src/HOL/Decision_Procs/Ferrack.thy
 author wenzelm Mon Mar 16 18:24:30 2009 +0100 (2009-03-16) changeset 30549 d2d7874648bd parent 30439 57c68b3af2ea child 30684 c98a64746c69 permissions -rw-r--r--
simplified method setup;
```     1 (*  Title:      HOL/Decision_Procs/Ferrack.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 theory Ferrack
```
```     6 imports Complex_Main Dense_Linear_Order Efficient_Nat
```
```     7 uses ("ferrack_tac.ML")
```
```     8 begin
```
```     9
```
```    10 section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}
```
```    11
```
```    12   (*********************************************************************************)
```
```    13   (*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
```
```    14   (*********************************************************************************)
```
```    15
```
```    16 consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
```
```    17 primrec
```
```    18   "alluopairs [] = []"
```
```    19   "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
```
```    20
```
```    21 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
```
```    22 by (induct xs, auto)
```
```    23
```
```    24 lemma alluopairs_set:
```
```    25   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
```
```    26 by (induct xs, auto)
```
```    27
```
```    28 lemma alluopairs_ex:
```
```    29   assumes Pc: "\<forall> x y. P x y = P y x"
```
```    30   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
```
```    31 proof
```
```    32   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
```
```    33   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
```
```    34   from alluopairs_set[OF x y] P Pc show"\<exists>(x, y)\<in>set (alluopairs xs). P x y"
```
```    35     by auto
```
```    36 next
```
```    37   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
```
```    38   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
```
```    39   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
```
```    40   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
```
```    41 qed
```
```    42
```
```    43 lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```    44 using Nat.gr0_conv_Suc
```
```    45 by clarsimp
```
```    46
```
```    47 lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
```
```    48   apply (induct xs, auto) done
```
```    49
```
```    50 consts remdps:: "'a list \<Rightarrow> 'a list"
```
```    51
```
```    52 recdef remdps "measure size"
```
```    53   "remdps [] = []"
```
```    54   "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
```
```    55 (hints simp add: filter_length[rule_format])
```
```    56
```
```    57 lemma remdps_set[simp]: "set (remdps xs) = set xs"
```
```    58   by (induct xs rule: remdps.induct, auto)
```
```    59
```
```    60
```
```    61
```
```    62   (*********************************************************************************)
```
```    63   (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
```
```    64   (*********************************************************************************)
```
```    65
```
```    66 datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
```
```    67   | Mul int num
```
```    68
```
```    69   (* A size for num to make inductive proofs simpler*)
```
```    70 consts num_size :: "num \<Rightarrow> nat"
```
```    71 primrec
```
```    72   "num_size (C c) = 1"
```
```    73   "num_size (Bound n) = 1"
```
```    74   "num_size (Neg a) = 1 + num_size a"
```
```    75   "num_size (Add a b) = 1 + num_size a + num_size b"
```
```    76   "num_size (Sub a b) = 3 + num_size a + num_size b"
```
```    77   "num_size (Mul c a) = 1 + num_size a"
```
```    78   "num_size (CN n c a) = 3 + num_size a "
```
```    79
```
```    80   (* Semantics of numeral terms (num) *)
```
```    81 consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
```
```    82 primrec
```
```    83   "Inum bs (C c) = (real c)"
```
```    84   "Inum bs (Bound n) = bs!n"
```
```    85   "Inum bs (CN n c a) = (real c) * (bs!n) + (Inum bs a)"
```
```    86   "Inum bs (Neg a) = -(Inum bs a)"
```
```    87   "Inum bs (Add a b) = Inum bs a + Inum bs b"
```
```    88   "Inum bs (Sub a b) = Inum bs a - Inum bs b"
```
```    89   "Inum bs (Mul c a) = (real c) * Inum bs a"
```
```    90     (* FORMULAE *)
```
```    91 datatype fm  =
```
```    92   T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
```
```    93   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
```
```    94
```
```    95
```
```    96   (* A size for fm *)
```
```    97 consts fmsize :: "fm \<Rightarrow> nat"
```
```    98 recdef fmsize "measure size"
```
```    99   "fmsize (NOT p) = 1 + fmsize p"
```
```   100   "fmsize (And p q) = 1 + fmsize p + fmsize q"
```
```   101   "fmsize (Or p q) = 1 + fmsize p + fmsize q"
```
```   102   "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
```
```   103   "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
```
```   104   "fmsize (E p) = 1 + fmsize p"
```
```   105   "fmsize (A p) = 4+ fmsize p"
```
```   106   "fmsize p = 1"
```
```   107   (* several lemmas about fmsize *)
```
```   108 lemma fmsize_pos: "fmsize p > 0"
```
```   109 by (induct p rule: fmsize.induct) simp_all
```
```   110
```
```   111   (* Semantics of formulae (fm) *)
```
```   112 consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
```
```   113 primrec
```
```   114   "Ifm bs T = True"
```
```   115   "Ifm bs F = False"
```
```   116   "Ifm bs (Lt a) = (Inum bs a < 0)"
```
```   117   "Ifm bs (Gt a) = (Inum bs a > 0)"
```
```   118   "Ifm bs (Le a) = (Inum bs a \<le> 0)"
```
```   119   "Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
```
```   120   "Ifm bs (Eq a) = (Inum bs a = 0)"
```
```   121   "Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
```
```   122   "Ifm bs (NOT p) = (\<not> (Ifm bs p))"
```
```   123   "Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
```
```   124   "Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
```
```   125   "Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
```
```   126   "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
```
```   127   "Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
```
```   128   "Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"
```
```   129
```
```   130 lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
```
```   131 apply simp
```
```   132 done
```
```   133
```
```   134 lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
```
```   135 apply simp
```
```   136 done
```
```   137 lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
```
```   138 apply simp
```
```   139 done
```
```   140 lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
```
```   141 apply simp
```
```   142 done
```
```   143 lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
```
```   144 apply simp
```
```   145 done
```
```   146 lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
```
```   147 apply simp
```
```   148 done
```
```   149 lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
```
```   150 apply simp
```
```   151 done
```
```   152 lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
```
```   153 apply simp
```
```   154 done
```
```   155
```
```   156 lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
```
```   157 apply simp
```
```   158 done
```
```   159 lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
```
```   160 apply simp
```
```   161 done
```
```   162
```
```   163 consts not:: "fm \<Rightarrow> fm"
```
```   164 recdef not "measure size"
```
```   165   "not (NOT p) = p"
```
```   166   "not T = F"
```
```   167   "not F = T"
```
```   168   "not p = NOT p"
```
```   169 lemma not[simp]: "Ifm bs (not p) = Ifm bs (NOT p)"
```
```   170 by (cases p) auto
```
```   171
```
```   172 constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
```
```   173   "conj p q \<equiv> (if (p = F \<or> q=F) then F else if p=T then q else if q=T then p else
```
```   174    if p = q then p else And p q)"
```
```   175 lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
```
```   176 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
```
```   177
```
```   178 constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
```
```   179   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
```
```   180        else if p=q then p else Or p q)"
```
```   181
```
```   182 lemma disj[simp]: "Ifm bs (disj p q) = Ifm bs (Or p q)"
```
```   183 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
```
```   184
```
```   185 constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
```
```   186   "imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p
```
```   187     else Imp p q)"
```
```   188 lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
```
```   189 by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   190
```
```   191 constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
```
```   192   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
```
```   193        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
```
```   194   Iff p q)"
```
```   195 lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
```
```   196   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
```
```   197
```
```   198 lemma conj_simps:
```
```   199   "conj F Q = F"
```
```   200   "conj P F = F"
```
```   201   "conj T Q = Q"
```
```   202   "conj P T = P"
```
```   203   "conj P P = P"
```
```   204   "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"
```
```   205   by (simp_all add: conj_def)
```
```   206
```
```   207 lemma disj_simps:
```
```   208   "disj T Q = T"
```
```   209   "disj P T = T"
```
```   210   "disj F Q = Q"
```
```   211   "disj P F = P"
```
```   212   "disj P P = P"
```
```   213   "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"
```
```   214   by (simp_all add: disj_def)
```
```   215 lemma imp_simps:
```
```   216   "imp F Q = T"
```
```   217   "imp P T = T"
```
```   218   "imp T Q = Q"
```
```   219   "imp P F = not P"
```
```   220   "imp P P = T"
```
```   221   "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"
```
```   222   by (simp_all add: imp_def)
```
```   223 lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
```
```   224 apply (induct p, auto)
```
```   225 done
```
```   226
```
```   227 lemma iff_simps:
```
```   228   "iff p p = T"
```
```   229   "iff p (NOT p) = F"
```
```   230   "iff (NOT p) p = F"
```
```   231   "iff p F = not p"
```
```   232   "iff F p = not p"
```
```   233   "p \<noteq> NOT T \<Longrightarrow> iff T p = p"
```
```   234   "p\<noteq> NOT T \<Longrightarrow> iff p T = p"
```
```   235   "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"
```
```   236   using trivNOT
```
```   237   by (simp_all add: iff_def, cases p, auto)
```
```   238   (* Quantifier freeness *)
```
```   239 consts qfree:: "fm \<Rightarrow> bool"
```
```   240 recdef qfree "measure size"
```
```   241   "qfree (E p) = False"
```
```   242   "qfree (A p) = False"
```
```   243   "qfree (NOT p) = qfree p"
```
```   244   "qfree (And p q) = (qfree p \<and> qfree q)"
```
```   245   "qfree (Or  p q) = (qfree p \<and> qfree q)"
```
```   246   "qfree (Imp p q) = (qfree p \<and> qfree q)"
```
```   247   "qfree (Iff p q) = (qfree p \<and> qfree q)"
```
```   248   "qfree p = True"
```
```   249
```
```   250   (* Boundedness and substitution *)
```
```   251 consts
```
```   252   numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
```
```   253   bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
```
```   254 primrec
```
```   255   "numbound0 (C c) = True"
```
```   256   "numbound0 (Bound n) = (n>0)"
```
```   257   "numbound0 (CN n c a) = (n\<noteq>0 \<and> numbound0 a)"
```
```   258   "numbound0 (Neg a) = numbound0 a"
```
```   259   "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
```
```   260   "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)"
```
```   261   "numbound0 (Mul i a) = numbound0 a"
```
```   262 lemma numbound0_I:
```
```   263   assumes nb: "numbound0 a"
```
```   264   shows "Inum (b#bs) a = Inum (b'#bs) a"
```
```   265 using nb
```
```   266 by (induct a rule: numbound0.induct,auto simp add: nth_pos2)
```
```   267
```
```   268 primrec
```
```   269   "bound0 T = True"
```
```   270   "bound0 F = True"
```
```   271   "bound0 (Lt a) = numbound0 a"
```
```   272   "bound0 (Le a) = numbound0 a"
```
```   273   "bound0 (Gt a) = numbound0 a"
```
```   274   "bound0 (Ge a) = numbound0 a"
```
```   275   "bound0 (Eq a) = numbound0 a"
```
```   276   "bound0 (NEq a) = numbound0 a"
```
```   277   "bound0 (NOT p) = bound0 p"
```
```   278   "bound0 (And p q) = (bound0 p \<and> bound0 q)"
```
```   279   "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
```
```   280   "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
```
```   281   "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
```
```   282   "bound0 (E p) = False"
```
```   283   "bound0 (A p) = False"
```
```   284
```
```   285 lemma bound0_I:
```
```   286   assumes bp: "bound0 p"
```
```   287   shows "Ifm (b#bs) p = Ifm (b'#bs) p"
```
```   288 using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
```
```   289 by (induct p rule: bound0.induct) (auto simp add: nth_pos2)
```
```   290
```
```   291 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
```
```   292 by (cases p, auto)
```
```   293 lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
```
```   294 by (cases p, auto)
```
```   295
```
```   296
```
```   297 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
```
```   298 using conj_def by auto
```
```   299 lemma conj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
```
```   300 using conj_def by auto
```
```   301
```
```   302 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
```
```   303 using disj_def by auto
```
```   304 lemma disj_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
```
```   305 using disj_def by auto
```
```   306
```
```   307 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
```
```   308 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   309 lemma imp_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
```
```   310 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
```
```   311
```
```   312 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
```
```   313   by (unfold iff_def,cases "p=q", auto)
```
```   314 lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
```
```   315 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   316
```
```   317 consts
```
```   318   decrnum:: "num \<Rightarrow> num"
```
```   319   decr :: "fm \<Rightarrow> fm"
```
```   320
```
```   321 recdef decrnum "measure size"
```
```   322   "decrnum (Bound n) = Bound (n - 1)"
```
```   323   "decrnum (Neg a) = Neg (decrnum a)"
```
```   324   "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
```
```   325   "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
```
```   326   "decrnum (Mul c a) = Mul c (decrnum a)"
```
```   327   "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
```
```   328   "decrnum a = a"
```
```   329
```
```   330 recdef decr "measure size"
```
```   331   "decr (Lt a) = Lt (decrnum a)"
```
```   332   "decr (Le a) = Le (decrnum a)"
```
```   333   "decr (Gt a) = Gt (decrnum a)"
```
```   334   "decr (Ge a) = Ge (decrnum a)"
```
```   335   "decr (Eq a) = Eq (decrnum a)"
```
```   336   "decr (NEq a) = NEq (decrnum a)"
```
```   337   "decr (NOT p) = NOT (decr p)"
```
```   338   "decr (And p q) = conj (decr p) (decr q)"
```
```   339   "decr (Or p q) = disj (decr p) (decr q)"
```
```   340   "decr (Imp p q) = imp (decr p) (decr q)"
```
```   341   "decr (Iff p q) = iff (decr p) (decr q)"
```
```   342   "decr p = p"
```
```   343
```
```   344 lemma decrnum: assumes nb: "numbound0 t"
```
```   345   shows "Inum (x#bs) t = Inum bs (decrnum t)"
```
```   346   using nb by (induct t rule: decrnum.induct, simp_all add: nth_pos2)
```
```   347
```
```   348 lemma decr: assumes nb: "bound0 p"
```
```   349   shows "Ifm (x#bs) p = Ifm bs (decr p)"
```
```   350   using nb
```
```   351   by (induct p rule: decr.induct, simp_all add: nth_pos2 decrnum)
```
```   352
```
```   353 lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr p)"
```
```   354 by (induct p, simp_all)
```
```   355
```
```   356 consts
```
```   357   isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
```
```   358 recdef isatom "measure size"
```
```   359   "isatom T = True"
```
```   360   "isatom F = True"
```
```   361   "isatom (Lt a) = True"
```
```   362   "isatom (Le a) = True"
```
```   363   "isatom (Gt a) = True"
```
```   364   "isatom (Ge a) = True"
```
```   365   "isatom (Eq a) = True"
```
```   366   "isatom (NEq a) = True"
```
```   367   "isatom p = False"
```
```   368
```
```   369 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
```
```   370 by (induct p, simp_all)
```
```   371
```
```   372 constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
```
```   373   "djf f p q \<equiv> (if q=T then T else if q=F then f p else
```
```   374   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
```
```   375 constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
```
```   376   "evaldjf f ps \<equiv> foldr (djf f) ps F"
```
```   377
```
```   378 lemma djf_Or: "Ifm bs (djf f p q) = Ifm bs (Or (f p) q)"
```
```   379 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
```
```   380 (cases "f p", simp_all add: Let_def djf_def)
```
```   381
```
```   382
```
```   383 lemma djf_simps:
```
```   384   "djf f p T = T"
```
```   385   "djf f p F = f p"
```
```   386   "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)"
```
```   387   by (simp_all add: djf_def)
```
```   388
```
```   389 lemma evaldjf_ex: "Ifm bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bs (f p))"
```
```   390   by(induct ps, simp_all add: evaldjf_def djf_Or)
```
```   391
```
```   392 lemma evaldjf_bound0:
```
```   393   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
```
```   394   shows "bound0 (evaldjf f xs)"
```
```   395   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   396
```
```   397 lemma evaldjf_qf:
```
```   398   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
```
```   399   shows "qfree (evaldjf f xs)"
```
```   400   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   401
```
```   402 consts disjuncts :: "fm \<Rightarrow> fm list"
```
```   403 recdef disjuncts "measure size"
```
```   404   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
```
```   405   "disjuncts F = []"
```
```   406   "disjuncts p = [p]"
```
```   407
```
```   408 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm bs q) = Ifm bs p"
```
```   409 by(induct p rule: disjuncts.induct, auto)
```
```   410
```
```   411 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
```
```   412 proof-
```
```   413   assume nb: "bound0 p"
```
```   414   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
```
```   415   thus ?thesis by (simp only: list_all_iff)
```
```   416 qed
```
```   417
```
```   418 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
```
```   419 proof-
```
```   420   assume qf: "qfree p"
```
```   421   hence "list_all qfree (disjuncts p)"
```
```   422     by (induct p rule: disjuncts.induct, auto)
```
```   423   thus ?thesis by (simp only: list_all_iff)
```
```   424 qed
```
```   425
```
```   426 constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
```
```   427   "DJ f p \<equiv> evaldjf f (disjuncts p)"
```
```   428
```
```   429 lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
```
```   430   and fF: "f F = F"
```
```   431   shows "Ifm bs (DJ f p) = Ifm bs (f p)"
```
```   432 proof-
```
```   433   have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
```
```   434     by (simp add: DJ_def evaldjf_ex)
```
```   435   also have "\<dots> = Ifm bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
```
```   436   finally show ?thesis .
```
```   437 qed
```
```   438
```
```   439 lemma DJ_qf: assumes
```
```   440   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
```
```   441   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
```
```   442 proof(clarify)
```
```   443   fix  p assume qf: "qfree p"
```
```   444   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
```
```   445   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
```
```   446   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
```
```   447
```
```   448   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
```
```   449 qed
```
```   450
```
```   451 lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
```
```   452   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
```
```   453 proof(clarify)
```
```   454   fix p::fm and bs
```
```   455   assume qf: "qfree p"
```
```   456   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
```
```   457   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
```
```   458   have "Ifm bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
```
```   459     by (simp add: DJ_def evaldjf_ex)
```
```   460   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
```
```   461   also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
```
```   462   finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
```
```   463 qed
```
```   464   (* Simplification *)
```
```   465 consts
```
```   466   numgcd :: "num \<Rightarrow> int"
```
```   467   numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
```
```   468   reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
```
```   469   reducecoeff :: "num \<Rightarrow> num"
```
```   470   dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
```
```   471 consts maxcoeff:: "num \<Rightarrow> int"
```
```   472 recdef maxcoeff "measure size"
```
```   473   "maxcoeff (C i) = abs i"
```
```   474   "maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
```
```   475   "maxcoeff t = 1"
```
```   476
```
```   477 lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
```
```   478   by (induct t rule: maxcoeff.induct, auto)
```
```   479
```
```   480 recdef numgcdh "measure size"
```
```   481   "numgcdh (C i) = (\<lambda>g. zgcd i g)"
```
```   482   "numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
```
```   483   "numgcdh t = (\<lambda>g. 1)"
```
```   484 defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"
```
```   485
```
```   486 recdef reducecoeffh "measure size"
```
```   487   "reducecoeffh (C i) = (\<lambda> g. C (i div g))"
```
```   488   "reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
```
```   489   "reducecoeffh t = (\<lambda>g. t)"
```
```   490
```
```   491 defs reducecoeff_def: "reducecoeff t \<equiv>
```
```   492   (let g = numgcd t in
```
```   493   if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"
```
```   494
```
```   495 recdef dvdnumcoeff "measure size"
```
```   496   "dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
```
```   497   "dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
```
```   498   "dvdnumcoeff t = (\<lambda>g. False)"
```
```   499
```
```   500 lemma dvdnumcoeff_trans:
```
```   501   assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
```
```   502   shows "dvdnumcoeff t g"
```
```   503   using dgt' gdg
```
```   504   by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])
```
```   505
```
```   506 declare dvd_trans [trans add]
```
```   507
```
```   508 lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
```
```   509 by arith
```
```   510
```
```   511 lemma numgcd0:
```
```   512   assumes g0: "numgcd t = 0"
```
```   513   shows "Inum bs t = 0"
```
```   514   using g0[simplified numgcd_def]
```
```   515   by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)
```
```   516
```
```   517 lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
```
```   518   using gp
```
```   519   by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)
```
```   520
```
```   521 lemma numgcd_pos: "numgcd t \<ge>0"
```
```   522   by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)
```
```   523
```
```   524 lemma reducecoeffh:
```
```   525   assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
```
```   526   shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
```
```   527   using gt
```
```   528 proof(induct t rule: reducecoeffh.induct)
```
```   529   case (1 i) hence gd: "g dvd i" by simp
```
```   530   from gp have gnz: "g \<noteq> 0" by simp
```
```   531   from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
```
```   532 next
```
```   533   case (2 n c t)  hence gd: "g dvd c" by simp
```
```   534   from gp have gnz: "g \<noteq> 0" by simp
```
```   535   from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
```
```   536 qed (auto simp add: numgcd_def gp)
```
```   537 consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
```
```   538 recdef ismaxcoeff "measure size"
```
```   539   "ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
```
```   540   "ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
```
```   541   "ismaxcoeff t = (\<lambda>x. True)"
```
```   542
```
```   543 lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
```
```   544 by (induct t rule: ismaxcoeff.induct, auto)
```
```   545
```
```   546 lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
```
```   547 proof (induct t rule: maxcoeff.induct)
```
```   548   case (2 n c t)
```
```   549   hence H:"ismaxcoeff t (maxcoeff t)" .
```
```   550   have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
```
```   551   from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
```
```   552 qed simp_all
```
```   553
```
```   554 lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
```
```   555   apply (cases "abs i = 0", simp_all add: zgcd_def)
```
```   556   apply (cases "abs j = 0", simp_all)
```
```   557   apply (cases "abs i = 1", simp_all)
```
```   558   apply (cases "abs j = 1", simp_all)
```
```   559   apply auto
```
```   560   done
```
```   561 lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
```
```   562   by (induct t rule: numgcdh.induct, auto simp add:zgcd0)
```
```   563
```
```   564 lemma dvdnumcoeff_aux:
```
```   565   assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
```
```   566   shows "dvdnumcoeff t (numgcdh t m)"
```
```   567 using prems
```
```   568 proof(induct t rule: numgcdh.induct)
```
```   569   case (2 n c t)
```
```   570   let ?g = "numgcdh t m"
```
```   571   from prems have th:"zgcd c ?g > 1" by simp
```
```   572   from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
```
```   573   have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
```
```   574   moreover {assume "abs c > 1" and gp: "?g > 1" with prems
```
```   575     have th: "dvdnumcoeff t ?g" by simp
```
```   576     have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
```
```   577     from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
```
```   578   moreover {assume "abs c = 0 \<and> ?g > 1"
```
```   579     with prems have th: "dvdnumcoeff t ?g" by simp
```
```   580     have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
```
```   581     from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
```
```   582     hence ?case by simp }
```
```   583   moreover {assume "abs c > 1" and g0:"?g = 0"
```
```   584     from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
```
```   585   ultimately show ?case by blast
```
```   586 qed(auto simp add: zgcd_zdvd1)
```
```   587
```
```   588 lemma dvdnumcoeff_aux2:
```
```   589   assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
```
```   590   using prems
```
```   591 proof (simp add: numgcd_def)
```
```   592   let ?mc = "maxcoeff t"
```
```   593   let ?g = "numgcdh t ?mc"
```
```   594   have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
```
```   595   have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
```
```   596   assume H: "numgcdh t ?mc > 1"
```
```   597   from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
```
```   598 qed
```
```   599
```
```   600 lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
```
```   601 proof-
```
```   602   let ?g = "numgcd t"
```
```   603   have "?g \<ge> 0"  by (simp add: numgcd_pos)
```
```   604   hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
```
```   605   moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
```
```   606   moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
```
```   607   moreover { assume g1:"?g > 1"
```
```   608     from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
```
```   609     from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
```
```   610       by (simp add: reducecoeff_def Let_def)}
```
```   611   ultimately show ?thesis by blast
```
```   612 qed
```
```   613
```
```   614 lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
```
```   615 by (induct t rule: reducecoeffh.induct, auto)
```
```   616
```
```   617 lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
```
```   618 using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)
```
```   619
```
```   620 consts
```
```   621   simpnum:: "num \<Rightarrow> num"
```
```   622   numadd:: "num \<times> num \<Rightarrow> num"
```
```   623   nummul:: "num \<Rightarrow> int \<Rightarrow> num"
```
```   624 recdef numadd "measure (\<lambda> (t,s). size t + size s)"
```
```   625   "numadd (CN n1 c1 r1,CN n2 c2 r2) =
```
```   626   (if n1=n2 then
```
```   627   (let c = c1 + c2
```
```   628   in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
```
```   629   else if n1 \<le> n2 then (CN n1 c1 (numadd (r1,CN n2 c2 r2)))
```
```   630   else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
```
```   631   "numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
```
```   632   "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))"
```
```   633   "numadd (C b1, C b2) = C (b1+b2)"
```
```   634   "numadd (a,b) = Add a b"
```
```   635
```
```   636 lemma numadd[simp]: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
```
```   637 apply (induct t s rule: numadd.induct, simp_all add: Let_def)
```
```   638 apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
```
```   639 apply (case_tac "n1 = n2", simp_all add: algebra_simps)
```
```   640 by (simp only: left_distrib[symmetric],simp)
```
```   641
```
```   642 lemma numadd_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
```
```   643 by (induct t s rule: numadd.induct, auto simp add: Let_def)
```
```   644
```
```   645 recdef nummul "measure size"
```
```   646   "nummul (C j) = (\<lambda> i. C (i*j))"
```
```   647   "nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
```
```   648   "nummul t = (\<lambda> i. Mul i t)"
```
```   649
```
```   650 lemma nummul[simp]: "\<And> i. Inum bs (nummul t i) = Inum bs (Mul i t)"
```
```   651 by (induct t rule: nummul.induct, auto simp add: algebra_simps)
```
```   652
```
```   653 lemma nummul_nb[simp]: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul t i)"
```
```   654 by (induct t rule: nummul.induct, auto )
```
```   655
```
```   656 constdefs numneg :: "num \<Rightarrow> num"
```
```   657   "numneg t \<equiv> nummul t (- 1)"
```
```   658
```
```   659 constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
```
```   660   "numsub s t \<equiv> (if s = t then C 0 else numadd (s,numneg t))"
```
```   661
```
```   662 lemma numneg[simp]: "Inum bs (numneg t) = Inum bs (Neg t)"
```
```   663 using numneg_def by simp
```
```   664
```
```   665 lemma numneg_nb[simp]: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
```
```   666 using numneg_def by simp
```
```   667
```
```   668 lemma numsub[simp]: "Inum bs (numsub a b) = Inum bs (Sub a b)"
```
```   669 using numsub_def by simp
```
```   670
```
```   671 lemma numsub_nb[simp]: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
```
```   672 using numsub_def by simp
```
```   673
```
```   674 recdef simpnum "measure size"
```
```   675   "simpnum (C j) = C j"
```
```   676   "simpnum (Bound n) = CN n 1 (C 0)"
```
```   677   "simpnum (Neg t) = numneg (simpnum t)"
```
```   678   "simpnum (Add t s) = numadd (simpnum t,simpnum s)"
```
```   679   "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
```
```   680   "simpnum (Mul i t) = (if i = 0 then (C 0) else nummul (simpnum t) i)"
```
```   681   "simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"
```
```   682
```
```   683 lemma simpnum_ci[simp]: "Inum bs (simpnum t) = Inum bs t"
```
```   684 by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)
```
```   685
```
```   686 lemma simpnum_numbound0[simp]:
```
```   687   "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
```
```   688 by (induct t rule: simpnum.induct, auto)
```
```   689
```
```   690 consts nozerocoeff:: "num \<Rightarrow> bool"
```
```   691 recdef nozerocoeff "measure size"
```
```   692   "nozerocoeff (C c) = True"
```
```   693   "nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
```
```   694   "nozerocoeff t = True"
```
```   695
```
```   696 lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"
```
```   697 by (induct a b rule: numadd.induct,auto simp add: Let_def)
```
```   698
```
```   699 lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"
```
```   700 by (induct a rule: nummul.induct,auto simp add: Let_def numadd_nz)
```
```   701
```
```   702 lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"
```
```   703 by (simp add: numneg_def nummul_nz)
```
```   704
```
```   705 lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"
```
```   706 by (simp add: numsub_def numneg_nz numadd_nz)
```
```   707
```
```   708 lemma simpnum_nz: "nozerocoeff (simpnum t)"
```
```   709 by(induct t rule: simpnum.induct, auto simp add: numadd_nz numneg_nz numsub_nz nummul_nz)
```
```   710
```
```   711 lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
```
```   712 proof (induct t rule: maxcoeff.induct)
```
```   713   case (2 n c t)
```
```   714   hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
```
```   715   have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
```
```   716   with cnz have "max (abs c) (maxcoeff t) > 0" by arith
```
```   717   with prems show ?case by simp
```
```   718 qed auto
```
```   719
```
```   720 lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
```
```   721 proof-
```
```   722   from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
```
```   723   from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
```
```   724   from maxcoeff_nz[OF nz th] show ?thesis .
```
```   725 qed
```
```   726
```
```   727 constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
```
```   728   "simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
```
```   729    (let t' = simpnum t ; g = numgcd t' in
```
```   730       if g > 1 then (let g' = zgcd n g in
```
```   731         if g' = 1 then (t',n)
```
```   732         else (reducecoeffh t' g', n div g'))
```
```   733       else (t',n))))"
```
```   734
```
```   735 lemma simp_num_pair_ci:
```
```   736   shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
```
```   737   (is "?lhs = ?rhs")
```
```   738 proof-
```
```   739   let ?t' = "simpnum t"
```
```   740   let ?g = "numgcd ?t'"
```
```   741   let ?g' = "zgcd n ?g"
```
```   742   {assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
```
```   743   moreover
```
```   744   { assume nnz: "n \<noteq> 0"
```
```   745     {assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
```
```   746     moreover
```
```   747     {assume g1:"?g>1" hence g0: "?g > 0" by simp
```
```   748       from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
```
```   749       hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
```
```   750       hence "?g'= 1 \<or> ?g' > 1" by arith
```
```   751       moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
```
```   752       moreover {assume g'1:"?g'>1"
```
```   753 	from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
```
```   754 	let ?tt = "reducecoeffh ?t' ?g'"
```
```   755 	let ?t = "Inum bs ?tt"
```
```   756 	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
```
```   757 	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
```
```   758 	have gpdgp: "?g' dvd ?g'" by simp
```
```   759 	from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
```
```   760 	have th2:"real ?g' * ?t = Inum bs ?t'" by simp
```
```   761 	from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
```
```   762 	also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
```
```   763 	also have "\<dots> = (Inum bs ?t' / real n)"
```
```   764 	  using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
```
```   765 	finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
```
```   766 	then have ?thesis using prems by (simp add: simp_num_pair_def)}
```
```   767       ultimately have ?thesis by blast}
```
```   768     ultimately have ?thesis by blast}
```
```   769   ultimately show ?thesis by blast
```
```   770 qed
```
```   771
```
```   772 lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
```
```   773   shows "numbound0 t' \<and> n' >0"
```
```   774 proof-
```
```   775     let ?t' = "simpnum t"
```
```   776   let ?g = "numgcd ?t'"
```
```   777   let ?g' = "zgcd n ?g"
```
```   778   {assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
```
```   779   moreover
```
```   780   { assume nnz: "n \<noteq> 0"
```
```   781     {assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
```
```   782     moreover
```
```   783     {assume g1:"?g>1" hence g0: "?g > 0" by simp
```
```   784       from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
```
```   785       hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
```
```   786       hence "?g'= 1 \<or> ?g' > 1" by arith
```
```   787       moreover {assume "?g'=1" hence ?thesis using prems
```
```   788 	  by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
```
```   789       moreover {assume g'1:"?g'>1"
```
```   790 	have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
```
```   791 	have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
```
```   792 	have gpdgp: "?g' dvd ?g'" by simp
```
```   793 	from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
```
```   794 	from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
```
```   795 	have "n div ?g' >0" by simp
```
```   796 	hence ?thesis using prems
```
```   797 	  by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
```
```   798       ultimately have ?thesis by blast}
```
```   799     ultimately have ?thesis by blast}
```
```   800   ultimately show ?thesis by blast
```
```   801 qed
```
```   802
```
```   803 consts simpfm :: "fm \<Rightarrow> fm"
```
```   804 recdef simpfm "measure fmsize"
```
```   805   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
```
```   806   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
```
```   807   "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
```
```   808   "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
```
```   809   "simpfm (NOT p) = not (simpfm p)"
```
```   810   "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F
```
```   811   | _ \<Rightarrow> Lt a')"
```
```   812   "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')"
```
```   813   "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
```
```   814   "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')"
```
```   815   "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
```
```   816   "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')"
```
```   817   "simpfm p = p"
```
```   818 lemma simpfm: "Ifm bs (simpfm p) = Ifm bs p"
```
```   819 proof(induct p rule: simpfm.induct)
```
```   820   case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   821   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   822   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   823       by (cases ?sa, simp_all add: Let_def)}
```
```   824   ultimately show ?case by blast
```
```   825 next
```
```   826   case (7 a)  let ?sa = "simpnum a"
```
```   827   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   828   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   829   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   830       by (cases ?sa, simp_all add: Let_def)}
```
```   831   ultimately show ?case by blast
```
```   832 next
```
```   833   case (8 a)  let ?sa = "simpnum a"
```
```   834   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   835   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   836   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   837       by (cases ?sa, simp_all add: Let_def)}
```
```   838   ultimately show ?case by blast
```
```   839 next
```
```   840   case (9 a)  let ?sa = "simpnum a"
```
```   841   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   842   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   843   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   844       by (cases ?sa, simp_all add: Let_def)}
```
```   845   ultimately show ?case by blast
```
```   846 next
```
```   847   case (10 a)  let ?sa = "simpnum a"
```
```   848   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   849   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   850   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   851       by (cases ?sa, simp_all add: Let_def)}
```
```   852   ultimately show ?case by blast
```
```   853 next
```
```   854   case (11 a)  let ?sa = "simpnum a"
```
```   855   from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
```
```   856   {fix v assume "?sa = C v" hence ?case using sa by simp }
```
```   857   moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa
```
```   858       by (cases ?sa, simp_all add: Let_def)}
```
```   859   ultimately show ?case by blast
```
```   860 qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)
```
```   861
```
```   862
```
```   863 lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
```
```   864 proof(induct p rule: simpfm.induct)
```
```   865   case (6 a) hence nb: "numbound0 a" by simp
```
```   866   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   867   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   868 next
```
```   869   case (7 a) hence nb: "numbound0 a" by simp
```
```   870   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   871   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   872 next
```
```   873   case (8 a) hence nb: "numbound0 a" by simp
```
```   874   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   875   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   876 next
```
```   877   case (9 a) hence nb: "numbound0 a" by simp
```
```   878   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   879   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   880 next
```
```   881   case (10 a) hence nb: "numbound0 a" by simp
```
```   882   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   883   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   884 next
```
```   885   case (11 a) hence nb: "numbound0 a" by simp
```
```   886   hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
```
```   887   thus ?case by (cases "simpnum a", auto simp add: Let_def)
```
```   888 qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)
```
```   889
```
```   890 lemma simpfm_qf: "qfree p \<Longrightarrow> qfree (simpfm p)"
```
```   891 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
```
```   892  (case_tac "simpnum a",auto)+
```
```   893
```
```   894 consts prep :: "fm \<Rightarrow> fm"
```
```   895 recdef prep "measure fmsize"
```
```   896   "prep (E T) = T"
```
```   897   "prep (E F) = F"
```
```   898   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
```
```   899   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
```
```   900   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
```
```   901   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
```
```   902   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
```
```   903   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
```
```   904   "prep (E p) = E (prep p)"
```
```   905   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
```
```   906   "prep (A p) = prep (NOT (E (NOT p)))"
```
```   907   "prep (NOT (NOT p)) = prep p"
```
```   908   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
```
```   909   "prep (NOT (A p)) = prep (E (NOT p))"
```
```   910   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
```
```   911   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
```
```   912   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
```
```   913   "prep (NOT p) = not (prep p)"
```
```   914   "prep (Or p q) = disj (prep p) (prep q)"
```
```   915   "prep (And p q) = conj (prep p) (prep q)"
```
```   916   "prep (Imp p q) = prep (Or (NOT p) q)"
```
```   917   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
```
```   918   "prep p = p"
```
```   919 (hints simp add: fmsize_pos)
```
```   920 lemma prep: "\<And> bs. Ifm bs (prep p) = Ifm bs p"
```
```   921 by (induct p rule: prep.induct, auto)
```
```   922
```
```   923   (* Generic quantifier elimination *)
```
```   924 consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
```
```   925 recdef qelim "measure fmsize"
```
```   926   "qelim (E p) = (\<lambda> qe. DJ qe (qelim p qe))"
```
```   927   "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
```
```   928   "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
```
```   929   "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))"
```
```   930   "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))"
```
```   931   "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
```
```   932   "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
```
```   933   "qelim p = (\<lambda> y. simpfm p)"
```
```   934
```
```   935 lemma qelim_ci:
```
```   936   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
```
```   937   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm bs p)"
```
```   938 using qe_inv DJ_qe[OF qe_inv]
```
```   939 by(induct p rule: qelim.induct)
```
```   940 (auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf
```
```   941   simpfm simpfm_qf simp del: simpfm.simps)
```
```   942
```
```   943 consts
```
```   944   plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
```
```   945   minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
```
```   946 recdef minusinf "measure size"
```
```   947   "minusinf (And p q) = conj (minusinf p) (minusinf q)"
```
```   948   "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
```
```   949   "minusinf (Eq  (CN 0 c e)) = F"
```
```   950   "minusinf (NEq (CN 0 c e)) = T"
```
```   951   "minusinf (Lt  (CN 0 c e)) = T"
```
```   952   "minusinf (Le  (CN 0 c e)) = T"
```
```   953   "minusinf (Gt  (CN 0 c e)) = F"
```
```   954   "minusinf (Ge  (CN 0 c e)) = F"
```
```   955   "minusinf p = p"
```
```   956
```
```   957 recdef plusinf "measure size"
```
```   958   "plusinf (And p q) = conj (plusinf p) (plusinf q)"
```
```   959   "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
```
```   960   "plusinf (Eq  (CN 0 c e)) = F"
```
```   961   "plusinf (NEq (CN 0 c e)) = T"
```
```   962   "plusinf (Lt  (CN 0 c e)) = F"
```
```   963   "plusinf (Le  (CN 0 c e)) = F"
```
```   964   "plusinf (Gt  (CN 0 c e)) = T"
```
```   965   "plusinf (Ge  (CN 0 c e)) = T"
```
```   966   "plusinf p = p"
```
```   967
```
```   968 consts
```
```   969   isrlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
```
```   970 recdef isrlfm "measure size"
```
```   971   "isrlfm (And p q) = (isrlfm p \<and> isrlfm q)"
```
```   972   "isrlfm (Or p q) = (isrlfm p \<and> isrlfm q)"
```
```   973   "isrlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   974   "isrlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   975   "isrlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   976   "isrlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   977   "isrlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   978   "isrlfm (Ge  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
```
```   979   "isrlfm p = (isatom p \<and> (bound0 p))"
```
```   980
```
```   981   (* splits the bounded from the unbounded part*)
```
```   982 consts rsplit0 :: "num \<Rightarrow> int \<times> num"
```
```   983 recdef rsplit0 "measure num_size"
```
```   984   "rsplit0 (Bound 0) = (1,C 0)"
```
```   985   "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b
```
```   986               in (ca+cb, Add ta tb))"
```
```   987   "rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
```
```   988   "rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
```
```   989   "rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
```
```   990   "rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
```
```   991   "rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
```
```   992   "rsplit0 t = (0,t)"
```
```   993 lemma rsplit0:
```
```   994   shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
```
```   995 proof (induct t rule: rsplit0.induct)
```
```   996   case (2 a b)
```
```   997   let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
```
```   998   let ?ca = "fst ?sa" let ?cb = "fst ?sb"
```
```   999   let ?ta = "snd ?sa" let ?tb = "snd ?sb"
```
```  1000   from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))"
```
```  1001     by(cases "rsplit0 a",auto simp add: Let_def split_def)
```
```  1002   have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
```
```  1003     Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
```
```  1004     by (simp add: Let_def split_def algebra_simps)
```
```  1005   also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
```
```  1006   finally show ?case using nb by simp
```
```  1007 qed(auto simp add: Let_def split_def algebra_simps , simp add: right_distrib[symmetric])
```
```  1008
```
```  1009     (* Linearize a formula*)
```
```  1010 definition
```
```  1011   lt :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1012 where
```
```  1013   "lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t))
```
```  1014     else (Gt (CN 0 (-c) (Neg t))))"
```
```  1015
```
```  1016 definition
```
```  1017   le :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1018 where
```
```  1019   "le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t))
```
```  1020     else (Ge (CN 0 (-c) (Neg t))))"
```
```  1021
```
```  1022 definition
```
```  1023   gt :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1024 where
```
```  1025   "gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t))
```
```  1026     else (Lt (CN 0 (-c) (Neg t))))"
```
```  1027
```
```  1028 definition
```
```  1029   ge :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1030 where
```
```  1031   "ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t))
```
```  1032     else (Le (CN 0 (-c) (Neg t))))"
```
```  1033
```
```  1034 definition
```
```  1035   eq :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1036 where
```
```  1037   "eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t))
```
```  1038     else (Eq (CN 0 (-c) (Neg t))))"
```
```  1039
```
```  1040 definition
```
```  1041   neq :: "int \<Rightarrow> num \<Rightarrow> fm"
```
```  1042 where
```
```  1043   "neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t))
```
```  1044     else (NEq (CN 0 (-c) (Neg t))))"
```
```  1045
```
```  1046 lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
```
```  1047 using rsplit0[where bs = "bs" and t="t"]
```
```  1048 by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1049
```
```  1050 lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
```
```  1051 using rsplit0[where bs = "bs" and t="t"]
```
```  1052 by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1053
```
```  1054 lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
```
```  1055 using rsplit0[where bs = "bs" and t="t"]
```
```  1056 by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1057
```
```  1058 lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
```
```  1059 using rsplit0[where bs = "bs" and t="t"]
```
```  1060 by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1061
```
```  1062 lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
```
```  1063 using rsplit0[where bs = "bs" and t="t"]
```
```  1064 by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1065
```
```  1066 lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
```
```  1067 using rsplit0[where bs = "bs" and t="t"]
```
```  1068 by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)
```
```  1069
```
```  1070 lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
```
```  1071 by (auto simp add: conj_def)
```
```  1072 lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"
```
```  1073 by (auto simp add: disj_def)
```
```  1074
```
```  1075 consts rlfm :: "fm \<Rightarrow> fm"
```
```  1076 recdef rlfm "measure fmsize"
```
```  1077   "rlfm (And p q) = conj (rlfm p) (rlfm q)"
```
```  1078   "rlfm (Or p q) = disj (rlfm p) (rlfm q)"
```
```  1079   "rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
```
```  1080   "rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
```
```  1081   "rlfm (Lt a) = split lt (rsplit0 a)"
```
```  1082   "rlfm (Le a) = split le (rsplit0 a)"
```
```  1083   "rlfm (Gt a) = split gt (rsplit0 a)"
```
```  1084   "rlfm (Ge a) = split ge (rsplit0 a)"
```
```  1085   "rlfm (Eq a) = split eq (rsplit0 a)"
```
```  1086   "rlfm (NEq a) = split neq (rsplit0 a)"
```
```  1087   "rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
```
```  1088   "rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
```
```  1089   "rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
```
```  1090   "rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
```
```  1091   "rlfm (NOT (NOT p)) = rlfm p"
```
```  1092   "rlfm (NOT T) = F"
```
```  1093   "rlfm (NOT F) = T"
```
```  1094   "rlfm (NOT (Lt a)) = rlfm (Ge a)"
```
```  1095   "rlfm (NOT (Le a)) = rlfm (Gt a)"
```
```  1096   "rlfm (NOT (Gt a)) = rlfm (Le a)"
```
```  1097   "rlfm (NOT (Ge a)) = rlfm (Lt a)"
```
```  1098   "rlfm (NOT (Eq a)) = rlfm (NEq a)"
```
```  1099   "rlfm (NOT (NEq a)) = rlfm (Eq a)"
```
```  1100   "rlfm p = p" (hints simp add: fmsize_pos)
```
```  1101
```
```  1102 lemma rlfm_I:
```
```  1103   assumes qfp: "qfree p"
```
```  1104   shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
```
```  1105   using qfp
```
```  1106 by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)
```
```  1107
```
```  1108     (* Operations needed for Ferrante and Rackoff *)
```
```  1109 lemma rminusinf_inf:
```
```  1110   assumes lp: "isrlfm p"
```
```  1111   shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
```
```  1112 using lp
```
```  1113 proof (induct p rule: minusinf.induct)
```
```  1114   case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
```
```  1115 next
```
```  1116   case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
```
```  1117 next
```
```  1118   case (3 c e)
```
```  1119   from prems have nb: "numbound0 e" by simp
```
```  1120   from prems have cp: "real c > 0" by simp
```
```  1121   fix a
```
```  1122   let ?e="Inum (a#bs) e"
```
```  1123   let ?z = "(- ?e) / real c"
```
```  1124   {fix x
```
```  1125     assume xz: "x < ?z"
```
```  1126     hence "(real c * x < - ?e)"
```
```  1127       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1128     hence "real c * x + ?e < 0" by arith
```
```  1129     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1130     with xz have "?P ?z x (Eq (CN 0 c e))"
```
```  1131       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
```
```  1132   hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
```
```  1133   thus ?case by blast
```
```  1134 next
```
```  1135   case (4 c e)
```
```  1136   from prems have nb: "numbound0 e" by simp
```
```  1137   from prems have cp: "real c > 0" by simp
```
```  1138   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     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1147     with xz have "?P ?z x (NEq (CN 0 c e))"
```
```  1148       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1149   hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
```
```  1150   thus ?case by blast
```
```  1151 next
```
```  1152   case (5 c e)
```
```  1153     from prems have nb: "numbound0 e" by simp
```
```  1154   from prems have cp: "real c > 0" by simp
```
```  1155   fix a
```
```  1156   let ?e="Inum (a#bs) e"
```
```  1157   let ?z = "(- ?e) / real c"
```
```  1158   {fix x
```
```  1159     assume xz: "x < ?z"
```
```  1160     hence "(real c * x < - ?e)"
```
```  1161       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1162     hence "real c * x + ?e < 0" by arith
```
```  1163     with xz have "?P ?z x (Lt (CN 0 c e))"
```
```  1164       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
```
```  1165   hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
```
```  1166   thus ?case by blast
```
```  1167 next
```
```  1168   case (6 c e)
```
```  1169     from prems have nb: "numbound0 e" by simp
```
```  1170   from prems have cp: "real c > 0" by simp
```
```  1171   fix a
```
```  1172   let ?e="Inum (a#bs) e"
```
```  1173   let ?z = "(- ?e) / real c"
```
```  1174   {fix x
```
```  1175     assume xz: "x < ?z"
```
```  1176     hence "(real c * x < - ?e)"
```
```  1177       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1178     hence "real c * x + ?e < 0" by arith
```
```  1179     with xz have "?P ?z x (Le (CN 0 c e))"
```
```  1180       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1181   hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
```
```  1182   thus ?case by blast
```
```  1183 next
```
```  1184   case (7 c e)
```
```  1185     from prems have nb: "numbound0 e" by simp
```
```  1186   from prems have cp: "real c > 0" by simp
```
```  1187   fix a
```
```  1188   let ?e="Inum (a#bs) e"
```
```  1189   let ?z = "(- ?e) / real c"
```
```  1190   {fix x
```
```  1191     assume xz: "x < ?z"
```
```  1192     hence "(real c * x < - ?e)"
```
```  1193       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1194     hence "real c * x + ?e < 0" by arith
```
```  1195     with xz have "?P ?z x (Gt (CN 0 c e))"
```
```  1196       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1197   hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
```
```  1198   thus ?case by blast
```
```  1199 next
```
```  1200   case (8 c e)
```
```  1201     from prems have nb: "numbound0 e" by simp
```
```  1202   from prems have cp: "real c > 0" by simp
```
```  1203   fix a
```
```  1204   let ?e="Inum (a#bs) e"
```
```  1205   let ?z = "(- ?e) / real c"
```
```  1206   {fix x
```
```  1207     assume xz: "x < ?z"
```
```  1208     hence "(real c * x < - ?e)"
```
```  1209       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
```
```  1210     hence "real c * x + ?e < 0" by arith
```
```  1211     with xz have "?P ?z x (Ge (CN 0 c e))"
```
```  1212       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1213   hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
```
```  1214   thus ?case by blast
```
```  1215 qed simp_all
```
```  1216
```
```  1217 lemma rplusinf_inf:
```
```  1218   assumes lp: "isrlfm p"
```
```  1219   shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
```
```  1220 using lp
```
```  1221 proof (induct p rule: isrlfm.induct)
```
```  1222   case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
```
```  1223 next
```
```  1224   case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
```
```  1225 next
```
```  1226   case (3 c e)
```
```  1227   from prems have nb: "numbound0 e" by simp
```
```  1228   from prems have cp: "real c > 0" by simp
```
```  1229   fix a
```
```  1230   let ?e="Inum (a#bs) e"
```
```  1231   let ?z = "(- ?e) / real c"
```
```  1232   {fix x
```
```  1233     assume xz: "x > ?z"
```
```  1234     with mult_strict_right_mono [OF xz cp] cp
```
```  1235     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1236     hence "real c * x + ?e > 0" by arith
```
```  1237     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1238     with xz have "?P ?z x (Eq (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 (Eq (CN 0 c e))" by simp
```
```  1241   thus ?case by blast
```
```  1242 next
```
```  1243   case (4 c e)
```
```  1244   from prems have nb: "numbound0 e" by simp
```
```  1245   from prems 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     hence "real c * x + ?e \<noteq> 0" by simp
```
```  1255     with xz have "?P ?z x (NEq (CN 0 c e))"
```
```  1256       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1257   hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
```
```  1258   thus ?case by blast
```
```  1259 next
```
```  1260   case (5 c e)
```
```  1261   from prems have nb: "numbound0 e" by simp
```
```  1262   from prems have cp: "real c > 0" by simp
```
```  1263   fix a
```
```  1264   let ?e="Inum (a#bs) e"
```
```  1265   let ?z = "(- ?e) / real c"
```
```  1266   {fix x
```
```  1267     assume xz: "x > ?z"
```
```  1268     with mult_strict_right_mono [OF xz cp] cp
```
```  1269     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1270     hence "real c * x + ?e > 0" by arith
```
```  1271     with xz have "?P ?z x (Lt (CN 0 c e))"
```
```  1272       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1273   hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
```
```  1274   thus ?case by blast
```
```  1275 next
```
```  1276   case (6 c e)
```
```  1277   from prems have nb: "numbound0 e" by simp
```
```  1278   from prems have cp: "real c > 0" by simp
```
```  1279   fix a
```
```  1280   let ?e="Inum (a#bs) e"
```
```  1281   let ?z = "(- ?e) / real c"
```
```  1282   {fix x
```
```  1283     assume xz: "x > ?z"
```
```  1284     with mult_strict_right_mono [OF xz cp] cp
```
```  1285     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1286     hence "real c * x + ?e > 0" by arith
```
```  1287     with xz have "?P ?z x (Le (CN 0 c e))"
```
```  1288       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1289   hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
```
```  1290   thus ?case by blast
```
```  1291 next
```
```  1292   case (7 c e)
```
```  1293   from prems have nb: "numbound0 e" by simp
```
```  1294   from prems have cp: "real c > 0" by simp
```
```  1295   fix a
```
```  1296   let ?e="Inum (a#bs) e"
```
```  1297   let ?z = "(- ?e) / real c"
```
```  1298   {fix x
```
```  1299     assume xz: "x > ?z"
```
```  1300     with mult_strict_right_mono [OF xz cp] cp
```
```  1301     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1302     hence "real c * x + ?e > 0" by arith
```
```  1303     with xz have "?P ?z x (Gt (CN 0 c e))"
```
```  1304       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
```
```  1305   hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
```
```  1306   thus ?case by blast
```
```  1307 next
```
```  1308   case (8 c e)
```
```  1309   from prems have nb: "numbound0 e" by simp
```
```  1310   from prems have cp: "real c > 0" by simp
```
```  1311   fix a
```
```  1312   let ?e="Inum (a#bs) e"
```
```  1313   let ?z = "(- ?e) / real c"
```
```  1314   {fix x
```
```  1315     assume xz: "x > ?z"
```
```  1316     with mult_strict_right_mono [OF xz cp] cp
```
```  1317     have "(real c * x > - ?e)" by (simp add: mult_ac)
```
```  1318     hence "real c * x + ?e > 0" by arith
```
```  1319     with xz have "?P ?z x (Ge (CN 0 c e))"
```
```  1320       using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
```
```  1321   hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
```
```  1322   thus ?case by blast
```
```  1323 qed simp_all
```
```  1324
```
```  1325 lemma rminusinf_bound0:
```
```  1326   assumes lp: "isrlfm p"
```
```  1327   shows "bound0 (minusinf p)"
```
```  1328   using lp
```
```  1329   by (induct p rule: minusinf.induct) simp_all
```
```  1330
```
```  1331 lemma rplusinf_bound0:
```
```  1332   assumes lp: "isrlfm p"
```
```  1333   shows "bound0 (plusinf p)"
```
```  1334   using lp
```
```  1335   by (induct p rule: plusinf.induct) simp_all
```
```  1336
```
```  1337 lemma rminusinf_ex:
```
```  1338   assumes lp: "isrlfm p"
```
```  1339   and ex: "Ifm (a#bs) (minusinf p)"
```
```  1340   shows "\<exists> x. Ifm (x#bs) p"
```
```  1341 proof-
```
```  1342   from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
```
```  1343   have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
```
```  1344   from rminusinf_inf[OF lp, where bs="bs"]
```
```  1345   obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
```
```  1346   from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
```
```  1347   moreover have "z - 1 < z" by simp
```
```  1348   ultimately show ?thesis using z_def by auto
```
```  1349 qed
```
```  1350
```
```  1351 lemma rplusinf_ex:
```
```  1352   assumes lp: "isrlfm p"
```
```  1353   and ex: "Ifm (a#bs) (plusinf p)"
```
```  1354   shows "\<exists> x. Ifm (x#bs) p"
```
```  1355 proof-
```
```  1356   from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
```
```  1357   have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
```
```  1358   from rplusinf_inf[OF lp, where bs="bs"]
```
```  1359   obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
```
```  1360   from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
```
```  1361   moreover have "z + 1 > z" by simp
```
```  1362   ultimately show ?thesis using z_def by auto
```
```  1363 qed
```
```  1364
```
```  1365 consts
```
```  1366   uset:: "fm \<Rightarrow> (num \<times> int) list"
```
```  1367   usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
```
```  1368 recdef uset "measure size"
```
```  1369   "uset (And p q) = (uset p @ uset q)"
```
```  1370   "uset (Or p q) = (uset p @ uset q)"
```
```  1371   "uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
```
```  1372   "uset (NEq (CN 0 c e)) = [(Neg e,c)]"
```
```  1373   "uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
```
```  1374   "uset (Le  (CN 0 c e)) = [(Neg e,c)]"
```
```  1375   "uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
```
```  1376   "uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
```
```  1377   "uset p = []"
```
```  1378 recdef usubst "measure size"
```
```  1379   "usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
```
```  1380   "usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
```
```  1381   "usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
```
```  1382   "usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
```
```  1383   "usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
```
```  1384   "usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
```
```  1385   "usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
```
```  1386   "usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
```
```  1387   "usubst p = (\<lambda> (t,n). p)"
```
```  1388
```
```  1389 lemma usubst_I: assumes lp: "isrlfm p"
```
```  1390   and np: "real n > 0" and nbt: "numbound0 t"
```
```  1391   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> _")
```
```  1392   using lp
```
```  1393 proof(induct p rule: usubst.induct)
```
```  1394   case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1395   have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
```
```  1396     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1397   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
```
```  1398     by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1399       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1400   also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
```
```  1401     using np by simp
```
```  1402   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1403 next
```
```  1404   case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1405   have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
```
```  1406     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1407   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
```
```  1408     by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1409       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1410   also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
```
```  1411     using np by simp
```
```  1412   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1413 next
```
```  1414   case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1415   have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
```
```  1416     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1417   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
```
```  1418     by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1419       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1420   also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
```
```  1421     using np by simp
```
```  1422   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1423 next
```
```  1424   case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1425   have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
```
```  1426     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1427   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
```
```  1428     by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1429       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1430   also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
```
```  1431     using np by simp
```
```  1432   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1433 next
```
```  1434   case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1435   from np have np: "real n \<noteq> 0" by simp
```
```  1436   have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
```
```  1437     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1438   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
```
```  1439     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1440       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1441   also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
```
```  1442     using np by simp
```
```  1443   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1444 next
```
```  1445   case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
```
```  1446   from np have np: "real n \<noteq> 0" by simp
```
```  1447   have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
```
```  1448     using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
```
```  1449   also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
```
```  1450     by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
```
```  1451       and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
```
```  1452   also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
```
```  1453     using np by simp
```
```  1454   finally show ?case using nbt nb by (simp add: algebra_simps)
```
```  1455 qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)
```
```  1456
```
```  1457 lemma uset_l:
```
```  1458   assumes lp: "isrlfm p"
```
```  1459   shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
```
```  1460 using lp
```
```  1461 by(induct p rule: uset.induct,auto)
```
```  1462
```
```  1463 lemma rminusinf_uset:
```
```  1464   assumes lp: "isrlfm p"
```
```  1465   and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
```
```  1466   and ex: "Ifm (x#bs) p" (is "?I x p")
```
```  1467   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")
```
```  1468 proof-
```
```  1469   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")
```
```  1470     using lp nmi ex
```
```  1471     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
```
```  1472   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
```
```  1473   from uset_l[OF lp] smU have mp: "real m > 0" by auto
```
```  1474   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"
```
```  1475     by (auto simp add: mult_commute)
```
```  1476   thus ?thesis using smU by auto
```
```  1477 qed
```
```  1478
```
```  1479 lemma rplusinf_uset:
```
```  1480   assumes lp: "isrlfm p"
```
```  1481   and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
```
```  1482   and ex: "Ifm (x#bs) p" (is "?I x p")
```
```  1483   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")
```
```  1484 proof-
```
```  1485   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")
```
```  1486     using lp nmi ex
```
```  1487     by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
```
```  1488   then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
```
```  1489   from uset_l[OF lp] smU have mp: "real m > 0" by auto
```
```  1490   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"
```
```  1491     by (auto simp add: mult_commute)
```
```  1492   thus ?thesis using smU by auto
```
```  1493 qed
```
```  1494
```
```  1495 lemma lin_dense:
```
```  1496   assumes lp: "isrlfm p"
```
```  1497   and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)"
```
```  1498   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
```
```  1499   and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
```
```  1500   and ly: "l < y" and yu: "y < u"
```
```  1501   shows "Ifm (y#bs) p"
```
```  1502 using lp px noS
```
```  1503 proof (induct p rule: isrlfm.induct)
```
```  1504   case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1505     from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
```
```  1506     hence pxc: "x < (- ?N x e) / real c"
```
```  1507       by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1508     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1509     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1510     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1511     moreover {assume y: "y < (-?N x e)/ real c"
```
```  1512       hence "y * real c < - ?N x e"
```
```  1513 	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1514       hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
```
```  1515       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1516     moreover {assume y: "y > (- ?N x e) / real c"
```
```  1517       with yu have eu: "u > (- ?N x e) / real c" by auto
```
```  1518       with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
```
```  1519       with lx pxc have "False" by auto
```
```  1520       hence ?case by simp }
```
```  1521     ultimately show ?case by blast
```
```  1522 next
```
```  1523   case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
```
```  1524     from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
```
```  1525     hence pxc: "x \<le> (- ?N x e) / real c"
```
```  1526       by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1527     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1528     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1529     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1530     moreover {assume y: "y < (-?N x e)/ real c"
```
```  1531       hence "y * real c < - ?N x e"
```
```  1532 	by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1533       hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
```
```  1534       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1535     moreover {assume y: "y > (- ?N x e) / real c"
```
```  1536       with yu have eu: "u > (- ?N x e) / real c" by auto
```
```  1537       with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
```
```  1538       with lx pxc have "False" by auto
```
```  1539       hence ?case by simp }
```
```  1540     ultimately show ?case by blast
```
```  1541 next
```
```  1542   case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1543     from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
```
```  1544     hence pxc: "x > (- ?N x e) / real c"
```
```  1545       by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1546     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1547     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1548     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1549     moreover {assume y: "y > (-?N x e)/ real c"
```
```  1550       hence "y * real c > - ?N x e"
```
```  1551 	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1552       hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
```
```  1553       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1554     moreover {assume y: "y < (- ?N x e) / real c"
```
```  1555       with ly have eu: "l < (- ?N x e) / real c" by auto
```
```  1556       with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
```
```  1557       with xu pxc have "False" by auto
```
```  1558       hence ?case by simp }
```
```  1559     ultimately show ?case by blast
```
```  1560 next
```
```  1561   case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1562     from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
```
```  1563     hence pxc: "x \<ge> (- ?N x e) / real c"
```
```  1564       by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
```
```  1565     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1566     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1567     hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
```
```  1568     moreover {assume y: "y > (-?N x e)/ real c"
```
```  1569       hence "y * real c > - ?N x e"
```
```  1570 	by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
```
```  1571       hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
```
```  1572       hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
```
```  1573     moreover {assume y: "y < (- ?N x e) / real c"
```
```  1574       with ly have eu: "l < (- ?N x e) / real c" by auto
```
```  1575       with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
```
```  1576       with xu pxc have "False" by auto
```
```  1577       hence ?case by simp }
```
```  1578     ultimately show ?case by blast
```
```  1579 next
```
```  1580   case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1581     from cp have cnz: "real c \<noteq> 0" by simp
```
```  1582     from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
```
```  1583     hence pxc: "x = (- ?N x e) / real c"
```
```  1584       by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
```
```  1585     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1586     with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
```
```  1587     with pxc show ?case by simp
```
```  1588 next
```
```  1589   case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
```
```  1590     from cp have cnz: "real c \<noteq> 0" by simp
```
```  1591     from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
```
```  1592     with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
```
```  1593     hence "y* real c \<noteq> -?N x e"
```
```  1594       by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
```
```  1595     hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
```
```  1596     thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
```
```  1597       by (simp add: algebra_simps)
```
```  1598 qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])
```
```  1599
```
```  1600 lemma finite_set_intervals:
```
```  1601   assumes px: "P (x::real)"
```
```  1602   and lx: "l \<le> x" and xu: "x \<le> u"
```
```  1603   and linS: "l\<in> S" and uinS: "u \<in> S"
```
```  1604   and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
```
```  1605   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"
```
```  1606 proof-
```
```  1607   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
```
```  1608   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
```
```  1609   let ?a = "Max ?Mx"
```
```  1610   let ?b = "Min ?xM"
```
```  1611   have MxS: "?Mx \<subseteq> S" by blast
```
```  1612   hence fMx: "finite ?Mx" using fS finite_subset by auto
```
```  1613   from lx linS have linMx: "l \<in> ?Mx" by blast
```
```  1614   hence Mxne: "?Mx \<noteq> {}" by blast
```
```  1615   have xMS: "?xM \<subseteq> S" by blast
```
```  1616   hence fxM: "finite ?xM" using fS finite_subset by auto
```
```  1617   from xu uinS have linxM: "u \<in> ?xM" by blast
```
```  1618   hence xMne: "?xM \<noteq> {}" by blast
```
```  1619   have ax:"?a \<le> x" using Mxne fMx by auto
```
```  1620   have xb:"x \<le> ?b" using xMne fxM by auto
```
```  1621   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
```
```  1622   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
```
```  1623   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
```
```  1624   proof(clarsimp)
```
```  1625     fix y
```
```  1626     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
```
```  1627     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
```
```  1628     moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
```
```  1629     moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
```
```  1630     ultimately show "False" by blast
```
```  1631   qed
```
```  1632   from ainS binS noy ax xb px show ?thesis by blast
```
```  1633 qed
```
```  1634
```
```  1635 lemma finite_set_intervals2:
```
```  1636   assumes px: "P (x::real)"
```
```  1637   and lx: "l \<le> x" and xu: "x \<le> u"
```
```  1638   and linS: "l\<in> S" and uinS: "u \<in> S"
```
```  1639   and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
```
```  1640   shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
```
```  1641 proof-
```
```  1642   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
```
```  1643   obtain a and b where
```
```  1644     as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
```
```  1645   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
```
```  1646   thus ?thesis using px as bs noS by blast
```
```  1647 qed
```
```  1648
```
```  1649 lemma rinf_uset:
```
```  1650   assumes lp: "isrlfm p"
```
```  1651   and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
```
```  1652   and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
```
```  1653   and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
```
```  1654   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"
```
```  1655 proof-
```
```  1656   let ?N = "\<lambda> x t. Inum (x#bs) t"
```
```  1657   let ?U = "set (uset p)"
```
```  1658   from ex obtain a where pa: "?I a p" by blast
```
```  1659   from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
```
```  1660   have nmi': "\<not> (?I a (?M p))" by simp
```
```  1661   from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
```
```  1662   have npi': "\<not> (?I a (?P p))" by simp
```
```  1663   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"
```
```  1664   proof-
```
```  1665     let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
```
```  1666     have fM: "finite ?M" by auto
```
```  1667     from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
```
```  1668     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
```
```  1669     then obtain "t" "n" "s" "m" where
```
```  1670       tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U"
```
```  1671       and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
```
```  1672     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
```
```  1673     from tnU have Mne: "?M \<noteq> {}" by auto
```
```  1674     hence Une: "?U \<noteq> {}" by simp
```
```  1675     let ?l = "Min ?M"
```
```  1676     let ?u = "Max ?M"
```
```  1677     have linM: "?l \<in> ?M" using fM Mne by simp
```
```  1678     have uinM: "?u \<in> ?M" using fM Mne by simp
```
```  1679     have tnM: "?N a t / real n \<in> ?M" using tnU by auto
```
```  1680     have smM: "?N a s / real m \<in> ?M" using smU by auto
```
```  1681     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
```
```  1682     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
```
```  1683     have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
```
```  1684     have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
```
```  1685     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
```
```  1686     have "(\<exists> s\<in> ?M. ?I s p) \<or>
```
```  1687       (\<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)" .
```
```  1688     moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
```
```  1689       hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
```
```  1690       then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
```
```  1691       have "(u + u) / 2 = u" by auto with pu tuu
```
```  1692       have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
```
```  1693       with tuU have ?thesis by blast}
```
```  1694     moreover{
```
```  1695       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"
```
```  1696       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
```
```  1697 	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"
```
```  1698 	by blast
```
```  1699       from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
```
```  1700       then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
```
```  1701       from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
```
```  1702       then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
```
```  1703       from t1x xt2 have t1t2: "t1 < t2" by simp
```
```  1704       let ?u = "(t1 + t2) / 2"
```
```  1705       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
```
```  1706       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
```
```  1707       with t1uU t2uU t1u t2u have ?thesis by blast}
```
```  1708     ultimately show ?thesis by blast
```
```  1709   qed
```
```  1710   then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U"
```
```  1711     and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
```
```  1712   from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
```
```  1713   from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
```
```  1714     numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
```
```  1715   have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
```
```  1716   with lnU smU
```
```  1717   show ?thesis by auto
```
```  1718 qed
```
```  1719     (* The Ferrante - Rackoff Theorem *)
```
```  1720
```
```  1721 theorem fr_eq:
```
```  1722   assumes lp: "isrlfm p"
```
```  1723   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))"
```
```  1724   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1725 proof
```
```  1726   assume px: "\<exists> x. ?I x p"
```
```  1727   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1728   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1729   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1730     from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
```
```  1731   ultimately show "?D" by blast
```
```  1732 next
```
```  1733   assume "?D"
```
```  1734   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
```
```  1735   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
```
```  1736   moreover {assume f:"?F" hence "?E" by blast}
```
```  1737   ultimately show "?E" by blast
```
```  1738 qed
```
```  1739
```
```  1740
```
```  1741 lemma fr_equsubst:
```
```  1742   assumes lp: "isrlfm p"
```
```  1743   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))))"
```
```  1744   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1745 proof
```
```  1746   assume px: "\<exists> x. ?I x p"
```
```  1747   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1748   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1749   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1750     let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
```
```  1751     let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1752     {fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
```
```  1753       with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
```
```  1754 	by auto
```
```  1755       let ?st = "Add (Mul m t) (Mul n s)"
```
```  1756       from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1757 	by (simp add: mult_commute)
```
```  1758       from tnb snb have st_nb: "numbound0 ?st" by simp
```
```  1759       have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1760 	using mnp mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1761       from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
```
```  1762       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])}
```
```  1763     with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
```
```  1764   ultimately show "?D" by blast
```
```  1765 next
```
```  1766   assume "?D"
```
```  1767   moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
```
```  1768   moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
```
```  1769   moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)"
```
```  1770     and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
```
```  1771     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
```
```  1772     let ?st = "Add (Mul l t) (Mul k s)"
```
```  1773     from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0"
```
```  1774       by (simp add: mult_commute)
```
```  1775     from tnb snb have st_nb: "numbound0 ?st" by simp
```
```  1776     from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
```
```  1777   ultimately show "?E" by blast
```
```  1778 qed
```
```  1779
```
```  1780
```
```  1781     (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
```
```  1782 constdefs ferrack:: "fm \<Rightarrow> fm"
```
```  1783   "ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
```
```  1784                 in if (mp = T \<or> pp = T) then T else
```
```  1785                    (let U = remdps(map simp_num_pair
```
```  1786                      (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
```
```  1787                            (alluopairs (uset p'))))
```
```  1788                     in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"
```
```  1789
```
```  1790 lemma uset_cong_aux:
```
```  1791   assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
```
```  1792   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))"
```
```  1793   (is "?lhs = ?rhs")
```
```  1794 proof(auto)
```
```  1795   fix t n s m
```
```  1796   assume "((t,n),(s,m)) \<in> set (alluopairs U)"
```
```  1797   hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
```
```  1798     using alluopairs_set1[where xs="U"] by blast
```
```  1799   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1800   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1801   from Ul th have mnz: "m \<noteq> 0" by auto
```
```  1802   from Ul th have  nnz: "n \<noteq> 0" by auto
```
```  1803   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1804    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
```
```  1805
```
```  1806   thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
```
```  1807        (2 * real n * real m)
```
```  1808        \<in> (\<lambda>((t, n), s, m).
```
```  1809              (Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
```
```  1810          (set U \<times> set U)"using mnz nnz th
```
```  1811     apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
```
```  1812     by (rule_tac x="(s,m)" in bexI,simp_all)
```
```  1813   (rule_tac x="(t,n)" in bexI,simp_all)
```
```  1814 next
```
```  1815   fix t n s m
```
```  1816   assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U"
```
```  1817   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1818   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1819   from Ul smU have mnz: "m \<noteq> 0" by auto
```
```  1820   from Ul tnU have  nnz: "n \<noteq> 0" by auto
```
```  1821   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1822    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
```
```  1823  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"
```
```  1824  have Pc:"\<forall> a b. ?P a b = ?P b a"
```
```  1825    by auto
```
```  1826  from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
```
```  1827  from alluopairs_ex[OF Pc, where xs="U"] tnU smU
```
```  1828  have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
```
```  1829    by blast
```
```  1830  then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
```
```  1831    and Pts': "?P (t',n') (s',m')" by blast
```
```  1832  from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
```
```  1833  let ?st' = "Add (Mul m' t') (Mul n' s')"
```
```  1834    have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
```
```  1835    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
```
```  1836  from Pts' have
```
```  1837    "(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
```
```  1838  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')
```
```  1839  finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
```
```  1840           \<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
```
```  1841             (\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
```
```  1842             set (alluopairs U)"
```
```  1843    using ts'_U by blast
```
```  1844 qed
```
```  1845
```
```  1846 lemma uset_cong:
```
```  1847   assumes lp: "isrlfm p"
```
```  1848   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)")
```
```  1849   and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
```
```  1850   and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
```
```  1851   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)))"
```
```  1852   (is "?lhs = ?rhs")
```
```  1853 proof
```
```  1854   assume ?lhs
```
```  1855   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
```
```  1856     Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
```
```  1857   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1858   from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
```
```  1859     and snb: "numbound0 s" and mp:"m > 0"  by auto
```
```  1860   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1861   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1862       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
```
```  1863     from tnb snb have stnb: "numbound0 ?st" by simp
```
```  1864   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1865    using mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1866   from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
```
```  1867   hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
```
```  1868     by auto (rule_tac x="(a,b)" in bexI, auto)
```
```  1869   then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
```
```  1870   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
```
```  1871   from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
```
```  1872   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
```
```  1873   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]]
```
```  1874   have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st)
```
```  1875   then show ?rhs using tnU' by auto
```
```  1876 next
```
```  1877   assume ?rhs
```
```  1878   then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))"
```
```  1879     by blast
```
```  1880   from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
```
```  1881   hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))"
```
```  1882     by auto (rule_tac x="(a,b)" in bexI, auto)
```
```  1883   then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
```
```  1884     th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
```
```  1885     let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1886   from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
```
```  1887     and snb: "numbound0 s" and mp:"m > 0"  by auto
```
```  1888   let ?st= "Add (Mul m t) (Mul n s)"
```
```  1889   from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
```
```  1890       by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
```
```  1891     from tnb snb have stnb: "numbound0 ?st" by simp
```
```  1892   have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
```
```  1893    using mp np by (simp add: algebra_simps add_divide_distrib)
```
```  1894   from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
```
```  1895   from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
```
```  1896   have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
```
```  1897   with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
```
```  1898 qed
```
```  1899
```
```  1900 lemma ferrack:
```
```  1901   assumes qf: "qfree p"
```
```  1902   shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
```
```  1903   (is "_ \<and> (?rhs = ?lhs)")
```
```  1904 proof-
```
```  1905   let ?I = "\<lambda> x p. Ifm (x#bs) p"
```
```  1906   fix x
```
```  1907   let ?N = "\<lambda> t. Inum (x#bs) t"
```
```  1908   let ?q = "rlfm (simpfm p)"
```
```  1909   let ?U = "uset ?q"
```
```  1910   let ?Up = "alluopairs ?U"
```
```  1911   let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
```
```  1912   let ?S = "map ?g ?Up"
```
```  1913   let ?SS = "map simp_num_pair ?S"
```
```  1914   let ?Y = "remdps ?SS"
```
```  1915   let ?f= "(\<lambda> (t,n). ?N t / real n)"
```
```  1916   let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
```
```  1917   let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
```
```  1918   let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
```
```  1919   from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
```
```  1920   from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
```
```  1921   from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
```
```  1922   from U_l UpU
```
```  1923   have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
```
```  1924   hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
```
```  1925     by (auto simp add: mult_pos_pos)
```
```  1926   have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0"
```
```  1927   proof-
```
```  1928     { fix t n assume tnY: "(t,n) \<in> set ?Y"
```
```  1929       hence "(t,n) \<in> set ?SS" by simp
```
```  1930       hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
```
```  1931 	by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
```
```  1932       then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
```
```  1933       from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
```
```  1934       from simp_num_pair_l[OF tnb np tns]
```
```  1935       have "numbound0 t \<and> n > 0" . }
```
```  1936     thus ?thesis by blast
```
```  1937   qed
```
```  1938
```
```  1939   have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
```
```  1940   proof-
```
```  1941      from simp_num_pair_ci[where bs="x#bs"] have
```
```  1942     "\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
```
```  1943      hence th: "?f o simp_num_pair = ?f" using ext by blast
```
```  1944     have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
```
```  1945     also have "\<dots> = (?f ` set ?S)" by (simp add: th)
```
```  1946     also have "\<dots> = ((?f o ?g) ` set ?Up)"
```
```  1947       by (simp only: set_map o_def image_compose[symmetric])
```
```  1948     also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
```
```  1949       using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
```
```  1950     finally show ?thesis .
```
```  1951   qed
```
```  1952   have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
```
```  1953   proof-
```
```  1954     { fix t n assume tnY: "(t,n) \<in> set ?Y"
```
```  1955       with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
```
```  1956       from usubst_I[OF lq np tnb]
```
```  1957     have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))"
```
```  1958       using simpfm_bound0 by simp}
```
```  1959     thus ?thesis by blast
```
```  1960   qed
```
```  1961   hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
```
```  1962   let ?mp = "minusinf ?q"
```
```  1963   let ?pp = "plusinf ?q"
```
```  1964   let ?M = "?I x ?mp"
```
```  1965   let ?P = "?I x ?pp"
```
```  1966   let ?res = "disj ?mp (disj ?pp ?ep)"
```
```  1967   from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
```
```  1968   have nbth: "bound0 ?res" by auto
```
```  1969
```
```  1970   from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm
```
```  1971
```
```  1972   have th: "?lhs = (\<exists> x. ?I x ?q)" by auto
```
```  1973   from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
```
```  1974     by (simp only: split_def fst_conv snd_conv)
```
```  1975   also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))"
```
```  1976     using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm)
```
```  1977   also have "\<dots> = (Ifm (x#bs) ?res)"
```
```  1978     using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
```
```  1979     by (simp add: split_def pair_collapse)
```
```  1980   finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
```
```  1981   hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
```
```  1982     by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
```
```  1983   from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
```
```  1984   with lr show ?thesis by blast
```
```  1985 qed
```
```  1986
```
```  1987 definition linrqe:: "fm \<Rightarrow> fm" where
```
```  1988   "linrqe p = qelim (prep p) ferrack"
```
```  1989
```
```  1990 theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
```
```  1991 using ferrack qelim_ci prep
```
```  1992 unfolding linrqe_def by auto
```
```  1993
```
```  1994 definition ferrack_test :: "unit \<Rightarrow> fm" where
```
```  1995   "ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
```
```  1996     (E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"
```
```  1997
```
```  1998 ML {* @{code ferrack_test} () *}
```
```  1999
```
```  2000 oracle linr_oracle = {*
```
```  2001 let
```
```  2002
```
```  2003 fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
```
```  2004      of NONE => error "Variable not found in the list!"
```
```  2005       | SOME n => @{code Bound} n)
```
```  2006   | num_of_term vs @{term "real (0::int)"} = @{code C} 0
```
```  2007   | num_of_term vs @{term "real (1::int)"} = @{code C} 1
```
```  2008   | num_of_term vs @{term "0::real"} = @{code C} 0
```
```  2009   | num_of_term vs @{term "1::real"} = @{code C} 1
```
```  2010   | num_of_term vs (Bound i) = @{code Bound} i
```
```  2011   | num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} \$ t') = @{code Neg} (num_of_term vs t')
```
```  2012   | num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
```
```  2013   | num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
```
```  2014   | num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = (case (num_of_term vs t1)
```
```  2015      of @{code C} i => @{code Mul} (i, num_of_term vs t2)
```
```  2016       | _ => error "num_of_term: unsupported Multiplication")
```
```  2017   | num_of_term vs (@{term "real :: int \<Rightarrow> real"} \$ (@{term "number_of :: int \<Rightarrow> int"} \$ t')) = @{code C} (HOLogic.dest_numeral t')
```
```  2018   | num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} \$ t') = @{code C} (HOLogic.dest_numeral t')
```
```  2019   | num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);
```
```  2020
```
```  2021 fun fm_of_term vs @{term True} = @{code T}
```
```  2022   | fm_of_term vs @{term False} = @{code F}
```
```  2023   | fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  2024   | fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  2025   | fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
```
```  2026   | fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
```
```  2027   | fm_of_term vs (@{term "op &"} \$ t1 \$ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
```
```  2028   | fm_of_term vs (@{term "op |"} \$ t1 \$ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
```
```  2029   | fm_of_term vs (@{term "op -->"} \$ t1 \$ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
```
```  2030   | fm_of_term vs (@{term "Not"} \$ t') = @{code NOT} (fm_of_term vs t')
```
```  2031   | fm_of_term vs (Const ("Ex", _) \$ Abs (xn, xT, p)) =
```
```  2032       @{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
```
```  2033   | fm_of_term vs (Const ("All", _) \$ Abs (xn, xT, p)) =
```
```  2034       @{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
```
```  2035   | fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);
```
```  2036
```
```  2037 fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} \$ HOLogic.mk_number HOLogic.intT i
```
```  2038   | term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
```
```  2039   | term_of_num vs (@{code Neg} t') = @{term "uminus :: real \<Rightarrow> real"} \$ term_of_num vs t'
```
```  2040   | term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  2041       term_of_num vs t1 \$ term_of_num vs t2
```
```  2042   | term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  2043       term_of_num vs t1 \$ term_of_num vs t2
```
```  2044   | term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$
```
```  2045       term_of_num vs (@{code C} i) \$ term_of_num vs t2
```
```  2046   | term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));
```
```  2047
```
```  2048 fun term_of_fm vs @{code T} = HOLogic.true_const
```
```  2049   | term_of_fm vs @{code F} = HOLogic.false_const
```
```  2050   | term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  2051       term_of_num vs t \$ @{term "0::real"}
```
```  2052   | term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  2053       term_of_num vs t \$ @{term "0::real"}
```
```  2054   | term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  2055       @{term "0::real"} \$ term_of_num vs t
```
```  2056   | term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  2057       @{term "0::real"} \$ term_of_num vs t
```
```  2058   | term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$
```
```  2059       term_of_num vs t \$ @{term "0::real"}
```
```  2060   | term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
```
```  2061   | term_of_fm vs (@{code NOT} t') = HOLogic.Not \$ term_of_fm vs t'
```
```  2062   | term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  2063   | term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  2064   | term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp \$ term_of_fm vs t1 \$ term_of_fm vs t2
```
```  2065   | term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$
```
```  2066       term_of_fm vs t1 \$ term_of_fm vs t2
```
```  2067   | term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";
```
```  2068
```
```  2069 in fn ct =>
```
```  2070   let
```
```  2071     val thy = Thm.theory_of_cterm ct;
```
```  2072     val t = Thm.term_of ct;
```
```  2073     val fs = OldTerm.term_frees t;
```
```  2074     val vs = fs ~~ (0 upto (length fs - 1));
```
```  2075     val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t))));
```
```  2076   in Thm.cterm_of thy res end
```
```  2077 end;
```
```  2078 *}
```
```  2079
```
```  2080 use "ferrack_tac.ML"
```
```  2081 setup Ferrack_Tac.setup
```
```  2082
```
```  2083 lemma
```
```  2084   fixes x :: real
```
```  2085   shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
```
```  2086 apply rferrack
```
```  2087 done
```
```  2088
```
```  2089 lemma
```
```  2090   fixes x :: real
```
```  2091   shows "\<exists>y \<le> x. x = y + 1"
```
```  2092 apply rferrack
```
```  2093 done
```
```  2094
```
```  2095 lemma
```
```  2096   fixes x :: real
```
```  2097   shows "\<not> (\<exists>z. x + z = x + z + 1)"
```
```  2098 apply rferrack
```
```  2099 done
```
```  2100
```
```  2101 end
```