src/HOL/Complex/ex/ReflectedFerrack.thy
author haftmann
Fri Aug 24 14:14:20 2007 +0200 (2007-08-24)
changeset 24423 ae9cd0e92423
parent 24348 c708ea5b109a
child 24783 5a3e336a2e37
permissions -rw-r--r--
overloaded definitions accompanied by explicit constants
     1 (*  Title:      Complex/ex/ReflectedFerrack.thy
     2     Author:     Amine Chaieb
     3 *)
     4 
     5 header {* Quatifier elimination for R(0,1,+,<) *}
     6 
     7 theory ReflectedFerrack
     8   imports GCD Real Efficient_Nat
     9   uses ("linreif.ML") ("linrtac.ML")
    10 begin
    11 
    12 
    13   (*********************************************************************************)
    14   (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
    15   (*********************************************************************************)
    16 
    17 consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
    18 primrec
    19   "alluopairs [] = []"
    20   "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
    21 
    22 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
    23 by (induct xs, auto)
    24 
    25 lemma alluopairs_set:
    26   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
    27 by (induct xs, auto)
    28 
    29 lemma alluopairs_ex:
    30   assumes Pc: "\<forall> x y. P x y = P y x"
    31   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
    32 proof
    33   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
    34   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
    35   from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
    36     by auto
    37 next
    38   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
    39   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
    40   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
    41   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
    42 qed
    43 
    44 lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
    45 using Nat.gr0_conv_Suc
    46 by clarsimp
    47 
    48 lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
    49   apply (induct xs, auto) done
    50 
    51 consts remdps:: "'a list \<Rightarrow> 'a list"
    52 
    53 recdef remdps "measure size"
    54   "remdps [] = []"
    55   "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
    56 (hints simp add: filter_length[rule_format])
    57 
    58 lemma remdps_set[simp]: "set (remdps xs) = set xs"
    59   by (induct xs rule: remdps.induct, auto)
    60 
    61 
    62 
    63   (*********************************************************************************)
    64   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
    65   (*********************************************************************************)
    66 
    67 datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
    68   | Mul int num 
    69 
    70   (* A size for num to make inductive proofs simpler*)
    71 consts num_size :: "num \<Rightarrow> nat" 
    72 primrec 
    73   "num_size (C c) = 1"
    74   "num_size (Bound n) = 1"
    75   "num_size (Neg a) = 1 + num_size a"
    76   "num_size (Add a b) = 1 + num_size a + num_size b"
    77   "num_size (Sub a b) = 3 + num_size a + num_size b"
    78   "num_size (Mul c a) = 1 + num_size a"
    79   "num_size (CN n c a) = 3 + num_size a "
    80 
    81   (* Semantics of numeral terms (num) *)
    82 consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
    83 primrec
    84   "Inum bs (C c) = (real c)"
    85   "Inum bs (Bound n) = bs!n"
    86   "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
    87   "Inum bs (Neg a) = -(Inum bs a)"
    88   "Inum bs (Add a b) = Inum bs a + Inum bs b"
    89   "Inum bs (Sub a b) = Inum bs a - Inum bs b"
    90   "Inum bs (Mul c a) = (real c) * Inum bs a"
    91     (* FORMULAE *)
    92 datatype fm  = 
    93   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
    94   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
    95 
    96 
    97   (* A size for fm *)
    98 consts fmsize :: "fm \<Rightarrow> nat"
    99 recdef fmsize "measure size"
   100   "fmsize (NOT p) = 1 + fmsize p"
   101   "fmsize (And p q) = 1 + fmsize p + fmsize q"
   102   "fmsize (Or p q) = 1 + fmsize p + fmsize q"
   103   "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
   104   "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
   105   "fmsize (E p) = 1 + fmsize p"
   106   "fmsize (A p) = 4+ fmsize p"
   107   "fmsize p = 1"
   108   (* several lemmas about fmsize *)
   109 lemma fmsize_pos: "fmsize p > 0"	
   110 by (induct p rule: fmsize.induct) simp_all
   111 
   112   (* Semantics of formulae (fm) *)
   113 consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
   114 primrec
   115   "Ifm bs T = True"
   116   "Ifm bs F = False"
   117   "Ifm bs (Lt a) = (Inum bs a < 0)"
   118   "Ifm bs (Gt a) = (Inum bs a > 0)"
   119   "Ifm bs (Le a) = (Inum bs a \<le> 0)"
   120   "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
   121   "Ifm bs (Eq a) = (Inum bs a = 0)"
   122   "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
   123   "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
   124   "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
   125   "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
   126   "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
   127   "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
   128   "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
   129   "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
   130 
   131 lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
   132 apply simp
   133 done
   134 
   135 lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
   136 apply simp
   137 done
   138 lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
   139 apply simp
   140 done
   141 lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
   142 apply simp
   143 done
   144 lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
   145 apply simp
   146 done
   147 lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
   148 apply simp
   149 done
   150 lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
   151 apply simp
   152 done
   153 lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
   154 apply simp
   155 done
   156 
   157 lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
   158 apply simp
   159 done
   160 lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
   161 apply simp
   162 done
   163 
   164 consts not:: "fm \<Rightarrow> fm"
   165 recdef not "measure size"
   166   "not (NOT p) = p"
   167   "not T = F"
   168   "not F = T"
   169   "not p = NOT p"
   170 lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
   171 by (cases p) auto
   172 
   173 constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
   174   "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 
   175    if p = q then p else And p q)"
   176 lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
   177 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
   178 
   179 constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
   180   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
   181        else if p=q then p else Or p q)"
   182 
   183 lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
   184 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
   185 
   186 constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
   187   "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 
   188     else Imp p q)"
   189 lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
   190 by (cases "p=F \<or> q=T",simp_all add: imp_def) 
   191 
   192 constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
   193   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
   194        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 
   195   Iff p q)"
   196 lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
   197   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
   198 
   199 lemma conj_simps:
   200   "conj F Q = F"
   201   "conj P F = F"
   202   "conj T Q = Q"
   203   "conj P T = P"
   204   "conj P P = P"
   205   "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"
   206   by (simp_all add: conj_def)
   207 
   208 lemma disj_simps:
   209   "disj T Q = T"
   210   "disj P T = T"
   211   "disj F Q = Q"
   212   "disj P F = P"
   213   "disj P P = P"
   214   "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
   215   by (simp_all add: disj_def)
   216 lemma imp_simps:
   217   "imp F Q = T"
   218   "imp P T = T"
   219   "imp T Q = Q"
   220   "imp P F = not P"
   221   "imp P P = T"
   222   "P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
   223   by (simp_all add: imp_def)
   224 lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
   225 apply (induct p, auto)
   226 done
   227 
   228 lemma iff_simps:
   229   "iff p p = T"
   230   "iff p (NOT p) = F"
   231   "iff (NOT p) p = F"
   232   "iff p F = not p"
   233   "iff F p = not p"
   234   "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
   235   "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
   236   "p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
   237   using trivNOT
   238   by (simp_all add: iff_def, cases p, auto)
   239   (* Quantifier freeness *)
   240 consts qfree:: "fm \<Rightarrow> bool"
   241 recdef qfree "measure size"
   242   "qfree (E p) = False"
   243   "qfree (A p) = False"
   244   "qfree (NOT p) = qfree p" 
   245   "qfree (And p q) = (qfree p \<and> qfree q)" 
   246   "qfree (Or  p q) = (qfree p \<and> qfree q)" 
   247   "qfree (Imp p q) = (qfree p \<and> qfree q)" 
   248   "qfree (Iff p q) = (qfree p \<and> qfree q)"
   249   "qfree p = True"
   250 
   251   (* Boundedness and substitution *)
   252 consts 
   253   numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
   254   bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
   255 primrec
   256   "numbound0 (C c) = True"
   257   "numbound0 (Bound n) = (n>0)"
   258   "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
   259   "numbound0 (Neg a) = numbound0 a"
   260   "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
   261   "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
   262   "numbound0 (Mul i a) = numbound0 a"
   263 lemma numbound0_I:
   264   assumes nb: "numbound0 a"
   265   shows "Inum (b#bs) a = Inum (b'#bs) a"
   266 using nb
   267 by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
   268 
   269 primrec
   270   "bound0 T = True"
   271   "bound0 F = True"
   272   "bound0 (Lt a) = numbound0 a"
   273   "bound0 (Le a) = numbound0 a"
   274   "bound0 (Gt a) = numbound0 a"
   275   "bound0 (Ge a) = numbound0 a"
   276   "bound0 (Eq a) = numbound0 a"
   277   "bound0 (NEq a) = numbound0 a"
   278   "bound0 (NOT p) = bound0 p"
   279   "bound0 (And p q) = (bound0 p \<and> bound0 q)"
   280   "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
   281   "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
   282   "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
   283   "bound0 (E p) = False"
   284   "bound0 (A p) = False"
   285 
   286 lemma bound0_I:
   287   assumes bp: "bound0 p"
   288   shows "Ifm (b#bs) p = Ifm (b'#bs) p"
   289 using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
   290 by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
   291 
   292 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
   293 by (cases p, auto)
   294 lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
   295 by (cases p, auto)
   296 
   297 
   298 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
   299 using conj_def by auto 
   300 lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
   301 using conj_def by auto 
   302 
   303 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
   304 using disj_def by auto 
   305 lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
   306 using disj_def by auto 
   307 
   308 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
   309 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
   310 lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
   311 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   312 
   313 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
   314   by (unfold iff_def,cases "p=q", auto)
   315 lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
   316 using iff_def by (unfold iff_def,cases "p=q", auto)
   317 
   318 consts 
   319   decrnum:: "num \<Rightarrow> num" 
   320   decr :: "fm \<Rightarrow> fm"
   321 
   322 recdef decrnum "measure size"
   323   "decrnum (Bound n) = Bound (n - 1)"
   324   "decrnum (Neg a) = Neg (decrnum a)"
   325   "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
   326   "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
   327   "decrnum (Mul c a) = Mul c (decrnum a)"
   328   "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
   329   "decrnum a = a"
   330 
   331 recdef decr "measure size"
   332   "decr (Lt a) = Lt (decrnum a)"
   333   "decr (Le a) = Le (decrnum a)"
   334   "decr (Gt a) = Gt (decrnum a)"
   335   "decr (Ge a) = Ge (decrnum a)"
   336   "decr (Eq a) = Eq (decrnum a)"
   337   "decr (NEq a) = NEq (decrnum a)"
   338   "decr (NOT p) = NOT (decr p)" 
   339   "decr (And p q) = conj (decr p) (decr q)"
   340   "decr (Or p q) = disj (decr p) (decr q)"
   341   "decr (Imp p q) = imp (decr p) (decr q)"
   342   "decr (Iff p q) = iff (decr p) (decr q)"
   343   "decr p = p"
   344 
   345 lemma decrnum: assumes nb: "numbound0 t"
   346   shows "Inum (x#bs) t = Inum bs (decrnum t)"
   347   using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
   348 
   349 lemma decr: assumes nb: "bound0 p"
   350   shows "Ifm (x#bs) p = Ifm bs (decr p)"
   351   using nb 
   352   by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
   353 
   354 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
   355 by (induct p, simp_all)
   356 
   357 consts 
   358   isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
   359 recdef isatom "measure size"
   360   "isatom T = True"
   361   "isatom F = True"
   362   "isatom (Lt a) = True"
   363   "isatom (Le a) = True"
   364   "isatom (Gt a) = True"
   365   "isatom (Ge a) = True"
   366   "isatom (Eq a) = True"
   367   "isatom (NEq a) = True"
   368   "isatom p = False"
   369 
   370 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
   371 by (induct p, simp_all)
   372 
   373 constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
   374   "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
   375   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
   376 constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
   377   "evaldjf f ps \<equiv> foldr (djf f) ps F"
   378 
   379 lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
   380 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
   381 (cases "f p", simp_all add: Let_def djf_def) 
   382 
   383 
   384 lemma djf_simps:
   385   "djf f p T = T"
   386   "djf f p F = f p"
   387   "q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"
   388   by (simp_all add: djf_def)
   389 
   390 lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
   391   by(induct ps, simp_all add: evaldjf_def djf_Or)
   392 
   393 lemma evaldjf_bound0: 
   394   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
   395   shows "bound0 (evaldjf f xs)"
   396   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   397 
   398 lemma evaldjf_qf: 
   399   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
   400   shows "qfree (evaldjf f xs)"
   401   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   402 
   403 consts disjuncts :: "fm \<Rightarrow> fm list"
   404 recdef disjuncts "measure size"
   405   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
   406   "disjuncts F = []"
   407   "disjuncts p = [p]"
   408 
   409 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
   410 by(induct p rule: disjuncts.induct, auto)
   411 
   412 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
   413 proof-
   414   assume nb: "bound0 p"
   415   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
   416   thus ?thesis by (simp only: list_all_iff)
   417 qed
   418 
   419 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
   420 proof-
   421   assume qf: "qfree p"
   422   hence "list_all qfree (disjuncts p)"
   423     by (induct p rule: disjuncts.induct, auto)
   424   thus ?thesis by (simp only: list_all_iff)
   425 qed
   426 
   427 constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
   428   "DJ f p \<equiv> evaldjf f (disjuncts p)"
   429 
   430 lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
   431   and fF: "f F = F"
   432   shows "Ifm bs (DJ f p) = Ifm bs (f p)"
   433 proof-
   434   have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
   435     by (simp add: DJ_def evaldjf_ex) 
   436   also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
   437   finally show ?thesis .
   438 qed
   439 
   440 lemma DJ_qf: assumes 
   441   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
   442   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
   443 proof(clarify)
   444   fix  p assume qf: "qfree p"
   445   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
   446   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
   447   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
   448   
   449   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
   450 qed
   451 
   452 lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
   453   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
   454 proof(clarify)
   455   fix p::fm and bs
   456   assume qf: "qfree p"
   457   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
   458   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
   459   have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
   460     by (simp add: DJ_def evaldjf_ex)
   461   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
   462   also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
   463   finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
   464 qed
   465   (* Simplification *)
   466 consts 
   467   numgcd :: "num \<Rightarrow> int"
   468   numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
   469   reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
   470   reducecoeff :: "num \<Rightarrow> num"
   471   dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
   472 consts maxcoeff:: "num \<Rightarrow> int"
   473 recdef maxcoeff "measure size"
   474   "maxcoeff (C i) = abs i"
   475   "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
   476   "maxcoeff t = 1"
   477 
   478 lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
   479   by (induct t rule: maxcoeff.induct, auto)
   480 
   481 recdef numgcdh "measure size"
   482   "numgcdh (C i) = (\<lambda>g. igcd i g)"
   483   "numgcdh (CN n c t) = (\<lambda>g. igcd c (numgcdh t g))"
   484   "numgcdh t = (\<lambda>g. 1)"
   485 defs numgcd_def [code func]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
   486 
   487 recdef reducecoeffh "measure size"
   488   "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
   489   "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
   490   "reducecoeffh t = (\<lambda>g. t)"
   491 
   492 defs reducecoeff_def: "reducecoeff t \<equiv> 
   493   (let g = numgcd t in 
   494   if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
   495 
   496 recdef dvdnumcoeff "measure size"
   497   "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
   498   "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
   499   "dvdnumcoeff t = (\<lambda>g. False)"
   500 
   501 lemma dvdnumcoeff_trans: 
   502   assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
   503   shows "dvdnumcoeff t g"
   504   using dgt' gdg 
   505   by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg zdvd_trans[OF gdg])
   506 
   507 declare zdvd_trans [trans add]
   508 
   509 lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
   510 by arith
   511 
   512 lemma numgcd0:
   513   assumes g0: "numgcd t = 0"
   514   shows "Inum bs t = 0"
   515   using g0[simplified numgcd_def] 
   516   by (induct t rule: numgcdh.induct, auto simp add: igcd_def gcd_zero natabs0 max_def maxcoeff_pos)
   517 
   518 lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
   519   using gp
   520   by (induct t rule: numgcdh.induct, auto simp add: igcd_def)
   521 
   522 lemma numgcd_pos: "numgcd t \<ge>0"
   523   by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
   524 
   525 lemma reducecoeffh:
   526   assumes gt: "dvdnumcoeff t g" and gp: "g > 0" 
   527   shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
   528   using gt
   529 proof(induct t rule: reducecoeffh.induct) 
   530   case (1 i) hence gd: "g dvd i" by simp
   531   from gp have gnz: "g \<noteq> 0" by simp
   532   from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
   533 next
   534   case (2 n c t)  hence gd: "g dvd c" by simp
   535   from gp have gnz: "g \<noteq> 0" by simp
   536   from prems show ?case by (simp add: real_of_int_div[OF gnz gd] ring_simps)
   537 qed (auto simp add: numgcd_def gp)
   538 consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
   539 recdef ismaxcoeff "measure size"
   540   "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
   541   "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
   542   "ismaxcoeff t = (\<lambda>x. True)"
   543 
   544 lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
   545 by (induct t rule: ismaxcoeff.induct, auto)
   546 
   547 lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
   548 proof (induct t rule: maxcoeff.induct)
   549   case (2 n c t)
   550   hence H:"ismaxcoeff t (maxcoeff t)" .
   551   have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
   552   from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
   553 qed simp_all
   554 
   555 lemma igcd_gt1: "igcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
   556   apply (cases "abs i = 0", simp_all add: igcd_def)
   557   apply (cases "abs j = 0", simp_all)
   558   apply (cases "abs i = 1", simp_all)
   559   apply (cases "abs j = 1", simp_all)
   560   apply auto
   561   done
   562 lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
   563   by (induct t rule: numgcdh.induct, auto simp add:igcd0)
   564 
   565 lemma dvdnumcoeff_aux:
   566   assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
   567   shows "dvdnumcoeff t (numgcdh t m)"
   568 using prems
   569 proof(induct t rule: numgcdh.induct)
   570   case (2 n c t) 
   571   let ?g = "numgcdh t m"
   572   from prems have th:"igcd c ?g > 1" by simp
   573   from igcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
   574   have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
   575   moreover {assume "abs c > 1" and gp: "?g > 1" with prems
   576     have th: "dvdnumcoeff t ?g" by simp
   577     have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2)
   578     from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)}
   579   moreover {assume "abs c = 0 \<and> ?g > 1"
   580     with prems have th: "dvdnumcoeff t ?g" by simp
   581     have th': "igcd c ?g dvd ?g" by (simp add:igcd_dvd2)
   582     from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: igcd_dvd1)
   583     hence ?case by simp }
   584   moreover {assume "abs c > 1" and g0:"?g = 0" 
   585     from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
   586   ultimately show ?case by blast
   587 qed(auto simp add: igcd_dvd1)
   588 
   589 lemma dvdnumcoeff_aux2:
   590   assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
   591   using prems 
   592 proof (simp add: numgcd_def)
   593   let ?mc = "maxcoeff t"
   594   let ?g = "numgcdh t ?mc"
   595   have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
   596   have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
   597   assume H: "numgcdh t ?mc > 1"
   598   from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
   599 qed
   600 
   601 lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
   602 proof-
   603   let ?g = "numgcd t"
   604   have "?g \<ge> 0"  by (simp add: numgcd_pos)
   605   hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
   606   moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)} 
   607   moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)} 
   608   moreover { assume g1:"?g > 1"
   609     from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
   610     from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis 
   611       by (simp add: reducecoeff_def Let_def)} 
   612   ultimately show ?thesis by blast
   613 qed
   614 
   615 lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
   616 by (induct t rule: reducecoeffh.induct, auto)
   617 
   618 lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
   619 using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
   620 
   621 consts
   622   simpnum:: "num \<Rightarrow> num"
   623   numadd:: "num \<times> num \<Rightarrow> num"
   624   nummul:: "num \<Rightarrow> int \<Rightarrow> num"
   625 recdef numadd "measure (\<lambda> (t,s). size t + size s)"
   626   "numadd (CN n1 c1 r1,CN n2 c2 r2) =
   627   (if n1=n2 then 
   628   (let c = c1 + c2
   629   in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
   630   else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2))) 
   631   else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
   632   "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"  
   633   "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
   634   "numadd (C b1, C b2) = C (b1+b2)"
   635   "numadd (a,b) = Add a b"
   636 
   637 lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
   638 apply (induct t s rule: numadd.induct, simp_all add: Let_def)
   639 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
   640 apply (case_tac "n1 = n2", simp_all add: ring_simps)
   641 by (simp only: left_distrib[symmetric],simp)
   642 
   643 lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
   644 by (induct t s rule: numadd.induct, auto simp add: Let_def)
   645 
   646 recdef nummul "measure size"
   647   "nummul (C j) = (\<lambda> i. C (i*j))"
   648   "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
   649   "nummul t = (\<lambda> i. Mul i t)"
   650 
   651 lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
   652 by (induct t rule: nummul.induct, auto simp add: ring_simps)
   653 
   654 lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
   655 by (induct t rule: nummul.induct, auto )
   656 
   657 constdefs numneg :: "num \<Rightarrow> num"
   658   "numneg t \<equiv> nummul t (- 1)"
   659 
   660 constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
   661   "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
   662 
   663 lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
   664 using numneg_def by simp
   665 
   666 lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
   667 using numneg_def by simp
   668 
   669 lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
   670 using numsub_def by simp
   671 
   672 lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
   673 using numsub_def by simp
   674 
   675 recdef simpnum "measure size"
   676   "simpnum (C j) = C j"
   677   "simpnum (Bound n) = CN n 1 (C 0)"
   678   "simpnum (Neg t) = numneg (simpnum t)"
   679   "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
   680   "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
   681   "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
   682   "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
   683 
   684 lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
   685 by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
   686 
   687 lemma simpnum_numbound0[simp]: 
   688   "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
   689 by (induct t rule: simpnum.induct, auto)
   690 
   691 consts nozerocoeff:: "num \<Rightarrow> bool"
   692 recdef nozerocoeff "measure size"
   693   "nozerocoeff (C c) = True"
   694   "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
   695   "nozerocoeff t = True"
   696 
   697 lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
   698 by (induct a b rule: numadd.induct,auto simp add: Let_def)
   699 
   700 lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
   701 by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
   702 
   703 lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
   704 by (simp add: numneg_def nummul_nz)
   705 
   706 lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
   707 by (simp add: numsub_def numneg_nz numadd_nz)
   708 
   709 lemma simpnum_nz: "nozerocoeff (simpnum t)"
   710 by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
   711 
   712 lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
   713 proof (induct t rule: maxcoeff.induct)
   714   case (2 n c t)
   715   hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
   716   have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
   717   with cnz have "max (abs c) (maxcoeff t) > 0" by arith
   718   with prems show ?case by simp
   719 qed auto
   720 
   721 lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
   722 proof-
   723   from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
   724   from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
   725   from maxcoeff_nz[OF nz th] show ?thesis .
   726 qed
   727 
   728 constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
   729   "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
   730    (let t' = simpnum t ; g = numgcd t' in 
   731       if g > 1 then (let g' = igcd n g in 
   732         if g' = 1 then (t',n) 
   733         else (reducecoeffh t' g', n div g')) 
   734       else (t',n))))"
   735 
   736 lemma simp_num_pair_ci:
   737   shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
   738   (is "?lhs = ?rhs")
   739 proof-
   740   let ?t' = "simpnum t"
   741   let ?g = "numgcd ?t'"
   742   let ?g' = "igcd n ?g"
   743   {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
   744   moreover
   745   { assume nnz: "n \<noteq> 0"
   746     {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
   747     moreover
   748     {assume g1:"?g>1" hence g0: "?g > 0" by simp
   749       from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
   750       hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith 
   751       hence "?g'= 1 \<or> ?g' > 1" by arith
   752       moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
   753       moreover {assume g'1:"?g'>1"
   754 	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
   755 	let ?tt = "reducecoeffh ?t' ?g'"
   756 	let ?t = "Inum bs ?tt"
   757 	have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2)
   758 	have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) 
   759 	have gpdgp: "?g' dvd ?g'" by simp
   760 	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p] 
   761 	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
   762 	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
   763 	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
   764 	also have "\<dots> = (Inum bs ?t' / real n)"
   765 	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
   766 	finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
   767 	then have ?thesis using prems by (simp add: simp_num_pair_def)}
   768       ultimately have ?thesis by blast}
   769     ultimately have ?thesis by blast} 
   770   ultimately show ?thesis by blast
   771 qed
   772 
   773 lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
   774   shows "numbound0 t' \<and> n' >0"
   775 proof-
   776     let ?t' = "simpnum t"
   777   let ?g = "numgcd ?t'"
   778   let ?g' = "igcd n ?g"
   779   {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
   780   moreover
   781   { assume nnz: "n \<noteq> 0"
   782     {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
   783     moreover
   784     {assume g1:"?g>1" hence g0: "?g > 0" by simp
   785       from igcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
   786       hence g'p: "?g' > 0" using igcd_pos[where i="n" and j="numgcd ?t'"] by arith
   787       hence "?g'= 1 \<or> ?g' > 1" by arith
   788       moreover {assume "?g'=1" hence ?thesis using prems 
   789 	  by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
   790       moreover {assume g'1:"?g'>1"
   791 	have gpdg: "?g' dvd ?g" by (simp add: igcd_dvd2)
   792 	have gpdd: "?g' dvd n" by (simp add: igcd_dvd1) 
   793 	have gpdgp: "?g' dvd ?g'" by simp
   794 	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
   795 	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
   796 	have "n div ?g' >0" by simp
   797 	hence ?thesis using prems 
   798 	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
   799       ultimately have ?thesis by blast}
   800     ultimately have ?thesis by blast} 
   801   ultimately show ?thesis by blast
   802 qed
   803 
   804 consts simpfm :: "fm \<Rightarrow> fm"
   805 recdef simpfm "measure fmsize"
   806   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
   807   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
   808   "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
   809   "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
   810   "simpfm (NOT p) = not (simpfm p)"
   811   "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
   812   | _ \<Rightarrow> Lt a')"
   813   "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
   814   "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
   815   "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
   816   "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
   817   "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
   818   "simpfm p = p"
   819 lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
   820 proof(induct p rule: simpfm.induct)
   821   case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   822   {fix v assume "?sa = C v" hence ?case using sa by simp }
   823   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   824       by (cases ?sa, simp_all add: Let_def)}
   825   ultimately show ?case by blast
   826 next
   827   case (7 a)  let ?sa = "simpnum a" 
   828   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   829   {fix v assume "?sa = C v" hence ?case using sa by simp }
   830   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   831       by (cases ?sa, simp_all add: Let_def)}
   832   ultimately show ?case by blast
   833 next
   834   case (8 a)  let ?sa = "simpnum a" 
   835   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   836   {fix v assume "?sa = C v" hence ?case using sa by simp }
   837   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   838       by (cases ?sa, simp_all add: Let_def)}
   839   ultimately show ?case by blast
   840 next
   841   case (9 a)  let ?sa = "simpnum a" 
   842   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   843   {fix v assume "?sa = C v" hence ?case using sa by simp }
   844   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   845       by (cases ?sa, simp_all add: Let_def)}
   846   ultimately show ?case by blast
   847 next
   848   case (10 a)  let ?sa = "simpnum a" 
   849   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   850   {fix v assume "?sa = C v" hence ?case using sa by simp }
   851   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   852       by (cases ?sa, simp_all add: Let_def)}
   853   ultimately show ?case by blast
   854 next
   855   case (11 a)  let ?sa = "simpnum a" 
   856   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
   857   {fix v assume "?sa = C v" hence ?case using sa by simp }
   858   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
   859       by (cases ?sa, simp_all add: Let_def)}
   860   ultimately show ?case by blast
   861 qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
   862 
   863 
   864 lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
   865 proof(induct p rule: simpfm.induct)
   866   case (6 a) hence nb: "numbound0 a" by simp
   867   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   868   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   869 next
   870   case (7 a) hence nb: "numbound0 a" by simp
   871   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   872   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   873 next
   874   case (8 a) hence nb: "numbound0 a" by simp
   875   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   876   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   877 next
   878   case (9 a) hence nb: "numbound0 a" by simp
   879   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   880   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   881 next
   882   case (10 a) hence nb: "numbound0 a" by simp
   883   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   884   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   885 next
   886   case (11 a) hence nb: "numbound0 a" by simp
   887   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
   888   thus ?case by (cases "simpnum a", auto simp add: Let_def)
   889 qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
   890 
   891 lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
   892 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
   893  (case_tac "simpnum a",auto)+
   894 
   895 consts prep :: "fm \<Rightarrow> fm"
   896 recdef prep "measure fmsize"
   897   "prep (E T) = T"
   898   "prep (E F) = F"
   899   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
   900   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
   901   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
   902   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
   903   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
   904   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
   905   "prep (E p) = E (prep p)"
   906   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
   907   "prep (A p) = prep (NOT (E (NOT p)))"
   908   "prep (NOT (NOT p)) = prep p"
   909   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
   910   "prep (NOT (A p)) = prep (E (NOT p))"
   911   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
   912   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
   913   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
   914   "prep (NOT p) = not (prep p)"
   915   "prep (Or p q) = disj (prep p) (prep q)"
   916   "prep (And p q) = conj (prep p) (prep q)"
   917   "prep (Imp p q) = prep (Or (NOT p) q)"
   918   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
   919   "prep p = p"
   920 (hints simp add: fmsize_pos)
   921 lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
   922 by (induct p rule: prep.induct, auto)
   923 
   924   (* Generic quantifier elimination *)
   925 consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
   926 recdef qelim "measure fmsize"
   927   "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
   928   "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
   929   "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
   930   "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
   931   "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
   932   "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
   933   "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
   934   "qelim p = (\<lambda> y. simpfm p)"
   935 
   936 lemma qelim_ci:
   937   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
   938   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
   939 using qe_inv DJ_qe[OF qe_inv] 
   940 by(induct p rule: qelim.induct) 
   941 (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
   942   simpfm simpfm_qf simp del: simpfm.simps)
   943 
   944 consts 
   945   plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
   946   minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
   947 recdef minusinf "measure size"
   948   "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
   949   "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
   950   "minusinf (Eq  (CN 0 c e)) = F"
   951   "minusinf (NEq (CN 0 c e)) = T"
   952   "minusinf (Lt  (CN 0 c e)) = T"
   953   "minusinf (Le  (CN 0 c e)) = T"
   954   "minusinf (Gt  (CN 0 c e)) = F"
   955   "minusinf (Ge  (CN 0 c e)) = F"
   956   "minusinf p = p"
   957 
   958 recdef plusinf "measure size"
   959   "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
   960   "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
   961   "plusinf (Eq  (CN 0 c e)) = F"
   962   "plusinf (NEq (CN 0 c e)) = T"
   963   "plusinf (Lt  (CN 0 c e)) = F"
   964   "plusinf (Le  (CN 0 c e)) = F"
   965   "plusinf (Gt  (CN 0 c e)) = T"
   966   "plusinf (Ge  (CN 0 c e)) = T"
   967   "plusinf p = p"
   968 
   969 consts
   970   isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
   971 recdef isrlfm "measure size"
   972   "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)" 
   973   "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)" 
   974   "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   975   "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   976   "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   977   "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   978   "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   979   "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
   980   "isrlfm p = (isatom p \<and> (bound0 p))"
   981 
   982   (* splits the bounded from the unbounded part*)
   983 consts rsplit0 :: "num \<Rightarrow> int \<times> num" 
   984 recdef rsplit0 "measure num_size"
   985   "rsplit0 (Bound 0) = (1,C 0)"
   986   "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b 
   987               in (ca+cb, Add ta tb))"
   988   "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
   989   "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
   990   "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
   991   "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
   992   "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
   993   "rsplit0 t = (0,t)"
   994 lemma rsplit0: 
   995   shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
   996 proof (induct t rule: rsplit0.induct)
   997   case (2 a b) 
   998   let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
   999   let ?ca = "fst ?sa" let ?cb = "fst ?sb"
  1000   let ?ta = "snd ?sa" let ?tb = "snd ?sb"
  1001   from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))" 
  1002     by(cases "rsplit0 a",auto simp add: Let_def split_def)
  1003   have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) = 
  1004     Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
  1005     by (simp add: Let_def split_def ring_simps)
  1006   also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
  1007   finally show ?case using nb by simp 
  1008 qed(auto simp add: Let_def split_def ring_simps , simp add: right_distrib[symmetric])
  1009 
  1010     (* Linearize a formula*)
  1011 definition
  1012   lt :: "int \<Rightarrow> num \<Rightarrow> fm"
  1013 where
  1014   "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t)) 
  1015     else (Gt (CN 0 (-c) (Neg t))))"
  1016 
  1017 definition
  1018   le :: "int \<Rightarrow> num \<Rightarrow> fm"
  1019 where
  1020   "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t)) 
  1021     else (Ge (CN 0 (-c) (Neg t))))"
  1022 
  1023 definition
  1024   gt :: "int \<Rightarrow> num \<Rightarrow> fm"
  1025 where
  1026   "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t)) 
  1027     else (Lt (CN 0 (-c) (Neg t))))"
  1028 
  1029 definition
  1030   ge :: "int \<Rightarrow> num \<Rightarrow> fm"
  1031 where
  1032   "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t)) 
  1033     else (Le (CN 0 (-c) (Neg t))))"
  1034 
  1035 definition
  1036   eq :: "int \<Rightarrow> num \<Rightarrow> fm"
  1037 where
  1038   "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t)) 
  1039     else (Eq (CN 0 (-c) (Neg t))))"
  1040 
  1041 definition
  1042   neq :: "int \<Rightarrow> num \<Rightarrow> fm"
  1043 where
  1044   "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t)) 
  1045     else (NEq (CN 0 (-c) (Neg t))))"
  1046 
  1047 lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
  1048 using rsplit0[where bs = "bs" and t="t"]
  1049 by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1050 
  1051 lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
  1052 using rsplit0[where bs = "bs" and t="t"]
  1053 by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1054 
  1055 lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
  1056 using rsplit0[where bs = "bs" and t="t"]
  1057 by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1058 
  1059 lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
  1060 using rsplit0[where bs = "bs" and t="t"]
  1061 by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1062 
  1063 lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
  1064 using rsplit0[where bs = "bs" and t="t"]
  1065 by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1066 
  1067 lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
  1068 using rsplit0[where bs = "bs" and t="t"]
  1069 by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
  1070 
  1071 lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
  1072 by (auto simp add: conj_def)
  1073 lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
  1074 by (auto simp add: disj_def)
  1075 
  1076 consts rlfm :: "fm \<Rightarrow> fm"
  1077 recdef rlfm "measure fmsize"
  1078   "rlfm (And p q) = conj (rlfm p) (rlfm q)"
  1079   "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
  1080   "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
  1081   "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
  1082   "rlfm (Lt a) = split lt (rsplit0 a)"
  1083   "rlfm (Le a) = split le (rsplit0 a)"
  1084   "rlfm (Gt a) = split gt (rsplit0 a)"
  1085   "rlfm (Ge a) = split ge (rsplit0 a)"
  1086   "rlfm (Eq a) = split eq (rsplit0 a)"
  1087   "rlfm (NEq a) = split neq (rsplit0 a)"
  1088   "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
  1089   "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
  1090   "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
  1091   "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
  1092   "rlfm (NOT (NOT p)) = rlfm p"
  1093   "rlfm (NOT T) = F"
  1094   "rlfm (NOT F) = T"
  1095   "rlfm (NOT (Lt a)) = rlfm (Ge a)"
  1096   "rlfm (NOT (Le a)) = rlfm (Gt a)"
  1097   "rlfm (NOT (Gt a)) = rlfm (Le a)"
  1098   "rlfm (NOT (Ge a)) = rlfm (Lt a)"
  1099   "rlfm (NOT (Eq a)) = rlfm (NEq a)"
  1100   "rlfm (NOT (NEq a)) = rlfm (Eq a)"
  1101   "rlfm p = p" (hints simp add: fmsize_pos)
  1102 
  1103 lemma rlfm_I:
  1104   assumes qfp: "qfree p"
  1105   shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
  1106   using qfp 
  1107 by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
  1108 
  1109     (* Operations needed for Ferrante and Rackoff *)
  1110 lemma rminusinf_inf:
  1111   assumes lp: "isrlfm p"
  1112   shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
  1113 using lp
  1114 proof (induct p rule: minusinf.induct)
  1115   case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto 
  1116 next
  1117   case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
  1118 next
  1119   case (3 c e) 
  1120   from prems have nb: "numbound0 e" by simp
  1121   from prems have cp: "real c > 0" by simp
  1122   let ?e="Inum (a#bs) e"
  1123   let ?z = "(- ?e) / real c"
  1124   {fix x
  1125     assume xz: "x < ?z"
  1126     hence "(real c * x < - ?e)" 
  1127       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1128     hence "real c * x + ?e < 0" by arith
  1129     hence "real c * x + ?e \<noteq> 0" by simp
  1130     with xz have "?P ?z x (Eq (CN 0 c e))"
  1131       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
  1132   hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
  1133   thus ?case by blast
  1134 next
  1135   case (4 c e)   
  1136   from prems have nb: "numbound0 e" by simp
  1137   from prems have cp: "real c > 0" by simp
  1138   let ?e="Inum (a#bs) e"
  1139   let ?z = "(- ?e) / real c"
  1140   {fix x
  1141     assume xz: "x < ?z"
  1142     hence "(real c * x < - ?e)" 
  1143       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1144     hence "real c * x + ?e < 0" by arith
  1145     hence "real c * x + ?e \<noteq> 0" by simp
  1146     with xz have "?P ?z x (NEq (CN 0 c e))"
  1147       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1148   hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
  1149   thus ?case by blast
  1150 next
  1151   case (5 c e) 
  1152     from prems have nb: "numbound0 e" by simp
  1153   from prems have cp: "real c > 0" by simp
  1154   let ?e="Inum (a#bs) e"
  1155   let ?z = "(- ?e) / real c"
  1156   {fix x
  1157     assume xz: "x < ?z"
  1158     hence "(real c * x < - ?e)" 
  1159       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1160     hence "real c * x + ?e < 0" by arith
  1161     with xz have "?P ?z x (Lt (CN 0 c e))"
  1162       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
  1163   hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
  1164   thus ?case by blast
  1165 next
  1166   case (6 c e)  
  1167     from prems have nb: "numbound0 e" by simp
  1168   from prems have cp: "real c > 0" by simp
  1169   let ?e="Inum (a#bs) e"
  1170   let ?z = "(- ?e) / real c"
  1171   {fix x
  1172     assume xz: "x < ?z"
  1173     hence "(real c * x < - ?e)" 
  1174       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1175     hence "real c * x + ?e < 0" by arith
  1176     with xz have "?P ?z x (Le (CN 0 c e))"
  1177       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1178   hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
  1179   thus ?case by blast
  1180 next
  1181   case (7 c e)  
  1182     from prems have nb: "numbound0 e" by simp
  1183   from prems have cp: "real c > 0" by simp
  1184   let ?e="Inum (a#bs) e"
  1185   let ?z = "(- ?e) / real c"
  1186   {fix x
  1187     assume xz: "x < ?z"
  1188     hence "(real c * x < - ?e)" 
  1189       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1190     hence "real c * x + ?e < 0" by arith
  1191     with xz have "?P ?z x (Gt (CN 0 c e))"
  1192       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1193   hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
  1194   thus ?case by blast
  1195 next
  1196   case (8 c e)  
  1197     from prems have nb: "numbound0 e" by simp
  1198   from prems have cp: "real c > 0" by simp
  1199   let ?e="Inum (a#bs) e"
  1200   let ?z = "(- ?e) / real c"
  1201   {fix x
  1202     assume xz: "x < ?z"
  1203     hence "(real c * x < - ?e)" 
  1204       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac) 
  1205     hence "real c * x + ?e < 0" by arith
  1206     with xz have "?P ?z x (Ge (CN 0 c e))"
  1207       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1208   hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
  1209   thus ?case by blast
  1210 qed simp_all
  1211 
  1212 lemma rplusinf_inf:
  1213   assumes lp: "isrlfm p"
  1214   shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
  1215 using lp
  1216 proof (induct p rule: isrlfm.induct)
  1217   case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
  1218 next
  1219   case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
  1220 next
  1221   case (3 c e) 
  1222   from prems have nb: "numbound0 e" by simp
  1223   from prems have cp: "real c > 0" by simp
  1224   let ?e="Inum (a#bs) e"
  1225   let ?z = "(- ?e) / real c"
  1226   {fix x
  1227     assume xz: "x > ?z"
  1228     with mult_strict_right_mono [OF xz cp] cp
  1229     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1230     hence "real c * x + ?e > 0" by arith
  1231     hence "real c * x + ?e \<noteq> 0" by simp
  1232     with xz have "?P ?z x (Eq (CN 0 c e))"
  1233       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1234   hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
  1235   thus ?case by blast
  1236 next
  1237   case (4 c e) 
  1238   from prems have nb: "numbound0 e" by simp
  1239   from prems have cp: "real c > 0" by simp
  1240   let ?e="Inum (a#bs) e"
  1241   let ?z = "(- ?e) / real c"
  1242   {fix x
  1243     assume xz: "x > ?z"
  1244     with mult_strict_right_mono [OF xz cp] cp
  1245     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1246     hence "real c * x + ?e > 0" by arith
  1247     hence "real c * x + ?e \<noteq> 0" by simp
  1248     with xz have "?P ?z x (NEq (CN 0 c e))"
  1249       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1250   hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
  1251   thus ?case by blast
  1252 next
  1253   case (5 c e) 
  1254   from prems have nb: "numbound0 e" by simp
  1255   from prems have cp: "real c > 0" by simp
  1256   let ?e="Inum (a#bs) e"
  1257   let ?z = "(- ?e) / real c"
  1258   {fix x
  1259     assume xz: "x > ?z"
  1260     with mult_strict_right_mono [OF xz cp] cp
  1261     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1262     hence "real c * x + ?e > 0" by arith
  1263     with xz have "?P ?z x (Lt (CN 0 c e))"
  1264       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1265   hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
  1266   thus ?case by blast
  1267 next
  1268   case (6 c e) 
  1269   from prems have nb: "numbound0 e" by simp
  1270   from prems have cp: "real c > 0" by simp
  1271   let ?e="Inum (a#bs) e"
  1272   let ?z = "(- ?e) / real c"
  1273   {fix x
  1274     assume xz: "x > ?z"
  1275     with mult_strict_right_mono [OF xz cp] cp
  1276     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1277     hence "real c * x + ?e > 0" by arith
  1278     with xz have "?P ?z x (Le (CN 0 c e))"
  1279       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1280   hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
  1281   thus ?case by blast
  1282 next
  1283   case (7 c e) 
  1284   from prems have nb: "numbound0 e" by simp
  1285   from prems have cp: "real c > 0" by simp
  1286   let ?e="Inum (a#bs) e"
  1287   let ?z = "(- ?e) / real c"
  1288   {fix x
  1289     assume xz: "x > ?z"
  1290     with mult_strict_right_mono [OF xz cp] cp
  1291     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1292     hence "real c * x + ?e > 0" by arith
  1293     with xz have "?P ?z x (Gt (CN 0 c e))"
  1294       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
  1295   hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
  1296   thus ?case by blast
  1297 next
  1298   case (8 c e) 
  1299   from prems have nb: "numbound0 e" by simp
  1300   from prems have cp: "real c > 0" by simp
  1301   let ?e="Inum (a#bs) e"
  1302   let ?z = "(- ?e) / real c"
  1303   {fix x
  1304     assume xz: "x > ?z"
  1305     with mult_strict_right_mono [OF xz cp] cp
  1306     have "(real c * x > - ?e)" by (simp add: mult_ac)
  1307     hence "real c * x + ?e > 0" by arith
  1308     with xz have "?P ?z x (Ge (CN 0 c e))"
  1309       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
  1310   hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
  1311   thus ?case by blast
  1312 qed simp_all
  1313 
  1314 lemma rminusinf_bound0:
  1315   assumes lp: "isrlfm p"
  1316   shows "bound0 (minusinf p)"
  1317   using lp
  1318   by (induct p rule: minusinf.induct) simp_all
  1319 
  1320 lemma rplusinf_bound0:
  1321   assumes lp: "isrlfm p"
  1322   shows "bound0 (plusinf p)"
  1323   using lp
  1324   by (induct p rule: plusinf.induct) simp_all
  1325 
  1326 lemma rminusinf_ex:
  1327   assumes lp: "isrlfm p"
  1328   and ex: "Ifm (a#bs) (minusinf p)"
  1329   shows "\<exists> x. Ifm (x#bs) p"
  1330 proof-
  1331   from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  1332   have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
  1333   from rminusinf_inf[OF lp, where bs="bs"] 
  1334   obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
  1335   from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
  1336   moreover have "z - 1 < z" by simp
  1337   ultimately show ?thesis using z_def by auto
  1338 qed
  1339 
  1340 lemma rplusinf_ex:
  1341   assumes lp: "isrlfm p"
  1342   and ex: "Ifm (a#bs) (plusinf p)"
  1343   shows "\<exists> x. Ifm (x#bs) p"
  1344 proof-
  1345   from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  1346   have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
  1347   from rplusinf_inf[OF lp, where bs="bs"] 
  1348   obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
  1349   from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
  1350   moreover have "z + 1 > z" by simp
  1351   ultimately show ?thesis using z_def by auto
  1352 qed
  1353 
  1354 consts 
  1355   uset:: "fm \<Rightarrow> (num \<times> int) list"
  1356   usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
  1357 recdef uset "measure size"
  1358   "uset (And p q) = (uset p @ uset q)" 
  1359   "uset (Or p q) = (uset p @ uset q)" 
  1360   "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
  1361   "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
  1362   "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
  1363   "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
  1364   "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
  1365   "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
  1366   "uset p = []"
  1367 recdef usubst "measure size"
  1368   "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
  1369   "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
  1370   "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
  1371   "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
  1372   "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
  1373   "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
  1374   "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
  1375   "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
  1376   "usubst p = (\<lambda> (t,n). p)"
  1377 
  1378 lemma usubst_I: assumes lp: "isrlfm p"
  1379   and np: "real n > 0" and nbt: "numbound0 t"
  1380   shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
  1381   using lp
  1382 proof(induct p rule: usubst.induct)
  1383   case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1384   have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
  1385     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1386   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
  1387     by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1388       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1389   also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
  1390     using np by simp 
  1391   finally show ?case using nbt nb by (simp add: ring_simps)
  1392 next
  1393   case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1394   have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
  1395     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1396   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
  1397     by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1398       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1399   also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
  1400     using np by simp 
  1401   finally show ?case using nbt nb by (simp add: ring_simps)
  1402 next
  1403   case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1404   have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
  1405     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1406   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
  1407     by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1408       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1409   also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
  1410     using np by simp 
  1411   finally show ?case using nbt nb by (simp add: ring_simps)
  1412 next
  1413   case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1414   have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
  1415     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1416   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
  1417     by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1418       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1419   also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
  1420     using np by simp 
  1421   finally show ?case using nbt nb by (simp add: ring_simps)
  1422 next
  1423   case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1424   from np have np: "real n \<noteq> 0" by simp
  1425   have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
  1426     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1427   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
  1428     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1429       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1430   also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
  1431     using np by simp 
  1432   finally show ?case using nbt nb by (simp add: ring_simps)
  1433 next
  1434   case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
  1435   from np have np: "real n \<noteq> 0" by simp
  1436   have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
  1437     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
  1438   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
  1439     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)" 
  1440       and b="0", simplified divide_zero_left]) (simp only: ring_simps)
  1441   also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
  1442     using np by simp 
  1443   finally show ?case using nbt nb by (simp add: ring_simps)
  1444 qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
  1445 
  1446 lemma uset_l:
  1447   assumes lp: "isrlfm p"
  1448   shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
  1449 using lp
  1450 by(induct p rule: uset.induct,auto)
  1451 
  1452 lemma rminusinf_uset:
  1453   assumes lp: "isrlfm p"
  1454   and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
  1455   and ex: "Ifm (x#bs) p" (is "?I x p")
  1456   shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
  1457 proof-
  1458   have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
  1459     using lp nmi ex
  1460     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
  1461   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
  1462   from uset_l[OF lp] smU have mp: "real m > 0" by auto
  1463   from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m" 
  1464     by (auto simp add: mult_commute)
  1465   thus ?thesis using smU by auto
  1466 qed
  1467 
  1468 lemma rplusinf_uset:
  1469   assumes lp: "isrlfm p"
  1470   and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
  1471   and ex: "Ifm (x#bs) p" (is "?I x p")
  1472   shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
  1473 proof-
  1474   have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
  1475     using lp nmi ex
  1476     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
  1477   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
  1478   from uset_l[OF lp] smU have mp: "real m > 0" by auto
  1479   from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m" 
  1480     by (auto simp add: mult_commute)
  1481   thus ?thesis using smU by auto
  1482 qed
  1483 
  1484 lemma lin_dense: 
  1485   assumes lp: "isrlfm p"
  1486   and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)" 
  1487   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
  1488   and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
  1489   and ly: "l < y" and yu: "y < u"
  1490   shows "Ifm (y#bs) p"
  1491 using lp px noS
  1492 proof (induct p rule: isrlfm.induct)
  1493   case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
  1494     from prems have "x * real c + ?N x e < 0" by (simp add: ring_simps)
  1495     hence pxc: "x < (- ?N x e) / real c" 
  1496       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
  1497     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1498     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
  1499     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
  1500     moreover {assume y: "y < (-?N x e)/ real c"
  1501       hence "y * real c < - ?N x e"
  1502 	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
  1503       hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
  1504       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
  1505     moreover {assume y: "y > (- ?N x e) / real c" 
  1506       with yu have eu: "u > (- ?N x e) / real c" by auto
  1507       with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
  1508       with lx pxc have "False" by auto
  1509       hence ?case by simp }
  1510     ultimately show ?case by blast
  1511 next
  1512   case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
  1513     from prems have "x * real c + ?N x e \<le> 0" by (simp add: ring_simps)
  1514     hence pxc: "x \<le> (- ?N x e) / real c" 
  1515       by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
  1516     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1517     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
  1518     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
  1519     moreover {assume y: "y < (-?N x e)/ real c"
  1520       hence "y * real c < - ?N x e"
  1521 	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
  1522       hence "real c * y + ?N x e < 0" by (simp add: ring_simps)
  1523       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
  1524     moreover {assume y: "y > (- ?N x e) / real c" 
  1525       with yu have eu: "u > (- ?N x e) / real c" by auto
  1526       with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
  1527       with lx pxc have "False" by auto
  1528       hence ?case by simp }
  1529     ultimately show ?case by blast
  1530 next
  1531   case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
  1532     from prems have "x * real c + ?N x e > 0" by (simp add: ring_simps)
  1533     hence pxc: "x > (- ?N x e) / real c" 
  1534       by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
  1535     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1536     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
  1537     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
  1538     moreover {assume y: "y > (-?N x e)/ real c"
  1539       hence "y * real c > - ?N x e"
  1540 	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
  1541       hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
  1542       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
  1543     moreover {assume y: "y < (- ?N x e) / real c" 
  1544       with ly have eu: "l < (- ?N x e) / real c" by auto
  1545       with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
  1546       with xu pxc have "False" by auto
  1547       hence ?case by simp }
  1548     ultimately show ?case by blast
  1549 next
  1550   case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
  1551     from prems have "x * real c + ?N x e \<ge> 0" by (simp add: ring_simps)
  1552     hence pxc: "x \<ge> (- ?N x e) / real c" 
  1553       by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
  1554     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1555     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
  1556     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
  1557     moreover {assume y: "y > (-?N x e)/ real c"
  1558       hence "y * real c > - ?N x e"
  1559 	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
  1560       hence "real c * y + ?N x e > 0" by (simp add: ring_simps)
  1561       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
  1562     moreover {assume y: "y < (- ?N x e) / real c" 
  1563       with ly have eu: "l < (- ?N x e) / real c" by auto
  1564       with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
  1565       with xu pxc have "False" by auto
  1566       hence ?case by simp }
  1567     ultimately show ?case by blast
  1568 next
  1569   case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
  1570     from cp have cnz: "real c \<noteq> 0" by simp
  1571     from prems have "x * real c + ?N x e = 0" by (simp add: ring_simps)
  1572     hence pxc: "x = (- ?N x e) / real c" 
  1573       by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
  1574     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1575     with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
  1576     with pxc show ?case by simp
  1577 next
  1578   case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
  1579     from cp have cnz: "real c \<noteq> 0" by simp
  1580     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
  1581     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
  1582     hence "y* real c \<noteq> -?N x e"      
  1583       by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
  1584     hence "y* real c + ?N x e \<noteq> 0" by (simp add: ring_simps)
  1585     thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] 
  1586       by (simp add: ring_simps)
  1587 qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
  1588 
  1589 lemma finite_set_intervals:
  1590   assumes px: "P (x::real)" 
  1591   and lx: "l \<le> x" and xu: "x \<le> u"
  1592   and linS: "l\<in> S" and uinS: "u \<in> S"
  1593   and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
  1594   shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
  1595 proof-
  1596   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
  1597   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
  1598   let ?a = "Max ?Mx"
  1599   let ?b = "Min ?xM"
  1600   have MxS: "?Mx \<subseteq> S" by blast
  1601   hence fMx: "finite ?Mx" using fS finite_subset by auto
  1602   from lx linS have linMx: "l \<in> ?Mx" by blast
  1603   hence Mxne: "?Mx \<noteq> {}" by blast
  1604   have xMS: "?xM \<subseteq> S" by blast
  1605   hence fxM: "finite ?xM" using fS finite_subset by auto
  1606   from xu uinS have linxM: "u \<in> ?xM" by blast
  1607   hence xMne: "?xM \<noteq> {}" by blast
  1608   have ax:"?a \<le> x" using Mxne fMx by auto
  1609   have xb:"x \<le> ?b" using xMne fxM by auto
  1610   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
  1611   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
  1612   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
  1613   proof(clarsimp)
  1614     fix y
  1615     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
  1616     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
  1617     moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
  1618     moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
  1619     ultimately show "False" by blast
  1620   qed
  1621   from ainS binS noy ax xb px show ?thesis by blast
  1622 qed
  1623 
  1624 lemma finite_set_intervals2:
  1625   assumes px: "P (x::real)" 
  1626   and lx: "l \<le> x" and xu: "x \<le> u"
  1627   and linS: "l\<in> S" and uinS: "u \<in> S"
  1628   and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
  1629   shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
  1630 proof-
  1631   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
  1632   obtain a and b where 
  1633     as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
  1634   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
  1635   thus ?thesis using px as bs noS by blast 
  1636 qed
  1637 
  1638 lemma rinf_uset:
  1639   assumes lp: "isrlfm p"
  1640   and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
  1641   and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
  1642   and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
  1643   shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p" 
  1644 proof-
  1645   let ?N = "\<lambda> x t. Inum (x#bs) t"
  1646   let ?U = "set (uset p)"
  1647   from ex obtain a where pa: "?I a p" by blast
  1648   from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
  1649   have nmi': "\<not> (?I a (?M p))" by simp
  1650   from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
  1651   have npi': "\<not> (?I a (?P p))" by simp
  1652   have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
  1653   proof-
  1654     let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
  1655     have fM: "finite ?M" by auto
  1656     from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa] 
  1657     have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
  1658     then obtain "t" "n" "s" "m" where 
  1659       tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U" 
  1660       and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
  1661     from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
  1662     from tnU have Mne: "?M \<noteq> {}" by auto
  1663     hence Une: "?U \<noteq> {}" by simp
  1664     let ?l = "Min ?M"
  1665     let ?u = "Max ?M"
  1666     have linM: "?l \<in> ?M" using fM Mne by simp
  1667     have uinM: "?u \<in> ?M" using fM Mne by simp
  1668     have tnM: "?N a t / real n \<in> ?M" using tnU by auto
  1669     have smM: "?N a s / real m \<in> ?M" using smU by auto 
  1670     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
  1671     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
  1672     have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
  1673     have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
  1674     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
  1675     have "(\<exists> s\<in> ?M. ?I s p) \<or> 
  1676       (\<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)" .
  1677     moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
  1678       hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
  1679       then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
  1680       have "(u + u) / 2 = u" by auto with pu tuu 
  1681       have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
  1682       with tuU have ?thesis by blast}
  1683     moreover{
  1684       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"
  1685       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
  1686 	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"
  1687 	by blast
  1688       from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
  1689       then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
  1690       from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
  1691       then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
  1692       from t1x xt2 have t1t2: "t1 < t2" by simp
  1693       let ?u = "(t1 + t2) / 2"
  1694       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
  1695       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
  1696       with t1uU t2uU t1u t2u have ?thesis by blast}
  1697     ultimately show ?thesis by blast
  1698   qed
  1699   then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U" 
  1700     and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
  1701   from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
  1702   from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
  1703     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
  1704   have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
  1705   with lnU smU
  1706   show ?thesis by auto
  1707 qed
  1708     (* The Ferrante - Rackoff Theorem *)
  1709 
  1710 theorem fr_eq: 
  1711   assumes lp: "isrlfm p"
  1712   shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
  1713   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  1714 proof
  1715   assume px: "\<exists> x. ?I x p"
  1716   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  1717   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  1718   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  1719     from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
  1720   ultimately show "?D" by blast
  1721 next
  1722   assume "?D" 
  1723   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
  1724   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
  1725   moreover {assume f:"?F" hence "?E" by blast}
  1726   ultimately show "?E" by blast
  1727 qed
  1728 
  1729 
  1730 lemma fr_equsubst: 
  1731   assumes lp: "isrlfm p"
  1732   shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))"
  1733   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  1734 proof
  1735   assume px: "\<exists> x. ?I x p"
  1736   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  1737   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  1738   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  1739     let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
  1740     let ?N = "\<lambda> t. Inum (x#bs) t"
  1741     {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
  1742       with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
  1743 	by auto
  1744       let ?st = "Add (Mul m t) (Mul n s)"
  1745       from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
  1746 	by (simp add: mult_commute)
  1747       from tnb snb have st_nb: "numbound0 ?st" by simp
  1748       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
  1749 	using mnp mp np by (simp add: ring_simps add_divide_distrib)
  1750       from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"] 
  1751       have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
  1752     with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
  1753   ultimately show "?D" by blast
  1754 next
  1755   assume "?D" 
  1756   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
  1757   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
  1758   moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)" 
  1759     and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
  1760     with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
  1761     let ?st = "Add (Mul l t) (Mul k s)"
  1762     from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0" 
  1763       by (simp add: mult_commute)
  1764     from tnb snb have st_nb: "numbound0 ?st" by simp
  1765     from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
  1766   ultimately show "?E" by blast
  1767 qed
  1768 
  1769 
  1770     (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
  1771 constdefs ferrack:: "fm \<Rightarrow> fm"
  1772   "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
  1773                 in if (mp = T \<or> pp = T) then T else 
  1774                    (let U = remdps(map simp_num_pair 
  1775                      (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
  1776                            (alluopairs (uset p')))) 
  1777                     in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
  1778 
  1779 lemma uset_cong_aux:
  1780   assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
  1781   shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
  1782   (is "?lhs = ?rhs")
  1783 proof(auto)
  1784   fix t n s m
  1785   assume "((t,n),(s,m)) \<in> set (alluopairs U)"
  1786   hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
  1787     using alluopairs_set1[where xs="U"] by blast
  1788   let ?N = "\<lambda> t. Inum (x#bs) t"
  1789   let ?st= "Add (Mul m t) (Mul n s)"
  1790   from Ul th have mnz: "m \<noteq> 0" by auto
  1791   from Ul th have  nnz: "n \<noteq> 0" by auto  
  1792   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
  1793    using mnz nnz by (simp add: ring_simps add_divide_distrib)
  1794  
  1795   thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
  1796        (2 * real n * real m)
  1797        \<in> (\<lambda>((t, n), s, m).
  1798              (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
  1799          (set U \<times> set U)"using mnz nnz th  
  1800     apply (auto simp add: th add_divide_distrib ring_simps split_def image_def)
  1801     by (rule_tac x="(s,m)" in bexI,simp_all) 
  1802   (rule_tac x="(t,n)" in bexI,simp_all)
  1803 next
  1804   fix t n s m
  1805   assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U" 
  1806   let ?N = "\<lambda> t. Inum (x#bs) t"
  1807   let ?st= "Add (Mul m t) (Mul n s)"
  1808   from Ul smU have mnz: "m \<noteq> 0" by auto
  1809   from Ul tnU have  nnz: "n \<noteq> 0" by auto  
  1810   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
  1811    using mnz nnz by (simp add: ring_simps add_divide_distrib)
  1812  let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
  1813  have Pc:"\<forall> a b. ?P a b = ?P b a"
  1814    by auto
  1815  from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
  1816  from alluopairs_ex[OF Pc, where xs="U"] tnU smU
  1817  have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
  1818    by blast
  1819  then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)" 
  1820    and Pts': "?P (t',n') (s',m')" by blast
  1821  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
  1822  let ?st' = "Add (Mul m' t') (Mul n' s')"
  1823    have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
  1824    using mnz' nnz' by (simp add: ring_simps add_divide_distrib)
  1825  from Pts' have 
  1826    "(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
  1827  also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
  1828  finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
  1829           \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
  1830             (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
  1831             set (alluopairs U)"
  1832    using ts'_U by blast
  1833 qed
  1834 
  1835 lemma uset_cong:
  1836   assumes lp: "isrlfm p"
  1837   and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
  1838   and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
  1839   and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
  1840   shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))"
  1841   (is "?lhs = ?rhs")
  1842 proof
  1843   assume ?lhs
  1844   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
  1845     Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
  1846   let ?N = "\<lambda> t. Inum (x#bs) t"
  1847   from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
  1848     and snb: "numbound0 s" and mp:"m > 0"  by auto
  1849   let ?st= "Add (Mul m t) (Mul n s)"
  1850   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
  1851       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
  1852     from tnb snb have stnb: "numbound0 ?st" by simp
  1853   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
  1854    using mp np by (simp add: ring_simps add_divide_distrib)
  1855   from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
  1856   hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
  1857     by auto (rule_tac x="(a,b)" in bexI, auto)
  1858   then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
  1859   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
  1860   from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst 
  1861   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
  1862   from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
  1863   have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st) 
  1864   then show ?rhs using tnU' by auto 
  1865 next
  1866   assume ?rhs
  1867   then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))" 
  1868     by blast
  1869   from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
  1870   hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))" 
  1871     by auto (rule_tac x="(a,b)" in bexI, auto)
  1872   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and 
  1873     th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
  1874     let ?N = "\<lambda> t. Inum (x#bs) t"
  1875   from tnU smU U have tnb: "numbound0 t" and np: "n > 0" 
  1876     and snb: "numbound0 s" and mp:"m > 0"  by auto
  1877   let ?st= "Add (Mul m t) (Mul n s)"
  1878   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0" 
  1879       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
  1880     from tnb snb have stnb: "numbound0 ?st" by simp
  1881   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
  1882    using mp np by (simp add: ring_simps add_divide_distrib)
  1883   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
  1884   from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
  1885   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
  1886   with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
  1887 qed
  1888 
  1889 lemma ferrack: 
  1890   assumes qf: "qfree p"
  1891   shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
  1892   (is "_ \<and> (?rhs = ?lhs)")
  1893 proof-
  1894   let ?I = "\<lambda> x p. Ifm (x#bs) p"
  1895   let ?N = "\<lambda> t. Inum (x#bs) t"
  1896   let ?q = "rlfm (simpfm p)" 
  1897   let ?U = "uset ?q"
  1898   let ?Up = "alluopairs ?U"
  1899   let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
  1900   let ?S = "map ?g ?Up"
  1901   let ?SS = "map simp_num_pair ?S"
  1902   let ?Y = "remdps ?SS"
  1903   let ?f= "(\<lambda> (t,n). ?N t / real n)"
  1904   let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
  1905   let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
  1906   let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
  1907   from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
  1908   from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
  1909   from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
  1910   from U_l UpU 
  1911   have Up_: "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
  1912   hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
  1913     by (auto simp add: mult_pos_pos)
  1914   have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0" 
  1915   proof-
  1916     { fix t n assume tnY: "(t,n) \<in> set ?Y" 
  1917       hence "(t,n) \<in> set ?SS" by simp
  1918       hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
  1919 	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
  1920       then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
  1921       from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
  1922       from simp_num_pair_l[OF tnb np tns]
  1923       have "numbound0 t \<and> n > 0" . }
  1924     thus ?thesis by blast
  1925   qed
  1926 
  1927   have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
  1928   proof-
  1929      from simp_num_pair_ci[where bs="x#bs"] have 
  1930     "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
  1931      hence th: "?f o simp_num_pair = ?f" using ext by blast
  1932     have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
  1933     also have "\<dots> = (?f ` set ?S)" by (simp add: th)
  1934     also have "\<dots> = ((?f o ?g) ` set ?Up)" 
  1935       by (simp only: set_map o_def image_compose[symmetric])
  1936     also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
  1937       using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
  1938     finally show ?thesis .
  1939   qed
  1940   have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
  1941   proof-
  1942     { fix t n assume tnY: "(t,n) \<in> set ?Y"
  1943       with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
  1944       from usubst_I[OF lq np tnb]
  1945     have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))" 
  1946       using simpfm_bound0 by simp}
  1947     thus ?thesis by blast
  1948   qed
  1949   hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
  1950   let ?mp = "minusinf ?q"
  1951   let ?pp = "plusinf ?q"
  1952   let ?M = "?I x ?mp"
  1953   let ?P = "?I x ?pp"
  1954   let ?res = "disj ?mp (disj ?pp ?ep)"
  1955   from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
  1956   have nbth: "bound0 ?res" by auto
  1957 
  1958   from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm  
  1959 
  1960   have th: "?lhs = (\<exists> x. ?I x ?q)" by auto 
  1961   from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
  1962     by (simp only: split_def fst_conv snd_conv)
  1963   also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))" 
  1964     using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm) 
  1965   also have "\<dots> = (Ifm (x#bs) ?res)"
  1966     using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
  1967     by (simp add: split_def pair_collapse)
  1968   finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
  1969   hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
  1970     by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
  1971   from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
  1972   with lr show ?thesis by blast
  1973 qed
  1974 
  1975 constdefs linrqe:: "fm \<Rightarrow> fm"
  1976   "linrqe \<equiv> (\<lambda> p. qelim (prep p) ferrack)"
  1977 
  1978 theorem linrqe: "(Ifm bs (linrqe p) = Ifm bs p) \<and> qfree (linrqe p)"
  1979 using ferrack qelim_ci prep
  1980 unfolding linrqe_def by auto
  1981 
  1982 definition
  1983   ferrack_test :: "unit \<Rightarrow> fm"
  1984 where
  1985   "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
  1986     (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
  1987 
  1988 export_code linrqe ferrack_test in SML module_name Ferrack
  1989 
  1990 (*code_module Ferrack
  1991   contains
  1992     linrqe = linrqe
  1993     test = ferrack_test*)
  1994 
  1995 ML {* Ferrack.ferrack_test () *}
  1996 
  1997 use "linreif.ML"
  1998 oracle linr_oracle ("term") = ReflectedFerrack.linrqe_oracle
  1999 use "linrtac.ML"
  2000 setup LinrTac.setup
  2001 
  2002 end