src/HOL/Complex/ex/ReflectedFerrack.thy

cvsps: back to non-verbose mode;

(*  Title:      Complex/ex/ReflectedFerrack.thy
    Author:     Amine Chaieb
*)

theory ReflectedFerrack
imports Main GCD Real Efficient_Nat
uses ("linrtac.ML")
begin

section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}

  (*********************************************************************************)
  (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
  (*********************************************************************************)

consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
primrec
  "alluopairs [] = []"
  "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"

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

lemma alluopairs_set:
  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
by (induct xs, auto)

lemma alluopairs_ex:
  assumes Pc: "\<forall> x y. P x y = P y x"
  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
proof
  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
  from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
    by auto
next
  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
qed

lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
using Nat.gr0_conv_Suc
by clarsimp

lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
  apply (induct xs, auto) done

consts remdps:: "'a list \<Rightarrow> 'a list"

recdef remdps "measure size"
  "remdps [] = []"
  "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
(hints simp add: filter_length[rule_format])

lemma remdps_set[simp]: "set (remdps xs) = set xs"
  by (induct xs rule: remdps.induct, auto)



  (*********************************************************************************)
  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
  (*********************************************************************************)

datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
  | Mul int num 

  (* A size for num to make inductive proofs simpler*)
consts num_size :: "num \<Rightarrow> nat" 
primrec 
  "num_size (C c) = 1"
  "num_size (Bound n) = 1"
  "num_size (Neg a) = 1 + num_size a"
  "num_size (Add a b) = 1 + num_size a + num_size b"
  "num_size (Sub a b) = 3 + num_size a + num_size b"
  "num_size (Mul c a) = 1 + num_size a"
  "num_size (CN n c a) = 3 + num_size a "

  (* Semantics of numeral terms (num) *)
consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
primrec
  "Inum bs (C c) = (real c)"
  "Inum bs (Bound n) = bs!n"
  "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
  "Inum bs (Neg a) = -(Inum bs a)"
  "Inum bs (Add a b) = Inum bs a + Inum bs b"
  "Inum bs (Sub a b) = Inum bs a - Inum bs b"
  "Inum bs (Mul c a) = (real c) * Inum bs a"
    (* FORMULAE *)
datatype fm  = 
  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm


  (* A size for fm *)
consts fmsize :: "fm \<Rightarrow> nat"
recdef fmsize "measure size"
  "fmsize (NOT p) = 1 + fmsize p"
  "fmsize (And p q) = 1 + fmsize p + fmsize q"
  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
  "fmsize (E p) = 1 + fmsize p"
  "fmsize (A p) = 4+ fmsize p"
  "fmsize p = 1"
  (* several lemmas about fmsize *)
lemma fmsize_pos: "fmsize p > 0"
by (induct p rule: fmsize.induct) simp_all

  (* Semantics of formulae (fm) *)
consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
primrec
  "Ifm bs T = True"
  "Ifm bs F = False"
  "Ifm bs (Lt a) = (Inum bs a < 0)"
  "Ifm bs (Gt a) = (Inum bs a > 0)"
  "Ifm bs (Le a) = (Inum bs a \<le> 0)"
  "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
  "Ifm bs (Eq a) = (Inum bs a = 0)"
  "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
  "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
  "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
  "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
  "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
  "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
  "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
  "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"

lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
apply simp
done

lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
apply simp
done
lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
apply simp
done
lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
apply simp
done
lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
apply simp
done
lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
apply simp
done
lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
apply simp
done
lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
apply simp
done

lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
apply simp
done
lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
apply simp
done

consts not:: "fm \<Rightarrow> fm"
recdef not "measure size"
  "not (NOT p) = p"
  "not T = F"
  "not F = T"
  "not p = NOT p"
lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
by (cases p) auto

constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
   if p = q then p else And p q)"
lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)

constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
       else if p=q then p else Or p q)"

lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)

constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
    else Imp p q)"
lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
by (cases "p=F \<or> q=T",simp_all add: imp_def) 

constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
  Iff p q)"
lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)

lemma conj_simps:
  "conj F Q = F"
  "conj P F = F"
  "conj T Q = Q"
  "conj P T = P"
  "conj P P = P"
  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"
  by (simp_all add: conj_def)

lemma disj_simps:
  "disj T Q = T"
  "disj P T = T"
  "disj F Q = Q"
  "disj P F = P"
  "disj P P = P"
  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
  by (simp_all add: disj_def)
lemma imp_simps:
  "imp F Q = T"
  "imp P T = T"
  "imp T Q = Q"
  "imp P F = not P"
  "imp P P = T"
  "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
  by (simp_all add: imp_def)
lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
apply (induct p, auto)
done

lemma iff_simps:
  "iff p p = T"
  "iff p (NOT p) = F"
  "iff (NOT p) p = F"
  "iff p F = not p"
  "iff F p = not p"
  "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
  "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
  "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
  using trivNOT
  by (simp_all add: iff_def, cases p, auto)
  (* Quantifier freeness *)
consts qfree:: "fm \<Rightarrow> bool"
recdef qfree "measure size"
  "qfree (E p) = False"
  "qfree (A p) = False"
  "qfree (NOT p) = qfree p" 
  "qfree (And p q) = (qfree p \<and> qfree q)" 
  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
  "qfree (Iff p q) = (qfree p \<and> qfree q)"
  "qfree p = True"

  (* Boundedness and substitution *)
consts 
  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
primrec
  "numbound0 (C c) = True"
  "numbound0 (Bound n) = (n>0)"
  "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
  "numbound0 (Neg a) = numbound0 a"
  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
  "numbound0 (Mul i a) = numbound0 a"
lemma numbound0_I:
  assumes nb: "numbound0 a"
  shows "Inum (b#bs) a = Inum (b'#bs) a"
using nb
by (induct a rule: numbound0.induct,auto simp add: nth_pos2)

primrec
  "bound0 T = True"
  "bound0 F = True"
  "bound0 (Lt a) = numbound0 a"
  "bound0 (Le a) = numbound0 a"
  "bound0 (Gt a) = numbound0 a"
  "bound0 (Ge a) = numbound0 a"
  "bound0 (Eq a) = numbound0 a"
  "bound0 (NEq a) = numbound0 a"
  "bound0 (NOT p) = bound0 p"
  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
  "bound0 (E p) = False"
  "bound0 (A p) = False"

lemma bound0_I:
  assumes bp: "bound0 p"
  shows "Ifm (b#bs) p = Ifm (b'#bs) p"
using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
by (induct p rule: bound0.induct) (auto simp add: nth_pos2)

lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
by (cases p, auto)
lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
by (cases p, auto)


lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
using conj_def by auto 
lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
using conj_def by auto 

lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
using disj_def by auto 
lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
using disj_def by auto 

lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)

lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
  by (unfold iff_def,cases "p=q", auto)
lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto)

consts 
  decrnum:: "num \<Rightarrow> num" 
  decr :: "fm \<Rightarrow> fm"

recdef decrnum "measure size"
  "decrnum (Bound n) = Bound (n - 1)"
  "decrnum (Neg a) = Neg (decrnum a)"
  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
  "decrnum (Mul c a) = Mul c (decrnum a)"
  "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
  "decrnum a = a"

recdef decr "measure size"
  "decr (Lt a) = Lt (decrnum a)"
  "decr (Le a) = Le (decrnum a)"
  "decr (Gt a) = Gt (decrnum a)"
  "decr (Ge a) = Ge (decrnum a)"
  "decr (Eq a) = Eq (decrnum a)"
  "decr (NEq a) = NEq (decrnum a)"
  "decr (NOT p) = NOT (decr p)" 
  "decr (And p q) = conj (decr p) (decr q)"
  "decr (Or p q) = disj (decr p) (decr q)"
  "decr (Imp p q) = imp (decr p) (decr q)"
  "decr (Iff p q) = iff (decr p) (decr q)"
  "decr p = p"

lemma decrnum: assumes nb: "numbound0 t"
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
  using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)

lemma decr: assumes nb: "bound0 p"
  shows "Ifm (x#bs) p = Ifm bs (decr p)"
  using nb 
  by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)

lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
by (induct p, simp_all)

consts 
  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
recdef isatom "measure size"
  "isatom T = True"
  "isatom F = True"
  "isatom (Lt a) = True"
  "isatom (Le a) = True"
  "isatom (Gt a) = True"
  "isatom (Ge a) = True"
  "isatom (Eq a) = True"
  "isatom (NEq a) = True"
  "isatom p = False"

lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
by (induct p, simp_all)

constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
  "evaldjf f ps \<equiv> foldr (djf f) ps F"

lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
(cases "f p", simp_all add: Let_def djf_def) 


lemma djf_simps:
  "djf f p T = T"
  "djf f p F = f p"
  "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
  by (simp_all add: djf_def)

lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
  by(induct ps, simp_all add: evaldjf_def djf_Or)

lemma evaldjf_bound0: 
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
  shows "bound0 (evaldjf f xs)"
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 

lemma evaldjf_qf: 
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
  shows "qfree (evaldjf f xs)"
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 

consts disjuncts :: "fm \<Rightarrow> fm list"
recdef disjuncts "measure size"
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
  "disjuncts F = []"
  "disjuncts p = [p]"

lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
by(induct p rule: disjuncts.induct, auto)

lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
proof-
  assume nb: "bound0 p"
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
  thus ?thesis by (simp only: list_all_iff)
qed

lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
proof-
  assume qf: "qfree p"
  hence "list_all qfree (disjuncts p)"
    by (induct p rule: disjuncts.induct, auto)
  thus ?thesis by (simp only: list_all_iff)
qed

constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
  "DJ f p \<equiv> evaldjf f (disjuncts p)"

lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
  and fF: "f F = F"
  shows "Ifm bs (DJ f p) = Ifm bs (f p)"
proof-
  have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
    by (simp add: DJ_def evaldjf_ex) 
  also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
  finally show ?thesis .
qed

lemma DJ_qf: assumes 
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
proof(clarify)
  fix  p assume qf: "qfree p"
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
  
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
qed

lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
proof(clarify)
  fix p::fm and bs
  assume qf: "qfree p"
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
  have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
    by (simp add: DJ_def evaldjf_ex)
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
  also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
qed
  (* Simplification *)
consts 
  numgcd :: "num \<Rightarrow> int"
  numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
  reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
  reducecoeff :: "num \<Rightarrow> num"
  dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
consts maxcoeff:: "num \<Rightarrow> int"
recdef maxcoeff "measure size"
  "maxcoeff (C i) = abs i"
  "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
  "maxcoeff t = 1"

lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
  by (induct t rule: maxcoeff.induct, auto)

recdef numgcdh "measure size"
  "numgcdh (C i) = (\<lambda>g. igcd i g)"
  "numgcdh (CN n c t) = (\<lambda>g. igcd c (numgcdh t g))"
  "numgcdh t = (\<lambda>g. 1)"
defs numgcd_def [code func]: "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_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_simps)
by (simp only: left_distrib[symmetric],simp)

lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
by (induct t s rule: numadd.induct, auto simp add: Let_def)

recdef nummul "measure size"
  "nummul (C j) = (\<lambda> i. C (i*j))"
  "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
  "nummul t = (\<lambda> i. Mul i t)"

lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
by (induct t rule: nummul.induct, auto simp add: ring_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_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_simps , simp add: right_distrib[symmetric])

    (* Linearize a formula*)
definition
  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
    else (Gt (CN 0 (-c) (Neg t))))"

definition
  le :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
    else (Ge (CN 0 (-c) (Neg t))))"

definition
  gt :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
    else (Lt (CN 0 (-c) (Neg t))))"

definition
  ge :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
    else (Le (CN 0 (-c) (Neg t))))"

definition
  eq :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
    else (Eq (CN 0 (-c) (Neg t))))"

definition
  neq :: "int \<Rightarrow> num \<Rightarrow> fm"
where
  "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
    else (NEq (CN 0 (-c) (Neg t))))"

lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
by (auto simp add: conj_def)
lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
by (auto simp add: disj_def)

consts rlfm :: "fm \<Rightarrow> fm"
recdef rlfm "measure fmsize"
  "rlfm (And p q) = conj (rlfm p) (rlfm q)"
  "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
  "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
  "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
  "rlfm (Lt a) = split lt (rsplit0 a)"
  "rlfm (Le a) = split le (rsplit0 a)"
  "rlfm (Gt a) = split gt (rsplit0 a)"
  "rlfm (Ge a) = split ge (rsplit0 a)"
  "rlfm (Eq a) = split eq (rsplit0 a)"
  "rlfm (NEq a) = split neq (rsplit0 a)"
  "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
  "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
  "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
  "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
  "rlfm (NOT (NOT p)) = rlfm p"
  "rlfm (NOT T) = F"
  "rlfm (NOT F) = T"
  "rlfm (NOT (Lt a)) = rlfm (Ge a)"
  "rlfm (NOT (Le a)) = rlfm (Gt a)"
  "rlfm (NOT (Gt a)) = rlfm (Le a)"
  "rlfm (NOT (Ge a)) = rlfm (Lt a)"
  "rlfm (NOT (Eq a)) = rlfm (NEq a)"
  "rlfm (NOT (NEq a)) = rlfm (Eq a)"
  "rlfm p = p" (hints simp add: fmsize_pos)

lemma rlfm_I:
  assumes qfp: "qfree p"
  shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
  using qfp 
by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)

    (* Operations needed for Ferrante and Rackoff *)
lemma rminusinf_inf:
  assumes lp: "isrlfm p"
  shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
using lp
proof (induct p rule: minusinf.induct)
  case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto 
next
  case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
next
  case (3 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    hence "real c * x + ?e \<noteq> 0" by simp
    with xz have "?P ?z x (Eq (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
  hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
  thus ?case by blast
next
  case (4 c e)   
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    hence "real c * x + ?e \<noteq> 0" by simp
    with xz have "?P ?z x (NEq (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
  thus ?case by blast
next
  case (5 c e) 
    from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    with xz have "?P ?z x (Lt (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
  hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
  thus ?case by blast
next
  case (6 c e)  
    from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    with xz have "?P ?z x (Le (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
  thus ?case by blast
next
  case (7 c e)  
    from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    with xz have "?P ?z x (Gt (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
  thus ?case by blast
next
  case (8 c e)  
    from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x < ?z"
    hence "(real c * x < - ?e)" 
      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
    hence "real c * x + ?e < 0" by arith
    with xz have "?P ?z x (Ge (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
  thus ?case by blast
qed simp_all

lemma rplusinf_inf:
  assumes lp: "isrlfm p"
  shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
using lp
proof (induct p rule: isrlfm.induct)
  case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
next
  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
next
  case (3 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    hence "real c * x + ?e \<noteq> 0" by simp
    with xz have "?P ?z x (Eq (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
  thus ?case by blast
next
  case (4 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    hence "real c * x + ?e \<noteq> 0" by simp
    with xz have "?P ?z x (NEq (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
  thus ?case by blast
next
  case (5 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    with xz have "?P ?z x (Lt (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
  thus ?case by blast
next
  case (6 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    with xz have "?P ?z x (Le (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
  thus ?case by blast
next
  case (7 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    with xz have "?P ?z x (Gt (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
  thus ?case by blast
next
  case (8 c e) 
  from prems have nb: "numbound0 e" by simp
  from prems have cp: "real c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real c"
  {fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real c * x > - ?e)" by (simp add: mult_ac)
    hence "real c * x + ?e > 0" by arith
    with xz have "?P ?z x (Ge (CN 0 c e))"
      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
  hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
  thus ?case by blast
qed simp_all

lemma rminusinf_bound0:
  assumes lp: "isrlfm p"
  shows "bound0 (minusinf p)"
  using lp
  by (induct p rule: minusinf.induct) simp_all

lemma rplusinf_bound0:
  assumes lp: "isrlfm p"
  shows "bound0 (plusinf p)"
  using lp
  by (induct p rule: plusinf.induct) simp_all

lemma rminusinf_ex:
  assumes lp: "isrlfm p"
  and ex: "Ifm (a#bs) (minusinf p)"
  shows "\<exists> x. Ifm (x#bs) p"
proof-
  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
  from rminusinf_inf[OF lp, where bs="bs"] 
  obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
  from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
  moreover have "z - 1 < z" by simp
  ultimately show ?thesis using z_def by auto
qed

lemma rplusinf_ex:
  assumes lp: "isrlfm p"
  and ex: "Ifm (a#bs) (plusinf p)"
  shows "\<exists> x. Ifm (x#bs) p"
proof-
  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
  from rplusinf_inf[OF lp, where bs="bs"] 
  obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
  from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
  moreover have "z + 1 > z" by simp
  ultimately show ?thesis using z_def by auto
qed

consts 
  uset:: "fm \<Rightarrow> (num \<times> int) list"
  usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
recdef uset "measure size"
  "uset (And p q) = (uset p @ uset q)" 
  "uset (Or p q) = (uset p @ uset q)" 
  "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
  "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
  "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
  "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
  "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
  "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
  "uset p = []"
recdef usubst "measure size"
  "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
  "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
  "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
  "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
  "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
  "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
  "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
  "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
  "usubst p = (\<lambda> (t,n). p)"

lemma usubst_I: assumes lp: "isrlfm p"
  and np: "real n > 0" and nbt: "numbound0 t"
  shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
  using lp
proof(induct p rule: usubst.induct)
  case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
    using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
      and b="0", simplified divide_zero_left]) (simp only: ring_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_simps)
    thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
      by (simp add: ring_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_simps add_divide_distrib)
      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
      have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
    with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
  ultimately show "?D" by blast
next
  assume "?D" 
  moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
  moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
  moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
    and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
    with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
    let ?st = "Add (Mul l t) (Mul k s)"
    from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
      by (simp add: mult_commute)
    from tnb snb have st_nb: "numbound0 ?st" by simp
    from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
  ultimately show "?E" by blast
qed


    (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
constdefs ferrack:: "fm \<Rightarrow> fm"
  "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
                in if (mp = T \<or> pp = T) then T else 
                   (let U = remdps(map simp_num_pair 
                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
                           (alluopairs (uset p')))) 
                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"

lemma uset_cong_aux:
  assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
  shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
  (is "?lhs = ?rhs")
proof(auto)
  fix t n s m
  assume "((t,n),(s,m)) \<in> set (alluopairs U)"
  hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
    using alluopairs_set1[where xs="U"] by blast
  let ?N = "\<lambda> t. Inum (x#bs) t"
  let ?st= "Add (Mul m t) (Mul n s)"
  from Ul th have mnz: "m \<noteq> 0" by auto
  from Ul th have  nnz: "n \<noteq> 0" by auto  
  have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
   using mnz nnz by (simp add: ring_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_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_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_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_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_simps add_divide_distrib)
  from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
  from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
  have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
  with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
qed

lemma ferrack: 
  assumes qf: "qfree p"
  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
  (is "_ \<and> (?rhs = ?lhs)")
proof-
  let ?I = "\<lambda> x p. Ifm (x#bs) p"
  fix x
  let ?N = "\<lambda> t. Inum (x#bs) t"
  let ?q = "rlfm (simpfm p)" 
  let ?U = "uset ?q"
  let ?Up = "alluopairs ?U"
  let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
  let ?S = "map ?g ?Up"
  let ?SS = "map simp_num_pair ?S"
  let ?Y = "remdps ?SS"
  let ?f= "(\<lambda> (t,n). ?N t / real n)"
  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
  let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
  from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
  from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
  from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
  from U_l UpU 
  have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
  hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
    by (auto simp add: mult_pos_pos)
  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
  proof-
    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
      hence "(t,n) \<in> set ?SS" by simp
      hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
      then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
      from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
      from simp_num_pair_l[OF tnb np tns]
      have "numbound0 t \<and> n > 0" . }
    thus ?thesis by blast
  qed

  have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
  proof-
     from simp_num_pair_ci[where bs="x#bs"] have 
    "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
     hence th: "?f o simp_num_pair = ?f" using ext by blast
    have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
    also have "\<dots> = (?f ` set ?S)" by (simp add: th)
    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
      by (simp only: set_map o_def image_compose[symmetric])
    also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
    finally show ?thesis .
  qed
  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
  proof-
    { fix t n assume tnY: "(t,n) \<in> set ?Y"
      with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
      from usubst_I[OF lq np tnb]
    have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
      using simpfm_bound0 by simp}
    thus ?thesis by blast
  qed
  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
  let ?mp = "minusinf ?q"
  let ?pp = "plusinf ?q"
  let ?M = "?I x ?mp"
  let ?P = "?I x ?pp"
  let ?res = "disj ?mp (disj ?pp ?ep)"
  from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
  have nbth: "bound0 ?res" by auto

  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  

  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
    by (simp only: split_def fst_conv snd_conv)
  also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
    using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
  also have "\<dots> = (Ifm (x#bs) ?res)"
    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
    by (simp add: split_def pair_collapse)
  finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
  hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
    by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
  from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
  with lr show ?thesis by blast
qed

definition linrqe:: "fm \<Rightarrow> fm" where
  "linrqe p = qelim (prep p) ferrack"

theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
using ferrack qelim_ci prep
unfolding linrqe_def by auto

definition ferrack_test :: "unit \<Rightarrow> fm" where
  "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
    (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"

ML {* @{code ferrack_test} () *}

oracle linr_oracle ("term") = {*
let

fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
     of NONE => error "Variable not found in the list!"
      | SOME n => @{code Bound} n)
  | num_of_term vs @{term "real (0::int)"} = @{code C} 0
  | num_of_term vs @{term "real (1::int)"} = @{code C} 1
  | num_of_term vs @{term "0::real"} = @{code C} 0
  | num_of_term vs @{term "1::real"} = @{code C} 1
  | num_of_term vs (Bound i) = @{code Bound} i
  | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} $ t') = @{code Neg} (num_of_term vs t')
  | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
  | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
  | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $ t1 $ t2) = (case (num_of_term vs t1)
     of @{code C} i => @{code Mul} (i, num_of_term vs t2)
      | _ => error "num_of_term: unsupported Multiplication")
  | num_of_term vs (@{term "real :: int \<Rightarrow> real"} $ (@{term "number_of :: int \<Rightarrow> int"} $ t')) = @{code C} (HOLogic.dest_numeral t')
  | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} $ t') = @{code C} (HOLogic.dest_numeral t')
  | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term_global @{theory} t);

fun fm_of_term vs @{term True} = @{code T}
  | fm_of_term vs @{term False} = @{code F}
  | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
  | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $ t1 $ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs (@{term "op &"} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs (@{term "op |"} $ t1 $ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs (@{term "op -->"} $ t1 $ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs (@{term "Not"} $ t') = @{code NOT} (fm_of_term vs t')
  | fm_of_term vs (Const ("Ex", _) $ Abs (xn, xT, p)) =
      @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
  | fm_of_term vs (Const ("All", _) $ Abs (xn, xT, p)) =
      @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
  | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term_global @{theory} t);

fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} $ HOLogic.mk_number HOLogic.intT i
  | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
  | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} $ term_of_num vs t'
  | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} $
      term_of_num vs t1 $ term_of_num vs t2
  | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} $
      term_of_num vs t1 $ term_of_num vs t2
  | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} $
      term_of_num vs (@{code C} i) $ term_of_num vs t2
  | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));

fun term_of_fm vs @{code T} = HOLogic.true_const 
  | term_of_fm vs @{code F} = HOLogic.false_const
  | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
      term_of_num vs t $ @{term "0::real"}
  | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
      term_of_num vs t $ @{term "0::real"}
  | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
      @{term "0::real"} $ term_of_num vs t
  | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $
      @{term "0::real"} $ term_of_num vs t
  | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $
      term_of_num vs t $ @{term "0::real"}
  | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
  | term_of_fm vs (@{code NOT} t') = HOLogic.Not $ term_of_fm vs t'
  | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj $ term_of_fm vs t1 $ term_of_fm vs t2
  | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj $ term_of_fm vs t1 $ term_of_fm vs t2
  | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp $ term_of_fm vs t1 $ term_of_fm vs t2
  | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} $
      term_of_fm vs t1 $ term_of_fm vs t2
  | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";

in fn thy => fn t =>
  let 
    val fs = term_frees t;
    val vs = fs ~~ (0 upto (length fs - 1));
  in HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t)))) end
end;
*}

use "linrtac.ML"
setup LinrTac.setup

lemma
  fixes x :: real
  shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
apply rferrack
done

lemma
  fixes x :: real
  shows "\<exists>y \<le> x. x = y + 1"
apply rferrack
done

lemma
  fixes x :: real
  shows "\<not> (\<exists>z. x + z = x + z + 1)"
apply rferrack
done

end