
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Decision_Procs/MIR.thy
--- a/src/HOL/Decision_Procs/MIR.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Decision_Procs/MIR.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -28,15 +28,15 @@
 definition rdvd:: "real \<Rightarrow> real \<Rightarrow> bool" (infixl "rdvd" 50)
   where "x rdvd y \<longleftrightarrow> (\<exists>k::int. y = x * real_of_int k)"
 
-lemma int_rdvd_real: 
+lemma int_rdvd_real:
   "real_of_int (i::int) rdvd x = (i dvd (floor x) \<and> real_of_int (floor x) = x)" (is "?l = ?r")
 proof
-  assume "?l" 
+  assume "?l"
   hence th: "\<exists> k. x=real_of_int (i*k)" by (simp add: rdvd_def)
   hence th': "real_of_int (floor x) = x" by (auto simp del: of_int_mult)
   with th have "\<exists> k. real_of_int (floor x) = real_of_int (i*k)" by simp
   hence "\<exists> k. floor x = i*k" by presburger
-  thus ?r  using th' by (simp add: dvd_def) 
+  thus ?r  using th' by (simp add: dvd_def)
 next
   assume "?r" hence "(i::int) dvd \<lfloor>x::real\<rfloor>" ..
   hence "\<exists> k. real_of_int (floor x) = real_of_int (i*k)"
@@ -55,7 +55,7 @@
     by auto
 
   from iffD2[OF abs_dvd_iff] d2 have "(abs d) dvd (floor t)" by blast
-  with ti int_rdvd_real[symmetric] have "real_of_int (abs d) rdvd t" by blast 
+  with ti int_rdvd_real[symmetric] have "real_of_int (abs d) rdvd t" by blast
   thus "abs (real_of_int d) rdvd t" by simp
 next
   assume "abs (real_of_int d) rdvd t" hence "real_of_int (abs d) rdvd t" by simp
@@ -67,14 +67,14 @@
 
 lemma rdvd_minus: "(real_of_int (d::int) rdvd t) = (real_of_int d rdvd -t)"
   apply (auto simp add: rdvd_def)
-  apply (rule_tac x="-k" in exI, simp) 
+  apply (rule_tac x="-k" in exI, simp)
   apply (rule_tac x="-k" in exI, simp)
   done
 
 lemma rdvd_left_0_eq: "(0 rdvd t) = (t=0)"
   by (auto simp add: rdvd_def)
 
-lemma rdvd_mult: 
+lemma rdvd_mult:
   assumes knz: "k\<noteq>0"
   shows "(real_of_int (n::int) * real_of_int (k::int) rdvd x * real_of_int k) = (real_of_int n rdvd x)"
   using knz by (simp add: rdvd_def)
@@ -83,7 +83,7 @@
   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
   (*********************************************************************************)
 
-datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
+datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
   | Mul int num | Floor num| CF int num num
 
   (* A size for num to make inductive proofs simpler*)
@@ -132,17 +132,17 @@
 proof-
   let ?I = "\<lambda> t. Inum bs t"
   assume ie: "isint e bs"
-  hence th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)  
+  hence th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)
   have "real_of_int (floor (?I (Neg e))) = real_of_int (floor (- (real_of_int (floor (?I e)))))" by (simp add: th)
   also have "\<dots> = - real_of_int (floor (?I e))" by simp
   finally show "isint (Neg e) bs" by (simp add: isint_def th)
 qed
 
-lemma isint_sub: 
+lemma isint_sub:
   assumes ie: "isint e bs" shows "isint (Sub (C c) e) bs"
 proof-
   let ?I = "\<lambda> t. Inum bs t"
-  from ie have th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)  
+  from ie have th: "real_of_int (floor (?I e)) = ?I e" by (simp add: isint_def)
   have "real_of_int (floor (?I (Sub (C c) e))) = real_of_int (floor ((real_of_int (c -floor (?I e)))))" by (simp add: th)
   also have "\<dots> = real_of_int (c- floor (?I e))" by simp
   finally show "isint (Sub (C c) e) bs" by (simp add: isint_def th)
@@ -166,7 +166,7 @@
 
 
     (* FORMULAE *)
-datatype fm  = 
+datatype fm  =
   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
 
@@ -213,7 +213,7 @@
   "prep (E F) = F"
   "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
   "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
-  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
+  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
   "prep (E (NOT (And p q))) = Or (prep (E (NOT p))) (prep (E(NOT q)))"
   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
   "prep (E (NOT (Iff p q))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
@@ -241,10 +241,10 @@
 fun qfree:: "fm \<Rightarrow> bool" where
   "qfree (E p) = False"
   | "qfree (A p) = False"
-  | "qfree (NOT p) = qfree p" 
-  | "qfree (And p q) = (qfree p \<and> qfree q)" 
-  | "qfree (Or  p q) = (qfree p \<and> qfree q)" 
-  | "qfree (Imp p q) = (qfree p \<and> qfree q)" 
+  | "qfree (NOT p) = qfree p"
+  | "qfree (And p q) = (qfree p \<and> qfree q)"
+  | "qfree (Or  p q) = (qfree p \<and> qfree q)"
+  | "qfree (Imp p q) = (qfree p \<and> qfree q)"
   | "qfree (Iff p q) = (qfree p \<and> qfree q)"
   | "qfree p = True"
 
@@ -255,20 +255,20 @@
   | "numbound0 (CN n i a) = (n > 0 \<and> numbound0 a)"
   | "numbound0 (Neg a) = numbound0 a"
   | "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
-  | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
+  | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
   | "numbound0 (Mul i a) = numbound0 a"
   | "numbound0 (Floor a) = numbound0 a"
-  | "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)" 
+  | "numbound0 (CF c a b) = (numbound0 a \<and> numbound0 b)"
 
 lemma numbound0_I:
   assumes nb: "numbound0 a"
   shows "Inum (b#bs) a = Inum (b'#bs) a"
   using nb by (induct a) auto
 
-lemma numbound0_gen: 
+lemma numbound0_gen:
   assumes nb: "numbound0 t" and ti: "isint t (x#bs)"
   shows "\<forall> y. isint t (y#bs)"
-  using nb ti 
+  using nb ti
 proof(clarify)
   fix y
   from numbound0_I[OF nb, where bs="bs" and b="y" and b'="x"] ti[simplified isint_def]
@@ -308,7 +308,7 @@
   | "numsubst0 t (CF i a b) = CF i (numsubst0 t a) (numsubst0 t b)"
   | "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
   | "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
-  | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" 
+  | "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)"
   | "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"
   | "numsubst0 t (Floor a) = Floor (numsubst0 t a)"
 
@@ -358,7 +358,7 @@
 | "decr (NEq a) = NEq (decrnum a)"
 | "decr (Dvd i a) = Dvd i (decrnum a)"
 | "decr (NDvd i a) = NDvd i (decrnum a)"
-| "decr (NOT p) = NOT (decr p)" 
+| "decr (NOT p) = NOT (decr p)"
 | "decr (And p q) = And (decr p) (decr q)"
 | "decr (Or p q) = Or (decr p) (decr q)"
 | "decr (Imp p q) = Imp (decr p) (decr q)"
@@ -404,20 +404,20 @@
 
 
 definition djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
-  "djf f p q = (if q=T then T else if q=F then f p else 
+  "djf f p q = (if q=T then T else if q=F then f p else
   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or fp q))"
 
 definition evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
   "evaldjf f ps = foldr (djf f) ps F"
 
 lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
-  by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
-  (cases "f p", simp_all add: Let_def djf_def) 
+  by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
+  (cases "f p", simp_all add: Let_def djf_def)
 
 lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
   by (induct ps) (simp_all add: evaldjf_def djf_Or)
 
-lemma evaldjf_bound0: 
+lemma evaldjf_bound0:
   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
   shows "bound0 (evaldjf f xs)"
   using nb
@@ -427,7 +427,7 @@
   apply auto
   done
 
-lemma evaldjf_qf: 
+lemma evaldjf_qf:
   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
   shows "qfree (evaldjf f xs)"
   using nb
@@ -474,12 +474,12 @@
   shows "Ifm bs (DJ f p) = Ifm bs (f p)"
 proof -
   have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
-    by (simp add: DJ_def evaldjf_ex) 
+    by (simp add: DJ_def evaldjf_ex)
   also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
   finally show ?thesis .
 qed
 
-lemma DJ_qf: assumes 
+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)
@@ -487,7 +487,7 @@
   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
-  
+
   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
 qed
 
@@ -551,7 +551,7 @@
 definition reducecoeff :: "num \<Rightarrow> num"
 where
   "reducecoeff t =
-    (let g = numgcd t in 
+    (let g = numgcd t in
      if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
 
 fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
@@ -560,10 +560,10 @@
 | "dvdnumcoeff (CF c s t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
 | "dvdnumcoeff t = (\<lambda>g. False)"
 
-lemma dvdnumcoeff_trans: 
+lemma dvdnumcoeff_trans:
   assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
   shows "dvdnumcoeff t g"
-  using dgt' gdg 
+  using dgt' gdg
   by (induct t rule: dvdnumcoeff.induct) (simp_all add: gdg dvd_trans[OF gdg])
 
 declare dvd_trans [trans add]
@@ -584,10 +584,10 @@
   by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
 
 lemma reducecoeffh:
-  assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
+  assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
   shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t"
   using gt
-proof(induct t rule: reducecoeffh.induct) 
+proof(induct t rule: reducecoeffh.induct)
   case (1 i) hence gd: "g dvd i" by simp
   from assms 1 show ?case by (simp add: real_of_int_div[OF gd])
 next
@@ -595,7 +595,7 @@
   from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
 next
   case (3 c s t)  hence gd: "g dvd c" by simp
-  from assms 3 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps) 
+  from assms 3 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
 qed (auto simp add: numgcd_def gp)
 
 fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
@@ -614,7 +614,7 @@
   have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by simp
   from ismaxcoeff_mono[OF H thh] show ?case by simp
 next
-  case (3 c t s) 
+  case (3 c t s)
   hence H1:"ismaxcoeff s (maxcoeff s)" by auto
   have thh1: "maxcoeff s \<le> max \<bar>c\<bar> (maxcoeff s)" by (simp add: max_def)
   from ismaxcoeff_mono[OF H1 thh1] show ?case by simp
@@ -637,7 +637,7 @@
   shows "dvdnumcoeff t (numgcdh t m)"
 using assms
 proof(induct t rule: numgcdh.induct)
-  case (2 n c t) 
+  case (2 n c t)
   let ?g = "numgcdh t m"
   from 2 have th:"gcd c ?g > 1" by simp
   from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
@@ -651,11 +651,11 @@
     have th': "gcd c ?g dvd ?g" by simp
     from dvdnumcoeff_trans[OF th' th] have ?case by simp
     hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
+  moreover {assume "abs c > 1" and g0:"?g = 0"
     from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
   ultimately show ?case by blast
 next
-  case (3 c s t) 
+  case (3 c s t)
   let ?g = "numgcdh t m"
   from 3 have th:"gcd c ?g > 1" by simp
   from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
@@ -669,14 +669,14 @@
     have th': "gcd c ?g dvd ?g" by simp
     from dvdnumcoeff_trans[OF th' th] have ?case by simp
     hence ?case by simp }
-  moreover {assume "abs c > 1" and g0:"?g = 0" 
+  moreover {assume "abs c > 1" and g0:"?g = 0"
     from numgcdh0[OF g0] have "m=0". with 3 g0 have ?case by simp }
   ultimately show ?case by blast
 qed auto
 
 lemma dvdnumcoeff_aux2:
   assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
-  using assms 
+  using assms
 proof (simp add: numgcd_def)
   let ?mc = "maxcoeff t"
   let ?g = "numgcdh t ?mc"
@@ -691,12 +691,12 @@
   let ?g = "numgcd t"
   have "?g \<ge> 0"  by (simp add: numgcd_pos)
   hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
-  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
-  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
+  moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
+  moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
   moreover { assume g1:"?g > 1"
     from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
-    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
-      by (simp add: reducecoeff_def Let_def)} 
+    from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
+      by (simp add: reducecoeff_def Let_def)}
   ultimately show ?thesis by blast
 qed
 
@@ -709,15 +709,15 @@
 consts numadd:: "num \<times> num \<Rightarrow> num"
 recdef numadd "measure (\<lambda> (t,s). size t + size s)"
   "numadd (CN n1 c1 r1,CN n2 c2 r2) =
-  (if n1=n2 then 
+  (if n1=n2 then
   (let c = c1 + c2
   in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
   else if n1 \<le> n2 then CN n1 c1 (numadd (r1,CN n2 c2 r2))
   else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
-  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
-  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
-  "numadd (CF c1 t1 r1,CF c2 t2 r2) = 
-   (if t1 = t2 then 
+  "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
+  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))"
+  "numadd (CF c1 t1 r1,CF c2 t2 r2) =
+   (if t1 = t2 then
     (let c=c1+c2; s= numadd(r1,r2) in (if c=0 then s else CF c t1 s))
    else if lex_bnd t1 t2 then CF c1 t1 (numadd(r1,CF c2 t2 r2))
    else CF c2 t2 (numadd(CF c1 t1 r1,r2)))"
@@ -775,7 +775,7 @@
 lemma isint_CF: assumes si: "isint s bs" shows "isint (CF c t s) bs"
 proof-
   have cti: "isint (Mul c (Floor t)) bs" by (simp add: isint_Mul isint_Floor)
-  
+
   have "?thesis = isint (Add (Mul c (Floor t)) s) bs" by (simp add: isint_def)
   also have "\<dots>" by (simp add: isint_add cti si)
   finally show ?thesis .
@@ -783,11 +783,11 @@
 
 fun split_int:: "num \<Rightarrow> num \<times> num" where
   "split_int (C c) = (C 0, C c)"
-| "split_int (CN n c b) = 
-     (let (bv,bi) = split_int b 
+| "split_int (CN n c b) =
+     (let (bv,bi) = split_int b
        in (CN n c bv, bi))"
-| "split_int (CF c a b) = 
-     (let (bv,bi) = split_int b 
+| "split_int (CF c a b) =
+     (let (bv,bi) = split_int b
        in (bv, CF c a bi))"
 | "split_int a = (a,C 0)"
 
@@ -801,7 +801,7 @@
   from 2(2) have tibi: "ti = ?bi" by (simp add: Let_def split_def)
   from 2(2) b[symmetric] bii show ?case by (auto simp add: Let_def split_def)
 next
-  case (3 c a b tv ti) 
+  case (3 c a b tv ti)
   let ?bv = "fst (split_int b)"
   let ?bi = "snd (split_int b)"
   have "split_int b = (?bv,?bi)" by simp
@@ -817,8 +817,8 @@
 
 definition numfloor:: "num \<Rightarrow> num"
 where
-  "numfloor t = (let (tv,ti) = split_int t in 
-  (case tv of C i \<Rightarrow> numadd (tv,ti) 
+  "numfloor t = (let (tv,ti) = split_int t in
+  (case tv of C i \<Rightarrow> numadd (tv,ti)
   | _ \<Rightarrow> numadd(CF 1 tv (C 0),ti)))"
 
 lemma numfloor[simp]: "Inum bs (numfloor t) = Inum bs (Floor t)" (is "?n t = ?N (Floor t)")
@@ -827,17 +827,17 @@
   let ?ti = "snd (split_int t)"
   have tvti:"split_int t = (?tv,?ti)" by simp
   {assume H: "\<forall> v. ?tv \<noteq> C v"
-    hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)" 
+    hence th1: "?n t = ?N (Add (Floor ?tv) ?ti)"
       by (cases ?tv) (auto simp add: numfloor_def Let_def split_def)
     from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp 
+    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp
     also have "\<dots> = real_of_int (floor (?N ?tv) + (floor (?N ?ti)))"
       by (simp,subst tii[simplified isint_iff, symmetric]) simp
     also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
     finally have ?thesis using th1 by simp}
-  moreover {fix v assume H:"?tv = C v" 
+  moreover {fix v assume H:"?tv = C v"
     from split_int[OF tvti] have "?N (Floor t) = ?N (Floor(Add ?tv ?ti))" and tii:"isint ?ti bs" by simp+
-    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp 
+    hence "?N (Floor t) = real_of_int (floor (?N (Add ?tv ?ti)))" by simp
     also have "\<dots> = real_of_int (floor (?N ?tv) + (floor (?N ?ti)))"
       by (simp,subst tii[simplified isint_iff, symmetric]) simp
     also have "\<dots> = ?N (Add (Floor ?tv) ?ti)" by (simp add: tii[simplified isint_iff])
@@ -906,7 +906,7 @@
   with cnz have "max (abs c) (maxcoeff t) > 0" by arith
   with 2 show ?case by simp
 next
-  case (3 c s t) 
+  case (3 c s t)
   hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
   have "max (abs c) (maxcoeff t) \<ge> abs c" by simp
   with cnz have "max (abs c) (maxcoeff t) > 0" by arith
@@ -922,10 +922,10 @@
 
 definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
   "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
-   (let t' = simpnum t ; g = numgcd t' in 
-      if g > 1 then (let g' = gcd n g in 
-        if g' = 1 then (t',n) 
-        else (reducecoeffh t' g', n div g')) 
+   (let t' = simpnum t ; g = numgcd t' in
+      if g > 1 then (let g' = gcd n g in
+        if g' = 1 then (t',n)
+        else (reducecoeffh t' g', n div g'))
       else (t',n))))"
 
 lemma simp_num_pair_ci:
@@ -952,7 +952,7 @@
         have gpdg: "?g' dvd ?g" by simp
         have gpdd: "?g' dvd n" by simp
         have gpdgp: "?g' dvd ?g'" by simp
-        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
+        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
         have th2:"real_of_int ?g' * ?t = Inum bs ?t'" by simp
         from nnz g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')" by (simp add: simp_num_pair_def Let_def)
         also have "\<dots> = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))" by simp
@@ -993,7 +993,7 @@
         hence ?thesis using assms g1 g'1
           by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)}
       ultimately have ?thesis by blast }
-    ultimately have ?thesis by blast } 
+    ultimately have ?thesis by blast }
   ultimately show ?thesis by blast
 qed
 
@@ -1020,29 +1020,29 @@
   by (induct p) auto
 
 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else 
+  "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else
    if p = q then p else And p q)"
 lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
   by (cases "p=F \<or> q=F", simp_all add: conj_def) (cases p, simp_all)
 
 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
-  using conj_def by auto 
+  using conj_def by auto
 lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
-  using conj_def by auto 
+  using conj_def by auto
 
 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
+  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
        else if p=q then p else Or p q)"
 
 lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
   by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
-  using disj_def by auto 
+  using disj_def by auto
 lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
-  using disj_def by auto 
+  using disj_def by auto
 
 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p 
+  "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p
     else Imp p q)"
 lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
   by (cases "p=F \<or> q=T",simp_all add: imp_def)
@@ -1050,8 +1050,8 @@
   using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
 
 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
-  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else 
-       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else 
+  "iff p q \<equiv> (if (p = q) then T else if (p = not q \<or> not p = q) then F else
+       if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else
   Iff p q)"
 
 lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
@@ -1074,7 +1074,7 @@
 lemma rdvd_left1_int: "real_of_int \<lfloor>t\<rfloor> = t \<Longrightarrow> 1 rdvd t"
   by (simp add: rdvd_def,rule_tac x="\<lfloor>t\<rfloor>" in exI) simp
 
-lemma rdvd_reduce: 
+lemma rdvd_reduce:
   assumes gd:"g dvd d" and gc:"g dvd c" and gp: "g > 0"
   shows "real_of_int (d::int) rdvd real_of_int (c::int)*t = (real_of_int (d div g) rdvd real_of_int (c div g)*t)"
 proof
@@ -1095,13 +1095,13 @@
 qed
 
 definition simpdvd :: "int \<Rightarrow> num \<Rightarrow> (int \<times> num)" where
-  "simpdvd d t \<equiv> 
-   (let g = numgcd t in 
-      if g > 1 then (let g' = gcd d g in 
-        if g' = 1 then (d, t) 
-        else (d div g',reducecoeffh t g')) 
+  "simpdvd d t \<equiv>
+   (let g = numgcd t in
+      if g > 1 then (let g' = gcd d g in
+        if g' = 1 then (d, t)
+        else (d div g',reducecoeffh t g'))
       else (d, t))"
-lemma simpdvd: 
+lemma simpdvd:
   assumes tnz: "nozerocoeff t" and dnz: "d \<noteq> 0"
   shows "Ifm bs (Dvd (fst (simpdvd d t)) (snd (simpdvd d t))) = Ifm bs (Dvd d t)"
 proof-
@@ -1121,13 +1121,13 @@
       have gpdg: "?g' dvd ?g" by simp
       have gpdd: "?g' dvd d" by simp
       have gpdgp: "?g' dvd ?g'" by simp
-      from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
+      from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
       have th2:"real_of_int ?g' * ?t = Inum bs t" by simp
       from assms g1 g0 g'1
       have "Ifm bs (Dvd (fst (simpdvd d t)) (snd(simpdvd d t))) = Ifm bs (Dvd (d div ?g') ?tt)"
         by (simp add: simpdvd_def Let_def)
       also have "\<dots> = (real_of_int d rdvd (Inum bs t))"
-        using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]] 
+        using rdvd_reduce[OF gpdd gpdgp g'p, where t="?t", simplified div_self[OF gp0]]
           th2[symmetric] by simp
       finally have ?thesis by simp  }
     ultimately have ?thesis by blast
@@ -1141,7 +1141,7 @@
 | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
 | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
 | "simpfm (NOT p) = not (simpfm p)"
-| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
+| "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F
   | _ \<Rightarrow> Lt (reducecoeff a'))"
 | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le (reducecoeff a'))"
 | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt (reducecoeff a'))"
@@ -1151,7 +1151,7 @@
 | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
              else if (abs i = 1) \<and> check_int a then T
              else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in Dvd d t))"
-| "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
+| "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a)
              else if (abs i = 1) \<and> check_int a then F
              else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> (let (d,t) = simpdvd i a' in NDvd d t))"
 | "simpfm p = p"
@@ -1265,22 +1265,22 @@
   case (12 i a)  let ?sa = "simpnum a"   have sa: "Inum bs ?sa = Inum bs a" by simp
   have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
   {assume "i=0" hence ?case using "12.hyps" by (simp add: rdvd_left_0_eq Let_def)}
-  moreover 
-  {assume ai1: "abs i = 1" and ai: "check_int a" 
+  moreover
+  {assume ai1: "abs i = 1" and ai: "check_int a"
     hence "i=1 \<or> i= - 1" by arith
-    moreover {assume i1: "i = 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
+    moreover {assume i1: "i = 1"
+      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
       have ?case using i1 ai by simp }
-    moreover {assume i1: "i = - 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
+    moreover {assume i1: "i = - 1"
+      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
         rdvd_abs1[where d="- 1" and t="Inum bs a"]
       have ?case using i1 ai by simp }
     ultimately have ?case by blast}
-  moreover   
+  moreover
   {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
     {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
         by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
-    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" 
+    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
       hence th: "simpfm (Dvd i a) = Dvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond by (cases ?sa, auto simp add: Let_def split_def)
       from simpnum_nz have nz:"nozerocoeff ?sa" by simp
       from simpdvd [OF nz inz] th have ?case using sa by simp}
@@ -1290,23 +1290,23 @@
   case (13 i a)  let ?sa = "simpnum a"   have sa: "Inum bs ?sa = Inum bs a" by simp
   have "i=0 \<or> (abs i = 1 \<and> check_int a) \<or> (i\<noteq>0 \<and> ((abs i \<noteq> 1) \<or> (\<not> check_int a)))" by auto
   {assume "i=0" hence ?case using "13.hyps" by (simp add: rdvd_left_0_eq Let_def)}
-  moreover 
-  {assume ai1: "abs i = 1" and ai: "check_int a" 
+  moreover
+  {assume ai1: "abs i = 1" and ai: "check_int a"
     hence "i=1 \<or> i= - 1" by arith
-    moreover {assume i1: "i = 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
+    moreover {assume i1: "i = 1"
+      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
       have ?case using i1 ai by simp }
-    moreover {assume i1: "i = - 1" 
-      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]] 
+    moreover {assume i1: "i = - 1"
+      from rdvd_left1_int[OF check_int[OF ai, simplified isint_iff]]
         rdvd_abs1[where d="- 1" and t="Inum bs a"]
       have ?case using i1 ai by simp }
     ultimately have ?case by blast}
-  moreover   
+  moreover
   {assume inz: "i\<noteq>0" and cond: "(abs i \<noteq> 1) \<or> (\<not> check_int a)"
     {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
         by (cases "abs i = 1", auto simp add: int_rdvd_iff) }
-    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)" 
-      hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond 
+    moreover {assume H:"\<not> (\<exists> v. ?sa = C v)"
+      hence th: "simpfm (NDvd i a) = NDvd (fst (simpdvd i ?sa)) (snd (simpdvd i ?sa))" using inz cond
         by (cases ?sa, auto simp add: Let_def split_def)
       from simpnum_nz have nz:"nozerocoeff ?sa" by simp
       from simpdvd [OF nz inz] th have ?case using sa by simp}
@@ -1371,7 +1371,7 @@
   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = List.partition bound0 cjs
                    in conj (decr (list_conj yes)) (f (list_conj no)))"
 
-lemma CJNB_qe: 
+lemma CJNB_qe:
   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm bs ((CJNB qe p)) = Ifm bs (E p))"
 proof(clarify)
@@ -1383,15 +1383,15 @@
   let ?cno = "list_conj ?no"
   let ?cyes = "list_conj ?yes"
   have part: "List.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) 
+  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 (decr ?cyes )" by (simp add: decr_qf)
-  from conjuncts_qf[OF qfp] partition_set[OF part] 
+  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 )" 
+  with qe have cno_qf:"qfree (qe ?cno )"
     and noE: "Ifm bs (qe ?cno) = Ifm bs (E ?cno)" by blast+
-  from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
+  from cno_qf yes_qf have qf: "qfree (CJNB qe p)"
     by (simp add: CJNB_def Let_def split_def)
   {fix bs
     from conjuncts have "Ifm bs p = (\<forall>q\<in> set ?cjs. Ifm bs q)" by blast
@@ -1405,7 +1405,7 @@
     by (auto simp add: decr[OF yes_nb] simp del: partition_filter_conv)
   also have "\<dots> = (Ifm bs (conj (decr ?cyes) (qe ?cno)))"
     using qe[rule_format, OF no_qf] by auto
-  finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)" 
+  finally have "Ifm bs (E p) = Ifm bs (CJNB qe p)"
     by (simp add: Let_def CJNB_def split_def)
   with qf show "qfree (CJNB qe p) \<and> Ifm bs (CJNB qe p) = Ifm bs (E p)" by blast
 qed
@@ -1414,8 +1414,8 @@
   "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 (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. disj (qelim (NOT p) qe) (qelim q qe))"
 | "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
 | "qelim p = (\<lambda> y. simpfm p)"
@@ -1425,7 +1425,7 @@
 lemma qelim_ci:
   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
-  using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]] 
+  using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
   by (induct p rule: qelim.induct) (auto simp del: simpfm.simps)
 
 
@@ -1438,11 +1438,11 @@
 | "zsplit0 (CN n c a) = zsplit0 (Add (Mul c (Bound n)) a)"
 | "zsplit0 (CF c a b) = zsplit0 (Add (Mul c (Floor a)) b)"
 | "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
-| "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
+| "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ;
+                            (ib,b') =  zsplit0 b
                             in (ia+ib, Add a' b'))"
-| "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
-                            (ib,b') =  zsplit0 b 
+| "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ;
+                            (ib,b') =  zsplit0 b
                             in (ia-ib, Sub a' b'))"
 | "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"
 | "zsplit0 (Floor a) = (let (i',a') =  zsplit0 a in (i',Floor a'))"
@@ -1453,18 +1453,18 @@
   shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((real_of_int (x::int)) #bs) (CN 0 n a) = Inum (real_of_int x #bs) t) \<and> numbound0 a"
   (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
 proof(induct t rule: zsplit0.induct)
-  case (1 c n a) thus ?case by auto 
+  case (1 c n a) thus ?case by auto
 next
   case (2 m n a) thus ?case by (cases "m=0") auto
 next
   case (3 n i a n a') thus ?case by auto
-next 
+next
   case (4 c a b n a') thus ?case by auto
 next
   case (5 t n a)
   let ?nt = "fst (zsplit0 t)"
   let ?at = "snd (zsplit0 t)"
-  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using 5 
+  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using 5
     by (simp add: Let_def split_def)
   from abj 5 have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from th2[simplified] th[simplified] show ?case by simp
@@ -1474,15 +1474,15 @@
   let ?as = "snd (zsplit0 s)"
   let ?nt = "fst (zsplit0 t)"
   let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
+  have abjs: "zsplit0 s = (?ns,?as)" by simp
+  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp
   ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using 6
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
   from 6 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
   with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs 6  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
+  from th3[simplified] th2[simplified] th[simplified] show ?case
     by (simp add: distrib_right)
 next
   case (7 s t n a)
@@ -1490,15 +1490,15 @@
   let ?as = "snd (zsplit0 s)"
   let ?nt = "fst (zsplit0 t)"
   let ?at = "snd (zsplit0 t)"
-  have abjs: "zsplit0 s = (?ns,?as)" by simp 
-  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
+  have abjs: "zsplit0 s = (?ns,?as)" by simp
+  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp
   ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using 7
     by (simp add: Let_def split_def)
   from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
   from 7 have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (real_of_int x # bs) (CN 0 xa xb) = Inum (real_of_int x # bs) t \<and> numbound0 xb)" by blast (*FIXME*)
   with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
   from abjs 7 have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
-  from th3[simplified] th2[simplified] th[simplified] show ?case 
+  from th3[simplified] th2[simplified] th[simplified] show ?case
     by (simp add: left_diff_distrib)
 next
   case (8 i t n a)
@@ -1522,8 +1522,8 @@
   have "?I x (Floor t) = ?I x (Floor (CN 0 ?nt ?at))" using th2 by simp
   also have "\<dots> = real_of_int (floor ((real_of_int ?nt)* real_of_int(x) + ?I x ?at))" by simp
   also have "\<dots> = real_of_int (floor (?I x ?at + real_of_int (?nt* x)))" by (simp add: ac_simps)
-  also have "\<dots> = real_of_int (floor (?I x ?at) + (?nt* x))" 
-    using floor_add_of_int[of "?I x ?at" "?nt* x"] by simp 
+  also have "\<dots> = real_of_int (floor (?I x ?at) + (?nt* x))"
+    using floor_add_of_int[of "?I x ?at" "?nt* x"] by simp
   also have "\<dots> = real_of_int (?nt)*(real_of_int x) + real_of_int (floor (?I x ?at))" by (simp add: ac_simps)
   finally have "?I x (Floor t) = ?I x (CN 0 n a)" using th by simp
   with na show ?case by simp
@@ -1533,17 +1533,17 @@
   iszlfm :: "fm \<Rightarrow> real list \<Rightarrow> bool"   (* Linearity test for fm *)
   zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
 recdef iszlfm "measure size"
-  "iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" 
-  "iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)" 
+  "iszlfm (And p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)"
+  "iszlfm (Or p q) = (\<lambda> bs. iszlfm p bs \<and> iszlfm q bs)"
   "iszlfm (Eq  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm (NEq (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm (Lt  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm (Le  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm (Gt  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm (Ge  (CN 0 c e)) = (\<lambda> bs. c>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (Dvd i (CN 0 c e)) = 
+  "iszlfm (Dvd i (CN 0 c e)) =
                  (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
-  "iszlfm (NDvd i (CN 0 c e))= 
+  "iszlfm (NDvd i (CN 0 c e))=
                  (\<lambda> bs. c>0 \<and> i>0 \<and> numbound0 e \<and> isint e bs)"
   "iszlfm p = (\<lambda> bs. isatom p \<and> (bound0 p))"
 
@@ -1570,39 +1570,39 @@
   "zlfm (Or p q) = disj (zlfm p) (zlfm q)"
   "zlfm (Imp p q) = disj (zlfm (NOT p)) (zlfm q)"
   "zlfm (Iff p q) = disj (conj (zlfm p) (zlfm q)) (conj (zlfm (NOT p)) (zlfm (NOT q)))"
-  "zlfm (Lt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Lt r else 
-     if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) 
+  "zlfm (Lt a) = (let (c,r) = zsplit0 a in
+     if c=0 then Lt r else
+     if c>0 then Or (Lt (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r)))
      else Or (Gt (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
-  "zlfm (Le a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Le r else 
-     if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r))) 
+  "zlfm (Le a) = (let (c,r) = zsplit0 a in
+     if c=0 then Le r else
+     if c>0 then Or (Le (CN 0 c (Neg (Floor (Neg r))))) (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Lt (Add (Floor (Neg r)) r)))
      else Or (Ge (CN 0 (-c) (Floor(Neg r)))) (And (Eq(CN 0 (-c) (Floor(Neg r)))) (Lt (Add (Floor (Neg r)) r))))"
-  "zlfm (Gt a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Gt r else 
-     if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) 
+  "zlfm (Gt a) = (let (c,r) = zsplit0 a in
+     if c=0 then Gt r else
+     if c>0 then Or (Gt (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r)))
      else Or (Lt (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
-  "zlfm (Ge a) = (let (c,r) = zsplit0 a in 
-     if c=0 then Ge r else 
-     if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r))) 
+  "zlfm (Ge a) = (let (c,r) = zsplit0 a in
+     if c=0 then Ge r else
+     if c>0 then Or (Ge (CN 0 c (Floor r))) (And (Eq (CN 0 c (Floor r))) (Lt (Sub (Floor r) r)))
      else Or (Le (CN 0 (-c) (Neg (Floor r)))) (And (Eq(CN 0 (-c) (Neg (Floor r)))) (Lt (Sub (Floor r) r))))"
-  "zlfm (Eq a) = (let (c,r) = zsplit0 a in 
-              if c=0 then Eq r else 
+  "zlfm (Eq a) = (let (c,r) = zsplit0 a in
+              if c=0 then Eq r else
       if c>0 then (And (Eq (CN 0 c (Neg (Floor (Neg r))))) (Eq (Add (Floor (Neg r)) r)))
       else (And (Eq (CN 0 (-c) (Floor (Neg r)))) (Eq (Add (Floor (Neg r)) r))))"
-  "zlfm (NEq a) = (let (c,r) = zsplit0 a in 
-              if c=0 then NEq r else 
+  "zlfm (NEq a) = (let (c,r) = zsplit0 a in
+              if c=0 then NEq r else
       if c>0 then (Or (NEq (CN 0 c (Neg (Floor (Neg r))))) (NEq (Add (Floor (Neg r)) r)))
       else (Or (NEq (CN 0 (-c) (Floor (Neg r)))) (NEq (Add (Floor (Neg r)) r))))"
-  "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) 
-  else (let (c,r) = zsplit0 a in 
-              if c=0 then Dvd (abs i) r else 
-      if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r))) 
+  "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a)
+  else (let (c,r) = zsplit0 a in
+              if c=0 then Dvd (abs i) r else
+      if c>0 then And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 c (Floor r)))
       else And (Eq (Sub (Floor r) r)) (Dvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
-  "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) 
-  else (let (c,r) = zsplit0 a in 
-              if c=0 then NDvd (abs i) r else 
-      if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r))) 
+  "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a)
+  else (let (c,r) = zsplit0 a in
+              if c=0 then NDvd (abs i) r else
+      if c>0 then Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 c (Floor r)))
       else Or (NEq (Sub (Floor r) r)) (NDvd (abs i) (CN 0 (-c) (Neg (Floor r))))))"
   "zlfm (NOT (And p q)) = disj (zlfm (NOT p)) (zlfm (NOT q))"
   "zlfm (NOT (Or p q)) = conj (zlfm (NOT p)) (zlfm (NOT q))"
@@ -1621,56 +1621,56 @@
   "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)"
   "zlfm p = p" (hints simp add: fmsize_pos)
 
-lemma split_int_less_real: 
+lemma split_int_less_real:
   "(real_of_int (a::int) < b) = (a < floor b \<or> (a = floor b \<and> real_of_int (floor b) < b))"
 proof( auto)
   assume alb: "real_of_int a < b" and agb: "\<not> a < floor b"
   from agb have "floor b \<le> a" by simp hence th: "b < real_of_int a + 1" by (simp only: floor_le_iff)
-  from floor_eq[OF alb th] show "a= floor b" by simp 
+  from floor_eq[OF alb th] show "a= floor b" by simp
 next
   assume alb: "a < floor b"
   hence "real_of_int a < real_of_int (floor b)" by simp
-  moreover have "real_of_int (floor b) \<le> b" by simp ultimately show  "real_of_int a < b" by arith 
+  moreover have "real_of_int (floor b) \<le> b" by simp ultimately show  "real_of_int a < b" by arith
 qed
 
-lemma split_int_less_real': 
+lemma split_int_less_real':
   "(real_of_int (a::int) + b < 0) = (real_of_int a - real_of_int (floor(-b)) < 0 \<or> (real_of_int a - real_of_int (floor (-b)) = 0 \<and> real_of_int (floor (-b)) + b < 0))"
-proof- 
+proof-
   have "(real_of_int a + b <0) = (real_of_int a < -b)" by arith
-  with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith  
+  with split_int_less_real[where a="a" and b="-b"] show ?thesis by arith
 qed
 
-lemma split_int_gt_real': 
+lemma split_int_gt_real':
   "(real_of_int (a::int) + b > 0) = (real_of_int a + real_of_int (floor b) > 0 \<or> (real_of_int a + real_of_int (floor b) = 0 \<and> real_of_int (floor b) - b < 0))"
-proof- 
+proof-
   have th: "(real_of_int a + b >0) = (real_of_int (-a) + (-b)< 0)" by arith
-  show ?thesis using myless[of _ "real_of_int (floor b)"] 
-    by (simp only:th split_int_less_real'[where a="-a" and b="-b"]) 
+  show ?thesis using myless[of _ "real_of_int (floor b)"]
+    by (simp only:th split_int_less_real'[where a="-a" and b="-b"])
     (simp add: algebra_simps,arith)
 qed
 
-lemma split_int_le_real: 
+lemma split_int_le_real:
   "(real_of_int (a::int) \<le> b) = (a \<le> floor b \<or> (a = floor b \<and> real_of_int (floor b) < b))"
 proof( auto)
   assume alb: "real_of_int a \<le> b" and agb: "\<not> a \<le> floor b"
-  from alb have "floor (real_of_int a) \<le> floor b " by (simp only: floor_mono) 
+  from alb have "floor (real_of_int a) \<le> floor b " by (simp only: floor_mono)
   hence "a \<le> floor b" by simp with agb show "False" by simp
 next
   assume alb: "a \<le> floor b"
   hence "real_of_int a \<le> real_of_int (floor b)" by (simp only: floor_mono)
-  also have "\<dots>\<le> b" by simp  finally show  "real_of_int a \<le> b" . 
+  also have "\<dots>\<le> b" by simp  finally show  "real_of_int a \<le> b" .
 qed
 
-lemma split_int_le_real': 
+lemma split_int_le_real':
   "(real_of_int (a::int) + b \<le> 0) = (real_of_int a - real_of_int (floor(-b)) \<le> 0 \<or> (real_of_int a - real_of_int (floor (-b)) = 0 \<and> real_of_int (floor (-b)) + b < 0))"
-proof- 
+proof-
   have "(real_of_int a + b \<le>0) = (real_of_int a \<le> -b)" by arith
-  with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith  
+  with split_int_le_real[where a="a" and b="-b"] show ?thesis by arith
 qed
 
-lemma split_int_ge_real': 
+lemma split_int_ge_real':
   "(real_of_int (a::int) + b \<ge> 0) = (real_of_int a + real_of_int (floor b) \<ge> 0 \<or> (real_of_int a + real_of_int (floor b) = 0 \<and> real_of_int (floor b) - b < 0))"
-proof- 
+proof-
   have th: "(real_of_int a + b \<ge>0) = (real_of_int (-a) + (-b) \<le> 0)" by arith
   show ?thesis by (simp only: th split_int_le_real'[where a="-a" and b="-b"])
     (simp add: algebra_simps ,arith)
@@ -1691,25 +1691,25 @@
   (is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
 using qfp
 proof(induct p rule: zlfm.induct)
-  case (5 a) 
+  case (5 a)
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def,rename_tac nat a b,case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Lt a)))" apply (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) by (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Lt a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Lt a) = (real_of_int (?c * i) + (?N ?r) < 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Lt a)))" by (simp only: split_int_less_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
@@ -1720,46 +1720,46 @@
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat",simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Le a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Le a) = (real_of_int (?c * i) + (?N ?r) \<le> 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Le a)))" by (simp only: split_int_le_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
     finally have ?case using l by simp}
   ultimately show ?case by blast
 next
-  case (7 a) 
+  case (7 a)
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Gt a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Gt a) = (real_of_int (?c * i) + (?N ?r) > 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Gt a)))" by (simp only: split_int_gt_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
@@ -1770,21 +1770,21 @@
    let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia cp cnz Let_def split_def)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Ge a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Ge a) = (real_of_int (?c * i) + (?N ?r) \<ge> 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Ge a)))" by (simp only: split_int_ge_real'[where a="?c*i" and b="?N ?r"]) (simp add: Ia Let_def split_def ac_simps, arith)
@@ -1795,21 +1795,21 @@
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (Eq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (Eq a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (Eq a) = (real_of_int (?c * i) + (?N ?r) = 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (Eq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith)
@@ -1820,21 +1820,21 @@
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "?c = 0 \<or> (?c >0 \<and> ?c\<noteq>0) \<or> (?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"] 
+  {assume "?c=0" hence ?case using zsplit0_I[OF spl, where x="i" and bs="bs"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
     also have "\<dots> = (?I (?l (NEq a)))" using cp cnz  by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult)
     finally have ?case using l by simp}
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" hence l: "?L (?l (NEq a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
     have "?I (NEq a) = (real_of_int (?c * i) + (?N ?r) \<noteq> 0)" using Ia by (simp add: Let_def split_def)
     also from cn cnz have "\<dots> = (?I (?l (NEq a)))" by (simp only: split_int_eq_real'[where a="?c*i" and b="?N ?r"]) (simp add: Let_def split_def Ia of_int_mult[symmetric] del: of_int_mult,arith)
@@ -1845,44 +1845,44 @@
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  { assume j: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) 
+  { assume j: "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def)
     hence ?case using 11 j by (simp del: zlfm.simps add: rdvd_left_0_eq)}
   moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
+  {assume "?c=0" and "j\<noteq>0" hence ?case
       using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" 
+    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))"
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))" 
+    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))"
       by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
-       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))" 
+    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and>
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))"
       by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
-    also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz  
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
+    also have "\<dots> = (?I (?l (Dvd j a)))" using cp cnz jnz
+      by (simp add: Let_def split_def int_rdvd_iff[symmetric]
         del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz  by simp }
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))" 
+    have "?I (Dvd j a) = (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r)))"
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))" 
+    also have "\<dots> = (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r))"
       by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
-       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))" 
+    also have "\<dots> = ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and>
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r))))"
       by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (Dvd j a)))" using cn cnz jnz
       using rdvd_minus [where d="abs j" and t="real_of_int (?c*i + floor (?N ?r))", simplified, symmetric]
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
+      by (simp add: Let_def split_def int_rdvd_iff[symmetric]
         del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz by blast }
   ultimately show ?case by blast
@@ -1891,44 +1891,44 @@
   let ?c = "fst (zsplit0 a)"
   let ?r = "snd (zsplit0 a)"
   have spl: "zsplit0 a = (?c,?r)" by simp
-  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
-  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
+  from zsplit0_I[OF spl, where x="i" and bs="bs"]
+  have Ia:"Inum (real_of_int i # bs) a = Inum (real_of_int i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto
   let ?N = "\<lambda> t. Inum (real_of_int i#bs) t"
   have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0 \<and> ?c\<noteq>0) \<or> (j\<noteq> 0 \<and> ?c<0 \<and> ?c\<noteq>0)" by arith
   moreover
-  {assume j: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) 
+  {assume j: "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def)
     hence ?case using 12 j by (simp del: zlfm.simps add: rdvd_left_0_eq)}
   moreover
-  {assume "?c=0" and "j\<noteq>0" hence ?case 
+  {assume "?c=0" and "j\<noteq>0" hence ?case
       using zsplit0_I[OF spl, where x="i" and bs="bs"] rdvd_abs1[where d="j"]
       by (cases "?r", simp_all add: Let_def split_def, rename_tac nat a b, case_tac "nat", simp_all)}
   moreover
-  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
+  {assume cp: "?c > 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" 
+    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))"
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))" 
+    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))"
       by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
-       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))" 
+    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and>
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))"
       by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
-    also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz  
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
+    also have "\<dots> = (?I (?l (NDvd j a)))" using cp cnz jnz
+      by (simp add: Let_def split_def int_rdvd_iff[symmetric]
         del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz  by simp }
   moreover
-  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))" 
+  {assume cn: "?c < 0" and cnz: "?c\<noteq>0" and jnz: "j\<noteq>0" hence l: "?L (?l (NDvd j a))"
       by (simp add: nb Let_def split_def isint_Floor isint_neg)
-    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))" 
+    have "?I (NDvd j a) = (\<not> (real_of_int j rdvd (real_of_int (?c * i) + (?N ?r))))"
       using Ia by (simp add: Let_def split_def)
-    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))" 
+    also have "\<dots> = (\<not> (real_of_int (abs j) rdvd real_of_int (?c*i) + (?N ?r)))"
       by (simp only: rdvd_abs1[where d="j" and t="real_of_int (?c*i) + ?N ?r", symmetric]) simp
-    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and> 
-       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))" 
+    also have "\<dots> = (\<not> ((abs j) dvd (floor ((?N ?r) + real_of_int (?c*i))) \<and>
+       (real_of_int (floor ((?N ?r) + real_of_int (?c*i))) = (real_of_int (?c*i) + (?N ?r)))))"
       by(simp only: int_rdvd_real[where i="abs j" and x="real_of_int (?c*i) + (?N ?r)"]) (simp only: ac_simps)
     also have "\<dots> = (?I (?l (NDvd j a)))" using cn cnz jnz
       using rdvd_minus [where d="abs j" and t="real_of_int (?c*i + floor (?N ?r))", simplified, symmetric]
-      by (simp add: Let_def split_def int_rdvd_iff[symmetric]  
+      by (simp add: Let_def split_def int_rdvd_iff[symmetric]
         del: of_int_mult) (auto simp add: ac_simps)
     finally have ?case using l jnz by blast }
   ultimately show ?case by blast
@@ -1940,8 +1940,8 @@
        \<open>d_\<delta>\<close> checks if a given l divides all the ds above\<close>
 
 fun minusinf:: "fm \<Rightarrow> fm" where
-  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
-| "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
+  "minusinf (And p q) = conj (minusinf p) (minusinf q)"
+| "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
 | "minusinf (Eq  (CN 0 c e)) = F"
 | "minusinf (NEq (CN 0 c e)) = T"
 | "minusinf (Lt  (CN 0 c e)) = T"
@@ -1954,8 +1954,8 @@
   by (induct p rule: minusinf.induct, auto)
 
 fun plusinf:: "fm \<Rightarrow> fm" where
-  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
-| "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
+  "plusinf (And p q) = conj (plusinf p) (plusinf q)"
+| "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
 | "plusinf (Eq  (CN 0 c e)) = F"
 | "plusinf (NEq (CN 0 c e)) = T"
 | "plusinf (Lt  (CN 0 c e)) = F"
@@ -1965,20 +1965,20 @@
 | "plusinf p = p"
 
 fun \<delta> :: "fm \<Rightarrow> int" where
-  "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)" 
-| "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)" 
+  "\<delta> (And p q) = lcm (\<delta> p) (\<delta> q)"
+| "\<delta> (Or p q) = lcm (\<delta> p) (\<delta> q)"
 | "\<delta> (Dvd i (CN 0 c e)) = i"
 | "\<delta> (NDvd i (CN 0 c e)) = i"
 | "\<delta> p = 1"
 
 fun d_\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" where
-  "d_\<delta> (And p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)" 
-| "d_\<delta> (Or p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)" 
+  "d_\<delta> (And p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)"
+| "d_\<delta> (Or p q) = (\<lambda> d. d_\<delta> p d \<and> d_\<delta> q d)"
 | "d_\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
 | "d_\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
 | "d_\<delta> p = (\<lambda> d. True)"
 
-lemma delta_mono: 
+lemma delta_mono:
   assumes lin: "iszlfm p bs"
   and d: "d dvd d'"
   and ad: "d_\<delta> p d"
@@ -1996,17 +1996,17 @@
   shows "d_\<delta> p (\<delta> p) \<and> \<delta> p >0"
 using lin
 proof (induct p rule: iszlfm.induct)
-  case (1 p q) 
+  case (1 p q)
   let ?d = "\<delta> (And p q)"
   from 1 lcm_pos_int have dp: "?d >0" by simp
-  have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp 
-  hence th: "d_\<delta> p ?d" 
+  have d1: "\<delta> p dvd \<delta> (And p q)" using 1 by simp
+  hence th: "d_\<delta> p ?d"
     using delta_mono 1 by (simp only: iszlfm.simps) blast
-  have "\<delta> q dvd \<delta> (And p q)" using 1 by simp 
+  have "\<delta> q dvd \<delta> (And p q)" using 1 by simp
   hence th': "d_\<delta> q ?d" using delta_mono 1 by (simp only: iszlfm.simps) blast
-  from th th' dp show ?case by simp 
+  from th th' dp show ?case by simp
 next
-  case (2 p q)  
+  case (2 p q)
   let ?d = "\<delta> (And p q)"
   from 2 lcm_pos_int have dp: "?d >0" by simp
   have "\<delta> p dvd \<delta> (And p q)" using 2 by simp
@@ -2041,101 +2041,101 @@
   from z1_def z2_def have "\<forall> x < ?z. ?I x (?M (Or f g)) = ?I x (Or f g)" by simp
   thus ?case by blast
 next
-  case (3 c e) 
+  case (3 c e)
   then have "c > 0" by simp
   hence rcpos: "real_of_int c > 0" by simp
   from 3 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Eq (CN 0 c e))) = ?I x (Eq (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int  x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
     hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" 
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"]  by simp
   qed
   thus ?case by blast
 next
-  case (4 c e) 
+  case (4 c e)
   then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 4 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (NEq (CN 0 c e))) = ?I x (NEq (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
     hence "real_of_int c * real_of_int x + Inum (y # bs) e \<noteq> 0"using rcpos  by simp
-    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0" 
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<noteq> 0"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"]  by simp
   qed
   thus ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 5 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Lt (CN 0 c e))) = ?I x (Lt (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0" 
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e < 0"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
-  case (6 c e) 
+  case (6 c e)
   then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 6 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Le (CN 0 c e))) = ?I x (Le (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0" 
+    thus "real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<le> 0"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
-  case (7 c e) 
+  case (7 c e)
   then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 7 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Gt (CN 0 c e))) = ?I x (Gt (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "\<not> (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e>0)" 
+    thus "\<not> (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e>0)"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
 next
-  case (8 c e) 
+  case (8 c e)
   then have "c > 0" by simp hence rcpos: "real_of_int c > 0" by simp
   from 8 have nbe: "numbound0 e" by simp
   fix y
   have "\<forall> x < (floor (- (Inum (y#bs) e) / (real_of_int c))). ?I x (?M (Ge (CN 0 c e))) = ?I x (Ge (CN 0 c e))"
-  proof (simp add: less_floor_iff , rule allI, rule impI) 
+  proof (simp add: less_floor_iff , rule allI, rule impI)
     fix x :: int
     assume A: "real_of_int x + 1 \<le> - (Inum (y # bs) e / real_of_int c)"
     hence th1:"real_of_int x < - (Inum (y # bs) e / real_of_int c)" by simp
     with rcpos  have "(real_of_int c)*(real_of_int x) < (real_of_int c)*(- (Inum (y # bs) e / real_of_int c))"
       by (simp only: mult_strict_left_mono [OF th1 rcpos])
-    thus "\<not> real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0" 
+    thus "\<not> real_of_int c * real_of_int x + Inum (real_of_int x # bs) e \<ge> 0"
       using numbound0_I[OF nbe, where b="y" and bs="bs" and b'="real_of_int x"] rcpos by simp
   qed
   thus ?case by blast
@@ -2145,24 +2145,24 @@
   assumes d: "d_\<delta> p d" and linp: "iszlfm p (a # bs)"
   shows "Ifm ((real_of_int(x - k*d))#bs) (minusinf p) = Ifm (real_of_int x #bs) (minusinf p)"
 using linp d
-proof(induct p rule: iszlfm.induct) 
+proof(induct p rule: iszlfm.induct)
   case (9 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
     hence "\<exists> k. d=i*k" by (simp add: dvd_def)
     then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
+    show ?case
     proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI)
-      assume 
+      assume
         "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e"
       (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
       hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" 
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))"
         by (simp add: algebra_simps di_def)
       hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))"
         by (simp add: algebra_simps)
       hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast
       thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp
     next
-      assume 
+      assume
         "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
       hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)" by (simp add: rdvd_def)
       hence "\<exists> (l::int). ?rc*?rx - real_of_int c * (real_of_int k * real_of_int d) +?e = ?ri * (real_of_int l) - real_of_int c * (real_of_int k * real_of_int d)" by simp
@@ -2176,20 +2176,20 @@
   case (10 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
     hence "\<exists> k. d=i*k" by (simp add: dvd_def)
     then obtain "di" where di_def: "d=i*di" by blast
-    show ?case 
+    show ?case
     proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="real_of_int x - real_of_int k * real_of_int d" and b'="real_of_int x"] right_diff_distrib, rule iffI)
-      assume 
+      assume
         "real_of_int i rdvd real_of_int c * real_of_int x - real_of_int c * (real_of_int k * real_of_int d) + Inum (real_of_int x # bs) e"
       (is "?ri rdvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri rdvd ?rt")
       hence "\<exists> (l::int). ?rt = ?ri * (real_of_int l)" by (simp add: rdvd_def)
-      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))" 
+      hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int l)+?rc*(?rk * (real_of_int i) * (real_of_int di))"
         by (simp add: algebra_simps di_def)
       hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri*(real_of_int (l + c*k*di))"
         by (simp add: algebra_simps)
       hence "\<exists> (l::int). ?rc*?rx+ ?I x e = ?ri* (real_of_int l)" by blast
       thus "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" using rdvd_def by simp
     next
-      assume 
+      assume
         "real_of_int i rdvd real_of_int c * real_of_int x + Inum (real_of_int x # bs) e" (is "?ri rdvd ?rc*?rx+?e")
       hence "\<exists> (l::int). ?rc*?rx+?e = ?ri * (real_of_int l)"
         by (simp add: rdvd_def)
@@ -2221,7 +2221,7 @@
 
 lemma minusinf_bex:
   assumes lin: "iszlfm p (real_of_int (a::int) #bs)"
-  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)) = 
+  shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (minusinf p)) =
          (\<exists> (x::int)\<in> {1..\<delta> p}. Ifm (real_of_int x#bs) (minusinf p))"
   (is "(\<exists> x. ?P x) = _")
 proof-
@@ -2234,7 +2234,7 @@
 
 lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)" by auto
 
-consts 
+consts
   a_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *)
   d_\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*)
   \<zeta>  :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*)
@@ -2242,8 +2242,8 @@
   \<alpha> :: "fm \<Rightarrow> num list"
 
 recdef a_\<beta> "measure size"
-  "a_\<beta> (And p q) = (\<lambda> k. And (a_\<beta> p k) (a_\<beta> q k))" 
-  "a_\<beta> (Or p q) = (\<lambda> k. Or (a_\<beta> p k) (a_\<beta> q k))" 
+  "a_\<beta> (And p q) = (\<lambda> k. And (a_\<beta> p k) (a_\<beta> q k))"
+  "a_\<beta> (Or p q) = (\<lambda> k. Or (a_\<beta> p k) (a_\<beta> q k))"
   "a_\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))"
   "a_\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))"
   "a_\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))"
@@ -2255,8 +2255,8 @@
   "a_\<beta> p = (\<lambda> k. p)"
 
 recdef d_\<beta> "measure size"
-  "d_\<beta> (And p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))" 
-  "d_\<beta> (Or p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))" 
+  "d_\<beta> (And p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))"
+  "d_\<beta> (Or p q) = (\<lambda> k. (d_\<beta> p k) \<and> (d_\<beta> q k))"
   "d_\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. c dvd k)"
   "d_\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)"
   "d_\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
@@ -2268,8 +2268,8 @@
   "d_\<beta> p = (\<lambda> k. True)"
 
 recdef \<zeta> "measure size"
-  "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)" 
-  "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)" 
+  "\<zeta> (And p q) = lcm (\<zeta> p) (\<zeta> q)"
+  "\<zeta> (Or p q) = lcm (\<zeta> p) (\<zeta> q)"
   "\<zeta> (Eq  (CN 0 c e)) = c"
   "\<zeta> (NEq (CN 0 c e)) = c"
   "\<zeta> (Lt  (CN 0 c e)) = c"
@@ -2281,8 +2281,8 @@
   "\<zeta> p = 1"
 
 recdef \<beta> "measure size"
-  "\<beta> (And p q) = (\<beta> p @ \<beta> q)" 
-  "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" 
+  "\<beta> (And p q) = (\<beta> p @ \<beta> q)"
+  "\<beta> (Or p q) = (\<beta> p @ \<beta> q)"
   "\<beta> (Eq  (CN 0 c e)) = [Sub (C (- 1)) e]"
   "\<beta> (NEq (CN 0 c e)) = [Neg e]"
   "\<beta> (Lt  (CN 0 c e)) = []"
@@ -2292,8 +2292,8 @@
   "\<beta> p = []"
 
 recdef \<alpha> "measure size"
-  "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" 
-  "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" 
+  "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)"
+  "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)"
   "\<alpha> (Eq  (CN 0 c e)) = [Add (C (- 1)) e]"
   "\<alpha> (NEq (CN 0 c e)) = [e]"
   "\<alpha> (Lt  (CN 0 c e)) = [e]"
@@ -2303,8 +2303,8 @@
   "\<alpha> p = []"
 consts mirror :: "fm \<Rightarrow> fm"
 recdef mirror "measure size"
-  "mirror (And p q) = And (mirror p) (mirror q)" 
-  "mirror (Or p q) = Or (mirror p) (mirror q)" 
+  "mirror (And p q) = And (mirror p) (mirror q)"
+  "mirror (Or p q) = Or (mirror p) (mirror q)"
   "mirror (Eq  (CN 0 c e)) = Eq (CN 0 c (Neg e))"
   "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))"
   "mirror (Lt  (CN 0 c e)) = Gt (CN 0 c (Neg e))"
@@ -2320,9 +2320,9 @@
   shows "(Inum (real_of_int (i::int)#bs)) ` set (\<alpha> p) = (Inum (real_of_int i#bs)) ` set (\<beta> (mirror p))"
   using lp by (induct p rule: mirror.induct) auto
 
-lemma mirror: 
+lemma mirror:
   assumes lp: "iszlfm p (a#bs)"
-  shows "Ifm (real_of_int (x::int)#bs) (mirror p) = Ifm (real_of_int (- x)#bs) p" 
+  shows "Ifm (real_of_int (x::int)#bs) (mirror p) = Ifm (real_of_int (- x)#bs) p"
   using lp
 proof(induct p rule: iszlfm.induct)
   case (9 j c e)
@@ -2345,7 +2345,7 @@
 lemma mirror_l: "iszlfm p (a#bs) \<Longrightarrow> iszlfm (mirror p) (a#bs)"
   by (induct p rule: mirror.induct) (auto simp add: isint_neg)
 
-lemma mirror_d_\<beta>: "iszlfm p (a#bs) \<and> d_\<beta> p 1 
+lemma mirror_d_\<beta>: "iszlfm p (a#bs) \<and> d_\<beta> p 1
   \<Longrightarrow> iszlfm (mirror p) (a#bs) \<and> d_\<beta> (mirror p) 1"
   by (induct p rule: mirror.induct) (auto simp add: isint_neg)
 
@@ -2353,7 +2353,7 @@
   by (induct p rule: mirror.induct) auto
 
 
-lemma mirror_ex: 
+lemma mirror_ex:
   assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
   shows "(\<exists> (x::int). Ifm (real_of_int x#bs) (mirror p)) = (\<exists> (x::int). Ifm (real_of_int x#bs) p)"
   (is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
@@ -2361,7 +2361,7 @@
   fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
   thus "\<exists> x. ?I x p" by blast
 next
-  fix x assume "?I x p" hence "?I (- x) ?mp" 
+  fix x assume "?I x p" hence "?I (- x) ?mp"
     using mirror[OF lp, where x="- x", symmetric] by auto
   thus "\<exists> x. ?I x ?mp" by blast
 qed
@@ -2370,7 +2370,7 @@
   shows "\<forall> b\<in> set (\<beta> p). numbound0 b"
   using lp by (induct p rule: \<beta>.induct,auto)
 
-lemma d_\<beta>_mono: 
+lemma d_\<beta>_mono:
   assumes linp: "iszlfm p (a #bs)"
   and dr: "d_\<beta> p l"
   and d: "l dvd l'"
@@ -2383,7 +2383,7 @@
 using lp
 by(induct p rule: \<alpha>.induct, auto simp add: isint_add isint_c)
 
-lemma \<zeta>: 
+lemma \<zeta>:
   assumes linp: "iszlfm p (a #bs)"
   shows "\<zeta> p > 0 \<and> d_\<beta> p (\<zeta> p)"
 using linp
@@ -2391,15 +2391,15 @@
   case (1 p q)
   then  have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
   from 1 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
-  from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] 
-    d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] 
+  from 1 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+    d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
     dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
 next
   case (2 p q)
   then have dl1: "\<zeta> p dvd lcm (\<zeta> p) (\<zeta> q)" by simp
   from 2 have dl2: "\<zeta> q dvd lcm (\<zeta> p) (\<zeta> q)" by simp
-  from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"] 
-    d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"] 
+  from 2 d_\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="lcm (\<zeta> p) (\<zeta> q)"]
+    d_\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="lcm (\<zeta> p) (\<zeta> q)"]
     dl1 dl2 show ?case by (auto simp add: lcm_pos_int)
 qed (auto simp add: lcm_pos_int)
 
@@ -2412,10 +2412,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e < 0)"
@@ -2430,10 +2430,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<le> 0)"
@@ -2448,10 +2448,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e > 0)"
@@ -2466,10 +2466,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<ge> 0)"
@@ -2484,10 +2484,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = 0)"
@@ -2502,10 +2502,10 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> (0::real)) =
           (real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e \<noteq> 0)"
@@ -2520,27 +2520,27 @@
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)"  by simp
     also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps)
     also fix k have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k = 0)"
     using zero_le_mult_iff [where a="real_of_int (l div c)" and b="real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k"] ldcp by simp
   also have "\<dots> = (\<exists> (k::int). real_of_int c * real_of_int x + Inum (real_of_int x # bs) e = real_of_int j * real_of_int k)" by simp
-  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp 
+  finally show ?case using numbound0_I[OF be,where b="real_of_int (l * x)" and b'="real_of_int x" and bs="bs"] rdvd_def  be  isint_Mul[OF ei] mult_strict_mono[OF ldcp jp ldcp ] by simp
 next
   case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and ei:"isint e (a#bs)" and jp: "j > 0" and d': "c dvd l" by simp+
     from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
     from cp have cnz: "c \<noteq> 0" by simp
     have "c div c\<le> l div c"
       by (simp add: zdiv_mono1[OF clel cp])
-    then have ldcp:"0 < l div c" 
+    then have ldcp:"0 < l div c"
       by (simp add: div_self[OF cnz])
     have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
-    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric] 
+    hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
       by simp
     hence "(\<exists> (k::int). real_of_int l * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k) = (\<exists> (k::int). real_of_int (c * (l div c)) * real_of_int x + real_of_int (l div c) * Inum (real_of_int x # bs) e = (real_of_int (l div c) * real_of_int j) * real_of_int k)"  by simp
     also have "\<dots> = (\<exists> (k::int). real_of_int (l div c) * (real_of_int c * real_of_int x + Inum (real_of_int x # bs) e - real_of_int j * real_of_int k) = real_of_int (l div c)*0)" by (simp add: algebra_simps)
@@ -2557,7 +2557,7 @@
   have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
     using unity_coeff_ex[where l="l" and P="?P", simplified] by simp
   also have "\<dots> = (\<exists> (x::int). ?P' x)" using a_\<beta>[OF linp d lp] by simp
-  finally show ?thesis  . 
+  finally show ?thesis  .
 qed
 
 lemma \<beta>:
@@ -2584,40 +2584,40 @@
   from ie1 have ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="a#bs"]
       numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]
     by (simp add: isint_iff)
-    {assume "real_of_int (x-d) +?e > 0" hence ?case using c1 
+    {assume "real_of_int (x-d) +?e > 0" hence ?case using c1
       numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"]
         by (simp del: of_int_minus)}
     moreover
-    {assume H: "\<not> real_of_int (x-d) + ?e > 0" 
+    {assume H: "\<not> real_of_int (x-d) + ?e > 0"
       let ?v="Neg e"
       have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
-      from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e + real_of_int j)" by auto 
+      from 7(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]]
+      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e + real_of_int j)" by auto
       from H p have "real_of_int x + ?e > 0 \<and> real_of_int x + ?e \<le> real_of_int d" by (simp add: c1)
       hence "real_of_int (x + floor ?e) > real_of_int (0::int) \<and> real_of_int (x + floor ?e) \<le> real_of_int d"
         using ie by simp
       hence "x + floor ?e \<ge> 1 \<and> x + floor ?e \<le> d"  by simp
       hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e" by simp
-      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = real_of_int (- floor ?e + j)" by force 
-      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = - ?e + real_of_int j" 
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = real_of_int (- floor ?e + j)" by force
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x = - ?e + real_of_int j"
         by (simp add: ie[simplified isint_iff])
       with nob have ?case by auto}
     ultimately show ?case by blast
 next
-  case (8 c e) hence p: "Ifm (real_of_int x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" 
+  case (8 c e) hence p: "Ifm (real_of_int x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e"
     and ie1:"isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
     let ?e = "Inum (real_of_int x # bs) e"
     from ie1 have ie: "real_of_int (floor ?e) = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"]
       by (simp add: isint_iff)
-    {assume "real_of_int (x-d) +?e \<ge> 0" hence ?case using  c1 
+    {assume "real_of_int (x-d) +?e \<ge> 0" hence ?case using  c1
       numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"]
         by (simp del: of_int_minus)}
     moreover
-    {assume H: "\<not> real_of_int (x-d) + ?e \<ge> 0" 
+    {assume H: "\<not> real_of_int (x-d) + ?e \<ge> 0"
       let ?v="Sub (C (- 1)) e"
       have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
-      from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]] 
-      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e - 1 + real_of_int j)" by auto 
+      from 8(5)[simplified simp_thms Inum.simps \<beta>.simps list.set bex_simps numbound0_I[OF bn,where b="a" and b'="real_of_int x" and bs="bs"]]
+      have nob: "\<not> (\<exists> j\<in> {1 ..d}. real_of_int x =  - ?e - 1 + real_of_int j)" by auto
       from H p have "real_of_int x + ?e \<ge> 0 \<and> real_of_int x + ?e < real_of_int d" by (simp add: c1)
       hence "real_of_int (x + floor ?e) \<ge> real_of_int (0::int) \<and> real_of_int (x + floor ?e) < real_of_int d"
         using ie by simp
@@ -2625,12 +2625,12 @@
       hence "\<exists> (j::int) \<in> {1 .. d}. j = x + floor ?e + 1" by simp
       hence "\<exists> (j::int) \<in> {1 .. d}. x= - floor ?e - 1 + j" by (simp add: algebra_simps)
       hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= real_of_int (- floor ?e - 1 + j)" by presburger
-      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= - ?e - 1 + real_of_int j" 
+      hence "\<exists> (j::int) \<in> {1 .. d}. real_of_int x= - ?e - 1 + real_of_int j"
         by (simp add: ie[simplified isint_iff])
       with nob have ?case by simp }
     ultimately show ?case by blast
 next
-  case (3 c e) hence p: "Ifm (real_of_int x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" 
+  case (3 c e) hence p: "Ifm (real_of_int x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1"
     and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
     let ?e = "Inum (real_of_int x # bs) e"
     let ?v="(Sub (C (- 1)) e)"
@@ -2639,12 +2639,12 @@
       by simp (erule ballE[where x="1"],
         simp_all add:algebra_simps numbound0_I[OF bn,where b="real_of_int x"and b'="a"and bs="bs"])
 next
-  case (4 c e)hence p: "Ifm (real_of_int x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" 
+  case (4 c e)hence p: "Ifm (real_of_int x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1"
     and bn:"numbound0 e" and ie1: "isint e (a #bs)" using dvd1_eq1[where x="c"] by simp+
     let ?e = "Inum (real_of_int x # bs) e"
     let ?v="Neg e"
     have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
-    {assume "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e \<noteq> 0" 
+    {assume "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e \<noteq> 0"
       hence ?case by (simp add: c1)}
     moreover
     {assume H: "real_of_int x - real_of_int d + Inum ((real_of_int (x -d)) # bs) e = 0"
@@ -2653,57 +2653,57 @@
         by (simp add: numbound0_I[OF bn,where b="real_of_int x - real_of_int d"and b'="a"and bs="bs"])
        with 4(5) have ?case using dp by simp}
   ultimately show ?case by blast
-next 
-  case (9 j c e) hence p: "Ifm (real_of_int x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" 
+next
+  case (9 j c e) hence p: "Ifm (real_of_int x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1"
     and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
   let ?e = "Inum (real_of_int x # bs) e"
-  from 9 have "isint e (a #bs)"  by simp 
+  from 9 have "isint e (a #bs)"  by simp
   hence ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real_of_int x)#bs"] numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"]
     by (simp add: isint_iff)
   from 9 have id: "j dvd d" by simp
   from c1 ie[symmetric] have "?p x = (real_of_int j rdvd real_of_int (x+ floor ?e))" by simp
-  also have "\<dots> = (j dvd x + floor ?e)" 
+  also have "\<dots> = (j dvd x + floor ?e)"
     using int_rdvd_real[where i="j" and x="real_of_int (x+ floor ?e)"] by simp
-  also have "\<dots> = (j dvd x - d + floor ?e)" 
+  also have "\<dots> = (j dvd x - d + floor ?e)"
     using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-  also have "\<dots> = (real_of_int j rdvd real_of_int (x - d + floor ?e))" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int (x - d + floor ?e))"
     using int_rdvd_real[where i="j" and x="real_of_int (x-d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (real_of_int j rdvd real_of_int x - real_of_int d + ?e)" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int x - real_of_int d + ?e)"
     using ie by simp
-  finally show ?case 
+  finally show ?case
     using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp
 next
   case (10 j c e) hence p: "Ifm (real_of_int x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" using dvd1_eq1[where x="c"] by simp+
   let ?e = "Inum (real_of_int x # bs) e"
-  from 10 have "isint e (a#bs)"  by simp 
+  from 10 have "isint e (a#bs)"  by simp
   hence ie: "real_of_int (floor ?e) = ?e" using numbound0_I[OF bn,where b="real_of_int x" and b'="a" and bs="bs"] isint_iff[where n="e" and bs="(real_of_int x)#bs"]
     by (simp add: isint_iff)
   from 10 have id: "j dvd d" by simp
   from c1 ie[symmetric] have "?p x = (\<not> real_of_int j rdvd real_of_int (x+ floor ?e))" by simp
-  also have "\<dots> = (\<not> j dvd x + floor ?e)" 
+  also have "\<dots> = (\<not> j dvd x + floor ?e)"
     using int_rdvd_real[where i="j" and x="real_of_int (x+ floor ?e)"] by simp
-  also have "\<dots> = (\<not> j dvd x - d + floor ?e)" 
+  also have "\<dots> = (\<not> j dvd x - d + floor ?e)"
     using dvd_period[OF id, where x="x" and c="-1" and t="floor ?e"] by simp
-  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int (x - d + floor ?e))" 
+  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int (x - d + floor ?e))"
     using int_rdvd_real[where i="j" and x="real_of_int (x-d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int x - real_of_int d + ?e)" 
+  also have "\<dots> = (\<not> real_of_int j rdvd real_of_int x - real_of_int d + ?e)"
     using ie by simp
   finally show ?case
     using numbound0_I[OF bn,where b="real_of_int (x-d)" and b'="real_of_int x" and bs="bs"] c1 p by simp
 qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int (x - d)" and b'="real_of_int x"]
   simp del: of_int_diff)
 
-lemma \<beta>':   
+lemma \<beta>':
   assumes lp: "iszlfm p (a #bs)"
   and u: "d_\<beta> p 1"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
   shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
 proof(clarify)
-  fix x 
-  assume nb:"?b" and px: "?P x" 
+  fix x
+  assume nb:"?b" and px: "?P x"
   hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). real_of_int x = b + real_of_int j)"
     by auto
   from  \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
@@ -2714,7 +2714,7 @@
 using lp by (induct p rule: iszlfm.induct) (auto simp add: isint_neg isint_sub)
 
 lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
-==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D) 
+==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
 ==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
 ==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
 apply(rule iffI)
@@ -2744,18 +2744,18 @@
   shows "(\<exists> (x::int). Ifm (real_of_int x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm (real_of_int j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm ((Inum (a#bs) b + real_of_int j) #bs) p))"
   (is "(\<exists> (x::int). ?P (real_of_int x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + real_of_int j)))")
 proof-
-  from minusinf_inf[OF lp] 
+  from minusinf_inf[OF lp]
   have th: "\<exists>(z::int). \<forall>x<z. ?P (real_of_int x) = ?M x" by blast
   let ?B' = "{floor (?I b) | b. b\<in> ?B}"
   from \<beta>_int[OF lp] isint_iff[where bs="a # bs"] have B: "\<forall> b\<in> ?B. real_of_int (floor (?I b)) = ?I b" by simp
-  from B[rule_format] 
-  have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (floor (?I b)) + real_of_int j))" 
+  from B[rule_format]
+  have "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (floor (?I b)) + real_of_int j))"
     by simp
   also have "\<dots> = (\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (real_of_int (floor (?I b) + j)))" by simp
   also have"\<dots> = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))"  by blast
-  finally have BB': 
-    "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))" 
-    by blast 
+  finally have BB':
+    "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b + real_of_int j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j)))"
+    by blast
   hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (real_of_int (b + j))) \<longrightarrow> ?P (real_of_int x) \<longrightarrow> ?P (real_of_int (x - d))" using \<beta>'[OF lp u d dp] by blast
   from minusinf_repeats[OF d lp]
   have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
@@ -2765,14 +2765,14 @@
     (* Reddy and Loveland *)
 
 
-consts 
+consts
   \<rho> :: "fm \<Rightarrow> (num \<times> int) list" (* Compute the Reddy and Loveland Bset*)
   \<sigma>_\<rho>:: "fm \<Rightarrow> num \<times> int \<Rightarrow> fm" (* Performs the modified substitution of Reddy and Loveland*)
   \<alpha>_\<rho> :: "fm \<Rightarrow> (num\<times>int) list"
   a_\<rho> :: "fm \<Rightarrow> int \<Rightarrow> fm"
 recdef \<rho> "measure size"
-  "\<rho> (And p q) = (\<rho> p @ \<rho> q)" 
-  "\<rho> (Or p q) = (\<rho> p @ \<rho> q)" 
+  "\<rho> (And p q) = (\<rho> p @ \<rho> q)"
+  "\<rho> (Or p q) = (\<rho> p @ \<rho> q)"
   "\<rho> (Eq  (CN 0 c e)) = [(Sub (C (- 1)) e,c)]"
   "\<rho> (NEq (CN 0 c e)) = [(Neg e,c)]"
   "\<rho> (Lt  (CN 0 c e)) = []"
@@ -2782,29 +2782,29 @@
   "\<rho> p = []"
 
 recdef \<sigma>_\<rho> "measure size"
-  "\<sigma>_\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))" 
-  "\<sigma>_\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))" 
-  "\<sigma>_\<rho> (Eq  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (And p q) = (\<lambda> (t,k). And (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))"
+  "\<sigma>_\<rho> (Or p q) = (\<lambda> (t,k). Or (\<sigma>_\<rho> p (t,k)) (\<sigma>_\<rho> q (t,k)))"
+  "\<sigma>_\<rho> (Eq  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Eq (Add (Mul (c div k) t) e))
                                             else (Eq (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (NEq (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (NEq (Add (Mul (c div k) t) e))
                                             else (NEq (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (Lt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (Lt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Lt (Add (Mul (c div k) t) e))
                                             else (Lt (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (Le  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (Le  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Le (Add (Mul (c div k) t) e))
                                             else (Le (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (Gt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (Gt  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Gt (Add (Mul (c div k) t) e))
                                             else (Gt (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (Ge  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (Ge  (CN 0 c e)) = (\<lambda> (t,k). if k dvd c then (Ge (Add (Mul (c div k) t) e))
                                             else (Ge (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (Dvd i (CN 0 c e)) =(\<lambda> (t,k). if k dvd c then (Dvd i (Add (Mul (c div k) t) e))
                                             else (Dvd (i*k) (Add (Mul c t) (Mul k e))))"
-  "\<sigma>_\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e)) 
+  "\<sigma>_\<rho> (NDvd i (CN 0 c e))=(\<lambda> (t,k). if k dvd c then (NDvd i (Add (Mul (c div k) t) e))
                                             else (NDvd (i*k) (Add (Mul c t) (Mul k e))))"
   "\<sigma>_\<rho> p = (\<lambda> (t,k). p)"
 
 recdef \<alpha>_\<rho> "measure size"
-  "\<alpha>_\<rho> (And p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)" 
-  "\<alpha>_\<rho> (Or p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)" 
+  "\<alpha>_\<rho> (And p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)"
+  "\<alpha>_\<rho> (Or p q) = (\<alpha>_\<rho> p @ \<alpha>_\<rho> q)"
   "\<alpha>_\<rho> (Eq  (CN 0 c e)) = [(Add (C (- 1)) e,c)]"
   "\<alpha>_\<rho> (NEq (CN 0 c e)) = [(e,c)]"
   "\<alpha>_\<rho> (Lt  (CN 0 c e)) = [(e,c)]"
@@ -2822,19 +2822,19 @@
   and tnb: "numbound0 t"
   and tint: "isint t (real_of_int x#bs)"
   and kdt: "k dvd floor (Inum (b'#bs) t)"
-  shows "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (t,k)) = 
-  (Ifm ((real_of_int ((floor (Inum (b'#bs) t)) div k))#bs) p)" 
+  shows "Ifm (real_of_int x#bs) (\<sigma>_\<rho> p (t,k)) =
+  (Ifm ((real_of_int ((floor (Inum (b'#bs) t)) div k))#bs) p)"
   (is "?I (real_of_int x) (?s p) = (?I (real_of_int ((floor (?N b' t)) div k)) p)" is "_ = (?I ?tk p)")
 using linp kpos tnb
 proof(induct p rule: \<sigma>_\<rho>.induct)
-  case (3 c e) 
+  case (3 c e)
   from 3 have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from kpos have knz': "real_of_int k \<noteq> 0" by simp
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t"
@@ -2851,16 +2851,16 @@
           numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
         by (simp add: ti)
       finally have ?case . }
-    ultimately show ?case by blast 
+    ultimately show ?case by blast
 next
-  case (4 c e)  
+  case (4 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from kpos have knz': "real_of_int k \<noteq> 0" by simp
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
@@ -2876,16 +2876,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from assms * have "?I (real_of_int x) (?s (Lt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k < 0)"
@@ -2900,16 +2900,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
   case (6 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from assms * have "?I (real_of_int x) (?s (Le (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<le> 0)"
@@ -2924,16 +2924,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
-  case (7 c e) 
+  case (7 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from assms * have "?I (real_of_int x) (?s (Gt (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k > 0)"
@@ -2948,16 +2948,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
-  case (8 c e)  
+  case (8 c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from assms * have "?I (real_of_int x) (?s (Ge (CN 0 c e))) = ((real_of_int c * (?N (real_of_int x) t / real_of_int k) + ?N (real_of_int x) e)* real_of_int k \<ge> 0)"
@@ -2972,16 +2972,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
   case (9 i c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
@@ -2996,16 +2996,16 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 next
   case (10 i c e)
   then have cp: "c > 0" and nb: "numbound0 e" by auto
-  { assume kdc: "k dvd c" 
+  { assume kdc: "k dvd c"
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
     from kdc have  ?case using real_of_int_div[OF kdc] real_of_int_div[OF kdt]
       numbound0_I[OF tnb, where bs="bs" and b="b'" and b'="real_of_int x"]
-      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) } 
-  moreover 
+      numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"] by (simp add: ti) }
+  moreover
   { assume *: "\<not> k dvd c"
     from kpos have knz: "k\<noteq>0" by simp hence knz': "real_of_int k \<noteq> 0" by simp
     from tint have ti: "real_of_int (floor (?N (real_of_int x) t)) = ?N (real_of_int x) t" using isint_def by simp
@@ -3020,7 +3020,7 @@
         numbound0_I[OF nb, where bs="bs" and b="?tk" and b'="real_of_int x"]
       by (simp add: ti)
     finally have ?case . }
-  ultimately show ?case by blast 
+  ultimately show ?case by blast
 qed (simp_all add: bound0_I[where bs="bs" and b="real_of_int ((floor (?N b' t)) div k)" and b'="real_of_int x"]
   numbound0_I[where bs="bs" and b="real_of_int ((floor (?N b' t)) div k)" and b'="real_of_int x"])
 
@@ -3056,7 +3056,7 @@
   from mult_strict_left_mono[OF dp cp]  have one:"1 \<in> {1 .. c*d}" by auto
   from nob[rule_format, where j="1", OF one] pi show ?case by simp
 next
-  case (4 c e)  
+  case (4 c e)
   hence cp: "c >0" and nb: "numbound0 e" and ei: "isint e (real_of_int i#bs)"
     and nob: "\<forall> j\<in> {1 .. c*d}. real_of_int (c*i) \<noteq> - ?N i e + real_of_int j"
     by simp+
@@ -3070,18 +3070,18 @@
   ultimately show ?case by blast
 next
   case (5 c e) hence cp: "c > 0" by simp
-  from 5 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] 
+  from 5 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric]
     of_int_mult]
-  show ?case using 5 dp 
-    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] 
+  show ?case using 5 dp
+    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"]
       algebra_simps del: mult_pos_pos)
      by (metis add.right_neutral of_int_0_less_iff of_int_mult pos_add_strict)
 next
   case (6 c e) hence cp: "c > 0" by simp
-  from 6 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric] 
+  from 6 mult_strict_left_mono[OF dp cp, simplified of_int_less_iff[symmetric]
     of_int_mult]
-  show ?case using 6 dp 
-    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"] 
+  show ?case using 6 dp
+    apply (simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"]
       algebra_simps del: mult_pos_pos)
       using order_trans by fastforce
 next
@@ -3096,9 +3096,9 @@
   have "real_of_int (c*i) + ?N i e > real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)" by auto
   moreover
   {assume "real_of_int (c*i) + ?N i e > real_of_int (c*d)" hence ?case
-      by (simp add: algebra_simps 
-        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} 
-  moreover 
+      by (simp add: algebra_simps
+        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])}
+  moreover
   {assume H:"real_of_int (c*i) + ?N i e \<le> real_of_int (c*d)"
     with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) \<le> real_of_int (c*d)" by simp
     hence pid: "c*i + ?fe \<le> c*d" by (simp only: of_int_le_iff)
@@ -3119,9 +3119,9 @@
   have "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d) \<or> real_of_int (c*i) + ?N i e < real_of_int (c*d)" by auto
   moreover
   {assume "real_of_int (c*i) + ?N i e \<ge> real_of_int (c*d)" hence ?case
-      by (simp add: algebra_simps 
-        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])} 
-  moreover 
+      by (simp add: algebra_simps
+        numbound0_I[OF nb,where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])}
+  moreover
   {assume H:"real_of_int (c*i) + ?N i e < real_of_int (c*d)"
     with ei[simplified isint_iff] have "real_of_int (c*i + ?fe) < real_of_int (c*d)" by simp
     hence pid: "c*i + 1 + ?fe \<le> c*d" by (simp only: of_int_le_iff)
@@ -3137,61 +3137,61 @@
 next
   case (9 j c e)  hence p: "real_of_int j rdvd real_of_int (c*i) + ?N i e" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"  by simp+
   let ?e = "Inum (real_of_int i # bs) e"
-  from 9 have "isint e (real_of_int i #bs)"  by simp 
+  from 9 have "isint e (real_of_int i #bs)"  by simp
   hence ie: "real_of_int (floor ?e) = ?e" using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"]
     by (simp add: isint_iff)
   from 9 have id: "j dvd d" by simp
   from ie[symmetric] have "?p i = (real_of_int j rdvd real_of_int (c*i+ floor ?e))" by simp
-  also have "\<dots> = (j dvd c*i + floor ?e)" 
+  also have "\<dots> = (j dvd c*i + floor ?e)"
     using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
-  also have "\<dots> = (j dvd c*i - c*d + floor ?e)" 
+  also have "\<dots> = (j dvd c*i - c*d + floor ?e)"
     using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-  also have "\<dots> = (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))"
     using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" 
+  also have "\<dots> = (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)"
     using ie by (simp add:algebra_simps)
-  finally show ?case 
-    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p 
+  finally show ?case
+    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p
     by (simp add: algebra_simps)
 next
   case (10 j c e)
   hence p: "\<not> (real_of_int j rdvd real_of_int (c*i) + ?N i e)" (is "?p x") and cp: "c > 0" and bn:"numbound0 e"
     by simp+
   let ?e = "Inum (real_of_int i # bs) e"
-  from 10 have "isint e (real_of_int i #bs)"  by simp 
+  from 10 have "isint e (real_of_int i #bs)"  by simp
   hence ie: "real_of_int (floor ?e) = ?e"
     using isint_iff[where n="e" and bs="(real_of_int i)#bs"] numbound0_I[OF bn,where b="real_of_int i" and b'="real_of_int i" and bs="bs"]
     by (simp add: isint_iff)
   from 10 have id: "j dvd d" by simp
   from ie[symmetric] have "?p i = (\<not> (real_of_int j rdvd real_of_int (c*i+ floor ?e)))" by simp
-  also have "\<dots> = Not (j dvd c*i + floor ?e)" 
+  also have "\<dots> = Not (j dvd c*i + floor ?e)"
     using int_rdvd_iff [where i="j" and t="c*i+ floor ?e"] by simp
-  also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)" 
+  also have "\<dots> = Not (j dvd c*i - c*d + floor ?e)"
     using dvd_period[OF id, where x="c*i" and c="-c" and t="floor ?e"] by simp
-  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))" 
+  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*i - c*d + floor ?e))"
     using int_rdvd_iff[where i="j" and t="(c*i - c*d + floor ?e)",symmetric, simplified]
       ie by simp
-  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)" 
+  also have "\<dots> = Not (real_of_int j rdvd real_of_int (c*(i - d)) + ?e)"
     using ie by (simp add:algebra_simps)
-  finally show ?case 
-    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p 
+  finally show ?case
+    using numbound0_I[OF bn,where b="real_of_int i - real_of_int d" and b'="real_of_int i" and bs="bs"] p
     by (simp add: algebra_simps)
 qed (auto simp add: numbound0_I[where bs="bs" and b="real_of_int i - real_of_int d" and b'="real_of_int i"])
 
 lemma \<sigma>_nb: assumes lp: "iszlfm p (a#bs)" and nb: "numbound0 t"
   shows "bound0 (\<sigma> p k t)"
   using \<sigma>_\<rho>_nb[OF lp nb] nb by (simp add: \<sigma>_def)
-  
+
 lemma \<rho>':   assumes lp: "iszlfm p (a #bs)"
   and d: "d_\<delta> p d"
   and dp: "d > 0"
   shows "\<forall> x. \<not>(\<exists> (e,c) \<in> set(\<rho> p). \<exists>(j::int) \<in> {1 .. c*d}. Ifm (a #bs) (\<sigma> p c (Add e (C j)))) \<longrightarrow> Ifm (real_of_int x#bs) p \<longrightarrow> Ifm (real_of_int (x - d)#bs) p" (is "\<forall> x. ?b x \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
 proof(clarify)
-  fix x 
-  assume nob1:"?b x" and px: "?P x" 
+  fix x
+  assume nob1:"?b x" and px: "?P x"
   from iszlfm_gen[OF lp, rule_format, where y="real_of_int x"] have lp': "iszlfm p (real_of_int x#bs)".
-  have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * x) \<noteq> Inum (real_of_int x # bs) e + real_of_int j" 
+  have nob: "\<forall>(e, c)\<in>set (\<rho> p). \<forall>j\<in>{1..c * d}. real_of_int (c * x) \<noteq> Inum (real_of_int x # bs) e + real_of_int j"
   proof(clarify)
     fix e c j assume ecR: "(e,c) \<in> set (\<rho> p)" and jD: "j\<in> {1 .. c*d}"
       and cx: "real_of_int (c*x) = Inum (real_of_int x#bs) e + real_of_int j"
@@ -3220,14 +3220,14 @@
     have "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" by blast
       with ecR jD nob1    show "False" by blast
   qed
-  from \<rho>[OF lp' px d dp nob] show "?P (x -d )" . 
+  from \<rho>[OF lp' px d dp nob] show "?P (x -d )" .
 qed
 
 
-lemma rl_thm: 
+lemma rl_thm:
   assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
   shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1 .. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
-  (is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))" 
+  (is "(\<exists>(x::int). ?P x) = ((\<exists> j\<in> {1.. \<delta> p}. ?MP j)\<or>(\<exists> (e,c) \<in> ?R. \<exists> j\<in> _. ?SP c e j))"
     is "?lhs = (?MD \<or> ?RD)"  is "?lhs = ?rhs")
 proof-
   let ?d= "\<delta> p"
@@ -3244,9 +3244,9 @@
     from nb have nb': "numbound0 (Add e (C j))" by simp
     from spx bound0_I[OF \<sigma>_nb[OF lp nb', where k="c"], where bs="bs" and b="a" and b'="real_of_int i"]
     have spx': "Ifm (real_of_int i # bs) (\<sigma> p c (Add e (C j)))" by blast
-    from spx' have rcdej:"real_of_int c rdvd (Inum (real_of_int i#bs) (Add e (C j)))" 
+    from spx' have rcdej:"real_of_int c rdvd (Inum (real_of_int i#bs) (Add e (C j)))"
       and sr:"Ifm (real_of_int i#bs) (\<sigma>_\<rho> p (Add e (C j),c))" by (simp add: \<sigma>_def)+
-    from rcdej eji[simplified isint_iff] 
+    from rcdej eji[simplified isint_iff]
     have "real_of_int c rdvd real_of_int (floor (Inum (real_of_int i#bs) (Add e (C j))))" by simp
     hence cdej:"c dvd floor (Inum (real_of_int i#bs) (Add e (C j)))" by (simp only: int_rdvd_iff)
     from cp have cp': "real_of_int c > 0" by simp
@@ -3260,7 +3260,7 @@
     from \<rho>'[OF lp' d dp, rule_format, OF nob] have th:"\<forall> x. ?P x \<longrightarrow> ?P (x - ?d)" by blast
     from minusinf_inf[OF lp] obtain z where z:"\<forall> x<z. ?MP x = ?P x" by blast
     have zp: "abs (x - z) + 1 \<ge> 0" by arith
-    from decr_lemma[OF dp,where x="x" and z="z"] 
+    from decr_lemma[OF dp,where x="x" and z="z"]
       decr_mult_lemma[OF dp th zp, rule_format, OF px] z have th:"\<exists> x. ?MP x" by auto
     with minusinf_bex[OF lp] px nob have ?thesis by blast}
   ultimately show ?thesis by blast
@@ -3270,15 +3270,15 @@
   shows "(\<lambda> (t,k). (Inum (a#bs) t, k)) ` set (\<alpha>_\<rho> p) = (\<lambda> (t,k). (Inum (a#bs) t,k)) ` set (\<rho> (mirror p))"
   using lp
   by (induct p rule: mirror.induct) (simp_all add: split_def image_Un)
-  
+
 text \<open>The \<open>\<real>\<close> part\<close>
 
 text\<open>Linearity for fm where Bound 0 ranges over \<open>\<real>\<close>\<close>
 consts
   isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
 recdef isrlfm "measure size"
-  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
-  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
+  "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
+  "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
   "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
@@ -3288,21 +3288,21 @@
   "isrlfm p = (isatom p \<and> (bound0 p))"
 
 definition fp :: "fm \<Rightarrow> int \<Rightarrow> num \<Rightarrow> int \<Rightarrow> fm" where
-  "fp p n s j \<equiv> (if n > 0 then 
+  "fp p n s j \<equiv> (if n > 0 then
             (And p (And (Ge (CN 0 n (Sub s (Add (Floor s) (C j)))))
                         (Lt (CN 0 n (Sub s (Add (Floor s) (C (j+1))))))))
-            else 
-            (And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j))))) 
+            else
+            (And p (And (Le (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C j)))))
                         (Gt (CN 0 (-n) (Add (Neg s) (Add (Floor s) (C (j + 1)))))))))"
 
   (* splits the bounded from the unbounded part*)
 function (sequential) rsplit0 :: "num \<Rightarrow> (fm \<times> int \<times> num) list" where
   "rsplit0 (Bound 0) = [(T,1,C 0)]"
-| "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b 
+| "rsplit0 (Add a b) = (let acs = rsplit0 a ; bcs = rsplit0 b
               in map (\<lambda> ((p,n,t),(q,m,s)). (And p q, n+m, Add t s)) [(a,b). a\<leftarrow>acs,b\<leftarrow>bcs])"
 | "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
 | "rsplit0 (Neg a) = map (\<lambda> (p,n,s). (p,-n,Neg s)) (rsplit0 a)"
-| "rsplit0 (Floor a) = concat (map 
+| "rsplit0 (Floor a) = concat (map
       (\<lambda> (p,n,s). if n=0 then [(p,0,Floor s)]
           else (map (\<lambda> j. (fp p n s j, 0, Add (Floor s) (C j))) (if n > 0 then [0 .. n] else [n .. 0])))
        (rsplit0 a))"
@@ -3321,22 +3321,22 @@
 
 
 lemma rsplit0_cs:
-  shows "\<forall> (p,n,s) \<in> set (rsplit0 t). 
-  (Ifm (x#bs) p \<longrightarrow>  (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p" 
+  shows "\<forall> (p,n,s) \<in> set (rsplit0 t).
+  (Ifm (x#bs) p \<longrightarrow>  (Inum (x#bs) t = Inum (x#bs) (CN 0 n s))) \<and> numbound0 s \<and> isrlfm p"
   (is "\<forall> (p,n,s) \<in> ?SS t. (?I p \<longrightarrow> ?N t = ?N (CN 0 n s)) \<and> _ \<and> _ ")
 proof(induct t rule: rsplit0.induct)
-  case (5 a) 
+  case (5 a)
   let ?p = "\<lambda> (p,n,s) j. fp p n s j"
   let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),Add (Floor s) (C j)))"
   let ?J = "\<lambda> n. if n>0 then [0..n] else [n..0]"
   let ?ff=" (\<lambda> (p,n,s). if n= 0 then [(p,0,Floor s)] else map (?f (p,n,s)) (?J n))"
   have int_cases: "\<forall> (i::int). i= 0 \<or> i < 0 \<or> i > 0" by arith
-  have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
+  have U1: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)]))" by auto
-  have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}. 
+  have U2': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0}.
     ?ff (p,n,s) = map (?f(p,n,s)) [0..n]" by auto
-  hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). 
+  hence U2: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s).
     set (map (?f(p,n,s)) [0..n])))"
   proof-
     fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
@@ -3346,7 +3346,7 @@
   qed
   have U3': "\<forall> (p,n,s) \<in> {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0}. ?ff (p,n,s) = map (?f(p,n,s)) [n..0]"
     by auto
-  hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) = 
+  hence U3: "(UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) =
     (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n<0} (\<lambda>(p,n,s). set (map (?f(p,n,s)) [n..0])))"
       proof-
     fix M :: "('a\<times>'b\<times>'c) set" and f :: "('a\<times>'b\<times>'c) \<Rightarrow> 'd list" and g
@@ -3357,29 +3357,29 @@
   have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))"
     by auto
   also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by blast
-  also have "\<dots> = 
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))"
     using int_cases[rule_format] by blast
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un 
-   (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) [0..n]))) Un 
-   (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un
+   (UNION {(p,n,s). (p,n,s)\<in> ?SS a\<and>n>0} (\<lambda>(p,n,s). set(map(?f(p,n,s)) [0..n]))) Un
+   (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).
     set (map (?f(p,n,s)) [n..0]))))" by (simp only: U1 U2 U3)
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
     by (simp only: set_map set_upto list.set)
-  also have "\<dots> =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))" by blast
-  finally 
-  have FS: "?SS (Floor a) =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
+  finally
+  have FS: "?SS (Floor a) =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"    by blast
   show ?case
   proof(simp only: FS, clarsimp simp del: Ifm.simps Inum.simps, -)
@@ -3396,17 +3396,17 @@
            ac < 0 \<and>
            (\<exists>j. p = fp ab ac ba j \<and>
                 n = 0 \<and> s = Add (Floor ba) (C j) \<and> ac \<le> j \<and> j \<le> 0))"
-    moreover 
+    moreover
     { fix s'
       assume "(p, 0, s') \<in> ?SS a" and "n = 0" and "s = Floor s'"
       hence ?ths using 5(1) by auto }
     moreover
     { fix p' n' s' j
-      assume pns: "(p', n', s') \<in> ?SS a" 
-        and np: "0 < n'" 
-        and p_def: "p = ?p (p',n',s') j" 
-        and n0: "n = 0" 
-        and s_def: "s = (Add (Floor s') (C j))" 
+      assume pns: "(p', n', s') \<in> ?SS a"
+        and np: "0 < n'"
+        and p_def: "p = ?p (p',n',s') j"
+        and n0: "n = 0"
+        and s_def: "s = (Add (Floor s') (C j))"
         and jp: "0 \<le> j" and jn: "j \<le> n'"
       from 5 pns have H:"(Ifm ((x::real) # (bs::real list)) p' \<longrightarrow>
           Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
@@ -3415,9 +3415,9 @@
       from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by simp
       let ?nxs = "CN 0 n' s'"
       let ?l = "floor (?N s') + j"
-      from H 
-      have "?I (?p (p',n',s') j) \<longrightarrow> 
-          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
+      from H
+      have "?I (?p (p',n',s') j) \<longrightarrow>
+          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))"
         by (simp add: fp_def np algebra_simps)
       also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
         using floor_unique_iff[where x="?N ?nxs" and a="?l"] by simp
@@ -3428,11 +3428,11 @@
       with s_def n0 p_def nb nf have ?ths by auto}
     moreover
     { fix p' n' s' j
-      assume pns: "(p', n', s') \<in> ?SS a" 
-        and np: "n' < 0" 
-        and p_def: "p = ?p (p',n',s') j" 
-        and n0: "n = 0" 
-        and s_def: "s = (Add (Floor s') (C j))" 
+      assume pns: "(p', n', s') \<in> ?SS a"
+        and np: "n' < 0"
+        and p_def: "p = ?p (p',n',s') j"
+        and n0: "n = 0"
+        and s_def: "s = (Add (Floor s') (C j))"
         and jp: "n' \<le> j" and jn: "j \<le> 0"
       from 5 pns have H:"(Ifm ((x::real) # (bs::real list)) p' \<longrightarrow>
           Inum (x # bs) a = Inum (x # bs) (CN 0 n' s')) \<and>
@@ -3441,9 +3441,9 @@
       from H have nf: "isrlfm (?p (p',n',s') j)" using fp_def np by simp
       let ?nxs = "CN 0 n' s'"
       let ?l = "floor (?N s') + j"
-      from H 
-      have "?I (?p (p',n',s') j) \<longrightarrow> 
-          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))" 
+      from H
+      have "?I (?p (p',n',s') j) \<longrightarrow>
+          (((?N ?nxs \<ge> real_of_int ?l) \<and> (?N ?nxs < real_of_int (?l + 1))) \<and> (?N a = ?N ?nxs ))"
         by (simp add: np fp_def algebra_simps)
       also have "\<dots> \<longrightarrow> ((floor (?N ?nxs) = ?l) \<and> (?N a = ?N ?nxs ))"
         using floor_unique_iff[where x="?N ?nxs" and a="?l"] by simp
@@ -3459,18 +3459,18 @@
     by auto
 qed (auto simp add: Let_def split_def algebra_simps)
 
-lemma real_in_int_intervals: 
+lemma real_in_int_intervals:
   assumes xb: "real_of_int m \<le> x \<and> x < real_of_int ((n::int) + 1)"
   shows "\<exists> j\<in> {m.. n}. real_of_int j \<le> x \<and> x < real_of_int (j+1)" (is "\<exists> j\<in> ?N. ?P j")
-by (rule bexI[where P="?P" and x="floor x" and A="?N"]) 
-(auto simp add: floor_less_iff[where x="x" and z="n+1", simplified] 
+by (rule bexI[where P="?P" and x="floor x" and A="?N"])
+(auto simp add: floor_less_iff[where x="x" and z="n+1", simplified]
   xb[simplified] floor_mono[where x="real_of_int m" and y="x", OF conjunct1[OF xb], simplified floor_of_int[where z="m"]])
 
 lemma rsplit0_complete:
   assumes xp:"0 \<le> x" and x1:"x < 1"
   shows "\<exists> (p,n,s) \<in> set (rsplit0 t). Ifm (x#bs) p" (is "\<exists> (p,n,s) \<in> ?SS t. ?I p")
 proof(induct t rule: rsplit0.induct)
-  case (2 a b) 
+  case (2 a b)
   then have "\<exists> (pa,na,sa) \<in> ?SS a. ?I pa" by auto
   then obtain "pa" "na" "sa" where pa: "(pa,na,sa)\<in> ?SS a \<and> ?I pa" by blast
   with 2 have "\<exists> (pb,nb,sb) \<in> ?SS b. ?I pb" by blast
@@ -3484,7 +3484,7 @@
   moreover from pa pb have "?I (And pa pb)" by simp
   ultimately show ?case by blast
 next
-  case (5 a) 
+  case (5 a)
   let ?p = "\<lambda> (p,n,s) j. fp p n s j"
   let ?f = "(\<lambda> (p,n,s) j. (?p (p,n,s) j, (0::int),(Add (Floor s) (C j))))"
   let ?J = "\<lambda> n. if n>0 then [0..n] else [n..0]"
@@ -3512,30 +3512,30 @@
 
   have "?SS (Floor a) = UNION (?SS a) (\<lambda>x. set (?ff x))" by auto
   also have "\<dots> = UNION (?SS a) (\<lambda> (p,n,s). set (?ff (p,n,s)))" by blast
-  also have "\<dots> = 
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))" 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (?ff (p,n,s)))) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (?ff (p,n,s)))))"
     using int_cases[rule_format] by blast
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [0..n]))) Un 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). set [(p,0,Floor s)])) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [0..n]))) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). set (map (?f(p,n,s)) [n..0]))))"
     by (simp only: U1 U2 U3)
-  also have "\<dots> =  
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {0 .. n})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s). (?f(p,n,s)) ` {n .. 0})))"
     by (simp only: set_map set_upto list.set)
-  also have "\<dots> =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
+  also have "\<dots> =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"
     by blast
-  finally 
-  have FS: "?SS (Floor a) =   
-    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un 
-    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un 
+  finally
+  have FS: "?SS (Floor a) =
+    ((UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n=0} (\<lambda> (p,n,s). {(p,0,Floor s)})) Un
+    (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n>0} (\<lambda> (p,n,s). {?f(p,n,s) j| j. j\<in> {0 .. n}})) Un
     (UNION {(p,n,s). (p,n,s) \<in> ?SS a \<and> n<0} (\<lambda> (p,n,s).  {?f(p,n,s) j| j. j\<in> {n .. 0}})))"
     by blast
   from 5 have "\<exists> (p,n,s) \<in> ?SS a. ?I p" by auto
@@ -3543,7 +3543,7 @@
   let ?N = "\<lambda> t. Inum (x#bs) t"
   from rsplit0_cs[rule_format] pns have ans:"(?N a = ?N (CN 0 n s)) \<and> numbound0 s \<and> isrlfm p"
     by auto
-  
+
   have "n=0 \<or> n >0 \<or> n <0" by arith
   moreover {assume "n=0" hence ?case using pns by (simp only: FS) auto }
   moreover
@@ -3554,21 +3554,21 @@
     finally have "?N (Floor s) \<le> real_of_int n * x + ?N s" .
     moreover
     {from x1 np have "real_of_int n *x + ?N s < real_of_int n + ?N s" by simp
-      also from real_of_int_floor_add_one_gt[where r="?N s"] 
+      also from real_of_int_floor_add_one_gt[where r="?N s"]
       have "\<dots> < real_of_int n + ?N (Floor s) + 1" by simp
       finally have "real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp}
     ultimately have "?N (Floor s) \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (n+1)" by simp
     hence th: "0 \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (n+1)" by simp
     from real_in_int_intervals th have  "\<exists> j\<in> {0 .. n}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp
-    
+
     hence "\<exists> j\<in> {0 .. n}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0"
-      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) 
+      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"])
     hence "\<exists> j\<in> {0.. n}. ?I (?p (p,n,s) j)"
       using pns by (simp add: fp_def np algebra_simps)
     then obtain "j" where j_def: "j\<in> {0 .. n} \<and> ?I (?p (p,n,s) j)" by blast
     hence "\<exists>x \<in> {?p (p,n,s) j |j. 0\<le> j \<and> j \<le> n }. ?I x" by auto
-    hence ?case using pns 
-      by (simp only: FS,simp add: bex_Un) 
+    hence ?case using pns
+      by (simp only: FS,simp add: bex_Un)
     (rule disjI2, rule disjI1,rule exI [where x="p"],
       rule exI [where x="n"],rule exI [where x="s"],simp_all add: np)
   }
@@ -3576,27 +3576,27 @@
   { assume nn: "n < 0" hence np: "-n >0" by simp
     from of_int_floor_le[of "?N s"] have "?N (Floor s) + 1 > ?N s" by simp
     moreover from mult_left_mono_neg[OF xp] nn have "?N s \<ge> real_of_int n * x + ?N s" by simp
-    ultimately have "?N (Floor s) + 1 > real_of_int n * x + ?N s" by arith 
+    ultimately have "?N (Floor s) + 1 > real_of_int n * x + ?N s" by arith
     moreover
     {from x1 nn have "real_of_int n *x + ?N s \<ge> real_of_int n + ?N s" by simp
       moreover from of_int_floor_le[of "?N s"]  have "real_of_int n + ?N s \<ge> real_of_int n + ?N (Floor s)" by simp
-      ultimately have "real_of_int n *x + ?N s \<ge> ?N (Floor s) + real_of_int n" 
+      ultimately have "real_of_int n *x + ?N s \<ge> ?N (Floor s) + real_of_int n"
         by (simp only: algebra_simps)}
     ultimately have "?N (Floor s) + real_of_int n \<le> real_of_int n *x + ?N s\<and> real_of_int n *x + ?N s < ?N (Floor s) + real_of_int (1::int)" by simp
     hence th: "real_of_int n \<le> real_of_int n *x + ?N s - ?N (Floor s) \<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (1::int)" by simp
     have th1: "\<forall> (a::real). (- a > 0) = (a < 0)" by auto
     have th2: "\<forall> (a::real). (0 \<ge> - a) = (a \<ge> 0)" by auto
     from real_in_int_intervals th  have  "\<exists> j\<in> {n .. 0}. real_of_int j \<le> real_of_int n *x + ?N s - ?N (Floor s)\<and> real_of_int n *x + ?N s - ?N (Floor s) < real_of_int (j+1)" by simp
-    
+
     hence "\<exists> j\<in> {n .. 0}. 0 \<le> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j \<and> real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1) < 0"
-      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"]) 
+      by(simp only: myle[of _ "real_of_int n * x + Inum (x # bs) s - Inum (x # bs) (Floor s)"] less_iff_diff_less_0[where a="real_of_int n *x + ?N s - ?N (Floor s)"])
     hence "\<exists> j\<in> {n .. 0}. 0 \<ge> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int j) \<and> - (real_of_int n *x + ?N s - ?N (Floor s) - real_of_int (j+1)) > 0" by (simp only: th1[rule_format] th2[rule_format])
     hence "\<exists> j\<in> {n.. 0}. ?I (?p (p,n,s) j)"
       using pns by (simp add: fp_def nn algebra_simps
-        del: diff_less_0_iff_less diff_le_0_iff_le) 
+        del: diff_less_0_iff_less diff_le_0_iff_le)
     then obtain "j" where j_def: "j\<in> {n .. 0} \<and> ?I (?p (p,n,s) j)" by blast
     hence "\<exists>x \<in> {?p (p,n,s) j |j. n\<le> j \<and> j \<le> 0 }. ?I x" by auto
-    hence ?case using pns 
+    hence ?case using pns
       by (simp only: FS,simp add: bex_Un)
     (rule disjI2, rule disjI2,rule exI [where x="p"],
       rule exI [where x="n"],rule exI [where x="s"],simp_all add: nn)
@@ -3615,7 +3615,7 @@
 lemma foldr_conj_map: "Ifm bs (foldr conj (map f xs) T) = (\<forall> x \<in> set xs. Ifm bs (f x))"
 by(induct xs, simp_all)
 
-lemma foldr_disj_map_rlfm: 
+lemma foldr_disj_map_rlfm:
   assumes lf: "\<forall> n s. numbound0 s \<longrightarrow> isrlfm (f n s)"
   and \<phi>: "\<forall> (\<phi>,n,s) \<in> set xs. numbound0 s \<and> isrlfm \<phi>"
   shows "isrlfm (foldr disj (map (\<lambda> (\<phi>, n, s). conj \<phi> (f n s)) xs) F)"
@@ -3631,7 +3631,7 @@
   from foldr_disj_map_rlfm[OF lf th] rsplit_def show ?thesis by simp
 qed
 
-lemma rsplit: 
+lemma rsplit:
   assumes xp: "x \<ge> 0" and x1: "x < 1"
   and f: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> (Ifm (x#bs) (f n s) = Ifm (x#bs) (g a))"
   shows "Ifm (x#bs) (rsplit f a) = Ifm (x#bs) (g a)"
@@ -3642,14 +3642,14 @@
   hence "\<exists> (\<phi>,n,s) \<in> set (rsplit0 a). ?I x (And \<phi> (f n s))" using rsplit_ex by simp
   then obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and "?I x (And \<phi> (f n s))" by blast
   hence \<phi>: "?I x \<phi>" and fns: "?I x (f n s)" by auto
-  from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi> 
+  from rsplit0_cs[where t="a" and bs="bs" and x="x", rule_format, OF fnsS] \<phi>
   have th: "(?N x a = ?N x (CN 0 n s)) \<and> numbound0 s" by auto
   from f[rule_format, OF th] fns show "?I x (g a)" by simp
 next
   let ?I = "\<lambda>x p. Ifm (x#bs) p"
   let ?N = "\<lambda> x t. Inum (x#bs) t"
   assume ga: "?I x (g a)"
-  from rsplit0_complete[OF xp x1, where bs="bs" and t="a"] 
+  from rsplit0_complete[OF xp x1, where bs="bs" and t="a"]
   obtain "\<phi>" "n" "s" where fnsS:"(\<phi>,n,s) \<in> set (rsplit0 a)" and fx: "?I x \<phi>" by blast
   from rsplit0_cs[where t="a" and x="x" and bs="bs"] fnsS fx
   have ans: "?N x a = ?N x (CN 0 n s)" and nb: "numbound0 s" by auto
@@ -3658,27 +3658,27 @@
 qed
 
 definition lt :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
+  lt_def: "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t))
                         else (Gt (CN 0 (-c) (Neg t))))"
 
 definition  le :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
+  le_def: "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t))
                         else (Ge (CN 0 (-c) (Neg t))))"
 
 definition  gt :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
+  gt_def: "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t))
                         else (Lt (CN 0 (-c) (Neg t))))"
 
 definition  ge :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
+  ge_def: "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t))
                         else (Le (CN 0 (-c) (Neg t))))"
 
 definition  eq :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
+  eq_def: "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t))
                         else (Eq (CN 0 (-c) (Neg t))))"
 
 definition neq :: "int \<Rightarrow> num \<Rightarrow> fm" where
-  neq_def: "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
+  neq_def: "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t))
                         else (NEq (CN 0 (-c) (Neg t))))"
 
 lemma lt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (lt n s) = Ifm (x#bs) (Lt a)"
@@ -3703,7 +3703,7 @@
 qed
 
 lemma le_l: "isrlfm (rsplit le a)"
-  by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def) 
+  by (rule rsplit_l[where f="le" and a="a"], auto simp add: le_def)
 (case_tac s, simp_all, rename_tac nat a b, case_tac "nat",simp_all)
 
 lemma gt_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (gt n s) = Ifm (x#bs) (Gt a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (gt n s) = ?I (Gt a)")
@@ -3714,28 +3714,28 @@
   (cases "n > 0", simp_all add: gt_def algebra_simps myless[of _ "0"])
 qed
 lemma gt_l: "isrlfm (rsplit gt a)"
-  by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def) 
+  by (rule rsplit_l[where f="gt" and a="a"], auto simp add: gt_def)
 (case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all)
 
 lemma ge_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (ge n s) = Ifm (x#bs) (Ge a)" (is "\<forall> a n s . ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (ge n s) = ?I (Ge a)")
 proof(clarify)
-  fix a n s 
+  fix a n s
   assume H: "?N a = ?N (CN 0 n s)"
   show "?I (ge n s) = ?I (Ge a)" using H by (cases "n=0", (simp add: ge_def))
   (cases "n > 0", simp_all add: ge_def algebra_simps myle[of _ "0"])
 qed
 lemma ge_l: "isrlfm (rsplit ge a)"
-  by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def) 
+  by (rule rsplit_l[where f="ge" and a="a"], auto simp add: ge_def)
 (case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all)
 
 lemma eq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (eq n s) = Ifm (x#bs) (Eq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (eq n s) = ?I (Eq a)")
 proof(clarify)
-  fix a n s 
+  fix a n s
   assume H: "?N a = ?N (CN 0 n s)"
   show "?I (eq n s) = ?I (Eq a)" using H by (auto simp add: eq_def algebra_simps)
 qed
 lemma eq_l: "isrlfm (rsplit eq a)"
-  by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def) 
+  by (rule rsplit_l[where f="eq" and a="a"], auto simp add: eq_def)
 (case_tac s, simp_all, rename_tac nat a b, case_tac"nat", simp_all)
 
 lemma neq_mono: "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (neq n s) = Ifm (x#bs) (NEq a)" (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (neq n s) = ?I (NEq a)")
@@ -3746,35 +3746,35 @@
 qed
 
 lemma neq_l: "isrlfm (rsplit neq a)"
-  by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def) 
+  by (rule rsplit_l[where f="neq" and a="a"], auto simp add: neq_def)
 (case_tac s, simp_all, rename_tac nat a b, case_tac"nat", simp_all)
 
-lemma small_le: 
+lemma small_le:
   assumes u0:"0 \<le> u" and u1: "u < 1"
   shows "(-u \<le> real_of_int (n::int)) = (0 \<le> n)"
 using u0 u1  by auto
 
-lemma small_lt: 
+lemma small_lt:
   assumes u0:"0 \<le> u" and u1: "u < 1"
   shows "(real_of_int (n::int) < real_of_int (m::int) - u) = (n < m)"
 using u0 u1  by auto
 
-lemma rdvd01_cs: 
+lemma rdvd01_cs:
   assumes up: "u \<ge> 0" and u1: "u<1" and np: "real_of_int n > 0"
   shows "(real_of_int (i::int) rdvd real_of_int (n::int) * u - s) = (\<exists> j\<in> {0 .. n - 1}. real_of_int n * u = s - real_of_int (floor s) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor s))" (is "?lhs = ?rhs")
 proof-
   let ?ss = "s - real_of_int (floor s)"
-  from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]] 
-    of_int_floor_le  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1" 
+  from real_of_int_floor_add_one_gt[where r="s", simplified myless[of "s"]]
+    of_int_floor_le  have ss0:"?ss \<ge> 0" and ss1:"?ss < 1"
     by (auto simp add: myle[of _ "s", symmetric] myless[of "?ss"])
   from np have n0: "real_of_int n \<ge> 0" by simp
-  from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np] 
-  have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto  
-  from int_rdvd_real[where i="i" and x="real_of_int (n::int) * u - s"] 
-  have "real_of_int i rdvd real_of_int n * u - s = 
-    (i dvd floor (real_of_int n * u -s) \<and> (real_of_int (floor (real_of_int n * u - s)) = real_of_int n * u - s ))" 
+  from mult_left_mono[OF up n0] mult_strict_left_mono[OF u1 np]
+  have nu0:"real_of_int n * u - s \<ge> -s" and nun:"real_of_int n * u -s < real_of_int n - s" by auto
+  from int_rdvd_real[where i="i" and x="real_of_int (n::int) * u - s"]
+  have "real_of_int i rdvd real_of_int n * u - s =
+    (i dvd floor (real_of_int n * u -s) \<and> (real_of_int (floor (real_of_int n * u - s)) = real_of_int n * u - s ))"
     (is "_ = (?DE)" is "_ = (?D \<and> ?E)") by simp
-  also have "\<dots> = (?DE \<and> real_of_int(floor (real_of_int n * u - s) + floor s)\<ge> -?ss 
+  also have "\<dots> = (?DE \<and> real_of_int(floor (real_of_int n * u - s) + floor s)\<ge> -?ss
     \<and> real_of_int(floor (real_of_int n * u - s) + floor s)< real_of_int n - ?ss)" (is "_=(?DE \<and>real_of_int ?a \<ge> _ \<and> real_of_int ?a < _)")
     using nu0 nun  by auto
   also have "\<dots> = (?DE \<and> ?a \<ge> 0 \<and> ?a < n)" by(simp only: small_le[OF ss0 ss1] small_lt[OF ss0 ss1])
@@ -3797,43 +3797,43 @@
 where
   NDVDJ_def: "NDVDJ i n s = (foldr conj (map (\<lambda> j. disj (NEq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor (Neg s))))) [0..n - 1]) T)"
 
-lemma DVDJ_DVD: 
+lemma DVDJ_DVD:
   assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0"
   shows "Ifm (x#bs) (DVDJ i n s) = Ifm (x#bs) (Dvd i (CN 0 n s))"
 proof-
   let ?f = "\<lambda> j. conj (Eq(CN 0 n (Add s (Sub(Floor (Neg s)) (C j))))) (Dvd i (Sub (C j) (Floor (Neg s))))"
   let ?s= "Inum (x#bs) s"
   from foldr_disj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"]
-  have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
+  have "Ifm (x#bs) (DVDJ i n s) = (\<exists> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
     by (simp add: np DVDJ_def)
   also have "\<dots> = (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int (floor (- ?s)) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor (- ?s)))"
     by (simp add: algebra_simps)
-  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
+  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
   have "\<dots> = (real_of_int i rdvd real_of_int n * x - (-?s))" by simp
   finally show ?thesis by simp
 qed
 
-lemma NDVDJ_NDVD: 
+lemma NDVDJ_NDVD:
   assumes xp:"x\<ge> 0" and x1: "x < 1" and np:"real_of_int n > 0"
   shows "Ifm (x#bs) (NDVDJ i n s) = Ifm (x#bs) (NDvd i (CN 0 n s))"
 proof-
   let ?f = "\<lambda> j. disj(NEq(CN 0 n (Add s (Sub (Floor (Neg s)) (C j))))) (NDvd i (Sub (C j) (Floor(Neg s))))"
   let ?s= "Inum (x#bs) s"
   from foldr_conj_map[where xs="[0..n - 1]" and bs="x#bs" and f="?f"]
-  have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))" 
+  have "Ifm (x#bs) (NDVDJ i n s) = (\<forall> j\<in> {0 .. (n - 1)}. Ifm (x#bs) (?f j))"
     by (simp add: np NDVDJ_def)
   also have "\<dots> = (\<not> (\<exists> j\<in> {0 .. (n - 1)}. real_of_int n * x = (- ?s) - real_of_int (floor (- ?s)) + real_of_int j \<and> real_of_int i rdvd real_of_int (j - floor (- ?s))))"
     by (simp add: algebra_simps)
-  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"] 
+  also from rdvd01_cs[OF xp x1 np, where i="i" and s="-?s"]
   have "\<dots> = (\<not> (real_of_int i rdvd real_of_int n * x - (-?s)))" by simp
   finally show ?thesis by simp
-qed  
-
-lemma foldr_disj_map_rlfm2: 
+qed
+
+lemma foldr_disj_map_rlfm2:
   assumes lf: "\<forall> n . isrlfm (f n)"
   shows "isrlfm (foldr disj (map f xs) F)"
 using lf by (induct xs, auto)
-lemma foldr_And_map_rlfm2: 
+lemma foldr_And_map_rlfm2:
   assumes lf: "\<forall> n . isrlfm (f n)"
   shows "isrlfm (foldr conj (map f xs) T)"
 using lf by (induct xs, auto)
@@ -3844,7 +3844,7 @@
   let ?f="\<lambda>j. conj (Eq (CN 0 n (Add s (Sub (Floor (Neg s)) (C j)))))
                          (Dvd i (Sub (C j) (Floor (Neg s))))"
   have th: "\<forall> j. isrlfm (?f j)" using nb np by auto
-  from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp 
+  from DVDJ_def foldr_disj_map_rlfm2[OF th] show ?thesis by simp
 qed
 
 lemma NDVDJ_l: assumes ip: "i >0" and np: "n>0" and nb: "numbound0 s"
@@ -3858,58 +3858,58 @@
 
 definition DVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
   DVD_def: "DVD i c t =
-  (if i=0 then eq c t else 
+  (if i=0 then eq c t else
   if c = 0 then (Dvd i t) else if c >0 then DVDJ (abs i) c t else DVDJ (abs i) (-c) (Neg t))"
 
 definition  NDVD :: "int \<Rightarrow> int \<Rightarrow> num \<Rightarrow> fm" where
   "NDVD i c t =
-  (if i=0 then neq c t else 
+  (if i=0 then neq c t else
   if c = 0 then (NDvd i t) else if c >0 then NDVDJ (abs i) c t else NDVDJ (abs i) (-c) (Neg t))"
 
-lemma DVD_mono: 
-  assumes xp: "0\<le> x" and x1: "x < 1" 
+lemma DVD_mono:
+  assumes xp: "0\<le> x" and x1: "x < 1"
   shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (DVD i n s) = Ifm (x#bs) (Dvd i a)"
   (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (DVD i n s) = ?I (Dvd i a)")
 proof(clarify)
-  fix a n s 
+  fix a n s
   assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
   let ?th = "?I (DVD i n s) = ?I (Dvd i a)"
   have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
-  moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]] 
+  moreover {assume iz: "i=0" hence ?th using eq_mono[rule_format, OF conjI[OF H nb]]
       by (simp add: DVD_def rdvd_left_0_eq)}
-  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) } 
-  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
-      by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1 
-        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } 
+  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H DVD_def) }
+  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th
+      by (simp add: DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1
+        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) }
   moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th by (simp add:DVD_def H DVDJ_DVD[OF xp x1] rdvd_abs1)}
   ultimately show ?th by blast
 qed
 
-lemma NDVD_mono:   assumes xp: "0\<le> x" and x1: "x < 1" 
+lemma NDVD_mono:   assumes xp: "0\<le> x" and x1: "x < 1"
   shows "\<forall> a n s. Inum (x#bs) a = Inum (x#bs) (CN 0 n s) \<and> numbound0 s \<longrightarrow> Ifm (x#bs) (NDVD i n s) = Ifm (x#bs) (NDvd i a)"
   (is "\<forall> a n s. ?N a = ?N (CN 0 n s) \<and> _ \<longrightarrow> ?I (NDVD i n s) = ?I (NDvd i a)")
 proof(clarify)
-  fix a n s 
+  fix a n s
   assume H: "?N a = ?N (CN 0 n s)" and nb: "numbound0 s"
   let ?th = "?I (NDVD i n s) = ?I (NDvd i a)"
   have "i=0 \<or> (i\<noteq>0 \<and> n=0) \<or> (i\<noteq>0 \<and> n < 0) \<or> (i\<noteq>0 \<and> n > 0)" by arith
-  moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]] 
+  moreover {assume iz: "i=0" hence ?th using neq_mono[rule_format, OF conjI[OF H nb]]
       by (simp add: NDVD_def rdvd_left_0_eq)}
-  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) } 
-  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th 
-      by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1 
-        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) } 
-  moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th 
+  moreover {assume inz: "i\<noteq>0" and "n=0" hence ?th by (simp add: H NDVD_def) }
+  moreover {assume inz: "i\<noteq>0" and "n<0" hence ?th
+      by (simp add: NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1
+        rdvd_minus[where d="i" and t="real_of_int n * x + Inum (x # bs) s"]) }
+  moreover {assume inz: "i\<noteq>0" and "n>0" hence ?th
       by (simp add:NDVD_def H NDVDJ_NDVD[OF xp x1] rdvd_abs1)}
   ultimately show ?th by blast
 qed
 
 lemma DVD_l: "isrlfm (rsplit (DVD i) a)"
-  by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l) 
+  by (rule rsplit_l[where f="DVD i" and a="a"], auto simp add: DVD_def eq_def DVDJ_l)
 (case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all)
 
 lemma NDVD_l: "isrlfm (rsplit (NDVD i) a)"
-  by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l) 
+  by (rule rsplit_l[where f="NDVD i" and a="a"], auto simp add: NDVD_def neq_def NDVDJ_l)
 (case_tac s, simp_all, rename_tac nat a b, case_tac "nat", simp_all)
 
 consts rlfm :: "fm \<Rightarrow> fm"
@@ -3948,20 +3948,20 @@
 
 lemma simpfm_rl: "isrlfm p \<Longrightarrow> isrlfm (simpfm p)"
 proof (induct p)
-  case (Lt a) 
+  case (Lt a)
   hence "bound0 (Lt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all)
   moreover
-  {assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))"  
+  {assume "bound0 (Lt a)" hence bn:"bound0 (simpfm (Lt a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (Lt a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -3972,20 +3972,20 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (Le a)   
+  case (Le a)
   hence "bound0 (Le a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all)
   moreover
-  { assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"  
+  { assume "bound0 (Le a)" hence bn:"bound0 (simpfm (Le a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (Le a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -3996,20 +3996,20 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (Gt a)   
+  case (Gt a)
   hence "bound0 (Gt a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a, simp_all, rename_tac nat a b,case_tac "nat", simp_all)
   moreover
-  {assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))"  
+  {assume "bound0 (Gt a)" hence bn:"bound0 (simpfm (Gt a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (Gt a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1: "numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -4020,20 +4020,20 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (Ge a)   
+  case (Ge a)
   hence "bound0 (Ge a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all)
   moreover
-  { assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"  
+  { assume "bound0 (Ge a)" hence bn:"bound0 (simpfm (Ge a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (Ge a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -4044,20 +4044,20 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (Eq a)   
+  case (Eq a)
   hence "bound0 (Eq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all)
   moreover
-  { assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"  
+  { assume "bound0 (Eq a)" hence bn:"bound0 (simpfm (Eq a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (Eq a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -4068,20 +4068,20 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (NEq a)  
+  case (NEq a)
   hence "bound0 (NEq a) \<or> (\<exists> c e. a = CN 0 c e \<and> c > 0 \<and> numbound0 e)"
     by (cases a,simp_all, rename_tac nat a b, case_tac "nat", simp_all)
   moreover
-  {assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))"  
+  {assume "bound0 (NEq a)" hence bn:"bound0 (simpfm (NEq a))"
       using simpfm_bound0 by blast
     have "isatom (simpfm (NEq a))" by (cases "simpnum a", auto simp add: Let_def)
     with bn bound0at_l have ?case by blast}
-  moreover 
+  moreover
   { fix c e assume a: "a = CN 0 c e" and "c>0" and "numbound0 e"
     { assume cn1:"numgcd (CN 0 c (simpnum e)) \<noteq> 1" and cnz:"numgcd (CN 0 c (simpnum e)) \<noteq> 0"
       with numgcd_pos[where t="CN 0 c (simpnum e)"]
       have th1:"numgcd (CN 0 c (simpnum e)) > 0" by simp
-      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c" 
+      from \<open>c > 0\<close> have th:"numgcd (CN 0 c (simpnum e)) \<le> c"
         by (simp add: numgcd_def)
       from \<open>c > 0\<close> have th': "c\<noteq>0" by auto
       from \<open>c > 0\<close> have cp: "c \<ge> 0" by simp
@@ -4092,12 +4092,12 @@
       by (simp add: Let_def reducecoeff_def reducecoeffh_numbound0)}
   ultimately show ?case by blast
 next
-  case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))"  
+  case (Dvd i a) hence "bound0 (Dvd i a)" by auto hence bn:"bound0 (simpfm (Dvd i a))"
     using simpfm_bound0 by blast
   have "isatom (simpfm (Dvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
   with bn bound0at_l show ?case by blast
 next
-  case (NDvd i a)  hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))"  
+  case (NDvd i a)  hence "bound0 (NDvd i a)" by auto hence bn:"bound0 (simpfm (NDvd i a))"
     using simpfm_bound0 by blast
   have "isatom (simpfm (NDvd i a))" by (cases "simpnum a", auto simp add: Let_def split_def)
   with bn bound0at_l show ?case by blast
@@ -4107,15 +4107,15 @@
   assumes qfp: "qfree p"
   and xp: "0 \<le> x" and x1: "x < 1"
   shows "(Ifm (x#bs) (rlfm p) = Ifm (x# bs) p) \<and> isrlfm (rlfm p)"
-  using qfp 
-by (induct p rule: rlfm.induct) 
+  using qfp
+by (induct p rule: rlfm.induct)
 (auto simp add: rsplit[OF xp x1 lt_mono] lt_l rsplit[OF xp x1 le_mono] le_l rsplit[OF xp x1 gt_mono] gt_l
                rsplit[OF xp x1 ge_mono] ge_l rsplit[OF xp x1 eq_mono] eq_l rsplit[OF xp x1 neq_mono] neq_l
                rsplit[OF xp x1 DVD_mono[OF xp x1]] DVD_l rsplit[OF xp x1 NDVD_mono[OF xp x1]] NDVD_l simpfm_rl)
 lemma rlfm_l:
   assumes qfp: "qfree p"
   shows "isrlfm (rlfm p)"
-  using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l 
+  using qfp lt_l gt_l ge_l le_l eq_l neq_l DVD_l NDVD_l
 by (induct p rule: rlfm.induct) (auto simp add: simpfm_rl)
 
     (* Operations needed for Ferrante and Rackoff *)
@@ -4128,7 +4128,7 @@
 next
   case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
 next
-  case (3 c e) 
+  case (3 c e)
   from 3 have nb: "numbound0 e" by simp
   from 3 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4136,8 +4136,8 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (Eq (CN 0 c e))"
@@ -4145,7 +4145,7 @@
   hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (4 c e)   
+  case (4 c e)
   from 4 have nb: "numbound0 e" by simp
   from 4 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4153,8 +4153,8 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     hence "real_of_int c * x + ?e \<noteq> 0" by simp
     with xz have "?P ?z x (NEq (CN 0 c e))"
@@ -4162,7 +4162,7 @@
   hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   from 5 have nb: "numbound0 e" by simp
   from 5 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4170,15 +4170,15 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Lt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
   hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (6 c e)  
+  case (6 c e)
   from 6 have nb: "numbound0 e" by simp
   from 6 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4186,15 +4186,15 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Le (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (7 c e)  
+  case (7 c e)
   from 7 have nb: "numbound0 e" by simp
   from 7 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4202,15 +4202,15 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Gt (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
   hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (8 c e)  
+  case (8 c e)
   from 8 have nb: "numbound0 e" by simp
   from 8 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4218,8 +4218,8 @@
   let ?z = "(- ?e) / real_of_int c"
   {fix x
     assume xz: "x < ?z"
-    hence "(real_of_int c * x < - ?e)" 
-      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps) 
+    hence "(real_of_int c * x < - ?e)"
+      by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] ac_simps)
     hence "real_of_int c * x + ?e < 0" by arith
     with xz have "?P ?z x (Ge (CN 0 c e))"
       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
@@ -4236,7 +4236,7 @@
 next
   case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
 next
-  case (3 c e) 
+  case (3 c e)
   from 3 have nb: "numbound0 e" by simp
   from 3 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4253,7 +4253,7 @@
   hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (4 c e) 
+  case (4 c e)
   from 4 have nb: "numbound0 e" by simp
   from 4 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4270,7 +4270,7 @@
   hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (5 c e) 
+  case (5 c e)
   from 5 have nb: "numbound0 e" by simp
   from 5 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4286,7 +4286,7 @@
   hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (6 c e) 
+  case (6 c e)
   from 6 have nb: "numbound0 e" by simp
   from 6 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4302,7 +4302,7 @@
   hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (7 c e) 
+  case (7 c e)
   from 7 have nb: "numbound0 e" by simp
   from 7 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4318,7 +4318,7 @@
   hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
   thus ?case by blast
 next
-  case (8 c e) 
+  case (8 c e)
   from 8 have nb: "numbound0 e" by simp
   from 8 have cp: "real_of_int c > 0" by simp
   fix a
@@ -4354,7 +4354,7 @@
 proof-
   from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
   have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
-  from rminusinf_inf[OF lp, where bs="bs"] 
+  from rminusinf_inf[OF lp, where bs="bs"]
   obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
   from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
   moreover have "z - 1 < z" by simp
@@ -4368,19 +4368,19 @@
 proof-
   from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
   have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
-  from rplusinf_inf[OF lp, where bs="bs"] 
+  from rplusinf_inf[OF lp, where bs="bs"]
   obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
   from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
   moreover have "z + 1 > z" by simp
   ultimately show ?thesis using z_def by auto
 qed
 
-consts 
+consts
   \<Upsilon>:: "fm \<Rightarrow> (num \<times> int) list"
   \<upsilon> :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
 recdef \<Upsilon> "measure size"
-  "\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)" 
-  "\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)" 
+  "\<Upsilon> (And p q) = (\<Upsilon> p @ \<Upsilon> q)"
+  "\<Upsilon> (Or p q) = (\<Upsilon> p @ \<Upsilon> q)"
   "\<Upsilon> (Eq  (CN 0 c e)) = [(Neg e,c)]"
   "\<Upsilon> (NEq (CN 0 c e)) = [(Neg e,c)]"
   "\<Upsilon> (Lt  (CN 0 c e)) = [(Neg e,c)]"
@@ -4410,10 +4410,10 @@
   have "?I ?u (Lt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) < 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) < 0)"
-    by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) < 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (6 c e)
@@ -4421,10 +4421,10 @@
   have "?I ?u (Le (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<le> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
-    by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<le> 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (7 c e)
@@ -4432,10 +4432,10 @@
   have "?I ?u (Gt (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) > 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) > 0)"
-    by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) > 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (8 c e)
@@ -4443,10 +4443,10 @@
   have "?I ?u (Ge (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<ge> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
-    by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<ge> 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (3 c e)
@@ -4455,10 +4455,10 @@
   have "?I ?u (Eq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) = 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) = 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) = 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 next
   case (4 c e)
@@ -4467,10 +4467,10 @@
   have "?I ?u (NEq (CN 0 c e)) = (real_of_int c *(?t/?n) + (?N x e) \<noteq> 0)"
     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
   also have "\<dots> = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
-    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)" 
+    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
   also have "\<dots> = (real_of_int c *?t + ?n* (?N x e) \<noteq> 0)"
-    using np by simp 
+    using np by simp
   finally show ?case using nbt nb by (simp add: algebra_simps)
 qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real_of_int n" and b'="x"])
 
@@ -4491,7 +4491,7 @@
     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
   then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<ge> ?N a s" by blast
   from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto
-  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real_of_int m" 
+  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real_of_int m"
     by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
@@ -4507,14 +4507,14 @@
     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
   then obtain s m where smU: "(s,m) \<in> set (\<Upsilon> p)" and mx: "real_of_int m * x \<le> ?N a s" by blast
   from \<Upsilon>_l[OF lp] smU have mp: "real_of_int m > 0" by auto
-  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real_of_int m" 
+  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real_of_int m"
     by (auto simp add: mult.commute)
   thus ?thesis using smU by auto
 qed
 
-lemma lin_dense: 
+lemma lin_dense:
   assumes lp: "isrlfm p"
-  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real_of_int n) ` set (\<Upsilon> p)" 
+  and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real_of_int n) ` set (\<Upsilon> p)"
   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real_of_int n ) ` (?U p)")
   and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
   and ly: "l < y" and yu: "y < u"
@@ -4523,7 +4523,7 @@
 proof (induct p rule: isrlfm.induct)
   case (5 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
   from 5 have "x * real_of_int c + ?N x e < 0" by (simp add: algebra_simps)
-  hence pxc: "x < (- ?N x e) / real_of_int c" 
+  hence pxc: "x < (- ?N x e) / real_of_int c"
     by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
   from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
@@ -4533,7 +4533,7 @@
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
     hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real_of_int c" 
+  moreover {assume y: "y > (- ?N x e) / real_of_int c"
     with yu have eu: "u > (- ?N x e) / real_of_int c" by auto
     with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with lx pxc have "False" by auto
@@ -4542,7 +4542,7 @@
 next
   case (6 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
   from 6 have "x * real_of_int c + ?N x e \<le> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<le> (- ?N x e) / real_of_int c" 
+  hence pxc: "x \<le> (- ?N x e) / real_of_int c"
     by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
   from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
@@ -4552,7 +4552,7 @@
       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
     hence "real_of_int c * y + ?N x e < 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y > (- ?N x e) / real_of_int c" 
+  moreover {assume y: "y > (- ?N x e) / real_of_int c"
     with yu have eu: "u > (- ?N x e) / real_of_int c" by auto
     with noSc ly yu have "(- ?N x e) / real_of_int c \<le> l" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with lx pxc have "False" by auto
@@ -4561,7 +4561,7 @@
 next
   case (7 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
   from 7 have "x * real_of_int c + ?N x e > 0" by (simp add: algebra_simps)
-  hence pxc: "x > (- ?N x e) / real_of_int c" 
+  hence pxc: "x > (- ?N x e) / real_of_int c"
     by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
   from 7 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
@@ -4571,7 +4571,7 @@
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
     hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real_of_int c" 
+  moreover {assume y: "y < (- ?N x e) / real_of_int c"
     with ly have eu: "l < (- ?N x e) / real_of_int c" by auto
     with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with xu pxc have "False" by auto
@@ -4580,7 +4580,7 @@
 next
   case (8 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
   from 8 have "x * real_of_int c + ?N x e \<ge> 0" by (simp add: algebra_simps)
-  hence pxc: "x \<ge> (- ?N x e) / real_of_int c" 
+  hence pxc: "x \<ge> (- ?N x e) / real_of_int c"
     by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
   from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
@@ -4590,7 +4590,7 @@
       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
     hence "real_of_int c * y + ?N x e > 0" by (simp add: algebra_simps)
     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
-  moreover {assume y: "y < (- ?N x e) / real_of_int c" 
+  moreover {assume y: "y < (- ?N x e) / real_of_int c"
     with ly have eu: "l < (- ?N x e) / real_of_int c" by auto
     with noSc ly yu have "(- ?N x e) / real_of_int c \<ge> u" by (cases "(- ?N x e) / real_of_int c > l", auto)
     with xu pxc have "False" by auto
@@ -4600,7 +4600,7 @@
   case (3 c e) hence cp: "real_of_int c > 0" and nb: "numbound0 e" by simp_all
   from cp have cnz: "real_of_int c \<noteq> 0" by simp
   from 3 have "x * real_of_int c + ?N x e = 0" by (simp add: algebra_simps)
-  hence pxc: "x = (- ?N x e) / real_of_int c" 
+  hence pxc: "x = (- ?N x e) / real_of_int c"
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
   from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with lx xu have yne: "x \<noteq> - ?N x e / real_of_int c" by auto
@@ -4610,10 +4610,10 @@
   from cp have cnz: "real_of_int c \<noteq> 0" by simp
   from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real_of_int c" by auto
   with ly yu have yne: "y \<noteq> - ?N x e / real_of_int c" by auto
-  hence "y* real_of_int c \<noteq> -?N x e"      
+  hence "y* real_of_int c \<noteq> -?N x e"
     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
   hence "y* real_of_int c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
-  thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
+  thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
     by (simp add: algebra_simps)
 qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
 
@@ -4623,7 +4623,7 @@
   and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
   and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
   shows "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p).
-    ?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p" 
+    ?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p"
 proof-
   let ?N = "\<lambda> x t. Inum (x#bs) t"
   let ?U = "set (\<Upsilon> p)"
@@ -4636,10 +4636,10 @@
   proof-
     let ?M = "(\<lambda> (t,c). ?N a t / real_of_int c) ` ?U"
     have fM: "finite ?M" by auto
-    from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa] 
+    from rminusinf_\<Upsilon>[OF lp nmi pa] rplusinf_\<Upsilon>[OF lp npi pa]
     have "\<exists> (l,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). a \<le> ?N x l / real_of_int n \<and> a \<ge> ?N x s / real_of_int m" by blast
-    then obtain "t" "n" "s" "m" where 
-      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
+    then obtain "t" "n" "s" "m" where
+      tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U"
       and xs1: "a \<le> ?N x s / real_of_int m" and tx1: "a \<ge> ?N x t / real_of_int n" by blast
     from \<Upsilon>_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real_of_int m" and tx: "a \<ge> ?N a t / real_of_int n" by auto
     from tnU have Mne: "?M \<noteq> {}" by auto
@@ -4649,23 +4649,23 @@
     have linM: "?l \<in> ?M" using fM Mne by simp
     have uinM: "?u \<in> ?M" using fM Mne by simp
     have tnM: "?N a t / real_of_int n \<in> ?M" using tnU by auto
-    have smM: "?N a s / real_of_int m \<in> ?M" using smU by auto 
+    have smM: "?N a s / real_of_int m \<in> ?M" using smU by auto
     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
     have "?l \<le> ?N a t / real_of_int n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
     have "?N a s / real_of_int m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
-    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
+    have "(\<exists> s\<in> ?M. ?I s p) \<or>
       (\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
     moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
       hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real_of_int nu" by auto
       then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real_of_int nu" by blast
-      have "(u + u) / 2 = u" by auto with pu tuu 
+      have "(u + u) / 2 = u" by auto with pu tuu
       have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int 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" 
+      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
         and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
         by blast
       from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real_of_int t1n" by auto
@@ -4679,10 +4679,10 @@
       with t1uU t2uU t1u t2u have ?thesis by blast}
     ultimately show ?thesis by blast
   qed
-  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
+  then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U"
     and pu: "?I ((?N a l / real_of_int n + ?N a s / real_of_int m) / 2) p" by blast
   from lnU smU \<Upsilon>_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
-  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
+  from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
   have "?I ((?N x l / real_of_int n + ?N x s / real_of_int m) / 2) p" by simp
   with lnU smU
@@ -4690,7 +4690,7 @@
 qed
     (* The Ferrante - Rackoff Theorem *)
 
-theorem fr_eq: 
+theorem fr_eq:
   assumes lp: "isrlfm p"
   shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> p). \<exists> (s,m) \<in> set (\<Upsilon> p). Ifm ((((Inum (x#bs) t)/  real_of_int n + (Inum (x#bs) s) / real_of_int m) /2)#bs) p))"
   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
@@ -4702,7 +4702,7 @@
     from rinf_\<Upsilon>[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
   ultimately show "?D" by blast
 next
-  assume "?D" 
+  assume "?D"
   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
   moreover {assume f:"?F" hence "?E" by blast}
@@ -4710,7 +4710,7 @@
 qed
 
 
-lemma fr_eq_\<upsilon>: 
+lemma fr_eq_\<upsilon>:
   assumes lp: "isrlfm p"
   shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (\<Upsilon> p). \<exists> (s,l) \<in> set (\<Upsilon> p). Ifm (x#bs) (\<upsilon> p (Add(Mul l t) (Mul k s) , 2*k*l))))"
   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
@@ -4729,15 +4729,15 @@
       from tnb snb have st_nb: "numbound0 ?st" by simp
       have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
-      from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
+      from \<upsilon>_I[OF lp mnp st_nb, where x="x" and bs="bs"]
       have "?I x (\<upsilon> p (?st,2*n*m)) = ?I ((?N t / real_of_int n + ?N s / real_of_int m) /2) p" by (simp only: st[symmetric])}
     with rinf_\<Upsilon>[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
   ultimately show "?D" by blast
 next
-  assume "?D" 
+  assume "?D"
   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
-  moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)" 
+  moreover {fix t k s l assume "(t,k) \<in> set (\<Upsilon> p)" and "(s,l) \<in> set (\<Upsilon> p)"
     and px:"?I x (\<upsilon> p (Add (Mul l t) (Mul k s), 2*k*l))"
     with \<Upsilon>_l[OF lp] have tnb: "numbound0 t" and np:"real_of_int k > 0" and snb: "numbound0 s" and mp:"real_of_int l > 0" by auto
     let ?st = "Add (Mul l t) (Mul k s)"
@@ -4759,7 +4759,7 @@
   have "x = real_of_int ?i + ?u" by simp
   hence "P (real_of_int ?i + ?u)" using Px by simp
   moreover have "real_of_int ?i \<le> x" using of_int_floor_le by simp hence "0 \<le> ?u" by arith
-  moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith 
+  moreover have "?u < 1" using real_of_int_floor_add_one_gt[where r="x"] by arith
   ultimately show "(\<exists> (i::int) (u::real). 0\<le> u \<and> u< 1 \<and> P (real_of_int i + u))" by blast
 qed
 
@@ -4792,11 +4792,11 @@
 | "exsplit (NOT p) = NOT (exsplit p)"
 | "exsplit p = p"
 
-lemma exsplitnum: 
+lemma exsplitnum:
   "Inum (x#y#bs) (exsplitnum t) = Inum ((x+y) #bs) t"
   by(induct t rule: exsplitnum.induct) (simp_all add: algebra_simps)
 
-lemma exsplit: 
+lemma exsplit:
   assumes qfp: "qfree p"
   shows "Ifm (x#y#bs) (exsplit p) = Ifm ((x+y)#bs) p"
 using qfp exsplitnum[where x="x" and y="y" and bs="bs"]
@@ -4810,7 +4810,7 @@
     by (simp add: myless[of _ "1"] myless[of _ "0"] ac_simps)
   also have "\<dots> = (\<exists> (i::int). \<exists> x. 0\<le> x \<and> x < 1 \<and> Ifm ((real_of_int i + x) #bs) p)"
     by (simp only: exsplit[OF qf] ac_simps)
-  also have "\<dots> = (\<exists> x. Ifm (x#bs) p)" 
+  also have "\<dots> = (\<exists> x. Ifm (x#bs) p)"
     by (simp only: real_ex_int_real01[where P="\<lambda> x. Ifm (x#bs) p"])
   finally show ?thesis by simp
 qed
@@ -4819,12 +4819,12 @@
 
 definition ferrack01 :: "fm \<Rightarrow> fm" where
   "ferrack01 p \<equiv> (let p' = rlfm(And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p);
-                    U = remdups(map simp_num_pair 
+                    U = remdups(map simp_num_pair
                      (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
-                           (alluopairs (\<Upsilon> p')))) 
+                           (alluopairs (\<Upsilon> p'))))
   in decr (evaldjf (\<upsilon> p') U ))"
 
-lemma fr_eq_01: 
+lemma fr_eq_01:
   assumes qf: "qfree p"
   shows "(\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (\<exists> (t,n) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). \<exists> (s,m) \<in> set (\<Upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p))). Ifm (x#bs) (\<upsilon> (rlfm (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) (Add (Mul m t) (Mul n s), 2*n*m)))"
   (is "(\<exists> x. ?I x ?q) = ?F")
@@ -4837,18 +4837,18 @@
     by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))", simp_all)
   have PF: "?P = False" apply (simp add: Let_def reducecoeff_def numgcd_def rsplit_def ge_def lt_def conj_def disj_def)
     by (cases "rlfm p = And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))", simp_all)
-  have "(\<exists> x. ?I x ?q ) = 
+  have "(\<exists> x. ?I x ?q ) =
     ((?I x (minusinf ?rq)) \<or> (?I x (plusinf ?rq )) \<or> (\<exists> (t,n) \<in> set (\<Upsilon> ?rq). \<exists> (s,m) \<in> set (\<Upsilon> ?rq ). ?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))))"
     (is "(\<exists> x. ?I x ?q) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
   proof
-    assume "\<exists> x. ?I x ?q"  
+    assume "\<exists> x. ?I x ?q"
     then obtain x where qx: "?I x ?q" by blast
-    hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p" 
+    hence xp: "0\<le> x" and x1: "x< 1" and px: "?I x p"
       by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf])
-    from qx have "?I x ?rq " 
+    from qx have "?I x ?rq "
       by (simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
     hence lqx: "?I x ?rq " using simpfm[where p="?rq" and bs="x#bs"] by auto
-    from qf have qfq:"isrlfm ?rq"  
+    from qf have qfq:"isrlfm ?rq"
       by (auto simp add: rsplit_def lt_def ge_def rlfm_I[OF qf xp x1])
     with lqx fr_eq_\<upsilon>[OF qfq] show "?M \<or> ?P \<or> ?F" by blast
   next
@@ -4856,7 +4856,7 @@
     let ?U = "set (\<Upsilon> ?rq )"
     from MF PF D have "?F" by auto
     then obtain t n s m where aU:"(t,n) \<in> ?U" and bU:"(s,m)\<in> ?U" and rqx: "?I x (\<upsilon> ?rq (Add (Mul m t) (Mul n s), 2*n*m))" by blast
-    from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf] 
+    from qf have lrq:"isrlfm ?rq"using rlfm_l[OF qf]
       by (auto simp add: rsplit_def lt_def ge_def)
     from aU bU \<Upsilon>_l[OF lrq] have tnb: "numbound0 t" and np:"real_of_int n > 0" and snb: "numbound0 s" and mp:"real_of_int m > 0" by (auto simp add: split_def)
     let ?st = "Add (Mul m t) (Mul n s)"
@@ -4864,7 +4864,7 @@
     from np mp have mnp: "real_of_int (2*n*m) > 0" by (simp add: mult.commute)
     from conjunct1[OF \<upsilon>_I[OF lrq mnp stnb, where bs="bs" and x="x"], symmetric] rqx
     have "\<exists> x. ?I x ?rq" by auto
-    thus "?E" 
+    thus "?E"
       using rlfm_I[OF qf] by (auto simp add: rsplit_def lt_def ge_def)
   qed
   with MF PF show ?thesis by blast
@@ -4882,25 +4882,25 @@
   let ?N = "\<lambda> t. Inum (x#bs) t"
   let ?st= "Add (Mul m t) (Mul n s)"
   from Ul th have mnz: "m \<noteq> 0" by auto
-  from Ul th have  nnz: "n \<noteq> 0" by auto  
+  from Ul th have  nnz: "n \<noteq> 0" by auto
   have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
- 
+
   thus "(real_of_int m *  Inum (x # bs) t + real_of_int n * Inum (x # bs) s) /
        (2 * real_of_int n * real_of_int m)
        \<in> (\<lambda>((t, n), s, m).
              (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) `
-         (set U \<times> set U)"using mnz nnz th  
+         (set U \<times> set U)"using mnz nnz th
     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
-    by (rule_tac x="(s,m)" in bexI,simp_all) 
+    by (rule_tac x="(s,m)" in bexI,simp_all)
   (rule_tac x="(t,n)" in bexI,simp_all add: mult.commute)
 next
   fix t n s m
-  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
+  assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U"
   let ?N = "\<lambda> t. Inum (x#bs) t"
   let ?st= "Add (Mul m t) (Mul n s)"
   from Ul smU have mnz: "m \<noteq> 0" by auto
-  from Ul tnU have  nnz: "n \<noteq> 0" by auto  
+  from Ul tnU have  nnz: "n \<noteq> 0" by auto
   have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
  let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2"
@@ -4910,13 +4910,13 @@
  from alluopairs_ex[OF Pc, where xs="U"] tnU smU
  have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
    by blast
- then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
+ then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
    and Pts': "?P (t',n') (s',m')" by blast
  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
  let ?st' = "Add (Mul m' t') (Mul n' s')"
    have st': "(?N t' / real_of_int n' + ?N s' / real_of_int m')/2 = ?N ?st' / real_of_int (2*n'*m')"
    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
- from Pts' have 
+ from Pts' have
    "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 = (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m')/2" by simp
  also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real_of_int n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
  finally show "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2
@@ -4935,13 +4935,13 @@
   (is "?lhs = ?rhs")
 proof
   assume ?lhs
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
+  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
     Pst: "Ifm (x#bs) (\<upsilon> p (Add (Mul m t) (Mul n s),2*n*m))" by blast
   let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
+  from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
-  from np mp have mnp: "real_of_int (2*n*m) > 0" 
+  from np mp have mnp: "real_of_int (2*n*m) > 0"
       by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
@@ -4951,25 +4951,25 @@
     by auto (rule_tac x="(a,b)" in bexI, auto)
   then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
   from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int n' > 0" by auto
-  from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
+  from \<upsilon>_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp
   from conjunct1[OF \<upsilon>_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
-  have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st) 
-  then show ?rhs using tnU' by auto 
+  have "Ifm (x # bs) (\<upsilon> p (t', n')) " by (simp only: st)
+  then show ?rhs using tnU' by auto
 next
   assume ?rhs
-  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (\<upsilon> p (t', n'))" 
+  then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (\<upsilon> p (t', n'))"
     by blast
   from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
-  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
+  hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))"
     by auto (rule_tac x="(a,b)" in bexI, auto)
-  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
+  then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
     th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
     let ?N = "\<lambda> t. Inum (x#bs) t"
-  from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
+  from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
     and snb: "numbound0 s" and mp:"m > 0"  by auto
   let ?st= "Add (Mul m t) (Mul n s)"
-  from np mp have mnp: "real_of_int (2*n*m) > 0" 
+  from np mp have mnp: "real_of_int (2*n*m) > 0"
       by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
     from tnb snb have stnb: "numbound0 ?st" by simp
   have st: "(?N t / real_of_int n + ?N s / real_of_int m)/2 = ?N ?st / real_of_int (2*n*m)"
@@ -4979,8 +4979,8 @@
   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real_of_int (2 * n * m) # bs) p" by simp
   with \<upsilon>_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
 qed
-  
-lemma ferrack01: 
+
+lemma ferrack01:
   assumes qf: "qfree p"
   shows "((\<exists> x. Ifm (x#bs) (And (And (Ge(CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) p)) = (Ifm bs (ferrack01 p))) \<and> qfree (ferrack01 p)" (is "(?lhs = ?rhs) \<and> _")
 proof-
@@ -4998,17 +4998,17 @@
   let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real_of_int n + ?N s/ real_of_int m) /2"
   let ?F = "\<lambda> p. \<exists> a \<in> set (\<Upsilon> p). \<exists> b \<in> set (\<Upsilon> p). ?I x (\<upsilon> p (?g(a,b)))"
   let ?ep = "evaldjf (\<upsilon> ?q) ?Y"
-  from rlfm_l[OF qf] have lq: "isrlfm ?q" 
+  from rlfm_l[OF qf] have lq: "isrlfm ?q"
     by (simp add: rsplit_def lt_def ge_def conj_def disj_def Let_def reducecoeff_def numgcd_def)
   from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
   from \<Upsilon>_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
-  from U_l UpU 
+  from U_l UpU
   have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
   hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
     by (auto)
-  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
+  have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0"
   proof-
-    { fix t n assume tnY: "(t,n) \<in> set ?Y" 
+    { fix t n assume tnY: "(t,n) \<in> set ?Y"
       hence "(t,n) \<in> set ?SS" by simp
       hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
         by (auto simp add: split_def simp del: map_map)
@@ -5022,12 +5022,12 @@
 
   have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
   proof-
-     from simp_num_pair_ci[where bs="x#bs"] have 
+     from simp_num_pair_ci[where bs="x#bs"] have
     "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
      hence th: "?f o simp_num_pair = ?f" using ext by blast
     have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_comp comp_assoc)
     also have "\<dots> = (?f ` set ?S)" by (simp add: th)
-    also have "\<dots> = ((?f o ?g) ` set ?Up)" 
+    also have "\<dots> = ((?f o ?g) ` set ?Up)"
       by (simp only: set_map o_def image_comp)
     also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
       using \<Upsilon>_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_comp] by blast
@@ -5047,8 +5047,8 @@
   from fr_eq_01[OF qf, where bs="bs" and x="x"] have "?lhs = ?F ?q"
     by (simp only: split_def fst_conv snd_conv)
   also have "\<dots> = (\<exists> (t,n) \<in> set ?Y. ?I x (\<upsilon> ?q (t,n)))" using \<Upsilon>_cong[OF lq YU U_l Y_l]
-    by (simp only: split_def fst_conv snd_conv) 
-  also have "\<dots> = (Ifm (x#bs) ?ep)" 
+    by (simp only: split_def fst_conv snd_conv)
+  also have "\<dots> = (Ifm (x#bs) ?ep)"
     using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="\<upsilon> ?q",symmetric]
     by (simp only: split_def prod.collapse)
   also have "\<dots> = (Ifm bs (decr ?ep))" using decr[OF ep_nb] by blast
@@ -5057,7 +5057,7 @@
   with lr show ?thesis by blast
 qed
 
-lemma cp_thm': 
+lemma cp_thm':
   assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
   and up: "d_\<beta> p 1" and dd: "d_\<delta> p d" and dp: "d > 0"
   shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (real_of_int i#bs)) ` set (\<beta> p). Ifm ((b+real_of_int j)#bs) p))"
@@ -5070,12 +5070,12 @@
 
 lemma unit: assumes qf: "qfree p"
   shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow>
-      ((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
-       (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> 
-       (Inum (real_of_int i#bs)) ` set B = (Inum (real_of_int i#bs)) ` set (\<beta> q) \<and> 
+      ((\<exists> (x::int). Ifm (real_of_int x#bs) p) =
+       (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and>
+       (Inum (real_of_int i#bs)) ` set B = (Inum (real_of_int i#bs)) ` set (\<beta> q) \<and>
        d_\<beta> q 1 \<and> d_\<delta> q d \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> b\<in> set B. numbound0 b)"
 proof-
-  fix q B d 
+  fix q B d
   assume qBd: "unit p = (q,B,d)"
   let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and>
     Inum (real_of_int i#bs) ` set B = Inum (real_of_int i#bs) ` set (\<beta> q) \<and>
@@ -5089,22 +5089,22 @@
   let ?B'= "remdups (map simpnum (\<beta> ?q))"
   let ?A = "set (\<alpha> ?q)"
   let ?A'= "remdups (map simpnum (\<alpha> ?q))"
-  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
+  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]]
   have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
   from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]]
-  have lp': "\<forall> (i::int). iszlfm ?p' (real_of_int i#bs)" by simp 
+  have lp': "\<forall> (i::int). iszlfm ?p' (real_of_int i#bs)" by simp
   hence lp'': "iszlfm ?p' (real_of_int (i::int)#bs)" by simp
   from lp' \<zeta>[where p="?p'" and bs="bs"] have lp: "?l >0" and dl: "d_\<beta> ?p' ?l" by auto
   from a_\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp'' dl lp] pp'
-  have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff) 
-  from lp'' lp a_\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real_of_int i#bs)" and uq: "d_\<beta> ?q 1" 
+  have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by (simp add: int_rdvd_iff)
+  from lp'' lp a_\<beta>[OF lp'' dl lp] have lq:"iszlfm ?q (real_of_int i#bs)" and uq: "d_\<beta> ?q 1"
     by (auto simp add: isint_def)
   from \<delta>[OF lq] have dp:"?d >0" and dd: "d_\<delta> ?q ?d" by blast+
   let ?N = "\<lambda> t. Inum (real_of_int (i::int)#bs) t"
-  have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_comp) 
+  have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by (simp add:image_comp)
   also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="real_of_int i #bs"] by auto
   finally have BB': "?N ` set ?B' = ?N ` ?B" .
-  have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_comp) 
+  have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by (simp add:image_comp)
   also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"] by auto
   finally have AA': "?N ` set ?A' = ?N ` ?A" .
   from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
@@ -5114,16 +5114,16 @@
   { assume "length ?B' \<le> length ?A'"
     hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
       using qBd by (auto simp add: Let_def unit_def)
-    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)" 
+    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)"
       and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
     with pq_ex dp uq dd lq q d have ?thes by simp }
-  moreover 
+  moreover
   { assume "\<not> (length ?B' \<le> length ?A')"
     hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
       using qBd by (auto simp add: Let_def unit_def)
-    with AA' mirror_\<alpha>_\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)" 
+    with AA' mirror_\<alpha>_\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)"
       and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
-    from mirror_ex[OF lq] pq_ex q 
+    from mirror_ex[OF lq] pq_ex q
     have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
     from lq uq q mirror_d_\<beta> [where p="?q" and bs="bs" and a="real_of_int i"]
     have lq': "iszlfm q (real_of_int i#bs)" and uq: "d_\<beta> q 1" by auto
@@ -5135,17 +5135,17 @@
     (* Cooper's Algorithm *)
 
 definition cooper :: "fm \<Rightarrow> fm" where
-  "cooper p \<equiv> 
+  "cooper p \<equiv>
   (let (q,B,d) = unit p; js = [1..d];
        mq = simpfm (minusinf q);
        md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
    in if md = T then T else
-    (let qd = evaldjf (\<lambda> t. simpfm (subst0 t q)) 
-                               (remdups (map (\<lambda> (b,j). simpnum (Add b (C j))) 
+    (let qd = evaldjf (\<lambda> t. simpfm (subst0 t q))
+                               (remdups (map (\<lambda> (b,j). simpnum (Add b (C j)))
                                             [(b,j). b\<leftarrow>B,j\<leftarrow>js]))
      in decr (disj md qd)))"
 lemma cooper: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)" 
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (cooper p))) \<and> qfree (cooper p)"
   (is "(?lhs = ?rhs) \<and> _")
 proof-
 
@@ -5163,65 +5163,65 @@
   let ?sbjs = "map (\<lambda> (b,j). simpnum (Add b (C j))) ?bjs"
   let ?qd = "evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs)"
   have qbf:"unit p = (?q,?B,?d)" by simp
-  from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
-    B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and 
-    uq:"d_\<beta> ?q 1" and dd: "d_\<delta> ?q ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real_of_int i#bs)" and 
+  from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and
+    B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and
+    uq:"d_\<beta> ?q 1" and dd: "d_\<delta> ?q ?d" and dp: "?d > 0" and
+    lq: "iszlfm ?q (real_of_int i#bs)" and
     Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
   from zlin_qfree[OF lq] have qfq: "qfree ?q" .
   from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
   have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
-  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" 
+  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)"
     by (auto simp only: subst0_bound0[OF qfmq])
   hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
     by auto
-  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
+  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp
   from Bn jsnb have "\<forall> (b,j) \<in> set ?bjs. numbound0 (Add b (C j))"
     by simp
   hence "\<forall> (b,j) \<in> set ?bjs. numbound0 (simpnum (Add b (C j)))"
     using simpnum_numbound0 by blast
   hence "\<forall> t \<in> set ?sbjs. numbound0 t" by simp
   hence "\<forall> t \<in> set (remdups ?sbjs). bound0 (subst0 t ?q)"
-    using subst0_bound0[OF qfq] by auto 
+    using subst0_bound0[OF qfq] by auto
   hence th': "\<forall> t \<in> set (remdups ?sbjs). bound0 (simpfm (subst0 t ?q))"
     using simpfm_bound0 by blast
   from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
-  from mdb qdb 
+  from mdb qdb
   have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
   from trans [OF pq_ex cp_thm'[OF lq uq dd dp]] B
   have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm ((b+ real_of_int j)#bs) ?q))" by auto
   also have "\<dots> = ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> (b,j) \<in> (?N ` set ?B \<times> set ?js). Ifm ((b+ real_of_int j)#bs) ?q))" by auto
   also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (Add b (C j))) ` set ?bjs. Ifm (t #bs) ?q))" by simp
   also have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> (\<lambda> (b,j). ?N (simpnum (Add b (C j)))) ` set ?bjs. Ifm (t #bs) ?q))" by (simp only: simpnum_ci)
-  also  have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))" 
-    by (auto simp add: split_def) 
+  also  have "\<dots>= ((\<exists> j\<in> set ?js. ?I j ?smq) \<or> (\<exists> t \<in> set ?sbjs. Ifm (?N t #bs) ?q))"
+    by (auto simp add: split_def)
   also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> t \<in> set (remdups ?sbjs). (\<lambda> t. ?I i (simpfm (subst0 t ?q))) t))"
     by (simp only: simpfm subst0_I[OF qfq] Inum.simps subst0_I[OF qfmq] set_remdups)
   also have "\<dots> = ((?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js)) \<or> (?I i (evaldjf (\<lambda> t. simpfm (subst0 t ?q)) (remdups ?sbjs))))" by (simp only: evaldjf_ex)
   finally have mdqd: "?lhs = (?I i (disj ?md ?qd))" by simp
   hence mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" using decr [OF mdqdb] by simp
   {assume mdT: "?md = T"
-    hence cT:"cooper p = T" 
+    hence cT:"cooper p = T"
       by (simp only: cooper_def unit_def split_def Let_def if_True) simp
     from mdT mdqd have lhs:"?lhs" by auto
     from mdT have "?rhs" by (simp add: cooper_def unit_def split_def)
     with lhs cT have ?thesis by simp }
   moreover
-  {assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)" 
-      by (simp only: cooper_def unit_def split_def Let_def if_False) 
+  {assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)"
+      by (simp only: cooper_def unit_def split_def Let_def if_False)
     with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
   ultimately show ?thesis by blast
 qed
 
-lemma DJcooper: 
+lemma DJcooper:
   assumes qf: "qfree p"
   shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ cooper p))) \<and> qfree (DJ cooper p)"
 proof-
   from cooper have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (cooper p)" by  blast
   from DJ_qf[OF cqf] qf have thqf:"qfree (DJ cooper p)" by blast
-  have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))" 
+  have "Ifm bs (DJ cooper p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (cooper q))"
      by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)" 
+  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)"
     using cooper disjuncts_qf[OF qf] by blast
   also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto)
   finally show ?thesis using thqf by blast
@@ -5231,20 +5231,20 @@
 
 lemma \<sigma>_\<rho>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
   shows "Ifm (a#bs) (\<sigma>_\<rho> p (t,c)) = Ifm (a#bs) (\<sigma>_\<rho> p (t',c))"
-  using lp 
+  using lp
   by (induct p rule: iszlfm.induct, auto simp add: tt')
 
 lemma \<sigma>_cong: assumes lp: "iszlfm p (a#bs)" and tt': "Inum (a#bs) t = Inum (a#bs) t'"
   shows "Ifm (a#bs) (\<sigma> p c t) = Ifm (a#bs) (\<sigma> p c t')"
   by (simp add: \<sigma>_def tt' \<sigma>_\<rho>_cong[OF lp tt'])
 
-lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)" 
+lemma \<rho>_cong: assumes lp: "iszlfm p (a#bs)"
   and RR: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R =  (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
   shows "(\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))) = (\<exists> (e,c) \<in> set (\<rho> p). \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j))))"
   (is "?lhs = ?rhs")
 proof
   let ?d = "\<delta> p"
-  assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}" 
+  assume ?lhs then obtain e c j where ecR: "(e,c) \<in> R" and jD:"j \<in> {1 .. c*?d}"
     and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
   from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" by auto
   hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" using RR by simp
@@ -5252,14 +5252,14 @@
   then obtain e' c' where ecRo:"(e',c') \<in> set (\<rho> p)" and ee':"Inum (a#bs) e = Inum (a#bs) e'"
     and cc':"c = c'" by blast
   from ee' have tt': "Inum (a#bs) (Add e (C j)) = Inum (a#bs) (Add e' (C j))" by simp
-  
+
   from \<sigma>_cong[OF lp tt', where c="c"] px have px':"?sp c e' j" by simp
   from ecRo jD px' show ?rhs apply (auto simp: cc')
     by (rule_tac x="(e', c')" in bexI,simp_all)
   (rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
 next
   let ?d = "\<delta> p"
-  assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}" 
+  assume ?rhs then obtain e c j where ecR: "(e,c) \<in> set (\<rho> p)" and jD:"j \<in> {1 .. c*?d}"
     and px: "Ifm (a#bs) (\<sigma> p c (Add e (C j)))" (is "?sp c e j") by blast
   from ecR have "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)" by auto
   hence "(Inum (a#bs) e,c) \<in> (\<lambda>(b,k). (Inum (a#bs) b,k)) ` R" using RR by simp
@@ -5273,30 +5273,30 @@
   (rule_tac x="j" in bexI, simp_all add: cc'[symmetric])
 qed
 
-lemma rl_thm': 
-  assumes lp: "iszlfm p (real_of_int (i::int)#bs)" 
+lemma rl_thm':
+  assumes lp: "iszlfm p (real_of_int (i::int)#bs)"
   and R: "(\<lambda>(b,k). (Inum (a#bs) b,k)) ` R =  (\<lambda>(b,k). (Inum (a#bs) b,k)) ` set (\<rho> p)"
   shows "(\<exists> (x::int). Ifm (real_of_int x#bs) p) = ((\<exists> j\<in> {1 .. \<delta> p}. Ifm (real_of_int j#bs) (minusinf p)) \<or> (\<exists> (e,c) \<in> R. \<exists> j\<in> {1.. c*(\<delta> p)}. Ifm (a#bs) (\<sigma> p c (Add e (C j)))))"
-  using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp 
+  using rl_thm[OF lp] \<rho>_cong[OF iszlfm_gen[OF lp, rule_format, where y="a"] R] by simp
 
 definition chooset :: "fm \<Rightarrow> fm \<times> ((num\<times>int) list) \<times> int" where
   "chooset p \<equiv> (let q = zlfm p ; d = \<delta> q;
-             B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ; 
+             B = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<rho> q)) ;
              a = remdups (map (\<lambda> (t,k). (simpnum t,k)) (\<alpha>_\<rho> q))
              in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"
 
 lemma chooset: assumes qf: "qfree p"
-  shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow> 
-     ((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
-      (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> 
+  shows "\<And> q B d. chooset p = (q,B,d) \<Longrightarrow>
+     ((\<exists> (x::int). Ifm (real_of_int x#bs) p) =
+      (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and>
       ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and>
       (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
 proof-
-  fix q B d 
+  fix q B d
   assume qBd: "chooset p = (q,B,d)"
-  let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = 
-             (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and> 
-             (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)" 
+  let ?thes = "((\<exists> (x::int). Ifm (real_of_int x#bs) p) =
+             (\<exists> (x::int). Ifm (real_of_int x#bs) q)) \<and> ((\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set B = (\<lambda>(t,k). (Inum (real_of_int i#bs) t,k)) ` set (\<rho> q)) \<and>
+             (\<delta> q = d) \<and> d >0 \<and> iszlfm q (real_of_int (i::int)#bs) \<and> (\<forall> (e,c)\<in> set B. numbound0 e \<and> c>0)"
   let ?I = "\<lambda> (x::int) p. Ifm (real_of_int x#bs) p"
   let ?q = "zlfm p"
   let ?d = "\<delta> ?q"
@@ -5305,20 +5305,20 @@
   let ?B'= "remdups (map ?f (\<rho> ?q))"
   let ?A = "set (\<alpha>_\<rho> ?q)"
   let ?A'= "remdups (map ?f (\<alpha>_\<rho> ?q))"
-  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]] 
+  from conjunct1[OF zlfm_I[OF qf, where bs="bs"]]
   have pp': "\<forall> i. ?I i ?q = ?I i p" by auto
-  hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp 
+  hence pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp
   from iszlfm_gen[OF conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]], rule_format, where y="real_of_int i"]
-  have lq: "iszlfm ?q (real_of_int (i::int)#bs)" . 
+  have lq: "iszlfm ?q (real_of_int (i::int)#bs)" .
   from \<delta>[OF lq] have dp:"?d >0" by blast
   let ?N = "\<lambda> (t,c). (Inum (real_of_int (i::int)#bs) t,c)"
   have "?N ` set ?B' = ((?N o ?f) ` ?B)" by (simp add: split_def image_comp)
   also have "\<dots> = ?N ` ?B"
     by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def)
   finally have BB': "?N ` set ?B' = ?N ` ?B" .
-  have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_comp) 
+  have "?N ` set ?A' = ((?N o ?f) ` ?A)" by (simp add: split_def image_comp)
   also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="real_of_int i #bs"]
-    by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def) 
+    by(simp add: split_def image_comp simpnum_ci[where bs="real_of_int i #bs"] image_def)
   finally have AA': "?N ` set ?A' = ?N ` ?A" .
   from \<rho>_l[OF lq] have B_nb:"\<forall> (e,c)\<in> set ?B'. numbound0 e \<and> c > 0"
     by (simp add: split_def)
@@ -5327,16 +5327,16 @@
     {assume "length ?B' \<le> length ?A'"
     hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
       using qBd by (auto simp add: Let_def chooset_def)
-    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)" 
+    with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<rho> q)"
       and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto
   with pq_ex dp lq q d have ?thes by simp}
-  moreover 
+  moreover
   {assume "\<not> (length ?B' \<le> length ?A')"
     hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
       using qBd by (auto simp add: Let_def chooset_def)
-    with AA' mirror_\<alpha>_\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)" 
-      and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto 
-    from mirror_ex[OF lq] pq_ex q 
+    with AA' mirror_\<alpha>_\<rho>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<rho> q)"
+      and bn: "\<forall>(e,c)\<in> set B. numbound0 e \<and> c > 0" by auto
+    from mirror_ex[OF lq] pq_ex q
     have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
     from lq q mirror_l [where p="?q" and bs="bs" and a="real_of_int i"]
     have lq': "iszlfm q (real_of_int i#bs)" by auto
@@ -5366,7 +5366,7 @@
 qed
 
 definition redlove :: "fm \<Rightarrow> fm" where
-  "redlove p \<equiv> 
+  "redlove p \<equiv>
   (let (q,B,d) = chooset p;
        mq = simpfm (minusinf q);
        md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) [1..d]
@@ -5375,7 +5375,7 @@
      in decr (disj md qd)))"
 
 lemma redlove: assumes qf: "qfree p"
-  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)" 
+  shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (redlove p))) \<and> qfree (redlove p)"
   (is "(?lhs = ?rhs) \<and> _")
 proof-
 
@@ -5391,53 +5391,53 @@
   let ?N = "\<lambda> (t,k). (Inum (real_of_int (i::int)#bs) t,k)"
   let ?qd = "evaldjf (stage ?q ?d) ?B"
   have qbf:"chooset p = (?q,?B,?d)" by simp
-  from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and 
-    B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and 
-    lq: "iszlfm ?q (real_of_int i#bs)" and 
+  from chooset[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and
+    B:"?N ` set ?B = ?N ` set (\<rho> ?q)" and dd: "\<delta> ?q = ?d" and dp: "?d > 0" and
+    lq: "iszlfm ?q (real_of_int i#bs)" and
     Bn: "\<forall> (e,c)\<in> set ?B. numbound0 e \<and> c > 0" by auto
   from zlin_qfree[OF lq] have qfq: "qfree ?q" .
   from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
   have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
-  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)" 
+  hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)"
     by (auto simp only: subst0_bound0[OF qfmq])
   hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
     by auto
-  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp 
+  from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp
   from Bn stage_nb[OF lq] have th:"\<forall> x \<in> set ?B. bound0 (stage ?q ?d x)" by auto
   from evaldjf_bound0[OF th]  have qdb: "bound0 ?qd" .
-  from mdb qdb 
+  from mdb qdb
   have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
   from trans [OF pq_ex rl_thm'[OF lq B]] dd
   have "?lhs = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq) \<or> (\<exists> (e,c)\<in> set ?B. \<exists> j\<in> {1 .. c*?d}. Ifm (real_of_int i#bs) (\<sigma> ?q c (Add e (C j)))))" by auto
-  also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))" 
+  also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq) \<or> (\<exists> (e,c)\<in> set ?B. ?I i (stage ?q ?d (e,c) )))"
     by (simp add: stage split_def)
   also have "\<dots> = ((\<exists> j\<in> {1 .. ?d}. ?I i (subst0 (C j) ?smq))  \<or> ?I i ?qd)"
     by (simp add: evaldjf_ex subst0_I[OF qfmq])
-  finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex set_upto simpfm) 
+  finally have mdqd:"?lhs = (?I i ?md \<or> ?I i ?qd)" by (simp only: evaldjf_ex set_upto simpfm)
   also have "\<dots> = (?I i (disj ?md ?qd))" by simp
-  also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb]) 
-  finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" . 
+  also have "\<dots> = (Ifm bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb])
+  finally have mdqd2: "?lhs = (Ifm bs (decr (disj ?md ?qd)))" .
   {assume mdT: "?md = T"
     hence cT:"redlove p = T" by (simp add: redlove_def Let_def chooset_def split_def)
-    from mdT have lhs:"?lhs" using mdqd by simp 
+    from mdT have lhs:"?lhs" using mdqd by simp
     from mdT have "?rhs" by (simp add: redlove_def chooset_def split_def)
     with lhs cT have ?thesis by simp }
   moreover
-  {assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)" 
+  {assume mdT: "?md \<noteq> T" hence "redlove p = decr (disj ?md ?qd)"
       by (simp add: redlove_def chooset_def split_def Let_def)
     with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
   ultimately show ?thesis by blast
 qed
 
-lemma DJredlove: 
+lemma DJredlove:
   assumes qf: "qfree p"
   shows "((\<exists> (x::int). Ifm (real_of_int x#bs) p) = (Ifm bs (DJ redlove p))) \<and> qfree (DJ redlove p)"
 proof-
   from redlove have cqf: "\<forall> p. qfree p \<longrightarrow> qfree (redlove p)" by  blast
   from DJ_qf[OF cqf] qf have thqf:"qfree (DJ redlove p)" by blast
-  have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))" 
+  have "Ifm bs (DJ redlove p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (redlove q))"
      by (simp add: DJ_def evaldjf_ex)
-  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)" 
+  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). \<exists> (x::int). Ifm (real_of_int x#bs)  q)"
     using redlove disjuncts_qf[OF qf] by blast
   also have "\<dots> = (\<exists> (x::int). Ifm (real_of_int x#bs) p)" by (induct p rule: disjuncts.induct, auto)
   finally show ?thesis using thqf by blast
@@ -5461,13 +5461,13 @@
   show "qfree (mircfr p)\<and>(Ifm bs (mircfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
   proof-
     let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" 
+    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)"
       using splitex[OF qf] by simp
     with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
     with DJcooper[OF qf'] show ?thesis by (simp add: mircfr_def)
   qed
 qed
-  
+
 lemma mirlfr: "\<forall> bs p. qfree p \<longrightarrow> qfree(mirlfr p) \<and> Ifm bs (mirlfr p) = Ifm bs (E p)"
 proof(clarsimp simp del: Ifm.simps)
   fix bs p
@@ -5475,13 +5475,13 @@
   show "qfree (mirlfr p)\<and>(Ifm bs (mirlfr p) = Ifm bs (E p))" (is "_ \<and> (?lhs = ?rhs)")
   proof-
     let ?es = "(And (And (Ge (CN 0 1 (C 0))) (Lt (CN 0 1 (C (- 1))))) (simpfm (exsplit p)))"
-    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)" 
+    have "?rhs = (\<exists> (i::int). \<exists> x. Ifm (x#real_of_int i#bs) ?es)"
       using splitex[OF qf] by simp
     with ferrack01[OF simpfm_qf[OF exsplit_qf[OF qf]]] have th1: "?rhs = (\<exists> (i::int). Ifm (real_of_int i#bs) (ferrack01 (simpfm (exsplit p))))" and qf':"qfree (ferrack01 (simpfm (exsplit p)))" by simp+
     with DJredlove[OF qf'] show ?thesis by (simp add: mirlfr_def)
   qed
 qed
-  
+
 definition mircfrqe:: "fm \<Rightarrow> fm" where
   "mircfrqe p = qelim (prep p) mircfr"
 
@@ -5566,7 +5566,7 @@
   | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
       @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
   | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} $ t1 $ t2) =
-      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2)) 
+      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
   | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
       mk_Dvd (HOLogic.dest_num t1, num_of_term vs t2)
   | fm_of_term vs (@{term "op rdvd"} $ (@{term "of_int :: int \<Rightarrow> real"} $ (@{term "- numeral :: _ \<Rightarrow> int"} $ t1)) $ t2) =
@@ -5606,7 +5606,7 @@
   | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t))
   | term_of_num vs (@{code CF} (c, t, s)) = term_of_num vs (@{code Add} (@{code Mul} (c, @{code Floor} t), s));
 
-fun term_of_fm vs @{code T} = @{term True} 
+fun term_of_fm vs @{code T} = @{term True}
   | term_of_fm vs @{code F} = @{term False}
   | term_of_fm vs (@{code Lt} t) =
       @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} $ term_of_num vs t $ @{term "0::real"}
@@ -5637,7 +5637,7 @@
 
 in
   fn (ctxt, t) =>
-  let 
+  let
     val fs = Misc_Legacy.term_frees t;
     val vs = map_index swap fs;
     (*If quick_and_dirty then run without proof generation as oracle*)
@@ -5647,8 +5647,8 @@
 end;
 \<close>
 
-lemmas iff_real_of_int = of_int_eq_iff [where 'a = real, symmetric] 
-                         of_int_less_iff [where 'a = real, symmetric] 
+lemmas iff_real_of_int = of_int_eq_iff [where 'a = real, symmetric]
+                         of_int_less_iff [where 'a = real, symmetric]
                          of_int_le_iff [where 'a = real, symmetric]
 
 ML_file "mir_tac.ML"
@@ -5665,7 +5665,7 @@
   by mir
 
 lemma "\<forall>x::real. 2*\<lfloor>x\<rfloor> \<le> \<lfloor>2*x\<rfloor> \<and> \<lfloor>2*x\<rfloor> \<le> 2*\<lfloor>x+1\<rfloor>"
-  by mir 
+  by mir
 
 lemma "\<forall>x::real. \<exists>y \<le> x. (\<lfloor>x\<rfloor> = \<lceil>y\<rceil>)"
   by mir




