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
```