src/HOL/Decision_Procs/Ferrack.thy
 author hoelzl Fri Mar 22 10:41:43 2013 +0100 (2013-03-22) changeset 51474 1e9e68247ad1 parent 51272 9c8d63b4b6be child 53168 d998de7f0efc permissions -rw-r--r--
generalize Bfun and Bseq to metric spaces; Bseq is an abbreviation for Bfun
```     1 (*  Title:      HOL/Decision_Procs/Ferrack.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 theory Ferrack
```
```     6 imports Complex_Main Dense_Linear_Order DP_Library
```
```     7   "~~/src/HOL/Library/Code_Target_Numeral" "~~/src/HOL/Library/Old_Recdef"
```
```     8 begin
```
```     9
```
```    10 section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}
```
```    11
```
```    12   (*********************************************************************************)
```
```    13   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
```
```    14   (*********************************************************************************)
```
```    15
```
```    16 datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
```
```    17   | Mul int num
```
```    18
```
```    19   (* A size for num to make inductive proofs simpler*)
```
```    20 primrec num_size :: "num \<Rightarrow> nat" where
```
```    21   "num_size (C c) = 1"
```
```    22 | "num_size (Bound n) = 1"
```
```    23 | "num_size (Neg a) = 1 + num_size a"
```
```    24 | "num_size (Add a b) = 1 + num_size a + num_size b"
```
```    25 | "num_size (Sub a b) = 3 + num_size a + num_size b"
```
```    26 | "num_size (Mul c a) = 1 + num_size a"
```
```    27 | "num_size (CN n c a) = 3 + num_size a "
```
```    28
```
```    29   (* Semantics of numeral terms (num) *)
```
```    30 primrec Inum :: "real list \<Rightarrow> num \<Rightarrow> real" where
```
```    31   "Inum bs (C c) = (real c)"
```
```    32 | "Inum bs (Bound n) = bs!n"
```
```    33 | "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
```
```    34 | "Inum bs (Neg a) = -(Inum bs a)"
```
```    35 | "Inum bs (Add a b) = Inum bs a + Inum bs b"
```
```    36 | "Inum bs (Sub a b) = Inum bs a - Inum bs b"
```
```    37 | "Inum bs (Mul c a) = (real c) * Inum bs a"
```
```    38     (* FORMULAE *)
```
```    39 datatype fm  =
```
```    40   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
```
```    41   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
```
```    42
```
```    43
```
```    44   (* A size for fm *)
```
```    45 fun fmsize :: "fm \<Rightarrow> nat" where
```
```    46   "fmsize (NOT p) = 1 + fmsize p"
```
```    47 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
```
```    48 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"
```
```    49 | "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
```
```    50 | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
```
```    51 | "fmsize (E p) = 1 + fmsize p"
```
```    52 | "fmsize (A p) = 4+ fmsize p"
```
```    53 | "fmsize p = 1"
```
```    54   (* several lemmas about fmsize *)
```
```    55 lemma fmsize_pos: "fmsize p > 0"
```
```    56 by (induct p rule: fmsize.induct) simp_all
```
```    57
```
```    58   (* Semantics of formulae (fm) *)
```
```    59 primrec Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool" where
```
```    60   "Ifm bs T = True"
```
```    61 | "Ifm bs F = False"
```
```    62 | "Ifm bs (Lt a) = (Inum bs a < 0)"
```
```    63 | "Ifm bs (Gt a) = (Inum bs a > 0)"
```
```    64 | "Ifm bs (Le a) = (Inum bs a \<le> 0)"
```
```    65 | "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
```
```    66 | "Ifm bs (Eq a) = (Inum bs a = 0)"
```
```    67 | "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
```
```    68 | "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
```
```    69 | "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
```
```    70 | "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
```
```    71 | "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
```
```    72 | "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
```
```    73 | "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
```
```    74 | "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
```
```    75
```
```    76 lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
```
```    77 apply simp
```
```    78 done
```
```    79
```
```    80 lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
```
```    81 apply simp
```
```    82 done
```
```    83 lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
```
```    84 apply simp
```
```    85 done
```
```    86 lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
```
```    87 apply simp
```
```    88 done
```
```    89 lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
```
```    90 apply simp
```
```    91 done
```
```    92 lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
```
```    93 apply simp
```
```    94 done
```
```    95 lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
```
```    96 apply simp
```
```    97 done
```
```    98 lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
```
```    99 apply simp
```
```   100 done
```
```   101
```
```   102 lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
```
```   103 apply simp
```
```   104 done
```
```   105 lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
```
```   106 apply simp
```
```   107 done
```
```   108
```
```   109 fun not:: "fm \<Rightarrow> fm" where
```
```   110   "not (NOT p) = p"
```
```   111 | "not T = F"
```
```   112 | "not F = T"
```
```   113 | "not p = NOT p"
```
```   114 lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
```
```   115 by (cases p) auto
```
```   116
```
```   117 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   118   "conj p q = (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else
```
```   119    if p = q then p else And p q)"
```
```   120 lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
```
```   121 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
```
```   122
```
```   123 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   124   "disj p q = (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
```
```   125        else if p=q then p else Or p q)"
```
```   126
```
```   127 lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
```
```   128 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
```
```   129
```
```   130 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   131   "imp p q = (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p
```
```   132     else Imp p q)"
```
```   133 lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
```
```   134 by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   135
```
```   136 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   137   "iff p q = (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
```
```   138        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
```
```   139   Iff p q)"
```
```   140 lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
```
```   141   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
```
```   142
```
```   143 lemma conj_simps:
```
```   144   "conj F Q = F"
```
```   145   "conj P F = F"
```
```   146   "conj T Q = Q"
```
```   147   "conj P T = P"
```
```   148   "conj P P = P"
```
```   149   "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"
```
```   150   by (simp_all add: conj_def)
```
```   151
```
```   152 lemma disj_simps:
```
```   153   "disj T Q = T"
```
```   154   "disj P T = T"
```
```   155   "disj F Q = Q"
```
```   156   "disj P F = P"
```
```   157   "disj P P = P"
```
```   158   "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"
```
```   159   by (simp_all add: disj_def)
```
```   160 lemma imp_simps:
```
```   161   "imp F Q = T"
```
```   162   "imp P T = T"
```
```   163   "imp T Q = Q"
```
```   164   "imp P F = not P"
```
```   165   "imp P P = T"
```
```   166   "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"
```
```   167   by (simp_all add: imp_def)
```
```   168 lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
```
```   169 apply (induct p, auto)
```
```   170 done
```
```   171
```
```   172 lemma iff_simps:
```
```   173   "iff p p = T"
```
```   174   "iff p (NOT p) = F"
```
```   175   "iff (NOT p) p = F"
```
```   176   "iff p F = not p"
```
```   177   "iff F p = not p"
```
```   178   "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
```
```   179   "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
```
```   180   "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"
```
```   181   using trivNOT
```
```   182   by (simp_all add: iff_def, cases p, auto)
```
```   183   (* Quantifier freeness *)
```
```   184 fun qfree:: "fm \<Rightarrow> bool" where
```
```   185   "qfree (E p) = False"
```
```   186 | "qfree (A p) = False"
```
```   187 | "qfree (NOT p) = qfree p"
```
```   188 | "qfree (And p q) = (qfree p \<and> qfree q)"
```
```   189 | "qfree (Or  p q) = (qfree p \<and> qfree q)"
```
```   190 | "qfree (Imp p q) = (qfree p \<and> qfree q)"
```
```   191 | "qfree (Iff p q) = (qfree p \<and> qfree q)"
```
```   192 | "qfree p = True"
```
```   193
```
```   194   (* Boundedness and substitution *)
```
```   195 primrec numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *) where
```
```   196   "numbound0 (C c) = True"
```
```   197 | "numbound0 (Bound n) = (n>0)"
```
```   198 | "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
```
```   199 | "numbound0 (Neg a) = numbound0 a"
```
```   200 | "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
```
```   201 | "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
```
```   202 | "numbound0 (Mul i a) = numbound0 a"
```
```   203
```
```   204 lemma numbound0_I:
```
```   205   assumes nb: "numbound0 a"
```
```   206   shows "Inum (b#bs) a = Inum (b'#bs) a"
```
```   207 using nb
```
```   208 by (induct a) simp_all
```
```   209
```
```   210 primrec bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
```
```   211   "bound0 T = True"
```
```   212 | "bound0 F = True"
```
```   213 | "bound0 (Lt a) = numbound0 a"
```
```   214 | "bound0 (Le a) = numbound0 a"
```
```   215 | "bound0 (Gt a) = numbound0 a"
```
```   216 | "bound0 (Ge a) = numbound0 a"
```
```   217 | "bound0 (Eq a) = numbound0 a"
```
```   218 | "bound0 (NEq a) = numbound0 a"
```
```   219 | "bound0 (NOT p) = bound0 p"
```
```   220 | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
```
```   221 | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
```
```   222 | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
```
```   223 | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
```
```   224 | "bound0 (E p) = False"
```
```   225 | "bound0 (A p) = False"
```
```   226
```
```   227 lemma bound0_I:
```
```   228   assumes bp: "bound0 p"
```
```   229   shows "Ifm (b#bs) p = Ifm (b'#bs) p"
```
```   230 using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
```
```   231 by (induct p) auto
```
```   232
```
```   233 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
```
```   234 by (cases p, auto)
```
```   235 lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
```
```   236 by (cases p, auto)
```
```   237
```
```   238
```
```   239 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
```
```   240 using conj_def by auto
```
```   241 lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
```
```   242 using conj_def by auto
```
```   243
```
```   244 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
```
```   245 using disj_def by auto
```
```   246 lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
```
```   247 using disj_def by auto
```
```   248
```
```   249 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
```
```   250 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   251 lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
```
```   252 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
```
```   253
```
```   254 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
```
```   255   by (unfold iff_def,cases "p=q", auto)
```
```   256 lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
```
```   257 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   258
```
```   259 fun decrnum:: "num \<Rightarrow> num"  where
```
```   260   "decrnum (Bound n) = Bound (n - 1)"
```
```   261 | "decrnum (Neg a) = Neg (decrnum a)"
```
```   262 | "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
```
```   263 | "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
```
```   264 | "decrnum (Mul c a) = Mul c (decrnum a)"
```
```   265 | "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
```
```   266 | "decrnum a = a"
```
```   267
```
```   268 fun decr :: "fm \<Rightarrow> fm" where
```
```   269   "decr (Lt a) = Lt (decrnum a)"
```
```   270 | "decr (Le a) = Le (decrnum a)"
```
```   271 | "decr (Gt a) = Gt (decrnum a)"
```
```   272 | "decr (Ge a) = Ge (decrnum a)"
```
```   273 | "decr (Eq a) = Eq (decrnum a)"
```
```   274 | "decr (NEq a) = NEq (decrnum a)"
```
```   275 | "decr (NOT p) = NOT (decr p)"
```
```   276 | "decr (And p q) = conj (decr p) (decr q)"
```
```   277 | "decr (Or p q) = disj (decr p) (decr q)"
```
```   278 | "decr (Imp p q) = imp (decr p) (decr q)"
```
```   279 | "decr (Iff p q) = iff (decr p) (decr q)"
```
```   280 | "decr p = p"
```
```   281
```
```   282 lemma decrnum: assumes nb: "numbound0 t"
```
```   283   shows "Inum (x#bs) t = Inum bs (decrnum t)"
```
```   284   using nb by (induct t rule: decrnum.induct, simp_all)
```
```   285
```
```   286 lemma decr: assumes nb: "bound0 p"
```
```   287   shows "Ifm (x#bs) p = Ifm bs (decr p)"
```
```   288   using nb
```
```   289   by (induct p rule: decr.induct, simp_all add: decrnum)
```
```   290
```
```   291 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
```
```   292 by (induct p, simp_all)
```
```   293
```
```   294 fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
```
```   295   "isatom T = True"
```
```   296 | "isatom F = True"
```
```   297 | "isatom (Lt a) = True"
```
```   298 | "isatom (Le a) = True"
```
```   299 | "isatom (Gt a) = True"
```
```   300 | "isatom (Ge a) = True"
```
```   301 | "isatom (Eq a) = True"
```
```   302 | "isatom (NEq a) = True"
```
```   303 | "isatom p = False"
```
```   304
```
```   305 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
```
```   306 by (induct p, simp_all)
```
```   307
```
```   308 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
```
```   309   "djf f p q = (if q=T then T else if q=F then f p else
```
```   310   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
```
```   311 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
```
```   312   "evaldjf f ps = foldr (djf f) ps F"
```
```   313
```
```   314 lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
```
```   315 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
```
```   316 (cases "f p", simp_all add: Let_def djf_def)
```
```   317
```
```   318
```
```   319 lemma djf_simps:
```
```   320   "djf f p T = T"
```
```   321   "djf f p F = f p"
```
```   322   "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)"
```
```   323   by (simp_all add: djf_def)
```
```   324
```
```   325 lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
```
```   326   by(induct ps, simp_all add: evaldjf_def djf_Or)
```
```   327
```
```   328 lemma evaldjf_bound0:
```
```   329   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
```
```   330   shows "bound0 (evaldjf f xs)"
```
```   331   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   332
```
```   333 lemma evaldjf_qf:
```
```   334   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
```
```   335   shows "qfree (evaldjf f xs)"
```
```   336   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   337
```
```   338 fun disjuncts :: "fm \<Rightarrow> fm list" where
```
```   339   "disjuncts (Or p q) = disjuncts p @ disjuncts q"
```
```   340 | "disjuncts F = []"
```
```   341 | "disjuncts p = [p]"
```
```   342
```
```   343 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
```
```   344 by(induct p rule: disjuncts.induct, auto)
```
```   345
```
```   346 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
```
```   347 proof-
```
```   348   assume nb: "bound0 p"
```
```   349   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
```
```   350   thus ?thesis by (simp only: list_all_iff)
```
```   351 qed
```
```   352
```
```   353 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
```
```   354 proof-
```
```   355   assume qf: "qfree p"
```
```   356   hence "list_all qfree (disjuncts p)"
```
```   357     by (induct p rule: disjuncts.induct, auto)
```
```   358   thus ?thesis by (simp only: list_all_iff)
```
```   359 qed
```
```   360
```
```   361 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
```
```   362   "DJ f p = evaldjf f (disjuncts p)"
```
```   363
```
```   364 lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
```
```   365   and fF: "f F = F"
```
```   366   shows "Ifm bs (DJ f p) = Ifm bs (f p)"
```
```   367 proof-
```
```   368   have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
```
```   369     by (simp add: DJ_def evaldjf_ex)
```
```   370   also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
```
```   371   finally show ?thesis .
```
```   372 qed
```
```   373
```
```   374 lemma DJ_qf: assumes
```
```   375   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
```
```   376   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
```
```   377 proof(clarify)
```
```   378   fix  p assume qf: "qfree p"
```
```   379   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
```
```   380   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
```
```   381   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
```
```   382
```
```   383   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
```
```   384 qed
```
```   385
```
```   386 lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
```
```   387   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
```
```   388 proof(clarify)
```
```   389   fix p::fm and bs
```
```   390   assume qf: "qfree p"
```
```   391   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
```
```   392   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
```
```   393   have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
```
```   394     by (simp add: DJ_def evaldjf_ex)
```
```   395   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
```
```   396   also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
```
```   397   finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
```
```   398 qed
```
```   399   (* Simplification *)
```
```   400
```
```   401 fun maxcoeff:: "num \<Rightarrow> int" where
```
```   402   "maxcoeff (C i) = abs i"
```
```   403 | "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
```
```   404 | "maxcoeff t = 1"
```
```   405
```
```   406 lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
```
```   407   by (induct t rule: maxcoeff.induct, auto)
```
```   408
```
```   409 fun numgcdh:: "num \<Rightarrow> int \<Rightarrow> int" where
```
```   410   "numgcdh (C i) = (\<lambda>g. gcd i g)"
```
```   411 | "numgcdh (CN n c t) = (\<lambda>g. gcd c (numgcdh t g))"
```
```   412 | "numgcdh t = (\<lambda>g. 1)"
```
```   413
```
```   414 definition numgcd :: "num \<Rightarrow> int" where
```
```   415   "numgcd t = numgcdh t (maxcoeff t)"
```
```   416
```
```   417 fun reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num" where
```
```   418   "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
```
```   419 | "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
```
```   420 | "reducecoeffh t = (\<lambda>g. t)"
```
```   421
```
```   422 definition reducecoeff :: "num \<Rightarrow> num" where
```
```   423   "reducecoeff t =
```
```   424   (let g = numgcd t in
```
```   425   if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
```
```   426
```
```   427 fun dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
```
```   428   "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
```
```   429 | "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
```
```   430 | "dvdnumcoeff t = (\<lambda>g. False)"
```
```   431
```
```   432 lemma dvdnumcoeff_trans:
```
```   433   assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
```
```   434   shows "dvdnumcoeff t g"
```
```   435   using dgt' gdg
```
```   436   by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
```
```   437
```
```   438 declare dvd_trans [trans add]
```
```   439
```
```   440 lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
```
```   441 by arith
```
```   442
```
```   443 lemma numgcd0:
```
```   444   assumes g0: "numgcd t = 0"
```
```   445   shows "Inum bs t = 0"
```
```   446   using g0[simplified numgcd_def]
```
```   447   by (induct t rule: numgcdh.induct, auto simp add: natabs0 maxcoeff_pos min_max.sup_absorb2)
```
```   448
```
```   449 lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
```
```   450   using gp
```
```   451   by (induct t rule: numgcdh.induct, auto)
```
```   452
```
```   453 lemma numgcd_pos: "numgcd t \<ge>0"
```
```   454   by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
```
```   455
```
```   456 lemma reducecoeffh:
```
```   457   assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
```
```   458   shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
```
```   459   using gt
```
```   460 proof (induct t rule: reducecoeffh.induct)
```
```   461   case (1 i)
```
```   462   hence gd: "g dvd i" by simp
```
```   463   with assms show ?case by (simp add: real_of_int_div[OF gd])
```
```   464 next
```
```   465   case (2 n c t)
```
```   466   hence gd: "g dvd c" by simp
```
```   467   from assms 2 show ?case by (simp add: real_of_int_div[OF gd] algebra_simps)
```
```   468 qed (auto simp add: numgcd_def gp)
```
```   469
```
```   470 fun ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool" where
```
```   471   "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
```
```   472 | "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
```
```   473 | "ismaxcoeff t = (\<lambda>x. True)"
```
```   474
```
```   475 lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
```
```   476   by (induct t rule: ismaxcoeff.induct) auto
```
```   477
```
```   478 lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
```
```   479 proof (induct t rule: maxcoeff.induct)
```
```   480   case (2 n c t)
```
```   481   hence H:"ismaxcoeff t (maxcoeff t)" .
```
```   482   have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by simp
```
```   483   from ismaxcoeff_mono[OF H thh] show ?case by simp
```
```   484 qed simp_all
```
```   485
```
```   486 lemma zgcd_gt1: "gcd i j > (1::int) \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
```
```   487   apply (cases "abs i = 0", simp_all add: gcd_int_def)
```
```   488   apply (cases "abs j = 0", simp_all)
```
```   489   apply (cases "abs i = 1", simp_all)
```
```   490   apply (cases "abs j = 1", simp_all)
```
```   491   apply auto
```
```   492   done
```
```   493 lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
```
```   494   by (induct t rule: numgcdh.induct, auto)
```
```   495
```
```   496 lemma dvdnumcoeff_aux:
```
```   497   assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
```
```   498   shows "dvdnumcoeff t (numgcdh t m)"
```
```   499 using assms
```
```   500 proof(induct t rule: numgcdh.induct)
```
```   501   case (2 n c t)
```
```   502   let ?g = "numgcdh t m"
```
```   503   from 2 have th:"gcd c ?g > 1" by simp
```
```   504   from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
```
```   505   have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
```
```   506   moreover {assume "abs c > 1" and gp: "?g > 1" with 2
```
```   507     have th: "dvdnumcoeff t ?g" by simp
```
```   508     have th': "gcd c ?g dvd ?g" by simp
```
```   509     from dvdnumcoeff_trans[OF th' th] have ?case by simp }
```
```   510   moreover {assume "abs c = 0 \<and> ?g > 1"
```
```   511     with 2 have th: "dvdnumcoeff t ?g" by simp
```
```   512     have th': "gcd c ?g dvd ?g" by simp
```
```   513     from dvdnumcoeff_trans[OF th' th] have ?case by simp
```
```   514     hence ?case by simp }
```
```   515   moreover {assume "abs c > 1" and g0:"?g = 0"
```
```   516     from numgcdh0[OF g0] have "m=0". with 2 g0 have ?case by simp }
```
```   517   ultimately show ?case by blast
```
```   518 qed auto
```
```   519
```
```   520 lemma dvdnumcoeff_aux2:
```
```   521   assumes "numgcd t > 1"
```
```   522   shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
```
```   523   using assms
```
```   524 proof (simp add: numgcd_def)
```
```   525   let ?mc = "maxcoeff t"
```
```   526   let ?g = "numgcdh t ?mc"
```
```   527   have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
```
```   528   have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
```
```   529   assume H: "numgcdh t ?mc > 1"
```
```   530   from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
```
```   531 qed
```
```   532
```
```   533 lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
```
```   534 proof-
```
```   535   let ?g = "numgcd t"
```
```   536   have "?g \<ge> 0"  by (simp add: numgcd_pos)
```
```   537   hence "?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
```
```   538   moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
```
```   539   moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
```
```   540   moreover { assume g1:"?g > 1"
```
```   541     from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
```
```   542     from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
```
```   543       by (simp add: reducecoeff_def Let_def)}
```
```   544   ultimately show ?thesis by blast
```
```   545 qed
```
```   546
```
```   547 lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
```
```   548 by (induct t rule: reducecoeffh.induct, auto)
```
```   549
```
```   550 lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
```
```   551 using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
```
```   552
```
```   553 consts
```
```   554   numadd:: "num \<times> num \<Rightarrow> num"
```
```   555
```
```   556 recdef numadd "measure (\<lambda> (t,s). size t + size s)"
```
```   557   "numadd (CN n1 c1 r1,CN n2 c2 r2) =
```
```   558   (if n1=n2 then
```
```   559   (let c = c1 + c2
```
```   560   in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
```
```   561   else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2)))
```
```   562   else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
```
```   563   "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
```
```   564   "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))"
```
```   565   "numadd (C b1, C b2) = C (b1+b2)"
```
```   566   "numadd (a,b) = Add a b"
```
```   567
```
```   568 lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
```
```   569 apply (induct t s rule: numadd.induct, simp_all add: Let_def)
```
```   570 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
```
```   571 apply (case_tac "n1 = n2", simp_all add: algebra_simps)
```
```   572 by (simp only: distrib_right[symmetric],simp)
```
```   573
```
```   574 lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
```
```   575 by (induct t s rule: numadd.induct, auto simp add: Let_def)
```
```   576
```
```   577 fun nummul:: "num \<Rightarrow> int \<Rightarrow> num" where
```
```   578   "nummul (C j) = (\<lambda> i. C (i*j))"
```
```   579 | "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
```
```   580 | "nummul t = (\<lambda> i. Mul i t)"
```
```   581
```
```   582 lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
```
```   583 by (induct t rule: nummul.induct, auto simp add: algebra_simps)
```
```   584
```
```   585 lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
```
```   586 by (induct t rule: nummul.induct, auto )
```
```   587
```
```   588 definition numneg :: "num \<Rightarrow> num" where
```
```   589   "numneg t = nummul t (- 1)"
```
```   590
```
```   591 definition numsub :: "num \<Rightarrow> num \<Rightarrow> num" where
```
```   592   "numsub s t = (if s = t then C 0 else numadd (s,numneg t))"
```
```   593
```
```   594 lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
```
```   595 using numneg_def by simp
```
```   596
```
```   597 lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
```
```   598 using numneg_def by simp
```
```   599
```
```   600 lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
```
```   601 using numsub_def by simp
```
```   602
```
```   603 lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
```
```   604 using numsub_def by simp
```
```   605
```
```   606 primrec simpnum:: "num \<Rightarrow> num" where
```
```   607   "simpnum (C j) = C j"
```
```   608 | "simpnum (Bound n) = CN n 1 (C 0)"
```
```   609 | "simpnum (Neg t) = numneg (simpnum t)"
```
```   610 | "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
```
```   611 | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
```
```   612 | "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
```
```   613 | "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
```
```   614
```
```   615 lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
```
```   616 by (induct t) simp_all
```
```   617
```
```   618 lemma simpnum_numbound0[simp]:
```
```   619   "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
```
```   620 by (induct t) simp_all
```
```   621
```
```   622 fun nozerocoeff:: "num \<Rightarrow> bool" where
```
```   623   "nozerocoeff (C c) = True"
```
```   624 | "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
```
```   625 | "nozerocoeff t = True"
```
```   626
```
```   627 lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
```
```   628 by (induct a b rule: numadd.induct,auto simp add: Let_def)
```
```   629
```
```   630 lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
```
```   631 by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
```
```   632
```
```   633 lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
```
```   634 by (simp add: numneg_def nummul_nz)
```
```   635
```
```   636 lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
```
```   637 by (simp add: numsub_def numneg_nz numadd_nz)
```
```   638
```
```   639 lemma simpnum_nz: "nozerocoeff (simpnum t)"
```
```   640 by(induct t) (simp_all add: numadd_nz numneg_nz numsub_nz nummul_nz)
```
```   641
```
```   642 lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
```
```   643 proof (induct t rule: maxcoeff.induct)
```
```   644   case (2 n c t)
```
```   645   hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp_all
```
```   646   have "max (abs c) (maxcoeff t) \<ge> abs c" by simp
```
```   647   with cnz have "max (abs c) (maxcoeff t) > 0" by arith
```
```   648   with 2 show ?case by simp
```
```   649 qed auto
```
```   650
```
```   651 lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
```
```   652 proof-
```
```   653   from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
```
```   654   from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
```
```   655   from maxcoeff_nz[OF nz th] show ?thesis .
```
```   656 qed
```
```   657
```
```   658 definition simp_num_pair :: "(num \<times> int) \<Rightarrow> num \<times> int" where
```
```   659   "simp_num_pair = (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
```
```   660    (let t' = simpnum t ; g = numgcd t' in
```
```   661       if g > 1 then (let g' = gcd n g in
```
```   662         if g' = 1 then (t',n)
```
```   663         else (reducecoeffh t' g', n div g'))
```
```   664       else (t',n))))"
```
```   665
```
```   666 lemma simp_num_pair_ci:
```
```   667   shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
```
```   668   (is "?lhs = ?rhs")
```
```   669 proof-
```
```   670   let ?t' = "simpnum t"
```
```   671   let ?g = "numgcd ?t'"
```
```   672   let ?g' = "gcd n ?g"
```
```   673   {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
```
```   674   moreover
```
```   675   { assume nnz: "n \<noteq> 0"
```
```   676     {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def) }
```
```   677     moreover
```
```   678     {assume g1:"?g>1" hence g0: "?g > 0" by simp
```
```   679       from g1 nnz have gp0: "?g' \<noteq> 0" by simp
```
```   680       hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
```
```   681       hence "?g'= 1 \<or> ?g' > 1" by arith
```
```   682       moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
```
```   683       moreover {assume g'1:"?g'>1"
```
```   684         from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
```
```   685         let ?tt = "reducecoeffh ?t' ?g'"
```
```   686         let ?t = "Inum bs ?tt"
```
```   687         have gpdg: "?g' dvd ?g" by simp
```
```   688         have gpdd: "?g' dvd n" by simp
```
```   689         have gpdgp: "?g' dvd ?g'" by simp
```
```   690         from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
```
```   691         have th2:"real ?g' * ?t = Inum bs ?t'" by simp
```
```   692         from g1 g'1 have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
```
```   693         also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
```
```   694         also have "\<dots> = (Inum bs ?t' / real n)"
```
```   695           using real_of_int_div[OF gpdd] th2 gp0 by simp
```
```   696         finally have "?lhs = Inum bs t / real n" by simp
```
```   697         then have ?thesis by (simp add: simp_num_pair_def) }
```
```   698       ultimately have ?thesis by blast }
```
```   699     ultimately have ?thesis by blast }
```
```   700   ultimately show ?thesis by blast
```
```   701 qed
```
```   702
```
```   703 lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
```
```   704   shows "numbound0 t' \<and> n' >0"
```
```   705 proof-
```
```   706   let ?t' = "simpnum t"
```
```   707   let ?g = "numgcd ?t'"
```
```   708   let ?g' = "gcd n ?g"
```
```   709   { assume nz: "n = 0" hence ?thesis using assms by (simp add: Let_def simp_num_pair_def) }
```
```   710   moreover
```
```   711   { assume nnz: "n \<noteq> 0"
```
```   712     { assume "\<not> ?g > 1" hence ?thesis using assms
```
```   713         by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
```
```   714     moreover
```
```   715     { assume g1:"?g>1" hence g0: "?g > 0" by simp
```
```   716       from g1 nnz have gp0: "?g' \<noteq> 0" by simp
```
```   717       hence g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
```
```   718       hence "?g'= 1 \<or> ?g' > 1" by arith
```
```   719       moreover {
```
```   720         assume "?g' = 1" hence ?thesis using assms g1
```
```   721           by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0) }
```
```   722       moreover {
```
```   723         assume g'1: "?g' > 1"
```
```   724         have gpdg: "?g' dvd ?g" by simp
```
```   725         have gpdd: "?g' dvd n" by simp
```
```   726         have gpdgp: "?g' dvd ?g'" by simp
```
```   727         from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
```
```   728         from zdiv_mono1[OF g'n g'p, simplified div_self[OF gp0]]
```
```   729         have "n div ?g' >0" by simp
```
```   730         hence ?thesis using assms g1 g'1
```
```   731           by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0) }
```
```   732       ultimately have ?thesis by blast }
```
```   733     ultimately have ?thesis by blast }
```
```   734   ultimately show ?thesis by blast
```
```   735 qed
```
```   736
```
```   737 fun simpfm :: "fm \<Rightarrow> fm" where
```
```   738   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
```
```   739 | "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
```
```   740 | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
```
```   741 | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
```
```   742 | "simpfm (NOT p) = not (simpfm p)"
```
```   743 | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F
```
```   744   | _ \<Rightarrow> Lt a')"
```
```   745 | "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')"
```
```   746 | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
```
```   747 | "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')"
```
```   748 | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
```
```   749 | "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')"
```
```   750 | "simpfm p = p"
```
```   751 lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
```
```   752 proof(induct p rule: simpfm.induct)
```
```   753   case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   754   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   755   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   756       by (cases ?sa, simp_all add: Let_def)}
```
```   757   ultimately show ?case by blast
```
```   758 next
```
```   759   case (7 a)  let ?sa = "simpnum a"
```
```   760   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   761   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   762   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   763       by (cases ?sa, simp_all add: Let_def)}
```
```   764   ultimately show ?case by blast
```
```   765 next
```
```   766   case (8 a)  let ?sa = "simpnum a"
```
```   767   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   768   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   769   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   770       by (cases ?sa, simp_all add: Let_def)}
```
```   771   ultimately show ?case by blast
```
```   772 next
```
```   773   case (9 a)  let ?sa = "simpnum a"
```
```   774   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   775   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   776   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   777       by (cases ?sa, simp_all add: Let_def)}
```
```   778   ultimately show ?case by blast
```
```   779 next
```
```   780   case (10 a)  let ?sa = "simpnum a"
```
```   781   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   782   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   783   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   784       by (cases ?sa, simp_all add: Let_def)}
```
```   785   ultimately show ?case by blast
```
```   786 next
```
```   787   case (11 a)  let ?sa = "simpnum a"
```
```   788   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   789   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   790   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   791       by (cases ?sa, simp_all add: Let_def)}
```
```   792   ultimately show ?case by blast
```
```   793 qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
```
```   794
```
```   795
```
```   796 lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
```
```   797 proof(induct p rule: simpfm.induct)
```
```   798   case (6 a) hence nb: "numbound0 a" by simp
```
```   799   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   800   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   801 next
```
```   802   case (7 a) hence nb: "numbound0 a" by simp
```
```   803   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   804   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   805 next
```
```   806   case (8 a) hence nb: "numbound0 a" by simp
```
```   807   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   808   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   809 next
```
```   810   case (9 a) hence nb: "numbound0 a" by simp
```
```   811   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   812   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   813 next
```
```   814   case (10 a) hence nb: "numbound0 a" by simp
```
```   815   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   816   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   817 next
```
```   818   case (11 a) hence nb: "numbound0 a" by simp
```
```   819   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   820   thus ?case by (cases "simpnum a") (auto simp add: Let_def)
```
```   821 qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
```
```   822
```
```   823 lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
```
```   824   apply (induct p rule: simpfm.induct)
```
```   825   apply (auto simp add: Let_def)
```
```   826   apply (case_tac "simpnum a", auto)+
```
```   827   done
```
```   828
```
```   829 consts prep :: "fm \<Rightarrow> fm"
```
```   830 recdef prep "measure fmsize"
```
```   831   "prep (E T) = T"
```
```   832   "prep (E F) = F"
```
```   833   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
```
```   834   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
```
```   835   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
```
```   836   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
```
```   837   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
```
```   838   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
```
```   839   "prep (E p) = E (prep p)"
```
```   840   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
```
```   841   "prep (A p) = prep (NOT (E (NOT p)))"
```
```   842   "prep (NOT (NOT p)) = prep p"
```
```   843   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
```
```   844   "prep (NOT (A p)) = prep (E (NOT p))"
```
```   845   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
```
```   846   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
```
```   847   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
```
```   848   "prep (NOT p) = not (prep p)"
```
```   849   "prep (Or p q) = disj (prep p) (prep q)"
```
```   850   "prep (And p q) = conj (prep p) (prep q)"
```
```   851   "prep (Imp p q) = prep (Or (NOT p) q)"
```
```   852   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
```
```   853   "prep p = p"
```
```   854 (hints simp add: fmsize_pos)
```
```   855 lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
```
```   856   by (induct p rule: prep.induct) auto
```
```   857
```
```   858   (* Generic quantifier elimination *)
```
```   859 function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
```
```   860   "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
```
```   861 | "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
```
```   862 | "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
```
```   863 | "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))"
```
```   864 | "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))"
```
```   865 | "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
```
```   866 | "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
```
```   867 | "qelim p = (\<lambda> y. simpfm p)"
```
```   868 by pat_completeness auto
```
```   869 termination qelim by (relation "measure fmsize") simp_all
```
```   870
```
```   871 lemma qelim_ci:
```
```   872   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
```
```   873   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
```
```   874 using qe_inv DJ_qe[OF qe_inv]
```
```   875 by(induct p rule: qelim.induct)
```
```   876 (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
```
```   877   simpfm simpfm_qf simp del: simpfm.simps)
```
```   878
```
```   879 fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
```
```   880   "minusinf (And p q) = conj (minusinf p) (minusinf q)"
```
```   881 | "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
```
```   882 | "minusinf (Eq  (CN 0 c e)) = F"
```
```   883 | "minusinf (NEq (CN 0 c e)) = T"
```
```   884 | "minusinf (Lt  (CN 0 c e)) = T"
```
```   885 | "minusinf (Le  (CN 0 c e)) = T"
```
```   886 | "minusinf (Gt  (CN 0 c e)) = F"
```
```   887 | "minusinf (Ge  (CN 0 c e)) = F"
```
```   888 | "minusinf p = p"
```
```   889
```
```   890 fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
```
```   891   "plusinf (And p q) = conj (plusinf p) (plusinf q)"
```
```   892 | "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
```
```   893 | "plusinf (Eq  (CN 0 c e)) = F"
```
```   894 | "plusinf (NEq (CN 0 c e)) = T"
```
```   895 | "plusinf (Lt  (CN 0 c e)) = F"
```
```   896 | "plusinf (Le  (CN 0 c e)) = F"
```
```   897 | "plusinf (Gt  (CN 0 c e)) = T"
```
```   898 | "plusinf (Ge  (CN 0 c e)) = T"
```
```   899 | "plusinf p = p"
```
```   900
```
```   901 fun isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *) where
```
```   902   "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
```
```   903 | "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
```
```   904 | "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   905 | "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   906 | "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   907 | "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   908 | "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   909 | "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   910 | "isrlfm p = (isatom p \<and> (bound0 p))"
```
```   911
```
```   912   (* splits the bounded from the unbounded part*)
```
```   913 function (sequential) rsplit0 :: "num \<Rightarrow> int \<times> num" where
```
```   914   "rsplit0 (Bound 0) = (1,C 0)"
```
```   915 | "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b
```
```   916               in (ca+cb, Add ta tb))"
```
```   917 | "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
```
```   918 | "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
```
```   919 | "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
```
```   920 | "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
```
```   921 | "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
```
```   922 | "rsplit0 t = (0,t)"
```
```   923 by pat_completeness auto
```
```   924 termination rsplit0 by (relation "measure num_size") simp_all
```
```   925
```
```   926 lemma rsplit0:
```
```   927   shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
```
```   928 proof (induct t rule: rsplit0.induct)
```
```   929   case (2 a b)
```
```   930   let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
```
```   931   let ?ca = "fst ?sa" let ?cb = "fst ?sb"
```
```   932   let ?ta = "snd ?sa" let ?tb = "snd ?sb"
```
```   933   from 2 have nb: "numbound0 (snd(rsplit0 (Add a b)))"
```
```   934     by (cases "rsplit0 a") (auto simp add: Let_def split_def)
```
```   935   have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
```
```   936     Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
```
```   937     by (simp add: Let_def split_def algebra_simps)
```
```   938   also have "\<dots> = Inum bs a + Inum bs b" using 2 by (cases "rsplit0 a") auto
```
```   939   finally show ?case using nb by simp
```
```   940 qed (auto simp add: Let_def split_def algebra_simps, simp add: distrib_left[symmetric])
```
```   941
```
```   942     (* Linearize a formula*)
```
```   943 definition
```
```   944   lt :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   945 where
```
```   946   "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t))
```
```   947     else (Gt (CN 0 (-c) (Neg t))))"
```
```   948
```
```   949 definition
```
```   950   le :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   951 where
```
```   952   "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t))
```
```   953     else (Ge (CN 0 (-c) (Neg t))))"
```
```   954
```
```   955 definition
```
```   956   gt :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   957 where
```
```   958   "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t))
```
```   959     else (Lt (CN 0 (-c) (Neg t))))"
```
```   960
```
```   961 definition
```
```   962   ge :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   963 where
```
```   964   "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t))
```
```   965     else (Le (CN 0 (-c) (Neg t))))"
```
```   966
```
```   967 definition
```
```   968   eq :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   969 where
```
```   970   "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t))
```
```   971     else (Eq (CN 0 (-c) (Neg t))))"
```
```   972
```
```   973 definition
```
```   974   neq :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```   975 where
```
```   976   "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t))
```
```   977     else (NEq (CN 0 (-c) (Neg t))))"
```
```   978
```
```   979 lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
```
```   980 using rsplit0[where bs = "bs" and t="t"]
```
```   981 by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```   982
```
```   983 lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
```
```   984 using rsplit0[where bs = "bs" and t="t"]
```
```   985 by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```   986
```
```   987 lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
```
```   988 using rsplit0[where bs = "bs" and t="t"]
```
```   989 by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```   990
```
```   991 lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
```
```   992 using rsplit0[where bs = "bs" and t="t"]
```
```   993 by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```   994
```
```   995 lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
```
```   996 using rsplit0[where bs = "bs" and t="t"]
```
```   997 by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```   998
```
```   999 lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
```
```  1000 using rsplit0[where bs = "bs" and t="t"]
```
```  1001 by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1002
```
```  1003 lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
```
```  1004 by (auto simp add: conj_def)
```
```  1005 lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
```
```  1006 by (auto simp add: disj_def)
```
```  1007
```
```  1008 consts rlfm :: "fm \<Rightarrow> fm"
```
```  1009 recdef rlfm "measure fmsize"
```
```  1010   "rlfm (And p q) = conj (rlfm p) (rlfm q)"
```
```  1011   "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
```
```  1012   "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
```
```  1013   "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
```
```  1014   "rlfm (Lt a) = split lt (rsplit0 a)"
```
```  1015   "rlfm (Le a) = split le (rsplit0 a)"
```
```  1016   "rlfm (Gt a) = split gt (rsplit0 a)"
```
```  1017   "rlfm (Ge a) = split ge (rsplit0 a)"
```
```  1018   "rlfm (Eq a) = split eq (rsplit0 a)"
```
```  1019   "rlfm (NEq a) = split neq (rsplit0 a)"
```
```  1020   "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
```
```  1021   "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
```
```  1022   "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
```
```  1023   "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
```
```  1024   "rlfm (NOT (NOT p)) = rlfm p"
```
```  1025   "rlfm (NOT T) = F"
```
```  1026   "rlfm (NOT F) = T"
```
```  1027   "rlfm (NOT (Lt a)) = rlfm (Ge a)"
```
```  1028   "rlfm (NOT (Le a)) = rlfm (Gt a)"
```
```  1029   "rlfm (NOT (Gt a)) = rlfm (Le a)"
```
```  1030   "rlfm (NOT (Ge a)) = rlfm (Lt a)"
```
```  1031   "rlfm (NOT (Eq a)) = rlfm (NEq a)"
```
```  1032   "rlfm (NOT (NEq a)) = rlfm (Eq a)"
```
```  1033   "rlfm p = p" (hints simp add: fmsize_pos)
```
```  1034
```
```  1035 lemma rlfm_I:
```
```  1036   assumes qfp: "qfree p"
```
```  1037   shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
```
```  1038   using qfp
```
```  1039 by (induct p rule: rlfm.induct) (auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
```
```  1040
```
```  1041     (* Operations needed for Ferrante and Rackoff *)
```
```  1042 lemma rminusinf_inf:
```
```  1043   assumes lp: "isrlfm p"
```
```  1044   shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
```
```  1045 using lp
```
```  1046 proof (induct p rule: minusinf.induct)
```
```  1047   case (1 p q)
```
```  1048   thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
```
```  1049 next
```
```  1050   case (2 p q)
```
```  1051   thus ?case apply auto apply (rule_tac x= "min z za" in exI) apply auto done
```
```  1052 next
```
```  1053   case (3 c e)
```
```  1054   from 3 have nb: "numbound0 e" by simp
```
```  1055   from 3 have cp: "real c > 0" by simp
```
```  1056   fix a
```
```  1057   let ?e="Inum (a#bs) e"
```
```  1058   let ?z = "(- ?e) / real c"
```
```  1059   {fix x
```
```  1060     assume xz: "x < ?z"
```
```  1061     hence "(real c * x < - ?e)"
```
```  1062       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1063     hence "real c * x + ?e < 0" by arith
```
```  1064     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1065     with xz have "?P ?z x (Eq (CN 0 c e))"
```
```  1066       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
```
```  1067   hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
```
```  1068   thus ?case by blast
```
```  1069 next
```
```  1070   case (4 c e)
```
```  1071   from 4 have nb: "numbound0 e" by simp
```
```  1072   from 4 have cp: "real c > 0" by simp
```
```  1073   fix a
```
```  1074   let ?e="Inum (a#bs) e"
```
```  1075   let ?z = "(- ?e) / real c"
```
```  1076   {fix x
```
```  1077     assume xz: "x < ?z"
```
```  1078     hence "(real c * x < - ?e)"
```
```  1079       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1080     hence "real c * x + ?e < 0" by arith
```
```  1081     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1082     with xz have "?P ?z x (NEq (CN 0 c e))"
```
```  1083       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1084   hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
```
```  1085   thus ?case by blast
```
```  1086 next
```
```  1087   case (5 c e)
```
```  1088   from 5 have nb: "numbound0 e" by simp
```
```  1089   from 5 have cp: "real c > 0" by simp
```
```  1090   fix a
```
```  1091   let ?e="Inum (a#bs) e"
```
```  1092   let ?z = "(- ?e) / real c"
```
```  1093   {fix x
```
```  1094     assume xz: "x < ?z"
```
```  1095     hence "(real c * x < - ?e)"
```
```  1096       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1097     hence "real c * x + ?e < 0" by arith
```
```  1098     with xz have "?P ?z x (Lt (CN 0 c e))"
```
```  1099       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
```
```  1100   hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
```
```  1101   thus ?case by blast
```
```  1102 next
```
```  1103   case (6 c e)
```
```  1104   from 6 have nb: "numbound0 e" by simp
```
```  1105   from lp 6 have cp: "real c > 0" by simp
```
```  1106   fix a
```
```  1107   let ?e="Inum (a#bs) e"
```
```  1108   let ?z = "(- ?e) / real c"
```
```  1109   {fix x
```
```  1110     assume xz: "x < ?z"
```
```  1111     hence "(real c * x < - ?e)"
```
```  1112       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1113     hence "real c * x + ?e < 0" by arith
```
```  1114     with xz have "?P ?z x (Le (CN 0 c e))"
```
```  1115       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1116   hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
```
```  1117   thus ?case by blast
```
```  1118 next
```
```  1119   case (7 c e)
```
```  1120   from 7 have nb: "numbound0 e" by simp
```
```  1121   from 7 have cp: "real c > 0" by simp
```
```  1122   fix a
```
```  1123   let ?e="Inum (a#bs) e"
```
```  1124   let ?z = "(- ?e) / real c"
```
```  1125   {fix x
```
```  1126     assume xz: "x < ?z"
```
```  1127     hence "(real c * x < - ?e)"
```
```  1128       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1129     hence "real c * x + ?e < 0" by arith
```
```  1130     with xz have "?P ?z x (Gt (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 (Gt (CN 0 c e))" by simp
```
```  1133   thus ?case by blast
```
```  1134 next
```
```  1135   case (8 c e)
```
```  1136   from 8 have nb: "numbound0 e" by simp
```
```  1137   from 8 have cp: "real c > 0" by simp
```
```  1138   fix a
```
```  1139   let ?e="Inum (a#bs) e"
```
```  1140   let ?z = "(- ?e) / real c"
```
```  1141   {fix x
```
```  1142     assume xz: "x < ?z"
```
```  1143     hence "(real c * x < - ?e)"
```
```  1144       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1145     hence "real c * x + ?e < 0" by arith
```
```  1146     with xz have "?P ?z x (Ge (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 (Ge (CN 0 c e))" by simp
```
```  1149   thus ?case by blast
```
```  1150 qed simp_all
```
```  1151
```
```  1152 lemma rplusinf_inf:
```
```  1153   assumes lp: "isrlfm p"
```
```  1154   shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
```
```  1155 using lp
```
```  1156 proof (induct p rule: isrlfm.induct)
```
```  1157   case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
```
```  1158 next
```
```  1159   case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
```
```  1160 next
```
```  1161   case (3 c e)
```
```  1162   from 3 have nb: "numbound0 e" by simp
```
```  1163   from 3 have cp: "real c > 0" by simp
```
```  1164   fix a
```
```  1165   let ?e="Inum (a#bs) e"
```
```  1166   let ?z = "(- ?e) / real c"
```
```  1167   {fix x
```
```  1168     assume xz: "x > ?z"
```
```  1169     with mult_strict_right_mono [OF xz cp] cp
```
```  1170     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1171     hence "real c * x + ?e > 0" by arith
```
```  1172     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1173     with xz have "?P ?z x (Eq (CN 0 c e))"
```
```  1174       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1175   hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
```
```  1176   thus ?case by blast
```
```  1177 next
```
```  1178   case (4 c e)
```
```  1179   from 4 have nb: "numbound0 e" by simp
```
```  1180   from 4 have cp: "real c > 0" by simp
```
```  1181   fix a
```
```  1182   let ?e="Inum (a#bs) e"
```
```  1183   let ?z = "(- ?e) / real c"
```
```  1184   {fix x
```
```  1185     assume xz: "x > ?z"
```
```  1186     with mult_strict_right_mono [OF xz cp] cp
```
```  1187     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1188     hence "real c * x + ?e > 0" by arith
```
```  1189     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1190     with xz have "?P ?z x (NEq (CN 0 c e))"
```
```  1191       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1192   hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
```
```  1193   thus ?case by blast
```
```  1194 next
```
```  1195   case (5 c e)
```
```  1196   from 5 have nb: "numbound0 e" by simp
```
```  1197   from 5 have cp: "real c > 0" by simp
```
```  1198   fix a
```
```  1199   let ?e="Inum (a#bs) e"
```
```  1200   let ?z = "(- ?e) / real c"
```
```  1201   {fix x
```
```  1202     assume xz: "x > ?z"
```
```  1203     with mult_strict_right_mono [OF xz cp] cp
```
```  1204     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1205     hence "real c * x + ?e > 0" by arith
```
```  1206     with xz have "?P ?z x (Lt (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 (Lt (CN 0 c e))" by simp
```
```  1209   thus ?case by blast
```
```  1210 next
```
```  1211   case (6 c e)
```
```  1212   from 6 have nb: "numbound0 e" by simp
```
```  1213   from 6 have cp: "real c > 0" by simp
```
```  1214   fix a
```
```  1215   let ?e="Inum (a#bs) e"
```
```  1216   let ?z = "(- ?e) / real c"
```
```  1217   {fix x
```
```  1218     assume xz: "x > ?z"
```
```  1219     with mult_strict_right_mono [OF xz cp] cp
```
```  1220     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1221     hence "real c * x + ?e > 0" by arith
```
```  1222     with xz have "?P ?z x (Le (CN 0 c e))"
```
```  1223       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1224   hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
```
```  1225   thus ?case by blast
```
```  1226 next
```
```  1227   case (7 c e)
```
```  1228   from 7 have nb: "numbound0 e" by simp
```
```  1229   from 7 have cp: "real c > 0" by simp
```
```  1230   fix a
```
```  1231   let ?e="Inum (a#bs) e"
```
```  1232   let ?z = "(- ?e) / real c"
```
```  1233   {fix x
```
```  1234     assume xz: "x > ?z"
```
```  1235     with mult_strict_right_mono [OF xz cp] cp
```
```  1236     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1237     hence "real c * x + ?e > 0" by arith
```
```  1238     with xz have "?P ?z x (Gt (CN 0 c e))"
```
```  1239       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1240   hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
```
```  1241   thus ?case by blast
```
```  1242 next
```
```  1243   case (8 c e)
```
```  1244   from 8 have nb: "numbound0 e" by simp
```
```  1245   from 8 have cp: "real c > 0" by simp
```
```  1246   fix a
```
```  1247   let ?e="Inum (a#bs) e"
```
```  1248   let ?z = "(- ?e) / real c"
```
```  1249   {fix x
```
```  1250     assume xz: "x > ?z"
```
```  1251     with mult_strict_right_mono [OF xz cp] cp
```
```  1252     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1253     hence "real c * x + ?e > 0" by arith
```
```  1254     with xz have "?P ?z x (Ge (CN 0 c e))"
```
```  1255       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
```
```  1256   hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
```
```  1257   thus ?case by blast
```
```  1258 qed simp_all
```
```  1259
```
```  1260 lemma rminusinf_bound0:
```
```  1261   assumes lp: "isrlfm p"
```
```  1262   shows "bound0 (minusinf p)"
```
```  1263   using lp
```
```  1264   by (induct p rule: minusinf.induct) simp_all
```
```  1265
```
```  1266 lemma rplusinf_bound0:
```
```  1267   assumes lp: "isrlfm p"
```
```  1268   shows "bound0 (plusinf p)"
```
```  1269   using lp
```
```  1270   by (induct p rule: plusinf.induct) simp_all
```
```  1271
```
```  1272 lemma rminusinf_ex:
```
```  1273   assumes lp: "isrlfm p"
```
```  1274   and ex: "Ifm (a#bs) (minusinf p)"
```
```  1275   shows "\<exists> x. Ifm (x#bs) p"
```
```  1276 proof-
```
```  1277   from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
```
```  1278   have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
```
```  1279   from rminusinf_inf[OF lp, where bs="bs"]
```
```  1280   obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
```
```  1281   from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
```
```  1282   moreover have "z - 1 < z" by simp
```
```  1283   ultimately show ?thesis using z_def by auto
```
```  1284 qed
```
```  1285
```
```  1286 lemma rplusinf_ex:
```
```  1287   assumes lp: "isrlfm p"
```
```  1288   and ex: "Ifm (a#bs) (plusinf p)"
```
```  1289   shows "\<exists> x. Ifm (x#bs) p"
```
```  1290 proof-
```
```  1291   from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
```
```  1292   have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
```
```  1293   from rplusinf_inf[OF lp, where bs="bs"]
```
```  1294   obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
```
```  1295   from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
```
```  1296   moreover have "z + 1 > z" by simp
```
```  1297   ultimately show ?thesis using z_def by auto
```
```  1298 qed
```
```  1299
```
```  1300 consts
```
```  1301   uset:: "fm \<Rightarrow> (num \<times> int) list"
```
```  1302   usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
```
```  1303 recdef uset "measure size"
```
```  1304   "uset (And p q) = (uset p @ uset q)"
```
```  1305   "uset (Or p q) = (uset p @ uset q)"
```
```  1306   "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
```
```  1307   "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
```
```  1308   "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
```
```  1309   "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
```
```  1310   "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
```
```  1311   "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
```
```  1312   "uset p = []"
```
```  1313 recdef usubst "measure size"
```
```  1314   "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
```
```  1315   "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
```
```  1316   "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
```
```  1317   "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
```
```  1318   "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
```
```  1319   "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
```
```  1320   "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
```
```  1321   "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
```
```  1322   "usubst p = (\<lambda> (t,n). p)"
```
```  1323
```
```  1324 lemma usubst_I: assumes lp: "isrlfm p"
```
```  1325   and np: "real n > 0" and nbt: "numbound0 t"
```
```  1326   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> _")
```
```  1327   using lp
```
```  1328 proof(induct p rule: usubst.induct)
```
```  1329   case (5 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1330   have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
```
```  1331     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1332   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
```
```  1333     by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1334       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1335   also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
```
```  1336     using np by simp
```
```  1337   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1338 next
```
```  1339   case (6 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1340   have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
```
```  1341     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1342   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
```
```  1343     by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1344       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1345   also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
```
```  1346     using np by simp
```
```  1347   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1348 next
```
```  1349   case (7 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1350   have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
```
```  1351     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1352   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
```
```  1353     by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1354       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1355   also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
```
```  1356     using np by simp
```
```  1357   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1358 next
```
```  1359   case (8 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1360   have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
```
```  1361     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1362   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
```
```  1363     by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1364       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1365   also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
```
```  1366     using np by simp
```
```  1367   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1368 next
```
```  1369   case (3 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1370   from np have np: "real n \<noteq> 0" by simp
```
```  1371   have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
```
```  1372     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1373   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
```
```  1374     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1375       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1376   also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
```
```  1377     using np by simp
```
```  1378   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1379 next
```
```  1380   case (4 c e) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
```
```  1381   from np have np: "real n \<noteq> 0" by simp
```
```  1382   have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
```
```  1383     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1384   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
```
```  1385     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1386       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1387   also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
```
```  1388     using np by simp
```
```  1389   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1390 qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"])
```
```  1391
```
```  1392 lemma uset_l:
```
```  1393   assumes lp: "isrlfm p"
```
```  1394   shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
```
```  1395 using lp
```
```  1396 by(induct p rule: uset.induct,auto)
```
```  1397
```
```  1398 lemma rminusinf_uset:
```
```  1399   assumes lp: "isrlfm p"
```
```  1400   and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
```
```  1401   and ex: "Ifm (x#bs) p" (is "?I x p")
```
```  1402   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")
```
```  1403 proof-
```
```  1404   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")
```
```  1405     using lp nmi ex
```
```  1406     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
```
```  1407   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
```
```  1408   from uset_l[OF lp] smU have mp: "real m > 0" by auto
```
```  1409   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"
```
```  1410     by (auto simp add: mult_commute)
```
```  1411   thus ?thesis using smU by auto
```
```  1412 qed
```
```  1413
```
```  1414 lemma rplusinf_uset:
```
```  1415   assumes lp: "isrlfm p"
```
```  1416   and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
```
```  1417   and ex: "Ifm (x#bs) p" (is "?I x p")
```
```  1418   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")
```
```  1419 proof-
```
```  1420   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")
```
```  1421     using lp nmi ex
```
```  1422     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"])
```
```  1423   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
```
```  1424   from uset_l[OF lp] smU have mp: "real m > 0" by auto
```
```  1425   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"
```
```  1426     by (auto simp add: mult_commute)
```
```  1427   thus ?thesis using smU by auto
```
```  1428 qed
```
```  1429
```
```  1430 lemma lin_dense:
```
```  1431   assumes lp: "isrlfm p"
```
```  1432   and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)"
```
```  1433   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
```
```  1434   and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
```
```  1435   and ly: "l < y" and yu: "y < u"
```
```  1436   shows "Ifm (y#bs) p"
```
```  1437 using lp px noS
```
```  1438 proof (induct p rule: isrlfm.induct)
```
```  1439   case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1440   from 5 have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
```
```  1441   hence pxc: "x < (- ?N x e) / real c"
```
```  1442     by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1443   from 5 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1444   with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1445   hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1446   moreover {assume y: "y < (-?N x e)/ real c"
```
```  1447     hence "y * real c < - ?N x e"
```
```  1448       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1449     hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
```
```  1450     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1451   moreover {assume y: "y > (- ?N x e) / real c"
```
```  1452     with yu have eu: "u > (- ?N x e) / real c" by auto
```
```  1453     with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
```
```  1454     with lx pxc have "False" by auto
```
```  1455     hence ?case by simp }
```
```  1456   ultimately show ?case by blast
```
```  1457 next
```
```  1458   case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
```
```  1459   from 6 have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
```
```  1460   hence pxc: "x \<le> (- ?N x e) / real c"
```
```  1461     by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1462   from 6 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1463   with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1464   hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1465   moreover {assume y: "y < (-?N x e)/ real c"
```
```  1466     hence "y * real c < - ?N x e"
```
```  1467       by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1468     hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
```
```  1469     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1470   moreover {assume y: "y > (- ?N x e) / real c"
```
```  1471     with yu have eu: "u > (- ?N x e) / real c" by auto
```
```  1472     with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
```
```  1473     with lx pxc have "False" by auto
```
```  1474     hence ?case by simp }
```
```  1475   ultimately show ?case by blast
```
```  1476 next
```
```  1477   case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1478   from 7 have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
```
```  1479   hence pxc: "x > (- ?N x e) / real c"
```
```  1480     by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1481   from 7 have noSc: "\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1482   with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1483   hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1484   moreover {assume y: "y > (-?N x e)/ real c"
```
```  1485     hence "y * real c > - ?N x e"
```
```  1486       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1487     hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
```
```  1488     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1489   moreover {assume y: "y < (- ?N x e) / real c"
```
```  1490     with ly have eu: "l < (- ?N x e) / real c" by auto
```
```  1491     with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
```
```  1492     with xu pxc have "False" by auto
```
```  1493     hence ?case by simp }
```
```  1494   ultimately show ?case by blast
```
```  1495 next
```
```  1496   case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1497   from 8 have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
```
```  1498   hence pxc: "x \<ge> (- ?N x e) / real c"
```
```  1499     by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1500   from 8 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1501   with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1502   hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1503   moreover {assume y: "y > (-?N x e)/ real c"
```
```  1504     hence "y * real c > - ?N x e"
```
```  1505       by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1506     hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
```
```  1507     hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1508   moreover {assume y: "y < (- ?N x e) / real c"
```
```  1509     with ly have eu: "l < (- ?N x e) / real c" by auto
```
```  1510     with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
```
```  1511     with xu pxc have "False" by auto
```
```  1512     hence ?case by simp }
```
```  1513   ultimately show ?case by blast
```
```  1514 next
```
```  1515   case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1516   from cp have cnz: "real c \<noteq> 0" by simp
```
```  1517   from 3 have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
```
```  1518   hence pxc: "x = (- ?N x e) / real c"
```
```  1519     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
```
```  1520   from 3 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1521   with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
```
```  1522   with pxc show ?case by simp
```
```  1523 next
```
```  1524   case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1525   from cp have cnz: "real c \<noteq> 0" by simp
```
```  1526   from 4 have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1527   with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1528   hence "y* real c \<noteq> -?N x e"
```
```  1529     by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
```
```  1530   hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
```
```  1531   thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
```
```  1532     by (simp add: algebra_simps)
```
```  1533 qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])
```
```  1534
```
```  1535 lemma finite_set_intervals:
```
```  1536   assumes px: "P (x::real)"
```
```  1537   and lx: "l \<le> x" and xu: "x \<le> u"
```
```  1538   and linS: "l\<in> S" and uinS: "u \<in> S"
```
```  1539   and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
```
```  1540   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"
```
```  1541 proof-
```
```  1542   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
```
```  1543   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
```
```  1544   let ?a = "Max ?Mx"
```
```  1545   let ?b = "Min ?xM"
```
```  1546   have MxS: "?Mx \<subseteq> S" by blast
```
```  1547   hence fMx: "finite ?Mx" using fS finite_subset by auto
```
```  1548   from lx linS have linMx: "l \<in> ?Mx" by blast
```
```  1549   hence Mxne: "?Mx \<noteq> {}" by blast
```
```  1550   have xMS: "?xM \<subseteq> S" by blast
```
```  1551   hence fxM: "finite ?xM" using fS finite_subset by auto
```
```  1552   from xu uinS have linxM: "u \<in> ?xM" by blast
```
```  1553   hence xMne: "?xM \<noteq> {}" by blast
```
```  1554   have ax:"?a \<le> x" using Mxne fMx by auto
```
```  1555   have xb:"x \<le> ?b" using xMne fxM by auto
```
```  1556   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
```
```  1557   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
```
```  1558   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
```
```  1559   proof(clarsimp)
```
```  1560     fix y
```
```  1561     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
```
```  1562     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
```
```  1563     moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
```
```  1564     moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
```
```  1565     ultimately show "False" by blast
```
```  1566   qed
```
```  1567   from ainS binS noy ax xb px show ?thesis by blast
```
```  1568 qed
```
```  1569
```
```  1570 lemma rinf_uset:
```
```  1571   assumes lp: "isrlfm p"
```
```  1572   and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
```
```  1573   and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
```
```  1574   and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
```
```  1575   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"
```
```  1576 proof-
```
```  1577   let ?N = "\<lambda> x t. Inum (x#bs) t"
```
```  1578   let ?U = "set (uset p)"
```
```  1579   from ex obtain a where pa: "?I a p" by blast
```
```  1580   from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
```
```  1581   have nmi': "\<not> (?I a (?M p))" by simp
```
```  1582   from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
```
```  1583   have npi': "\<not> (?I a (?P p))" by simp
```
```  1584   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"
```
```  1585   proof-
```
```  1586     let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
```
```  1587     have fM: "finite ?M" by auto
```
```  1588     from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
```
```  1589     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
```
```  1590     then obtain "t" "n" "s" "m" where
```
```  1591       tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U"
```
```  1592       and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
```
```  1593     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
```
```  1594     from tnU have Mne: "?M \<noteq> {}" by auto
```
```  1595     hence Une: "?U \<noteq> {}" by simp
```
```  1596     let ?l = "Min ?M"
```
```  1597     let ?u = "Max ?M"
```
```  1598     have linM: "?l \<in> ?M" using fM Mne by simp
```
```  1599     have uinM: "?u \<in> ?M" using fM Mne by simp
```
```  1600     have tnM: "?N a t / real n \<in> ?M" using tnU by auto
```
```  1601     have smM: "?N a s / real m \<in> ?M" using smU by auto
```
```  1602     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
```
```  1603     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
```
```  1604     have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
```
```  1605     have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
```
```  1606     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
```
```  1607     have "(\<exists> s\<in> ?M. ?I s p) \<or>
```
```  1608       (\<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)" .
```
```  1609     moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
```
```  1610       hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
```
```  1611       then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
```
```  1612       have "(u + u) / 2 = u" by auto with pu tuu
```
```  1613       have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
```
```  1614       with tuU have ?thesis by blast}
```
```  1615     moreover{
```
```  1616       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"
```
```  1617       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
```
```  1618         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"
```
```  1619         by blast
```
```  1620       from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
```
```  1621       then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
```
```  1622       from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
```
```  1623       then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
```
```  1624       from t1x xt2 have t1t2: "t1 < t2" by simp
```
```  1625       let ?u = "(t1 + t2) / 2"
```
```  1626       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
```
```  1627       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
```
```  1628       with t1uU t2uU t1u t2u have ?thesis by blast}
```
```  1629     ultimately show ?thesis by blast
```
```  1630   qed
```
```  1631   then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U"
```
```  1632     and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
```
```  1633   from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
```
```  1634   from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
```
```  1635     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
```
```  1636   have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
```
```  1637   with lnU smU
```
```  1638   show ?thesis by auto
```
```  1639 qed
```
```  1640     (* The Ferrante - Rackoff Theorem *)
```
```  1641
```
```  1642 theorem fr_eq:
```
```  1643   assumes lp: "isrlfm p"
```
```  1644   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))"
```
```  1645   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1646 proof
```
```  1647   assume px: "\<exists> x. ?I x p"
```
```  1648   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1649   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1650   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1651     from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
```
```  1652   ultimately show "?D" by blast
```
```  1653 next
```
```  1654   assume "?D"
```
```  1655   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
```
```  1656   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
```
```  1657   moreover {assume f:"?F" hence "?E" by blast}
```
```  1658   ultimately show "?E" by blast
```
```  1659 qed
```
```  1660
```
```  1661
```
```  1662 lemma fr_equsubst:
```
```  1663   assumes lp: "isrlfm p"
```
```  1664   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))))"
```
```  1665   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1666 proof
```
```  1667   assume px: "\<exists> x. ?I x p"
```
```  1668   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1669   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1670   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1671     let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
```
```  1672     let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1673     {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
```
```  1674       with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
```
```  1675         by auto
```
```  1676       let ?st = "Add (Mul m t) (Mul n s)"
```
```  1677       from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1678         by (simp add: mult_commute)
```
```  1679       from tnb snb have st_nb: "numbound0 ?st" by simp
```
```  1680       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1681         using mnp mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1682       from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
```
```  1683       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])}
```
```  1684     with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
```
```  1685   ultimately show "?D" by blast
```
```  1686 next
```
```  1687   assume "?D"
```
```  1688   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
```
```  1689   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
```
```  1690   moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)"
```
```  1691     and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
```
```  1692     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
```
```  1693     let ?st = "Add (Mul l t) (Mul k s)"
```
```  1694     from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0"
```
```  1695       by (simp add: mult_commute)
```
```  1696     from tnb snb have st_nb: "numbound0 ?st" by simp
```
```  1697     from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
```
```  1698   ultimately show "?E" by blast
```
```  1699 qed
```
```  1700
```
```  1701
```
```  1702     (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
```
```  1703 definition ferrack :: "fm \<Rightarrow> fm" where
```
```  1704   "ferrack p = (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
```
```  1705                 in if (mp = T \<or> pp = T) then T else
```
```  1706                    (let U = remdups(map simp_num_pair
```
```  1707                      (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
```
```  1708                            (alluopairs (uset p'))))
```
```  1709                     in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
```
```  1710
```
```  1711 lemma uset_cong_aux:
```
```  1712   assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
```
```  1713   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))"
```
```  1714   (is "?lhs = ?rhs")
```
```  1715 proof(auto)
```
```  1716   fix t n s m
```
```  1717   assume "((t,n),(s,m)) \<in> set (alluopairs U)"
```
```  1718   hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
```
```  1719     using alluopairs_set1[where xs="U"] by blast
```
```  1720   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1721   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1722   from Ul th have mnz: "m \<noteq> 0" by auto
```
```  1723   from Ul th have  nnz: "n \<noteq> 0" by auto
```
```  1724   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1725    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
```
```  1726
```
```  1727   thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
```
```  1728        (2 * real n * real m)
```
```  1729        \<in> (\<lambda>((t, n), s, m).
```
```  1730              (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
```
```  1731          (set U \<times> set U)"using mnz nnz th
```
```  1732     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
```
```  1733     by (rule_tac x="(s,m)" in bexI,simp_all)
```
```  1734   (rule_tac x="(t,n)" in bexI,simp_all add: mult_commute)
```
```  1735 next
```
```  1736   fix t n s m
```
```  1737   assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U"
```
```  1738   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1739   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1740   from Ul smU have mnz: "m \<noteq> 0" by auto
```
```  1741   from Ul tnU have  nnz: "n \<noteq> 0" by auto
```
```  1742   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1743    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
```
```  1744  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"
```
```  1745  have Pc:"\<forall> a b. ?P a b = ?P b a"
```
```  1746    by auto
```
```  1747  from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
```
```  1748  from alluopairs_ex[OF Pc, where xs="U"] tnU smU
```
```  1749  have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
```
```  1750    by blast
```
```  1751  then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
```
```  1752    and Pts': "?P (t',n') (s',m')" by blast
```
```  1753  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
```
```  1754  let ?st' = "Add (Mul m' t') (Mul n' s')"
```
```  1755    have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
```
```  1756    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
```
```  1757  from Pts' have
```
```  1758    "(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
```
```  1759  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')
```
```  1760  finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
```
```  1761           \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
```
```  1762             (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
```
```  1763             set (alluopairs U)"
```
```  1764    using ts'_U by blast
```
```  1765 qed
```
```  1766
```
```  1767 lemma uset_cong:
```
```  1768   assumes lp: "isrlfm p"
```
```  1769   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)")
```
```  1770   and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
```
```  1771   and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
```
```  1772   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)))"
```
```  1773   (is "?lhs = ?rhs")
```
```  1774 proof
```
```  1775   assume ?lhs
```
```  1776   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
```
```  1777     Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
```
```  1778   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1779   from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
```
```  1780     and snb: "numbound0 s" and mp:"m > 0"  by auto
```
```  1781   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1782   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1783       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
```
```  1784     from tnb snb have stnb: "numbound0 ?st" by simp
```
```  1785   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1786    using mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1787   from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
```
```  1788   hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
```
```  1789     by auto (rule_tac x="(a,b)" in bexI, auto)
```
```  1790   then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
```
```  1791   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
```
```  1792   from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
```
```  1793   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
```
```  1794   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]]
```
```  1795   have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st)
```
```  1796   then show ?rhs using tnU' by auto
```
```  1797 next
```
```  1798   assume ?rhs
```
```  1799   then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))"
```
```  1800     by blast
```
```  1801   from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
```
```  1802   hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))"
```
```  1803     by auto (rule_tac x="(a,b)" in bexI, auto)
```
```  1804   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
```
```  1805     th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
```
```  1806     let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1807   from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
```
```  1808     and snb: "numbound0 s" and mp:"m > 0"  by auto
```
```  1809   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1810   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1811       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
```
```  1812     from tnb snb have stnb: "numbound0 ?st" by simp
```
```  1813   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1814    using mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1815   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
```
```  1816   from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
```
```  1817   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
```
```  1818   with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
```
```  1819 qed
```
```  1820
```
```  1821 lemma ferrack:
```
```  1822   assumes qf: "qfree p"
```
```  1823   shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
```
```  1824   (is "_ \<and> (?rhs = ?lhs)")
```
```  1825 proof-
```
```  1826   let ?I = "\<lambda> x p. Ifm (x#bs) p"
```
```  1827   fix x
```
```  1828   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1829   let ?q = "rlfm (simpfm p)"
```
```  1830   let ?U = "uset ?q"
```
```  1831   let ?Up = "alluopairs ?U"
```
```  1832   let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
```
```  1833   let ?S = "map ?g ?Up"
```
```  1834   let ?SS = "map simp_num_pair ?S"
```
```  1835   let ?Y = "remdups ?SS"
```
```  1836   let ?f= "(\<lambda> (t,n). ?N t / real n)"
```
```  1837   let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
```
```  1838   let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
```
```  1839   let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
```
```  1840   from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
```
```  1841   from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
```
```  1842   from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
```
```  1843   from U_l UpU
```
```  1844   have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
```
```  1845   hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
```
```  1846     by (auto simp add: mult_pos_pos)
```
```  1847   have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0"
```
```  1848   proof-
```
```  1849     { fix t n assume tnY: "(t,n) \<in> set ?Y"
```
```  1850       hence "(t,n) \<in> set ?SS" by simp
```
```  1851       hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
```
```  1852         by (auto simp add: split_def simp del: map_map)
```
```  1853            (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
```
```  1854       then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
```
```  1855       from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
```
```  1856       from simp_num_pair_l[OF tnb np tns]
```
```  1857       have "numbound0 t \<and> n > 0" . }
```
```  1858     thus ?thesis by blast
```
```  1859   qed
```
```  1860
```
```  1861   have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
```
```  1862   proof-
```
```  1863      from simp_num_pair_ci[where bs="x#bs"] have
```
```  1864     "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
```
```  1865      hence th: "?f o simp_num_pair = ?f" using ext by blast
```
```  1866     have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
```
```  1867     also have "\<dots> = (?f ` set ?S)" by (simp add: th)
```
```  1868     also have "\<dots> = ((?f o ?g) ` set ?Up)"
```
```  1869       by (simp only: set_map o_def image_compose[symmetric])
```
```  1870     also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
```
```  1871       using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
```
```  1872     finally show ?thesis .
```
```  1873   qed
```
```  1874   have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
```
```  1875   proof-
```
```  1876     { fix t n assume tnY: "(t,n) \<in> set ?Y"
```
```  1877       with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
```
```  1878       from usubst_I[OF lq np tnb]
```
```  1879     have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))"
```
```  1880       using simpfm_bound0 by simp}
```
```  1881     thus ?thesis by blast
```
```  1882   qed
```
```  1883   hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
```
```  1884   let ?mp = "minusinf ?q"
```
```  1885   let ?pp = "plusinf ?q"
```
```  1886   let ?M = "?I x ?mp"
```
```  1887   let ?P = "?I x ?pp"
```
```  1888   let ?res = "disj ?mp (disj ?pp ?ep)"
```
```  1889   from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
```
```  1890   have nbth: "bound0 ?res" by auto
```
```  1891
```
```  1892   from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm
```
```  1893
```
```  1894   have th: "?lhs = (\<exists> x. ?I x ?q)" by auto
```
```  1895   from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
```
```  1896     by (simp only: split_def fst_conv snd_conv)
```
```  1897   also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))"
```
```  1898     using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm)
```
```  1899   also have "\<dots> = (Ifm (x#bs) ?res)"
```
```  1900     using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
```
```  1901     by (simp add: split_def pair_collapse)
```
```  1902   finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
```
```  1903   hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
```
```  1904     by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
```
```  1905   from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
```
```  1906   with lr show ?thesis by blast
```
```  1907 qed
```
```  1908
```
```  1909 definition linrqe:: "fm \<Rightarrow> fm" where
```
```  1910   "linrqe p = qelim (prep p) ferrack"
```
```  1911
```
```  1912 theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
```
```  1913 using ferrack qelim_ci prep
```
```  1914 unfolding linrqe_def by auto
```
```  1915
```
```  1916 definition ferrack_test :: "unit \<Rightarrow> fm" where
```
```  1917   "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
```
```  1918     (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
```
```  1919
```
```  1920 ML_val {* @{code ferrack_test} () *}
```
```  1921
```
```  1922 oracle linr_oracle = {*
```
```  1923 let
```
```  1924
```
```  1925 val mk_C = @{code C} o @{code int_of_integer};
```
```  1926 val mk_Bound = @{code Bound} o @{code nat_of_integer};
```
```  1927
```
```  1928 fun num_of_term vs (Free vT) = mk_Bound (find_index (fn vT' => vT = vT') vs)
```
```  1929   | num_of_term vs @{term "real (0::int)"} = mk_C 0
```
```  1930   | num_of_term vs @{term "real (1::int)"} = mk_C 1
```
```  1931   | num_of_term vs @{term "0::real"} = mk_C 0
```
```  1932   | num_of_term vs @{term "1::real"} = mk_C 1
```
```  1933   | num_of_term vs (Bound i) = mk_Bound i
```
```  1934   | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} \$ t') = @{code Neg} (num_of_term vs t')
```
```  1935   | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) =
```
```  1936      @{code Add} (num_of_term vs t1, num_of_term vs t2)
```
```  1937   | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) =
```
```  1938      @{code Sub} (num_of_term vs t1, num_of_term vs t2)
```
```  1939   | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = (case num_of_term vs t1
```
```  1940      of @{code C} i => @{code Mul} (i, num_of_term vs t2)
```
```  1941       | _ => error "num_of_term: unsupported multiplication")
```
```  1942   | num_of_term vs (@{term "real :: int \<Rightarrow> real"} \$ t') =
```
```  1943      (mk_C (snd (HOLogic.dest_number t'))
```
```  1944        handle TERM _ => error ("num_of_term: unknown term"))
```
```  1945   | num_of_term vs t' =
```
```  1946      (mk_C (snd (HOLogic.dest_number t'))
```
```  1947        handle TERM _ => error ("num_of_term: unknown term"));
```
```  1948
```
```  1949 fun fm_of_term vs @{term True} = @{code T}
```
```  1950   | fm_of_term vs @{term False} = @{code F}
```
```  1951   | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) =
```
```  1952       @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  1953   | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) =
```
```  1954       @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  1955   | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) =
```
```  1956       @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  1957   | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$ t1 \$ t2) =
```
```  1958       @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
```
```  1959   | fm_of_term vs (@{term HOL.conj} \$ t1 \$ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
```
```  1960   | fm_of_term vs (@{term HOL.disj} \$ t1 \$ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
```
```  1961   | fm_of_term vs (@{term HOL.implies} \$ t1 \$ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
```
```  1962   | fm_of_term vs (@{term "Not"} \$ t') = @{code NOT} (fm_of_term vs t')
```
```  1963   | fm_of_term vs (Const (@{const_name Ex}, _) \$ Abs (xn, xT, p)) =
```
```  1964       @{code E} (fm_of_term (("", dummyT) :: vs) p)
```
```  1965   | fm_of_term vs (Const (@{const_name All}, _) \$ Abs (xn, xT, p)) =
```
```  1966       @{code A} (fm_of_term (("", dummyT) ::  vs) p)
```
```  1967   | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
```
```  1968
```
```  1969 fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} \$
```
```  1970       HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
```
```  1971   | term_of_num vs (@{code Bound} n) = Free (nth vs (@{code integer_of_nat} n))
```
```  1972   | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} \$ term_of_num vs t'
```
```  1973   | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  1974       term_of_num vs t1 \$ term_of_num vs t2
```
```  1975   | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  1976       term_of_num vs t1 \$ term_of_num vs t2
```
```  1977   | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  1978       term_of_num vs (@{code C} i) \$ term_of_num vs t2
```
```  1979   | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
```
```  1980
```
```  1981 fun term_of_fm vs @{code T} = @{term True}
```
```  1982   | term_of_fm vs @{code F} = @{term False}
```
```  1983   | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  1984       term_of_num vs t \$ @{term "0::real"}
```
```  1985   | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  1986       term_of_num vs t \$ @{term "0::real"}
```
```  1987   | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  1988       @{term "0::real"} \$ term_of_num vs t
```
```  1989   | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  1990       @{term "0::real"} \$ term_of_num vs t
```
```  1991   | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  1992       term_of_num vs t \$ @{term "0::real"}
```
```  1993   | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
```
```  1994   | term_of_fm vs (@{code NOT} t') = HOLogic.Not \$ term_of_fm vs t'
```
```  1995   | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  1996   | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  1997   | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  1998   | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$
```
```  1999       term_of_fm vs t1 \$ term_of_fm vs t2;
```
```  2000
```
```  2001 in fn (ctxt, t) =>
```
```  2002   let
```
```  2003     val vs = Term.add_frees t [];
```
```  2004     val t' = (term_of_fm vs o @{code linrqe} o fm_of_term vs) t;
```
```  2005   in (Thm.cterm_of (Proof_Context.theory_of ctxt) o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
```
```  2006 end;
```
```  2007 *}
```
```  2008
```
```  2009 ML_file "ferrack_tac.ML"
```
```  2010
```
```  2011 method_setup rferrack = {*
```
```  2012   Args.mode "no_quantify" >>
```
```  2013     (fn q => fn ctxt => SIMPLE_METHOD' (Ferrack_Tac.linr_tac ctxt (not q)))
```
```  2014 *} "decision procedure for linear real arithmetic"
```
```  2015
```
```  2016
```
```  2017 lemma
```
```  2018   fixes x :: real
```
```  2019   shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
```
```  2020   by rferrack
```
```  2021
```
```  2022 lemma
```
```  2023   fixes x :: real
```
```  2024   shows "\<exists>y \<le> x. x = y + 1"
```
```  2025   by rferrack
```
```  2026
```
```  2027 lemma
```
```  2028   fixes x :: real
```
```  2029   shows "\<not> (\<exists>z. x + z = x + z + 1)"
```
```  2030   by rferrack
```
```  2031
```
```  2032 end
```
```  2033
```