# HG changeset patch
# User paulson <lp15@cam.ac.uk>
# Date 1447763528 0
# Node ID 6571c78c966714772cb5feb20dc77f661cc73a3b
# Parent  f6b9f528c89c78214301057a14912cfc78e6bf7a
Removed some legacy theorems; minor adjustments to simplification rules; new material on homotopic paths

diff -r f6b9f528c89c -r 6571c78c9667 NEWS
--- a/NEWS	Tue Nov 17 12:01:19 2015 +0100
+++ b/NEWS	Tue Nov 17 12:32:08 2015 +0000
@@ -406,6 +406,36 @@
 constant and its defining fact become qualified, e.g. Option.is_none and
 Option.is_none_def. Occasional INCOMPATIBILITY in applications.
 
+* The coercions to type 'real' have been reorganised. The function 'real'
+is no longer overloaded, but has type 'nat => real' and abbreviates
+of_nat for that type. Also 'real_of_int :: int => real' abbreviates
+of_int for that type. Other overloaded instances of 'real' have been
+replaced by 'real_of_ereal' and 'real_of_float'.
+Consolidated facts (among others):
+  real_of_nat_le_iff -> of_nat_le_iff
+  real_of_nat_numeral of_nat_numeral
+  real_of_int_zero of_int_0
+  real_of_nat_zero of_nat_0
+  real_of_one of_int_1
+  real_of_int_add of_int_add
+  real_of_nat_add of_nat_add
+  real_of_int_diff of_int_diff
+  real_of_nat_diff of_nat_diff
+  floor_subtract floor_diff_of_int
+  real_of_int_inject of_int_eq_iff
+  real_of_int_gt_zero_cancel_iff of_int_0_less_iff
+  real_of_int_ge_zero_cancel_iff of_int_0_le_iff
+  real_of_nat_ge_zero of_nat_0_le_iff
+  real_of_int_ceiling_ge le_of_int_ceiling
+  ceiling_less_eq ceiling_less_iff
+  ceiling_le_eq ceiling_le_iff
+  less_floor_eq less_floor_iff
+  floor_less_eq floor_less_iff
+  floor_divide_eq_div floor_divide_of_int_eq
+  real_of_int_zero_cancel of_nat_eq_0_iff
+  ceiling_real_of_int ceiling_of_int
+INCOMPATIBILITY.
+
 * Some old and rarely used ASCII replacement syntax has been removed.
 INCOMPATIBILITY, standard syntax with symbols should be used instead.
 The subsequent commands help to reproduce the old forms, e.g. to
diff -r f6b9f528c89c -r 6571c78c9667 src/Doc/Tutorial/Types/Numbers.thy
--- a/src/Doc/Tutorial/Types/Numbers.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/Doc/Tutorial/Types/Numbers.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -22,8 +22,8 @@
 *}
 
 text{*
-@{thm[display] numeral_1_eq_1[no_vars]}
-\rulename{numeral_1_eq_1}
+@{thm[display] numeral_One[no_vars]}
+\rulename{numeral_One}
 
 @{thm[display] add_2_eq_Suc[no_vars]}
 \rulename{add_2_eq_Suc}
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
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Decision_Procs/cooper_tac.ML
--- a/src/HOL/Decision_Procs/cooper_tac.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Decision_Procs/cooper_tac.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -61,8 +61,8 @@
     (* Simp rules for changing (n::int) to int n *)
     val simpset1 =
       put_simpset HOL_basic_ss ctxt
-      addsimps @{thms zdvd_int} @
-        map (fn r => r RS sym) @{thms int_numeral int_int_eq zle_int zless_int zadd_int zmult_int}
+      addsimps @{thms zdvd_int} @ [@{thm "of_nat_add"}, @{thm "of_nat_mult"}] @
+        map (fn r => r RS sym) @{thms int_numeral int_int_eq zle_int zless_int}
       |> Splitter.add_split @{thm zdiff_int_split}
     (*simp rules for elimination of int n*)
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Decision_Procs/mir_tac.ML
--- a/src/HOL/Decision_Procs/mir_tac.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Decision_Procs/mir_tac.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -19,7 +19,7 @@
 
   val comp_arith = [@{thm "Let_def"}, @{thm "if_False"}, @{thm "if_True"}, @{thm "add_0"},
                  @{thm "add_Suc"}, @{thm add_numeral_left}, @{thm mult_numeral_left(1)}] @
-                 (map (fn th => th RS sym) [@{thm "numeral_1_eq_1"}])
+                 (map (fn th => th RS sym) [@{thm "numeral_One"}])
                  @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} 
   val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, 
              @{thm of_nat_numeral},
@@ -87,9 +87,8 @@
             @{thm "split_min"}, @{thm "split_max"}]
     (* Simp rules for changing (n::int) to int n *)
     val simpset1 = put_simpset HOL_basic_ss ctxt
-      addsimps [@{thm "zdvd_int"}] @ map (fn r => r RS sym)
-        [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"}, 
-         @{thm nat_numeral}, @{thm "zmult_int"}]
+      addsimps [@{thm "zdvd_int"}, @{thm "of_nat_add"}, @{thm "of_nat_mult"}] @ 
+          map (fn r => r RS sym) [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm nat_numeral}]
       |> Splitter.add_split @{thm "zdiff_int_split"}
     (*simp rules for elimination of int n*)
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Inequalities.thy
--- a/src/HOL/Inequalities.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Inequalities.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -34,7 +34,7 @@
     by (auto simp: Setsum_Icc_int[transferred, OF assms] zdiv_int int_mult simp del: of_nat_setsum
           split: zdiff_int_split)
   thus ?thesis
-    using int_int_eq by blast
+    using of_nat_eq_iff by blast
 qed
 
 lemma Setsum_Ico_nat: assumes "(m::nat) \<le> n"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Int.thy
--- a/src/HOL/Int.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Int.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -524,9 +524,6 @@
 lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
 by simp
 
-lemma int_Suc0_eq_1: "int (Suc 0) = 1"
-by simp
-
 text\<open>This version is proved for all ordered rings, not just integers!
       It is proved here because attribute @{text arith_split} is not available
       in theory @{text Rings}.
@@ -843,7 +840,6 @@
 
 lemmas zle_int = of_nat_le_iff [where 'a=int]
 lemmas int_int_eq = of_nat_eq_iff [where 'a=int]
-lemmas numeral_1_eq_1 = numeral_One
 
 subsection \<open>Setting up simplification procedures\<close>
 
@@ -1678,7 +1674,6 @@
 lemmas inj_int = inj_of_nat [where 'a=int]
 lemmas zadd_int = of_nat_add [where 'a=int, symmetric]
 lemmas int_mult = of_nat_mult [where 'a=int]
-lemmas zmult_int = of_nat_mult [where 'a=int, symmetric]
 lemmas int_eq_0_conv = of_nat_eq_0_iff [where 'a=int and m="n"] for n
 lemmas zless_int = of_nat_less_iff [where 'a=int]
 lemmas int_less_0_conv = of_nat_less_0_iff [where 'a=int and m="k"] for k
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Library/Convex.thy
--- a/src/HOL/Library/Convex.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Library/Convex.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -884,19 +884,19 @@
   assume t: "t > 0" "t < 1" and xy: "x \<in> A" "y \<in> A" "x < y"
   def z \<equiv> "(1 - t) * x + t * y"
   with \<open>connected A\<close> and xy have ivl: "{x..y} \<subseteq> A" using connected_contains_Icc by blast
-  
+
   from xy t have xz: "z > x" by (simp add: z_def algebra_simps)
   have "y - z = (1 - t) * (y - x)" by (simp add: z_def algebra_simps)
   also from xy t have "... > 0" by (intro mult_pos_pos) simp_all
   finally have yz: "z < y" by simp
-    
+
   from assms xz yz ivl t have "\<exists>\<xi>. \<xi> > x \<and> \<xi> < z \<and> f z - f x = (z - x) * f' \<xi>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<xi> where \<xi>: "\<xi> > x" "\<xi> < z" "f' \<xi> = (f z - f x) / (z - x)" by auto
   from assms xz yz ivl t have "\<exists>\<eta>. \<eta> > z \<and> \<eta> < y \<and> f y - f z = (y - z) * f' \<eta>"
     by (intro MVT2) (auto intro!: assms(2))
   then obtain \<eta> where \<eta>: "\<eta> > z" "\<eta> < y" "f' \<eta> = (f y - f z) / (y - z)" by auto
-  
+
   from \<eta>(3) have "(f y - f z) / (y - z) = f' \<eta>" ..
   also from \<xi> \<eta> ivl have "\<xi> \<in> A" "\<eta> \<in> A" by auto
   with \<xi> \<eta> have "f' \<eta> \<ge> f' \<xi>" by (intro assms(3)) auto
@@ -926,11 +926,11 @@
 proof (cases y x rule: linorder_cases)
   assume less: "x < y"
   hence d: "d > 0" by (simp add: d_def)
-  from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1" 
+  from assms(2) less have A: "0 \<le> (c - x) / d" "(c - x) / d \<le> 1"
     by (simp_all add: d_def divide_simps)
   have "f c = f (x + (c - x) * 1)" by simp
   also from less have "1 = ((y - x) / d)" by (simp add: d_def)
-  also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y" 
+  also from d have "x + (c - x) * \<dots> = (1 - (c - x) / d) *\<^sub>R x + ((c - x) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (c - x) / d) * f x + (c - x) / d * f y" using assms less
     by (intro convex_onD_Icc) simp_all
@@ -945,11 +945,11 @@
 proof (cases y x rule: linorder_cases)
   assume less: "x < y"
   hence d: "d > 0" by (simp add: d_def)
-  from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1" 
+  from assms(2) less have A: "0 \<le> (y - c) / d" "(y - c) / d \<le> 1"
     by (simp_all add: d_def divide_simps)
   have "f c = f (y - (y - c) * 1)" by simp
   also from less have "1 = ((y - x) / d)" by (simp add: d_def)
-  also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y" 
+  also from d have "y - (y - c) * \<dots> = (1 - (1 - (y - c) / d)) *\<^sub>R x + (1 - (y - c) / d) *\<^sub>R y"
     by (simp add: field_simps)
   also have "f \<dots> \<le> (1 - (1 - (y - c) / d)) * f x + (1 - (y - c) / d) * f y" using assms less
     by (intro convex_onD_Icc) (simp_all add: field_simps)
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Limits.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -635,6 +635,9 @@
   shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> continuous_on s (\<lambda>x. f x - g x)"
   unfolding continuous_on_def by (auto intro: tendsto_diff)
 
+lemma continuous_on_op_minus: "continuous_on (s::'a::real_normed_vector set) (op - x)"
+  by (rule continuous_intros | simp)+
+
 lemma tendsto_setsum [tendsto_intros]:
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
   assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) F"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Matrix_LP/ComputeNumeral.thy
--- a/src/HOL/Matrix_LP/ComputeNumeral.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Matrix_LP/ComputeNumeral.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -28,11 +28,11 @@
 lemmas compute_natarith =
   arith_simps rel_simps
   diff_nat_numeral nat_numeral nat_0 nat_neg_numeral
-  numeral_1_eq_1 [symmetric]
+  numeral_One [symmetric]
   numeral_1_eq_Suc_0 [symmetric]
   Suc_numeral natfac.simps
 
-lemmas number_norm = numeral_1_eq_1[symmetric]
+lemmas number_norm = numeral_One[symmetric]
 
 lemmas compute_numberarith =
   arith_simps rel_simps number_norm
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy
--- a/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Cauchy_Integral_Thm.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -3997,7 +3997,7 @@
         apply (rule power_strict_mono)
         using `pe>0` by auto
       then have pe2: "pe^2 < 4 * cmod (w - x) ^ 2"
-        by auto
+        by (simp add: power_divide)
       have "8 * L * cmod (w - z) < e * pe\<^sup>2"
         using w `L>0` by (simp add: field_simps)
       also have "... < e * 4 * cmod (w - x) * cmod (w - x)"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Complex_Transcendental.thy
--- a/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Complex_Transcendental.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -400,7 +400,7 @@
     "norm(cos z) ^ 2 = cos(Re z) ^ 2 + (exp(Im z) - inverse(exp(Im z))) ^ 2 / 4"
   apply (cases z)
   apply (simp add: cos_add cmod_power2 cos_of_real sin_of_real)
-  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
+  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
   apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
   apply (simp add: sin_squared_eq)
   apply (simp add: power2_eq_square algebra_simps divide_simps)
@@ -410,7 +410,7 @@
     "norm(sin z) ^ 2 = (exp(2 * Im z) + inverse(exp(2 * Im z)) - 2 * cos(2 * Re z)) / 4"
   apply (cases z)
   apply (simp add: sin_add cmod_power2 cos_of_real sin_of_real cos_double_cos exp_double)
-  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide)
+  apply (simp add: cos_exp_eq sin_exp_eq exp_minus exp_of_real Re_divide Im_divide power_divide)
   apply (simp only: left_diff_distrib [symmetric] power_mult_distrib)
   apply (simp add: cos_squared_eq)
   apply (simp add: power2_eq_square algebra_simps divide_simps)
@@ -2140,7 +2140,7 @@
       shows "Arcsin(sin z) = z"
 proof -
   have "Arcsin(sin z) = - (\<i> * Ln (csqrt (1 - (\<i> * (exp (\<i>*z) - inverse (exp (\<i>*z))))\<^sup>2 / 4) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
-    by (simp add: sin_exp_eq Arcsin_def exp_minus)
+    by (simp add: sin_exp_eq Arcsin_def exp_minus power_divide)
   also have "... = - (\<i> * Ln (csqrt (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2)\<^sup>2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
     by (simp add: field_simps power2_eq_square)
   also have "... = - (\<i> * Ln (((exp (\<i>*z) + inverse (exp (\<i>*z)))/2) - (inverse (exp (\<i>*z)) - exp (\<i>*z)) / 2))"
@@ -2308,7 +2308,7 @@
     by (simp add: field_simps power2_eq_square)
   then have "Arccos(cos z) = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
                            \<i> * csqrt (((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z)))\<^sup>2)))"
-    by (simp add: cos_exp_eq Arccos_def exp_minus)
+    by (simp add: cos_exp_eq Arccos_def exp_minus power_divide)
   also have "... = - (\<i> * Ln ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2 +
                               \<i> * ((\<i> - (exp (\<i> * z))\<^sup>2 * \<i>) / (2 * exp (\<i> * z)))))"
     apply (subst csqrt_square)
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -130,6 +130,10 @@
 lemma setsum_not_0: "setsum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
   by (rule ccontr) auto
 
+lemma subset_translation_eq [simp]:
+    fixes a :: "'a::real_vector" shows "op + a ` s \<subseteq> op + a ` t \<longleftrightarrow> s \<subseteq> t"
+  by auto
+
 lemma translate_inj_on:
   fixes A :: "'a::ab_group_add set"
   shows "inj_on (\<lambda>x. a + x) A"
@@ -160,6 +164,9 @@
   apply auto
   done
 
+lemma convex_translation_eq [simp]: "convex ((\<lambda>x. a + x) ` s) \<longleftrightarrow> convex s"
+  by (metis convex_translation translation_galois)
+
 lemma translation_inverse_subset:
   assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
   shows "V \<le> ((\<lambda>x. a + x) ` S)"
@@ -178,6 +185,11 @@
   then show ?thesis by auto
 qed
 
+lemma convex_linear_image_eq [simp]:
+    fixes f :: "'a::real_vector \<Rightarrow> 'b::real_vector"
+    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> convex (f ` s) \<longleftrightarrow> convex s"
+    by (metis (no_types) convex_linear_image convex_linear_vimage inj_vimage_image_eq)
+
 lemma basis_to_basis_subspace_isomorphism:
   assumes s: "subspace (S:: ('n::euclidean_space) set)"
     and t: "subspace (T :: ('m::euclidean_space) set)"
@@ -7254,7 +7266,7 @@
     using subset_rel_interior[of "convex hull insert 0 B" S] ** by auto
 qed
 
-lemma rel_interior_convex_nonempty:
+lemma rel_interior_eq_empty:
   fixes S :: "'n::euclidean_space set"
   assumes "convex S"
   shows "rel_interior S = {} \<longleftrightarrow> S = {}"
@@ -7313,7 +7325,7 @@
   proof (cases "S = {}")
     case False
     then obtain a where a: "a \<in> rel_interior S"
-      using rel_interior_convex_nonempty assms by auto
+      using rel_interior_eq_empty assms by auto
     { fix x
       assume x: "x \<in> closure S"
       {
@@ -7421,7 +7433,7 @@
 proof (cases "S = {}")
   case True
   then show ?thesis
-    using assms rel_interior_convex_nonempty by auto
+    using assms rel_interior_eq_empty by auto
 next
   case False
   have "rel_interior (closure S) \<supseteq> rel_interior S"
@@ -7432,7 +7444,7 @@
     fix z
     assume z: "z \<in> rel_interior (closure S)"
     obtain x where x: "x \<in> rel_interior S"
-      using \<open>S \<noteq> {}\<close> assms rel_interior_convex_nonempty by auto
+      using \<open>S \<noteq> {}\<close> assms rel_interior_eq_empty by auto
     have "z \<in> rel_interior S"
     proof (cases "x = z")
       case True
@@ -7566,7 +7578,7 @@
        apply auto
        done
     obtain a where a: "a \<in> rel_interior S1"
-      using \<open>S1 \<noteq> {}\<close> rel_interior_convex_nonempty assms by auto
+      using \<open>S1 \<noteq> {}\<close> rel_interior_eq_empty assms by auto
     obtain T where T: "open T" "a \<in> T \<inter> S1" "T \<inter> affine hull S1 \<subseteq> S1"
        using mem_rel_interior[of a S1] a by auto
     then have "a \<in> T \<inter> closure S2"
@@ -7668,7 +7680,7 @@
   shows "z \<in> rel_interior S"
 proof -
   obtain x where x: "x \<in> rel_interior S"
-    using rel_interior_convex_nonempty assms by auto
+    using rel_interior_eq_empty assms by auto
   then have "x \<in> S"
     using rel_interior_subset by auto
   then obtain e where e: "e > 1 \<and> (1 - e) *\<^sub>R x + e *\<^sub>R z \<in> S"
@@ -8085,7 +8097,7 @@
 proof (cases "S = {}")
   case True
   then show ?thesis
-    using assms rel_interior_empty rel_interior_convex_nonempty by auto
+    using assms rel_interior_empty rel_interior_eq_empty by auto
 next
   case False
   have *: "f ` (rel_interior S) \<subseteq> f ` S"
@@ -8219,7 +8231,7 @@
   {
     assume "S \<noteq> {}" "T \<noteq> {}"
     then have ri: "rel_interior S \<noteq> {}" "rel_interior T \<noteq> {}"
-      using rel_interior_convex_nonempty assms by auto
+      using rel_interior_eq_empty assms by auto
     then have "fst -` rel_interior S \<noteq> {}"
       using fst_vimage_eq_Times[of "rel_interior S"] by auto
     then have "rel_interior ((fst :: 'n * 'm \<Rightarrow> 'n) -` S) = fst -` rel_interior S"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Ordered_Euclidean_Space.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -211,15 +211,6 @@
   using convex_box[of a b]
   by (metis interval_cbox)
 
-lemma image_affinity_interval: fixes m::real
-  fixes a b c :: "'a::ordered_euclidean_space"
-  shows "(\<lambda>x. m *\<^sub>R x + c) ` {a..b} =
-    (if {a..b} = {} then {}
-     else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
-     else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
-  using image_affinity_cbox[of m c a b]
-  by (simp add: cbox_interval)
-
 lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space)) ` {a .. b} =
   (if {a .. b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a .. m *\<^sub>R b} else {m *\<^sub>R b .. m *\<^sub>R a})"
   using image_smult_cbox[of m a b]
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Path_Connected.thy
--- a/src/HOL/Multivariate_Analysis/Path_Connected.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Path_Connected.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -428,9 +428,6 @@
 lemma path_image_subset_reversepath: "path_image(reversepath g) \<le> path_image g"
   by (auto simp: path_image_def reversepath_def)
 
-lemma continuous_on_op_minus: "continuous_on (s::real set) (op - x)"
-  by (rule continuous_intros | simp)+
-
 lemma path_imp_reversepath: "path g \<Longrightarrow> path(reversepath g)"
   apply (auto simp: path_def reversepath_def)
   using continuous_on_compose [of "{0..1}" "\<lambda>x. 1 - x" g]
@@ -1141,7 +1138,7 @@
   unfolding path_connected_def path_component_def by auto
 
 lemma path_connected_component_set: "path_connected s \<longleftrightarrow> (\<forall>x\<in>s. path_component_set s x = s)"
-  unfolding path_connected_component path_component_subset 
+  unfolding path_connected_component path_component_subset
   using path_component_mem by blast
 
 lemma path_component_maximal:
@@ -2455,4 +2452,461 @@
   apply (metis cobounded_unique_unbounded_components connected_outside double_compl outside_bounded_nonempty outside_in_components unbounded_outside)
   by (simp add: connected_outside outside_bounded_nonempty outside_in_components unbounded_outside)
 
+section\<open> Homotopy of maps p,q : X=>Y with property P of all intermediate maps.\<close>
+
+text\<open> We often just want to require that it fixes some subset, but to take in
+  the case of a loop homotopy, it's convenient to have a general property P.\<close>
+
+definition homotopic_with ::
+  "[('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> bool, 'a set, 'b set, 'a \<Rightarrow> 'b, 'a \<Rightarrow> 'b] \<Rightarrow> bool"
+where
+ "homotopic_with P X Y p q \<equiv>
+   (\<exists>h:: real \<times> 'a \<Rightarrow> 'b.
+       continuous_on ({0..1} \<times> X) h \<and>
+       h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+       (\<forall>x. h(0, x) = p x) \<and>
+       (\<forall>x. h(1, x) = q x) \<and>
+       (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+
+
+text\<open> We often want to just localize the ending function equality or whatever.\<close>
+proposition homotopic_with:
+  fixes X :: "'a::topological_space set" and Y :: "'b::topological_space set"
+  assumes "\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k)"
+  shows "homotopic_with P X Y p q \<longleftrightarrow>
+           (\<exists>h :: real \<times> 'a \<Rightarrow> 'b.
+              continuous_on ({0..1} \<times> X) h \<and>
+              h ` ({0..1} \<times> X) \<subseteq> Y \<and>
+              (\<forall>x \<in> X. h(0,x) = p x) \<and>
+              (\<forall>x \<in> X. h(1,x) = q x) \<and>
+              (\<forall>t \<in> {0..1}. P(\<lambda>x. h(t, x))))"
+  unfolding homotopic_with_def
+  apply (rule iffI, blast, clarify)
+  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then p v else q v" in exI)
+  apply (auto simp:)
+  apply (force elim: continuous_on_eq)
+  apply (drule_tac x=t in bspec, force)
+  apply (subst assms; simp)
+  done
+
+proposition homotopic_with_eq:
+   assumes h: "homotopic_with P X Y f g"
+       and f': "\<And>x. x \<in> X \<Longrightarrow> f' x = f x"
+       and g': "\<And>x. x \<in> X \<Longrightarrow> g' x = g x"
+       and P:  "(\<And>h k. (\<And>x. x \<in> X \<Longrightarrow> h x = k x) \<Longrightarrow> (P h \<longleftrightarrow> P k))"
+   shows "homotopic_with P X Y f' g'"
+  using h unfolding homotopic_with_def
+  apply safe
+  apply (rule_tac x="\<lambda>(u,v). if v \<in> X then h(u,v) else if u = 0 then f' v else g' v" in exI)
+  apply (simp add: f' g', safe)
+  apply (fastforce intro: continuous_on_eq)
+  apply fastforce
+  apply (subst P; fastforce)
+  done
+
+proposition homotopic_with_equal:
+   assumes contf: "continuous_on X f" and fXY: "f ` X \<subseteq> Y"
+       and gf: "\<And>x. x \<in> X \<Longrightarrow> g x = f x"
+       and P:  "P f" "P g"
+   shows "homotopic_with P X Y f g"
+  unfolding homotopic_with_def
+  apply (rule_tac x="\<lambda>(u,v). if u = 1 then g v else f v" in exI)
+  using assms
+  apply (intro conjI)
+  apply (rule continuous_on_eq [where f = "f o snd"])
+  apply (rule continuous_intros | force)+
+  apply clarify
+  apply (case_tac "t=1"; force)
+  done
+
+
+lemma image_Pair_const: "(\<lambda>x. (x, c)) ` A = A \<times> {c}"
+  by (auto simp:)
+
+lemma homotopic_constant_maps:
+   "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b) \<longleftrightarrow> s = {} \<or> path_component t a b"
+proof (cases "s = {} \<or> t = {}")
+  case True with continuous_on_const show ?thesis
+    by (auto simp: homotopic_with path_component_def)
+next
+  case False
+  then obtain c where "c \<in> s" by blast
+  show ?thesis
+  proof
+    assume "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+    then obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+        where conth: "continuous_on ({0..1} \<times> s) h"
+          and h: "h ` ({0..1} \<times> s) \<subseteq> t" "(\<forall>x\<in>s. h (0, x) = a)" "(\<forall>x\<in>s. h (1, x) = b)"
+      by (auto simp: homotopic_with)
+    have "continuous_on {0..1} (h \<circ> (\<lambda>t. (t, c)))"
+      apply (rule continuous_intros conth | simp add: image_Pair_const)+
+      apply (blast intro:  \<open>c \<in> s\<close> continuous_on_subset [OF conth] )
+      done
+    with \<open>c \<in> s\<close> h show "s = {} \<or> path_component t a b"
+      apply (simp_all add: homotopic_with path_component_def)
+      apply (auto simp:)
+      apply (drule_tac x="h o (\<lambda>t. (t, c))" in spec)
+      apply (auto simp: pathstart_def pathfinish_def path_image_def path_def)
+      done
+  next
+    assume "s = {} \<or> path_component t a b"
+    with False show "homotopic_with (\<lambda>x. True) s t (\<lambda>x. a) (\<lambda>x. b)"
+      apply (clarsimp simp: homotopic_with path_component_def pathstart_def pathfinish_def path_image_def path_def)
+      apply (rule_tac x="g o fst" in exI)
+      apply (rule conjI continuous_intros | force)+
+      done
+  qed
+qed
+
+
+subsection\<open> Trivial properties.\<close>
+
+lemma homotopic_with_imp_property: "homotopic_with P X Y f g \<Longrightarrow> P f \<and> P g"
+  unfolding homotopic_with_def Ball_def
+  apply clarify
+  apply (frule_tac x=0 in spec)
+  apply (drule_tac x=1 in spec)
+  apply (auto simp:)
+  done
+
+lemma continuous_on_o_Pair: "\<lbrakk>continuous_on (T \<times> X) h; t \<in> T\<rbrakk> \<Longrightarrow> continuous_on X (h o Pair t)"
+  by (fast intro: continuous_intros elim!: continuous_on_subset)
+
+lemma homotopic_with_imp_continuous:
+    assumes "homotopic_with P X Y f g"
+    shows "continuous_on X f \<and> continuous_on X g"
+proof -
+  obtain h :: "real \<times> 'a \<Rightarrow> 'b"
+    where conth: "continuous_on ({0..1} \<times> X) h"
+      and h: "\<forall>x. h (0, x) = f x" "\<forall>x. h (1, x) = g x"
+    using assms by (auto simp: homotopic_with_def)
+  have *: "t \<in> {0..1} \<Longrightarrow> continuous_on X (h o (\<lambda>x. (t,x)))" for t
+    by (rule continuous_intros continuous_on_subset [OF conth] | force)+
+  show ?thesis
+    using h *[of 0] *[of 1] by auto
+qed
+
+proposition homotopic_with_imp_subset1:
+     "homotopic_with P X Y f g \<Longrightarrow> f ` X \<subseteq> Y"
+  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_imp_subset2:
+     "homotopic_with P X Y f g \<Longrightarrow> g ` X \<subseteq> Y"
+  by (simp add: homotopic_with_def image_subset_iff) (metis atLeastAtMost_iff order_refl zero_le_one)
+
+proposition homotopic_with_mono:
+    assumes hom: "homotopic_with P X Y f g"
+        and Q: "\<And>h. \<lbrakk>continuous_on X h; image h X \<subseteq> Y \<and> P h\<rbrakk> \<Longrightarrow> Q h"
+      shows "homotopic_with Q X Y f g"
+  using hom
+  apply (simp add: homotopic_with_def)
+  apply (erule ex_forward)
+  apply (force simp: intro!: Q dest: continuous_on_o_Pair)
+  done
+
+proposition homotopic_with_subset_left:
+     "\<lbrakk>homotopic_with P X Y f g; Z \<subseteq> X\<rbrakk> \<Longrightarrow> homotopic_with P Z Y f g"
+  apply (simp add: homotopic_with_def)
+  apply (fast elim!: continuous_on_subset ex_forward)
+  done
+
+proposition homotopic_with_subset_right:
+     "\<lbrakk>homotopic_with P X Y f g; Y \<subseteq> Z\<rbrakk> \<Longrightarrow> homotopic_with P X Z f g"
+  apply (simp add: homotopic_with_def)
+  apply (fast elim!: continuous_on_subset ex_forward)
+  done
+
+proposition homotopic_with_compose_continuous_right:
+    "\<lbrakk>homotopic_with (\<lambda>f. p (f \<circ> h)) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+     \<Longrightarrow> homotopic_with p W Y (f o h) (g o h)"
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k)
+  apply (rule_tac x="k o (\<lambda>y. (fst y, h (snd y)))" in exI)
+  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+  apply (erule continuous_on_subset)
+  apply (fastforce simp: o_def)+
+  done
+
+proposition homotopic_compose_continuous_right:
+     "\<lbrakk>homotopic_with (\<lambda>f. True) X Y f g; continuous_on W h; h ` W \<subseteq> X\<rbrakk>
+      \<Longrightarrow> homotopic_with (\<lambda>f. True) W Y (f o h) (g o h)"
+  using homotopic_with_compose_continuous_right by fastforce
+
+proposition homotopic_with_compose_continuous_left:
+     "\<lbrakk>homotopic_with (\<lambda>f. p (h \<circ> f)) X Y f g; continuous_on Y h; h ` Y \<subseteq> Z\<rbrakk>
+      \<Longrightarrow> homotopic_with p X Z (h o f) (h o g)"
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k)
+  apply (rule_tac x="h o k" in exI)
+  apply (rule conjI continuous_intros continuous_on_compose [where f=snd and g=h, unfolded o_def] | simp)+
+  apply (erule continuous_on_subset)
+  apply (fastforce simp: o_def)+
+  done
+
+proposition homotopic_compose_continuous_left:
+   "homotopic_with (\<lambda>f. True) X Y f g \<and>
+        continuous_on Y h \<and> image h Y \<subseteq> Z
+        \<Longrightarrow> homotopic_with (\<lambda>f. True) X Z (h o f) (h o g)"
+  using homotopic_with_compose_continuous_left by fastforce
+
+proposition homotopic_with_Pair:
+   assumes hom: "homotopic_with p s t f g" "homotopic_with p' s' t' f' g'"
+       and q: "\<And>f g. \<lbrakk>p f; p' g\<rbrakk> \<Longrightarrow> q(\<lambda>(x,y). (f x, g y))"
+     shows "homotopic_with q (s \<times> s') (t \<times> t')
+                  (\<lambda>(x,y). (f x, f' y)) (\<lambda>(x,y). (g x, g' y))"
+  using hom
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac k k')
+  apply (rule_tac x="\<lambda>z. ((k o (\<lambda>x. (fst x, fst (snd x)))) z, (k' o (\<lambda>x. (fst x, snd (snd x)))) z)" in exI)
+  apply (rule conjI continuous_intros | erule continuous_on_subset | clarsimp)+
+  apply (auto intro!: q [unfolded case_prod_unfold])
+  done
+
+lemma homotopic_on_empty: "homotopic_with (\<lambda>x. True) {} t f g"
+  by (metis continuous_on_def empty_iff homotopic_with_equal image_subset_iff)
+
+
+text\<open>Homotopy with P is an equivalence relation (on continuous functions mapping X into Y that satisfy P,
+     though this only affects reflexivity.\<close>
+
+
+proposition homotopic_with_refl:
+   "homotopic_with P X Y f f \<longleftrightarrow> continuous_on X f \<and> image f X \<subseteq> Y \<and> P f"
+  apply (rule iffI)
+  using homotopic_with_imp_continuous homotopic_with_imp_property homotopic_with_imp_subset2 apply blast
+  apply (simp add: homotopic_with_def)
+  apply (rule_tac x="f o snd" in exI)
+  apply (rule conjI continuous_intros | force)+
+  done
+
+lemma homotopic_with_symD:
+  fixes X :: "'a::real_normed_vector set"
+    assumes "homotopic_with P X Y f g"
+      shows "homotopic_with P X Y g f"
+  using assms
+  apply (clarsimp simp add: homotopic_with_def)
+  apply (rename_tac h)
+  apply (rule_tac x="h o (\<lambda>y. (1 - fst y, snd y))" in exI)
+  apply (rule conjI continuous_intros | erule continuous_on_subset | force simp add: image_subset_iff)+
+  done
+
+proposition homotopic_with_sym:
+    fixes X :: "'a::real_normed_vector set"
+    shows "homotopic_with P X Y f g \<longleftrightarrow> homotopic_with P X Y g f"
+  using homotopic_with_symD by blast
+
+lemma split_01_prod: "{0..1::real} \<times> X = ({0..1/2} \<times> X) \<union> ({1/2..1} \<times> X)"
+  by force
+
+proposition homotopic_with_trans:
+    fixes X :: "'a::real_normed_vector set"
+    assumes "homotopic_with P X Y f g" and "homotopic_with P X Y g h"
+      shows "homotopic_with P X Y f h"
+proof -
+  have clo1: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({0..1/2::real} \<times> X)"
+    apply (simp add: closedin_closed split_01_prod [symmetric])
+    apply (rule_tac x="{0..1/2} \<times> UNIV" in exI)
+    apply (force simp add: closed_Times)
+    done
+  have clo2: "closedin (subtopology euclidean ({0..1/2} \<times> X \<union> {1/2..1} \<times> X)) ({1/2..1::real} \<times> X)"
+    apply (simp add: closedin_closed split_01_prod [symmetric])
+    apply (rule_tac x="{1/2..1} \<times> UNIV" in exI)
+    apply (force simp add: closed_Times)
+    done
+  { fix k1 k2:: "real \<times> 'a \<Rightarrow> 'b"
+    assume cont: "continuous_on ({0..1} \<times> X) k1" "continuous_on ({0..1} \<times> X) k2"
+       and Y: "k1 ` ({0..1} \<times> X) \<subseteq> Y" "k2 ` ({0..1} \<times> X) \<subseteq> Y"
+       and geq: "\<forall>x. k1 (1, x) = g x" "\<forall>x. k2 (0, x) = g x"
+       and k12: "\<forall>x. k1 (0, x) = f x" "\<forall>x. k2 (1, x) = h x"
+       and P:   "\<forall>t\<in>{0..1}. P (\<lambda>x. k1 (t, x))" "\<forall>t\<in>{0..1}. P (\<lambda>x. k2 (t, x))"
+    def k \<equiv> "\<lambda>y. if fst y \<le> 1 / 2 then (k1 o (\<lambda>x. (2 *\<^sub>R fst x, snd x))) y
+                                   else (k2 o (\<lambda>x. (2 *\<^sub>R fst x -1, snd x))) y"
+    have keq: "k1 (2 * u, v) = k2 (2 * u - 1, v)" if "u = 1/2"  for u v
+      by (simp add: geq that)
+    have "continuous_on ({0..1} \<times> X) k"
+      using cont
+      apply (simp add: split_01_prod k_def)
+      apply (rule clo1 clo2 continuous_on_cases_local continuous_intros | erule continuous_on_subset | simp add: linear image_subset_iff)+
+      apply (force simp add: keq)
+      done
+    moreover have "k ` ({0..1} \<times> X) \<subseteq> Y"
+      using Y by (force simp add: k_def)
+    moreover have "\<forall>x. k (0, x) = f x"
+      by (simp add: k_def k12)
+    moreover have "(\<forall>x. k (1, x) = h x)"
+      by (simp add: k_def k12)
+    moreover have "\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x))"
+      using P
+      apply (clarsimp simp add: k_def)
+      apply (case_tac "t \<le> 1/2")
+      apply (auto simp:)
+      done
+    ultimately have *: "\<exists>k :: real \<times> 'a \<Rightarrow> 'b.
+                       continuous_on ({0..1} \<times> X) k \<and> k ` ({0..1} \<times> X) \<subseteq> Y \<and>
+                       (\<forall>x. k (0, x) = f x) \<and> (\<forall>x. k (1, x) = h x) \<and> (\<forall>t\<in>{0..1}. P (\<lambda>x. k (t, x)))"
+      by blast
+  } note * = this
+  show ?thesis
+    using assms by (auto intro: * simp add: homotopic_with_def)
+qed
+
+proposition homotopic_compose:
+      fixes s :: "'a::real_normed_vector set"
+      shows "\<lbrakk>homotopic_with (\<lambda>x. True) s t f f'; homotopic_with (\<lambda>x. True) t u g g'\<rbrakk>
+             \<Longrightarrow> homotopic_with (\<lambda>x. True) s u (g o f) (g' o f')"
+  apply (rule homotopic_with_trans [where g = "g o f'"])
+  apply (metis homotopic_compose_continuous_left homotopic_with_imp_continuous homotopic_with_imp_subset1)
+  by (metis homotopic_compose_continuous_right homotopic_with_imp_continuous homotopic_with_imp_subset2)
+
+
+subsection\<open>Homotopy of paths, maintaining the same endpoints.\<close>
+
+
+definition homotopic_paths :: "['a set, real \<Rightarrow> 'a, real \<Rightarrow> 'a::topological_space] \<Rightarrow> bool"
+  where
+     "homotopic_paths s p q \<equiv>
+       homotopic_with (\<lambda>r. pathstart r = pathstart p \<and> pathfinish r = pathfinish p) {0..1} s p q"
+
+lemma homotopic_paths:
+   "homotopic_paths s p q \<longleftrightarrow>
+      (\<exists>h. continuous_on ({0..1} \<times> {0..1}) h \<and>
+          h ` ({0..1} \<times> {0..1}) \<subseteq> s \<and>
+          (\<forall>x \<in> {0..1}. h(0,x) = p x) \<and>
+          (\<forall>x \<in> {0..1}. h(1,x) = q x) \<and>
+          (\<forall>t \<in> {0..1::real}. pathstart(h o Pair t) = pathstart p \<and>
+                        pathfinish(h o Pair t) = pathfinish p))"
+  by (auto simp: homotopic_paths_def homotopic_with pathstart_def pathfinish_def)
+
+proposition homotopic_paths_imp_pathstart:
+     "homotopic_paths s p q \<Longrightarrow> pathstart p = pathstart q"
+  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
+
+proposition homotopic_paths_imp_pathfinish:
+     "homotopic_paths s p q \<Longrightarrow> pathfinish p = pathfinish q"
+  by (metis (mono_tags, lifting) homotopic_paths_def homotopic_with_imp_property)
+
+lemma homotopic_paths_imp_path:
+     "homotopic_paths s p q \<Longrightarrow> path p \<and> path q"
+  using homotopic_paths_def homotopic_with_imp_continuous path_def by blast
+
+lemma homotopic_paths_imp_subset:
+     "homotopic_paths s p q \<Longrightarrow> path_image p \<subseteq> s \<and> path_image q \<subseteq> s"
+  by (simp add: homotopic_paths_def homotopic_with_imp_subset1 homotopic_with_imp_subset2 path_image_def)
+
+proposition homotopic_paths_refl [simp]: "homotopic_paths s p p \<longleftrightarrow> path p \<and> path_image p \<subseteq> s"
+by (simp add: homotopic_paths_def homotopic_with_refl path_def path_image_def)
+
+proposition homotopic_paths_sym: "homotopic_paths s p q \<longleftrightarrow> homotopic_paths s q p"
+  by (metis (mono_tags) homotopic_paths_def homotopic_paths_imp_pathfinish homotopic_paths_imp_pathstart homotopic_with_symD)+
+
+proposition homotopic_paths_trans:
+     "\<lbrakk>homotopic_paths s p q; homotopic_paths s q r\<rbrakk> \<Longrightarrow> homotopic_paths s p r"
+  apply (simp add: homotopic_paths_def)
+  apply (rule homotopic_with_trans, assumption)
+  by (metis (mono_tags, lifting) homotopic_with_imp_property homotopic_with_mono)
+
+proposition homotopic_paths_eq:
+     "\<lbrakk>path p; path_image p \<subseteq> s; \<And>t. t \<in> {0..1} \<Longrightarrow> p t = q t\<rbrakk> \<Longrightarrow> homotopic_paths s p q"
+  apply (simp add: homotopic_paths_def)
+  apply (rule homotopic_with_eq)
+  apply (auto simp: path_def homotopic_with_refl pathstart_def pathfinish_def path_image_def elim: continuous_on_eq)
+  done
+
+proposition homotopic_paths_reparametrize:
+  assumes "path p"
+      and pips: "path_image p \<subseteq> s"
+      and contf: "continuous_on {0..1} f"
+      and f01:"f ` {0..1} \<subseteq> {0..1}"
+      and [simp]: "f(0) = 0" "f(1) = 1"
+      and q: "\<And>t. t \<in> {0..1} \<Longrightarrow> q(t) = p(f t)"
+    shows "homotopic_paths s p q"
+proof -
+  have contp: "continuous_on {0..1} p"
+    by (metis \<open>path p\<close> path_def)
+  then have "continuous_on {0..1} (p o f)"
+    using contf continuous_on_compose continuous_on_subset f01 by blast
+  then have "path q"
+    by (simp add: path_def) (metis q continuous_on_cong)
+  have piqs: "path_image q \<subseteq> s"
+    by (metis (no_types, hide_lams) pips f01 image_subset_iff path_image_def q)
+  have fb0: "\<And>a b. \<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> 0 \<le> (1 - a) * f b + a * b"
+    using f01 by force
+  have fb1: "\<lbrakk>0 \<le> a; a \<le> 1; 0 \<le> b; b \<le> 1\<rbrakk> \<Longrightarrow> (1 - a) * f b + a * b \<le> 1" for a b
+    using f01 [THEN subsetD, of "f b"] by (simp add: convex_bound_le)
+  have "homotopic_paths s q p"
+  proof (rule homotopic_paths_trans)
+    show "homotopic_paths s q (p \<circ> f)"
+      using q by (force intro: homotopic_paths_eq [OF  \<open>path q\<close> piqs])
+  next
+    show "homotopic_paths s (p \<circ> f) p"
+      apply (simp add: homotopic_paths_def homotopic_with_def)
+      apply (rule_tac x="p o (\<lambda>y. (1 - (fst y)) *\<^sub>R ((f o snd) y) + (fst y) *\<^sub>R snd y)"  in exI)
+      apply (rule conjI contf continuous_intros continuous_on_subset [OF contp] | simp)+
+      using pips [unfolded path_image_def]
+      apply (auto simp: fb0 fb1 pathstart_def pathfinish_def)
+      done
+  qed
+  then show ?thesis
+    by (simp add: homotopic_paths_sym)
+qed
+
+lemma homotopic_paths_subset: "\<lbrakk>homotopic_paths s p q; s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t p q"
+  using homotopic_paths_def homotopic_with_subset_right by blast
+
+
+text\<open> A slightly ad-hoc but useful lemma in constructing homotopies.\<close>
+lemma homotopic_join_lemma:
+  fixes q :: "[real,real] \<Rightarrow> 'a::topological_space"
+  assumes p: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. p (fst y) (snd y))"
+      and q: "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. q (fst y) (snd y))"
+      and pf: "\<And>t. t \<in> {0..1} \<Longrightarrow> pathfinish(p t) = pathstart(q t)"
+    shows "continuous_on ({0..1} \<times> {0..1}) (\<lambda>y. (p(fst y) +++ q(fst y)) (snd y))"
+proof -
+  have 1: "(\<lambda>y. p (fst y) (2 * snd y)) = (\<lambda>y. p (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y))"
+    by (rule ext) (simp )
+  have 2: "(\<lambda>y. q (fst y) (2 * snd y - 1)) = (\<lambda>y. q (fst y) (snd y)) o (\<lambda>y. (fst y, 2 * snd y - 1))"
+    by (rule ext) (simp )
+  show ?thesis
+    apply (simp add: joinpaths_def)
+    apply (rule continuous_on_cases_le)
+    apply (simp_all only: 1 2)
+    apply (rule continuous_intros continuous_on_subset [OF p] continuous_on_subset [OF q] | force)+
+    using pf
+    apply (auto simp: mult.commute pathstart_def pathfinish_def)
+    done
+qed
+
+text\<open> Congruence properties of homotopy w.r.t. path-combining operations.\<close>
+
+lemma homotopic_paths_reversepath_D:
+      assumes "homotopic_paths s p q"
+      shows   "homotopic_paths s (reversepath p) (reversepath q)"
+  using assms
+  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
+  apply (rule_tac x="h o (\<lambda>x. (fst x, 1 - snd x))" in exI)
+  apply (rule conjI continuous_intros)+
+  apply (auto simp: reversepath_def pathstart_def pathfinish_def elim!: continuous_on_subset)
+  done
+
+proposition homotopic_paths_reversepath:
+     "homotopic_paths s (reversepath p) (reversepath q) \<longleftrightarrow> homotopic_paths s p q"
+  using homotopic_paths_reversepath_D by force
+
+
+proposition homotopic_paths_join:
+    "\<lbrakk>homotopic_paths s p p'; homotopic_paths s q q'; pathfinish p = pathstart q\<rbrakk> \<Longrightarrow> homotopic_paths s (p +++ q) (p' +++ q')"
+  apply (simp add: homotopic_paths_def homotopic_with_def, clarify)
+  apply (rename_tac k1 k2)
+  apply (rule_tac x="(\<lambda>y. ((k1 o Pair (fst y)) +++ (k2 o Pair (fst y))) (snd y))" in exI)
+  apply (rule conjI continuous_intros homotopic_join_lemma)+
+  apply (auto simp: joinpaths_def pathstart_def pathfinish_def path_image_def)
+  done
+
+proposition homotopic_paths_continuous_image:
+    "\<lbrakk>homotopic_paths s f g; continuous_on s h; h ` s \<subseteq> t\<rbrakk> \<Longrightarrow> homotopic_paths t (h o f) (h o g)"
+  unfolding homotopic_paths_def
+  apply (rule homotopic_with_compose_continuous_left [of _ _ _ s])
+  apply (auto simp: pathstart_def pathfinish_def elim!: homotopic_with_mono)
+  done
+
 end
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -888,7 +888,7 @@
   apply (metis zero_le_dist order_trans dist_self)
   done
 
-lemma ball_empty[intro]: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
+lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
 
 lemma euclidean_dist_l2:
   fixes x y :: "'a :: euclidean_space"
@@ -6856,6 +6856,38 @@
       \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
   by (rule continuous_on_union_local) (auto intro: continuous_on_eq)
 
+lemma continuous_on_cases_le:
+  fixes h :: "'a :: topological_space \<Rightarrow> real"
+  assumes "continuous_on {t \<in> s. h t \<le> a} f"
+      and "continuous_on {t \<in> s. a \<le> h t} g"
+      and h: "continuous_on s h"
+      and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
+    shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
+proof -
+  have s: "s = {t \<in> s. h t \<in> atMost a} \<union> {t \<in> s. h t \<in> atLeast a}"
+    by force
+  have 1: "closedin (subtopology euclidean s) {t \<in> s. h t \<in> atMost a}"
+    by (rule continuous_closedin_preimage [OF h closed_atMost])
+  have 2: "closedin (subtopology euclidean s) {t \<in> s. h t \<in> atLeast a}"
+    by (rule continuous_closedin_preimage [OF h closed_atLeast])
+  show ?thesis
+    apply (rule continuous_on_subset [of s, OF _ order_refl])
+    apply (subst s)
+    apply (rule continuous_on_cases_local)
+    using 1 2 s assms apply auto
+    done
+qed
+
+lemma continuous_on_cases_1:
+  fixes s :: "real set"
+  assumes "continuous_on {t \<in> s. t \<le> a} f"
+      and "continuous_on {t \<in> s. a \<le> t} g"
+      and "continuous_on s h"
+      and "a \<in> s \<Longrightarrow> f a = g a"
+    shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
+using assms
+by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
+
 text\<open>Some more convenient intermediate-value theorem formulations.\<close>
 
 lemma connected_ivt_hyperplane:
@@ -8136,6 +8168,216 @@
   ultimately show "\<exists>!x\<in>s. g x = x" using \<open>a \<in> s\<close> by blast
 qed
 
+
+lemma cball_subset_cball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs 
+  proof (cases "r < 0")
+    case True then show ?rhs by simp
+  next
+    case False
+    then have [simp]: "r \<ge> 0" by simp
+    have "norm (a - a') + r \<le> r'"
+    proof (cases "a = a'")
+      case True then show ?thesis
+        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>]  subsetD [where c = "a", OF \<open>?lhs\<close>]
+        by (force simp add: SOME_Basis dist_norm)
+    next
+      case False
+      have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
+        by (simp add: algebra_simps)
+      also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
+        by (simp add: algebra_simps)
+      also have "... = \<bar>- norm (a - a') - r\<bar>"
+        using  \<open>a \<noteq> a'\<close> by (simp add: abs_mult_pos field_simps)
+      finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>" by linarith
+      show ?thesis
+        using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close> 
+        by (simp add: dist_norm scaleR_add_left)
+    qed
+    then show ?rhs by (simp add: dist_norm)
+  qed
+next
+  assume ?rhs then show ?lhs 
+    apply (auto simp: ball_def dist_norm )
+    apply (metis add.commute add_le_cancel_right dist_norm dist_triangle_alt order_trans)
+    using le_less_trans apply fastforce
+    done
+qed
+
+lemma cball_subset_ball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs 
+  proof (cases "r < 0")
+    case True then show ?rhs by simp
+  next
+    case False
+    then have [simp]: "r \<ge> 0" by simp
+    have "norm (a - a') + r < r'"
+    proof (cases "a = a'")
+      case True then show ?thesis
+        using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>]  subsetD [where c = "a", OF \<open>?lhs\<close>]
+        by (force simp add: SOME_Basis dist_norm)
+    next
+      case False
+      { assume "norm (a - a') + r \<ge> r'"
+        then have "\<bar>r' - norm (a - a')\<bar> \<le> r"
+          apply (simp split: abs_split)
+          by (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
+        then have False
+        using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
+        apply (simp add: dist_norm field_simps)
+        apply (simp add: diff_divide_distrib scaleR_left_diff_distrib)
+        done
+      }
+      then show ?thesis by force
+    qed
+    then show ?rhs by (simp add: dist_norm)
+  qed
+next
+  assume ?rhs then show ?lhs 
+    apply (auto simp: ball_def dist_norm )
+    apply (metis add.commute add_le_cancel_right dist_norm dist_triangle_alt le_less_trans)
+    using le_less_trans apply fastforce
+    done
+qed
+
+lemma ball_subset_cball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
+        (is "?lhs = ?rhs")
+proof (cases "r \<le> 0")
+  case True then show ?thesis   
+    using dist_not_less_zero less_le_trans by force
+next
+  case False show ?thesis  
+  proof
+    assume ?lhs
+    then have "(cball a r \<subseteq> cball a' r')"
+      by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
+    then show ?rhs
+      using False cball_subset_cball_iff by fastforce
+  next
+    assume ?rhs with False show ?lhs
+      using ball_subset_cball cball_subset_cball_iff by blast 
+  qed
+qed
+
+lemma ball_subset_ball_iff:
+  fixes a :: "'a :: euclidean_space"
+  shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
+        (is "?lhs = ?rhs")
+proof (cases "r \<le> 0")
+  case True then show ?thesis   
+    using dist_not_less_zero less_le_trans by force
+next
+  case False show ?thesis  
+  proof
+    assume ?lhs
+    then have "0 < r'"
+      by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
+    then have "(cball a r \<subseteq> cball a' r')"
+      by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
+    then show ?rhs
+      using False cball_subset_cball_iff by fastforce
+  next
+  assume ?rhs then show ?lhs
+    apply (auto simp: ball_def)
+    apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
+    using dist_not_less_zero order.strict_trans2 apply blast
+    done
+  qed
+qed
+
+
+lemma ball_eq_ball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs 
+  proof (cases "d \<le> 0 \<or> e \<le> 0")
+    case True 
+      with \<open>?lhs\<close> show ?rhs
+        by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
+  next
+    case False
+    with \<open>?lhs\<close> show ?rhs 
+      apply (auto simp add: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
+      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
+      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
+      done
+  qed
+next
+  assume ?rhs then show ?lhs 
+    by (auto simp add: set_eq_subset ball_subset_ball_iff)
+qed
+
+lemma cball_eq_cball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
+        (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs 
+  proof (cases "d < 0 \<or> e < 0")
+    case True 
+      with \<open>?lhs\<close> show ?rhs
+        by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
+  next
+    case False
+    with \<open>?lhs\<close> show ?rhs 
+      apply (auto simp add: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
+      apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
+      apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
+      done
+  qed
+next
+  assume ?rhs then show ?lhs 
+    by (auto simp add: set_eq_subset cball_subset_cball_iff)
+qed
+
+lemma ball_eq_cball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs
+    apply (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
+    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
+    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
+    using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
+    done
+next
+  assume ?rhs then show ?lhs 
+    by (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff)
+qed
+
+lemma cball_eq_ball_iff:
+  fixes x :: "'a :: euclidean_space"
+  shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0" (is "?lhs = ?rhs")
+proof
+  assume ?lhs 
+  then show ?rhs
+    apply (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
+    apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
+    apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
+    using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
+    done
+next
+  assume ?rhs then show ?lhs 
+    by (auto simp add: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff)
+qed
+
 no_notation
   eucl_less (infix "<e" 50)
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Multivariate_Analysis/ex/Approximations.thy
--- a/src/HOL/Multivariate_Analysis/ex/Approximations.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/ex/Approximations.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -28,10 +28,10 @@
   apply (simp only: abs_diff_le_iff)
   apply (rule sin_pi6_straddle, simp_all)
    using Taylor_sin [of "1686629713/3221225472" 11]
-  apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral)
+  apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral power_divide)
    apply (simp only: pos_le_divide_eq [symmetric])
   using Taylor_sin [of "6746518853/12884901888" 11]
-  apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral)
+  apply (simp add: in_Reals_norm sin_coeff_def Re_sin atMost_nat_numeral fact_numeral power_divide)
   apply (simp only: pos_le_divide_eq [symmetric] pos_divide_le_eq [symmetric])
   done
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Old_Number_Theory/Fib.thy
--- a/src/HOL/Old_Number_Theory/Fib.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Old_Number_Theory/Fib.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -85,7 +85,7 @@
  "fib (Suc (Suc n)) * fib n =
   (if n mod 2 = 0 then fib (Suc n) * fib (Suc n) - 1
    else fib (Suc n) * fib (Suc n) + 1)"
-  apply (rule int_int_eq [THEN iffD1]) 
+  apply (rule of_nat_eq_iff [where 'a = int, THEN iffD1]) 
   using fib_Cassini_int apply (auto simp add: Suc_leI fib_Suc_gr_0 of_nat_diff)
   done
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Old_Number_Theory/WilsonRuss.thy
--- a/src/HOL/Old_Number_Theory/WilsonRuss.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Old_Number_Theory/WilsonRuss.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -29,7 +29,7 @@
 text \<open>\medskip @{term [source] inv}\<close>
 
 lemma inv_is_inv_aux: "1 < m ==> Suc (nat (m - 2)) = nat (m - 1)"
-  by (subst int_int_eq [symmetric]) auto
+  by simp
 
 lemma inv_is_inv:
     "zprime p \<Longrightarrow> 0 < a \<Longrightarrow> a < p ==> [a * inv p a = 1] (mod p)"
@@ -38,10 +38,7 @@
   apply (subst mod_mult_right_eq [symmetric])
   apply (subst zcong_zmod [symmetric])
   apply (subst power_Suc [symmetric])
-  apply (subst inv_is_inv_aux)
-   apply (erule_tac [2] Little_Fermat)
-   apply (erule_tac [2] zdvd_not_zless)
-   apply (unfold zprime_def, auto)
+  using Little_Fermat inv_is_inv_aux zdvd_not_zless apply auto
   done
 
 lemma inv_distinct:
@@ -116,8 +113,7 @@
 
 lemma inv_inv_aux: "5 \<le> p ==>
     nat (p - 2) * nat (p - 2) = Suc (nat (p - 1) * nat (p - 3))"
-  apply (subst int_int_eq [symmetric])
-  apply (simp add: of_nat_mult)
+  apply (subst of_nat_eq_iff [where 'a = int, symmetric])
   apply (simp add: left_diff_distrib right_diff_distrib)
   done
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Power.thy
--- a/src/HOL/Power.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Power.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -391,8 +391,6 @@
   "(a / b) ^ n = a ^ n / b ^ n"
   by (induct n) simp_all
 
-declare power_divide [where b = "numeral w" for w, simp]
-
 end
 
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Probability/Projective_Limit.thy
--- a/src/HOL/Probability/Projective_Limit.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Probability/Projective_Limit.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -298,7 +298,7 @@
     also have "\<dots> < ereal (1 * real_of_ereal ?a)" unfolding less_ereal.simps
     proof (rule mult_strict_right_mono)
       have "(\<Sum>i = 1..n. 2 powr - real i) = (\<Sum>i = 1..<Suc n. (1/2) ^ i)"
-        by (rule setsum.cong) (auto simp: powr_realpow powr_divide powr_minus_divide)  
+        by (rule setsum.cong) (auto simp: powr_realpow powr_divide power_divide powr_minus_divide)  
       also have "{1..<Suc n} = {..<Suc n} - {0}" by auto
       also have "setsum (op ^ (1 / 2::real)) ({..<Suc n} - {0}) =
         setsum (op ^ (1 / 2)) ({..<Suc n}) - 1" by (auto simp: setsum_diff1)
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Real.thy
--- a/src/HOL/Real.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Real.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -1044,7 +1044,7 @@
   have "(0::real) \<le> 1"
     by (metis less_eq_real_def zero_less_one)
   thus ?thesis
-    by (metis floor_unique less_add_one less_imp_le not_less of_int_le_iff order_trans)
+    by (metis floor_of_int less_floor_iff)
 qed
 
 lemma int_le_real_less: "(n \<le> m) = (real_of_int n < real_of_int m + 1)"
@@ -1509,29 +1509,23 @@
 lemma floor_minus_divide_eq_div_numeral[simp]: "\<lfloor>- (numeral a / numeral b)::real\<rfloor> = - numeral a div numeral b"
   by (metis divide_minus_left floor_divide_of_int_eq of_int_neg_numeral of_int_numeral)
 
-lemma real_of_int_ceiling_cancel [simp]:
-     "(real_of_int (ceiling x) = x) = (\<exists>n::int. x = real_of_int n)"
+lemma of_int_ceiling_cancel [simp]:
+     "(of_int (ceiling x) = x) = (\<exists>n::int. x = of_int n)"
   using ceiling_of_int by metis
 
-lemma ceiling_eq: "[| real_of_int n < x; x < real_of_int n + 1 |] ==> ceiling x = n + 1"
-  by linarith
+lemma ceiling_eq: "[| of_int n < x; x \<le> of_int n + 1 |] ==> ceiling x = n + 1"
+  by (simp add: ceiling_unique)
 
-lemma ceiling_eq2: "[| real_of_int n < x; x \<le> real_of_int n + 1 |] ==> ceiling x = n + 1"
+lemma of_int_ceiling_diff_one_le [simp]: "of_int (ceiling r) - 1 \<le> r"
   by linarith
 
-lemma real_of_int_ceiling_diff_one_le [simp]: "real_of_int (ceiling r) - 1 \<le> r"
-  by linarith
-
-lemma real_of_int_ceiling_le_add_one [simp]: "real_of_int (ceiling r) \<le> r + 1"
+lemma of_int_ceiling_le_add_one [simp]: "of_int (ceiling r) \<le> r + 1"
   by linarith
 
-lemma ceiling_le: "x <= real_of_int a ==> ceiling x <= a"
-  by linarith
+lemma ceiling_le: "x <= of_int a ==> ceiling x <= a"
+  by (simp add: ceiling_le_iff)
 
-lemma ceiling_le_real: "ceiling x <= a ==> x <= real_of_int a"
-  by linarith
-
-lemma ceiling_divide_eq_div: "\<lceil>real_of_int a / real_of_int b\<rceil> = - (- a div b)"
+lemma ceiling_divide_eq_div: "\<lceil>of_int a / of_int b\<rceil> = - (- a div b)"
   by (metis ceiling_def floor_divide_of_int_eq minus_divide_left of_int_minus)
 
 lemma ceiling_divide_eq_div_numeral [simp]:
@@ -1574,25 +1568,13 @@
 subsection \<open>Exponentiation with floor\<close>
 
 lemma floor_power:
-  assumes "x = real_of_int (floor x)"
+  assumes "x = of_int (floor x)"
   shows "floor (x ^ n) = floor x ^ n"
 proof -
-  have "x ^ n = real_of_int (floor x ^ n)"
+  have "x ^ n = of_int (floor x ^ n)"
     using assms by (induct n arbitrary: x) simp_all
-  then show ?thesis  by linarith
+  then show ?thesis by (metis floor_of_int) 
 qed
-(*
-lemma natfloor_power:
-  assumes "x = real (natfloor x)"
-  shows "natfloor (x ^ n) = natfloor x ^ n"
-proof -
-  from assms have "0 \<le> floor x" by auto
-  note assms[unfolded natfloor_def of_nat_nat[OF `0 \<le> floor x`]]
-  from floor_power[OF this]
-  show ?thesis unfolding natfloor_def nat_power_eq[OF `0 \<le> floor x`, symmetric]
-    by simp
-qed
-*)
 
 lemma floor_numeral_power[simp]:
   "\<lfloor>numeral x ^ n\<rfloor> = numeral x ^ n"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Tools/Qelim/cooper.ML
--- a/src/HOL/Tools/Qelim/cooper.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Tools/Qelim/cooper.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -810,10 +810,9 @@
 local
 val ss1 =
   simpset_of (put_simpset comp_ss @{context}
-    addsimps @{thms simp_thms} @ [@{thm "nat_numeral"} RS sym, @{thm "zdvd_int"}] 
-        @ map (fn r => r RS sym) 
-          [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}, @{thm "zadd_int"}, 
-           @{thm "zmult_int"}]
+    addsimps @{thms simp_thms} @ 
+            [@{thm "nat_numeral"} RS sym, @{thm "zdvd_int"}, @{thm "of_nat_add"}, @{thm "of_nat_mult"}] 
+        @ map (fn r => r RS sym) [@{thm "int_int_eq"}, @{thm "zle_int"}, @{thm "zless_int"}]
     |> Splitter.add_split @{thm "zdiff_int_split"})
 
 val ss2 =
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Tools/nat_numeral_simprocs.ML
--- a/src/HOL/Tools/nat_numeral_simprocs.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Tools/nat_numeral_simprocs.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -26,7 +26,7 @@
 struct
 
 (*Maps n to #n for n = 1, 2*)
-val numeral_syms = [@{thm numeral_1_eq_1} RS sym, @{thm numeral_2_eq_2} RS sym];
+val numeral_syms = [@{thm numeral_One} RS sym, @{thm numeral_2_eq_2} RS sym];
 val numeral_sym_ss =
   simpset_of (put_simpset HOL_basic_ss @{context} addsimps numeral_syms);
 
@@ -59,7 +59,7 @@
 (** Other simproc items **)
 
 val bin_simps =
-     [@{thm numeral_1_eq_1} RS sym,
+     [@{thm numeral_One} RS sym,
       @{thm numeral_plus_numeral}, @{thm add_numeral_left},
       @{thm diff_nat_numeral}, @{thm diff_0_eq_0}, @{thm diff_0},
       @{thm numeral_times_numeral}, @{thm mult_numeral_left(1)},
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Tools/numeral_simprocs.ML
--- a/src/HOL/Tools/numeral_simprocs.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Tools/numeral_simprocs.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -169,7 +169,7 @@
     |> Simplifier.set_termless numtermless);
 
 (*Maps 1 to Numeral1 so that arithmetic isn't complicated by the abstract 1.*)
-val numeral_syms = [@{thm numeral_1_eq_1} RS sym];
+val numeral_syms = [@{thm numeral_One} RS sym];
 
 (*Simplify 0+n, n+0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
 val add_0s =  @{thms add_0_left add_0_right};
@@ -177,7 +177,7 @@
 
 (* For post-simplification of the rhs of simproc-generated rules *)
 val post_simps =
-    [@{thm numeral_1_eq_1},
+    [@{thm numeral_One},
      @{thm add_0_left}, @{thm add_0_right},
      @{thm mult_zero_left}, @{thm mult_zero_right},
      @{thm mult_1_left}, @{thm mult_1_right},
@@ -192,7 +192,7 @@
 (*To perform binary arithmetic.  The "left" rewriting handles patterns
   created by the Numeral_Simprocs, such as 3 * (5 * x). *)
 val simps =
-    [@{thm numeral_1_eq_1} RS sym] @
+    [@{thm numeral_One} RS sym] @
     @{thms add_numeral_left} @
     @{thms add_neg_numeral_left} @
     @{thms mult_numeral_left} @
@@ -475,7 +475,7 @@
 
 (** Final simplification for the CancelFactor simprocs **)
 val simplify_one = Arith_Data.simplify_meta_eq
-  [@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_by_1}, @{thm numeral_1_eq_1}];
+  [@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_by_1}, @{thm numeral_One}];
 
 fun cancel_simplify_meta_eq ctxt cancel_th th =
     simplify_one ctxt (([th, cancel_th]) MRS trans);
@@ -739,7 +739,7 @@
 fun field_comp_conv ctxt =
   Simplifier.rewrite (put_simpset field_comp_ss ctxt)
   then_conv
-  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_1_eq_1}])
+  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_One}])
 
 end
 
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Tools/semiring_normalizer.ML
--- a/src/HOL/Tools/semiring_normalizer.ML	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Tools/semiring_normalizer.ML	Tue Nov 17 12:32:08 2015 +0000
@@ -122,7 +122,7 @@
       (case Rat.quotient_of_rat x of (i, 1) => i | _ => error "int_of_rat: bad int")),
     conv = (fn ctxt =>
       Simplifier.rewrite (put_simpset semiring_norm_ss ctxt)
-      then_conv Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms numeral_1_eq_1}))};
+      then_conv Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps @{thms numeral_One}))};
 
 val divide_const = Thm.cterm_of @{context} (Logic.varify_global @{term "op /"});
 val [divide_tvar] = Term.add_tvars (Thm.term_of divide_const) [];
@@ -257,10 +257,10 @@
 val is_number = can dest_number;
 
 fun numeral01_conv ctxt =
-  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_1_eq_1}]);
+  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_One}]);
 
 fun zero1_numeral_conv ctxt =
-  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_1_eq_1} RS sym]);
+  Simplifier.rewrite (put_simpset HOL_basic_ss ctxt addsimps [@{thm numeral_One} RS sym]);
 
 fun zerone_conv ctxt cv =
   zero1_numeral_conv ctxt then_conv cv then_conv numeral01_conv ctxt;
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/Transcendental.thy
--- a/src/HOL/Transcendental.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/Transcendental.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -256,6 +256,7 @@
 qed
 
 subsection \<open>Alternating series test / Leibniz formula\<close>
+text\<open>FIXME: generalise these results from the reals via type classes?\<close>
 
 lemma sums_alternating_upper_lower:
   fixes a :: "nat \<Rightarrow> real"
@@ -3549,82 +3550,80 @@
     done
 qed
 
-
-lemma reals_Archimedean4:
-  assumes "0 < y" "0 \<le> x"
-  obtains n where "real n * y \<le> x" "x < real (Suc n) * y"
-  using floor_correct [of "x/y"] assms
-  apply (auto simp: field_simps intro!: that [of "nat (floor (x/y))"])
-by (metis (no_types, hide_lams)  divide_inverse divide_self_if floor_correct mult.left_commute mult.right_neutral mult_eq_0_iff order_refl order_trans real_mult_le_cancel_iff2)
-
 lemma cos_zero_lemma:
-     "\<lbrakk>0 \<le> x; cos x = 0\<rbrakk> \<Longrightarrow>
-      \<exists>n::nat. odd n & x = real n * (pi/2)"
-apply (erule reals_Archimedean4 [OF pi_gt_zero])
-apply (auto simp: )
-apply (subgoal_tac "0 \<le> x - real n * pi &
-                    (x - real n * pi) \<le> pi & (cos (x - real n * pi) = 0) ")
-apply (auto simp: algebra_simps of_nat_Suc)
- prefer 2 apply (simp add: cos_diff)
-apply (simp add: cos_diff)
-apply (subgoal_tac "EX! x. 0 \<le> x & x \<le> pi & cos x = 0")
-apply (rule_tac [2] cos_total, safe)
-apply (drule_tac x = "x - real n * pi" in spec)
-apply (drule_tac x = "pi/2" in spec)
-apply (simp add: cos_diff)
-apply (rule_tac x = "Suc (2 * n)" in exI)
-apply (simp add: of_nat_Suc algebra_simps, auto)
-done
+  assumes "0 \<le> x" "cos x = 0"
+  shows "\<exists>n. odd n \<and> x = of_nat n * (pi/2) \<and> n > 0"
+proof -
+  have xle: "x < (1 + real_of_int \<lfloor>x/pi\<rfloor>) * pi"
+    using floor_correct [of "x/pi"]
+    by (simp add: add.commute divide_less_eq)
+  obtain n where "real n * pi \<le> x" "x < real (Suc n) * pi"
+    apply (rule that [of "nat (floor (x/pi))"] )
+    using assms
+    apply (simp_all add: xle)
+    apply (metis floor_less_iff less_irrefl mult_imp_div_pos_less not_le pi_gt_zero)
+    done
+  then have x: "0 \<le> x - n * pi" "(x - n * pi) \<le> pi" "cos (x - n * pi) = 0"
+    by (auto simp: algebra_simps cos_diff assms)
+  then have "\<exists>!x. 0 \<le> x \<and> x \<le> pi \<and> cos x = 0"
+    by (auto simp: intro!: cos_total)
+  then obtain \<theta> where \<theta>: "0 \<le> \<theta>" "\<theta> \<le> pi" "cos \<theta> = 0" 
+                  and uniq: "\<And>\<phi>. \<lbrakk>0 \<le> \<phi>; \<phi> \<le> pi; cos \<phi> = 0\<rbrakk> \<Longrightarrow> \<phi> = \<theta>"
+    by blast
+  then have "x - real n * pi = \<theta>"
+    using x by blast
+  moreover have "pi/2 = \<theta>"
+    using pi_half_ge_zero uniq by fastforce 
+  ultimately show ?thesis
+    by (rule_tac x = "Suc (2 * n)" in exI) (simp add: algebra_simps)
+qed
 
 lemma sin_zero_lemma:
-     "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow>
-      \<exists>n::nat. even n & x = real n * (pi/2)"
-apply (subgoal_tac "\<exists>n::nat. ~ even n & x + pi/2 = real n * (pi/2) ")
- apply (clarify, rule_tac x = "n - 1" in exI)
- apply (auto elim!: oddE simp add: of_nat_Suc field_simps)[1]
- apply (rule cos_zero_lemma)
- apply (auto simp: cos_add)
-done
+     "\<lbrakk>0 \<le> x; sin x = 0\<rbrakk> \<Longrightarrow> \<exists>n::nat. even n & x = real n * (pi/2)"
+  using cos_zero_lemma [of "x + pi/2"]
+  apply (clarsimp simp add: cos_add)
+  apply (rule_tac x = "n - 1" in exI)
+  apply (simp add: algebra_simps of_nat_diff)
+  done
 
 lemma cos_zero_iff:
-     "(cos x = 0) =
-      ((\<exists>n::nat. odd n & (x = real n * (pi/2))) |
-       (\<exists>n::nat. odd n & (x = -(real n * (pi/2)))))"
+     "(cos x = 0) \<longleftrightarrow>
+      ((\<exists>n. odd n & (x = real n * (pi/2))) \<or> (\<exists>n. odd n & (x = -(real n * (pi/2)))))"
+      (is "?lhs = ?rhs")
 proof -
   { fix n :: nat
     assume "odd n"
     then obtain m where "n = 2 * m + 1" ..
     then have "cos (real n * pi / 2) = 0"
-      by (simp add: field_simps of_nat_Suc) (simp add: cos_add add_divide_distrib)
+      by (simp add: field_simps) (simp add: cos_add add_divide_distrib)
   } note * = this
   show ?thesis
-  apply (rule iffI)
-  apply (cut_tac linorder_linear [of 0 x], safe)
-  apply (drule cos_zero_lemma, assumption+)
-  apply (cut_tac x="-x" in cos_zero_lemma, simp, simp)
-  apply (auto dest: *)
-  done
+  proof
+    assume "cos x = 0" then show ?rhs
+      using cos_zero_lemma [of x] cos_zero_lemma [of "-x"] by force
+  next
+    assume ?rhs then show "cos x = 0" 
+    by (auto dest: * simp del: eq_divide_eq_numeral1)
+  qed
 qed
 
-(* ditto: but to a lesser extent *)
 lemma sin_zero_iff:
-     "(sin x = 0) =
-      ((\<exists>n::nat. even n & (x = real n * (pi/2))) |
-       (\<exists>n::nat. even n & (x = -(real n * (pi/2)))))"
-apply (rule iffI)
-apply (cut_tac linorder_linear [of 0 x], safe)
-apply (drule sin_zero_lemma, assumption+)
-apply (cut_tac x="-x" in sin_zero_lemma, simp, simp, safe)
-apply (force simp add: minus_equation_iff [of x])
-apply (auto elim: evenE)
-done
-
+     "(sin x = 0) \<longleftrightarrow>
+      ((\<exists>n. even n & (x = real n * (pi/2))) \<or> (\<exists>n. even n & (x = -(real n * (pi/2)))))"
+      (is "?lhs = ?rhs")
+proof
+  assume "sin x = 0" then show ?rhs
+    using sin_zero_lemma [of x] sin_zero_lemma [of "-x"] by force
+next
+  assume ?rhs then show "sin x = 0" 
+    by (auto elim: evenE)
+qed
 
 lemma cos_zero_iff_int:
-     "cos x = 0 \<longleftrightarrow> (\<exists>n::int. odd n \<and> x = of_int n * (pi/2))"
+     "cos x = 0 \<longleftrightarrow> (\<exists>n. odd n \<and> x = of_int n * (pi/2))"
 proof safe
   assume "cos x = 0"
-  then show "\<exists>n::int. odd n & x = of_int n * (pi/2)"
+  then show "\<exists>n. odd n & x = of_int n * (pi/2)"
     apply (simp add: cos_zero_iff, safe)
     apply (metis even_int_iff of_int_of_nat_eq)
     apply (rule_tac x="- (int n)" in exI, simp)
@@ -3634,18 +3633,16 @@
   assume "odd n"
   then show "cos (of_int n * (pi / 2)) = 0"
     apply (simp add: cos_zero_iff)
-    apply (case_tac n rule: int_cases2, simp)
-    apply (rule disjI2)
-    apply (rule_tac x="nat (-n)" in exI, simp)
+    apply (case_tac n rule: int_cases2, simp_all)
     done
 qed
 
 lemma sin_zero_iff_int:
-     "sin x = 0 \<longleftrightarrow> (\<exists>n::int. even n & (x = of_int n * (pi/2)))"
+     "sin x = 0 \<longleftrightarrow> (\<exists>n. even n & (x = of_int n * (pi/2)))"
 proof safe
   assume "sin x = 0"
-  then show "\<exists>n::int. even n \<and> x = of_int n * (pi / 2)"
-    apply (simp add: sin_zero_iff, safe)
+  then show "\<exists>n. even n \<and> x = of_int n * (pi / 2)"
+    apply (simp add: sin_zero_iff, safe)    
     apply (metis even_int_iff of_int_of_nat_eq)
     apply (rule_tac x="- (int n)" in exI, simp)
     done
@@ -3654,9 +3651,7 @@
   assume "even n"
   then show "sin (of_int n * (pi / 2)) = 0"
     apply (simp add: sin_zero_iff)
-    apply (case_tac n rule: int_cases2, simp)
-    apply (rule disjI2)
-    apply (rule_tac x="nat (-n)" in exI, simp)
+    apply (case_tac n rule: int_cases2, simp_all)
     done
 qed
 
@@ -4907,20 +4902,16 @@
 lemma machin_Euler: "5 * arctan(1/7) + 2 * arctan(3/79) = pi/4"
 proof -
   have 17: "\<bar>1/7\<bar> < (1 :: real)" by auto
-  with arctan_double
-  have "2 * arctan (1/7) = arctan (7/24)"  by auto
-  moreover
-  have "\<bar>7/24\<bar> < (1 :: real)" by auto
-  with arctan_double
-  have "2 * arctan (7/24) = arctan (336/527)"  by auto
-  moreover
-  have "\<bar>336/527\<bar> < (1 :: real)" by auto
+  with arctan_double have "2 * arctan (1/7) = arctan (7/24)" 
+    by simp (simp add: field_simps)
+  moreover have "\<bar>7/24\<bar> < (1 :: real)" by auto
+  with arctan_double have "2 * arctan (7/24) = arctan (336/527)"  by simp (simp add: field_simps)
+  moreover have "\<bar>336/527\<bar> < (1 :: real)" by auto
   from arctan_add[OF less_imp_le[OF 17] this]
   have "arctan(1/7) + arctan (336/527) = arctan (2879/3353)"  by auto
   ultimately have I: "5 * arctan(1/7) = arctan (2879/3353)"  by auto
   have 379: "\<bar>3/79\<bar> < (1 :: real)" by auto
-  with arctan_double
-  have II: "2 * arctan (3/79) = arctan (237/3116)"  by auto
+  with arctan_double have II: "2 * arctan (3/79) = arctan (237/3116)"  by simp (simp add: field_simps)
   have *: "\<bar>2879/3353\<bar> < (1 :: real)" by auto
   have "\<bar>237/3116\<bar> < (1 :: real)" by auto
   from arctan_add[OF less_imp_le[OF *] this]
@@ -5011,9 +5002,8 @@
   qed
 qed
 
-text\<open>FIXME: generalise from the reals via type classes?\<close>
 lemma summable_arctan_series:
-  fixes x :: real and n :: nat
+  fixes n :: nat
   assumes "\<bar>x\<bar> \<le> 1"
   shows "summable (\<lambda> k. (-1)^k * (1 / real (k*2+1) * x ^ (k*2+1)))"
   (is "summable (?c x)")
@@ -5467,7 +5457,7 @@
   shows "(\<Sum>i\<le>m. (a i) * x ^ i) * (\<Sum>j\<le>n. (b j) * x ^ j) =
          (\<Sum>r\<le>m + n. (\<Sum>k\<le>r. (a k) * (b (r - k))) * x ^ r)"
   using polynomial_product [of m a n b x] assms
-  by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] int_int_eq Int.int_setsum [symmetric])
+  by (simp only: of_nat_mult [symmetric] of_nat_power [symmetric] of_nat_eq_iff Int.int_setsum [symmetric])
 
 lemma polyfun_diff: (*COMPLEX_SUB_POLYFUN in HOL Light*)
     fixes x :: "'a::idom"
diff -r f6b9f528c89c -r 6571c78c9667 src/HOL/ex/Sum_of_Powers.thy
--- a/src/HOL/ex/Sum_of_Powers.thy	Tue Nov 17 12:01:19 2015 +0100
+++ b/src/HOL/ex/Sum_of_Powers.thy	Tue Nov 17 12:32:08 2015 +0000
@@ -189,7 +189,7 @@
     by (auto simp add: sum_of_powers)
   also have "... = ((n ^ 2 + n) / 2) ^ 2"
     by (simp add: unroll algebra_simps power2_eq_square power4_eq power3_eq_cube)
-  finally show ?thesis by simp
+  finally show ?thesis by (simp add: power_divide)
 qed
                        
 lemma sum_of_cubes_nat: "(\<Sum>k\<le>n::nat. k ^ 3) = (n ^ 2 + n) ^ 2 div 4"