# HG changeset patch
# User wenzelm
# Date 1393431127 -3600
# Node ID 72c6ce5aea2ac46a6516616ff3a633a6decd7c85
# Parent  96ddf9bf12ac8c4933481acaabe19a3fff92293d
tuned specifications and proofs;

diff -r 96ddf9bf12ac -r 72c6ce5aea2a src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
--- a/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Wed Feb 26 14:59:24 2014 +0100
+++ b/src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy	Wed Feb 26 17:12:07 2014 +0100
@@ -158,30 +158,46 @@
      else []!(m - (length xs - 1)))"
 proof (induct xs arbitrary: n m)
   case Nil
-  thus ?case by simp
+  then show ?case by simp
 next
   case (Cons x xs n m)
-  {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
+  {
+    assume nxs: "n \<ge> length (x#xs)"
+    then have ?case using removen_same[OF nxs] by simp
+  }
   moreover
-  {assume nxs: "\<not> (n \<ge> length (x#xs))"
-    {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
+  {
+    assume nxs: "\<not> (n \<ge> length (x#xs))"
+    {
+      assume mln: "m < n"
+      then have ?case using Cons by (cases m) auto
+    }
     moreover
-    {assume mln: "\<not> (m < n)"
-      {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
+    {
+      assume mln: "\<not> (m < n)"
+      {
+        assume mxs: "m \<le> length (x#xs)"
+        then have ?case using Cons by (cases m) auto
+      }
       moreover
-      {assume mxs: "\<not> (m \<le> length (x#xs))"
+      {
+        assume mxs: "\<not> (m \<le> length (x#xs))"
         have th: "length (removen n (x#xs)) = length xs"
           using removen_length[where n="n" and xs="x#xs"] nxs by simp
-        with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
-        hence "(removen n (x#xs))!m = [] ! (m - length xs)"
+        with mxs have mxs':"m \<ge> length (removen n (x#xs))"
+          by auto
+        then have "(removen n (x#xs))!m = [] ! (m - length xs)"
           using th nth_length_exceeds[OF mxs'] by auto
-        hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))"
+        then have th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))"
           by auto
-        hence ?case using nxs mln mxs by auto }
+        then have ?case
+          using nxs mln mxs by auto
+      }
       ultimately have ?case by blast
     }
     ultimately have ?case by blast
-  } ultimately show ?case by blast
+  }
+  ultimately show ?case by blast
 qed
 
 lemma decrtm:
@@ -243,7 +259,7 @@
 (* Simplification *)
 
 consts tmadd:: "tm \<times> tm \<Rightarrow> tm"
-recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
+recdef tmadd "measure (\<lambda>(t,s). size t + size s)"
   "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
     (if n1 = n2 then
       let c = c1 +\<^sub>p c2
@@ -278,9 +294,9 @@
 
 fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm"
 where
-  "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
-| "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
-| "tmmul t = (\<lambda> i. Mul i t)"
+  "tmmul (CP j) = (\<lambda>i. CP (i *\<^sub>p j))"
+| "tmmul (CNP n c a) = (\<lambda>i. CNP n (i *\<^sub>p c) (tmmul a i))"
+| "tmmul t = (\<lambda>i. Mul i t)"
 
 lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
   by (induct t arbitrary: i rule: tmmul.induct) (simp_all add: field_simps)
@@ -362,8 +378,7 @@
   done
 
 lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
-  by (induct t rule: simptm.induct)
-    (auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid)
+  by (induct t rule: simptm.induct) (auto simp add: Let_def polynate_stupid)
 
 lemma simptm_tmbound0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
   by (induct t rule: simptm.induct) (auto simp add: Let_def)
@@ -422,9 +437,10 @@
 proof -
   fix c' t'
   assume "split0 t = (c', t')"
-  hence "c' = fst (split0 t)" and "t' = snd (split0 t)"
+  then have "c' = fst (split0 t)" and "t' = snd (split0 t)"
     by auto
-  with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')"
+  with split0[where t="t" and bs="bs"]
+  show "Itm vs bs t = Itm vs bs (CNP 0 c' t')"
     by simp
 qed
 
@@ -434,7 +450,7 @@
 proof -
   fix c' t'
   assume "split0 t = (c', t')"
-  hence "c' = fst (split0 t)" and "t' = snd (split0 t)"
+  then have "c' = fst (split0 t)" and "t' = snd (split0 t)"
     by auto
   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'"
     by simp
@@ -498,7 +514,8 @@
   by (induct p rule: fmsize.induct) simp_all
 
   (* Semantics of formulae (fm) *)
-primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
+primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool"
+where
   "Ifm vs bs T = True"
 | "Ifm vs bs F = False"
 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
@@ -513,7 +530,8 @@
 | "Ifm vs bs (E p) = (\<exists>x. Ifm vs (x#bs) p)"
 | "Ifm vs bs (A p) = (\<forall>x. Ifm vs (x#bs) p)"
 
-fun not:: "fm \<Rightarrow> fm" where
+fun not:: "fm \<Rightarrow> fm"
+where
   "not (NOT (NOT p)) = not p"
 | "not (NOT p) = p"
 | "not T = F"
@@ -549,7 +567,7 @@
      else Or p q)"
 
 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs 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
@@ -560,7 +578,7 @@
      else Imp p q)"
 
 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs 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
@@ -574,7 +592,7 @@
     else Iff p q)"
 
 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs 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)
 
 (* Quantifier freeness *)
 fun qfree:: "fm \<Rightarrow> bool"
@@ -593,10 +611,10 @@
 where
   "boundslt n T = True"
 | "boundslt n F = True"
-| "boundslt n (Lt t) = (tmboundslt n t)"
-| "boundslt n (Le t) = (tmboundslt n t)"
-| "boundslt n (Eq t) = (tmboundslt n t)"
-| "boundslt n (NEq t) = (tmboundslt n t)"
+| "boundslt n (Lt t) = tmboundslt n t"
+| "boundslt n (Le t) = tmboundslt n t"
+| "boundslt n (Eq t) = tmboundslt n t"
+| "boundslt n (NEq t) = tmboundslt n t"
 | "boundslt n (NOT p) = boundslt n p"
 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
@@ -656,7 +674,7 @@
       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .
   }
-  thus ?case by simp
+  then show ?case by simp
 next
   case (A p bs n)
   {
@@ -667,10 +685,11 @@
       by simp_all
     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .
   }
-  thus ?case by simp
+  then show ?case by simp
 qed auto
 
-fun decr0 :: "fm \<Rightarrow> fm" where
+fun decr0 :: "fm \<Rightarrow> fm"
+where
   "decr0 (Lt a) = Lt (decrtm0 a)"
 | "decr0 (Le a) = Le (decrtm0 a)"
 | "decr0 (Eq a) = Eq (decrtm0 a)"
@@ -721,7 +740,7 @@
     from E(1)[OF bnd nb nle]
     have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" .
   }
-  thus ?case by auto
+  then show ?case by auto
 next
   case (A p bs m)
   { fix x
@@ -733,7 +752,7 @@
     from A(1)[OF bnd nb nle]
     have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p" .
   }
-  thus ?case by auto
+  then show ?case by auto
 qed (auto simp add: decrtm removen_nth)
 
 primrec subst0 :: "tm \<Rightarrow> fm \<Rightarrow> fm"
@@ -795,12 +814,14 @@
     from E have nlm: "Suc n \<le> length (x#bs)"
       by simp
     from E(1)[OF bn nlm]
-    have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
+    have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
+        Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
       by simp
-    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
+    then have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
+        Ifm vs (x#bs[n:= Itm vs bs t]) p"
       by (simp add: incrtm0[where x="x" and bs="bs" and t="t"])
   }
-  thus ?case by simp
+  then show ?case by simp
 next
   case (A p bs n)
   {
@@ -810,12 +831,14 @@
     from A have nlm: "Suc n \<le> length (x#bs)"
       by simp
     from A(1)[OF bn nlm]
-    have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
+    have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
+        Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p"
       by simp
-    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
+    then have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) =
+        Ifm vs (x#bs[n:= Itm vs bs t]) p"
       by (simp add: incrtm0[where x="x" and bs="bs" and t="t"])
   }
-  thus ?case by simp
+  then show ?case by simp
 qed (auto simp add: tmsubst)
 
 lemma subst_nb:
@@ -852,11 +875,11 @@
   using disj_def by auto
 
 lemma imp_qf[simp]: "qfree p \<Longrightarrow> qfree q \<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_nb0[simp]: "bound0 p \<Longrightarrow> bound0 q \<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 imp_nb[simp]: "bound n p \<Longrightarrow> bound n q \<Longrightarrow> bound n (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 imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
   using imp_def by auto
 
@@ -895,7 +918,7 @@
   where "evaldjf f ps \<equiv> foldr (djf f) ps F"
 
 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
-  by (cases "q=T", simp add: djf_def,cases "q=F",simp add: 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 evaldjf_ex: "Ifm vs bs (evaldjf f ps) \<longleftrightarrow> (\<exists>p \<in> set ps. Ifm vs bs (f p))"
@@ -923,89 +946,109 @@
 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-
   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 \<equiv> evaldjf f (disjuncts p)"
-
-lemma DJ: assumes fdj: "\<forall>p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
-  and fF: "f F = F"
+definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
+  where "DJ f p \<equiv> evaldjf f (disjuncts p)"
+
+lemma DJ:
+  assumes fdj: "\<forall>p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
+    and fF: "f F = F"
   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
-proof-
+proof -
   have "Ifm vs bs (DJ f p) = (\<exists>q \<in> set (disjuncts p). Ifm vs bs (f q))"
     by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
+  also have "\<dots> = Ifm vs bs (f p)"
+    using fdj fF by (induct p rule: disjuncts.induct) auto
   finally show ?thesis .
 qed
 
-lemma DJ_qf: assumes
-  fqf: "\<forall>p. qfree p \<longrightarrow> qfree (f p)"
-  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
-proof(clarify)
-  fix  p assume qf: "qfree p"
-  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
+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
+  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 vs bs (qe p) = Ifm vs bs (E p))"
+lemma DJ_qe:
+  assumes qe: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
   shows "\<forall>bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
-proof(clarify)
-  fix p::fm and bs
+proof clarify
+  fix p :: fm and bs
   assume qf: "qfree p"
-  from qe have qth: "\<forall>p. qfree p \<longrightarrow> qfree (qe p)" by blast
-  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
-  have "Ifm vs bs (DJ qe p) = (\<exists>q\<in> set (disjuncts p). Ifm vs 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 vs bs (DJ qe p) \<longleftrightarrow> (\<exists>q\<in> set (disjuncts p). Ifm vs bs (qe q))"
     by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists>q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
-  also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
-  finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
+  also have "\<dots> = (\<exists>q \<in> set(disjuncts p). Ifm vs bs (E q))"
+    using qe disjuncts_qf[OF qf] by auto
+  also have "\<dots> = Ifm vs bs (E p)"
+    by (induct p rule: disjuncts.induct) auto
+  finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)"
+    using qfth by blast
 qed
 
-fun conjuncts :: "fm \<Rightarrow> fm list" where
-  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
+fun conjuncts :: "fm \<Rightarrow> fm list"
+where
+  "conjuncts (And p q) = conjuncts p @ conjuncts q"
 | "conjuncts T = []"
 | "conjuncts p = [p]"
 
-definition list_conj :: "fm list \<Rightarrow> fm" where
-  "list_conj ps \<equiv> foldr conj ps T"
-
-definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
-  "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
-                   in conj (decr0 (list_conj yes)) (f (list_conj no)))"
+definition list_conj :: "fm list \<Rightarrow> fm"
+  where "list_conj ps \<equiv> foldr conj ps T"
+
+definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
+where
+  "CJNB f p \<equiv>
+    (let cjs = conjuncts p;
+      (yes, no) = partition bound0 cjs
+     in conj (decr0 (list_conj yes)) (f (list_conj no)))"
 
 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall>q\<in> set (conjuncts p). qfree q"
-proof-
+proof -
   assume qf: "qfree p"
-  hence "list_all qfree (conjuncts p)"
-    by (induct p rule: conjuncts.induct, auto)
-  thus ?thesis by (simp only: list_all_iff)
+  then have "list_all qfree (conjuncts p)"
+    by (induct p rule: conjuncts.induct) auto
+  then show ?thesis
+    by (simp only: list_all_iff)
 qed
 
 lemma conjuncts: "(\<forall>q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
-by(induct p rule: conjuncts.induct, auto)
+  by (induct p rule: conjuncts.induct) auto
 
 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall>q\<in> set (conjuncts p). bound0 q"
-proof-
+proof -
   assume nb: "bound0 p"
-  hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
-  thus ?thesis by (simp only: list_all_iff)
+  then have "list_all bound0 (conjuncts p)"
+    by (induct p rule:conjuncts.induct) auto
+  then show ?thesis
+    by (simp only: list_all_iff)
 qed
 
-fun islin :: "fm \<Rightarrow> bool" where
+fun islin :: "fm \<Rightarrow> bool"
+where
   "islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
 | "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
@@ -1017,62 +1060,84 @@
 | "islin (Iff p q) = False"
 | "islin p = bound0 p"
 
-lemma islin_stupid: assumes nb: "tmbound0 p"
-  shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
+lemma islin_stupid:
+  assumes nb: "tmbound0 p"
+  shows "islin (Lt p)"
+    and "islin (Le p)"
+    and "islin (Eq p)"
+    and "islin (NEq p)"
   using nb by (cases p, auto, case_tac nat, auto)+
 
 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
-definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
+definition "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
 definition "neq p = not (eq p)"
 
 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
-  apply(simp add: lt_def)
-  apply(cases p, simp_all)
-  apply (case_tac poly, simp_all add: isnpoly_def)
+  apply (simp add: lt_def)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply (simp_all add: isnpoly_def)
   done
 
 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
-  apply(simp add: le_def)
-  apply(cases p, simp_all)
-  apply (case_tac poly, simp_all add: isnpoly_def)
+  apply (simp add: le_def)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply (simp_all add: isnpoly_def)
   done
 
 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
-  apply(simp add: eq_def)
-  apply(cases p, simp_all)
-  apply (case_tac poly, simp_all add: isnpoly_def)
+  apply (simp add: eq_def)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply (simp_all add: isnpoly_def)
   done
 
 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
-  by(simp add: neq_def eq)
+  by (simp add: neq_def eq)
 
 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
   apply (simp add: lt_def)
-  apply (cases p, simp_all)
-  apply (case_tac poly, simp_all)
-  apply (case_tac nat, simp_all)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply simp_all
+  apply (case_tac nat)
+  apply simp_all
   done
 
 lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
   apply (simp add: le_def)
-  apply (cases p, simp_all)
-  apply (case_tac poly, simp_all)
-  apply (case_tac nat, simp_all)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply simp_all
+  apply (case_tac nat)
+  apply simp_all
   done
 
 lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
   apply (simp add: eq_def)
-  apply (cases p, simp_all)
-  apply (case_tac poly, simp_all)
-  apply (case_tac nat, simp_all)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply simp_all
+  apply (case_tac nat)
+  apply simp_all
   done
 
 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
   apply (simp add: neq_def eq_def)
-  apply (cases p, simp_all)
-  apply (case_tac poly, simp_all)
-  apply (case_tac nat, simp_all)
+  apply (cases p)
+  apply simp_all
+  apply (case_tac poly)
+  apply simp_all
+  apply (case_tac nat)
+  apply simp_all
   done
 
 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
@@ -1080,203 +1145,269 @@
 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
 
-lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simplt_islin[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "islin (simplt t)"
   unfolding simplt_def
   using split0_nb0'
-by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly])
-
-lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly]
+      islin_stupid allpolys_split0[OF simptm_allpolys_npoly])
+
+lemma simple_islin[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "islin (simple t)"
   unfolding simple_def
   using split0_nb0'
-by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin)
-lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly]
+      islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin)
+
+lemma simpeq_islin[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "islin (simpeq t)"
   unfolding simpeq_def
   using split0_nb0'
-by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin)
-
-lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly]
+      islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin)
+
+lemma simpneq_islin[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "islin (simpneq t)"
   unfolding simpneq_def
   using split0_nb0'
-by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin)
+  by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly]
+      islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin)
 
 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
-  by (cases "split0 s", auto)
-lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
-  and n: "allpolys isnpoly t"
-  shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
+  by (cases "split0 s") auto
+
+lemma split0_npoly:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and n: "allpolys isnpoly t"
+  shows "isnpoly (fst (split0 t))"
+    and "allpolys isnpoly (snd (split0 t))"
   using n
-  by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
-lemma simplt[simp]:
-  shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
-proof-
-  have n: "allpolys isnpoly (simptm t)" by simp
+  by (induct t rule: split0.induct)
+    (auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm
+      polysub_norm really_stupid)
+
+lemma simplt[simp]: "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
+proof -
+  have n: "allpolys isnpoly (simptm t)"
+    by simp
   let ?t = "simptm t"
-  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
+  {
+    assume "fst (split0 ?t) = 0\<^sub>p"
+    then have ?thesis
       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
-      by (simp add: simplt_def Let_def split_def lt)}
+      by (simp add: simplt_def Let_def split_def lt)
+  }
   moreover
-  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
-    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
+  {
+    assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
+    then have ?thesis
+      using  split0[of "simptm t" vs bs]
+      by (simp add: simplt_def Let_def split_def)
   }
   ultimately show ?thesis by blast
 qed
 
-lemma simple[simp]:
-  shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
-proof-
-  have n: "allpolys isnpoly (simptm t)" by simp
+lemma simple[simp]: "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
+proof -
+  have n: "allpolys isnpoly (simptm t)"
+    by simp
   let ?t = "simptm t"
-  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
+  {
+    assume "fst (split0 ?t) = 0\<^sub>p"
+    then have ?thesis
       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
-      by (simp add: simple_def Let_def split_def le)}
+      by (simp add: simple_def Let_def split_def le)
+  }
   moreover
-  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
-    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
+  {
+    assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
+    then have ?thesis
+      using split0[of "simptm t" vs bs]
+      by (simp add: simple_def Let_def split_def)
   }
   ultimately show ?thesis by blast
 qed
 
-lemma simpeq[simp]:
-  shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
-proof-
-  have n: "allpolys isnpoly (simptm t)" by simp
+lemma simpeq[simp]: "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
+proof -
+  have n: "allpolys isnpoly (simptm t)"
+    by simp
   let ?t = "simptm t"
-  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
+  {
+    assume "fst (split0 ?t) = 0\<^sub>p"
+    then have ?thesis
       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
-      by (simp add: simpeq_def Let_def split_def)}
+      by (simp add: simpeq_def Let_def split_def)
+  }
   moreover
-  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
-    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
+  {
+    assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
+    then have ?thesis using  split0[of "simptm t" vs bs]
+      by (simp add: simpeq_def Let_def split_def)
   }
   ultimately show ?thesis by blast
 qed
 
-lemma simpneq[simp]:
-  shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
-proof-
-  have n: "allpolys isnpoly (simptm t)" by simp
+lemma simpneq[simp]: "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
+proof -
+  have n: "allpolys isnpoly (simptm t)"
+    by simp
   let ?t = "simptm t"
-  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
+  {
+    assume "fst (split0 ?t) = 0\<^sub>p"
+    then have ?thesis
       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
-      by (simp add: simpneq_def Let_def split_def )}
+      by (simp add: simpneq_def Let_def split_def)
+  }
   moreover
-  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
-    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
+  {
+    assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
+    then have ?thesis
+      using split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
   }
   ultimately show ?thesis by blast
 qed
 
 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
   apply (simp add: lt_def)
-  apply (cases t, auto)
-  apply (case_tac poly, auto)
+  apply (cases t)
+  apply auto
+  apply (case_tac poly)
+  apply auto
   done
 
 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
   apply (simp add: le_def)
-  apply (cases t, auto)
-  apply (case_tac poly, auto)
+  apply (cases t)
+  apply auto
+  apply (case_tac poly)
+  apply auto
   done
 
 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
   apply (simp add: eq_def)
-  apply (cases t, auto)
-  apply (case_tac poly, auto)
+  apply (cases t)
+  apply auto
+  apply (case_tac poly)
+  apply auto
   done
 
 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
   apply (simp add: neq_def eq_def)
-  apply (cases t, auto)
-  apply (case_tac poly, auto)
+  apply (cases t)
+  apply auto
+  apply (case_tac poly)
+  apply auto
   done
 
-lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simplt_nb[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
-  using split0 [of "simptm t" "vs::'a list" bs]
-proof(simp add: simplt_def Let_def split_def)
+proof (simp add: simplt_def Let_def split_def)
   assume nb: "tmbound0 t"
-  hence nb': "tmbound0 (simptm t)" by simp
+  then have nb': "tmbound0 (simptm t)"
+    by simp
   let ?c = "fst (split0 (simptm t))"
   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
-  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
+  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p"
+    by auto
   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
-  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
+  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0"
+    by (simp_all add: isnpoly_def)
   from iffD1[OF isnpolyh_unique[OF ths] th]
   have "fst (split0 (simptm t)) = 0\<^sub>p" .
-  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
-       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
+  then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
+      fst (split0 (simptm t)) = 0\<^sub>p"
+    by (simp add: simplt_def Let_def split_def lt_nb)
 qed
 
-lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simple_nb[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
-  using split0 [of "simptm t" "vs::'a list" bs]
 proof(simp add: simple_def Let_def split_def)
   assume nb: "tmbound0 t"
-  hence nb': "tmbound0 (simptm t)" by simp
+  then have nb': "tmbound0 (simptm t)"
+    by simp
   let ?c = "fst (split0 (simptm t))"
   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
-  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
+  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p"
+    by auto
   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
-  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
+  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0"
+    by (simp_all add: isnpoly_def)
   from iffD1[OF isnpolyh_unique[OF ths] th]
   have "fst (split0 (simptm t)) = 0\<^sub>p" .
-  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
-       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
+  then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
+      fst (split0 (simptm t)) = 0\<^sub>p"
+    by (simp add: simplt_def Let_def split_def le_nb)
 qed
 
-lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simpeq_nb[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
-  using split0 [of "simptm t" "vs::'a list" bs]
-proof(simp add: simpeq_def Let_def split_def)
+proof (simp add: simpeq_def Let_def split_def)
   assume nb: "tmbound0 t"
-  hence nb': "tmbound0 (simptm t)" by simp
+  then have nb': "tmbound0 (simptm t)"
+    by simp
   let ?c = "fst (split0 (simptm t))"
   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
-  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
+  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p"
+    by auto
   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
-  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
+  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0"
+    by (simp_all add: isnpoly_def)
   from iffD1[OF isnpolyh_unique[OF ths] th]
   have "fst (split0 (simptm t)) = 0\<^sub>p" .
-  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
-       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
+  then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
+      fst (split0 (simptm t)) = 0\<^sub>p"
+    by (simp add: simpeq_def Let_def split_def eq_nb)
 qed
 
-lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simpneq_nb[simp]:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
-  using split0 [of "simptm t" "vs::'a list" bs]
-proof(simp add: simpneq_def Let_def split_def)
+proof (simp add: simpneq_def Let_def split_def)
   assume nb: "tmbound0 t"
-  hence nb': "tmbound0 (simptm t)" by simp
+  then have nb': "tmbound0 (simptm t)"
+    by simp
   let ?c = "fst (split0 (simptm t))"
   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
-  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
+  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p"
+    by auto
   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
-  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
+  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0"
+    by (simp_all add: isnpoly_def)
   from iffD1[OF isnpolyh_unique[OF ths] th]
   have "fst (split0 (simptm t)) = 0\<^sub>p" .
-  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
-       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
+  then show "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
+      fst (split0 (simptm t)) = 0\<^sub>p"
+    by (simp add: simpneq_def Let_def split_def neq_nb)
 qed
 
-fun conjs   :: "fm \<Rightarrow> fm list" where
-  "conjs (And p q) = (conjs p)@(conjs q)"
+fun conjs :: "fm \<Rightarrow> fm list"
+where
+  "conjs (And p q) = conjs p @ conjs q"
 | "conjs T = []"
 | "conjs p = [p]"
+
 lemma conjs_ci: "(\<forall>q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
-by (induct p rule: conjs.induct, auto)
-definition list_disj :: "fm list \<Rightarrow> fm" where
-  "list_disj ps \<equiv> foldr disj ps F"
+  by (induct p rule: conjs.induct) auto
+
+definition list_disj :: "fm list \<Rightarrow> fm"
+  where "list_disj ps \<equiv> foldr disj ps F"
 
 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
-  by (induct ps, auto simp add: list_conj_def)
+  by (induct ps) (auto simp add: list_conj_def)
+
 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
-  by (induct ps, auto simp add: list_conj_def)
+  by (induct ps) (auto simp add: list_conj_def)
+
 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
-  by (induct ps, auto simp add: list_disj_def)
+  by (induct ps) (auto simp add: list_disj_def)
 
 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
   unfolding conj_def by auto
@@ -1297,25 +1428,27 @@
   apply (unfold ball_Un)
   apply (unfold boundslt.simps)
   apply blast
-by simp_all
+  apply simp_all
+  done
 
 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
   unfolding list_conj_def
-  by (induct ps, auto simp add: conj_boundslt)
-
-lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
+  by (induct ps) auto
+
+lemma list_conj_nb:
+  assumes bnd: "\<forall>p\<in> set ps. bound n p"
   shows "bound n (list_conj ps)"
   using bnd
   unfolding list_conj_def
-  by (induct ps, auto simp add: conj_nb)
+  by (induct ps) auto
 
 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
-unfolding list_conj_def by (induct ps , auto)
+  unfolding list_conj_def by (induct ps) auto
 
 lemma CJNB_qe:
   assumes qe: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
   shows "\<forall>bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
-proof(clarify)
+proof clarify
   fix bs p
   assume qfp: "qfree p"
   let ?cjs = "conjuncts p"
@@ -1323,24 +1456,36 @@
   let ?no = "snd (partition bound0 ?cjs)"
   let ?cno = "list_conj ?no"
   let ?cyes = "list_conj ?yes"
-  have part: "partition bound0 ?cjs = (?yes,?no)" by simp
-  from partition_P[OF part] have "\<forall>q\<in> set ?yes. bound0 q" by blast
-  hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb')
-  hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
+  have part: "partition bound0 ?cjs = (?yes,?no)"
+    by simp
+  from partition_P[OF part] have "\<forall>q\<in> set ?yes. bound0 q"
+    by blast
+  then have yes_nb: "bound0 ?cyes"
+    by (simp add: list_conj_nb')
+  then have yes_qf: "qfree (decr0 ?cyes)"
+    by (simp add: decr0_qf)
   from conjuncts_qf[OF qfp] partition_set[OF part]
-  have " \<forall>q\<in> set ?no. qfree q" by auto
-  hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
-  with qe have cno_qf:"qfree (qe ?cno )"
-    and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
+  have " \<forall>q\<in> set ?no. qfree q"
+    by auto
+  then have no_qf: "qfree ?cno"
+    by (simp add: list_conj_qf)
+  with qe have cno_qf:"qfree (qe ?cno)"
+    and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)"
+    by blast+
   from cno_qf yes_qf have qf: "qfree (CJNB qe p)"
-    by (simp add: CJNB_def Let_def conj_qf split_def)
-  {fix bs
-    from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
+    by (simp add: CJNB_def Let_def split_def)
+  {
+    fix bs
+    from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)"
+      by blast
     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
       using partition_set[OF part] by auto
-    finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
-  hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
-  also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
+    finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))"
+      using list_conj[of vs bs] by simp
+  }
+  then have "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
+    by simp
+  also fix y have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
@@ -1348,7 +1493,8 @@
     using qe[rule_format, OF no_qf] by auto
   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)"
     by (simp add: Let_def CJNB_def split_def)
-  with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
+  with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)"
+    by blast
 qed
 
 consts simpfm :: "fm \<Rightarrow> fm"
@@ -1360,11 +1506,13 @@
   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
-  "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
+  "simpfm (Iff p q) =
+    disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
-  "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
+  "simpfm (NOT (Iff p q)) =
+    disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
   "simpfm (NOT (Eq t)) = simpneq t"
   "simpfm (NOT (NEq t)) = simpeq t"
   "simpfm (NOT (Le t)) = simplt (Neg t)"
@@ -1375,23 +1523,33 @@
   "simpfm p = p"
 
 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
-by(induct p arbitrary: bs rule: simpfm.induct, auto)
-
-lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  by (induct p arbitrary: bs rule: simpfm.induct) auto
+
+lemma simpfm_bound0:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
-by (induct p rule: simpfm.induct, auto)
+  by (induct p rule: simpfm.induct) auto
 
 lemma lt_qf[simp]: "qfree (lt t)"
-  apply (cases t, auto simp add: lt_def)
-  by (case_tac poly, auto)
+  apply (cases t)
+  apply (auto simp add: lt_def)
+  apply (case_tac poly)
+  apply auto
+  done
 
 lemma le_qf[simp]: "qfree (le t)"
-  apply (cases t, auto simp add: le_def)
-  by (case_tac poly, auto)
+  apply (cases t)
+  apply (auto simp add: le_def)
+  apply (case_tac poly)
+  apply auto
+  done
 
 lemma eq_qf[simp]: "qfree (eq t)"
-  apply (cases t, auto simp add: eq_def)
-  by (case_tac poly, auto)
+  apply (cases t)
+  apply (auto simp add: eq_def)
+  apply (case_tac poly)
+  apply auto
+  done
 
 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
 
@@ -1401,16 +1559,17 @@
 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
 
 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
-by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
-
-lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
-lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
-
-lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+  by (induct p rule: simpfm.induct) auto
+
+lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)"
+  by (simp add: disj_def)
+lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)"
+  by (simp add: conj_def)
+
+lemma
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "qfree p \<Longrightarrow> islin (simpfm p)"
-  apply (induct p rule: simpfm.induct)
-  apply (simp_all add: conj_lin disj_lin)
-  done
+  by (induct p rule: simpfm.induct) (simp_all add: conj_lin disj_lin)
 
 consts prep :: "fm \<Rightarrow> fm"
 recdef prep "measure fmsize"
@@ -1438,33 +1597,36 @@
   "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: "Ifm vs bs (prep p) = Ifm vs bs p"
-by (induct p arbitrary: bs rule: prep.induct, auto)
-
-
-
-  (* Generic quantifier elimination *)
-function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
-  "qelim (E p) = (\<lambda> qe. DJ (CJNB 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 (induct p arbitrary: bs rule: prep.induct) auto
+
+
+(* Generic quantifier elimination *)
+function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
+where
+  "qelim (E p) = (\<lambda>qe. DJ (CJNB 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 simp_all
 termination by (relation "measure fmsize") auto
 
 lemma qelim:
   assumes qe_inv: "\<forall>bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
-using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
-by (induct p rule: qelim.induct) auto
-
-subsection{* Core Procedure *}
-
-fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
+  using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
+  by (induct p rule: qelim.induct) auto
+
+
+subsection {* Core Procedure *}
+
+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  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
@@ -1473,7 +1635,8 @@
 | "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
 | "minusinf p = p"
 
-fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
+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  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
@@ -1482,443 +1645,811 @@
 | "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
 | "plusinf p = p"
 
-lemma minusinf_inf: assumes lp:"islin p"
+lemma minusinf_inf:
+  assumes lp: "islin p"
   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
   using lp
 proof (induct p rule: minusinf.induct)
-  case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
+  case 1
+  then show ?case
+    apply auto
+    apply (rule_tac x="min z za" in exI)
+    apply auto
+    done
 next
-  case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
+  case 2
+  then show ?case
+    apply auto
+    apply (rule_tac x="min z za" in exI)
+    apply auto
+    done
 next
-  case (3 c e) hence nbe: "tmbound0 e" by simp
-  from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
+  case (3 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case
-      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
-        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
-        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
-  ultimately show ?case by blast
-next
-  case (4 c e)  hence nbe: "tmbound0 e" by simp
-  from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
-  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
-  let ?c = "Ipoly vs c"
-  let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
-        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
-        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
+  have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case
+      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x < - ?e"
+        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x > - ?e"
+        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
 next
-  case (5 c e)  hence nbe: "tmbound0 e" by simp
-  from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
-  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
-  note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
+  case (4 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
+  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
-        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
-        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
+  have "?c=0 \<or> ?c > 0 \<or> ?c < 0"
+    by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case
+      using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x < - ?e"
+        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x > - ?e"
+        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
 next
-  case (6 c e)  hence nbe: "tmbound0 e" by simp
-  from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
-  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
+  case (5 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
+  then have nc': "allpolys isnpoly (CP (~\<^sub>p c))"
+    by (simp add: polyneg_norm)
+  note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
+  let ?c = "Ipoly vs c"
+  fix y
+  let ?e = "Itm vs (y#bs) e"
+  have "?c=0 \<or> ?c > 0 \<or> ?c < 0"
+    by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x < - ?e"
+        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0" by simp
+      then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x > - ?e"
+        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto
+    }
+    then have ?case by auto
+  }
+  ultimately show ?case by blast
+next
+  case (6 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
+  then have nc': "allpolys isnpoly (CP (~\<^sub>p c))"
+    by (simp add: polyneg_norm)
   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
-        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
-        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
+  have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x < - ?e"
+        using pos_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs
+        by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x < -?e / ?c"
+      then have "?c * x > - ?e"
+        using neg_less_divide_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs
+        by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
-qed (auto)
-
-lemma plusinf_inf: assumes lp:"islin p"
+qed auto
+
+lemma plusinf_inf:
+  assumes lp: "islin p"
   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
   using lp
 proof (induct p rule: plusinf.induct)
-  case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
+  case 1
+  then show ?case
+    apply auto
+    apply (rule_tac x="max z za" in exI)
+    apply auto
+    done
 next
-  case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
+  case 2
+  then show ?case
+    apply auto
+    apply (rule_tac x="max z za" in exI)
+    apply auto
+    done
 next
-  case (3 c e) hence nbe: "tmbound0 e" by simp
-  from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
+  case (3 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case
-      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
-        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
-        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
+  moreover {
+    assume "?c = 0"
+    then have ?case
+      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x > - ?e"
+        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x < - ?e"
+        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0" by simp
+      then have "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
 next
-  case (4 c e) hence nbe: "tmbound0 e" by simp
-  from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
+  case (4 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
-  let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
-        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
-        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
-  ultimately show ?case by blast
-next
-  case (5 c e) hence nbe: "tmbound0 e" by simp
-  from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
-  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
-  note eqs = lt[OF nc(1), where ?'a = 'a] lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
-  let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
-        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
-        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
-        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
+  moreover {
+    assume "?c = 0"
+    then have ?case using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x > - ?e"
+        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x < - ?e"
+        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
 next
-  case (6 c e)  hence nbe: "tmbound0 e" by simp
-  from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
-  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
+  case (5 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
+  then have nc': "allpolys isnpoly (CP (~\<^sub>p c))"
+    by (simp add: polyneg_norm)
+  note eqs = lt[OF nc(1), where ?'a = 'a] lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
+  let ?c = "Ipoly vs c"
+  fix y
+  let ?e = "Itm vs (y#bs) e"
+  have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x > - ?e"
+        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x < - ?e"
+        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
+        using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto
+    }
+    then have ?case by auto
+  }
+  ultimately show ?case by blast
+next
+  case (6 c e)
+  then have nbe: "tmbound0 e"
+    by simp
+  from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e"
+    by simp_all
+  then have nc': "allpolys isnpoly (CP (~\<^sub>p c))"
+    by (simp add: polyneg_norm)
   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
   let ?c = "Ipoly vs c"
+  fix y
   let ?e = "Itm vs (y#bs) e"
-  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
-  moreover {assume "?c = 0" hence ?case using eqs by auto}
-  moreover {assume cp: "?c > 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
-        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e > 0" by simp
-      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
-  moreover {assume cp: "?c < 0"
-    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
-        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
-      hence "?c * x + ?e < 0" by simp
-      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
-        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
+  have "?c = 0 \<or> ?c > 0 \<or> ?c < 0" by arith
+  moreover {
+    assume "?c = 0"
+    then have ?case using eqs by auto
+  }
+  moreover {
+    assume cp: "?c > 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x > - ?e"
+        using pos_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e > 0"
+        by simp
+      then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto
+    }
+    then have ?case by auto
+  }
+  moreover {
+    assume cp: "?c < 0"
+    {
+      fix x
+      assume xz: "x > -?e / ?c"
+      then have "?c * x < - ?e"
+        using neg_divide_less_eq[OF cp, where a="x" and b="-?e"]
+        by (simp add: mult_commute)
+      then have "?c * x + ?e < 0"
+        by simp
+      then have "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
+        using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto
+    }
+    then have ?case by auto
+  }
   ultimately show ?case by blast
-qed (auto)
+qed auto
 
 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)"
-  by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
+  by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb)
+
 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)"
-  by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
-
-lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
+  by (induct p rule: minusinf.induct) (auto simp add: eq_nb lt_nb le_nb)
+
+lemma minusinf_ex:
+  assumes lp: "islin p"
+    and ex: "Ifm vs (x#bs) (minusinf p)"
   shows "\<exists>x. Ifm vs (x#bs) p"
-proof-
-  from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall>x. Ifm vs (x#bs) (minusinf p)" by auto
+proof -
+  from bound0_I [OF minusinf_nb[OF lp], where bs ="bs"] ex
+  have th: "\<forall>x. Ifm vs (x#bs) (minusinf p)"
+    by auto
   from minusinf_inf[OF lp, where bs="bs"]
-  obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
-  from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
-  moreover have "z - 1 < z" by simp
-  ultimately show ?thesis using z_def by auto
+  obtain z where z: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p"
+    by blast
+  from th have "Ifm vs ((z - 1)#bs) (minusinf p)"
+    by simp
+  moreover have "z - 1 < z"
+    by simp
+  ultimately show ?thesis
+    using z by auto
 qed
 
-lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
+lemma plusinf_ex:
+  assumes lp: "islin p"
+    and ex: "Ifm vs (x#bs) (plusinf p)"
   shows "\<exists>x. Ifm vs (x#bs) p"
-proof-
-  from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
-  have th: "\<forall>x. Ifm vs (x#bs) (plusinf p)" by auto
+proof -
+  from bound0_I [OF plusinf_nb[OF lp], where bs ="bs"] ex
+  have th: "\<forall>x. Ifm vs (x#bs) (plusinf p)"
+    by auto
   from plusinf_inf[OF lp, where bs="bs"]
-  obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
-  from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
-  moreover have "z + 1 > z" by simp
-  ultimately show ?thesis using z_def by auto
+  obtain z where z: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p"
+    by blast
+  from th have "Ifm vs ((z + 1)#bs) (plusinf p)"
+    by simp
+  moreover have "z + 1 > z"
+    by simp
+  ultimately show ?thesis
+    using z by auto
 qed
 
-fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
+fun uset :: "fm \<Rightarrow> (poly \<times> tm) list"
+where
   "uset (And p q) = uset p @ uset q"
 | "uset (Or p q) = uset p @ uset q"
-| "uset (Eq (CNP 0 a e))  = [(a,e)]"
-| "uset (Le (CNP 0 a e))  = [(a,e)]"
-| "uset (Lt (CNP 0 a e))  = [(a,e)]"
-| "uset (NEq (CNP 0 a e)) = [(a,e)]"
+| "uset (Eq (CNP 0 a e)) = [(a, e)]"
+| "uset (Le (CNP 0 a e)) = [(a, e)]"
+| "uset (Lt (CNP 0 a e)) = [(a, e)]"
+| "uset (NEq (CNP 0 a e)) = [(a, e)]"
 | "uset p = []"
 
 lemma uset_l:
   assumes lp: "islin p"
   shows "\<forall>(c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
-using lp by(induct p rule: uset.induct,auto)
+  using lp by (induct p rule: uset.induct) auto
 
 lemma minusinf_uset0:
   assumes lp: "islin p"
-  and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
-  and ex: "Ifm vs (x#bs) p" (is "?I x p")
-  shows "\<exists>(c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c"
-proof-
-  have "\<exists>(c,s) \<in> set (uset p). (Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)"
+    and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
+    and ex: "Ifm vs (x#bs) p" (is "?I x p")
+  shows "\<exists>(c, s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c"
+proof -
+  have "\<exists>(c, s) \<in> set (uset p).
+      Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s \<or>
+      Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s"
     using lp nmi ex
-    apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
+    apply (induct p rule: minusinf.induct)
+    apply (auto simp add: eq le lt polyneg_norm)
     apply (auto simp add: linorder_not_less order_le_less)
     done
-  then obtain c s where csU: "(c,s) \<in> set (uset p)" and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" by blast
-  hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
+  then obtain c s where csU: "(c, s) \<in> set (uset p)"
+    and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or>
+      (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" by blast
+  then have "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
-      by (auto simp add: mult_commute)
-  thus ?thesis using csU by auto
+    by (auto simp add: mult_commute)
+  then show ?thesis
+    using csU by auto
 qed
 
 lemma minusinf_uset:
   assumes lp: "islin p"
-  and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
-  and ex: "Ifm vs (x#bs) p" (is "?I x p")
+    and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
+    and ex: "Ifm vs (x#bs) p" (is "?I x p")
   shows "\<exists>(c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c"
-proof-
-  from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))"
+proof -
+  from nmi have nmi': "\<not> Ifm vs (x#bs) (minusinf p)"
     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
   from minusinf_uset0[OF lp nmi' ex]
-  obtain c s where csU: "(c,s) \<in> set (uset p)" and th: "x \<ge> - Itm vs (x#bs) s / Ipoly vs c" by blast
-  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
-  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
+  obtain c s where csU: "(c,s) \<in> set (uset p)"
+    and th: "x \<ge> - Itm vs (x#bs) s / Ipoly vs c"
+    by blast
+  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s"
+    by simp
+  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis
+    by auto
 qed
 
 
 lemma plusinf_uset0:
   assumes lp: "islin p"
-  and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
-  and ex: "Ifm vs (x#bs) p" (is "?I x p")
-  shows "\<exists>(c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c"
+    and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
+    and ex: "Ifm vs (x#bs) p" (is "?I x p")
+  shows "\<exists>(c, s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c"
 proof-
-  have "\<exists>(c,s) \<in> set (uset p). (Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s)"
+  have "\<exists>(c, s) \<in> set (uset p).
+      Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s \<or>
+      Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s"
     using lp nmi ex
-    apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
+    apply (induct p rule: minusinf.induct)
+    apply (auto simp add: eq le lt polyneg_norm)
     apply (auto simp add: linorder_not_less order_le_less)
     done
-  then obtain c s where csU: "(c,s) \<in> set (uset p)" and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s)" by blast
-  hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
+  then obtain c s where csU: "(c, s) \<in> set (uset p)"
+    and x: "Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s \<or>
+      Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s"
+    by blast
+  then have "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
-    by (auto simp add: mult_commute del: divide_minus_left)
-  thus ?thesis using csU by auto
+    by (auto simp add: mult_commute)
+  then show ?thesis
+    using csU by auto
 qed
 
 lemma plusinf_uset:
   assumes lp: "islin p"
-  and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
-  and ex: "Ifm vs (x#bs) p" (is "?I x p")
+    and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
+    and ex: "Ifm vs (x#bs) p" (is "?I x p")
   shows "\<exists>(c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c"
-proof-
+proof -
   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))"
     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
   from plusinf_uset0[OF lp nmi' ex]
-  obtain c s where csU: "(c,s) \<in> set (uset p)" and th: "x \<le> - Itm vs (x#bs) s / Ipoly vs c" by blast
-  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
-  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
+  obtain c s where csU: "(c,s) \<in> set (uset p)"
+    and th: "x \<le> - Itm vs (x#bs) s / Ipoly vs c"
+    by blast
+  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s"
+    by simp
+  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis
+    by auto
 qed
 
 lemma lin_dense:
   assumes lp: "islin p"
-  and noS: "\<forall>t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (c,t). - Itm vs (x#bs) t / Ipoly vs c) ` set (uset p)"
-  (is "\<forall>t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
-  and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
-  and ly: "l < y" and yu: "y < u"
+    and noS: "\<forall>t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda>(c,t). - Itm vs (x#bs) t / Ipoly vs c) ` set (uset p)"
+      (is "\<forall>t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda>(c,t). - ?Nt x t / ?N c) ` ?U p")
+    and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
+    and ly: "l < y" and yu: "y < u"
   shows "Ifm vs (y#bs) p"
-using lp px noS
+  using lp px noS
 proof (induct p rule: islin.induct)
   case (5 c s)
   from "5.prems"
   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
-    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
-  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
-  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
-  have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
-  moreover
-  {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
+    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp_all
+  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp
+  then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c"
+    by auto
+  have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0"
+    by dlo
   moreover
-  {assume c: "?N c > 0"
-      from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]
-      have px': "x < - ?Nt x s / ?N c"
-        by (auto simp add: not_less field_simps)
-    {assume y: "y < - ?Nt x s / ?N c"
-      hence "y * ?N c < - ?Nt x s"
-        by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
-      hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
-      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
-    moreover
-    {assume y: "y > -?Nt x s / ?N c"
-      with yu have eu: "u > - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
-      with lx px' have "False" by simp  hence ?case by simp }
-    ultimately have ?case using ycs by blast
+  {
+    assume "?N c = 0"
+    then have ?case
+      using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])
   }
   moreover
-  {assume c: "?N c < 0"
-      from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]
-      have px': "x > - ?Nt x s / ?N c"
-        by (auto simp add: not_less field_simps)
-    {assume y: "y > - ?Nt x s / ?N c"
-      hence "y * ?N c < - ?Nt x s"
+  {
+    assume c: "?N c > 0"
+    from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]
+    have px': "x < - ?Nt x s / ?N c"
+      by (auto simp add: not_less field_simps)
+    {
+      assume y: "y < - ?Nt x s / ?N c"
+      then have "y * ?N c < - ?Nt x s"
+        by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
+      then have "?N c * y + ?Nt x s < 0"
+        by (simp add: field_simps)
+      then have ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]
+        by simp
+    }
+    moreover
+    {
+      assume y: "y > -?Nt x s / ?N c"
+      with yu have eu: "u > - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l"
+        by (cases "- ?Nt x s / ?N c > l") auto
+      with lx px' have False
+        by simp
+      then have ?case ..
+    }
+    ultimately have ?case
+      using ycs by blast
+  }
+  moreover
+  {
+    assume c: "?N c < 0"
+    from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]
+    have px': "x > - ?Nt x s / ?N c"
+      by (auto simp add: not_less field_simps)
+    {
+      assume y: "y > - ?Nt x s / ?N c"
+      then have "y * ?N c < - ?Nt x s"
         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
-      hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
-      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
+      then have "?N c * y + ?Nt x s < 0"
+        by (simp add: field_simps)
+      then have ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"]
+        by simp
+    }
     moreover
-    {assume y: "y < -?Nt x s / ?N c"
-      with ly have eu: "l < - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
-      with xu px' have "False" by simp  hence ?case by simp }
-    ultimately have ?case using ycs by blast
+    {
+      assume y: "y < -?Nt x s / ?N c"
+      with ly have eu: "l < - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u"
+        by (cases "- ?Nt x s / ?N c < u") auto
+      with xu px' have False
+        by simp
+      then have ?case ..
+    }
+    ultimately have ?case
+      using ycs by blast
   }
-  ultimately show ?case by blast
+  ultimately show ?case
+    by blast
 next
   case (6 c s)
   from "6.prems"
   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
-    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
-  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
-  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
+    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp_all
+  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp
+  then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c"
+    by auto
   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
   moreover
-  {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
-  moreover
-  {assume c: "?N c > 0"
-      from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]
-      have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps)
-    {assume y: "y < - ?Nt x s / ?N c"
-      hence "y * ?N c < - ?Nt x s"
-        by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
-      hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
-      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
-    moreover
-    {assume y: "y > -?Nt x s / ?N c"
-      with yu have eu: "u > - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
-      with lx px' have "False" by simp  hence ?case by simp }
-    ultimately have ?case using ycs by blast
+  {
+    assume "?N c = 0"
+    then have ?case
+      using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])
   }
   moreover
-  {assume c: "?N c < 0"
-      from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]
-      have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps)
-    {assume y: "y > - ?Nt x s / ?N c"
-      hence "y * ?N c < - ?Nt x s"
-        by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
-      hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
-      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
+  {
+    assume c: "?N c > 0"
+    from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]
+    have px': "x \<le> - ?Nt x s / ?N c"
+      by (simp add: not_less field_simps)
+    {
+      assume y: "y < - ?Nt x s / ?N c"
+      then have "y * ?N c < - ?Nt x s"
+        by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
+      then have "?N c * y + ?Nt x s < 0"
+        by (simp add: field_simps)
+      then have ?case
+        using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp
+    }
     moreover
-    {assume y: "y < -?Nt x s / ?N c"
-      with ly have eu: "l < - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
-      with xu px' have "False" by simp  hence ?case by simp }
-    ultimately have ?case using ycs by blast
+    {
+      assume y: "y > -?Nt x s / ?N c"
+      with yu have eu: "u > - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l"
+        by (cases "- ?Nt x s / ?N c > l") auto
+      with lx px' have False
+        by simp
+      then have ?case ..
+    }
+    ultimately have ?case
+      using ycs by blast
   }
-  ultimately show ?case by blast
+  moreover
+  {
+    assume c: "?N c < 0"
+    from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]
+    have px': "x >= - ?Nt x s / ?N c"
+      by (simp add: field_simps)
+    {
+      assume y: "y > - ?Nt x s / ?N c"
+      then have "y * ?N c < - ?Nt x s"
+        by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
+      then have "?N c * y + ?Nt x s < 0"
+        by (simp add: field_simps)
+      then have ?case
+        using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp
+    }
+    moreover
+    {
+      assume y: "y < -?Nt x s / ?N c"
+      with ly have eu: "l < - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u"
+        by (cases "- ?Nt x s / ?N c < u") auto
+      with xu px' have False by simp
+      then have ?case ..
+    }
+    ultimately have ?case
+      using ycs by blast
+  }
+  ultimately show ?case
+    by blast
 next
-    case (3 c s)
+  case (3 c s)
   from "3.prems"
   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
-    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
-  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
-  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
+    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp_all
+  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp
+  then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c"
+    by auto
   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
   moreover
-  {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
-  moreover
-  {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
-    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
-    have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
-    {assume y: "y < -?Nt x s / ?N c"
-      with ly have eu: "l < - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
-      with xu px' have "False" by simp  hence ?case by simp }
-    moreover
-    {assume y: "y > -?Nt x s / ?N c"
-      with yu have eu: "u > - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
-      with lx px' have "False" by simp  hence ?case by simp }
-    ultimately have ?case using ycs by blast
+  {
+    assume "?N c = 0"
+    then have ?case
+      using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])
   }
   moreover
-  {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
-    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
-    have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
-    {assume y: "y < -?Nt x s / ?N c"
-      with ly have eu: "l < - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
-      with xu px' have "False" by simp  hence ?case by simp }
+  {
+    assume c: "?N c > 0"
+    then have cnz: "?N c \<noteq> 0"
+      by simp
+    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz
+    have px': "x = - ?Nt x s / ?N c"
+      by (simp add: field_simps)
+    {
+      assume y: "y < -?Nt x s / ?N c"
+      with ly have eu: "l < - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u"
+        by (cases "- ?Nt x s / ?N c < u") auto
+      with xu px' have False by simp
+      then have ?case ..
+    }
     moreover
-    {assume y: "y > -?Nt x s / ?N c"
-      with yu have eu: "u > - ?Nt x s / ?N c" by auto
-      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
-      with lx px' have "False" by simp  hence ?case by simp }
+    {
+      assume y: "y > -?Nt x s / ?N c"
+      with yu have eu: "u > - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l"
+        by (cases "- ?Nt x s / ?N c > l") auto
+      with lx px' have False by simp
+      then have ?case ..
+    }
+    ultimately have ?case
+      using ycs by blast
+  }
+  moreover
+  {
+    assume c: "?N c < 0"
+    then have cnz: "?N c \<noteq> 0"
+      by simp
+    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"] cnz
+    have px': "x = - ?Nt x s / ?N c"
+      by (simp add: field_simps)
+    {
+      assume y: "y < -?Nt x s / ?N c"
+      with ly have eu: "l < - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u"
+        by (cases "- ?Nt x s / ?N c < u") auto
+      with xu px' have False by simp
+      then have ?case ..
+    }
+    moreover
+    {
+      assume y: "y > -?Nt x s / ?N c"
+      with yu have eu: "u > - ?Nt x s / ?N c"
+        by auto
+      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l"
+        by (cases "- ?Nt x s / ?N c > l") auto
+      with lx px' have False by simp
+      then have ?case ..
+    }
     ultimately have ?case using ycs by blast
   }
   ultimately show ?case by blast
@@ -1927,136 +2458,228 @@
   from "4.prems"
   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
-    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
-  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
-  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
+    and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp_all
+  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
+    by simp
+  then have ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c"
+    by auto
   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
   moreover
-  {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
+  {
+    assume "?N c = 0"
+    then have ?case
+      using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])
+  }
   moreover
-  {assume c: "?N c \<noteq> 0"
-    from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
-      by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
-  ultimately show ?case by blast
+  {
+    assume c: "?N c \<noteq> 0"
+    from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"]
+    have ?case
+      by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric])
+  }
+  ultimately show ?case
+    by blast
 qed (auto simp add: tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])
 
 lemma inf_uset:
   assumes lp: "islin p"
-  and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
-  and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
-  and ex: "\<exists>x.  Ifm vs (x#bs) p" (is "\<exists>x. ?I x p")
-  shows "\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). ?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2) p"
-proof-
-  let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
+    and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
+    and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
+    and ex: "\<exists>x.  Ifm vs (x#bs) p" (is "\<exists>x. ?I x p")
+  shows "\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+    ?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2) p"
+proof -
+  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?N = "Ipoly vs"
   let ?U = "set (uset p)"
-  from ex obtain a where pa: "?I a p" by blast
+  from ex obtain a where pa: "?I a p"
+    by blast
   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
-  have nmi': "\<not> (?I a (?M p))" by simp
+  have nmi': "\<not> (?I a (?M p))"
+    by simp
   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
-  have npi': "\<not> (?I a (?P p))" by simp
+  have npi': "\<not> (?I a (?P p))"
+    by simp
   have "\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). ?I ((- ?Nt a t/?N c + - ?Nt a s /?N d) / 2) p"
-  proof-
-    let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
-    have fM: "finite ?M" by auto
+  proof -
+    let ?M = "(\<lambda>(c,t). - ?Nt a t / ?N c) ` ?U"
+    have fM: "finite ?M"
+      by auto
     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa]
-    have "\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). a \<le> - ?Nt x t / ?N c \<and> a \<ge> - ?Nt x s / ?N d" by blast
-    then obtain "c" "t" "d" "s" where
-      ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U"
-      and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
+    have "\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+        a \<le> - ?Nt x t / ?N c \<and> a \<ge> - ?Nt x s / ?N d"
+      by blast
+    then obtain "c" "t" "d" "s"
+      where ctU: "(c,t) \<in> ?U"
+        and dsU: "(d,s) \<in> ?U"
+        and xs1: "a \<le> - ?Nt x s / ?N d"
+        and tx1: "a \<ge> - ?Nt x t / ?N c"
+      by blast
     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1
-    have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
+    have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c"
+      by auto
     from ctU 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 ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
-    have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU 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> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
-    have "- ?Nt a s / ?N d \<le> ?u" using dsM 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 linM: "?l \<in> ?M"
+      using fM Mne by simp
+    have uinM: "?u \<in> ?M"
+      using fM Mne by simp
+    have ctM: "- ?Nt a t / ?N c \<in> ?M"
+      using ctU by auto
+    have dsM: "- ?Nt a s / ?N d \<in> ?M"
+      using dsU 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> - ?Nt a t / ?N c"
+      using ctM Mne by simp
+    then have lx: "?l \<le> a"
+      using tx by simp
+    have "- ?Nt a s / ?N d \<le> ?u"
+      using dsM 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>(nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
-      then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
-      from pu tuu
-      have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / 2) p" by simp
-      with tuU have ?thesis by blast}
-    moreover{
+    moreover {
+      fix u
+      assume um: "u\<in> ?M"
+        and pu: "?I u p"
+      then have "\<exists>(nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu"
+        by auto
+      then obtain tu nu where tuU: "(nu, tu) \<in> ?U"
+        and tuu:"u= - ?Nt a tu / ?N nu"
+        by blast
+      from pu tuu have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N 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"
+      then obtain t1 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>(t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
-      then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
-      from t2M have "\<exists>(t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
-      then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
+      from t1M have "\<exists>(t1n, t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n"
+        by auto
+      then obtain t1u t1n where t1uU: "(t1n, t1u) \<in> ?U"
+        and t1u: "t1 = - ?Nt a t1u / ?N t1n"
+        by blast
+      from t2M have "\<exists>(t2n, t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n"
+        by auto
+      then obtain t2u t2n where t2uU: "(t2n, t2u) \<in> ?U"
+        and t2u: "t2 = - ?Nt a t2u / ?N t2n"
+        by blast
       from t1x xt2 have t1t2: "t1 < t2" by simp
       let ?u = "(t1 + t2) / 2"
-      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
+      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2"
+        by auto
       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
-      with t1uU t2uU t1u t2u have ?thesis by blast}
+      with t1uU t2uU t1u t2u have ?thesis by blast
+    }
     ultimately show ?thesis by blast
   qed
-  then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U"
-    and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / 2) p" by blast
-  from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
+  then obtain l n s  m
+    where lnU: "(n, l) \<in> ?U"
+      and smU:"(m,s) \<in> ?U"
+      and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / 2) p"
+    by blast
+  from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s"
+    by auto
   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
-  have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / 2) p" by simp
-  with lnU smU
-  show ?thesis by auto
+  have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / 2) p"
+    by simp
+  with lnU smU show ?thesis by auto
 qed
 
-    (* The Ferrante - Rackoff Theorem *)
+(* The Ferrante - Rackoff Theorem *)
 
 theorem fr_eq:
   assumes lp: "islin p"
-  shows "(\<exists>x. Ifm vs (x#bs) p) = ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> (\<exists>(n,t) \<in> set (uset p). \<exists>(m,s) \<in> set (uset p). Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) / 2)#bs) p))"
+  shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow>
+    (Ifm vs (x#bs) (minusinf p) \<or>
+     Ifm vs (x#bs) (plusinf p) \<or>
+     (\<exists>(n, t) \<in> set (uset p). \<exists>(m, s) \<in> set (uset p).
+       Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) / 2)#bs) p))"
   (is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
 proof
   assume px: "\<exists>x. ?I x p"
-  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
-    from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
-  ultimately show "?D" by blast
+  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)"
+    by blast
+  moreover {
+    assume "?M \<or> ?P"
+    then have "?D" by blast
+  }
+  moreover {
+    assume nmi: "\<not> ?M"
+      and npi: "\<not> ?P"
+    from inf_uset[OF lp nmi npi] have ?F
+      using px by blast
+    then have ?D by blast
+  }
+  ultimately show ?D by blast
 next
-  assume "?D"
-  moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
-  moreover {assume f:"?F" hence "?E" by blast}
-  ultimately show "?E" by blast
+  assume ?D
+  moreover {
+    assume m: ?M
+    from minusinf_ex[OF lp m] have ?E .
+  }
+  moreover {
+    assume p: ?P
+    from plusinf_ex[OF lp p] have ?E .
+  }
+  moreover {
+    assume f: ?F
+    then have ?E by blast
+  }
+  ultimately show ?E by blast
 qed
 
-section{* First implementation : Naive by encoding all case splits locally *}
+
+section {* First implementation : Naive by encoding all case splits locally *}
+
 definition "msubsteq c t d s a r =
   evaldjf (split conj)
-  [(let cd = c *\<^sub>p d in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
-
-lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
+  [(let cd = c *\<^sub>p d
+    in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (Eq (CP c)) (Eq (CP d)), Eq r)]"
+
+lemma msubsteq_nb:
+  assumes lp: "islin (Eq (CNP 0 a r))"
+    and t: "tmbound0 t"
+    and s: "tmbound0 s"
   shows "bound0 (msubsteq c t d s a r)"
-proof-
-  have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
-    using lp by (simp add: Let_def t s )
-  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
+proof -
+  have th: "\<forall>x \<in> set
+    [(let cd = c *\<^sub>p d
+      in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+     (conj (Eq (CP c)) (NEq (CP d)), Eq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+     (conj (NEq (CP c)) (Eq (CP d)), Eq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+     (conj (Eq (CP c)) (Eq (CP d)), Eq r)]. bound0 (split conj x)"
+    using lp by (simp add: Let_def t s)
+  from evaldjf_bound0[OF th] show ?thesis
+    by (simp add: msubsteq_def)
 qed
 
-lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
-  shows "Ifm vs (x#bs) (msubsteq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2)#bs) (Eq (CNP 0 a r))" (is "?lhs = ?rhs")
-proof-
-  let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
+lemma msubsteq:
+  assumes lp: "islin (Eq (CNP 0 a r))"
+  shows "Ifm vs (x#bs) (msubsteq c t d s a r) =
+    Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / 2)#bs) (Eq (CNP 0 a r))"
+  (is "?lhs = ?rhs")
+proof -
+  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?N = "\<lambda>p. Ipoly vs p"
   let ?c = "?N c"
   let ?d = "?N d"
@@ -2064,81 +2687,116 @@
   let ?s = "?Nt x s"
   let ?a = "?N a"
   let ?r = "?Nt x r"
-  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
+  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r"
+    by simp_all
   note r= tmbound0_I[OF lin(3), of vs _ bs x]
-  have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" by auto
+  have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)"
+    by auto
   moreover
-  {assume c: "?c = 0" and d: "?d=0"
-    hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
+  {
+    assume c: "?c = 0"
+      and d: "?d=0"
+    then have ?thesis
+      by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)
+  }
   moreover
-  {assume c: "?c = 0" and d: "?d\<noteq>0"
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" by simp
-    have "?rhs = Ifm vs (-?s / (2*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r = 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
-    also have "\<dots> \<longleftrightarrow> 2*?d * (?a * (-?s / (2*?d)) + ?r) = 0"
+  {
+    assume c: "?c = 0"
+      and d: "?d\<noteq>0"
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)"
+      by simp
+    have "?rhs = Ifm vs (-?s / (2*?d) # bs) (Eq (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r = 0"
+      by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (-?s / (2*?d)) + ?r) = 0"
       using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp
-    also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r= 0"
+    also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2 * ?d * ?r= 0"
       by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left)
-
-    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r = 0" using d by simp
-    finally have ?thesis using c d
+    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r = 0"
+      using d by simp
+    finally have ?thesis
+      using c d
       by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubsteq_def Let_def evaldjf_ex)
   }
   moreover
-  {assume c: "?c \<noteq> 0" and d: "?d=0"
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" by simp
-    have "?rhs = Ifm vs (-?t / (2*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r = 0" by (simp add: r[of "- (?t/ (2 * ?c))"])
-    also have "\<dots> \<longleftrightarrow> 2*?c * (?a * (-?t / (2*?c)) + ?r) = 0"
-      using c mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp
-    also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r= 0"
-      by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left)
-    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r = 0" using c by simp
+  {
+    assume c: "?c \<noteq> 0"
+      and d: "?d = 0"
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2 * ?c)"
+      by simp
+    have "?rhs = Ifm vs (-?t / (2*?c) # bs) (Eq (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r = 0"
+      by (simp add: r[of "- (?t/ (2 * ?c))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (-?t / (2 * ?c)) + ?r) = 0"
+      using c mult_cancel_left[of "2 * ?c" "(?a * (-?t / (2 * ?c)) + ?r)" 0] by simp
+    also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2 * ?c) / (2 * ?c) + 2 * ?c * ?r= 0"
+      by (simp add: field_simps distrib_left[of "2 * ?c"] del: distrib_left)
+    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2 * ?c * ?r = 0" using c by simp
     finally have ?thesis using c d
-      by (simp add: r[of "- (?t/ (2*?c))"] msubsteq_def Let_def evaldjf_ex)
+      by (simp add: r[of "- (?t/ (2 * ?c))"] msubsteq_def Let_def evaldjf_ex)
   }
   moreover
-  {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *2 \<noteq> 0" by simp
+  {
+    assume c: "?c \<noteq> 0"
+      and d: "?d \<noteq> 0"
+    then have dc: "?c * ?d * 2 \<noteq> 0"
+      by simp
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Eq (CNP 0 a r))"
+      by (simp only: th)
     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r = 0"
       by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
-    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) =0 "
-      using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
-    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r =0"
+    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) = 0"
+      using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2 * ?c * ?d)) + ?r" 0]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2 * ?c * ?d * ?r = 0"
       using nonzero_mult_divide_cancel_left [OF dc] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally  have ?thesis using c d
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"]
+          msubsteq_def Let_def evaldjf_ex field_simps)
   }
-  ultimately show ?thesis by blast
+  ultimately show ?thesis
+    by blast
 qed
 
 
 definition "msubstneq c t d s a r =
   evaldjf (split conj)
-  [(let cd = c *\<^sub>p d in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
-
-lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
+  [(let cd = c *\<^sub>p d
+    in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (Eq (CP c)) (Eq (CP d)), NEq r)]"
+
+lemma msubstneq_nb:
+  assumes lp: "islin (NEq (CNP 0 a r))"
+    and t: "tmbound0 t"
+    and s: "tmbound0 s"
   shows "bound0 (msubstneq c t d s a r)"
-proof-
-  have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
-    using lp by (simp add: Let_def t s )
-  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
+proof -
+  have th: "\<forall>x\<in> set
+   [(let cd = c *\<^sub>p d
+     in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+    (conj (Eq (CP c)) (NEq (CP d)), NEq (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+    (conj (NEq (CP c)) (Eq (CP d)), NEq (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+    (conj (Eq (CP c)) (Eq (CP d)), NEq r)]. bound0 (split conj x)"
+    using lp by (simp add: Let_def t s)
+  from evaldjf_bound0[OF th] show ?thesis
+    by (simp add: msubstneq_def)
 qed
 
-lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
-  shows "Ifm vs (x#bs) (msubstneq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (NEq (CNP 0 a r))" (is "?lhs = ?rhs")
-proof-
-  let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
+lemma msubstneq:
+  assumes lp: "islin (Eq (CNP 0 a r))"
+  shows "Ifm vs (x#bs) (msubstneq c t d s a r) =
+    Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (NEq (CNP 0 a r))"
+  (is "?lhs = ?rhs")
+proof -
+  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?N = "\<lambda>p. Ipoly vs p"
   let ?c = "?N c"
   let ?d = "?N d"
@@ -2146,87 +2804,125 @@
   let ?s = "?Nt x s"
   let ?a = "?N a"
   let ?r = "?Nt x r"
-  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
-  note r= tmbound0_I[OF lin(3), of vs _ bs x]
-  have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" by auto
+  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r"
+    by simp_all
+  note r = tmbound0_I[OF lin(3), of vs _ bs x]
+  have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)"
+    by auto
   moreover
-  {assume c: "?c = 0" and d: "?d=0"
-    hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
+  {
+    assume c: "?c = 0"
+      and d: "?d=0"
+    then have ?thesis
+      by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)
+  }
   moreover
-  {assume c: "?c = 0" and d: "?d\<noteq>0"
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" by simp
-    have "?rhs = Ifm vs (-?s / (2*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r \<noteq> 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
+  {
+    assume c: "?c = 0"
+      and d: "?d\<noteq>0"
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2 * ?d)"
+      by simp
+    have "?rhs = Ifm vs (-?s / (2*?d) # bs) (NEq (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r \<noteq> 0"
+      by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
     also have "\<dots> \<longleftrightarrow> 2*?d * (?a * (-?s / (2*?d)) + ?r) \<noteq> 0"
       using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp
     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r\<noteq> 0"
       by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left)
-
-    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r \<noteq> 0" using d by simp
-    finally have ?thesis using c d
+    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r \<noteq> 0"
+      using d by simp
+    finally have ?thesis
+      using c d
       by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubstneq_def Let_def evaldjf_ex)
   }
   moreover
-  {assume c: "?c \<noteq> 0" and d: "?d=0"
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" by simp
-    have "?rhs = Ifm vs (-?t / (2*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ (2 * ?c))"])
+  {
+    assume c: "?c \<noteq> 0"
+      and d: "?d=0"
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)"
+      by simp
+    have "?rhs = Ifm vs (-?t / (2*?c) # bs) (NEq (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r \<noteq> 0"
+      by (simp add: r[of "- (?t/ (2 * ?c))"])
     also have "\<dots> \<longleftrightarrow> 2*?c * (?a * (-?t / (2*?c)) + ?r) \<noteq> 0"
       using c mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp
     also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r \<noteq> 0"
       by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left)
-    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r \<noteq> 0" using c by simp
-    finally have ?thesis using c d
-      by (simp add: r[of "- (?t/ (2*?c))"] msubstneq_def Let_def evaldjf_ex)
+    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r \<noteq> 0"
+      using c by simp
+    finally have ?thesis
+      using c d by (simp add: r[of "- (?t/ (2*?c))"] msubstneq_def Let_def evaldjf_ex)
   }
   moreover
-  {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *2 \<noteq> 0" by simp
+  {
+    assume c: "?c \<noteq> 0"
+      and d: "?d \<noteq> 0"
+    then have dc: "?c * ?d *2 \<noteq> 0"
+      by simp
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (NEq (CNP 0 a r))"
+      by (simp only: th)
     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<noteq> 0"
       by (simp add: r [of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
-    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<noteq> 0 "
-      using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
+    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<noteq> 0"
+      using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0]
+      by simp
     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r \<noteq> 0"
       using nonzero_mult_divide_cancel_left[OF dc] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
-    finally  have ?thesis using c d
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
+    finally have ?thesis
+      using c d
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"]
+          msubstneq_def Let_def evaldjf_ex field_simps)
   }
   ultimately show ?thesis by blast
 qed
 
 definition "msubstlt c t d s a r =
   evaldjf (split conj)
-  [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-  (let cd = c *\<^sub>p d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"
-
-lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
+  [(let cd = c *\<^sub>p d
+    in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (let cd = c *\<^sub>p d
+    in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (Eq (CP c)) (Eq (CP d)), Lt r)]"
+
+lemma msubstlt_nb:
+  assumes lp: "islin (Lt (CNP 0 a r))"
+    and t: "tmbound0 t"
+    and s: "tmbound0 s"
   shows "bound0 (msubstlt c t d s a r)"
-proof-
-  have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-  (let cd = c *\<^sub>p d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
-    using lp by (simp add: Let_def t s lt_nb )
-  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
+proof -
+  have th: "\<forall>x\<in> set
+  [(let cd = c *\<^sub>p d
+    in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (let cd = c *\<^sub>p d
+    in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Lt (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (lt (CP c)) (Eq (CP d)), Lt (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Lt (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (lt (CP d)) (Eq (CP c)), Lt (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+   (conj (Eq (CP c)) (Eq (CP d)), Lt r)]. bound0 (split conj x)"
+    using lp by (simp add: Let_def t s lt_nb)
+  from evaldjf_bound0[OF th] show ?thesis
+    by (simp add: msubstlt_def)
 qed
 
-
-lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))"
+lemma msubstlt:
+  assumes nc: "isnpoly c"
+    and nd: "isnpoly d"
+    and lp: "islin (Lt (CNP 0 a r))"
   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow>
-  Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Lt (CNP 0 a r))" (is "?lhs = ?rhs")
-proof-
+    Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Lt (CNP 0 a r))"
+  (is "?lhs = ?rhs")
+proof -
   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?N = "\<lambda>p. Ipoly vs p"
   let ?c = "?N c"
@@ -2235,143 +2931,210 @@
   let ?s = "?Nt x s"
   let ?a = "?N a"
   let ?r = "?Nt x r"
-  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
-  note r= tmbound0_I[OF lin(3), of vs _ bs x]
-  have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" by auto
+  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r"
+    by simp_all
+  note r = tmbound0_I[OF lin(3), of vs _ bs x]
+  have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)"
+    by auto
   moreover
-  {assume c: "?c=0" and d: "?d=0"
-    hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
+  {
+    assume c: "?c=0" and d: "?d=0"
+    then have ?thesis
+      using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)
+  }
   moreover
-  {assume dc: "?c*?d > 0"
-    from dc have dc': "2*?c *?d > 0" by simp
-    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
+  {
+    assume dc: "?c*?d > 0"
+    from dc have dc': "2*?c *?d > 0"
+      by simp
+    then have c:"?c \<noteq> 0" and d: "?d \<noteq> 0"
+      by auto
     from dc' have dc'': "\<not> 2*?c *?d < 0" by simp
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0"
       by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) < 0"
-
-      using dc' dc'' mult_less_cancel_left_disj[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
+      using dc' dc''
+        mult_less_cancel_left_disj[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0]
+      by simp
     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r < 0"
       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally  have ?thesis using dc c d  nc nd dc'
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"]
+          msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume dc: "?c*?d < 0"
-
+  {
+    assume dc: "?c * ?d < 0"
     from dc have dc': "2*?c *?d < 0"
       by (simp add: mult_less_0_iff field_simps)
-    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
+    then have c:"?c \<noteq> 0" and d: "?d \<noteq> 0"
+      by auto
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0"
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)) + ?r < 0"
       by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
-
-    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) > 0"
-
-      using dc' order_less_not_sym[OF dc'] mult_less_cancel_left_disj[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp
-    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2*?c*?d*?r < 0"
-      using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
+    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)) + ?r) > 0"
+      using dc' order_less_not_sym[OF dc']
+        mult_less_cancel_left_disj[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r < 0"
+      using nonzero_mult_divide_cancel_left[of "2 * ?c * ?d"] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
-    finally  have ?thesis using dc c d  nc nd
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+    finally have ?thesis using dc c d nc nd
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"]
+          msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c > 0" and d: "?d=0"
-    from c have c'': "2*?c > 0" by (simp add: zero_less_mult_iff)
-    from c have c': "2*?c \<noteq> 0" by simp
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r < 0" by (simp add: r[of "- (?t / (2*?c))"])
-    also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) < 0"
-      using c mult_less_cancel_left_disj[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
-    also have "\<dots> \<longleftrightarrow> - ?a*?t+  2*?c *?r < 0"
+  {
+    assume c: "?c > 0" and d: "?d = 0"
+    from c have c'': "2*?c > 0"
+      by (simp add: zero_less_mult_iff)
+    from c have c': "2 * ?c \<noteq> 0"
+      by simp
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2 * ?c))+ ?r < 0"
+      by (simp add: r[of "- (?t / (2 * ?c))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) < 0"
+      using c mult_less_cancel_left_disj[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c''
+        order_less_not_sym[OF c'']
+      by simp
+    also have "\<dots> \<longleftrightarrow> - ?a * ?t + 2 * ?c * ?r < 0"
       using nonzero_mult_divide_cancel_left[OF c'] c
       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps
+          lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c < 0" and d: "?d=0"  hence c': "2*?c \<noteq> 0" by simp
-    from c have c'': "2*?c < 0" by (simp add: mult_less_0_iff)
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r < 0" by (simp add: r[of "- (?t / (2*?c))"])
-    also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) > 0"
-      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_less_cancel_left_disj[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp
+  {
+    assume c: "?c < 0" and d: "?d = 0"
+    then have c': "2 * ?c \<noteq> 0"
+      by simp
+    from c have c'': "2 * ?c < 0"
+      by (simp add: mult_less_0_iff)
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2 * ?c)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2*?c))+ ?r < 0"
+      by (simp add: r[of "- (?t / (2*?c))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) > 0"
+      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
+        mult_less_cancel_left_disj[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"]
+      by simp
     also have "\<dots> \<longleftrightarrow> ?a*?t -  2*?c *?r < 0"
-      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
-        by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
+      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c'']
+          less_imp_neq[OF c''] c''
+        by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps
+          lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c = 0" and d: "?d>0"
-    from d have d'': "2*?d > 0" by (simp add: zero_less_mult_iff)
-    from d have d': "2*?d \<noteq> 0" by simp
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r < 0" by (simp add: r[of "- (?s / (2*?d))"])
-    also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) < 0"
-      using d mult_less_cancel_left_disj[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
-    also have "\<dots> \<longleftrightarrow> - ?a*?s+  2*?d *?r < 0"
+  {
+    assume c: "?c = 0" and d: "?d > 0"
+    from d have d'': "2 * ?d > 0"
+      by (simp add: zero_less_mult_iff)
+    from d have d': "2 * ?d \<noteq> 0"
+      by simp
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d))+ ?r < 0"
+      by (simp add: r[of "- (?s / (2 * ?d))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d))+ ?r) < 0"
+      using d mult_less_cancel_left_disj[of "2 * ?d" "?a * (- ?s / (2 * ?d))+ ?r" 0] d' d''
+        order_less_not_sym[OF d'']
+      by simp
+    also have "\<dots> \<longleftrightarrow> - ?a * ?s+  2 * ?d * ?r < 0"
       using nonzero_mult_divide_cancel_left[OF d'] d
       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps
+          lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c = 0" and d: "?d<0"  hence d': "2*?d \<noteq> 0" by simp
-    from d have d'': "2*?d < 0" by (simp add: mult_less_0_iff)
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r < 0" by (simp add: r[of "- (?s / (2*?d))"])
-    also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) > 0"
-      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_less_cancel_left_disj[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp
-    also have "\<dots> \<longleftrightarrow> ?a*?s -  2*?d *?r < 0"
-      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
-        by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
+  {
+    assume c: "?c = 0" and d: "?d < 0"
+    then have d': "2 * ?d \<noteq> 0"
+      by simp
+    from d have d'': "2 * ?d < 0"
+      by (simp add: mult_less_0_iff)
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Lt (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d)) + ?r < 0"
+      by (simp add: r[of "- (?s / (2 * ?d))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) > 0"
+      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
+        mult_less_cancel_left_disj[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * ?s -  2 * ?d * ?r < 0"
+      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d'']
+          less_imp_neq[OF d''] d''
+        by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps
+          lt polyneg_norm polymul_norm)
   }
-ultimately show ?thesis by blast
+  ultimately show ?thesis by blast
 qed
 
 definition "msubstle c t d s a r =
   evaldjf (split conj)
-  [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-  (let cd = c *\<^sub>p d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"
-
-lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
+   [(let cd = c *\<^sub>p d
+     in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+    (let cd = c *\<^sub>p d
+     in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)), Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+    (conj (lt (CP c)) (Eq (CP d)), Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)), Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+    (conj (lt (CP d)) (Eq (CP c)), Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+    (conj (Eq (CP c)) (Eq (CP d)), Le r)]"
+
+lemma msubstle_nb:
+  assumes lp: "islin (Le (CNP 0 a r))"
+    and t: "tmbound0 t"
+    and s: "tmbound0 s"
   shows "bound0 (msubstle c t d s a r)"
-proof-
-  have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-  (let cd = c *\<^sub>p d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
-   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
-   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
-   (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
-    using lp by (simp add: Let_def t s lt_nb )
-  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
+proof -
+  have th: "\<forall>x\<in> set
+   [(let cd = c *\<^sub>p d
+     in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+    (let cd = c *\<^sub>p d
+     in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul ((2)\<^sub>p *\<^sub>p cd) r)))),
+    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul ((2)\<^sub>p *\<^sub>p c) r))),
+    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul ((2)\<^sub>p *\<^sub>p d) r))),
+    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
+    using lp by (simp add: Let_def t s lt_nb)
+  from evaldjf_bound0[OF th] show ?thesis
+    by (simp add: msubstle_def)
 qed
 
-lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))"
+lemma msubstle:
+  assumes nc: "isnpoly c"
+    and nd: "isnpoly d"
+    and lp: "islin (Le (CNP 0 a r))"
   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow>
-  Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Le (CNP 0 a r))" (is "?lhs = ?rhs")
-proof-
+    Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Le (CNP 0 a r))"
+  (is "?lhs = ?rhs")
+proof -
   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?N = "\<lambda>p. Ipoly vs p"
   let ?c = "?N c"
@@ -2380,269 +3143,424 @@
   let ?s = "?Nt x s"
   let ?a = "?N a"
   let ?r = "?Nt x r"
-  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
-  note r= tmbound0_I[OF lin(3), of vs _ bs x]
-  have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" by auto
+  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r"
+    by simp_all
+  note r = tmbound0_I[OF lin(3), of vs _ bs x]
+  have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)"
+    by auto
   moreover
-  {assume c: "?c=0" and d: "?d=0"
-    hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
+  {
+    assume c: "?c = 0" and d: "?d = 0"
+    then have ?thesis
+      using nc nd
+      by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)
+  }
   moreover
-  {assume dc: "?c*?d > 0"
-    from dc have dc': "2*?c *?d > 0" by simp
-    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
-    from dc' have dc'': "\<not> 2*?c *?d < 0" by simp
+  {
+    assume dc: "?c * ?d > 0"
+    from dc have dc': "2 * ?c * ?d > 0"
+      by simp
+    then have c: "?c \<noteq> 0" and d: "?d \<noteq> 0"
+      by auto
+    from dc' have dc'': "\<not> 2 * ?c * ?d < 0"
+      by simp
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c * ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r <= 0"
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<le> 0"
       by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
-    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) <= 0"
-
-      using dc' dc'' mult_le_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
-    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r <= 0"
+    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<le> 0"
+      using dc' dc''
+        mult_le_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r \<le> 0"
       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally  have ?thesis using dc c d  nc nd dc'
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def
+          Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume dc: "?c*?d < 0"
-
-    from dc have dc': "2*?c *?d < 0"
+  {
+    assume dc: "?c * ?d < 0"
+    from dc have dc': "2 * ?c * ?d < 0"
       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
-    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
+    then have c: "?c \<noteq> 0" and d: "?d \<noteq> 0"
+      by auto
     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
-    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
+    have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2 * ?c * ?d)"
       by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r <= 0"
+    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<le> 0"
       by (simp add: r[of "(- (?d * ?t) - (?c *?s)) / (2 * ?c * ?d)"])
-
-    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) >= 0"
-
-      using dc' order_less_not_sym[OF dc'] mult_le_cancel_left[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp
-    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2*?c*?d*?r <= 0"
-      using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
+    also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<ge> 0"
+      using dc' order_less_not_sym[OF dc']
+        mult_le_cancel_left[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2 * ?c * ?d * ?r \<le> 0"
+      using nonzero_mult_divide_cancel_left[of "2 * ?c * ?d"] c d
       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally  have ?thesis using dc c d  nc nd
-      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def
+          Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c > 0" and d: "?d=0"
-    from c have c'': "2*?c > 0" by (simp add: zero_less_mult_iff)
-    from c have c': "2*?c \<noteq> 0" by simp
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r <= 0" by (simp add: r[of "- (?t / (2*?c))"])
-    also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) <= 0"
-      using c mult_le_cancel_left[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
-    also have "\<dots> \<longleftrightarrow> - ?a*?t+  2*?c *?r <= 0"
+  {
+    assume c: "?c > 0" and d: "?d = 0"
+    from c have c'': "2 * ?c > 0"
+      by (simp add: zero_less_mult_iff)
+    from c have c': "2 * ?c \<noteq> 0"
+      by simp
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2 * ?c))+ ?r \<le> 0"
+      by (simp add: r[of "- (?t / (2 * ?c))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) \<le> 0"
+      using c mult_le_cancel_left[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c''
+        order_less_not_sym[OF c'']
+      by simp
+    also have "\<dots> \<longleftrightarrow> - ?a*?t+  2*?c *?r \<le> 0"
       using nonzero_mult_divide_cancel_left[OF c'] c
       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def
+          evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c < 0" and d: "?d=0"  hence c': "2*?c \<noteq> 0" by simp
-    from c have c'': "2*?c < 0" by (simp add: mult_less_0_iff)
-    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r <= 0" by (simp add: r[of "- (?t / (2*?c))"])
-    also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) >= 0"
-      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_le_cancel_left[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp
-    also have "\<dots> \<longleftrightarrow> ?a*?t -  2*?c *?r <= 0"
-      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
-        by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
+  {
+    assume c: "?c < 0" and d: "?d = 0"
+    then have c': "2 * ?c \<noteq> 0"
+      by simp
+    from c have c'': "2 * ?c < 0"
+      by (simp add: mult_less_0_iff)
+    from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2 * ?c) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?t / (2*?c))+ ?r \<le> 0"
+      by (simp add: r[of "- (?t / (2*?c))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?c * (?a * (- ?t / (2 * ?c))+ ?r) \<ge> 0"
+      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
+        mult_le_cancel_left[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * ?t - 2 * ?c * ?r \<le> 0"
+      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c'']
+          less_imp_neq[OF c''] c''
+        by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def
+          evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c = 0" and d: "?d>0"
-    from d have d'': "2*?d > 0" by (simp add: zero_less_mult_iff)
-    from d have d': "2*?d \<noteq> 0" by simp
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r <= 0" by (simp add: r[of "- (?s / (2*?d))"])
-    also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) <= 0"
-      using d mult_le_cancel_left[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
-    also have "\<dots> \<longleftrightarrow> - ?a*?s+  2*?d *?r <= 0"
+  {
+    assume c: "?c = 0" and d: "?d > 0"
+    from d have d'': "2 * ?d > 0"
+      by (simp add: zero_less_mult_iff)
+    from d have d': "2 * ?d \<noteq> 0"
+      by simp
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2 * ?d)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2 * ?d) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a * (- ?s / (2 * ?d))+ ?r \<le> 0"
+      by (simp add: r[of "- (?s / (2*?d))"])
+    also have "\<dots> \<longleftrightarrow> 2 * ?d * (?a * (- ?s / (2 * ?d)) + ?r) \<le> 0"
+      using d mult_le_cancel_left[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d''
+        order_less_not_sym[OF d'']
+      by simp
+    also have "\<dots> \<longleftrightarrow> - ?a * ?s + 2 * ?d * ?r \<le> 0"
       using nonzero_mult_divide_cancel_left[OF d'] d
       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def
+          evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
   moreover
-  {assume c: "?c = 0" and d: "?d<0"  hence d': "2*?d \<noteq> 0" by simp
-    from d have d'': "2*?d < 0" by (simp add: mult_less_0_iff)
-    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
-    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
-    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r <= 0" by (simp add: r[of "- (?s / (2*?d))"])
-    also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) >= 0"
-      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_le_cancel_left[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp
-    also have "\<dots> \<longleftrightarrow> ?a*?s -  2*?d *?r <= 0"
-      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
-        by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
+  {
+    assume c: "?c = 0" and d: "?d < 0"
+    then have d': "2 * ?d \<noteq> 0"
+      by simp
+    from d have d'': "2 * ?d < 0"
+      by (simp add: mult_less_0_iff)
+    from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"
+      by (simp add: field_simps)
+    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))"
+      by (simp only: th)
+    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r \<le> 0"
+      by (simp add: r[of "- (?s / (2*?d))"])
+    also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) \<ge> 0"
+      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
+        mult_le_cancel_left[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"]
+      by simp
+    also have "\<dots> \<longleftrightarrow> ?a * ?s -  2 * ?d * ?r \<le> 0"
+      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d'']
+          less_imp_neq[OF d''] d''
+        by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
     finally have ?thesis using c d nc nd
-      by(simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
+      by (simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def
+          evaldjf_ex field_simps lt polyneg_norm polymul_norm)
   }
-ultimately show ?thesis by blast
+  ultimately show ?thesis by blast
 qed
 
-
-fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
-  "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
-| "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
-| "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
-| "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
-| "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
-| "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
-| "msubst p ((c,t),(d,s)) = p"
-
-lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
-  shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) p"
-  using lp by (induct p rule: islin.induct)
+fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm"
+where
+  "msubst (And p q) ((c, t), (d, s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))"
+| "msubst (Or p q) ((c, t), (d, s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c, t), (d, s)))"
+| "msubst (Eq (CNP 0 a r)) ((c, t), (d, s)) = msubsteq c t d s a r"
+| "msubst (NEq (CNP 0 a r)) ((c, t), (d, s)) = msubstneq c t d s a r"
+| "msubst (Lt (CNP 0 a r)) ((c, t), (d, s)) = msubstlt c t d s a r"
+| "msubst (Le (CNP 0 a r)) ((c, t), (d, s)) = msubstle c t d s a r"
+| "msubst p ((c, t), (d, s)) = p"
+
+lemma msubst_I:
+  assumes lp: "islin p"
+    and nc: "isnpoly c"
+    and nd: "isnpoly d"
+  shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) =
+    Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) p"
+  using lp
+  by (induct p rule: islin.induct)
     (auto simp add: tmbound0_I
-    [where b = "(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) / 2"
-      and b' = x and bs = bs and vs = vs]
-    msubsteq msubstneq msubstlt [OF nc nd] msubstle [OF nc nd])
-
-lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
+      [where b = "(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) / 2"
+        and b' = x and bs = bs and vs = vs]
+      msubsteq msubstneq msubstlt [OF nc nd] msubstle [OF nc nd])
+
+lemma msubst_nb:
+  assumes lp: "islin p"
+    and t: "tmbound0 t"
+    and s: "tmbound0 s"
   shows "bound0 (msubst p ((c,t),(d,s)))"
   using lp t s
   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
 
 lemma fr_eq_msubst:
   assumes lp: "islin p"
-  shows "(\<exists>x. Ifm vs (x#bs) p) = ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> (\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst p ((c,t),(d,s)))))"
+  shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow>
+    (Ifm vs (x#bs) (minusinf p) \<or>
+     Ifm vs (x#bs) (plusinf p) \<or>
+     (\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+      Ifm vs (x#bs) (msubst p ((c, t), (d, s)))))"
   (is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
-proof-
-from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
-{fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
-  and pts: "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
-  from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
-  from msubst_I[OF lp norm, of vs x bs t s] pts
-  have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
-moreover
-{fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
-  and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
-  from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
-  from msubst_I[OF lp norm, of vs x bs t s] pts
-  have "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" ..}
-ultimately have th': "(\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p) \<longleftrightarrow> ?F" by blast
-from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
+proof -
+  from uset_l[OF lp]
+  have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s"
+    by blast
+  {
+    fix c t d s
+    assume ctU: "(c, t) \<in>set (uset p)"
+      and dsU: "(d,s) \<in>set (uset p)"
+      and pts: "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
+    from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d"
+      by simp_all
+    from msubst_I[OF lp norm, of vs x bs t s] pts
+    have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..
+  }
+  moreover
+  {
+    fix c t d s
+    assume ctU: "(c, t) \<in> set (uset p)"
+      and dsU: "(d,s) \<in>set (uset p)"
+      and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
+    from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d"
+      by simp_all
+    from msubst_I[OF lp norm, of vs x bs t s] pts
+    have "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p" ..
+  }
+  ultimately have th': "(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+      Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p) \<longleftrightarrow> ?F"
+    by blast
+  from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
 qed
 
-lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+lemma simpfm_lin:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   shows "qfree p \<Longrightarrow> islin (simpfm p)"
-  by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
-
-definition
-  "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
-  in if (mp = T \<or> pp = T) then T
-     else (let U = alluopairs (remdups (uset  q))
-           in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
+  by (induct p rule: simpfm.induct) (auto simp add: conj_lin disj_lin)
+
+definition "ferrack p \<equiv>
+  let
+    q = simpfm p;
+    mp = minusinf q;
+    pp = plusinf q
+  in
+    if (mp = T \<or> pp = T) then T
+    else
+     (let U = alluopairs (remdups (uset  q))
+      in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
 
 lemma ferrack:
   assumes qf: "qfree p"
-  shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
-  (is "_ \<and> (?rhs = ?lhs)")
-proof-
-  let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
-  let ?N = "\<lambda> t. Ipoly vs t"
+  shows "qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)"
+  (is "_ \<and> ?rhs = ?lhs")
+proof -
+  let ?I = "\<lambda>x p. Ifm vs (x#bs) p"
+  let ?N = "\<lambda>t. Ipoly vs t"
   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?q = "simpfm p"
   let ?U = "remdups(uset ?q)"
   let ?Up = "alluopairs ?U"
   let ?mp = "minusinf ?q"
   let ?pp = "plusinf ?q"
+  fix x
   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
   from uset_l[OF lq] have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
     by simp
-  {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
-    from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
+  {
+    fix c t d s
+    assume ctU: "(c, t) \<in> set ?U"
+      and dsU: "(d,s) \<in> set ?U"
+    from U_l ctU dsU have norm: "isnpoly c" "isnpoly d"
+      by auto
     from msubst_I[OF lq norm, of vs x bs t s] msubst_I[OF lq norm(2,1), of vs x bs s t]
-    have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
-  hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (msubst ?q (x, y)) \<longleftrightarrow> ?I (msubst ?q (y, x))" by clarsimp
-  {fix x assume xUp: "x \<in> set ?Up"
-    then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
-      and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto
+    have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))"
+      by (simp add: field_simps)
+  }
+  then have th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (msubst ?q (x, y)) \<longleftrightarrow> ?I (msubst ?q (y, x))"
+    by auto
+  {
+    fix x
+    assume xUp: "x \<in> set ?Up"
+    then obtain c t d s
+      where ctU: "(c, t) \<in> set ?U"
+        and dsU: "(d,s) \<in> set ?U"
+        and x: "x = ((c, t),(d, s))"
+      using alluopairs_set1[of ?U] by auto
     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU]
     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]]
-    have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
+    have "bound0 ((simpfm o (msubst (simpfm p))) x)"
+      using x by simp
+  }
   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
   with mp_nb pp_nb
-  have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by simp
-  from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
-  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists>(c, t)\<in>set ?U. \<exists>(d, s)\<in>set ?U. ?I (msubst (simpfm p) ((c, t), d, s)))" using fr_eq_msubst[OF lq, of vs bs x] by simp
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists>(x,y) \<in> set ?Up. ?I ((simpfm o (msubst ?q)) (x,y)))" using alluopairs_bex[OF th0] by simp
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)"
+  have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))"
+    by simp
+  from decr0_qf[OF th1] have thqf: "qfree (ferrack p)"
+    by (simp add: ferrack_def Let_def)
+  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)"
+    by simp
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or>
+      (\<exists>(c, t)\<in>set ?U. \<exists>(d, s)\<in>set ?U. ?I (msubst (simpfm p) ((c, t), d, s)))"
+    using fr_eq_msubst[OF lq, of vs bs x] by simp
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or>
+      (\<exists>(x, y) \<in> set ?Up. ?I ((simpfm \<circ> msubst ?q) (x, y)))"
+    using alluopairs_bex[OF th0] by simp
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm \<circ> msubst ?q) ?Up)"
     by (simp add: evaldjf_ex)
-  also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
-  also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
+  also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm \<circ> msubst ?q) ?Up)))"
+    by simp
+  also have "\<dots> \<longleftrightarrow> ?rhs"
+    using decr0[OF th1, of vs x bs]
     apply (simp add: ferrack_def Let_def)
-    by (cases "?mp = T \<or> ?pp = T", auto)
-  finally show ?thesis using thqf by blast
+    apply (cases "?mp = T \<or> ?pp = T")
+    apply auto
+    done
+  finally show ?thesis
+    using thqf by blast
 qed
 
 definition "frpar p = simpfm (qelim p ferrack)"
+
 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
-proof-
-  from ferrack have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" by blast
-  from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
+proof -
+  from ferrack
+  have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)"
+    by blast
+  from qelim[OF th, of p bs] show ?thesis
+    unfolding frpar_def by auto
 qed
 
 
-section{* Second implemenation: Case splits not local *}
-
-lemma fr_eq2:  assumes lp: "islin p"
+section {* Second implemenation: Case splits not local *}
+
+lemma fr_eq2:
+  assumes lp: "islin p"
   shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow>
-   ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or>
-    (Ifm vs (0#bs) p) \<or>
-    (\<exists>(n,t) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t /  (Ipoly vs n * 2))#bs) p) \<or>
-    (\<exists>(n,t) \<in> set (uset p). \<exists>(m,s) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ipoly vs m \<noteq> 0 \<and> Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /2)#bs) p))"
+   (Ifm vs (x#bs) (minusinf p) \<or>
+    Ifm vs (x#bs) (plusinf p) \<or>
+    Ifm vs (0#bs) p \<or>
+    (\<exists>(n, t) \<in> set (uset p).
+      Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t /  (Ipoly vs n * 2))#bs) p) \<or>
+      (\<exists>(n, t) \<in> set (uset p). \<exists>(m, s) \<in> set (uset p).
+        Ipoly vs n \<noteq> 0 \<and>
+        Ipoly vs m \<noteq> 0 \<and>
+        Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /2)#bs) p))"
   (is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
 proof
   assume px: "\<exists>x. ?I x p"
   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
-  moreover {assume "?M \<or> ?P" hence "?D" by blast}
-  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
+  moreover {
+    assume "?M \<or> ?P"
+    then have "?D" by blast
+  }
+  moreover {
+    assume nmi: "\<not> ?M"
+      and npi: "\<not> ?P"
     from inf_uset[OF lp nmi npi, OF px]
-    obtain c t d s where ct: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" "?I ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / ((1\<Colon>'a) + (1\<Colon>'a))) p"
+    obtain c t d s where ct:
+      "(c, t) \<in> set (uset p)"
+      "(d, s) \<in> set (uset p)"
+      "?I ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1)) p"
       by auto
     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
     let ?s = "Itm vs (x # bs) s"
     let ?t = "Itm vs (x # bs) t"
     have eq2: "\<And>(x::'a). x + x = 2 * x"
-      by  (simp add: field_simps)
-    {assume "?c = 0 \<and> ?d = 0"
-      with ct have ?D by simp}
+      by (simp add: field_simps)
+    {
+      assume "?c = 0 \<and> ?d = 0"
+      with ct have ?D by simp
+    }
     moreover
-    {assume z: "?c = 0" "?d \<noteq> 0"
-      from z have ?D using ct by auto}
+    {
+      assume z: "?c = 0" "?d \<noteq> 0"
+      from z have ?D using ct by auto
+    }
     moreover
-    {assume z: "?c \<noteq> 0" "?d = 0"
-      with ct have ?D by auto }
+    {
+      assume z: "?c \<noteq> 0" "?d = 0"
+      with ct have ?D by auto
+    }
     moreover
-    {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
+    {
+      assume z: "?c \<noteq> 0" "?d \<noteq> 0"
       from z have ?F using ct
-        apply - apply (rule bexI[where x = "(c,t)"], simp_all)
-        by (rule bexI[where x = "(d,s)"], simp_all)
-      hence ?D by blast}
-    ultimately have ?D by auto}
+        apply -
+        apply (rule bexI[where x = "(c,t)"])
+        apply simp_all
+        apply (rule bexI[where x = "(d,s)"])
+        apply simp_all
+        done
+      then have ?D by blast
+    }
+    ultimately have ?D by auto
+  }
   ultimately show "?D" by blast
 next
   assume "?D"
-  moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
-  moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
-  moreover {assume f:"?F" hence "?E" by blast}
+  moreover {
+    assume m: "?M"
+    from minusinf_ex[OF lp m] have "?E" .
+  }
+  moreover {
+    assume p: "?P"
+    from plusinf_ex[OF lp p] have "?E" .
+  }
+  moreover {
+    assume f:"?F"
+    then have "?E" by blast
+  }
   ultimately show "?E" by blast
 qed
 
@@ -2653,36 +3571,47 @@
 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
 
 lemma msubsteq2:
-  assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
-  shows "Ifm vs (x#bs) (msubsteq2 c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Eq (CNP 0 a b))" (is "?lhs = ?rhs")
-  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
+  assumes nz: "Ipoly vs c \<noteq> 0"
+    and l: "islin (Eq (CNP 0 a b))"
+  shows "Ifm vs (x#bs) (msubsteq2 c t a b) =
+    Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Eq (CNP 0 a b))"
+  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric]
   by (simp add: msubsteq2_def field_simps)
 
 lemma msubstltpos:
-  assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
-  shows "Ifm vs (x#bs) (msubstltpos c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Lt (CNP 0 a b))" (is "?lhs = ?rhs")
-  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
+  assumes nz: "Ipoly vs c > 0"
+    and l: "islin (Lt (CNP 0 a b))"
+  shows "Ifm vs (x#bs) (msubstltpos c t a b) =
+    Ifm vs (((Itm vs (x#bs) t / Ipoly vs c ))#bs) (Lt (CNP 0 a b))"
+  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric]
   by (simp add: msubstltpos_def field_simps)
 
 lemma msubstlepos:
-  assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
-  shows "Ifm vs (x#bs) (msubstlepos c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))" (is "?lhs = ?rhs")
-  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
+  assumes nz: "Ipoly vs c > 0"
+    and l: "islin (Le (CNP 0 a b))"
+  shows "Ifm vs (x#bs) (msubstlepos c t a b) =
+    Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))"
+  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric]
   by (simp add: msubstlepos_def field_simps)
 
 lemma msubstltneg:
-  assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
-  shows "Ifm vs (x#bs) (msubstltneg c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Lt (CNP 0 a b))" (is "?lhs = ?rhs")
-  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
+  assumes nz: "Ipoly vs c < 0"
+    and l: "islin (Lt (CNP 0 a b))"
+  shows "Ifm vs (x#bs) (msubstltneg c t a b) =
+    Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Lt (CNP 0 a b))"
+  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric]
   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
 
 lemma msubstleneg:
-  assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
-  shows "Ifm vs (x#bs) (msubstleneg c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))" (is "?lhs = ?rhs")
-  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
+  assumes nz: "Ipoly vs c < 0"
+    and l: "islin (Le (CNP 0 a b))"
+  shows "Ifm vs (x#bs) (msubstleneg c t a b) =
+    Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))"
+  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>", symmetric]
   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
 
-fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
+fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm"
+where
   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
@@ -2692,12 +3621,18 @@
 | "msubstpos p c t = p"
 
 lemma msubstpos_I:
-  assumes lp: "islin p" and pos: "Ipoly vs c > 0"
-  shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
+  assumes lp: "islin p"
+    and pos: "Ipoly vs c > 0"
+  shows "Ifm vs (x#bs) (msubstpos p c t) =
+    Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
   using lp pos
-  by (induct p rule: islin.induct, auto simp add: msubsteq2 msubstltpos[OF pos] msubstlepos[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
-
-fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
+  by (induct p rule: islin.induct)
+    (auto simp add: msubsteq2 msubstltpos[OF pos] msubstlepos[OF pos]
+      tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x]
+      bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
+
+fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm"
+where
   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
@@ -2707,33 +3642,46 @@
 | "msubstneg p c t = p"
 
 lemma msubstneg_I:
-  assumes lp: "islin p" and pos: "Ipoly vs c < 0"
+  assumes lp: "islin p"
+    and pos: "Ipoly vs c < 0"
   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
   using lp pos
-  by (induct p rule: islin.induct, auto simp add: msubsteq2 msubstltneg[OF pos] msubstleneg[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
-
-
-definition "msubst2 p c t = disj (conj (lt (CP (polyneg c))) (simpfm (msubstpos p c t))) (conj (lt (CP c)) (simpfm (msubstneg p c t)))"
-
-lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
+  by (induct p rule: islin.induct)
+    (auto simp add: msubsteq2 msubstltneg[OF pos] msubstleneg[OF pos]
+      tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x]
+      bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
+
+definition
+  "msubst2 p c t =
+    disj
+      (conj (lt (CP (polyneg c))) (simpfm (msubstpos p c t)))
+      (conj (lt (CP c)) (simpfm (msubstneg p c t)))"
+
+lemma msubst2:
+  assumes lp: "islin p"
+    and nc: "isnpoly c"
+    and nz: "Ipoly vs c \<noteq> 0"
   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
-proof-
+proof -
   let ?c = "Ipoly vs c"
   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))"
     by (simp_all add: polyneg_norm)
   from nz have "?c > 0 \<or> ?c < 0" by arith
   moreover
-  {assume c: "?c < 0"
+  {
+    assume c: "?c < 0"
     from c msubstneg_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"]
-    have ?thesis by (auto simp add: msubst2_def)}
+    have ?thesis by (auto simp add: msubst2_def)
+  }
   moreover
-  {assume c: "?c > 0"
+  {
+    assume c: "?c > 0"
     from c msubstpos_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"]
-    have ?thesis by (auto simp add: msubst2_def)}
+    have ?thesis by (auto simp add: msubst2_def)
+  }
   ultimately show ?thesis by blast
 qed
 
-term msubsteq2
 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
   by (simp add: msubsteq2_def)
 
@@ -2747,20 +3695,30 @@
 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
   by (simp add: msubstleneg_def)
 
-lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
+lemma msubstpos_nb:
+  assumes lp: "islin p"
+    and tnb: "tmbound0 t"
   shows "bound0 (msubstpos p c t)"
-using lp tnb
-by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
-
-lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
+  using lp tnb
+  by (induct p c t rule: msubstpos.induct)
+    (auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
+
+lemma msubstneg_nb:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and lp: "islin p"
+    and tnb: "tmbound0 t"
   shows "bound0 (msubstneg p c t)"
-using lp tnb
-by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
-
-lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
+  using lp tnb
+  by (induct p c t rule: msubstneg.induct)
+    (auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
+
+lemma msubst2_nb:
+  assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
+    and lp: "islin p"
+    and tnb: "tmbound0 t"
   shows "bound0 (msubst2 p c t)"
-using lp tnb
-by (simp add: msubst2_def msubstneg_nb msubstpos_nb lt_nb simpfm_bound0)
+  using lp tnb
+  by (simp add: msubst2_def msubstneg_nb msubstpos_nb lt_nb simpfm_bound0)
 
 lemma mult_minus2_left: "-2 * (x::'a::comm_ring_1) = - (2 * x)"
   by simp
@@ -2769,58 +3727,104 @@
   by simp
 
 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
-  by (induct p rule: islin.induct, auto simp add: bound0_qf)
+  by (induct p rule: islin.induct) (auto simp add: bound0_qf)
+
 lemma fr_eq_msubst2:
   assumes lp: "islin p"
-  shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow> ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> Ifm vs (x#bs) (subst0 (CP 0\<^sub>p) p) \<or> (\<exists>(n, t)\<in>set (uset p). Ifm vs (x# bs) (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<or> (\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))))"
+  shows "(\<exists>x. Ifm vs (x#bs) p) \<longleftrightarrow>
+    ((Ifm vs (x#bs) (minusinf p)) \<or>
+     (Ifm vs (x#bs) (plusinf p)) \<or>
+     Ifm vs (x#bs) (subst0 (CP 0\<^sub>p) p) \<or>
+     (\<exists>(n, t) \<in> set (uset p).
+      Ifm vs (x# bs) (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<or>
+      (\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+        Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))))"
   (is "(\<exists>x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
-proof-
-  from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
+proof -
+  from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s"
+    by blast
   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
-  have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
+  have n2: "isnpoly (C (-2,1))"
+    by (simp add: isnpoly_def)
   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
 
-  have eq1: "(\<exists>(n, t)\<in>set (uset p). ?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<longleftrightarrow> (\<exists>(n, t)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p)"
-  proof-
-    {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
-      from H(1) th have "isnpoly n" by blast
-      hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
+  have eq1: "(\<exists>(n, t) \<in> set (uset p). ?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<longleftrightarrow>
+    (\<exists>(n, t) \<in> set (uset p).
+      \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and>
+      Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p)"
+  proof -
+    {
+      fix n t
+      assume H: "(n, t) \<in> set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
+      from H(1) th have "isnpoly n"
+        by blast
+      then have nn: "isnpoly (n *\<^sub>p (C (-2,1)))"
+        by (simp_all add: polymul_norm n2)
       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
         by (simp add: polyneg_norm nn)
-      hence nn2: "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>n \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" using H(2) nn' nn
+      then have nn2: "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>n \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        using H(2) nn' nn
         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
       from msubst2[OF lp nn nn2(1), of x bs t]
       have "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p"
-        using H(2) nn2 by (simp add: mult_minus2_right)}
+        using H(2) nn2 by (simp add: mult_minus2_right)
+    }
     moreover
-    {fix n t assume H: "(n, t)\<in>set (uset p)" "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p"
-      from H(1) th have "isnpoly n" by blast
-      hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+    {
+      fix n t
+      assume H:
+        "(n, t) \<in> set (uset p)" "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        "Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * 2) # bs) p"
+      from H(1) th have "isnpoly n"
+        by blast
+      then have nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
         using H(2) by (simp_all add: polymul_norm n2)
-      from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)" using H(2,3) by (simp add: mult_minus2_right)}
+      from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)"
+        using H(2,3) by (simp add: mult_minus2_right)
+    }
     ultimately show ?thesis by blast
   qed
-  have eq2: "(\<exists>(c,t) \<in> set (uset p). \<exists>(d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))) \<longleftrightarrow> (\<exists>(n, t)\<in>set (uset p).
-     \<exists>(m, s)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p)"
-  proof-
-    {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
-     "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
-      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
-      hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)"
+  have eq2: "(\<exists>(c, t) \<in> set (uset p). \<exists>(d, s) \<in> set (uset p).
+      Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))) \<longleftrightarrow>
+    (\<exists>(n, t)\<in>set (uset p). \<exists>(m, s)\<in>set (uset p).
+      \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and>
+      \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and>
+      Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p)"
+  proof -
+    {
+      fix c t d s
+      assume H:
+        "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
+        "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
+      from H(1,2) th have "isnpoly c" "isnpoly d"
+        by blast+
+      then have nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)"
         by (simp_all add: polymul_norm n2)
-      have stupid: "allpolys isnpoly (CP (~\<^sub>p (C (-2, 1) *\<^sub>p c *\<^sub>p d)))" "allpolys isnpoly (CP ((C (-2, 1) *\<^sub>p c *\<^sub>p d)))"
+      have stupid:
+          "allpolys isnpoly (CP (~\<^sub>p (C (-2, 1) *\<^sub>p c *\<^sub>p d)))"
+          "allpolys isnpoly (CP ((C (-2, 1) *\<^sub>p c *\<^sub>p d)))"
         by (simp_all add: polyneg_norm nn)
       have nn': "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
-        using H(3) by (auto simp add: msubst2_def lt[OF stupid(1)]  lt[OF stupid(2)] zero_less_mult_iff mult_less_0_iff)
+        using H(3)
+        by (auto simp add: msubst2_def lt[OF stupid(1)]
+            lt[OF stupid(2)] zero_less_mult_iff mult_less_0_iff)
       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
-      have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
+      have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and>
+          Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
         by (simp add: add_divide_distrib diff_divide_distrib mult_minus2_left mult_commute)
     }
     moreover
-    {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
-      "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
-     from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
-      hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+    {
+      fix c t d s
+      assume H:
+        "(c, t) \<in> set (uset p)"
+        "(d, s) \<in> set (uset p)"
+        "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / 2 # bs) p"
+      from H(1,2) th have "isnpoly c" "isnpoly d"
+        by blast+
+      then have nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
         using H(3,4) by (simp_all add: polymul_norm n2)
       from msubst2[OF lp nn, of x bs ] H(3,4,5)
       have "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
@@ -2833,20 +3837,34 @@
 qed
 
 definition
-"ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
- in if (mp = T \<or> pp = T) then T
-  else (let U = remdups (uset  q)
-    in decr0 (list_disj [mp, pp, simpfm (subst0 (CP 0\<^sub>p) q), evaldjf (\<lambda>(c,t). msubst2 q (c *\<^sub>p C (-2, 1)) t) U,
-   evaldjf (\<lambda>((b,a),(d,c)). msubst2 q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs U)]))"
+  "ferrack2 p \<equiv>
+    let
+      q = simpfm p;
+      mp = minusinf q;
+      pp = plusinf q
+    in
+      if (mp = T \<or> pp = T) then T
+      else
+       (let U = remdups (uset  q)
+        in
+          decr0
+            (list_disj
+              [mp,
+               pp,
+               simpfm (subst0 (CP 0\<^sub>p) q),
+               evaldjf (\<lambda>(c, t). msubst2 q (c *\<^sub>p C (-2, 1)) t) U,
+               evaldjf (\<lambda>((b, a),(d, c)).
+                msubst2 q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs U)]))"
 
 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
 
-lemma ferrack2: assumes qf: "qfree p"
-  shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
+lemma ferrack2:
+  assumes qf: "qfree p"
+  shows "qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)"
   (is "_ \<and> (?rhs = ?lhs)")
-proof-
-  let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
-  let ?N = "\<lambda> t. Ipoly vs t"
+proof -
+  let ?J = "\<lambda>x p. Ifm vs (x#bs) p"
+  let ?N = "\<lambda>t. Ipoly vs t"
   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
   let ?q = "simpfm p"
   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
@@ -2854,68 +3872,109 @@
   let ?Up = "alluopairs ?U"
   let ?mp = "minusinf ?q"
   let ?pp = "plusinf ?q"
+  fix x
   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
-  from uset_l[OF lq] have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
+  from uset_l[OF lq]
+  have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
     by simp
   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)"
-  proof-
-    {fix c t assume ct: "(c,t) \<in> set ?U"
-      hence tnb: "tmbound0 t" using U_l by blast
+  proof -
+    {
+      fix c t
+      assume ct: "(c, t) \<in> set ?U"
+      then have tnb: "tmbound0 t"
+        using U_l by blast
       from msubst2_nb[OF lq tnb]
-      have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
-    thus ?thesis by auto
+      have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp
+    }
+    then show ?thesis by auto
   qed
-  have bnd1: "\<forall>x \<in> set ?Up. bound0 ((\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) x)"
-  proof-
-    {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
+  have bnd1: "\<forall>x \<in> set ?Up. bound0 ((\<lambda>((b, a), (d, c)).
+    msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) x)"
+  proof -
+    {
+      fix b a d c
+      assume badc: "((b,a),(d,c)) \<in> set ?Up"
       from badc U_l alluopairs_set1[of ?U]
-      have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
-      from msubst2_nb[OF lq nb] have "bound0 ((\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) ((b,a),(d,c)))" by simp}
-    thus ?thesis by auto
+      have nb: "tmbound0 (Add (Mul d a) (Mul b c))"
+        by auto
+      from msubst2_nb[OF lq nb]
+      have "bound0 ((\<lambda>((b, a),(d, c)).
+          msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) ((b,a),(d,c)))"
+        by simp
+    }
+    then show ?thesis by auto
   qed
-  have stupid: "bound0 F" by simp
-  let ?R = "list_disj [?mp, ?pp, simpfm (subst0 (CP 0\<^sub>p) ?q), evaldjf (\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) ?U,
-   evaldjf (\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs ?U)]"
-  from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
-  have nb: "bound0 ?R "
+  have stupid: "bound0 F"
+    by simp
+  let ?R =
+    "list_disj
+     [?mp,
+      ?pp,
+      simpfm (subst0 (CP 0\<^sub>p) ?q),
+      evaldjf (\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) ?U,
+      evaldjf (\<lambda>((b,a),(d,c)).
+        msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs ?U)]"
+  from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf
+    evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
+  have nb: "bound0 ?R"
     by (simp add: list_disj_def simpfm_bound0)
-  let ?s = "\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))"
-
-  {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
+  let ?s = "\<lambda>((b, a),(d, c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))"
+
+  {
+    fix b a d c
+    assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))"
       by auto (simp add: isnpoly_def)
     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
       using norm by (simp_all add: polymul_norm)
-    have stupid: "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p b*\<^sub>p d))" "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p d*\<^sub>p b))" "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p b*\<^sub>p d)))" "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p d*\<^sub>p b)))"
+    have stupid:
+        "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p b *\<^sub>p d))"
+        "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p d *\<^sub>p b))"
+        "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p b *\<^sub>p d)))"
+        "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p d*\<^sub>p b)))"
       by (simp_all add: polyneg_norm norm2)
-    have "?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) = ?I (msubst2 ?q (C (-2, 1) *\<^sub>p d*\<^sub>p b) (Add (Mul b c) (Mul d a)))" (is "?lhs \<longleftrightarrow> ?rhs")
+    have "?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) =
+        ?I (msubst2 ?q (C (-2, 1) *\<^sub>p d*\<^sub>p b) (Add (Mul b c) (Mul d a)))"
+      (is "?lhs \<longleftrightarrow> ?rhs")
     proof
       assume H: ?lhs
-      hence z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
-        by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
-      from msubst2[OF lq norm2(1) z(1), of x bs]
-        msubst2[OF lq norm2(2) z(2), of x bs] H
-      show ?rhs by (simp add: field_simps)
+      then have z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)]
+            mult_less_0_iff zero_less_mult_iff)
+      from msubst2[OF lq norm2(1) z(1), of x bs] msubst2[OF lq norm2(2) z(2), of x bs] H
+      show ?rhs
+        by (simp add: field_simps)
     next
       assume H: ?rhs
-      hence z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
-        by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
-      from msubst2[OF lq norm2(1) z(1), of x bs]
-        msubst2[OF lq norm2(2) z(2), of x bs] H
-      show ?lhs by (simp add: field_simps)
-    qed}
-  hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
-    by clarsimp
-
-  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists>(b, a)\<in>set ?U. \<exists>(d, c)\<in>set ?U. ?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))))"
+      then have z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
+        by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)]
+            mult_less_0_iff zero_less_mult_iff)
+      from msubst2[OF lq norm2(1) z(1), of x bs] msubst2[OF lq norm2(2) z(2), of x bs] H
+      show ?lhs
+        by (simp add: field_simps)
+    qed
+  }
+  then have th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
+    by auto
+
+  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)"
+    by simp
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or>
+      (\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or>
+      (\<exists>(b, a) \<in> set ?U. \<exists>(d, c) \<in> set ?U.
+        ?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))))"
     using fr_eq_msubst2[OF lq, of vs bs x] by simp
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists>x\<in>set ?U. \<exists>y \<in>set ?U. ?I (?s (x,y)))"
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or>
+      (\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or>
+      (\<exists>x \<in> set ?U. \<exists>y \<in>set ?U. ?I (?s (x, y)))"
     by (simp add: split_def)
-  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists>(x,y) \<in> set ?Up. ?I (?s (x,y)))"
+  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or>
+      (\<exists>(n, t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or>
+      (\<exists>(x, y) \<in> set ?Up. ?I (?s (x, y)))"
     using alluopairs_bex[OF th0] by simp
   also have "\<dots> \<longleftrightarrow> ?I ?R"
     by (simp add: list_disj_def evaldjf_ex split_def)
@@ -2925,9 +3984,10 @@
     apply (simp add: list_disj_def)
     apply (cases "?pp = T")
     apply (simp add: list_disj_def)
-    by (simp_all add: Let_def decr0[OF nb])
+    apply (simp_all add: Let_def decr0[OF nb])
+    done
   finally show ?thesis using decr0_qf[OF nb]
-    by (simp  add: ferrack2_def Let_def)
+    by (simp add: ferrack2_def Let_def)
 qed
 
 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
@@ -2936,7 +3996,8 @@
   have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)"
     by blast
   from qelim[OF th, of "prep p" bs]
-  show ?thesis unfolding frpar2_def by (auto simp add: prep)
+  show ?thesis
+    unfolding frpar2_def by (auto simp add: prep)
 qed
 
 oracle frpar_oracle = {*
@@ -2962,31 +4023,47 @@
     val k = find_index (fn t' => t aconv t') ts;
   in if k < 0 then raise General.Subscript else k end;
 
-fun num_of_term ps (Const (@{const_name Groups.uminus}, _) $ t) = @{code poly.Neg} (num_of_term ps t)
-  | num_of_term ps (Const (@{const_name Groups.plus}, _) $ a $ b) = @{code poly.Add} (num_of_term ps a, num_of_term ps b)
-  | num_of_term ps (Const (@{const_name Groups.minus}, _) $ a $ b) = @{code poly.Sub} (num_of_term ps a, num_of_term ps b)
-  | num_of_term ps (Const (@{const_name Groups.times}, _) $ a $ b) = @{code poly.Mul} (num_of_term ps a, num_of_term ps b)
-  | num_of_term ps (Const (@{const_name Power.power}, _) $ a $ n) = @{code poly.Pw} (num_of_term ps a, @{code nat_of_integer} (dest_nat n))
-  | num_of_term ps (Const (@{const_name Fields.divide}, _) $ a $ b) = mk_C (dest_num a, dest_num b)
-  | num_of_term ps t = (case try_dest_num t
-     of SOME k => mk_C (k, 1)
+fun num_of_term ps (Const (@{const_name Groups.uminus}, _) $ t) =
+      @{code poly.Neg} (num_of_term ps t)
+  | num_of_term ps (Const (@{const_name Groups.plus}, _) $ a $ b) =
+      @{code poly.Add} (num_of_term ps a, num_of_term ps b)
+  | num_of_term ps (Const (@{const_name Groups.minus}, _) $ a $ b) =
+      @{code poly.Sub} (num_of_term ps a, num_of_term ps b)
+  | num_of_term ps (Const (@{const_name Groups.times}, _) $ a $ b) =
+      @{code poly.Mul} (num_of_term ps a, num_of_term ps b)
+  | num_of_term ps (Const (@{const_name Power.power}, _) $ a $ n) =
+      @{code poly.Pw} (num_of_term ps a, @{code nat_of_integer} (dest_nat n))
+  | num_of_term ps (Const (@{const_name Fields.divide}, _) $ a $ b) =
+      mk_C (dest_num a, dest_num b)
+  | num_of_term ps t =
+      (case try_dest_num t of
+        SOME k => mk_C (k, 1)
       | NONE => mk_poly_Bound (the_index ps t));
 
-fun tm_of_term fs ps (Const(@{const_name Groups.uminus}, _) $ t) = @{code Neg} (tm_of_term fs ps t)
-  | tm_of_term fs ps (Const(@{const_name Groups.plus}, _) $ a $ b) = @{code Add} (tm_of_term fs ps a, tm_of_term fs ps b)
-  | tm_of_term fs ps (Const(@{const_name Groups.minus}, _) $ a $ b) = @{code Sub} (tm_of_term fs ps a, tm_of_term fs ps b)
-  | tm_of_term fs ps (Const(@{const_name Groups.times}, _) $ a $ b) = @{code Mul} (num_of_term ps a, tm_of_term fs ps b)
+fun tm_of_term fs ps (Const(@{const_name Groups.uminus}, _) $ t) =
+      @{code Neg} (tm_of_term fs ps t)
+  | tm_of_term fs ps (Const(@{const_name Groups.plus}, _) $ a $ b) =
+      @{code Add} (tm_of_term fs ps a, tm_of_term fs ps b)
+  | tm_of_term fs ps (Const(@{const_name Groups.minus}, _) $ a $ b) =
+      @{code Sub} (tm_of_term fs ps a, tm_of_term fs ps b)
+  | tm_of_term fs ps (Const(@{const_name Groups.times}, _) $ a $ b) =
+      @{code Mul} (num_of_term ps a, tm_of_term fs ps b)
   | tm_of_term fs ps t = (@{code CP} (num_of_term ps t)
       handle TERM _ => mk_Bound (the_index fs t)
-           | General.Subscript => mk_Bound (the_index fs t));
+        | General.Subscript => mk_Bound (the_index fs t));
 
 fun fm_of_term fs ps @{term True} = @{code T}
   | fm_of_term fs ps @{term False} = @{code F}
-  | fm_of_term fs ps (Const (@{const_name Not}, _) $ p) = @{code NOT} (fm_of_term fs ps p)
-  | fm_of_term fs ps (Const (@{const_name HOL.conj}, _) $ p $ q) = @{code And} (fm_of_term fs ps p, fm_of_term fs ps q)
-  | fm_of_term fs ps (Const (@{const_name HOL.disj}, _) $ p $ q) = @{code Or} (fm_of_term fs ps p, fm_of_term fs ps q)
-  | fm_of_term fs ps (Const (@{const_name HOL.implies}, _) $ p $ q) = @{code Imp} (fm_of_term fs ps p, fm_of_term fs ps q)
-  | fm_of_term fs ps (@{term HOL.iff} $ p $ q) = @{code Iff} (fm_of_term fs ps p, fm_of_term fs ps q)
+  | fm_of_term fs ps (Const (@{const_name Not}, _) $ p) =
+      @{code NOT} (fm_of_term fs ps p)
+  | fm_of_term fs ps (Const (@{const_name HOL.conj}, _) $ p $ q) =
+      @{code And} (fm_of_term fs ps p, fm_of_term fs ps q)
+  | fm_of_term fs ps (Const (@{const_name HOL.disj}, _) $ p $ q) =
+      @{code Or} (fm_of_term fs ps p, fm_of_term fs ps q)
+  | fm_of_term fs ps (Const (@{const_name HOL.implies}, _) $ p $ q) =
+      @{code Imp} (fm_of_term fs ps p, fm_of_term fs ps q)
+  | fm_of_term fs ps (@{term HOL.iff} $ p $ q) =
+      @{code Iff} (fm_of_term fs ps p, fm_of_term fs ps q)
   | fm_of_term fs ps (Const (@{const_name HOL.eq}, T) $ p $ q) =
       @{code Eq} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q))
   | fm_of_term fs ps (Const (@{const_name Orderings.less}, _) $ p $ q) =
@@ -3009,38 +4086,62 @@
       in
         (if d = 1 then HOLogic.mk_number T c
         else if d = 0 then Const (@{const_name Groups.zero}, T)
-        else Const (@{const_name Fields.divide}, binopT T) $ HOLogic.mk_number T c $ HOLogic.mk_number T d)
+        else
+          Const (@{const_name Fields.divide}, binopT T) $
+            HOLogic.mk_number T c $ HOLogic.mk_number T d)
       end
   | term_of_num T ps (@{code poly.Bound} i) = nth ps (@{code integer_of_nat} i)
-  | term_of_num T ps (@{code poly.Add} (a, b)) = Const (@{const_name Groups.plus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
-  | term_of_num T ps (@{code poly.Mul} (a, b)) = Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
-  | term_of_num T ps (@{code poly.Sub} (a, b)) = Const (@{const_name Groups.minus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
-  | term_of_num T ps (@{code poly.Neg} a) = Const (@{const_name Groups.uminus}, T --> T) $ term_of_num T ps a
-  | term_of_num T ps (@{code poly.Pw} (a, n)) = Const (@{const_name Power.power},
-      T --> @{typ nat} --> T) $ term_of_num T ps a $ HOLogic.mk_number HOLogic.natT (@{code integer_of_nat} n)
+  | term_of_num T ps (@{code poly.Add} (a, b)) =
+      Const (@{const_name Groups.plus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
+  | term_of_num T ps (@{code poly.Mul} (a, b)) =
+      Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
+  | term_of_num T ps (@{code poly.Sub} (a, b)) =
+      Const (@{const_name Groups.minus}, binopT T) $ term_of_num T ps a $ term_of_num T ps b
+  | term_of_num T ps (@{code poly.Neg} a) =
+      Const (@{const_name Groups.uminus}, T --> T) $ term_of_num T ps a
+  | term_of_num T ps (@{code poly.Pw} (a, n)) =
+      Const (@{const_name Power.power}, T --> @{typ nat} --> T) $
+        term_of_num T ps a $ HOLogic.mk_number HOLogic.natT (@{code integer_of_nat} n)
   | term_of_num T ps (@{code poly.CN} (c, n, p)) =
       term_of_num T ps (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)));
 
 fun term_of_tm T fs ps (@{code CP} p) = term_of_num T ps p
   | term_of_tm T fs ps (@{code Bound} i) = nth fs (@{code integer_of_nat} i)
-  | term_of_tm T fs ps (@{code Add} (a, b)) = Const (@{const_name Groups.plus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b
-  | term_of_tm T fs ps (@{code Mul} (a, b)) = Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_tm T fs ps b
-  | term_of_tm T fs ps (@{code Sub} (a, b)) = Const (@{const_name Groups.minus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b
-  | term_of_tm T fs ps (@{code Neg} a) = Const (@{const_name Groups.uminus}, T --> T) $ term_of_tm T fs ps a
-  | term_of_tm T fs ps (@{code CNP} (n, c, p)) = term_of_tm T fs ps
-     (@{code Add} (@{code Mul} (c, @{code Bound} n), p));
+  | term_of_tm T fs ps (@{code Add} (a, b)) =
+      Const (@{const_name Groups.plus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b
+  | term_of_tm T fs ps (@{code Mul} (a, b)) =
+      Const (@{const_name Groups.times}, binopT T) $ term_of_num T ps a $ term_of_tm T fs ps b
+  | term_of_tm T fs ps (@{code Sub} (a, b)) =
+      Const (@{const_name Groups.minus}, binopT T) $ term_of_tm T fs ps a $ term_of_tm T fs ps b
+  | term_of_tm T fs ps (@{code Neg} a) =
+      Const (@{const_name Groups.uminus}, T --> T) $ term_of_tm T fs ps a
+  | term_of_tm T fs ps (@{code CNP} (n, c, p)) =
+      term_of_tm T fs ps (@{code Add} (@{code Mul} (c, @{code Bound} n), p));
 
 fun term_of_fm T fs ps @{code T} = @{term True}
   | term_of_fm T fs ps @{code F} = @{term False}
   | term_of_fm T fs ps (@{code NOT} p) = @{term Not} $ term_of_fm T fs ps p
-  | term_of_fm T fs ps (@{code And} (p, q)) = @{term HOL.conj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
-  | term_of_fm T fs ps (@{code Or} (p, q)) = @{term HOL.disj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
-  | term_of_fm T fs ps (@{code Imp} (p, q)) = @{term HOL.implies} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
-  | term_of_fm T fs ps (@{code Iff} (p, q)) = @{term HOL.iff} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
-  | term_of_fm T fs ps (@{code Lt} p) = Const (@{const_name Orderings.less}, relT T) $ term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
-  | term_of_fm T fs ps (@{code Le} p) = Const (@{const_name Orderings.less_eq}, relT T) $ term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
-  | term_of_fm T fs ps (@{code Eq} p) = Const (@{const_name HOL.eq}, relT T) $ term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
-  | term_of_fm T fs ps (@{code NEq} p) = @{term Not} $ (Const (@{const_name HOL.eq}, relT T) $ term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T))
+  | term_of_fm T fs ps (@{code And} (p, q)) =
+      @{term HOL.conj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
+  | term_of_fm T fs ps (@{code Or} (p, q)) =
+      @{term HOL.disj} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
+  | term_of_fm T fs ps (@{code Imp} (p, q)) =
+      @{term HOL.implies} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
+  | term_of_fm T fs ps (@{code Iff} (p, q)) =
+      @{term HOL.iff} $ term_of_fm T fs ps p $ term_of_fm T fs ps q
+  | term_of_fm T fs ps (@{code Lt} p) =
+      Const (@{const_name Orderings.less}, relT T) $
+        term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
+  | term_of_fm T fs ps (@{code Le} p) =
+      Const (@{const_name Orderings.less_eq}, relT T) $
+        term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
+  | term_of_fm T fs ps (@{code Eq} p) =
+      Const (@{const_name HOL.eq}, relT T) $
+        term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T)
+  | term_of_fm T fs ps (@{code NEq} p) =
+      @{term Not} $
+        (Const (@{const_name HOL.eq}, relT T) $
+          term_of_tm T fs ps p $ Const (@{const_name Groups.zero}, T))
   | term_of_fm T fs ps _ = error "term_of_fm: quantifiers";
 
 fun frpar_procedure alternative T ps fm =
@@ -3049,9 +4150,7 @@
     val fs = subtract (op aconv) (map Free (Term.add_frees fm [])) ps;
     val eval = term_of_fm T fs ps o frpar o fm_of_term fs ps;
     val t = HOLogic.dest_Trueprop fm;
-  in
-    (HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, eval t)
-  end;
+  in HOLogic.mk_Trueprop (HOLogic.mk_eq (t, eval t)) end;
 
 in
 
@@ -3070,21 +4169,21 @@
   Object_Logic.full_atomize_tac ctxt
   THEN' CSUBGOAL (fn (g, i) =>
     let
-      val th = frpar_oracle (ctxt, alternative, T, ps, g)
+      val th = frpar_oracle (ctxt, alternative, T, ps, g);
     in rtac (th RS iffD2) i end);
 
 fun method alternative =
   let
-    fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
-    val parsN = "pars"
-    val typN = "type"
-    val any_keyword = keyword parsN || keyword typN
-    val terms = Scan.repeat (Scan.unless any_keyword Args.term)
-    val typ = Scan.unless any_keyword Args.typ
+    fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ();
+    val parsN = "pars";
+    val typN = "type";
+    val any_keyword = keyword parsN || keyword typN;
+    val terms = Scan.repeat (Scan.unless any_keyword Args.term);
+    val typ = Scan.unless any_keyword Args.typ;
   in
     (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
       (fn (T, ps) => fn ctxt => SIMPLE_METHOD' (tactic ctxt alternative T ps))
-  end
+  end;
 
 end;
 *}
@@ -3097,19 +4196,19 @@
   Parametric_Ferrante_Rackoff.method true
 *} "parametric QE for linear Arithmetic over fields, Version 2"
 
-lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
-  apply (frpar type: "'a::{linordered_field_inverse_zero}" pars: "y::'a::{linordered_field_inverse_zero}")
+lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1) * x < 0"
+  apply (frpar type: 'a pars: y)
   apply (simp add: field_simps)
   apply (rule spec[where x=y])
-  apply (frpar type: "'a::{linordered_field_inverse_zero}" pars: "z::'a::{linordered_field_inverse_zero}")
+  apply (frpar type: 'a pars: "z::'a")
   apply simp
   done
 
 lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
-  apply (frpar2 type: "'a::{linordered_field_inverse_zero}" pars: "y::'a::{linordered_field_inverse_zero}")
+  apply (frpar2 type: 'a pars: y)
   apply (simp add: field_simps)
   apply (rule spec[where x=y])
-  apply (frpar2 type: "'a::{linordered_field_inverse_zero}" pars: "z::'a::{linordered_field_inverse_zero}")
+  apply (frpar2 type: 'a pars: "z::'a")
   apply simp
   done