
diff -r 0c227412b285 -r 324622260d29 src/HOL/Complex/ex/ReflectedFerrack.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Complex/ex/ReflectedFerrack.thy	Tue Jun 05 20:44:12 2007 +0200
@@ -0,0 +1,1991 @@
+(*  Title:      Complex/ex/ReflectedFerrack.thy
+    Author:     Amine Chaieb
+*)
+
+header {* Quatifier elimination for R(0,1,+,<) *}
+
+theory ReflectedFerrack
+  imports GCD Real 
+  uses ("linreif.ML") ("linrtac.ML")
+begin
+
+
+  (*********************************************************************************)
+  (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
+  (*********************************************************************************)
+
+consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
+primrec
+  "alluopairs [] = []"
+  "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
+
+lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
+by (induct xs, auto)
+
+lemma alluopairs_set:
+  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
+by (induct xs, auto)
+
+lemma alluopairs_ex:
+  assumes Pc: "\<forall> x y. P x y = P y x"
+  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
+proof
+  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
+  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
+  from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
+    by auto
+next
+  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
+  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
+  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
+  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
+qed
+
+lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
+using Nat.gr0_conv_Suc
+by clarsimp
+
+lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
+  apply (induct xs, auto) done
+
+consts remdps:: "'a list \<Rightarrow> 'a list"
+
+recdef remdps "measure size"
+  "remdps [] = []"
+  "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
+(hints simp add: filter_length[rule_format])
+
+lemma remdps_set[simp]: "set (remdps xs) = set xs"
+  by (induct xs rule: remdps.induct, auto)
+
+
+
+  (*********************************************************************************)
+  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
+  (*********************************************************************************)
+
+datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
+  | Mul int num 
+
+  (* A size for num to make inductive proofs simpler*)
+consts num_size :: "num \<Rightarrow> nat" 
+primrec 
+  "num_size (C c) = 1"
+  "num_size (Bound n) = 1"
+  "num_size (Neg a) = 1 + num_size a"
+  "num_size (Add a b) = 1 + num_size a + num_size b"
+  "num_size (Sub a b) = 3 + num_size a + num_size b"
+  "num_size (Mul c a) = 1 + num_size a"
+  "num_size (CN n c a) = 3 + num_size a "
+
+  (* Semantics of numeral terms (num) *)
+consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
+primrec
+  "Inum bs (C c) = (real c)"
+  "Inum bs (Bound n) = bs!n"
+  "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
+  "Inum bs (Neg a) = -(Inum bs a)"
+  "Inum bs (Add a b) = Inum bs a + Inum bs b"
+  "Inum bs (Sub a b) = Inum bs a - Inum bs b"
+  "Inum bs (Mul c a) = (real c) * Inum bs a"
+    (* FORMULAE *)
+datatype fm  = 
+  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
+  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
+
+
+  (* A size for fm *)
+consts fmsize :: "fm \<Rightarrow> nat"
+recdef fmsize "measure size"
+  "fmsize (NOT p) = 1 + fmsize p"
+  "fmsize (And p q) = 1 + fmsize p + fmsize q"
+  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
+  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
+  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
+  "fmsize (E p) = 1 + fmsize p"
+  "fmsize (A p) = 4+ fmsize p"
+  "fmsize p = 1"
+  (* several lemmas about fmsize *)
+lemma fmsize_pos: "fmsize p > 0"	
+by (induct p rule: fmsize.induct) simp_all
+
+  (* Semantics of formulae (fm) *)
+consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
+primrec
+  "Ifm bs T = True"
+  "Ifm bs F = False"
+  "Ifm bs (Lt a) = (Inum bs a < 0)"
+  "Ifm bs (Gt a) = (Inum bs a > 0)"
+  "Ifm bs (Le a) = (Inum bs a \<le> 0)"
+  "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
+  "Ifm bs (Eq a) = (Inum bs a = 0)"
+  "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
+  "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
+  "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
+  "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
+  "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
+  "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
+  "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
+  "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
+
+lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
+apply simp
+done
+
+lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
+apply simp
+done
+lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
+apply simp
+done
+lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
+apply simp
+done
+lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
+apply simp
+done
+lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
+apply simp
+done
+lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
+apply simp
+done
+lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
+apply simp
+done
+
+lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
+apply simp
+done
+lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
+apply simp
+done
+
+consts not:: "fm \<Rightarrow> fm"
+recdef not "measure size"
+  "not (NOT p) = p"
+  "not T = F"
+  "not F = T"
+  "not p = NOT p"
+lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
+by (cases p) auto
+
+constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
+   if p = q then p else And p q)"
+lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
+by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
+
+constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
+       else if p=q then p else Or p q)"
+
+lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
+by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
+
+constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
+    else Imp p q)"
+lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
+by (cases "p=F \<or> q=T",simp_all add: imp_def) 
+
+constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
+       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
+  Iff p q)"
+lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
+  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
+
+lemma conj_simps:
+  "conj F Q = F"
+  "conj P F = F"
+  "conj T Q = Q"
+  "conj P T = P"
+  "conj P P = P"
+  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"
+  by (simp_all add: conj_def)
+
+lemma disj_simps:
+  "disj T Q = T"
+  "disj P T = T"
+  "disj F Q = Q"
+  "disj P F = P"
+  "disj P P = P"
+  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
+  by (simp_all add: disj_def)
+lemma imp_simps:
+  "imp F Q = T"
+  "imp P T = T"
+  "imp T Q = Q"
+  "imp P F = not P"
+  "imp P P = T"
+  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
+  by (simp_all add: imp_def)
+lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
+apply (induct p, auto)
+done
+
+lemma iff_simps:
+  "iff p p = T"
+  "iff p (NOT p) = F"
+  "iff (NOT p) p = F"
+  "iff p F = not p"
+  "iff F p = not p"
+  "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
+  "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
+  "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
+  using trivNOT
+  by (simp_all add: iff_def, cases p, auto)
+  (* Quantifier freeness *)
+consts qfree:: "fm \<Rightarrow> bool"
+recdef qfree "measure size"
+  "qfree (E p) = False"
+  "qfree (A p) = False"
+  "qfree (NOT p) = qfree p" 
+  "qfree (And p q) = (qfree p \<and> qfree q)" 
+  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
+  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
+  "qfree (Iff p q) = (qfree p \<and> qfree q)"
+  "qfree p = True"
+
+  (* Boundedness and substitution *)
+consts 
+  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
+  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
+primrec
+  "numbound0 (C c) = True"
+  "numbound0 (Bound n) = (n>0)"
+  "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
+  "numbound0 (Neg a) = numbound0 a"
+  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
+  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
+  "numbound0 (Mul i a) = numbound0 a"
+lemma numbound0_I:
+  assumes nb: "numbound0 a"
+  shows "Inum (b#bs) a = Inum (b'#bs) a"
+using nb
+by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
+
+primrec
+  "bound0 T = True"
+  "bound0 F = True"
+  "bound0 (Lt a) = numbound0 a"
+  "bound0 (Le a) = numbound0 a"
+  "bound0 (Gt a) = numbound0 a"
+  "bound0 (Ge a) = numbound0 a"
+  "bound0 (Eq a) = numbound0 a"
+  "bound0 (NEq a) = numbound0 a"
+  "bound0 (NOT p) = bound0 p"
+  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
+  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
+  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
+  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
+  "bound0 (E p) = False"
+  "bound0 (A p) = False"
+
+lemma bound0_I:
+  assumes bp: "bound0 p"
+  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
+using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
+by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
+
+lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
+by (cases p, auto)
+lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
+by (cases p, auto)
+
+
+lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
+using conj_def by auto 
+lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
+using conj_def by auto 
+
+lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
+using disj_def by auto 
+lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
+using disj_def by auto 
+
+lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
+using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
+lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
+using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
+
+lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
+  by (unfold iff_def,cases "p=q", auto)
+lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
+using iff_def by (unfold iff_def,cases "p=q", auto)
+
+consts 
+  decrnum:: "num \<Rightarrow> num" 
+  decr :: "fm \<Rightarrow> fm"
+
+recdef decrnum "measure size"
+  "decrnum (Bound n) = Bound (n - 1)"
+  "decrnum (Neg a) = Neg (decrnum a)"
+  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
+  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
+  "decrnum (Mul c a) = Mul c (decrnum a)"
+  "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
+  "decrnum a = a"
+
+recdef decr "measure size"
+  "decr (Lt a) = Lt (decrnum a)"
+  "decr (Le a) = Le (decrnum a)"
+  "decr (Gt a) = Gt (decrnum a)"
+  "decr (Ge a) = Ge (decrnum a)"
+  "decr (Eq a) = Eq (decrnum a)"
+  "decr (NEq a) = NEq (decrnum a)"
+  "decr (NOT p) = NOT (decr p)" 
+  "decr (And p q) = conj (decr p) (decr q)"
+  "decr (Or p q) = disj (decr p) (decr q)"
+  "decr (Imp p q) = imp (decr p) (decr q)"
+  "decr (Iff p q) = iff (decr p) (decr q)"
+  "decr p = p"
+
+lemma decrnum: assumes nb: "numbound0 t"
+  shows "Inum (x#bs) t = Inum bs (decrnum t)"
+  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
+
+lemma decr: assumes nb: "bound0 p"
+  shows "Ifm (x#bs) p = Ifm bs (decr p)"
+  using nb 
+  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
+
+lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
+by (induct p, simp_all)
+
+consts 
+  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
+recdef isatom "measure size"
+  "isatom T = True"
+  "isatom F = True"
+  "isatom (Lt a) = True"
+  "isatom (Le a) = True"
+  "isatom (Gt a) = True"
+  "isatom (Ge a) = True"
+  "isatom (Eq a) = True"
+  "isatom (NEq a) = True"
+  "isatom p = False"
+
+lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
+by (induct p, simp_all)
+
+constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
+  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
+  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
+constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
+  "evaldjf f ps \<equiv> foldr (djf f) ps F"
+
+lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
+by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
+(cases "f p", simp_all add: Let_def djf_def) 
+
+
+lemma djf_simps:
+  "djf f p T = T"
+  "djf f p F = f p"
+  "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
+  by (simp_all add: djf_def)
+
+lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
+  by(induct ps, simp_all add: evaldjf_def djf_Or)
+
+lemma evaldjf_bound0: 
+  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
+  shows "bound0 (evaldjf f xs)"
+  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
+
+lemma evaldjf_qf: 
+  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
+  shows "qfree (evaldjf f xs)"
+  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
+
+consts disjuncts :: "fm \<Rightarrow> fm list"
+recdef disjuncts "measure size"
+  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
+  "disjuncts F = []"
+  "disjuncts p = [p]"
+
+lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
+by(induct p rule: disjuncts.induct, auto)
+
+lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
+proof-
+  assume nb: "bound0 p"
+  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
+  thus ?thesis by (simp only: list_all_iff)
+qed
+
+lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
+proof-
+  assume qf: "qfree p"
+  hence "list_all qfree (disjuncts p)"
+    by (induct p rule: disjuncts.induct, auto)
+  thus ?thesis by (simp only: list_all_iff)
+qed
+
+constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
+  "DJ f p \<equiv> evaldjf f (disjuncts p)"
+
+lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
+  and fF: "f F = F"
+  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
+proof-
+  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
+    by (simp add: DJ_def evaldjf_ex) 
+  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
+  finally show ?thesis .
+qed
+
+lemma DJ_qf: assumes 
+  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
+  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
+proof(clarify)
+  fix  p assume qf: "qfree p"
+  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
+  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
+  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
+  
+  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
+qed
+
+lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
+  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
+proof(clarify)
+  fix p::fm and bs
+  assume qf: "qfree p"
+  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
+  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
+  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
+    by (simp add: DJ_def evaldjf_ex)
+  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
+  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
+  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
+qed
+  (* Simplification *)
+consts 
+  numgcd :: "num \<Rightarrow> int"
+  numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
+  reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
+  reducecoeff :: "num \<Rightarrow> num"
+  dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
+consts maxcoeff:: "num \<Rightarrow> int"
+recdef maxcoeff "measure size"
+  "maxcoeff (C i) = abs i"
+  "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
+  "maxcoeff t = 1"
+
+lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
+  by (induct t rule: maxcoeff.induct, auto)
+
+recdef numgcdh "measure size"
+  "numgcdh (C i) = (\<lambda>g. igcd i g)"
+  "numgcdh (CN n c t) = (\<lambda>g. igcd c (numgcdh t g))"
+  "numgcdh t = (\<lambda>g. 1)"
+defs numgcd_def: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
+
+recdef reducecoeffh "measure size"
+  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
+  "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
+  "reducecoeffh t = (\<lambda>g. t)"
+
+defs reducecoeff_def: "reducecoeff t \<equiv> 
+  (let g = numgcd t in 
+  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
+
+recdef dvdnumcoeff "measure size"
+  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
+  "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
+  "dvdnumcoeff t = (\<lambda>g. False)"
+
+lemma dvdnumcoeff_trans: 
+  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
+  shows "dvdnumcoeff t g"
+  using dgt' gdg 
+  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg])
+
+declare zdvd_trans [trans add]
+
+lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
+by arith
+
+lemma numgcd0:
+  assumes g0: "numgcd t = 0"
+  shows "Inum bs t = 0"
+  using g0[simplified numgcd_def] 
+  by (induct t rule: numgcdh.induct, auto simp add: igcd_def gcd_zero natabs0 max_def maxcoeff_pos)
+
+lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
+  using gp
+  by (induct t rule: numgcdh.induct, auto simp add: igcd_def)
+
+lemma numgcd_pos: "numgcd t \<ge>0"
+  by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
+
+lemma reducecoeffh:
+  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
+  shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
+  using gt
+proof(induct t rule: reducecoeffh.induct) 
+  case (1 i) hence gd: "g dvd i" by simp
+  from gp have gnz: "g \<noteq> 0" by simp
+  from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
+next
+  case (2 n c t)  hence gd: "g dvd c" by simp
+  from gp have gnz: "g \<noteq> 0" by simp
+  from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_eq_simps)
+qed (auto simp add: numgcd_def gp)
+consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
+recdef ismaxcoeff "measure size"
+  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
+  "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
+  "ismaxcoeff t = (\<lambda>x. True)"
+
+lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
+by (induct t rule: ismaxcoeff.induct, auto)
+
+lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
+proof (induct t rule: maxcoeff.induct)
+  case (2 n c t)
+  hence H:"ismaxcoeff t (maxcoeff t)" .
+  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
+  from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
+qed simp_all
+
+lemma igcd_gt1: "igcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
+  apply (cases "abs i = 0", simp_all add: igcd_def)
+  apply (cases "abs j = 0", simp_all)
+  apply (cases "abs i = 1", simp_all)
+  apply (cases "abs j = 1", simp_all)
+  apply auto
+  done
+lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
+  by (induct t rule: numgcdh.induct, auto simp add:igcd0)
+
+lemma dvdnumcoeff_aux:
+  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
+  shows "dvdnumcoeff t (numgcdh t m)"
+using prems
+proof(induct t rule: numgcdh.induct)
+  case (2 n c t) 
+  let ?g = "numgcdh t m"
+  from prems have th:"igcd c ?g > 1" by simp
+  from igcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
+  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
+  moreover {assume "abs c > 1" and gp: "?g > 1" with prems
+    have th: "dvdnumcoeff t ?g" by simp
+    have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2)
+    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)}
+  moreover {assume "abs c = 0 \<and> ?g > 1"
+    with prems have th: "dvdnumcoeff t ?g" by simp
+    have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2)
+    from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)
+    hence ?case by simp }
+  moreover {assume "abs c > 1" and g0:"?g = 0" 
+    from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
+  ultimately show ?case by blast
+qed(auto simp add: igcd_dvd1)
+
+lemma dvdnumcoeff_aux2:
+  assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
+  using prems 
+proof (simp add: numgcd_def)
+  let ?mc = "maxcoeff t"
+  let ?g = "numgcdh t ?mc"
+  have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
+  have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
+  assume H: "numgcdh t ?mc > 1"
+  from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
+qed
+
+lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
+proof-
+  let ?g = "numgcd t"
+  have "?g \<ge> 0"  by (simp add: numgcd_pos)
+  hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
+  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
+  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
+  moreover { assume g1:"?g > 1"
+    from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
+    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
+      by (simp add: reducecoeff_def Let_def)} 
+  ultimately show ?thesis by blast
+qed
+
+lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
+by (induct t rule: reducecoeffh.induct, auto)
+
+lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
+using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
+
+consts
+  simpnum:: "num \<Rightarrow> num"
+  numadd:: "num \<times> num \<Rightarrow> num"
+  nummul:: "num \<Rightarrow> int \<Rightarrow> num"
+recdef numadd "measure (\<lambda> (t,s). size t + size s)"
+  "numadd (CN n1 c1 r1,CN n2 c2 r2) =
+  (if n1=n2 then 
+  (let c = c1 + c2
+  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
+  else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
+  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
+  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
+  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
+  "numadd (C b1, C b2) = C (b1+b2)"
+  "numadd (a,b) = Add a b"
+
+lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
+apply (induct t s rule: numadd.induct, simp_all add: Let_def)
+apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
+apply (case_tac "n1 = n2", simp_all add: ring_eq_simps)
+by (simp only: ring_eq_simps(1)[symmetric],simp)
+
+lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
+by (induct t s rule: numadd.induct, auto simp add: Let_def)
+
+recdef nummul "measure size"
+  "nummul (C j) = (\<lambda> i. C (i*j))"
+  "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
+  "nummul t = (\<lambda> i. Mul i t)"
+
+lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
+by (induct t rule: nummul.induct, auto simp add: ring_eq_simps)
+
+lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
+by (induct t rule: nummul.induct, auto )
+
+constdefs numneg :: "num \<Rightarrow> num"
+  "numneg t \<equiv> nummul t (- 1)"
+
+constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
+  "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
+
+lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
+using numneg_def by simp
+
+lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
+using numneg_def by simp
+
+lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
+using numsub_def by simp
+
+lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
+using numsub_def by simp
+
+recdef simpnum "measure size"
+  "simpnum (C j) = C j"
+  "simpnum (Bound n) = CN n 1 (C 0)"
+  "simpnum (Neg t) = numneg (simpnum t)"
+  "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
+  "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
+  "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
+  "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
+
+lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
+by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
+
+lemma simpnum_numbound0[simp]: 
+  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
+by (induct t rule: simpnum.induct, auto)
+
+consts nozerocoeff:: "num \<Rightarrow> bool"
+recdef nozerocoeff "measure size"
+  "nozerocoeff (C c) = True"
+  "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
+  "nozerocoeff t = True"
+
+lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
+by (induct a b rule: numadd.induct,auto simp add: Let_def)
+
+lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
+by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
+
+lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
+by (simp add: numneg_def nummul_nz)
+
+lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
+by (simp add: numsub_def numneg_nz numadd_nz)
+
+lemma simpnum_nz: "nozerocoeff (simpnum t)"
+by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
+
+lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
+proof (induct t rule: maxcoeff.induct)
+  case (2 n c t)
+  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
+  have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
+  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
+  with prems show ?case by simp
+qed auto
+
+lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
+proof-
+  from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
+  from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
+  from maxcoeff_nz[OF nz th] show ?thesis .
+qed
+
+constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
+  "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
+   (let t' = simpnum t ; g = numgcd t' in 
+      if g > 1 then (let g' = igcd n g in 
+        if g' = 1 then (t',n) 
+        else (reducecoeffh t' g', n div g')) 
+      else (t',n))))"
+
+lemma simp_num_pair_ci:
+  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
+  (is "?lhs = ?rhs")
+proof-
+  let ?t' = "simpnum t"
+  let ?g = "numgcd ?t'"
+  let ?g' = "igcd n ?g"
+  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
+  moreover
+  { assume nnz: "n \<noteq> 0"
+    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
+    moreover
+    {assume g1:"?g>1" hence g0: "?g > 0" by simp
+      from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
+      hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith 
+      hence "?g'= 1 \<or> ?g' > 1" by arith
+      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
+      moreover {assume g'1:"?g'>1"
+	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
+	let ?tt = "reducecoeffh ?t' ?g'"
+	let ?t = "Inum bs ?tt"
+	have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2)
+	have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) 
+	have gpdgp: "?g' dvd ?g'" by simp
+	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
+	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
+	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
+	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
+	also have "\<dots> = (Inum bs ?t' / real n)"
+	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
+	finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
+	then have ?thesis using prems by (simp add: simp_num_pair_def)}
+      ultimately have ?thesis by blast}
+    ultimately have ?thesis by blast} 
+  ultimately show ?thesis by blast
+qed
+
+lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
+  shows "numbound0 t' \<and> n' >0"
+proof-
+    let ?t' = "simpnum t"
+  let ?g = "numgcd ?t'"
+  let ?g' = "igcd n ?g"
+  {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
+  moreover
+  { assume nnz: "n \<noteq> 0"
+    {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
+    moreover
+    {assume g1:"?g>1" hence g0: "?g > 0" by simp
+      from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
+      hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith
+      hence "?g'= 1 \<or> ?g' > 1" by arith
+      moreover {assume "?g'=1" hence ?thesis using prems 
+	  by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
+      moreover {assume g'1:"?g'>1"
+	have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2)
+	have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) 
+	have gpdgp: "?g' dvd ?g'" by simp
+	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
+	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
+	have "n div ?g' >0" by simp
+	hence ?thesis using prems 
+	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
+      ultimately have ?thesis by blast}
+    ultimately have ?thesis by blast} 
+  ultimately show ?thesis by blast
+qed
+
+consts simpfm :: "fm \<Rightarrow> fm"
+recdef simpfm "measure fmsize"
+  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
+  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
+  "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
+  "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
+  "simpfm (NOT p) = not (simpfm p)"
+  "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
+  | _ \<Rightarrow> Lt a')"
+  "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
+  "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
+  "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
+  "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
+  "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
+  "simpfm p = p"
+lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
+proof(induct p rule: simpfm.induct)
+  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+next
+  case (7 a)  let ?sa = "simpnum a" 
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+next
+  case (8 a)  let ?sa = "simpnum a" 
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+next
+  case (9 a)  let ?sa = "simpnum a" 
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+next
+  case (10 a)  let ?sa = "simpnum a" 
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+next
+  case (11 a)  let ?sa = "simpnum a" 
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
+  {fix v assume "?sa = C v" hence ?case using sa by simp }
+  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
+      by (cases ?sa, simp_all add: Let_def)}
+  ultimately show ?case by blast
+qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
+
+
+lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
+proof(induct p rule: simpfm.induct)
+  case (6 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+next
+  case (7 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+next
+  case (8 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+next
+  case (9 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+next
+  case (10 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+next
+  case (11 a) hence nb: "numbound0 a" by simp
+  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  thus ?case by (cases "simpnum a", auto simp add: Let_def)
+qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
+
+lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
+by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
+ (case_tac "simpnum a",auto)+
+
+consts prep :: "fm \<Rightarrow> fm"
+recdef prep "measure fmsize"
+  "prep (E T) = T"
+  "prep (E F) = F"
+  "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
+  "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
+  "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
+  "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
+  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
+  "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
+  "prep (E p) = E (prep p)"
+  "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
+  "prep (A p) = prep (NOT (E (NOT p)))"
+  "prep (NOT (NOT p)) = prep p"
+  "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
+  "prep (NOT (A p)) = prep (E (NOT p))"
+  "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
+  "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
+  "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
+  "prep (NOT p) = not (prep p)"
+  "prep (Or p q) = disj (prep p) (prep q)"
+  "prep (And p q) = conj (prep p) (prep q)"
+  "prep (Imp p q) = prep (Or (NOT p) q)"
+  "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
+  "prep p = p"
+(hints simp add: fmsize_pos)
+lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
+by (induct p rule: prep.induct, auto)
+
+  (* Generic quantifier elimination *)
+consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
+recdef qelim "measure fmsize"
+  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
+  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
+  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
+  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
+  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
+  "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
+  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
+  "qelim p = (\<lambda> y. simpfm p)"
+
+lemma qelim_ci:
+  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
+  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
+using qe_inv DJ_qe[OF qe_inv] 
+by(induct p rule: qelim.induct) 
+(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
+  simpfm simpfm_qf simp del: simpfm.simps)
+
+consts 
+  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
+  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
+recdef minusinf "measure size"
+  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
+  "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
+  "minusinf (Eq  (CN 0 c e)) = F"
+  "minusinf (NEq (CN 0 c e)) = T"
+  "minusinf (Lt  (CN 0 c e)) = T"
+  "minusinf (Le  (CN 0 c e)) = T"
+  "minusinf (Gt  (CN 0 c e)) = F"
+  "minusinf (Ge  (CN 0 c e)) = F"
+  "minusinf p = p"
+
+recdef plusinf "measure size"
+  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
+  "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
+  "plusinf (Eq  (CN 0 c e)) = F"
+  "plusinf (NEq (CN 0 c e)) = T"
+  "plusinf (Lt  (CN 0 c e)) = F"
+  "plusinf (Le  (CN 0 c e)) = F"
+  "plusinf (Gt  (CN 0 c e)) = T"
+  "plusinf (Ge  (CN 0 c e)) = T"
+  "plusinf p = p"
+
+consts
+  isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
+recdef isrlfm "measure size"
+  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
+  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
+  "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
+  "isrlfm p = (isatom p \<and> (bound0 p))"
+
+  (* splits the bounded from the unbounded part*)
+consts rsplit0 :: "num \<Rightarrow> int \<times> num" 
+recdef rsplit0 "measure num_size"
+  "rsplit0 (Bound 0) = (1,C 0)"
+  "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
+              in (ca+cb, Add ta tb))"
+  "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
+  "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
+  "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
+  "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
+  "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
+  "rsplit0 t = (0,t)"
+lemma rsplit0: 
+  shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
+proof (induct t rule: rsplit0.induct)
+  case (2 a b) 
+  let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
+  let ?ca = "fst ?sa" let ?cb = "fst ?sb"
+  let ?ta = "snd ?sa" let ?tb = "snd ?sb"
+  from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
+    by(cases "rsplit0 a",auto simp add: Let_def split_def)
+  have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = 
+    Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
+    by (simp add: Let_def split_def ring_eq_simps)
+  also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
+  finally show ?case using nb by simp 
+qed(auto simp add: Let_def split_def ring_eq_simps , simp add: ring_eq_simps(2)[symmetric])
+
+    (* Linearize a formula*)
+consts 
+  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
+  le :: "int \<Rightarrow> num \<Rightarrow> fm"
+  gt :: "int \<Rightarrow> num \<Rightarrow> fm"
+  ge :: "int \<Rightarrow> num \<Rightarrow> fm"
+  eq :: "int \<Rightarrow> num \<Rightarrow> fm"
+  neq :: "int \<Rightarrow> num \<Rightarrow> fm"
+
+defs lt_def: "lt c t \<equiv> (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
+                        else (Gt (CN 0 (-c) (Neg t))))"
+defs le_def: "le c t \<equiv> (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
+                        else (Ge (CN 0 (-c) (Neg t))))"
+defs gt_def: "gt c t \<equiv> (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
+                        else (Lt (CN 0 (-c) (Neg t))))"
+defs ge_def: "ge c t \<equiv> (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
+                        else (Le (CN 0 (-c) (Neg t))))"
+defs eq_def: "eq c t \<equiv> (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
+                        else (Eq (CN 0 (-c) (Neg t))))"
+defs neq_def: "neq c t \<equiv> (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
+                        else (NEq (CN 0 (-c) (Neg t))))"
+
+lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
+using rsplit0[where bs = "bs" and t="t"]
+by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
+
+lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
+by (auto simp add: conj_def)
+lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
+by (auto simp add: disj_def)
+
+consts rlfm :: "fm \<Rightarrow> fm"
+recdef rlfm "measure fmsize"
+  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
+  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
+  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
+  "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
+  "rlfm (Lt a) = split lt (rsplit0 a)"
+  "rlfm (Le a) = split le (rsplit0 a)"
+  "rlfm (Gt a) = split gt (rsplit0 a)"
+  "rlfm (Ge a) = split ge (rsplit0 a)"
+  "rlfm (Eq a) = split eq (rsplit0 a)"
+  "rlfm (NEq a) = split neq (rsplit0 a)"
+  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
+  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
+  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
+  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
+  "rlfm (NOT (NOT p)) = rlfm p"
+  "rlfm (NOT T) = F"
+  "rlfm (NOT F) = T"
+  "rlfm (NOT (Lt a)) = rlfm (Ge a)"
+  "rlfm (NOT (Le a)) = rlfm (Gt a)"
+  "rlfm (NOT (Gt a)) = rlfm (Le a)"
+  "rlfm (NOT (Ge a)) = rlfm (Lt a)"
+  "rlfm (NOT (Eq a)) = rlfm (NEq a)"
+  "rlfm (NOT (NEq a)) = rlfm (Eq a)"
+  "rlfm p = p" (hints simp add: fmsize_pos)
+
+lemma rlfm_I:
+  assumes qfp: "qfree p"
+  shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
+  using qfp 
+by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
+
+    (* Operations needed for Ferrante and Rackoff *)
+lemma rminusinf_inf:
+  assumes lp: "isrlfm p"
+  shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
+using lp
+proof (induct p rule: minusinf.induct)
+  case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto 
+next
+  case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
+next
+  case (3 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    hence "real c * x + ?e \<noteq> 0" by simp
+    with xz have "?P ?z x (Eq (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
+  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (4 c e)   
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    hence "real c * x + ?e \<noteq> 0" by simp
+    with xz have "?P ?z x (NEq (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (5 c e) 
+    from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    with xz have "?P ?z x (Lt (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
+  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (6 c e)  
+    from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    with xz have "?P ?z x (Le (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (7 c e)  
+    from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    with xz have "?P ?z x (Gt (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (8 c e)  
+    from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x < ?z"
+    hence "(real c * x < - ?e)" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
+    hence "real c * x + ?e < 0" by arith
+    with xz have "?P ?z x (Ge (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
+  thus ?case by blast
+qed simp_all
+
+lemma rplusinf_inf:
+  assumes lp: "isrlfm p"
+  shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
+using lp
+proof (induct p rule: isrlfm.induct)
+  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
+next
+  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
+next
+  case (3 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    hence "real c * x + ?e \<noteq> 0" by simp
+    with xz have "?P ?z x (Eq (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (4 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    hence "real c * x + ?e \<noteq> 0" by simp
+    with xz have "?P ?z x (NEq (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (5 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    with xz have "?P ?z x (Lt (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (6 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    with xz have "?P ?z x (Le (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (7 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    with xz have "?P ?z x (Gt (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
+  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
+  thus ?case by blast
+next
+  case (8 c e) 
+  from prems have nb: "numbound0 e" by simp
+  from prems have cp: "real c > 0" by simp
+  let ?e="Inum (a#bs) e"
+  let ?z = "(- ?e) / real c"
+  {fix x
+    assume xz: "x > ?z"
+    with mult_strict_right_mono [OF xz cp] cp
+    have "(real c * x > - ?e)" by (simp add: mult_ac)
+    hence "real c * x + ?e > 0" by arith
+    with xz have "?P ?z x (Ge (CN 0 c e))"
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
+  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
+  thus ?case by blast
+qed simp_all
+
+lemma rminusinf_bound0:
+  assumes lp: "isrlfm p"
+  shows "bound0 (minusinf p)"
+  using lp
+  by (induct p rule: minusinf.induct) simp_all
+
+lemma rplusinf_bound0:
+  assumes lp: "isrlfm p"
+  shows "bound0 (plusinf p)"
+  using lp
+  by (induct p rule: plusinf.induct) simp_all
+
+lemma rminusinf_ex:
+  assumes lp: "isrlfm p"
+  and ex: "Ifm (a#bs) (minusinf p)"
+  shows "\<exists> x. Ifm (x#bs) p"
+proof-
+  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
+  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
+  from rminusinf_inf[OF lp, where bs="bs"] 
+  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
+  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
+  moreover have "z - 1 < z" by simp
+  ultimately show ?thesis using z_def by auto
+qed
+
+lemma rplusinf_ex:
+  assumes lp: "isrlfm p"
+  and ex: "Ifm (a#bs) (plusinf p)"
+  shows "\<exists> x. Ifm (x#bs) p"
+proof-
+  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
+  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
+  from rplusinf_inf[OF lp, where bs="bs"] 
+  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
+  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
+  moreover have "z + 1 > z" by simp
+  ultimately show ?thesis using z_def by auto
+qed
+
+consts 
+  uset:: "fm \<Rightarrow> (num \<times> int) list"
+  usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
+recdef uset "measure size"
+  "uset (And p q) = (uset p @ uset q)" 
+  "uset (Or p q) = (uset p @ uset q)" 
+  "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
+  "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
+  "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
+  "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
+  "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
+  "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
+  "uset p = []"
+recdef usubst "measure size"
+  "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
+  "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
+  "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
+  "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
+  "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
+  "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
+  "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
+  "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
+  "usubst p = (\<lambda> (t,n). p)"
+
+lemma usubst_I: assumes lp: "isrlfm p"
+  and np: "real n > 0" and nbt: "numbound0 t"
+  shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
+  using lp
+proof(induct p rule: usubst.induct)
+  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
+    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+next
+  case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
+    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+next
+  case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
+    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+next
+  case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
+    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+next
+  case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  from np have np: "real n \<noteq> 0" by simp
+  have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+next
+  case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
+  from np have np: "real n \<noteq> 0" by simp
+  have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
+    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
+  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
+      and b="0", simplified divide_zero_left]) (simp only: ring_eq_simps)
+  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
+    using np by simp 
+  finally show ?case using nbt nb by (simp add: ring_eq_simps)
+qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
+
+lemma uset_l:
+  assumes lp: "isrlfm p"
+  shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
+using lp
+by(induct p rule: uset.induct,auto)
+
+lemma rminusinf_uset:
+  assumes lp: "isrlfm p"
+  and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
+  and ex: "Ifm (x#bs) p" (is "?I x p")
+  shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
+proof-
+  have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
+    using lp nmi ex
+    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
+  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
+  from uset_l[OF lp] smU have mp: "real m > 0" by auto
+  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
+    by (auto simp add: mult_commute)
+  thus ?thesis using smU by auto
+qed
+
+lemma rplusinf_uset:
+  assumes lp: "isrlfm p"
+  and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
+  and ex: "Ifm (x#bs) p" (is "?I x p")
+  shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
+proof-
+  have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
+    using lp nmi ex
+    by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
+  then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
+  from uset_l[OF lp] smU have mp: "real m > 0" by auto
+  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
+    by (auto simp add: mult_commute)
+  thus ?thesis using smU by auto
+qed
+
+lemma lin_dense: 
+  assumes lp: "isrlfm p"
+  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" 
+  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
+  and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
+  and ly: "l < y" and yu: "y < u"
+  shows "Ifm (y#bs) p"
+using lp px noS
+proof (induct p rule: isrlfm.induct)
+  case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+    from prems have "x * real c + ?N x e < 0" by (simp add: ring_eq_simps)
+    hence pxc: "x < (- ?N x e) / real c" 
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
+    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
+    moreover {assume y: "y < (-?N x e)/ real c"
+      hence "y * real c < - ?N x e"
+	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
+      hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps)
+      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
+    moreover {assume y: "y > (- ?N x e) / real c" 
+      with yu have eu: "u > (- ?N x e) / real c" by auto
+      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
+      with lx pxc have "False" by auto
+      hence ?case by simp }
+    ultimately show ?case by blast
+next
+  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
+    from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_eq_simps)
+    hence pxc: "x \<le> (- ?N x e) / real c" 
+      by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
+    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
+    moreover {assume y: "y < (-?N x e)/ real c"
+      hence "y * real c < - ?N x e"
+	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
+      hence "real c * y + ?N x e < 0" by (simp add: ring_eq_simps)
+      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
+    moreover {assume y: "y > (- ?N x e) / real c" 
+      with yu have eu: "u > (- ?N x e) / real c" by auto
+      with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
+      with lx pxc have "False" by auto
+      hence ?case by simp }
+    ultimately show ?case by blast
+next
+  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+    from prems have "x * real c + ?N x e > 0" by (simp add: ring_eq_simps)
+    hence pxc: "x > (- ?N x e) / real c" 
+      by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
+    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
+    moreover {assume y: "y > (-?N x e)/ real c"
+      hence "y * real c > - ?N x e"
+	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
+      hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps)
+      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
+    moreover {assume y: "y < (- ?N x e) / real c" 
+      with ly have eu: "l < (- ?N x e) / real c" by auto
+      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
+      with xu pxc have "False" by auto
+      hence ?case by simp }
+    ultimately show ?case by blast
+next
+  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+    from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_eq_simps)
+    hence pxc: "x \<ge> (- ?N x e) / real c" 
+      by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
+    hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
+    moreover {assume y: "y > (-?N x e)/ real c"
+      hence "y * real c > - ?N x e"
+	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
+      hence "real c * y + ?N x e > 0" by (simp add: ring_eq_simps)
+      hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
+    moreover {assume y: "y < (- ?N x e) / real c" 
+      with ly have eu: "l < (- ?N x e) / real c" by auto
+      with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
+      with xu pxc have "False" by auto
+      hence ?case by simp }
+    ultimately show ?case by blast
+next
+  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+    from cp have cnz: "real c \<noteq> 0" by simp
+    from prems have "x * real c + ?N x e = 0" by (simp add: ring_eq_simps)
+    hence pxc: "x = (- ?N x e) / real c" 
+      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
+    with pxc show ?case by simp
+next
+  case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+    from cp have cnz: "real c \<noteq> 0" by simp
+    from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+    with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
+    hence "y* real c \<noteq> -?N x e"      
+      by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
+    hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_eq_simps)
+    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
+      by (simp add: ring_eq_simps)
+qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
+
+lemma finite_set_intervals:
+  assumes px: "P (x::real)" 
+  and lx: "l \<le> x" and xu: "x \<le> u"
+  and linS: "l\<in> S" and uinS: "u \<in> S"
+  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
+  shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
+proof-
+  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
+  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
+  let ?a = "Max ?Mx"
+  let ?b = "Min ?xM"
+  have MxS: "?Mx \<subseteq> S" by blast
+  hence fMx: "finite ?Mx" using fS finite_subset by auto
+  from lx linS have linMx: "l \<in> ?Mx" by blast
+  hence Mxne: "?Mx \<noteq> {}" by blast
+  have xMS: "?xM \<subseteq> S" by blast
+  hence fxM: "finite ?xM" using fS finite_subset by auto
+  from xu uinS have linxM: "u \<in> ?xM" by blast
+  hence xMne: "?xM \<noteq> {}" by blast
+  have ax:"?a \<le> x" using Mxne fMx by auto
+  have xb:"x \<le> ?b" using xMne fxM by auto
+  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
+  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
+  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
+  proof(clarsimp)
+    fix y
+    assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
+    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
+    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
+    moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
+    ultimately show "False" by blast
+  qed
+  from ainS binS noy ax xb px show ?thesis by blast
+qed
+
+lemma finite_set_intervals2:
+  assumes px: "P (x::real)" 
+  and lx: "l \<le> x" and xu: "x \<le> u"
+  and linS: "l\<in> S" and uinS: "u \<in> S"
+  and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
+  shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
+proof-
+  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
+  obtain a and b where 
+    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
+  from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
+  thus ?thesis using px as bs noS by blast 
+qed
+
+lemma rinf_uset:
+  assumes lp: "isrlfm p"
+  and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
+  and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
+  and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
+  shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
+proof-
+  let ?N = "\<lambda> x t. Inum (x#bs) t"
+  let ?U = "set (uset p)"
+  from ex obtain a where pa: "?I a p" by blast
+  from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
+  have nmi': "\<not> (?I a (?M p))" by simp
+  from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
+  have npi': "\<not> (?I a (?P p))" by simp
+  have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
+  proof-
+    let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
+    have fM: "finite ?M" by auto
+    from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] 
+    have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
+    then obtain "t" "n" "s" "m" where 
+      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
+      and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
+    from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
+    from tnU have Mne: "?M \<noteq> {}" by auto
+    hence Une: "?U \<noteq> {}" by simp
+    let ?l = "Min ?M"
+    let ?u = "Max ?M"
+    have linM: "?l \<in> ?M" using fM Mne by simp
+    have uinM: "?u \<in> ?M" using fM Mne by simp
+    have tnM: "?N a t / real n \<in> ?M" using tnU by auto
+    have smM: "?N a s / real m \<in> ?M" using smU by auto 
+    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
+    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
+    have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
+    have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
+    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
+    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
+      (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
+    moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
+      hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
+      then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
+      have "(u + u) / 2 = u" by auto with pu tuu 
+      have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
+      with tuU have ?thesis by blast}
+    moreover{
+      assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
+      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
+	and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
+	by blast
+      from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
+      then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
+      from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
+      then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
+      from t1x xt2 have t1t2: "t1 < t2" by simp
+      let ?u = "(t1 + t2) / 2"
+      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
+      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
+      with t1uU t2uU t1u t2u have ?thesis by blast}
+    ultimately show ?thesis by blast
+  qed
+  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
+    and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
+  from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
+  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
+    numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
+  have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
+  with lnU smU
+  show ?thesis by auto
+qed
+    (* The Ferrante - Rackoff Theorem *)
+
+theorem fr_eq: 
+  assumes lp: "isrlfm p"
+  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
+  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
+proof
+  assume px: "\<exists> x. ?I x p"
+  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
+  moreover {assume "?M \<or> ?P" hence "?D" by blast}
+  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
+    from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
+  ultimately show "?D" by blast
+next
+  assume "?D" 
+  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
+  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
+  moreover {assume f:"?F" hence "?E" by blast}
+  ultimately show "?E" by blast
+qed
+
+
+lemma fr_equsubst: 
+  assumes lp: "isrlfm p"
+  shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))"
+  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
+proof
+  assume px: "\<exists> x. ?I x p"
+  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
+  moreover {assume "?M \<or> ?P" hence "?D" by blast}
+  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
+    let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
+    let ?N = "\<lambda> t. Inum (x#bs) t"
+    {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
+      with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
+	by auto
+      let ?st = "Add (Mul m t) (Mul n s)"
+      from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
+	by (simp add: mult_commute)
+      from tnb snb have st_nb: "numbound0 ?st" by simp
+      have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+	using mnp mp np by (simp add: ring_eq_simps add_divide_distrib)
+      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
+      have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
+    with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
+  ultimately show "?D" by blast
+next
+  assume "?D" 
+  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
+  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
+  moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
+    and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
+    with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
+    let ?st = "Add (Mul l t) (Mul k s)"
+    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
+      by (simp add: mult_commute)
+    from tnb snb have st_nb: "numbound0 ?st" by simp
+    from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
+  ultimately show "?E" by blast
+qed
+
+consts allpairs:: "'a list \<Rightarrow> 'b list \<Rightarrow> ('a \<times> 'b) list"
+primrec
+  "allpairs [] ys = []"
+  "allpairs (x#xs) ys = (map (Pair x) ys)@(allpairs xs ys)"
+
+lemma allpairs_set: "set (allpairs xs ys) = {(x,y). x\<in> set xs \<and> y \<in> set ys}"
+by (induct xs) auto
+
+    (* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
+constdefs ferrack:: "fm \<Rightarrow> fm"
+  "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
+                in if (mp = T \<or> pp = T) then T else 
+                   (let U = remdps(map simp_num_pair 
+                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
+                           (alluopairs (uset p')))) 
+                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
+
+lemma uset_cong_aux:
+  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
+  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
+  (is "?lhs = ?rhs")
+proof(auto)
+  fix t n s m
+  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
+  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
+    using alluopairs_set1[where xs="U"] by blast
+  let ?N = "\<lambda> t. Inum (x#bs) t"
+  let ?st= "Add (Mul m t) (Mul n s)"
+  from Ul th have mnz: "m \<noteq> 0" by auto
+  from Ul th have  nnz: "n \<noteq> 0" by auto  
+  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+   using mnz nnz by (simp add: ring_eq_simps add_divide_distrib)
+ 
+  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
+       (2 * real n * real m)
+       \<in> (\<lambda>((t, n), s, m).
+             (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
+         (set U \<times> set U)"using mnz nnz th  
+    apply (auto simp add: th add_divide_distrib ring_eq_simps split_def image_def)
+    by (rule_tac x="(s,m)" in bexI,simp_all) 
+  (rule_tac x="(t,n)" in bexI,simp_all)
+next
+  fix t n s m
+  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
+  let ?N = "\<lambda> t. Inum (x#bs) t"
+  let ?st= "Add (Mul m t) (Mul n s)"
+  from Ul smU have mnz: "m \<noteq> 0" by auto
+  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
+  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+   using mnz nnz by (simp add: ring_eq_simps add_divide_distrib)
+ let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
+ have Pc:"\<forall> a b. ?P a b = ?P b a"
+   by auto
+ from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
+ from alluopairs_ex[OF Pc, where xs="U"] tnU smU
+ have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
+   by blast
+ then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
+   and Pts': "?P (t',n') (s',m')" by blast
+ from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
+ let ?st' = "Add (Mul m' t') (Mul n' s')"
+   have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
+   using mnz' nnz' by (simp add: ring_eq_simps add_divide_distrib)
+ from Pts' have 
+   "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
+ also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
+ finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
+          \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
+            (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
+            set (alluopairs U)"
+   using ts'_U by blast
+qed
+
+lemma uset_cong:
+  assumes lp: "isrlfm p"
+  and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
+  and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
+  and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
+  shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))"
+  (is "?lhs = ?rhs")
+proof
+  assume ?lhs
+  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
+    Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
+  let ?N = "\<lambda> t. Inum (x#bs) t"
+  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
+    and snb: "numbound0 s" and mp:"m > 0"  by auto
+  let ?st= "Add (Mul m t) (Mul n s)"
+  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
+      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+    from tnb snb have stnb: "numbound0 ?st" by simp
+  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+   using mp np by (simp add: ring_eq_simps add_divide_distrib)
+  from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
+  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
+    by auto (rule_tac x="(a,b)" in bexI, auto)
+  then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
+  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
+  from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
+  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
+  from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
+  have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) 
+  then show ?rhs using tnU' by auto 
+next
+  assume ?rhs
+  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" 
+    by blast
+  from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
+  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
+    by auto (rule_tac x="(a,b)" in bexI, auto)
+  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
+    th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
+    let ?N = "\<lambda> t. Inum (x#bs) t"
+  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
+    and snb: "numbound0 s" and mp:"m > 0"  by auto
+  let ?st= "Add (Mul m t) (Mul n s)"
+  from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
+      by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
+    from tnb snb have stnb: "numbound0 ?st" by simp
+  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
+   using mp np by (simp add: ring_eq_simps add_divide_distrib)
+  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
+  from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
+  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
+  with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
+qed
+
+lemma ferrack: 
+  assumes qf: "qfree p"
+  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
+  (is "_ \<and> (?rhs = ?lhs)")
+proof-
+  let ?I = "\<lambda> x p. Ifm (x#bs) p"
+  let ?N = "\<lambda> t. Inum (x#bs) t"
+  let ?q = "rlfm (simpfm p)" 
+  let ?U = "uset ?q"
+  let ?Up = "alluopairs ?U"
+  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
+  let ?S = "map ?g ?Up"
+  let ?SS = "map simp_num_pair ?S"
+  let ?Y = "remdps ?SS"
+  let ?f= "(\<lambda> (t,n). ?N t / real n)"
+  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
+  let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
+  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
+  from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
+  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
+  from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
+  from U_l UpU 
+  have Up_: "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
+  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
+    by (auto simp add: mult_pos_pos)
+  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
+  proof-
+    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
+      hence "(t,n) \<in> set ?SS" by simp
+      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
+	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
+      then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
+      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
+      from simp_num_pair_l[OF tnb np tns]
+      have "numbound0 t \<and> n > 0" . }
+    thus ?thesis by blast
+  qed
+
+  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
+  proof-
+     from simp_num_pair_ci[where bs="x#bs"] have 
+    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
+     hence th: "?f o simp_num_pair = ?f" using ext by blast
+    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
+    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
+    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
+      by (simp only: set_map o_def image_compose[symmetric])
+    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
+      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
+    finally show ?thesis .
+  qed
+  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
+  proof-
+    { fix t n assume tnY: "(t,n) \<in> set ?Y"
+      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
+      from usubst_I[OF lq np tnb]
+    have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
+      using simpfm_bound0 by simp}
+    thus ?thesis by blast
+  qed
+  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
+  let ?mp = "minusinf ?q"
+  let ?pp = "plusinf ?q"
+  let ?M = "?I x ?mp"
+  let ?P = "?I x ?pp"
+  let ?res = "disj ?mp (disj ?pp ?ep)"
+  from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
+  have nbth: "bound0 ?res" by auto
+  thm rlfm_I[OF simpfm_qf[OF qf]]
+
+  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  
+
+  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
+  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
+    by (simp only: split_def fst_conv snd_conv)
+  also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
+    using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
+  also have "\<dots> = (Ifm (x#bs) ?res)"
+    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
+    by (simp add: split_def pair_collapse)
+  finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
+  hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
+    by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
+  from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
+  with lr show ?thesis by blast
+qed
+
+constdefs linrqe:: "fm \<Rightarrow> fm"
+  "linrqe \<equiv> (\<lambda> p. qelim (prep p) ferrack)"
+
+theorem linrqe: "(Ifm bs (linrqe p) = Ifm bs p) \<and> qfree (linrqe p)"
+using ferrack qelim_ci prep
+unfolding linrqe_def by auto
+
+declare max_def [code unfold]
+
+code_module Ferrack
+file "generated_ferrack.ML"
+contains linrqe = "linrqe"
+test = "%x . linrqe (A(A(Imp (Lt(Sub (Bound 1) (Bound 0))) (E(Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
+
+ML{* use "generated_ferrack.ML"*}
+ML "Ferrack.test ()"
+
+use "linreif.ML"
+oracle linr_oracle ("term") = ReflectedFerrack.linrqe_oracle
+
+use"linrtac.ML"
+setup "LinrTac.setup"
+
+end




