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