# HG changeset patch
# User wenzelm
# Date 1436566494 -7200
# Node ID 07089a750d2a89a09ecc6e3ed0f581cbffbbcb7b
# Parent  c7bdbf3f1aec51b92716e985be94ac0bab9a2582
tuned proofs;

diff -r c7bdbf3f1aec -r 07089a750d2a src/HOL/Decision_Procs/Ferrack.thy
--- a/src/HOL/Decision_Procs/Ferrack.thy	Thu Jul 09 23:48:55 2015 +0200
+++ b/src/HOL/Decision_Procs/Ferrack.thy	Sat Jul 11 00:14:54 2015 +0200
@@ -13,11 +13,12 @@
   (****                            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 
+datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
+  | Mul int num
 
   (* A size for num to make inductive proofs simpler*)
-primrec num_size :: "num \<Rightarrow> nat" where
+primrec num_size :: "num \<Rightarrow> nat"
+where
   "num_size (C c) = 1"
 | "num_size (Bound n) = 1"
 | "num_size (Neg a) = 1 + num_size a"
@@ -27,7 +28,8 @@
 | "num_size (CN n c a) = 3 + num_size a "
 
   (* Semantics of numeral terms (num) *)
-primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
+primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
+where
   "Inum bs (C c) = (real c)"
 | "Inum bs (Bound n) = bs!n"
 | "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
@@ -36,13 +38,14 @@
 | "Inum bs (Sub a b) = Inum bs a - Inum bs b"
 | "Inum bs (Mul c a) = (real c) * Inum bs a"
     (* FORMULAE *)
-datatype fm  = 
+datatype fm  =
   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
 
 
   (* A size for fm *)
-fun fmsize :: "fm \<Rightarrow> nat" where
+fun fmsize :: "fm \<Rightarrow> nat"
+where
   "fmsize (NOT p) = 1 + fmsize p"
 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"
@@ -52,11 +55,13 @@
 | "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
+  by (induct p rule: fmsize.induct) simp_all
 
   (* Semantics of formulae (fm) *)
-primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
+primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
+where
   "Ifm bs T = True"
 | "Ifm bs F = False"
 | "Ifm bs (Lt a) = (Inum bs a < 0)"
@@ -70,75 +75,95 @@
 | "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)"
+| "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
+  by simp
 
 lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
-apply simp
-done
+  by simp
+
 lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
-apply simp
-done
+  by simp
+
 lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
-apply simp
-done
+  by simp
+
 lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
-apply simp
-done
+  by simp
+
 lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
-apply simp
-done
+  by simp
+
 lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
-apply simp
-done
+  by simp
+
 lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
-apply simp
-done
+  by simp
 
 lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
-apply simp
-done
+  by simp
+
 lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
-apply simp
-done
+  by simp
 
-fun not:: "fm \<Rightarrow> fm" where
+fun not:: "fm \<Rightarrow> fm"
+where
   "not (NOT p) = p"
 | "not T = F"
 | "not F = T"
 | "not p = NOT p"
+
 lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
-by (cases p) auto
+  by (cases p) auto
 
-definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "conj p q = (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
-   if p = q then p else And p q)"
+definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "conj p q =
+   (if p = F \<or> q = F then F
+    else if p = T then q
+    else if q = T then p
+    else if p = q then p else And p q)"
+
 lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
-by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
+  by (cases "p = F \<or> q = F", simp_all add: conj_def) (cases p, simp_all)
 
-definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "disj p q = (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
-       else if p=q then p else Or p q)"
+definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "disj p q =
+   (if p = T \<or> q = T then T
+    else if p = F then q
+    else if q = F then p
+    else if p = q then p else Or p q)"
 
 lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
-by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
+  by (cases "p = T \<or> q = T", simp_all add: disj_def) (cases p, simp_all)
 
-definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "imp p q = (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
+definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "imp p q =
+   (if p = F \<or> q = T \<or> p = q then T
+    else if p = T then q
+    else if q = F then not p
     else Imp p q)"
+
 lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
-by (cases "p=F \<or> q=T",simp_all add: imp_def) 
+  by (cases "p = F \<or> q = T") (simp_all add: imp_def)
 
-definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "iff p q = (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
-       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
-  Iff p q)"
+definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "iff p q =
+   (if p = q then T
+    else if p = NOT q \<or> NOT p = q then F
+    else if p = F then not q
+    else if q = F then not p
+    else if p = T then q
+    else if q = T then p
+    else Iff p q)"
+
 lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
-  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
+  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"
@@ -157,6 +182,7 @@
   "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"
@@ -165,9 +191,9 @@
   "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
+  by (induct p) auto
 
 lemma iff_simps:
   "iff p p = T"
@@ -180,34 +206,37 @@
   "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
   using trivNOT
   by (simp_all add: iff_def, cases p, auto)
+
   (* Quantifier freeness *)
-fun qfree:: "fm \<Rightarrow> bool" where
+fun qfree:: "fm \<Rightarrow> bool"
+where
   "qfree (E p) = False"
 | "qfree (A p) = False"
-| "qfree (NOT p) = qfree p" 
-| "qfree (And p q) = (qfree p \<and> qfree q)" 
-| "qfree (Or  p q) = (qfree p \<and> qfree q)" 
-| "qfree (Imp p q) = (qfree p \<and> qfree q)" 
+| "qfree (NOT p) = qfree p"
+| "qfree (And p q) = (qfree p \<and> qfree q)"
+| "qfree (Or  p q) = (qfree p \<and> qfree q)"
+| "qfree (Imp p q) = (qfree p \<and> qfree q)"
 | "qfree (Iff p q) = (qfree p \<and> qfree q)"
 | "qfree p = True"
 
   (* Boundedness and substitution *)
-primrec numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
+primrec numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
+where
   "numbound0 (C c) = True"
-| "numbound0 (Bound n) = (n>0)"
-| "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
+| "numbound0 (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 (Sub a b) = (numbound0 a \<and> numbound0 b)"
 | "numbound0 (Mul i a) = numbound0 a"
 
 lemma numbound0_I:
   assumes nb: "numbound0 a"
   shows "Inum (b#bs) a = Inum (b'#bs) a"
-using nb
-by (induct a) simp_all
+  using nb by (induct a) simp_all
 
-primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
+primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
+where
   "bound0 T = True"
 | "bound0 F = True"
 | "bound0 (Lt a) = numbound0 a"
@@ -227,36 +256,38 @@
 lemma bound0_I:
   assumes bp: "bound0 p"
   shows "Ifm (b#bs) p = Ifm (b'#bs) p"
-using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
-by (induct p) auto
+  using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
+  by (induct p) auto
 
 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
-by (cases p, auto)
+  by (cases p) auto
+
 lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
-by (cases p, auto)
+  by (cases p) auto
 
 
 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-using conj_def by auto 
+  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 
+  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 
+  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 
+  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)
+  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)
+  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)
+  unfolding iff_def by (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)
+  using iff_def unfolding iff_def by (cases "p = q") auto
 
-fun decrnum:: "num \<Rightarrow> num"  where
+fun decrnum:: "num \<Rightarrow> num"
+where
   "decrnum (Bound n) = Bound (n - 1)"
 | "decrnum (Neg a) = Neg (decrnum a)"
 | "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
@@ -265,33 +296,36 @@
 | "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
 | "decrnum a = a"
 
-fun decr :: "fm \<Rightarrow> fm" where
+fun decr :: "fm \<Rightarrow> fm"
+where
   "decr (Lt a) = Lt (decrnum a)"
 | "decr (Le a) = Le (decrnum a)"
 | "decr (Gt a) = Gt (decrnum a)"
 | "decr (Ge a) = Ge (decrnum a)"
 | "decr (Eq a) = Eq (decrnum a)"
 | "decr (NEq a) = NEq (decrnum a)"
-| "decr (NOT p) = NOT (decr p)" 
+| "decr (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)
+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
 
-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: decrnum)
+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: decrnum)
 
 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
-by (induct p, simp_all)
+  by (induct p) simp_all
 
-fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
+fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
+where
   "isatom T = True"
 | "isatom F = True"
 | "isatom (Lt a) = True"
@@ -303,102 +337,124 @@
 | "isatom p = False"
 
 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
-by (induct p, simp_all)
+  by (induct p) simp_all
 
-definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
-  "djf f p q = (if q=T then T else if q=F then f p else 
-  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
-definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
-  "evaldjf f ps = foldr (djf f) ps F"
+definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "djf f p q =
+   (if q = T then T
+    else if q = F then f p
+    else (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
+
+definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
+  where "evaldjf f ps = foldr (djf f) ps F"
 
 lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
-by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
-(cases "f p", simp_all add: Let_def djf_def) 
+  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)"
+  "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_ex: "Ifm bs (evaldjf f ps) \<longleftrightarrow> (\<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)"
+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) 
+  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)"
+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) 
+  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
 
-fun disjuncts :: "fm \<Rightarrow> fm list" where
+fun disjuncts :: "fm \<Rightarrow> fm list"
+where
   "disjuncts (Or p q) = disjuncts p @ disjuncts q"
 | "disjuncts F = []"
 | "disjuncts p = [p]"
 
-lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
-by(induct p rule: disjuncts.induct, auto)
+lemma disjuncts: "(\<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-
+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)
+  then have "list_all bound0 (disjuncts p)"
+    by (induct p rule: disjuncts.induct) auto
+  then show ?thesis
+    by (simp only: list_all_iff)
 qed
 
-lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
-proof-
+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)
+  then have "list_all qfree (disjuncts p)"
+    by (induct p rule: disjuncts.induct) auto
+  then show ?thesis
+    by (simp only: list_all_iff)
 qed
 
-definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
-  "DJ f p = evaldjf f (disjuncts p)"
+definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
+  where "DJ f p = evaldjf f (disjuncts p)"
 
-lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
-  and fF: "f F = F"
+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)
+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)"
+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
+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
+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
+  fix 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))"
+  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) \<longleftrightarrow> (\<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
+  also have "\<dots> \<longleftrightarrow> (\<exists>q \<in> set(disjuncts p). Ifm bs (E q))"
+    using qe disjuncts_qf[OF qf] by auto
+  also have "\<dots> = Ifm bs (E p)"
+    by (induct p rule: disjuncts.induct) auto
+  finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)"
+    using qfth by blast
 qed
+
   (* Simplification *)
 
-fun maxcoeff:: "num \<Rightarrow> int" where
+fun maxcoeff:: "num \<Rightarrow> int"
+where
   "maxcoeff (C i) = abs i"
 | "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
 | "maxcoeff t = 1"
@@ -406,70 +462,82 @@
 lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
   by (induct t rule: maxcoeff.induct, auto)
 
-fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where
+fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
+where
   "numgcdh (C i) = (\<lambda>g. gcd i g)"
 | "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
 | "numgcdh t = (\<lambda>g. 1)"
 
-definition numgcd :: "num \<Rightarrow> int" where
-  "numgcd t = numgcdh t (maxcoeff t)"
+definition numgcd :: "num \<Rightarrow> int"
+  where "numgcd t = numgcdh t (maxcoeff t)"
 
-fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where
-  "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
-| "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
+fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
+where
+  "reducecoeffh (C i) = (\<lambda>g. C (i div g))"
+| "reducecoeffh (CN n c t) = (\<lambda>g. CN n (c div g) (reducecoeffh t g))"
 | "reducecoeffh t = (\<lambda>g. t)"
 
-definition reducecoeff :: "num \<Rightarrow> num" where
+definition reducecoeff :: "num \<Rightarrow> num"
+where
   "reducecoeff t =
-  (let g = numgcd t in 
-  if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
+   (let g = numgcd t
+    in if g = 0 then C 0 else if g = 1 then t else reducecoeffh t g)"
 
-fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
-  "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
-| "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
+fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
+where
+  "dvdnumcoeff (C i) = (\<lambda>g. g dvd i)"
+| "dvdnumcoeff (CN n c t) = (\<lambda>g. g dvd c \<and> dvdnumcoeff t g)"
 | "dvdnumcoeff t = (\<lambda>g. False)"
 
-lemma dvdnumcoeff_trans: 
-  assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
+lemma dvdnumcoeff_trans:
+  assumes gdg: "g dvd g'"
+    and dgt':"dvdnumcoeff t g'"
   shows "dvdnumcoeff t g"
-  using dgt' gdg 
-  by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
+  using dgt' gdg
+  by (induct t rule: dvdnumcoeff.induct) (simp_all add: gdg dvd_trans[OF gdg])
 
 declare dvd_trans [trans add]
 
-lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
-by arith
+lemma natabs0: "nat (abs x) = 0 \<longleftrightarrow> x = 0"
+  by arith
 
 lemma numgcd0:
   assumes g0: "numgcd t = 0"
   shows "Inum bs t = 0"
-  using g0[simplified numgcd_def] 
-  by (induct t rule: numgcdh.induct, auto simp add: natabs0 maxcoeff_pos max.absorb2)
+  using g0[simplified numgcd_def]
+  by (induct t rule: numgcdh.induct) (auto simp add: natabs0 maxcoeff_pos max.absorb2)
 
-lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
-  using gp
-  by (induct t rule: numgcdh.induct, auto)
+lemma numgcdh_pos:
+  assumes gp: "g \<ge> 0"
+  shows "numgcdh t g \<ge> 0"
+  using gp by (induct t rule: numgcdh.induct) auto
 
 lemma numgcd_pos: "numgcd t \<ge>0"
   by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
 
 lemma reducecoeffh:
-  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
+  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) 
+proof (induct t rule: reducecoeffh.induct)
   case (1 i)
-  hence gd: "g dvd i" by simp
-  with assms show ?case by (simp add: real_of_int_div[OF gd])
+  then have gd: "g dvd i"
+    by simp
+  with assms show ?case
+    by (simp add: real_of_int_div[OF gd])
 next
   case (2 n c t)
-  hence gd: "g dvd c" by simp
-  from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
+  then have gd: "g dvd c"
+    by simp
+  from assms 2 show ?case
+    by (simp add: real_of_int_div[OF gd] algebra_simps)
 qed (auto simp add: numgcd_def gp)
 
-fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
-  "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
-| "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
+fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
+where
+  "ismaxcoeff (C i) = (\<lambda>x. abs i \<le> x)"
+| "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> ismaxcoeff t x)"
 | "ismaxcoeff t = (\<lambda>x. True)"
 
 lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
@@ -478,43 +546,61 @@
 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
-  from ismaxcoeff_mono[OF H thh] show ?case by simp
+  then have H:"ismaxcoeff t (maxcoeff t)" .
+  have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)"
+    by simp
+  from ismaxcoeff_mono[OF H thh] show ?case
+    by simp
 qed simp_all
 
-lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
+lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow>
+  abs i > 1 \<and> abs j > 1 \<or> abs i = 0 \<and> abs j > 1 \<or> abs i > 1 \<and> abs j = 0"
   apply (cases "abs i = 0", simp_all add: gcd_int_def)
   apply (cases "abs j = 0", simp_all)
   apply (cases "abs i = 1", simp_all)
   apply (cases "abs j = 1", simp_all)
   apply auto
   done
+
 lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
-  by (induct t rule: numgcdh.induct, auto)
+  by (induct t rule: numgcdh.induct) auto
 
 lemma dvdnumcoeff_aux:
-  assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
+  assumes "ismaxcoeff t m"
+    and mp: "m \<ge> 0"
+    and "numgcdh t m > 1"
   shows "dvdnumcoeff t (numgcdh t m)"
-using assms
-proof(induct t rule: numgcdh.induct)
-  case (2 n c t) 
+  using assms
+proof (induct t rule: numgcdh.induct)
+  case (2 n c t)
   let ?g = "numgcdh t m"
-  from 2 have th:"gcd c ?g > 1" by simp
+  from 2 have th: "gcd c ?g > 1"
+    by simp
   from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
-  have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
-  moreover {assume "abs c > 1" and gp: "?g > 1" with 2
-    have th: "dvdnumcoeff t ?g" by simp
-    have th': "gcd c ?g dvd ?g" by simp
-    from dvdnumcoeff_trans[OF th' th] have ?case by simp }
-  moreover {assume "abs c = 0 \<and> ?g > 1"
-    with 2 have th: "dvdnumcoeff t ?g" by simp
-    have th': "gcd c ?g dvd ?g" by simp
-    from dvdnumcoeff_trans[OF th' th] have ?case by simp
-    hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
-    from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
-  ultimately show ?case by blast
+  consider "abs c > 1" "?g > 1" | "abs c = 0" "?g > 1" | "?g = 0"
+    by auto
+  then show ?case
+  proof cases
+    case 1
+    with 2 have th: "dvdnumcoeff t ?g"
+      by simp
+    have th': "gcd c ?g dvd ?g"
+      by simp
+    from dvdnumcoeff_trans[OF th' th] show ?thesis
+      by simp
+  next
+    case "2'": 2
+    with 2 have th: "dvdnumcoeff t ?g"
+      by simp
+    have th': "gcd c ?g dvd ?g"
+      by simp
+    from dvdnumcoeff_trans[OF th' th] show ?thesis
+      by simp
+  next
+    case 3
+    then have "m = 0" by (rule numgcdh0)
+    with 2 3 show ?thesis by simp
+  qed
 qed auto
 
 lemma dvdnumcoeff_aux2:
@@ -524,301 +610,416 @@
 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)
+  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" .
+  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-
+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
+  have "?g \<ge> 0"
+    by (simp add: numgcd_pos)
+  then consider "?g = 0" | "?g = 1" | "?g > 1" by atomize_elim auto
+  then show ?thesis
+  proof cases
+    case 1
+    then show ?thesis by (simp add: numgcd0)
+  next
+    case 2
+    then show ?thesis by (simp add: reducecoeff_def)
+  next
+    case g1: 3
+    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 show ?thesis
+      by (simp add: reducecoeff_def Let_def)
+  qed
 qed
 
 lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
-by (induct t rule: reducecoeffh.induct, auto)
+  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)
+  using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
 
-consts
-  numadd:: "num \<times> num \<Rightarrow> num"
-
-recdef numadd "measure (\<lambda> (t,s). size t + size s)"
+consts numadd:: "num \<times> num \<Rightarrow> num"
+recdef numadd "measure (\<lambda>(t,s). size t + size s)"
   "numadd (CN n1 c1 r1,CN n2 c2 r2) =
-  (if n1=n2 then 
-  (let c = c1 + c2
-  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
-  else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
-  else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
-  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
-  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
-  "numadd (C b1, C b2) = C (b1+b2)"
+   (if n1 = n2 then
+    (let c = c1 + c2
+     in (if c = 0 then numadd(r1,r2) else CN n1 c (numadd (r1, r2))))
+    else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2)))
+    else (CN n2 c2 (numadd (CN n1 c1 r1, r2))))"
+  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
+  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t, r2))"
+  "numadd (C b1, C b2) = C (b1 + b2)"
   "numadd (a,b) = Add a b"
 
 lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
-apply (induct t s rule: numadd.induct, simp_all add: Let_def)
-apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
-apply (case_tac "n1 = n2", simp_all add: algebra_simps)
-by (simp only: distrib_right[symmetric],simp)
+  apply (induct t s rule: numadd.induct)
+  apply (simp_all add: Let_def)
+  apply (case_tac "c1 + c2 = 0")
+  apply (case_tac "n1 \<le> n2")
+  apply simp_all
+  apply (case_tac "n1 = n2")
+  apply (simp_all add: algebra_simps)
+  apply (simp only: distrib_right[symmetric])
+  apply simp
+  done
 
 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)
+  by (induct t s rule: numadd.induct) (auto simp add: Let_def)
 
-fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where
-  "nummul (C j) = (\<lambda> i. C (i*j))"
-| "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
-| "nummul t = (\<lambda> i. Mul i t)"
+fun nummul:: "num \<Rightarrow> int \<Rightarrow> num"
+where
+  "nummul (C j) = (\<lambda>i. C (i * j))"
+| "nummul (CN n c a) = (\<lambda>i. CN n (i * c) (nummul a i))"
+| "nummul t = (\<lambda>i. Mul i t)"
 
-lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
-by (induct t rule: nummul.induct, auto simp add: algebra_simps)
+lemma nummul[simp]: "\<And>i. Inum bs (nummul t i) = Inum bs (Mul i t)"
+  by (induct t rule: nummul.induct) (auto simp add: algebra_simps)
 
-lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
-by (induct t rule: nummul.induct, auto )
+lemma nummul_nb[simp]: "\<And>i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
+  by (induct t rule: nummul.induct) auto
 
-definition numneg :: "num \<Rightarrow> num" where
-  "numneg t = nummul t (- 1)"
+definition numneg :: "num \<Rightarrow> num"
+  where "numneg t = nummul t (- 1)"
 
-definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where
-  "numsub s t = (if s = t then C 0 else numadd (s,numneg t))"
+definition numsub :: "num \<Rightarrow> num \<Rightarrow> num"
+  where "numsub s t = (if s = t then C 0 else numadd (s, numneg t))"
 
 lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
-using numneg_def by simp
+  using numneg_def by simp
 
 lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
-using numneg_def by simp
+  using numneg_def by simp
 
 lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
-using numsub_def by simp
+  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
+  using numsub_def by simp
 
-primrec simpnum:: "num \<Rightarrow> num" where
+primrec simpnum:: "num \<Rightarrow> num"
+where
   "simpnum (C j) = C j"
 | "simpnum (Bound n) = CN n 1 (C 0)"
 | "simpnum (Neg t) = numneg (simpnum t)"
 | "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
 | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
-| "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
-| "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
+| "simpnum (Mul i t) = (if i = 0 then C 0 else nummul (simpnum t) i)"
+| "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0), simpnum t))"
 
 lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
-by (induct t) simp_all
+  by (induct t) simp_all
+
+lemma simpnum_numbound0[simp]: "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
+  by (induct t) simp_all
 
-lemma simpnum_numbound0[simp]: 
-  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
-by (induct t) simp_all
-
-fun nozerocoeff:: "num \<Rightarrow> bool" where
+fun nozerocoeff:: "num \<Rightarrow> bool"
+where
   "nozerocoeff (C c) = True"
-| "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
+| "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)
+  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 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)
+  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)
+  by (simp add: numsub_def numneg_nz numadd_nz)
 
 lemma simpnum_nz: "nozerocoeff (simpnum t)"
-by(induct t) (simp_all add: numadd_nz numneg_nz numsub_nz nummul_nz)
+  by (induct t) (simp_all add: numadd_nz numneg_nz numsub_nz nummul_nz)
 
 lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
 proof (induct t rule: maxcoeff.induct)
   case (2 n c t)
-  hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp_all
-  have "max (abs c) (maxcoeff t) \<ge> abs c" by simp
-  with cnz have "max (abs c) (maxcoeff t) > 0" by arith
-  with 2 show ?case by simp
+  then have cnz: "c \<noteq> 0" and mx: "max (abs c) (maxcoeff t) = 0"
+    by simp_all
+  have "max (abs c) (maxcoeff t) \<ge> abs c"
+    by simp
+  with cnz have "max (abs c) (maxcoeff t) > 0"
+    by arith
+  with 2 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" .
+lemma numgcd_nz:
+  assumes nz: "nozerocoeff t"
+    and g0: "numgcd t = 0"
+  shows "t = C 0"
+proof -
+  from g0 have th:"numgcdh t (maxcoeff t) = 0"
+    by (simp add: numgcd_def)
+  from numgcdh0[OF th] have th:"maxcoeff t = 0" .
   from maxcoeff_nz[OF nz th] show ?thesis .
 qed
 
-definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
-  "simp_num_pair = (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
-   (let t' = simpnum t ; g = numgcd t' in 
-      if g > 1 then (let g' = gcd n g in 
-        if g' = 1 then (t',n) 
-        else (reducecoeffh t' g', n div g')) 
-      else (t',n))))"
+definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int"
+where
+  "simp_num_pair =
+    (\<lambda>(t,n).
+     (if n = 0 then (C 0, 0)
+      else
+       (let t' = simpnum t ; g = numgcd t' in
+         if g > 1 then
+          (let g' = gcd n g
+           in if g' = 1 then (t', n) else (reducecoeffh t' g', n div g'))
+         else (t', n))))"
 
 lemma simp_num_pair_ci:
-  shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
+  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-
+proof -
   let ?t' = "simpnum t"
   let ?g = "numgcd ?t'"
   let ?g' = "gcd n ?g"
-  {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def) }
-    moreover
-    {assume g1:"?g>1" hence g0: "?g > 0" by simp
-      from g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith 
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
-      moreover {assume g'1:"?g'>1"
-        from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
+  show ?thesis
+  proof (cases "n = 0")
+    case True
+    then show ?thesis
+      by (simp add: Let_def simp_num_pair_def)
+  next
+    case nnz: False
+    show ?thesis
+    proof (cases "?g > 1")
+      case False
+      then show ?thesis by (simp add: Let_def simp_num_pair_def)
+    next
+      case g1: True
+      then have g0: "?g > 0"
+        by simp
+      from g1 nnz have gp0: "?g' \<noteq> 0"
+        by simp
+      then have g'p: "?g' > 0"
+        using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
+      then consider "?g' = 1" | "?g' > 1" by arith
+      then show ?thesis
+      proof cases
+        case 1
+        then show ?thesis
+          by (simp add: Let_def simp_num_pair_def)
+      next
+        case g'1: 2
+        from dvdnumcoeff_aux2[OF g1] have th1: "dvdnumcoeff ?t' ?g" ..
         let ?tt = "reducecoeffh ?t' ?g'"
         let ?t = "Inum bs ?tt"
         have gpdg: "?g' dvd ?g" by simp
-        have gpdd: "?g' dvd n" by simp 
+        have gpdd: "?g' dvd n" by simp
         have gpdgp: "?g' dvd ?g'" by simp
-        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
-        have th2:"real ?g' * ?t = Inum bs ?t'" by simp
-        from g1 g'1 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
+        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
+        have th2:"real ?g' * ?t = Inum bs ?t'"
+          by simp
+        from g1 g'1 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 gpdd] th2 gp0 by simp
-        finally have "?lhs = Inum bs t / real n" by simp
-        then have ?thesis by (simp add: simp_num_pair_def) }
-      ultimately have ?thesis by blast }
-    ultimately have ?thesis by blast }
-  ultimately show ?thesis by blast
+        finally have "?lhs = Inum bs t / real n"
+          by simp
+        then show ?thesis
+          by (simp add: simp_num_pair_def)
+      qed
+    qed
+  qed
 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-
+lemma simp_num_pair_l:
+  assumes tnb: "numbound0 t"
+    and np: "n > 0"
+    and tn: "simp_num_pair (t, n) = (t', n')"
+  shows "numbound0 t' \<and> n' > 0"
+proof -
   let ?t' = "simpnum t"
   let ?g = "numgcd ?t'"
   let ?g' = "gcd n ?g"
-  { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) }
-  moreover
-  { assume nnz: "n \<noteq> 0"
-    { assume "\<not> ?g > 1" hence ?thesis using assms
-        by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
-    moreover
-    { assume g1:"?g>1" hence g0: "?g > 0" by simp
+  show ?thesis
+  proof (cases "n = 0")
+    case True
+    then show ?thesis
+      using assms by (simp add: Let_def simp_num_pair_def)
+  next
+    case nnz: False
+    show ?thesis
+    proof (cases "?g > 1")
+      case False
+      then show ?thesis
+        using assms by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)
+    next
+      case g1: True
+      then have g0: "?g > 0" by simp
       from g1 nnz have gp0: "?g' \<noteq> 0" by simp
-      hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
-      hence "?g'= 1 \<or> ?g' > 1" by arith
-      moreover {
-        assume "?g' = 1" hence ?thesis using assms g1
-          by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
-      moreover {
-        assume g'1: "?g' > 1"
+      then have g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"]
+        by arith
+      then consider "?g'= 1" | "?g' > 1" by arith
+      then show ?thesis
+      proof cases
+        case 1
+        then show ?thesis
+          using assms g1 by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)
+      next
+        case g'1: 2
         have gpdg: "?g' dvd ?g" by simp
         have gpdd: "?g' dvd n" by simp
         have gpdgp: "?g' dvd ?g'" by simp
         from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
-        from zdiv_mono1[OF g'n g'p, simplified div_self[OF gp0]]
-        have "n div ?g' >0" by simp
-        hence ?thesis using assms g1 g'1
-          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
+        from zdiv_mono1[OF g'n g'p, simplified div_self[OF gp0]] have "n div ?g' > 0"
+          by simp
+        then show ?thesis
+          using assms g1 g'1
+          by (auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)
+      qed
+    qed
+  qed
 qed
 
-fun simpfm :: "fm \<Rightarrow> fm" where
+fun simpfm :: "fm \<Rightarrow> fm"
+where
   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
 | "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
 | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
 | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
 | "simpfm (NOT p) = not (simpfm p)"
-| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
-  | _ \<Rightarrow> Lt a')"
+| "simpfm (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
+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
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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
+  case (7 a)
+  let ?sa = "simpnum a"
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
+    by simp
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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
+  case (8 a)
+  let ?sa = "simpnum a"
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
+    by simp
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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
+  case (9 a)
+  let ?sa = "simpnum a"
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
+    by simp
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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
+  case (10 a)
+  let ?sa = "simpnum a"
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
+    by simp
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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
+  case (11 a)
+  let ?sa = "simpnum a"
+  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
+    by simp
+  consider v where "?sa = C v" | "\<not> (\<exists>v. ?sa = C v)" by blast
+  then show ?case
+  proof cases
+    case 1
+    then show ?thesis using sa by simp
+  next
+    case 2
+    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
+  qed
 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)
+proof (induct p rule: simpfm.induct)
+  case (6 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)
+  case (7 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)
+  case (8 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)
+  case (9 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)
+  case (10 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)
+  case (11 a)
+  then have nb: "numbound0 a" by simp
+  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
+  then show ?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)"
   apply (induct p rule: simpfm.induct)
@@ -832,7 +1033,7 @@
   "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 (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)))"
@@ -851,34 +1052,37 @@
   "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"
+  (hints simp add: fmsize_pos)
+
+lemma prep: "\<And>bs. Ifm bs (prep p) = Ifm bs p"
   by (induct p rule: prep.induct) auto
 
   (* Generic quantifier elimination *)
-function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
-  "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
-| "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
-| "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
-| "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
-| "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
-| "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
-| "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
-| "qelim p = (\<lambda> y. simpfm p)"
-by pat_completeness auto
+function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
+where
+  "qelim (E p) = (\<lambda>qe. DJ qe (qelim p qe))"
+| "qelim (A p) = (\<lambda>qe. not (qe ((qelim (NOT p) qe))))"
+| "qelim (NOT p) = (\<lambda>qe. not (qelim p qe))"
+| "qelim (And p q) = (\<lambda>qe. conj (qelim p qe) (qelim q qe))"
+| "qelim (Or  p q) = (\<lambda>qe. disj (qelim p qe) (qelim q qe))"
+| "qelim (Imp p q) = (\<lambda>qe. imp (qelim p qe) (qelim q qe))"
+| "qelim (Iff p q) = (\<lambda>qe. iff (qelim p qe) (qelim q qe))"
+| "qelim p = (\<lambda>y. simpfm p)"
+  by pat_completeness auto
 termination qelim by (relation "measure fmsize") simp_all
 
 lemma qelim_ci:
-  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
-  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
-using qe_inv DJ_qe[OF qe_inv] 
-by(induct p rule: qelim.induct) 
-(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
-  simpfm simpfm_qf simp del: simpfm.simps)
+  assumes qe_inv: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
+  shows "\<And>bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
+  using qe_inv DJ_qe[OF qe_inv]
+  by (induct p rule: qelim.induct)
+    (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
+      simpfm simpfm_qf simp del: simpfm.simps)
 
-fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
-  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
-| "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
+fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
+where
+  "minusinf (And p q) = conj (minusinf p) (minusinf q)"
+| "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
 | "minusinf (Eq  (CN 0 c e)) = F"
 | "minusinf (NEq (CN 0 c e)) = T"
 | "minusinf (Lt  (CN 0 c e)) = T"
@@ -887,9 +1091,10 @@
 | "minusinf (Ge  (CN 0 c e)) = F"
 | "minusinf p = p"
 
-fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
-  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
-| "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
+fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
+where
+  "plusinf (And p q) = conj (plusinf p) (plusinf q)"
+| "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
 | "plusinf (Eq  (CN 0 c e)) = F"
 | "plusinf (NEq (CN 0 c e)) = T"
 | "plusinf (Lt  (CN 0 c e)) = F"
@@ -898,9 +1103,10 @@
 | "plusinf (Ge  (CN 0 c e)) = T"
 | "plusinf p = p"
 
-fun isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *) where
-  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
-| "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
+fun isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
+where
+  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
+| "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
 | "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
 | "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
 | "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
@@ -910,100 +1116,111 @@
 | "isrlfm p = (isatom p \<and> (bound0 p))"
 
   (* splits the bounded from the unbounded part*)
-function (sequential) rsplit0 :: "num \<Rightarrow> int \<times> num" where
+function (sequential) rsplit0 :: "num \<Rightarrow> int \<times> num"
+where
   "rsplit0 (Bound 0) = (1,C 0)"
-| "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
-              in (ca+cb, Add ta tb))"
+| "rsplit0 (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 (Neg a) = (let (c,t) = rsplit0 a in (- c, Neg t))"
+| "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c * ca, Mul c ta))"
+| "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c + ca, ta))"
+| "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca, CN n c ta))"
 | "rsplit0 t = (0,t)"
-by pat_completeness auto
+  by pat_completeness auto
 termination rsplit0 by (relation "measure num_size") simp_all
 
-lemma rsplit0: 
-  shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
+lemma rsplit0: "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 2 have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
+  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 2 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))) = 
+  have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
     Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
     by (simp add: Let_def split_def algebra_simps)
-  also have "\<dots> = Inum bs a + Inum bs b" using 2 by (cases "rsplit0 a") auto
-  finally show ?case using nb by simp 
+  also have "\<dots> = Inum bs a + Inum bs b"
+    using 2 by (cases "rsplit0 a") auto
+  finally show ?case
+    using nb by simp
 qed (auto simp add: Let_def split_def algebra_simps, simp add: distrib_left[symmetric])
 
     (* Linearize a formula*)
-definition
-  lt :: "int \<Rightarrow> num \<Rightarrow> fm"
+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)) 
+  "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"
+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)) 
+  "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"
+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)) 
+  "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"
+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)) 
+  "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"
+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)) 
+  "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"
+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)) 
+  "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,rename_tac nat a b,case_tac "nat",auto)
+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,
+    rename_tac nat a b, 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,rename_tac nat a b,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,
+    rename_tac nat a b, 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,rename_tac nat a b,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,
+    rename_tac nat a b, 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,rename_tac nat a b,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,
+    rename_tac nat a b, 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,rename_tac nat a b,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,
+    rename_tac nat a b, 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,rename_tac nat a b,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,
+    rename_tac nat a b, case_tac "nat", auto)
 
 lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
-by (auto simp add: conj_def)
+  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)
+  by (auto simp add: disj_def)
 
 consts rlfm :: "fm \<Rightarrow> fm"
 recdef rlfm "measure fmsize"
@@ -1030,279 +1247,320 @@
   "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)
+  "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)
+  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
+  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 apply auto apply (rule_tac x= "min z za" in exI) apply auto done
+  then show ?case
+    apply auto
+    apply (rule_tac x= "min z za" in exI)
+    apply auto
+    done
 next
   case (2 p q)
-  thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
+  then show ?case
+    apply auto
+    apply (rule_tac x= "min z za" in exI)
+    apply auto
+    done
 next
-  case (3 c e) 
+  case (3 c e)
   from 3 have nb: "numbound0 e" by simp
   from 3 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a#bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "real c * x + ?e < 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (4 c e)   
+  case (4 c e)
   from 4 have nb: "numbound0 e" by simp
   from 4 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "real c * x + ?e < 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   from 5 have nb: "numbound0 e" by simp
   from 5 have cp: "real c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (6 c e)  
+  case (6 c e)
   from 6 have nb: "numbound0 e" by simp
   from lp 6 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (7 c e)  
+  case (7 c e)
   from 7 have nb: "numbound0 e" by simp
   from 7 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (8 c e)  
+  case (8 c e)
   from 8 have nb: "numbound0 e" by simp
   from 8 have cp: "real c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    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"] ac_simps) 
-    hence "real c * x + ?e < 0" by arith
+    then have "(real c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
+  then show ?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")
+  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
+  case (1 p q)
+  then show ?case
+    apply auto
+    apply (rule_tac x= "max z za" in exI)
+    apply auto
+    done
 next
-  case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
+  case (2 p q)
+  then show ?case
+    apply auto
+    apply (rule_tac x= "max z za" in exI)
+    apply auto
+    done
 next
-  case (3 c e) 
+  case (3 c e)
   from 3 have nb: "numbound0 e" by simp
   from 3 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    then have "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (4 c e) 
+  case (4 c e)
   from 4 have nb: "numbound0 e" by simp
   from 4 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
-    hence "real c * x + ?e \<noteq> 0" by simp
+    then have "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   from 5 have nb: "numbound0 e" by simp
   from 5 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (6 c e) 
+  case (6 c e)
   from 6 have nb: "numbound0 e" by simp
   from 6 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (7 c e) 
+  case (7 c e)
   from 7 have nb: "numbound0 e" by simp
   from 7 have cp: "real c > 0" by simp
   fix a
-  let ?e="Inum (a#bs) e"
+  let ?e = "Inum (a # bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
+  then show ?case by blast
 next
-  case (8 c e) 
+  case (8 c e)
   from 8 have nb: "numbound0 e" by simp
   from 8 have cp: "real c > 0" by simp
   fix a
   let ?e="Inum (a#bs) e"
   let ?z = "(- ?e) / real c"
-  {fix x
+  {
+    fix x
     assume xz: "x > ?z"
     with mult_strict_right_mono [OF xz cp] cp
     have "(real c * x > - ?e)" by (simp add: ac_simps)
-    hence "real c * x + ?e > 0" by arith
+    then have "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
+      using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp
+  }
+  then have "\<forall>x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
+  then show ?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
+  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
+  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-
+    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"] 
+  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
+  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-
+    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"] 
+  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
+  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 
+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 (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)]"
@@ -1311,257 +1569,272 @@
   "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)"
+  "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> _")
+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) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
+proof (induct p rule: usubst.induct)
+  case (5 c e)
+  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
+  have "?I ?u (Lt (CN 0 c e)) \<longleftrightarrow> 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)" 
+  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n*(?N x e) < 0"
+    by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
-    using np by simp 
+  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * (?N x e) < 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
-  case (6 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
+  case (6 c e)
+  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
+  have "?I ?u (Le (CN 0 c e)) \<longleftrightarrow> 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)" 
+    by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
-    using np by simp 
+  also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
-  case (7 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
+  case (7 c e)
+  with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
+  have "?I ?u (Gt (CN 0 c e)) \<longleftrightarrow> 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)" 
+  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e > 0"
+    by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
-    using np by simp 
+  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e > 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
-  case (8 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
-  have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
+  case (8 c e)
+  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
+  have "?I ?u (Ge (CN 0 c e)) \<longleftrightarrow> 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)" 
+  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e \<ge> 0"
+    by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
-    using np by simp 
+  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e \<ge> 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
-  case (3 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
+  case (3 c e)
+  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
   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)"
+  have "?I ?u (Eq (CN 0 c e)) \<longleftrightarrow> 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)" 
+  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e = 0"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
-    using np by simp 
+  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e = 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (4 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
   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)"
+  have "?I ?u (NEq (CN 0 c e)) \<longleftrightarrow> 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)" 
+  also have "\<dots> \<longleftrightarrow> ?n * (real c * (?t / ?n)) + ?n * ?N x e \<noteq> 0"
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
-  also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
-    using np by simp 
+  also have "\<dots> \<longleftrightarrow> real c * ?t + ?n * ?N x e \<noteq> 0" using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"])
 
 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)
+  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")
+    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"])
-  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 (induct p rule: minusinf.induct) (auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
+  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
+  then show ?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")
+    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"])
-  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 (induct p rule: minusinf.induct)
+      (auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
+  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
+  then show ?thesis
+    using smU by auto
 qed
 
-lemma lin_dense: 
+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 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+
+  case (5 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp+
   from 5 have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
-  hence pxc: "x < (- ?N x e) / real c" 
+  then have 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 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 5 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
+  then have "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"
+    then have "y * real c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real c" 
+    then have "real c * y + ?N x e < 0" by (simp add: algebra_simps)
+    then have ?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 }
+    then have ?case by simp }
   ultimately show ?case by blast
 next
-  case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
+  case (6 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp +
   from 6 have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<le> (- ?N x e) / real c" 
+  then have 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 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 6 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
+  then have "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"
+    then have "y * real c < - ?N x e"
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
-    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real c" 
+    then have "real c * y + ?N x e < 0" by (simp add: algebra_simps)
+    then have ?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 }
+    then have ?case by simp }
   ultimately show ?case by blast
 next
-  case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+  case (7 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp+
   from 7 have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
-  hence pxc: "x > (- ?N x e) / real c" 
+  then have 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 7 have noSc: "\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 7 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
+  then have "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"
+    then have "y * real c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real c" 
+    then have "real c * y + ?N x e > 0" by (simp add: algebra_simps)
+    then have ?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 }
+    then have ?case by simp }
   ultimately show ?case by blast
 next
-  case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+  case (8 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp+
   from 8 have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<ge> (- ?N x e) / real c" 
+  then have 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 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 8 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
+  then have "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"
+    then have "y * real c > - ?N x e"
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
-    hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
-    hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real c" 
+    then have "real c * y + ?N x e > 0" by (simp add: algebra_simps)
+    then have ?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 }
+    then have ?case by simp }
   ultimately show ?case by blast
 next
-  case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
+  case (3 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp+
   from cp have cnz: "real c \<noteq> 0" by simp
   from 3 have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
-  hence pxc: "x = (- ?N x e) / real c" 
+  then have 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 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 3 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+
+  case (4 c e) then have cp: "real c > 0" and nb: "numbound0 e" by simp+
   from cp have cnz: "real c \<noteq> 0" by simp
-  from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
+  from 4 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"      
+  then have "y* real c \<noteq> -?N x e"
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
-  hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
-  thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
+  then have "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
+  then show ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
     by (simp add: algebra_simps)
 qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
 
 lemma finite_set_intervals:
-  assumes px: "P (x::real)" 
+  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-
+  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
+  then have 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
+  then have Mxne: "?Mx \<noteq> {}" by blast
   have xMS: "?xM \<subseteq> S" by blast
-  hence fxM: "finite ?xM" using fS finite_subset by auto
+  then have 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
+  then have 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"
+  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp then have ainS: "?a \<in> S" using MxS by blast
+  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp then have 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}
+    moreover {assume "y \<in> ?Mx" then have "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
+    moreover {assume "y \<in> ?xM" then have "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
@@ -1571,55 +1844,55 @@
   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"
+  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 "\<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" 
+    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
+    then have 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)" .
+    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 then have lx: "?l \<le> a" using tx by simp
+    have "?N a s / real m \<le> ?u" using smM Mne by simp then have 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 have "\<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 "(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"
+      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
+      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
+      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"
@@ -1628,10 +1901,10 @@
       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" 
+  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"] 
+  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
@@ -1639,37 +1912,37 @@
 qed
     (* The Ferrante - Rackoff Theorem *)
 
-theorem fr_eq: 
+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")
+  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"
+  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 "?M \<or> ?P" then have "?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}
+    from rinf_uset[OF lp nmi npi] have "?F" using px by blast then have "?D" by blast}
   ultimately show "?D" by blast
 next
-  assume "?D" 
+  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}
+  moreover {assume f:"?F" then have "?E" by blast}
   ultimately show "?E" by blast
 qed
 
 
-lemma fr_equsubst: 
+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")
+  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"
+  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 "?M \<or> ?P" then have "?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"
+    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
@@ -1678,15 +1951,15 @@
       from tnb snb have st_nb: "numbound0 ?st" by simp
       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
-      from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
+      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}
+    with rinf_uset[OF lp nmi npi px] have "?F" by blast then have "?D" by blast}
   ultimately show "?D" by blast
 next
-  assume "?D" 
+  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)" 
+  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)"
@@ -1700,59 +1973,59 @@
     (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
 definition ferrack :: "fm \<Rightarrow> fm" where
   "ferrack p = (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
-                in if (mp = T \<or> pp = T) then T else 
-                   (let U = remdups(map simp_num_pair 
-                     (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
-                           (alluopairs (uset p')))) 
-                    in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
+                in if (mp = T \<or> pp = T) then T else
+                   (let U = remdups(map simp_num_pair
+                     (map (\<lambda>((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
+                           (alluopairs (uset p'))))
+                    in decr (disj mp (disj pp (evaldjf (simpfm \<circ> (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))"
+  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)"
+  then have 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 ?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  
+  from Ul th have  nnz: "n \<noteq> 0" by auto
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- 
-  thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
+
+  then show "(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  
+         (set U \<times> set U)"using mnz nnz th
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
-    by (rule_tac x="(s,m)" in bexI,simp_all) 
+    by (rule_tac x="(s,m)" in bexI,simp_all)
   (rule_tac x="(t,n)" in bexI,simp_all add: mult.commute)
 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"
+  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  
+  from Ul tnU have  nnz: "n \<noteq> 0" by auto
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
- have Pc:"\<forall> a b. ?P a b = ?P b a"
+ 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 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')"
+ 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)" 
+ then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
    and Pts': "?P (t',n') (s',m')" by blast
  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
  let ?st' = "Add (Mul m' t') (Mul n' s')"
    have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
- from Pts' have 
+ 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
@@ -1764,48 +2037,48 @@
 
 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)))"
+  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 
+  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" 
+  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 np mp have mnp: "real (2*n*m) > 0" 
+  from np mp have mnp: "real (2*n*m) > 0"
       by (simp add: mult.commute real_of_int_mult[symmetric] del: real_of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
    using mp np by (simp add: algebra_simps add_divide_distrib)
   from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
-  hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
+  then have "\<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 
+  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 
+  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'))" 
+  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))" 
+  then have "\<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 
+  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" 
+    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 np mp have mnp: "real (2*n*m) > 0" 
+  from 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)"
@@ -1818,66 +2091,66 @@
 
 lemma ferrack:
   assumes qf: "qfree p"
-  shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) 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"
+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 ?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 ?g = "\<lambda>((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
   let ?S = "map ?g ?Up"
   let ?SS = "map simp_num_pair ?S"
   let ?Y = "remdups ?SS"
-  let ?f= "(\<lambda> (t,n). ?N t / real n)"
-  let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
-  let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
-  let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
+  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 \<circ> (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
-  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)"
+  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
+  then have Snb: "\<forall>(t,n) \<in> set ?S. numbound0 t \<and> n > 0 " by auto
+  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"
+      then have "(t,n) \<in> set ?SS" by simp
+      then have "\<exists>(t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
         by (auto simp add: split_def simp del: map_map)
            (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
       then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
       from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
       from simp_num_pair_l[OF tnb np tns]
       have "numbound0 t \<and> n > 0" . }
-    thus ?thesis by blast
+    then show ?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: comp_assoc image_comp)
+  proof -
+     from simp_num_pair_ci[where bs="x#bs"] have
+    "\<forall>x. (?f \<circ> simp_num_pair) x = ?f x" by auto
+     then have th: "?f \<circ> simp_num_pair = ?f" using ext by blast
+    have "(?f ` set ?Y) = ((?f \<circ> simp_num_pair) ` set ?S)" by (simp add: comp_assoc image_comp)
     also have "\<dots> = (?f ` set ?S)" by (simp add: th)
-    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
+    also have "\<dots> = ((?f \<circ> ?g) ` set ?Up)"
       by (simp only: set_map o_def image_comp)
     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_comp] by blast
     finally show ?thesis .
   qed
-  have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
-  proof-
+  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)))" 
+    have "bound0 (usubst ?q (t,n))"  by simp then have "bound0 (simpfm (usubst ?q (t,n)))"
       using simpfm_bound0 by simp}
-    thus ?thesis by blast
+    then show ?thesis by blast
   qed
-  hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
+  then have ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm \<circ> (usubst ?q)"] by auto
   let ?mp = "minusinf ?q"
   let ?pp = "plusinf ?q"
   let ?M = "?I x ?mp"
@@ -1886,18 +2159,18 @@
   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  
+  from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm
 
-  have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
+  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> = (?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]
+    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm \<circ> (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)
+  then have 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
@@ -1950,7 +2223,7 @@
   | 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)) 
+      @{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 HOL.conj} $ t1 $ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
@@ -1975,7 +2248,7 @@
       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} = @{term True} 
+fun term_of_fm vs @{code T} = @{term True}
   | term_of_fm vs @{code F} = @{term False}
   | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $
       term_of_num vs t $ @{term "0::real"}
@@ -1996,7 +2269,7 @@
       term_of_fm vs t1 $ term_of_fm vs t2;
 
 in fn (ctxt, t) =>
-  let 
+  let
     val vs = Term.add_frees t [];
     val t' = (term_of_fm vs o @{code linrqe} o fm_of_term vs) t;
   in (Thm.cterm_of ctxt o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end