(*  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                      *)

   (*********************************************************************************)

 primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where

   "alluopairs [] = []"

 | "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"

 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"

 by (induct xs, auto)

 lemma alluopairs_set:

   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "

 by (induct xs, auto)

 lemma alluopairs_ex:

   assumes Pc: "\<forall> x y. P x y = P y x"

   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"

 proof

   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"

   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast

   from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 

     by auto

 next

   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"

   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+

   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast

   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast

 qed

 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

   (*********************************************************************************)

   (****                            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*)

 primrec num_size :: "num \<Rightarrow> nat" where

   "num_size (C c) = 1"

 | "num_size (Bound n) = 1"

 | "num_size (Neg a) = 1 + num_size a"

 | "num_size (Add a b) = 1 + num_size a + num_size b"

 | "num_size (Sub a b) = 3 + num_size a + num_size b"

 | "num_size (Mul c a) = 1 + num_size a"

 | "num_size (CN n c a) = 3 + num_size a "

   (* Semantics of numeral terms (num) *)

 primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where

   "Inum bs (C c) = (real c)"

 | "Inum bs (Bound n) = bs!n"

 | "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"

 | "Inum bs (Neg a) = -(Inum bs a)"

 | "Inum bs (Add a b) = Inum bs a + Inum bs b"

 | "Inum bs (Sub a b) = Inum bs a - Inum bs b"

 | "Inum bs (Mul c a) = (real c) * Inum bs a"

     (* 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 *)

 fun fmsize :: "fm \<Rightarrow> nat" where

   "fmsize (NOT p) = 1 + fmsize p"

 | "fmsize (And p q) = 1 + fmsize p + fmsize q"

 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"

 | "fmsize (Imp p q) = 3 + fmsize p + fmsize q"

 | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"

 | "fmsize (E p) = 1 + fmsize p"

 | "fmsize (A p) = 4+ fmsize p"

 | "fmsize p = 1"

   (* several lemmas about fmsize *)

 lemma fmsize_pos: "fmsize p > 0"

 by (induct p rule: fmsize.induct) simp_all

   (* Semantics of formulae (fm) *)

 primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where

   "Ifm bs T = True"

 | "Ifm bs F = False"

 | "Ifm bs (Lt a) = (Inum bs a < 0)"

 | "Ifm bs (Gt a) = (Inum bs a > 0)"

 | "Ifm bs (Le a) = (Inum bs a \<le> 0)"

 | "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"

 | "Ifm bs (Eq a) = (Inum bs a = 0)"

 | "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"

 | "Ifm bs (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

 fun not:: "fm \<Rightarrow> fm" where

   "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

 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where

   "conj p q = (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)

 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where

   "disj p q = (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)

 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where

   "imp p q = (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) 

 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where

   "iff p q = (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 *)

 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 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) (simp_all add: nth_pos2)

 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 (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 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)

 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 (CN n c a) = CN (n - 1) c (decrnum a)"

 | "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 (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)

 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 p = False"

 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 (f p) 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 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) 

 fun disjuncts :: "fm \<Rightarrow> fm list" where

   "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

 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where

   "DJ f p = 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 *)

 fun maxcoeff:: "num \<Rightarrow> int" where

   "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)

 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 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 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 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: natabs0 maxcoeff_pos min_max.sup_absorb2)

 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 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)

 fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where

   "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: "gcd i j > (1::int) \<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: gcd_int_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)

 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:"gcd 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': "gcd c ?g dvd ?g" by simp

     from dvdnumcoeff_trans[OF th' th] have ?case by simp }

   moreover {assume "abs c = 0 \<and> ?g > 1"

     with prems 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 "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

 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

   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 (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)

 fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where

   "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 )

 definition numneg :: "num \<Rightarrow> num" where

   "numneg t = nummul t (- 1)"

 definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where

   "numsub s t = (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

 primrec 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 (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) simp_all

 lemma simpnum_numbound0[simp]: 

   "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"

 by (induct t) simp_all

 fun nozerocoeff:: "num \<Rightarrow> bool" where

   "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) (simp_all 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

 definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where

   "simp_num_pair = (\<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 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' = "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 simpnum_ci)}

     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 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

         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 ?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' = "gcd 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 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 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

         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 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

 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 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 *)

 function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where

   "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)"

 by pat_completeness auto

 termination qelim by (relation "measure fmsize") simp_all

 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)

 fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) 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"

 fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) 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 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))"

   (* splits the bounded from the unbounded part*)

 function (sequential) rsplit0 :: "num \<Rightarrow> int \<times> num" where

   "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)"

 by pat_completeness auto

 termination rsplit0 by (relation "measure num_size") simp_all

 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") auto

   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. *)

 definition ferrack :: "fm \<Rightarrow> fm" where

   "ferrack p = (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'

                 in if (mp = T \<or> pp = T) then T else 

                    (let U = remdups(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 = "remdups ?SS"

   let ?f= "(\<lambda> (t,n). ?N t / real n)"

   let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"

   let ?F = "\<lambda> p. \<exists> a \<in> set (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 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_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 (Free vT) = @{code Bound} (find_index (fn vT' => vT = vT') vs)

   | 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");

 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 HOL.implies} $ 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 (@{const_name Ex}, _) $ Abs (xn, xT, p)) =

       @{code E} (fm_of_term (("", dummyT) :: vs) p)

   | fm_of_term vs (Const (@{const_name All}, _) $ Abs (xn, xT, p)) =

       @{code A} (fm_of_term (("", dummyT) ::  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) = Free (nth vs n)

   | 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;

 in fn (ctxt, t) =>

   let 

     val vs = Term.add_frees t [];

     val t' = (term_of_fm vs o @{code linrqe} o fm_of_term vs) t;

   in (Thm.cterm_of (ProofContext.theory_of ctxt) o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') 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