x" proof- let ?Mx = "{y. y\<in> S \<and> y \<le> x}" let ?xM = "{y. y\<in> S \<and> x \<le> y}" let ?a = "Max ?Mx" let ?b = "Min ?xM" have MxS: "?Mx \<subseteq> S" by blast hence fMx: "finite ?Mx" using fS finite_subset by auto from lx linS have linMx: "l \<in> ?Mx" by blast hence Mxne: "?Mx \<noteq> {}" by blast have xMS: "?xM \<subseteq> S" by blast hence fxM: "finite ?xM" using fS finite_subset by auto from xu uinS have linxM: "u \<in> ?xM" by blast hence xMne: "?xM \<noteq> {}" by blast have ax:"?a \<le> x" using Mxne fMx by auto have xb:"x \<le> ?b" using xMne fxM by auto have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S" proof(clarsimp) fix y assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp} moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp} ultimately show "False" by blast qed from ainS binS noy ax xb px show ?thesis by blast qed lemma finite_set_intervals2: assumes px: "P (x::real)" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" and uinS: "u \<in> S" and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u" shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)" proof- from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] obtain a and b where as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto thus ?thesis using px as bs noS by blast qed lemma rinf_uset: assumes lp: "isrlfm p" and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))") and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))") and ex: "\<exists> x. Ifm (x#bs) p" (is "\<exists> x. ?I x p") shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" proof- let ?N = "\<lambda> x t. Inum (x#bs) t" let ?U = "set (uset p)" from ex obtain a where pa: "?I a p" by blast from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi have nmi': "\<not> (?I a (?M p))" by simp from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi have npi': "\<not> (?I a (?P p))" by simp have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p" proof- let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U" have fM: "finite ?M" by auto from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast then obtain "t" "n" "s" "m" where tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto from tnU have Mne: "?M \<noteq> {}" by auto hence Une: "?U \<noteq> {}" by simp let ?l = "Min ?M" let ?u = "Max ?M" have linM: "?l \<in> ?M" using fM Mne by simp have uinM: "?u \<in> ?M" using fM Mne by simp have tnM: "?N a t / real n \<in> ?M" using tnU by auto have smM: "?N a s / real m \<in> ?M" using smU by auto have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu] have "(\<exists> s\<in> ?M. ?I s p) \<or> (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" . moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p" hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast have "(u + u) / 2 = u" by auto with pu tuu have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp with tuU have ?thesis by blast} moreover{ assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p" then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p" by blast from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast from t1x xt2 have t1t2: "t1 < t2" by simp let ?u = "(t1 + t2) / 2" from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" . with t1uU t2uU t1u t2u have ?thesis by blast} ultimately show ?thesis by blast qed then obtain "l" "n" "s" "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp with lnU smU show ?thesis by auto qed (* The Ferrante - Rackoff Theorem *) theorem fr_eq: assumes lp: "isrlfm p" shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/ real n + (Inum (x#bs) s) / real m) /2)#bs) p))" (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") proof assume px: "\<exists> x. ?I x p" have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast moreover {assume "?M \<or> ?P" hence "?D" by blast} moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast} ultimately show "?D" by blast next assume "?D" moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } moreover {assume f:"?F" hence "?E" by blast} ultimately show "?E" by blast qed lemma fr_equsubst: assumes lp: "isrlfm p" shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))" (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D") proof assume px: "\<exists> x. ?I x p" have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast moreover {assume "?M \<or> ?P" hence "?D" by blast} moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P" let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n" let ?N = "\<lambda> t. Inum (x#bs) t" {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)" with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0" by auto let ?st = "Add (Mul m t) (Mul n s)" from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" by (simp add: mult_commute) from tnb snb have st_nb: "numbound0 ?st" by simp have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" using mnp mp np by (simp add: algebra_simps add_divide_distrib) from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])} with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast} ultimately show "?D" by blast next assume "?D" moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .} moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . } moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))" with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto let ?st = "Add (Mul l t) (Mul k s)" from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" by (simp add: mult_commute) from tnb snb have st_nb: "numbound0 ?st" by simp from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto} ultimately show "?E" by blast qed (* Implement the right hand side of Ferrante and Rackoff's Theorem. *) constdefs ferrack:: "fm \<Rightarrow> fm" "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p' in if (mp = T \<or> pp = T) then T else (let U = remdps(map simp_num_pair (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs (uset p')))) in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))" lemma uset_cong_aux: assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0" shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))" (is "?lhs = ?rhs") proof(auto) fix t n s m assume "((t,n),(s,m)) \<in> set (alluopairs U)" hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)" using alluopairs_set1[where xs="U"] by blast let ?N = "\<lambda> t. Inum (x#bs) t" let ?st= "Add (Mul m t) (Mul n s)" from Ul th have mnz: "m \<noteq> 0" by auto from Ul th have nnz: "n \<noteq> 0" by auto have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" using mnz nnz by (simp add: algebra_simps add_divide_distrib) thus "(real m * Inum (x # bs) t + real n * Inum (x # bs) s) / (2 * real n * real m) \<in> (\<lambda>((t, n), s, m). (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) ` (set U \<times> set U)"using mnz nnz th apply (auto simp add: th add_divide_distrib 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) next fix t n s m assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" let ?N = "\<lambda> t. Inum (x#bs) t" let ?st= "Add (Mul m t) (Mul n s)" from Ul smU have mnz: "m \<noteq> 0" by auto from Ul tnU have nnz: "n \<noteq> 0" by auto have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" using mnz nnz by (simp add: algebra_simps add_divide_distrib) let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" have Pc:"\<forall> a b. ?P a b = ?P b a" by auto from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast from alluopairs_ex[OF Pc, where xs="U"] tnU smU have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')" by blast then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" and Pts': "?P (t',n') (s',m')" by blast from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto let ?st' = "Add (Mul m' t') (Mul n' s')" have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')" using mnz' nnz' by (simp add: algebra_simps add_divide_distrib) from Pts' have "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st') finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2 \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) ` (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ` set (alluopairs U)" using ts'_U by blast qed lemma uset_cong: assumes lp: "isrlfm p" and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)") and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0" and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0" shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))" (is "?lhs = ?rhs") proof assume ?lhs then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast let ?N = "\<lambda> t. Inum (x#bs) t" from tnU smU U have tnb: "numbound0 t" and np: "n > 0" and snb: "numbound0 s" and mp:"m > 0" by auto let ?st= "Add (Mul m t) (Mul n s)" from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) from tnb snb have stnb: "numbound0 ?st" by simp have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" using mp np by (simp add: 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 n' > 0" by auto from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]] have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) then show ?rhs using tnU' by auto next assume ?rhs then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" by blast from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" by auto (rule_tac x="(a,b)" in bexI, auto) then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and th: "?f (t',n') = ?g((t,n),(s,m)) "by blast let ?N = "\<lambda> t. Inum (x#bs) t" from tnU smU U have tnb: "numbound0 t" and np: "n > 0" and snb: "numbound0 s" and mp:"m > 0" by auto let ?st= "Add (Mul m t) (Mul n s)" from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult) from tnb snb have stnb: "numbound0 ?st" by simp have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)" using mp np by (simp add: algebra_simps add_divide_distrib) from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt' have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast qed lemma ferrack: assumes qf: "qfree p" shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))" (is "_ \<and> (?rhs = ?lhs)") proof- let ?I = "\<lambda> x p. Ifm (x#bs) p" fix x let ?N = "\<lambda> t. Inum (x#bs) t" let ?q = "rlfm (simpfm p)" let ?U = "uset ?q" let ?Up = "alluopairs ?U" let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)" let ?S = "map ?g ?Up" let ?SS = "map simp_num_pair ?S" let ?Y = "remdps ?SS" let ?f= "(\<lambda> (t,n). ?N t / real n)" let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2" let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))" let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y" from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" . from U_l UpU have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 " by (auto simp add: mult_pos_pos) have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" proof- { fix t n assume tnY: "(t,n) \<in> set ?Y" hence "(t,n) \<in> set ?SS" by simp hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)" by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all) then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto from simp_num_pair_l[OF tnb np tns] have "numbound0 t \<and> n > 0" . } thus ?thesis by blast qed have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))" proof- from simp_num_pair_ci[where bs="x#bs"] have "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto hence th: "?f o simp_num_pair = ?f" using ext by blast have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose) also have "\<dots> = (?f ` set ?S)" by (simp add: th) also have "\<dots> = ((?f o ?g) ` set ?Up)" by (simp only: set_map o_def image_compose[symmetric]) also have "\<dots> = (?h ` (set ?U \<times> set ?U))" using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast finally show ?thesis . qed have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))" proof- { fix t n assume tnY: "(t,n) \<in> set ?Y" with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto from usubst_I[OF lq np tnb] have "bound0 (usubst ?q (t,n))" by simp hence "bound0 (simpfm (usubst ?q (t,n)))" using simpfm_bound0 by simp} thus ?thesis by blast qed hence ep_nb: "bound0 ?ep" using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto let ?mp = "minusinf ?q" let ?pp = "plusinf ?q" let ?M = "?I x ?mp" let ?P = "?I x ?pp" let ?res = "disj ?mp (disj ?pp ?ep)" from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb have nbth: "bound0 ?res" by auto from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm have th: "?lhs = (\<exists> x. ?I x ?q)" by auto from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)" by (simp only: split_def fst_conv snd_conv) also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" using uset_cong[OF lq YU U_l Y_l] by (simp only: split_def fst_conv snd_conv simpfm) also have "\<dots> = (Ifm (x#bs) ?res)" using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric] by (simp add: split_def pair_collapse) finally have lheq: "?lhs = (Ifm bs (decr ?res))" using decr[OF nbth] by blast hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def) by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+ from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def) with lr show ?thesis by blast qed definition linrqe:: "fm \<Rightarrow> fm" where "linrqe p = qelim (prep p) ferrack" theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)" using ferrack qelim_ci prep unfolding linrqe_def by auto definition ferrack_test :: "unit \<Rightarrow> fm" where "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0))) (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))" ML {* @{code ferrack_test} () *} oracle linr_oracle = {* let fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t of NONE => error "Variable not found in the list!" | SOME n => @{code Bound} n) | num_of_term vs @{term "real (0::int)"} = @{code C} 0 | num_of_term vs @{term "real (1::int)"} = @{code C} 1 | num_of_term vs @{term "0::real"} = @{code C} 0 | num_of_term vs @{term "1::real"} = @{code C} 1 | num_of_term vs (Bound i) = @{code Bound} i | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t') | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2) | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2) | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (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 (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) = @{code C} (HOLogic.dest_numeral t') | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') = @{code 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 @{term True} = @{code T} | fm_of_term vs @{term False} = @{code F} | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2) | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t') | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) = @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p) | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) = @{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) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs)) | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t' | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ term_of_num vs t1 $ term_of_num vs t2 | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ term_of_num vs t1 $ term_of_num vs t2 | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ term_of_num vs (@{code C} i) $ term_of_num vs t2 | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t)); fun term_of_fm vs @{code T} = HOLogic.true_const | term_of_fm vs @{code F} = HOLogic.false_const | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ @{term "0::real"} $ term_of_num vs t | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"} | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} 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)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ term_of_fm vs t1 $ term_of_fm vs t2 | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent."; in fn ct => let val thy = Thm.theory_of_cterm ct; val t = Thm.term_of ct; val fs = OldTerm.term_frees t; val vs = fs ~~ (0 upto (length fs - 1)); val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t)))); in Thm.cterm_of thy res end end; *} use "ferrack_tac.ML" setup Ferrack_Tac.setup lemma fixes x :: real shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1" apply rferrack done lemma fixes x :: real shows "\<exists>y \<le> x. x = y + 1" apply rferrack done lemma fixes x :: real shows "\<not> (\<exists>z. x + z = x + z + 1)" apply rferrack done end