src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
author haftmann
Wed Sep 08 19:21:46 2010 +0200 (2010-09-08)
changeset 39246 9e58f0499f57
parent 38864 4abe644fcea5
child 41413 64cd30d6b0b8
permissions -rw-r--r--
modernized primrec
     1 (*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
     2     Author:     Amine Chaieb
     3 *)
     4 
     5 header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *}
     6 
     7 theory Parametric_Ferrante_Rackoff
     8 imports Reflected_Multivariate_Polynomial
     9   Dense_Linear_Order
    10   Efficient_Nat
    11 begin
    12 
    13 subsection {* Terms *}
    14 
    15 datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
    16   | Neg tm | Sub tm tm | CNP nat poly tm
    17   (* A size for poly to make inductive proofs simpler*)
    18 
    19 primrec tmsize :: "tm \<Rightarrow> nat" where
    20   "tmsize (CP c) = polysize c"
    21 | "tmsize (Bound n) = 1"
    22 | "tmsize (Neg a) = 1 + tmsize a"
    23 | "tmsize (Add a b) = 1 + tmsize a + tmsize b"
    24 | "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
    25 | "tmsize (Mul c a) = 1 + polysize c + tmsize a"
    26 | "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
    27 
    28   (* Semantics of terms tm *)
    29 primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
    30   "Itm vs bs (CP c) = (Ipoly vs c)"
    31 | "Itm vs bs (Bound n) = bs!n"
    32 | "Itm vs bs (Neg a) = -(Itm vs bs a)"
    33 | "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
    34 | "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
    35 | "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
    36 | "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"   
    37 
    38 
    39 fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
    40   "allpolys P (CP c) = P c"
    41 | "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
    42 | "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
    43 | "allpolys P (Neg p) = allpolys P p"
    44 | "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
    45 | "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
    46 | "allpolys P p = True"
    47 
    48 primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
    49   "tmboundslt n (CP c) = True"
    50 | "tmboundslt n (Bound m) = (m < n)"
    51 | "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
    52 | "tmboundslt n (Neg a) = tmboundslt n a"
    53 | "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
    54 | "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
    55 | "tmboundslt n (Mul i a) = tmboundslt n a"
    56 
    57 primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
    58   "tmbound0 (CP c) = True"
    59 | "tmbound0 (Bound n) = (n>0)"
    60 | "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
    61 | "tmbound0 (Neg a) = tmbound0 a"
    62 | "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
    63 | "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
    64 | "tmbound0 (Mul i a) = tmbound0 a"
    65 lemma tmbound0_I:
    66   assumes nb: "tmbound0 a"
    67   shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
    68 using nb
    69 by (induct a rule: tm.induct,auto simp add: nth_pos2)
    70 
    71 primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
    72   "tmbound n (CP c) = True"
    73 | "tmbound n (Bound m) = (n \<noteq> m)"
    74 | "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
    75 | "tmbound n (Neg a) = tmbound n a"
    76 | "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
    77 | "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
    78 | "tmbound n (Mul i a) = tmbound n a"
    79 lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
    80 
    81 lemma tmbound_I: 
    82   assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
    83   shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
    84   using nb le bnd
    85   by (induct t rule: tm.induct , auto)
    86 
    87 consts 
    88   incrtm0:: "tm \<Rightarrow> tm"
    89   decrtm0:: "tm \<Rightarrow> tm" 
    90 
    91 recdef decrtm0 "measure size"
    92   "decrtm0 (Bound n) = Bound (n - 1)"
    93   "decrtm0 (Neg a) = Neg (decrtm0 a)"
    94   "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
    95   "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
    96   "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
    97   "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
    98   "decrtm0 a = a"
    99 
   100 recdef incrtm0 "measure size"
   101   "incrtm0 (Bound n) = Bound (n + 1)"
   102   "incrtm0 (Neg a) = Neg (incrtm0 a)"
   103   "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
   104   "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
   105   "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
   106   "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
   107   "incrtm0 a = a"
   108 
   109 lemma decrtm0: assumes nb: "tmbound0 t"
   110   shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
   111   using nb by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)
   112 
   113 lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
   114   by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)
   115 
   116 primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
   117   "decrtm m (CP c) = (CP c)"
   118 | "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
   119 | "decrtm m (Neg a) = Neg (decrtm m a)"
   120 | "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
   121 | "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
   122 | "decrtm m (Mul c a) = Mul c (decrtm m a)"
   123 | "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
   124 
   125 primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   126   "removen n [] = []"
   127 | "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
   128 
   129 lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
   130   by (induct xs arbitrary: n, auto)
   131 
   132 lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
   133   by (induct xs arbitrary: n, auto)
   134 
   135 lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
   136   by (induct xs arbitrary: n, auto)
   137 lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
   138   else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
   139 proof(induct xs arbitrary: n m)
   140   case Nil thus ?case by simp
   141 next
   142   case (Cons x xs n m)
   143   {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
   144   moreover
   145   {assume nxs: "\<not> (n \<ge> length (x#xs))" 
   146     {assume mln: "m < n" hence ?case using prems by (cases m, auto)}
   147     moreover
   148     {assume mln: "\<not> (m < n)" 
   149       
   150       {assume mxs: "m \<le> length (x#xs)" hence ?case using prems by (cases m, auto)}
   151       moreover
   152       {assume mxs: "\<not> (m \<le> length (x#xs))" 
   153         have th: "length (removen n (x#xs)) = length xs" 
   154           using removen_length[where n="n" and xs="x#xs"] nxs by simp
   155         with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
   156         hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
   157           using th nth_length_exceeds[OF mxs'] by auto
   158         hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
   159           by auto
   160         hence ?case using nxs mln mxs by auto }
   161       ultimately have ?case by blast
   162     }
   163     ultimately have ?case by blast
   164     
   165   }      ultimately show ?case by blast
   166 qed
   167 
   168 lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
   169   and nle: "m \<le> length bs" 
   170   shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
   171   using bnd nb nle
   172   by (induct t rule: tm.induct, auto simp add: removen_nth)
   173 
   174 primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   175   "tmsubst0 t (CP c) = CP c"
   176 | "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
   177 | "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
   178 | "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
   179 | "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
   180 | "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
   181 | "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
   182 lemma tmsubst0:
   183   shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
   184 by (induct a rule: tm.induct,auto simp add: nth_pos2)
   185 
   186 lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
   187 by (induct a rule: tm.induct,auto simp add: nth_pos2)
   188 
   189 primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
   190   "tmsubst n t (CP c) = CP c"
   191 | "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
   192 | "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
   193              else CNP m c (tmsubst n t a))"
   194 | "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
   195 | "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
   196 | "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
   197 | "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
   198 
   199 lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
   200   shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
   201 using nb nlt
   202 by (induct a rule: tm.induct,auto simp add: nth_pos2)
   203 
   204 lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
   205 shows "tmbound0 (tmsubst 0 t a)"
   206 using tnb
   207 by (induct a rule: tm.induct, auto)
   208 
   209 lemma tmsubst_nb: assumes tnb: "tmbound m t"
   210 shows "tmbound m (tmsubst m t a)"
   211 using tnb
   212 by (induct a rule: tm.induct, auto)
   213 lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
   214   by (induct t, auto)
   215   (* Simplification *)
   216 
   217 consts
   218   simptm:: "tm \<Rightarrow> tm"
   219   tmadd:: "tm \<times> tm \<Rightarrow> tm"
   220   tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm"
   221 recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
   222   "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
   223   (if n1=n2 then 
   224   (let c = c1 +\<^sub>p c2
   225   in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
   226   else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
   227   else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
   228   "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
   229   "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
   230   "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
   231   "tmadd (a,b) = Add a b"
   232 
   233 lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
   234 apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
   235 apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
   236 apply (case_tac "n1 = n2", simp_all add: field_simps)
   237 apply (simp only: right_distrib[symmetric]) 
   238 by (auto simp del: polyadd simp add: polyadd[symmetric])
   239 
   240 lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
   241 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   242 
   243 lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
   244 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   245 lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
   246 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   247 
   248 lemma tmadd_allpolys_npoly[simp]: "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmadd(t,s))" by (induct t s rule: tmadd.induct, simp_all add: Let_def polyadd_norm)
   249 
   250 recdef tmmul "measure size"
   251   "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
   252   "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
   253   "tmmul t = (\<lambda> i. Mul i t)"
   254 
   255 lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
   256 by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
   257 
   258 lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
   259 by (induct t arbitrary: i rule: tmmul.induct, auto )
   260 
   261 lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
   262 by (induct t arbitrary: n rule: tmmul.induct, auto )
   263 lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
   264 by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
   265 
   266 lemma tmmul_allpolys_npoly[simp]: 
   267   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   268   shows "allpolys isnpoly t \<Longrightarrow> isnpoly c \<Longrightarrow> allpolys isnpoly (tmmul t c)" by (induct t rule: tmmul.induct, simp_all add: Let_def polymul_norm)
   269 
   270 definition tmneg :: "tm \<Rightarrow> tm" where
   271   "tmneg t \<equiv> tmmul t (C (- 1,1))"
   272 
   273 definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   274   "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
   275 
   276 lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
   277 using tmneg_def[of t] 
   278 apply simp
   279 apply (subst number_of_Min)
   280 apply (simp only: of_int_minus)
   281 apply simp
   282 done
   283 
   284 lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
   285 using tmneg_def by simp
   286 
   287 lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
   288 using tmneg_def by simp
   289 lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
   290 using tmneg_def by simp
   291 lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
   292 lemma tmneg_allpolys_npoly[simp]: 
   293   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   294   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
   295   unfolding tmneg_def by auto
   296 
   297 lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
   298 using tmsub_def by simp
   299 
   300 lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
   301 using tmsub_def by simp
   302 lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
   303 using tmsub_def by simp
   304 lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
   305 using tmsub_def by simp
   306 lemma tmsub_allpolys_npoly[simp]: 
   307   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   308   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
   309   unfolding tmsub_def by (simp add: isnpoly_def)
   310 
   311 recdef simptm "measure size"
   312   "simptm (CP j) = CP (polynate j)"
   313   "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
   314   "simptm (Neg t) = tmneg (simptm t)"
   315   "simptm (Add t s) = tmadd (simptm t,simptm s)"
   316   "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
   317   "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
   318   "simptm (CNP n c t) = (let c' = polynate c in if c' = 0\<^sub>p then simptm t else tmadd (CNP n c' (CP 0\<^sub>p ), simptm t))"
   319 
   320 lemma polynate_stupid: 
   321   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   322   shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a::{field_char_0, field_inverse_zero})" 
   323 apply (subst polynate[symmetric])
   324 apply simp
   325 done
   326 
   327 lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
   328 by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 
   329 
   330 lemma simptm_tmbound0[simp]: 
   331   "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
   332 by (induct t rule: simptm.induct, auto simp add: Let_def)
   333 
   334 lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
   335 by (induct t rule: simptm.induct, auto simp add: Let_def)
   336 lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
   337 by (induct t rule: simptm.induct, auto simp add: Let_def)
   338 
   339 lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
   340   by (simp_all add: isnpoly_def)
   341 lemma simptm_allpolys_npoly[simp]: 
   342   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   343   shows "allpolys isnpoly (simptm p)"
   344   by (induct p rule: simptm.induct, auto simp add: Let_def)
   345 
   346 consts split0 :: "tm \<Rightarrow> (poly \<times> tm)"
   347 recdef split0 "measure tmsize"
   348   "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
   349   "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
   350   "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
   351   "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
   352   "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
   353   "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
   354   "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
   355   "split0 t = (0\<^sub>p, t)"
   356 
   357 lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
   358   apply (rule exI[where x="fst (split0 p)"])
   359   apply (rule exI[where x="snd (split0 p)"])
   360   by simp
   361 
   362 lemma split0:
   363   "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
   364   apply (induct t rule: split0.induct)
   365   apply simp
   366   apply (simp add: Let_def split_def field_simps)
   367   apply (simp add: Let_def split_def field_simps)
   368   apply (simp add: Let_def split_def field_simps)
   369   apply (simp add: Let_def split_def field_simps)
   370   apply (simp add: Let_def split_def field_simps)
   371   apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
   372   apply (simp add: Let_def split_def field_simps)
   373   apply (simp add: Let_def split_def field_simps)
   374   done
   375 
   376 lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
   377 proof-
   378   fix c' t'
   379   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   380   with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
   381 qed
   382 
   383 lemma split0_nb0: 
   384   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   385   shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
   386 proof-
   387   fix c' t'
   388   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   389   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
   390 qed
   391 
   392 lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   393   shows "tmbound0 (snd (split0 t))"
   394   using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
   395 
   396 
   397 lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
   398   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   399 
   400 lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
   401   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   402 
   403 lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   404  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   405 
   406 lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
   407 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   408 
   409 lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   410  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   411 
   412 lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
   413 by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
   414 
   415 lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   416   shows 
   417   "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
   418   by (induct p rule: split0.induct, 
   419     auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
   420     Let_def split_def split0_stupid)
   421 
   422 subsection{* Formulae *}
   423 
   424 datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
   425   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
   426 
   427 
   428   (* A size for fm *)
   429 consts fmsize :: "fm \<Rightarrow> nat"
   430 recdef fmsize "measure size"
   431   "fmsize (NOT p) = 1 + fmsize p"
   432   "fmsize (And p q) = 1 + fmsize p + fmsize q"
   433   "fmsize (Or p q) = 1 + fmsize p + fmsize q"
   434   "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
   435   "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
   436   "fmsize (E p) = 1 + fmsize p"
   437   "fmsize (A p) = 4+ fmsize p"
   438   "fmsize p = 1"
   439   (* several lemmas about fmsize *)
   440 lemma fmsize_pos: "fmsize p > 0"        
   441 by (induct p rule: fmsize.induct) simp_all
   442 
   443   (* Semantics of formulae (fm) *)
   444 primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
   445   "Ifm vs bs T = True"
   446 | "Ifm vs bs F = False"
   447 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
   448 | "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
   449 | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
   450 | "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
   451 | "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
   452 | "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
   453 | "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
   454 | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
   455 | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
   456 | "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
   457 | "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
   458 
   459 consts not:: "fm \<Rightarrow> fm"
   460 recdef not "measure size"
   461   "not (NOT (NOT p)) = not p"
   462   "not (NOT p) = p"
   463   "not T = F"
   464   "not F = T"
   465   "not (Lt t) = Le (tmneg t)"
   466   "not (Le t) = Lt (tmneg t)"
   467   "not (Eq t) = NEq t"
   468   "not (NEq t) = Eq t"
   469   "not p = NOT p"
   470 lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
   471 by (induct p rule: not.induct) auto
   472 
   473 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   474   "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 
   475    if p = q then p else And p q)"
   476 lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
   477 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
   478 
   479 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   480   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
   481        else if p=q then p else Or p q)"
   482 
   483 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
   484 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
   485 
   486 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   487   "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 
   488     else Imp p q)"
   489 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
   490 by (cases "p=F \<or> q=T",simp_all add: imp_def) 
   491 
   492 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   493   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
   494        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 
   495   Iff p q)"
   496 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
   497   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
   498   (* Quantifier freeness *)
   499 consts qfree:: "fm \<Rightarrow> bool"
   500 recdef qfree "measure size"
   501   "qfree (E p) = False"
   502   "qfree (A p) = False"
   503   "qfree (NOT p) = qfree p" 
   504   "qfree (And p q) = (qfree p \<and> qfree q)" 
   505   "qfree (Or  p q) = (qfree p \<and> qfree q)" 
   506   "qfree (Imp p q) = (qfree p \<and> qfree q)" 
   507   "qfree (Iff p q) = (qfree p \<and> qfree q)"
   508   "qfree p = True"
   509 
   510   (* Boundedness and substitution *)
   511 
   512 primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
   513   "boundslt n T = True"
   514 | "boundslt n F = True"
   515 | "boundslt n (Lt t) = (tmboundslt n t)"
   516 | "boundslt n (Le t) = (tmboundslt n t)"
   517 | "boundslt n (Eq t) = (tmboundslt n t)"
   518 | "boundslt n (NEq t) = (tmboundslt n t)"
   519 | "boundslt n (NOT p) = boundslt n p"
   520 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
   521 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
   522 | "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
   523 | "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
   524 | "boundslt n (E p) = boundslt (Suc n) p"
   525 | "boundslt n (A p) = boundslt (Suc n) p"
   526 
   527 consts 
   528   bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
   529   decr0 :: "fm \<Rightarrow> fm"
   530 recdef bound0 "measure size"
   531   "bound0 T = True"
   532   "bound0 F = True"
   533   "bound0 (Lt a) = tmbound0 a"
   534   "bound0 (Le a) = tmbound0 a"
   535   "bound0 (Eq a) = tmbound0 a"
   536   "bound0 (NEq a) = tmbound0 a"
   537   "bound0 (NOT p) = bound0 p"
   538   "bound0 (And p q) = (bound0 p \<and> bound0 q)"
   539   "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
   540   "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
   541   "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
   542   "bound0 p = False"
   543 lemma bound0_I:
   544   assumes bp: "bound0 p"
   545   shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
   546 using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
   547 by (induct p rule: bound0.induct,auto simp add: nth_pos2)
   548 
   549 primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
   550   "bound m T = True"
   551 | "bound m F = True"
   552 | "bound m (Lt t) = tmbound m t"
   553 | "bound m (Le t) = tmbound m t"
   554 | "bound m (Eq t) = tmbound m t"
   555 | "bound m (NEq t) = tmbound m t"
   556 | "bound m (NOT p) = bound m p"
   557 | "bound m (And p q) = (bound m p \<and> bound m q)"
   558 | "bound m (Or p q) = (bound m p \<and> bound m q)"
   559 | "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
   560 | "bound m (Iff p q) = (bound m p \<and> bound m q)"
   561 | "bound m (E p) = bound (Suc m) p"
   562 | "bound m (A p) = bound (Suc m) p"
   563 
   564 lemma bound_I:
   565   assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
   566   shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
   567   using bnd nb le tmbound_I[where bs=bs and vs = vs]
   568 proof(induct p arbitrary: bs n rule: fm.induct)
   569   case (E p bs n) 
   570   {fix y
   571     from prems have bnd: "boundslt (length (y#bs)) p" 
   572       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   573     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   574   thus ?case by simp 
   575 next
   576   case (A p bs n) {fix y
   577     from prems have bnd: "boundslt (length (y#bs)) p" 
   578       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   579     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   580   thus ?case by simp 
   581 qed auto
   582 
   583 recdef decr0 "measure size"
   584   "decr0 (Lt a) = Lt (decrtm0 a)"
   585   "decr0 (Le a) = Le (decrtm0 a)"
   586   "decr0 (Eq a) = Eq (decrtm0 a)"
   587   "decr0 (NEq a) = NEq (decrtm0 a)"
   588   "decr0 (NOT p) = NOT (decr0 p)" 
   589   "decr0 (And p q) = conj (decr0 p) (decr0 q)"
   590   "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
   591   "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
   592   "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
   593   "decr0 p = p"
   594 
   595 lemma decr0: assumes nb: "bound0 p"
   596   shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
   597   using nb 
   598   by (induct p rule: decr0.induct, simp_all add: decrtm0)
   599 
   600 primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
   601   "decr m T = T"
   602 | "decr m F = F"
   603 | "decr m (Lt t) = (Lt (decrtm m t))"
   604 | "decr m (Le t) = (Le (decrtm m t))"
   605 | "decr m (Eq t) = (Eq (decrtm m t))"
   606 | "decr m (NEq t) = (NEq (decrtm m t))"
   607 | "decr m (NOT p) = NOT (decr m p)" 
   608 | "decr m (And p q) = conj (decr m p) (decr m q)"
   609 | "decr m (Or p q) = disj (decr m p) (decr m q)"
   610 | "decr m (Imp p q) = imp (decr m p) (decr m q)"
   611 | "decr m (Iff p q) = iff (decr m p) (decr m q)"
   612 | "decr m (E p) = E (decr (Suc m) p)"
   613 | "decr m (A p) = A (decr (Suc m) p)"
   614 
   615 lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
   616   and nle: "m < length bs" 
   617   shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
   618   using bnd nb nle
   619 proof(induct p arbitrary: bs m rule: fm.induct)
   620   case (E p bs m) 
   621   {fix x
   622     from prems have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   623   and nle: "Suc m < length (x#bs)" by auto
   624     from prems(4)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   625   } thus ?case by auto 
   626 next
   627   case (A p bs m)  
   628   {fix x
   629     from prems have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   630   and nle: "Suc m < length (x#bs)" by auto
   631     from prems(4)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   632   } thus ?case by auto
   633 qed (auto simp add: decrtm removen_nth)
   634 
   635 primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
   636   "subst0 t T = T"
   637 | "subst0 t F = F"
   638 | "subst0 t (Lt a) = Lt (tmsubst0 t a)"
   639 | "subst0 t (Le a) = Le (tmsubst0 t a)"
   640 | "subst0 t (Eq a) = Eq (tmsubst0 t a)"
   641 | "subst0 t (NEq a) = NEq (tmsubst0 t a)"
   642 | "subst0 t (NOT p) = NOT (subst0 t p)"
   643 | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
   644 | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
   645 | "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
   646 | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
   647 | "subst0 t (E p) = E p"
   648 | "subst0 t (A p) = A p"
   649 
   650 lemma subst0: assumes qf: "qfree p"
   651   shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
   652 using qf tmsubst0[where x="x" and bs="bs" and t="t"]
   653 by (induct p rule: fm.induct, auto)
   654 
   655 lemma subst0_nb:
   656   assumes bp: "tmbound0 t" and qf: "qfree p"
   657   shows "bound0 (subst0 t p)"
   658 using qf tmsubst0_nb[OF bp] bp
   659 by (induct p rule: fm.induct, auto)
   660 
   661 primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
   662   "subst n t T = T"
   663 | "subst n t F = F"
   664 | "subst n t (Lt a) = Lt (tmsubst n t a)"
   665 | "subst n t (Le a) = Le (tmsubst n t a)"
   666 | "subst n t (Eq a) = Eq (tmsubst n t a)"
   667 | "subst n t (NEq a) = NEq (tmsubst n t a)"
   668 | "subst n t (NOT p) = NOT (subst n t p)"
   669 | "subst n t (And p q) = And (subst n t p) (subst n t q)"
   670 | "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
   671 | "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
   672 | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
   673 | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
   674 | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
   675 
   676 lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
   677   shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
   678   using nb nlm
   679 proof (induct p arbitrary: bs n t rule: fm.induct)
   680   case (E p bs n) 
   681   {fix x 
   682     from prems have bn: "boundslt (length (x#bs)) p" by simp 
   683       from prems have nlm: "Suc n \<le> length (x#bs)" by simp
   684     from prems(3)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
   685     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   686     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   687 thus ?case by simp 
   688 next
   689   case (A p bs n)   
   690   {fix x 
   691     from prems have bn: "boundslt (length (x#bs)) p" by simp 
   692       from prems have nlm: "Suc n \<le> length (x#bs)" by simp
   693     from prems(3)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
   694     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   695     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   696 thus ?case by simp 
   697 qed(auto simp add: tmsubst)
   698 
   699 lemma subst_nb: assumes tnb: "tmbound m t"
   700 shows "bound m (subst m t p)"
   701 using tnb tmsubst_nb incrtm0_tmbound
   702 by (induct p arbitrary: m t rule: fm.induct, auto)
   703 
   704 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
   705 by (induct p rule: not.induct, auto)
   706 lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
   707 by (induct p rule: not.induct, auto)
   708 lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
   709 by (induct p rule: not.induct, auto)
   710 lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
   711  by (induct p rule: not.induct, auto)
   712 
   713 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
   714 using conj_def by auto 
   715 lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
   716 using conj_def by auto 
   717 lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
   718 using conj_def by auto 
   719 lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
   720 using conj_def by auto 
   721 
   722 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
   723 using disj_def by auto 
   724 lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
   725 using disj_def by auto 
   726 lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
   727 using disj_def by auto 
   728 lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
   729 using disj_def by auto 
   730 
   731 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
   732 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
   733 lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
   734 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   735 lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
   736 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   737 lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
   738 using imp_def by auto 
   739 
   740 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
   741   by (unfold iff_def,cases "p=q", auto)
   742 lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
   743 using iff_def by (unfold iff_def,cases "p=q", auto)
   744 lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
   745 using iff_def by (unfold iff_def,cases "p=q", auto)
   746 lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
   747 using iff_def by auto 
   748 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
   749 by (induct p, simp_all)
   750 
   751 consts 
   752   isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
   753 recdef isatom "measure size"
   754   "isatom T = True"
   755   "isatom F = True"
   756   "isatom (Lt a) = True"
   757   "isatom (Le a) = True"
   758   "isatom (Eq a) = True"
   759   "isatom (NEq a) = True"
   760   "isatom p = False"
   761 
   762 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
   763 by (induct p, simp_all)
   764 
   765 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
   766   "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
   767   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
   768 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
   769   "evaldjf f ps \<equiv> foldr (djf f) ps F"
   770 
   771 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
   772 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
   773 (cases "f p", simp_all add: Let_def djf_def) 
   774 
   775 lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
   776   by(induct ps, simp_all add: evaldjf_def djf_Or)
   777 
   778 lemma evaldjf_bound0: 
   779   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
   780   shows "bound0 (evaldjf f xs)"
   781   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   782 
   783 lemma evaldjf_qf: 
   784   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
   785   shows "qfree (evaldjf f xs)"
   786   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   787 
   788 consts disjuncts :: "fm \<Rightarrow> fm list"
   789 recdef disjuncts "measure size"
   790   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
   791   "disjuncts F = []"
   792   "disjuncts p = [p]"
   793 
   794 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
   795 by(induct p rule: disjuncts.induct, auto)
   796 
   797 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
   798 proof-
   799   assume nb: "bound0 p"
   800   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
   801   thus ?thesis by (simp only: list_all_iff)
   802 qed
   803 
   804 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
   805 proof-
   806   assume qf: "qfree p"
   807   hence "list_all qfree (disjuncts p)"
   808     by (induct p rule: disjuncts.induct, auto)
   809   thus ?thesis by (simp only: list_all_iff)
   810 qed
   811 
   812 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   813   "DJ f p \<equiv> evaldjf f (disjuncts p)"
   814 
   815 lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
   816   and fF: "f F = F"
   817   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
   818 proof-
   819   have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
   820     by (simp add: DJ_def evaldjf_ex) 
   821   also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
   822   finally show ?thesis .
   823 qed
   824 
   825 lemma DJ_qf: assumes 
   826   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
   827   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
   828 proof(clarify)
   829   fix  p assume qf: "qfree p"
   830   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
   831   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
   832   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
   833   
   834   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
   835 qed
   836 
   837 lemma DJ_qe: assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
   838   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
   839 proof(clarify)
   840   fix p::fm and bs
   841   assume qf: "qfree p"
   842   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
   843   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
   844   have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
   845     by (simp add: DJ_def evaldjf_ex)
   846   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
   847   also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
   848   finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
   849 qed
   850 
   851 consts conjuncts :: "fm \<Rightarrow> fm list"
   852 
   853 recdef conjuncts "measure size"
   854   "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
   855   "conjuncts T = []"
   856   "conjuncts p = [p]"
   857 
   858 definition list_conj :: "fm list \<Rightarrow> fm" where
   859   "list_conj ps \<equiv> foldr conj ps T"
   860 
   861 definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   862   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
   863                    in conj (decr0 (list_conj yes)) (f (list_conj no)))"
   864 
   865 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
   866 proof-
   867   assume qf: "qfree p"
   868   hence "list_all qfree (conjuncts p)"
   869     by (induct p rule: conjuncts.induct, auto)
   870   thus ?thesis by (simp only: list_all_iff)
   871 qed
   872 
   873 lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
   874 by(induct p rule: conjuncts.induct, auto)
   875 
   876 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
   877 proof-
   878   assume nb: "bound0 p"
   879   hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
   880   thus ?thesis by (simp only: list_all_iff)
   881 qed
   882 
   883 fun islin :: "fm \<Rightarrow> bool" where
   884   "islin (And p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
   885 | "islin (Or p q) = (islin p \<and> islin q \<and> p \<noteq> T \<and> p \<noteq> F \<and> q \<noteq> T \<and> q \<noteq> F)"
   886 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   887 | "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   888 | "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   889 | "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   890 | "islin (NOT p) = False"
   891 | "islin (Imp p q) = False"
   892 | "islin (Iff p q) = False"
   893 | "islin p = bound0 p"
   894 
   895 lemma islin_stupid: assumes nb: "tmbound0 p"
   896   shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
   897   using nb by (cases p, auto, case_tac nat, auto)+
   898 
   899 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
   900 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
   901 definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
   902 definition "neq p = not (eq p)"
   903 
   904 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
   905   apply(simp add: lt_def)
   906   apply(cases p, simp_all)
   907   apply (case_tac poly, simp_all add: isnpoly_def)
   908   done
   909 
   910 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
   911   apply(simp add: le_def)
   912   apply(cases p, simp_all)
   913   apply (case_tac poly, simp_all add: isnpoly_def)
   914   done
   915 
   916 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
   917   apply(simp add: eq_def)
   918   apply(cases p, simp_all)
   919   apply (case_tac poly, simp_all add: isnpoly_def)
   920   done
   921 
   922 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
   923   by(simp add: neq_def eq)
   924 
   925 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
   926   apply (simp add: lt_def)
   927   apply (cases p, simp_all)
   928   apply (case_tac poly, simp_all)
   929   apply (case_tac nat, simp_all)
   930   done
   931 
   932 lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
   933   apply (simp add: le_def)
   934   apply (cases p, simp_all)
   935   apply (case_tac poly, simp_all)
   936   apply (case_tac nat, simp_all)
   937   done
   938 
   939 lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
   940   apply (simp add: eq_def)
   941   apply (cases p, simp_all)
   942   apply (case_tac poly, simp_all)
   943   apply (case_tac nat, simp_all)
   944   done
   945 
   946 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
   947   apply (simp add: neq_def eq_def)
   948   apply (cases p, simp_all)
   949   apply (case_tac poly, simp_all)
   950   apply (case_tac nat, simp_all)
   951   done
   952 
   953 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
   954 definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
   955 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
   956 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
   957 
   958 lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   959   shows "islin (simplt t)"
   960   unfolding simplt_def 
   961   using split0_nb0'
   962 by (auto simp add: lt_lin Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly])
   963   
   964 lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   965   shows "islin (simple t)"
   966   unfolding simple_def 
   967   using split0_nb0'
   968 by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] le_lin)
   969 lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   970   shows "islin (simpeq t)"
   971   unfolding simpeq_def 
   972   using split0_nb0'
   973 by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin)
   974 
   975 lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   976   shows "islin (simpneq t)"
   977   unfolding simpneq_def 
   978   using split0_nb0'
   979 by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] neq_lin)
   980 
   981 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
   982   by (cases "split0 s", auto)
   983 lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   984   and n: "allpolys isnpoly t"
   985   shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
   986   using n
   987   by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
   988 lemma simplt[simp]:
   989   shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
   990 proof-
   991   have n: "allpolys isnpoly (simptm t)" by simp
   992   let ?t = "simptm t"
   993   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
   994       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
   995       by (simp add: simplt_def Let_def split_def lt)}
   996   moreover
   997   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
   998     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
   999   }
  1000   ultimately show ?thesis by blast
  1001 qed
  1002 
  1003 lemma simple[simp]:
  1004   shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
  1005 proof-
  1006   have n: "allpolys isnpoly (simptm t)" by simp
  1007   let ?t = "simptm t"
  1008   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1009       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
  1010       by (simp add: simple_def Let_def split_def le)}
  1011   moreover
  1012   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1013     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
  1014   }
  1015   ultimately show ?thesis by blast
  1016 qed
  1017 
  1018 lemma simpeq[simp]:
  1019   shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
  1020 proof-
  1021   have n: "allpolys isnpoly (simptm t)" by simp
  1022   let ?t = "simptm t"
  1023   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1024       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
  1025       by (simp add: simpeq_def Let_def split_def)}
  1026   moreover
  1027   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1028     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
  1029   }
  1030   ultimately show ?thesis by blast
  1031 qed
  1032 
  1033 lemma simpneq[simp]:
  1034   shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
  1035 proof-
  1036   have n: "allpolys isnpoly (simptm t)" by simp
  1037   let ?t = "simptm t"
  1038   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1039       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
  1040       by (simp add: simpneq_def Let_def split_def )}
  1041   moreover
  1042   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1043     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
  1044   }
  1045   ultimately show ?thesis by blast
  1046 qed
  1047 
  1048 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
  1049   apply (simp add: lt_def)
  1050   apply (cases t, auto)
  1051   apply (case_tac poly, auto)
  1052   done
  1053 
  1054 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
  1055   apply (simp add: le_def)
  1056   apply (cases t, auto)
  1057   apply (case_tac poly, auto)
  1058   done
  1059 
  1060 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
  1061   apply (simp add: eq_def)
  1062   apply (cases t, auto)
  1063   apply (case_tac poly, auto)
  1064   done
  1065 
  1066 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
  1067   apply (simp add: neq_def eq_def)
  1068   apply (cases t, auto)
  1069   apply (case_tac poly, auto)
  1070   done
  1071 
  1072 lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1073   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
  1074   using split0 [of "simptm t" vs bs]
  1075 proof(simp add: simplt_def Let_def split_def)
  1076   assume nb: "tmbound0 t"
  1077   hence nb': "tmbound0 (simptm t)" by simp
  1078   let ?c = "fst (split0 (simptm t))"
  1079   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1080   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1081   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1082   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1083   from iffD1[OF isnpolyh_unique[OF ths] th]
  1084   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1085   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
  1086        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
  1087 qed
  1088 
  1089 lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1090   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
  1091   using split0 [of "simptm t" vs bs]
  1092 proof(simp add: simple_def Let_def split_def)
  1093   assume nb: "tmbound0 t"
  1094   hence nb': "tmbound0 (simptm t)" by simp
  1095   let ?c = "fst (split0 (simptm t))"
  1096   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1097   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1098   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1099   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1100   from iffD1[OF isnpolyh_unique[OF ths] th]
  1101   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1102   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
  1103        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
  1104 qed
  1105 
  1106 lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1107   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
  1108   using split0 [of "simptm t" vs bs]
  1109 proof(simp add: simpeq_def Let_def split_def)
  1110   assume nb: "tmbound0 t"
  1111   hence nb': "tmbound0 (simptm t)" by simp
  1112   let ?c = "fst (split0 (simptm t))"
  1113   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1114   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1115   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1116   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1117   from iffD1[OF isnpolyh_unique[OF ths] th]
  1118   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1119   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
  1120        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
  1121 qed
  1122 
  1123 lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1124   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
  1125   using split0 [of "simptm t" vs bs]
  1126 proof(simp add: simpneq_def Let_def split_def)
  1127   assume nb: "tmbound0 t"
  1128   hence nb': "tmbound0 (simptm t)" by simp
  1129   let ?c = "fst (split0 (simptm t))"
  1130   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1131   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1132   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1133   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1134   from iffD1[OF isnpolyh_unique[OF ths] th]
  1135   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1136   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
  1137        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
  1138 qed
  1139 
  1140 consts conjs   :: "fm \<Rightarrow> fm list"
  1141 recdef conjs "measure size"
  1142   "conjs (And p q) = (conjs p)@(conjs q)"
  1143   "conjs T = []"
  1144   "conjs p = [p]"
  1145 lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
  1146 by (induct p rule: conjs.induct, auto)
  1147 definition list_disj :: "fm list \<Rightarrow> fm" where
  1148   "list_disj ps \<equiv> foldr disj ps F"
  1149 
  1150 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
  1151   by (induct ps, auto simp add: list_conj_def)
  1152 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
  1153   by (induct ps, auto simp add: list_conj_def conj_qf)
  1154 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
  1155   by (induct ps, auto simp add: list_disj_def)
  1156 
  1157 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
  1158   unfolding conj_def by auto
  1159 
  1160 lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
  1161   apply (induct p rule: conjs.induct) 
  1162   apply (unfold conjs.simps)
  1163   apply (unfold set_append)
  1164   apply (unfold ball_Un)
  1165   apply (unfold bound.simps)
  1166   apply auto
  1167   done
  1168 
  1169 lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
  1170   apply (induct p rule: conjs.induct) 
  1171   apply (unfold conjs.simps)
  1172   apply (unfold set_append)
  1173   apply (unfold ball_Un)
  1174   apply (unfold boundslt.simps)
  1175   apply blast
  1176 by simp_all
  1177 
  1178 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
  1179   unfolding list_conj_def
  1180   by (induct ps, auto simp add: conj_boundslt)
  1181 
  1182 lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
  1183   shows "bound n (list_conj ps)"
  1184   using bnd
  1185   unfolding list_conj_def
  1186   by (induct ps, auto simp add: conj_nb)
  1187 
  1188 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
  1189 unfolding list_conj_def by (induct ps , auto)
  1190 
  1191 lemma CJNB_qe: 
  1192   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1193   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
  1194 proof(clarify)
  1195   fix bs p
  1196   assume qfp: "qfree p"
  1197   let ?cjs = "conjuncts p"
  1198   let ?yes = "fst (partition bound0 ?cjs)"
  1199   let ?no = "snd (partition bound0 ?cjs)"
  1200   let ?cno = "list_conj ?no"
  1201   let ?cyes = "list_conj ?yes"
  1202   have part: "partition bound0 ?cjs = (?yes,?no)" by simp
  1203   from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
  1204   hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
  1205   hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
  1206   from conjuncts_qf[OF qfp] partition_set[OF part] 
  1207   have " \<forall>q\<in> set ?no. qfree q" by auto
  1208   hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
  1209   with qe have cno_qf:"qfree (qe ?cno )" 
  1210     and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
  1211   from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
  1212     by (simp add: CJNB_def Let_def conj_qf split_def)
  1213   {fix bs
  1214     from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
  1215     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
  1216       using partition_set[OF part] by auto
  1217     finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
  1218   hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
  1219   also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
  1220     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
  1221   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
  1222     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
  1223   also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
  1224     using qe[rule_format, OF no_qf] by auto
  1225   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" 
  1226     by (simp add: Let_def CJNB_def split_def)
  1227   with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
  1228 qed
  1229 
  1230 consts simpfm :: "fm \<Rightarrow> fm"
  1231 recdef simpfm "measure fmsize"
  1232   "simpfm (Lt t) = simplt (simptm t)"
  1233   "simpfm (Le t) = simple (simptm t)"
  1234   "simpfm (Eq t) = simpeq(simptm t)"
  1235   "simpfm (NEq t) = simpneq(simptm t)"
  1236   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
  1237   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
  1238   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
  1239   "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
  1240   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
  1241   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
  1242   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
  1243   "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
  1244   "simpfm (NOT (Eq t)) = simpneq t"
  1245   "simpfm (NOT (NEq t)) = simpeq t"
  1246   "simpfm (NOT (Le t)) = simplt (Neg t)"
  1247   "simpfm (NOT (Lt t)) = simple (Neg t)"
  1248   "simpfm (NOT (NOT p)) = simpfm p"
  1249   "simpfm (NOT T) = F"
  1250   "simpfm (NOT F) = T"
  1251   "simpfm p = p"
  1252 
  1253 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
  1254 by(induct p arbitrary: bs rule: simpfm.induct, auto)
  1255 
  1256 lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1257   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
  1258 by (induct p rule: simpfm.induct, auto)
  1259 
  1260 lemma lt_qf[simp]: "qfree (lt t)"
  1261   apply (cases t, auto simp add: lt_def)
  1262   by (case_tac poly, auto)
  1263 
  1264 lemma le_qf[simp]: "qfree (le t)"
  1265   apply (cases t, auto simp add: le_def)
  1266   by (case_tac poly, auto)
  1267 
  1268 lemma eq_qf[simp]: "qfree (eq t)"
  1269   apply (cases t, auto simp add: eq_def)
  1270   by (case_tac poly, auto)
  1271 
  1272 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
  1273 
  1274 lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
  1275 lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
  1276 lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
  1277 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
  1278 
  1279 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
  1280 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
  1281 
  1282 lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
  1283 lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
  1284 
  1285 lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1286   shows "qfree p \<Longrightarrow> islin (simpfm p)" 
  1287   apply (induct p rule: simpfm.induct)
  1288   apply (simp_all add: conj_lin disj_lin)
  1289   done
  1290 
  1291 consts prep :: "fm \<Rightarrow> fm"
  1292 recdef prep "measure fmsize"
  1293   "prep (E T) = T"
  1294   "prep (E F) = F"
  1295   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
  1296   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
  1297   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
  1298   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
  1299   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
  1300   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
  1301   "prep (E p) = E (prep p)"
  1302   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
  1303   "prep (A p) = prep (NOT (E (NOT p)))"
  1304   "prep (NOT (NOT p)) = prep p"
  1305   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
  1306   "prep (NOT (A p)) = prep (E (NOT p))"
  1307   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
  1308   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
  1309   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
  1310   "prep (NOT p) = not (prep p)"
  1311   "prep (Or p q) = disj (prep p) (prep q)"
  1312   "prep (And p q) = conj (prep p) (prep q)"
  1313   "prep (Imp p q) = prep (Or (NOT p) q)"
  1314   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
  1315   "prep p = p"
  1316 (hints simp add: fmsize_pos)
  1317 lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
  1318 by (induct p arbitrary: bs rule: prep.induct, auto)
  1319 
  1320 
  1321 
  1322   (* Generic quantifier elimination *)
  1323 consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
  1324 recdef qelim "measure fmsize"
  1325   "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
  1326   "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
  1327   "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
  1328   "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
  1329   "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
  1330   "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
  1331   "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
  1332   "qelim p = (\<lambda> y. simpfm p)"
  1333 
  1334 
  1335 lemma qelim:
  1336   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1337   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
  1338 using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
  1339 by (induct p rule: qelim.induct) auto
  1340 
  1341 subsection{* Core Procedure *}
  1342 
  1343 consts 
  1344   plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
  1345   minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
  1346 recdef minusinf "measure size"
  1347   "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
  1348   "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
  1349   "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1350   "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1351   "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
  1352   "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
  1353   "minusinf p = p"
  1354 
  1355 recdef plusinf "measure size"
  1356   "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
  1357   "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
  1358   "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1359   "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1360   "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
  1361   "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
  1362   "plusinf p = p"
  1363 
  1364 lemma minusinf_inf: assumes lp:"islin p"
  1365   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1366   using lp
  1367 proof (induct p rule: minusinf.induct)
  1368   case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1369 next
  1370   case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1371 next
  1372   case (3 c e) hence nbe: "tmbound0 e" by simp
  1373   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1374   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1375   let ?c = "Ipoly vs c"
  1376   let ?e = "Itm vs (y#bs) e"
  1377   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1378   moreover {assume "?c = 0" hence ?case 
  1379       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1380   moreover {assume cp: "?c > 0"
  1381     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1382         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1383       hence "?c * x + ?e < 0" by simp
  1384       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1385         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1386   moreover {assume cp: "?c < 0"
  1387     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1388         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1389       hence "?c * x + ?e > 0" by simp
  1390       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1391         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1392   ultimately show ?case by blast
  1393 next
  1394   case (4 c e)  hence nbe: "tmbound0 e" by simp
  1395   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1396   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1397   let ?c = "Ipoly vs c"
  1398   let ?e = "Itm vs (y#bs) e"
  1399   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1400   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1401   moreover {assume cp: "?c > 0"
  1402     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1403         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1404       hence "?c * x + ?e < 0" by simp
  1405       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1406         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1407   moreover {assume cp: "?c < 0"
  1408     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1409         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1410       hence "?c * x + ?e > 0" by simp
  1411       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1412         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1413   ultimately show ?case by blast
  1414 next
  1415   case (5 c e)  hence nbe: "tmbound0 e" by simp
  1416   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1417   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1418   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
  1419   let ?c = "Ipoly vs c"
  1420   let ?e = "Itm vs (y#bs) e"
  1421   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1422   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1423   moreover {assume cp: "?c > 0"
  1424     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1425         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1426       hence "?c * x + ?e < 0" by simp
  1427       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1428         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1429   moreover {assume cp: "?c < 0"
  1430     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1431         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1432       hence "?c * x + ?e > 0" by simp
  1433       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1434         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1435   ultimately show ?case by blast
  1436 next
  1437   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1438   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1439   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1440   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1441   let ?c = "Ipoly vs c"
  1442   let ?e = "Itm vs (y#bs) e"
  1443   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1444   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1445   moreover {assume cp: "?c > 0"
  1446     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1447         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1448       hence "?c * x + ?e < 0" by simp
  1449       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1450         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1451   moreover {assume cp: "?c < 0"
  1452     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1453         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1454       hence "?c * x + ?e > 0" by simp
  1455       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1456         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1457   ultimately show ?case by blast
  1458 qed (auto)
  1459 
  1460 lemma plusinf_inf: assumes lp:"islin p"
  1461   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1462   using lp
  1463 proof (induct p rule: plusinf.induct)
  1464   case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1465 next
  1466   case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1467 next
  1468   case (3 c e) hence nbe: "tmbound0 e" by simp
  1469   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1470   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1471   let ?c = "Ipoly vs c"
  1472   let ?e = "Itm vs (y#bs) e"
  1473   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1474   moreover {assume "?c = 0" hence ?case 
  1475       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1476   moreover {assume cp: "?c > 0"
  1477     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e" 
  1478         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1479       hence "?c * x + ?e > 0" by simp
  1480       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1481         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1482   moreover {assume cp: "?c < 0"
  1483     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1484         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1485       hence "?c * x + ?e < 0" by simp
  1486       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1487         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1488   ultimately show ?case by blast
  1489 next
  1490   case (4 c e)  hence nbe: "tmbound0 e" by simp
  1491   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1492   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1493   let ?c = "Ipoly vs c"
  1494   let ?e = "Itm vs (y#bs) e"
  1495   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1496   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1497   moreover {assume cp: "?c > 0"
  1498     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1499         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1500       hence "?c * x + ?e > 0" by simp
  1501       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1502         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1503   moreover {assume cp: "?c < 0"
  1504     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1505         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1506       hence "?c * x + ?e < 0" by simp
  1507       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1508         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1509   ultimately show ?case by blast
  1510 next
  1511   case (5 c e)  hence nbe: "tmbound0 e" by simp
  1512   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1513   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1514   note eqs = lt[OF nc(1), where ?'a = 'a] lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
  1515   let ?c = "Ipoly vs c"
  1516   let ?e = "Itm vs (y#bs) e"
  1517   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1518   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1519   moreover {assume cp: "?c > 0"
  1520     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1521         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1522       hence "?c * x + ?e > 0" by simp
  1523       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1524         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1525   moreover {assume cp: "?c < 0"
  1526     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1527         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1528       hence "?c * x + ?e < 0" by simp
  1529       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1530         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1531   ultimately show ?case by blast
  1532 next
  1533   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1534   from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1535   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1536   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1537   let ?c = "Ipoly vs c"
  1538   let ?e = "Itm vs (y#bs) e"
  1539   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1540   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1541   moreover {assume cp: "?c > 0"
  1542     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1543         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1544       hence "?c * x + ?e > 0" by simp
  1545       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1546         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1547   moreover {assume cp: "?c < 0"
  1548     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1549         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1550       hence "?c * x + ?e < 0" by simp
  1551       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1552         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1553   ultimately show ?case by blast
  1554 qed (auto)
  1555 
  1556 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)" 
  1557   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1558 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)" 
  1559   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1560 
  1561 lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
  1562   shows "\<exists>x. Ifm vs (x#bs) p"
  1563 proof-
  1564   from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1565   have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
  1566   from minusinf_inf[OF lp, where bs="bs"] 
  1567   obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
  1568   from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
  1569   moreover have "z - 1 < z" by simp
  1570   ultimately show ?thesis using z_def by auto
  1571 qed
  1572 
  1573 lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
  1574   shows "\<exists>x. Ifm vs (x#bs) p"
  1575 proof-
  1576   from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1577   have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
  1578   from plusinf_inf[OF lp, where bs="bs"] 
  1579   obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
  1580   from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
  1581   moreover have "z + 1 > z" by simp
  1582   ultimately show ?thesis using z_def by auto
  1583 qed
  1584 
  1585 fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
  1586   "uset (And p q) = uset p @ uset q"
  1587 | "uset (Or p q) = uset p @ uset q"
  1588 | "uset (Eq (CNP 0 a e))  = [(a,e)]"
  1589 | "uset (Le (CNP 0 a e))  = [(a,e)]"
  1590 | "uset (Lt (CNP 0 a e))  = [(a,e)]"
  1591 | "uset (NEq (CNP 0 a e)) = [(a,e)]"
  1592 | "uset p = []"
  1593 
  1594 lemma uset_l:
  1595   assumes lp: "islin p"
  1596   shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
  1597 using lp by(induct p rule: uset.induct,auto)
  1598 
  1599 lemma minusinf_uset0:
  1600   assumes lp: "islin p"
  1601   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
  1602   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1603   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c" 
  1604 proof-
  1605   have "\<exists> (c,s) \<in> set (uset p). (Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" 
  1606     using lp nmi ex
  1607     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
  1608     apply (auto simp add: linorder_not_less order_le_less)
  1609     done 
  1610   then obtain c s where csU: "(c,s) \<in> set (uset p)" and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s)" by blast
  1611   hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
  1612     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
  1613     by (auto simp add: mult_commute del: divide_minus_left)
  1614   thus ?thesis using csU by auto
  1615 qed
  1616 
  1617 lemma minusinf_uset:
  1618   assumes lp: "islin p"
  1619   and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
  1620   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1621   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c" 
  1622 proof-
  1623   from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))" 
  1624     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
  1625   from minusinf_uset0[OF lp nmi' ex] 
  1626   obtain c s where csU: "(c,s) \<in> set (uset p)" and th: "x \<ge> - Itm vs (x#bs) s / Ipoly vs c" by blast
  1627   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1628   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1629 qed
  1630 
  1631 
  1632 lemma plusinf_uset0:
  1633   assumes lp: "islin p"
  1634   and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
  1635   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1636   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c" 
  1637 proof-
  1638   have "\<exists> (c,s) \<in> set (uset p). (Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s)" 
  1639     using lp nmi ex
  1640     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
  1641     apply (auto simp add: linorder_not_less order_le_less)
  1642     done 
  1643   then obtain c s where csU: "(c,s) \<in> set (uset p)" and x: "(Ipoly vs c < 0 \<and> Ipoly vs c * x \<ge> - Itm vs (x#bs) s) \<or>  (Ipoly vs c > 0 \<and> Ipoly vs c * x \<le> - Itm vs (x#bs) s)" by blast
  1644   hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
  1645     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
  1646     by (auto simp add: mult_commute del: divide_minus_left)
  1647   thus ?thesis using csU by auto
  1648 qed
  1649 
  1650 lemma plusinf_uset:
  1651   assumes lp: "islin p"
  1652   and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
  1653   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1654   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c" 
  1655 proof-
  1656   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))" 
  1657     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
  1658   from plusinf_uset0[OF lp nmi' ex] 
  1659   obtain c s where csU: "(c,s) \<in> set (uset p)" and th: "x \<le> - Itm vs (x#bs) s / Ipoly vs c" by blast
  1660   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1661   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1662 qed
  1663 
  1664 lemma lin_dense: 
  1665   assumes lp: "islin p"
  1666   and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (c,t). - Itm vs (x#bs) t / Ipoly vs c) ` set (uset p)" 
  1667   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
  1668   and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
  1669   and ly: "l < y" and yu: "y < u"
  1670   shows "Ifm vs (y#bs) p"
  1671 using lp px noS
  1672 proof (induct p rule: islin.induct) 
  1673   case (5 c s)
  1674   from "5.prems" 
  1675   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1676     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
  1677     and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
  1678   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1679   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1680   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1681   moreover
  1682   {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
  1683   moreover
  1684   {assume c: "?N c > 0"
  1685       from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1686       have px': "x < - ?Nt x s / ?N c" 
  1687         by (auto simp add: not_less field_simps) 
  1688     {assume y: "y < - ?Nt x s / ?N c" 
  1689       hence "y * ?N c < - ?Nt x s"
  1690         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1691       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1692       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1693     moreover
  1694     {assume y: "y > -?Nt x s / ?N c" 
  1695       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1696       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1697       with lx px' have "False" by simp  hence ?case by simp }
  1698     ultimately have ?case using ycs by blast
  1699   }
  1700   moreover
  1701   {assume c: "?N c < 0"
  1702       from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]  
  1703       have px': "x > - ?Nt x s / ?N c" 
  1704         by (auto simp add: not_less field_simps) 
  1705     {assume y: "y > - ?Nt x s / ?N c" 
  1706       hence "y * ?N c < - ?Nt x s"
  1707         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1708       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1709       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1710     moreover
  1711     {assume y: "y < -?Nt x s / ?N c" 
  1712       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1713       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1714       with xu px' have "False" by simp  hence ?case by simp }
  1715     ultimately have ?case using ycs by blast
  1716   }
  1717   ultimately show ?case by blast
  1718 next
  1719   case (6 c s)
  1720   from "6.prems" 
  1721   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1722     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
  1723     and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
  1724   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1725   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1726   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1727   moreover
  1728   {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
  1729   moreover
  1730   {assume c: "?N c > 0"
  1731       from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1732       have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps) 
  1733     {assume y: "y < - ?Nt x s / ?N c" 
  1734       hence "y * ?N c < - ?Nt x s"
  1735         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1736       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1737       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1738     moreover
  1739     {assume y: "y > -?Nt x s / ?N c" 
  1740       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1741       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1742       with lx px' have "False" by simp  hence ?case by simp }
  1743     ultimately have ?case using ycs by blast
  1744   }
  1745   moreover
  1746   {assume c: "?N c < 0"
  1747       from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]  
  1748       have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps) 
  1749     {assume y: "y > - ?Nt x s / ?N c" 
  1750       hence "y * ?N c < - ?Nt x s"
  1751         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1752       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1753       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1754     moreover
  1755     {assume y: "y < -?Nt x s / ?N c" 
  1756       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1757       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1758       with xu px' have "False" by simp  hence ?case by simp }
  1759     ultimately have ?case using ycs by blast
  1760   }
  1761   ultimately show ?case by blast
  1762 next
  1763     case (3 c s)
  1764   from "3.prems" 
  1765   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1766     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
  1767     and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
  1768   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1769   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1770   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1771   moreover
  1772   {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
  1773   moreover
  1774   {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
  1775     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1776     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1777     {assume y: "y < -?Nt x s / ?N c" 
  1778       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1779       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1780       with xu px' have "False" by simp  hence ?case by simp }
  1781     moreover
  1782     {assume y: "y > -?Nt x s / ?N c" 
  1783       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1784       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1785       with lx px' have "False" by simp  hence ?case by simp }
  1786     ultimately have ?case using ycs by blast
  1787   }
  1788   moreover
  1789   {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
  1790     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1791     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1792     {assume y: "y < -?Nt x s / ?N c" 
  1793       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1794       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1795       with xu px' have "False" by simp  hence ?case by simp }
  1796     moreover
  1797     {assume y: "y > -?Nt x s / ?N c" 
  1798       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1799       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1800       with lx px' have "False" by simp  hence ?case by simp }
  1801     ultimately have ?case using ycs by blast
  1802   }
  1803   ultimately show ?case by blast
  1804 next
  1805     case (4 c s)
  1806   from "4.prems" 
  1807   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1808     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
  1809     and noS: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<noteq> - Itm vs (x # bs) s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp_all
  1810   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1811   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1812   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
  1813   moreover
  1814   {assume "?N c = 0" hence ?case using px by (simp add: tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"])}
  1815   moreover
  1816   {assume c: "?N c \<noteq> 0"
  1817     from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
  1818       by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
  1819   ultimately show ?case by blast
  1820 qed (auto simp add: nth_pos2 tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])
  1821 
  1822 lemma one_plus_one_pos[simp]: "(1::'a::{linordered_field}) + 1 > 0"
  1823 proof-
  1824   have op: "(1::'a) > 0" by simp
  1825   from add_pos_pos[OF op op] show ?thesis . 
  1826 qed
  1827 
  1828 lemma one_plus_one_nonzero[simp]: "(1::'a::{linordered_field}) + 1 \<noteq> 0" 
  1829   using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le) 
  1830 
  1831 lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{linordered_field})" 
  1832 proof-
  1833   have "(u + u) = (1 + 1) * u" by (simp add: field_simps)
  1834   hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
  1835   with nonzero_mult_divide_cancel_left[OF one_plus_one_nonzero, of u] show ?thesis by simp
  1836 qed
  1837 
  1838 lemma inf_uset:
  1839   assumes lp: "islin p"
  1840   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
  1841   and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
  1842   and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
  1843   shows "\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). ?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / (1 + 1)) p" 
  1844 proof-
  1845   let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
  1846   let ?N = "Ipoly vs"
  1847   let ?U = "set (uset p)"
  1848   from ex obtain a where pa: "?I a p" by blast
  1849   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
  1850   have nmi': "\<not> (?I a (?M p))" by simp
  1851   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
  1852   have npi': "\<not> (?I a (?P p))" by simp
  1853   have "\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). ?I ((- ?Nt a t/?N c + - ?Nt a s /?N d) / (1 + 1)) p"
  1854   proof-
  1855     let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
  1856     have fM: "finite ?M" by auto
  1857     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] 
  1858     have "\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). a \<le> - ?Nt x t / ?N c \<and> a \<ge> - ?Nt x s / ?N d" by blast
  1859     then obtain "c" "t" "d" "s" where 
  1860       ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U" 
  1861       and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
  1862     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 
  1863     have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
  1864     from ctU have Mne: "?M \<noteq> {}" by auto
  1865     hence Une: "?U \<noteq> {}" by simp
  1866     let ?l = "Min ?M"
  1867     let ?u = "Max ?M"
  1868     have linM: "?l \<in> ?M" using fM Mne by simp
  1869     have uinM: "?u \<in> ?M" using fM Mne by simp
  1870     have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
  1871     have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto 
  1872     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
  1873     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
  1874     have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
  1875     have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
  1876     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
  1877     have "(\<exists> s\<in> ?M. ?I s p) \<or> 
  1878       (\<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)" .
  1879     moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
  1880       hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
  1881       then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
  1882       from half_sum_eq[of u] pu tuu 
  1883       have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / (1 + 1)) p" by simp
  1884       with tuU have ?thesis by blast}
  1885     moreover{
  1886       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"
  1887       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
  1888         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"
  1889         by blast
  1890       from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
  1891       then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
  1892       from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
  1893       then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
  1894       from t1x xt2 have t1t2: "t1 < t2" by simp
  1895       let ?u = "(t1 + t2) / (1 + 1)"
  1896       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
  1897       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
  1898       with t1uU t2uU t1u t2u have ?thesis by blast}
  1899     ultimately show ?thesis by blast
  1900   qed
  1901   then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U" 
  1902     and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / (1 + 1)) p" by blast
  1903   from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
  1904   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
  1905     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
  1906   have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / (1 + 1)) p" by simp
  1907   with lnU smU
  1908   show ?thesis by auto
  1909 qed
  1910 
  1911     (* The Ferrante - Rackoff Theorem *)
  1912 
  1913 theorem fr_eq: 
  1914   assumes lp: "islin p"
  1915   shows "(\<exists> x. Ifm vs (x#bs) p) = ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> (\<exists> (n,t) \<in> set (uset p). \<exists> (m,s) \<in> set (uset p). Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /(1 + 1))#bs) p))"
  1916   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  1917 proof
  1918   assume px: "\<exists> x. ?I x p"
  1919   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  1920   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  1921   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  1922     from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
  1923   ultimately show "?D" by blast
  1924 next
  1925   assume "?D" 
  1926   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  1927   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  1928   moreover {assume f:"?F" hence "?E" by blast}
  1929   ultimately show "?E" by blast
  1930 qed
  1931 
  1932 section{* First implementation : Naive by encoding all case splits locally *}
  1933 definition "msubsteq c t d s a r = 
  1934   evaldjf (split conj) 
  1935   [(let cd = c *\<^sub>p d in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  1936    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1937    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1938    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
  1939 
  1940 lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  1941   shows "bound0 (msubsteq c t d s a r)"
  1942 proof-
  1943   have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (NEq (CP cd), Eq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  1944    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1945    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1946    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
  1947     using lp by (simp add: Let_def t s )
  1948   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
  1949 qed
  1950 
  1951 lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
  1952   shows "Ifm vs (x#bs) (msubsteq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Eq (CNP 0 a r))" (is "?lhs = ?rhs")
  1953 proof-
  1954   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  1955   let ?N = "\<lambda>p. Ipoly vs p"
  1956   let ?c = "?N c"
  1957   let ?d = "?N d"
  1958   let ?t = "?Nt x t"
  1959   let ?s = "?Nt x s"
  1960   let ?a = "?N a"
  1961   let ?r = "?Nt x r"
  1962   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  1963   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  1964   have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" by auto
  1965   moreover
  1966   {assume c: "?c = 0" and d: "?d=0"
  1967     hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
  1968   moreover 
  1969   {assume c: "?c = 0" and d: "?d\<noteq>0"
  1970     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  1971     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1972     also have "\<dots> \<longleftrightarrow> ?a * (-?s / ((1 + 1)*?d)) + ?r = 0" by (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
  1973     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) = 0" 
  1974       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  1975     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r= 0"
  1976       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  1977     
  1978     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r = 0" using d by simp 
  1979     finally have ?thesis using c d 
  1980       apply (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
  1981       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1982       apply simp
  1983       done}
  1984   moreover
  1985   {assume c: "?c \<noteq> 0" and d: "?d=0"
  1986     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  1987     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1988     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r = 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  1989     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) = 0" 
  1990       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  1991     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r= 0"
  1992       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  1993     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r = 0" using c by simp 
  1994     finally have ?thesis using c d 
  1995       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
  1996       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1997       apply simp
  1998       done }
  1999   moreover
  2000   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  2001     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2002     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2003       by (simp add: field_simps)
  2004     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  2005     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r = 0" 
  2006       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2007     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) =0 "
  2008       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2009     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r =0" 
  2010       using nonzero_mult_divide_cancel_left [OF dc] c d
  2011       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2012     finally  have ?thesis using c d 
  2013       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
  2014       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2015       apply (simp add: field_simps)
  2016       done }
  2017   ultimately show ?thesis by blast
  2018 qed
  2019 
  2020 
  2021 definition "msubstneq c t d s a r = 
  2022   evaldjf (split conj) 
  2023   [(let cd = c *\<^sub>p d in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2024    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2025    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2026    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
  2027 
  2028 lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2029   shows "bound0 (msubstneq c t d s a r)"
  2030 proof-
  2031   have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (NEq (CP cd), NEq (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))), 
  2032     (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2033     (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2034     (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
  2035     using lp by (simp add: Let_def t s )
  2036   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
  2037 qed
  2038 
  2039 lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
  2040   shows "Ifm vs (x#bs) (msubstneq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (NEq (CNP 0 a r))" (is "?lhs = ?rhs")
  2041 proof-
  2042   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  2043   let ?N = "\<lambda>p. Ipoly vs p"
  2044   let ?c = "?N c"
  2045   let ?d = "?N d"
  2046   let ?t = "?Nt x t"
  2047   let ?s = "?Nt x s"
  2048   let ?a = "?N a"
  2049   let ?r = "?Nt x r"
  2050   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2051   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2052   have cd_cs: "?c * ?d \<noteq> 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d \<noteq> 0) \<or> (?c \<noteq> 0 \<and> ?d = 0)" by auto
  2053   moreover
  2054   {assume c: "?c = 0" and d: "?d=0"
  2055     hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
  2056   moreover 
  2057   {assume c: "?c = 0" and d: "?d\<noteq>0"
  2058     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  2059     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2060     also have "\<dots> \<longleftrightarrow> ?a * (-?s / ((1 + 1)*?d)) + ?r \<noteq> 0" by (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
  2061     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) \<noteq> 0" 
  2062       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  2063     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r\<noteq> 0"
  2064       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  2065     
  2066     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r \<noteq> 0" using d by simp 
  2067     finally have ?thesis using c d 
  2068       apply (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
  2069       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2070       apply simp
  2071       done}
  2072   moreover
  2073   {assume c: "?c \<noteq> 0" and d: "?d=0"
  2074     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  2075     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2076     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  2077     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) \<noteq> 0" 
  2078       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  2079     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r \<noteq> 0"
  2080       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  2081     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r \<noteq> 0" using c by simp 
  2082     finally have ?thesis using c d 
  2083       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
  2084       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2085       apply simp
  2086       done }
  2087   moreover
  2088   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  2089     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2090     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2091       by (simp add: field_simps)
  2092     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2093     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r \<noteq> 0" 
  2094       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2095     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) \<noteq> 0 "
  2096       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2097     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r \<noteq> 0" 
  2098       using nonzero_mult_divide_cancel_left[OF dc] c d
  2099       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2100     finally  have ?thesis using c d 
  2101       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
  2102       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2103       apply (simp add: field_simps)
  2104       done }
  2105   ultimately show ?thesis by blast
  2106 qed
  2107 
  2108 definition "msubstlt c t d s a r = 
  2109   evaldjf (split conj) 
  2110   [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2111   (let cd = c *\<^sub>p d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2112    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2113    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2114    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2115    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2116    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"
  2117 
  2118 lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2119   shows "bound0 (msubstlt c t d s a r)"
  2120 proof-
  2121   have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Lt (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2122   (let cd = c *\<^sub>p d in (lt (CP cd), Lt (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2123    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2124    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2125    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2126    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2127    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
  2128     using lp by (simp add: Let_def t s lt_nb )
  2129   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
  2130 qed
  2131 
  2132 
  2133 lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))" 
  2134   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow> 
  2135   Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Lt (CNP 0 a r))" (is "?lhs = ?rhs")
  2136 proof-
  2137   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2138   let ?N = "\<lambda>p. Ipoly vs p"
  2139   let ?c = "?N c"
  2140   let ?d = "?N d"
  2141   let ?t = "?Nt x t"
  2142   let ?s = "?Nt x s"
  2143   let ?a = "?N a"
  2144   let ?r = "?Nt x r"
  2145   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2146   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2147   have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" by auto
  2148   moreover
  2149   {assume c: "?c=0" and d: "?d=0"
  2150     hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
  2151   moreover
  2152   {assume dc: "?c*?d > 0" 
  2153     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2154     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2155     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2156     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2157     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2158       by (simp add: field_simps)
  2159     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2160     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2161       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2162     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) < 0"
  2163       
  2164       using dc' dc'' mult_less_cancel_left_disj[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2165     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r < 0" 
  2166       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2167       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2168     finally  have ?thesis using dc c d  nc nd dc'
  2169       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) 
  2170     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2171     by (simp add: field_simps order_less_not_sym[OF dc])}
  2172   moreover
  2173   {assume dc: "?c*?d < 0" 
  2174 
  2175     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2176       by (simp add: mult_less_0_iff field_simps) 
  2177     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2178     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2179     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2180       by (simp add: field_simps)
  2181     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2182     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2183       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2184 
  2185     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) > 0"
  2186       
  2187       using dc' order_less_not_sym[OF dc'] mult_less_cancel_left_disj[of "(1 + 1) * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r"] by simp
  2188     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r < 0" 
  2189       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2190       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2191     finally  have ?thesis using dc c d  nc nd
  2192       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) 
  2193       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2194       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2195   moreover
  2196   {assume c: "?c > 0" and d: "?d=0"  
  2197     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2198     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2199     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2200     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2201     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2202     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) < 0"
  2203       using c mult_less_cancel_left_disj[of "(1 + 1) * ?c" "?a* (- ?t / ((1 + 1)*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
  2204     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r < 0" 
  2205       using nonzero_mult_divide_cancel_left[OF c'] c
  2206       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2207     finally have ?thesis using c d nc nd 
  2208       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2209       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2210       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2211       by (simp add: field_simps )  }
  2212   moreover
  2213   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2214     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2215     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2216     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2217     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2218     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) > 0"
  2219       using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_less_cancel_left_disj[of "(1 + 1) * ?c" 0 "?a* (- ?t / ((1 + 1)*?c))+ ?r"] by simp
  2220     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r < 0" 
  2221       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2222         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2223     finally have ?thesis using c d nc nd 
  2224       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2225       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2226       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2227       by (simp add: field_simps )    }
  2228   moreover
  2229   moreover
  2230   {assume c: "?c = 0" and d: "?d>0"  
  2231     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2232     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2233     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2234     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2235     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2236     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) < 0"
  2237       using d mult_less_cancel_left_disj[of "(1 + 1) * ?d" "?a* (- ?s / ((1 + 1)*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
  2238     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r < 0" 
  2239       using nonzero_mult_divide_cancel_left[OF d'] d
  2240       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2241     finally have ?thesis using c d nc nd 
  2242       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2243       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2244       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2245       by (simp add: field_simps)  }
  2246   moreover
  2247   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2248     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2249     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2250     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2251     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2252     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) > 0"
  2253       using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_less_cancel_left_disj[of "(1 + 1) * ?d" 0 "?a* (- ?s / ((1 + 1)*?d))+ ?r"] by simp
  2254     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r < 0" 
  2255       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2256         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2257     finally have ?thesis using c d nc nd 
  2258       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2259       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2260       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2261       by (simp add: field_simps )    }
  2262 ultimately show ?thesis by blast
  2263 qed
  2264 
  2265 definition "msubstle c t d s a r = 
  2266   evaldjf (split conj) 
  2267   [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2268   (let cd = c *\<^sub>p d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2269    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2270    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2271    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2272    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2273    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"
  2274 
  2275 lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2276   shows "bound0 (msubstle c t d s a r)"
  2277 proof-
  2278   have th: "\<forall>x\<in> set [(let cd = c *\<^sub>p d in (lt (CP (~\<^sub>p cd)), Le (Add (Mul (~\<^sub>p a) (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2279   (let cd = c *\<^sub>p d in (lt (CP cd), Le (Sub (Mul a (Add (Mul d t) (Mul c s))) (Mul (2\<^sub>p *\<^sub>p cd) r)))),
  2280    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2281    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2282    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2283    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2284    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
  2285     using lp by (simp add: Let_def t s lt_nb )
  2286   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
  2287 qed
  2288 
  2289 lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))" 
  2290   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow> 
  2291   Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Le (CNP 0 a r))" (is "?lhs = ?rhs")
  2292 proof-
  2293   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2294   let ?N = "\<lambda>p. Ipoly vs p"
  2295   let ?c = "?N c"
  2296   let ?d = "?N d"
  2297   let ?t = "?Nt x t"
  2298   let ?s = "?Nt x s"
  2299   let ?a = "?N a"
  2300   let ?r = "?Nt x r"
  2301   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2302   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2303   have cd_cs: "?c * ?d < 0 \<or> ?c * ?d > 0 \<or> (?c = 0 \<and> ?d = 0) \<or> (?c = 0 \<and> ?d < 0) \<or> (?c = 0 \<and> ?d > 0) \<or> (?c < 0 \<and> ?d = 0) \<or> (?c > 0 \<and> ?d = 0)" by auto
  2304   moreover
  2305   {assume c: "?c=0" and d: "?d=0"
  2306     hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
  2307   moreover
  2308   {assume dc: "?c*?d > 0" 
  2309     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2310     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2311     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2312     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2313     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2314       by (simp add: field_simps)
  2315     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2316     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2317       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2318     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) <= 0"
  2319       
  2320       using dc' dc'' mult_le_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2321     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r <= 0" 
  2322       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2323       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2324     finally  have ?thesis using dc c d  nc nd dc'
  2325       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) 
  2326     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2327     by (simp add: field_simps order_less_not_sym[OF dc])}
  2328   moreover
  2329   {assume dc: "?c*?d < 0" 
  2330 
  2331     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2332       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
  2333     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2334     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2335     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2336       by (simp add: field_simps)
  2337     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2338     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2339       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2340 
  2341     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) >= 0"
  2342       
  2343       using dc' order_less_not_sym[OF dc'] mult_le_cancel_left[of "(1 + 1) * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r"] by simp
  2344     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r <= 0" 
  2345       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2346       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2347     finally  have ?thesis using dc c d  nc nd
  2348       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm) 
  2349       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2350       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2351   moreover
  2352   {assume c: "?c > 0" and d: "?d=0"  
  2353     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2354     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2355     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2356     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2357     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2358     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) <= 0"
  2359       using c mult_le_cancel_left[of "(1 + 1) * ?c" "?a* (- ?t / ((1 + 1)*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
  2360     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r <= 0" 
  2361       using nonzero_mult_divide_cancel_left[OF c'] c
  2362       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2363     finally have ?thesis using c d nc nd 
  2364       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2365       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2366       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2367       by (simp add: field_simps )  }
  2368   moreover
  2369   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2370     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2371     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2372     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2373     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2374     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) >= 0"
  2375       using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_le_cancel_left[of "(1 + 1) * ?c" 0 "?a* (- ?t / ((1 + 1)*?c))+ ?r"] by simp
  2376     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r <= 0" 
  2377       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2378         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2379     finally have ?thesis using c d nc nd 
  2380       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2381       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2382       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2383       by (simp add: field_simps )    }
  2384   moreover
  2385   moreover
  2386   {assume c: "?c = 0" and d: "?d>0"  
  2387     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2388     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2389     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2390     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2391     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2392     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) <= 0"
  2393       using d mult_le_cancel_left[of "(1 + 1) * ?d" "?a* (- ?s / ((1 + 1)*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
  2394     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r <= 0" 
  2395       using nonzero_mult_divide_cancel_left[OF d'] d
  2396       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2397     finally have ?thesis using c d nc nd 
  2398       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2399       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2400       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2401       by (simp add: field_simps )  }
  2402   moreover
  2403   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2404     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2405     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2406     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2407     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2408     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) >= 0"
  2409       using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_le_cancel_left[of "(1 + 1) * ?d" 0 "?a* (- ?s / ((1 + 1)*?d))+ ?r"] by simp
  2410     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r <= 0" 
  2411       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2412         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2413     finally have ?thesis using c d nc nd 
  2414       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2415       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2416       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2417       by (simp add: field_simps )    }
  2418 ultimately show ?thesis by blast
  2419 qed
  2420 
  2421 
  2422 fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
  2423   "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
  2424 | "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
  2425 | "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
  2426 | "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
  2427 | "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
  2428 | "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
  2429 | "msubst p ((c,t),(d,s)) = p"
  2430 
  2431 lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
  2432   shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) p"
  2433   using lp
  2434 by (induct p rule: islin.induct, auto simp add: tmbound0_I[where b="(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) + - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) /(1 + 1)" and b'=x and bs = bs and vs=vs] bound0_I[where b="(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) + - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) /(1 + 1)" and b'=x and bs = bs and vs=vs] msubsteq msubstneq msubstlt[OF nc nd] msubstle[OF nc nd])
  2435 
  2436 lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
  2437   shows "bound0 (msubst p ((c,t),(d,s)))"
  2438   using lp t s
  2439   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
  2440 
  2441 lemma fr_eq_msubst: 
  2442   assumes lp: "islin p"
  2443   shows "(\<exists> x. Ifm vs (x#bs) p) = ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> (\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst p ((c,t),(d,s)))))"
  2444   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  2445 proof-
  2446 from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2447 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2448   and pts: "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p"
  2449   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2450   from msubst_I[OF lp norm, of vs x bs t s] pts
  2451   have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
  2452 moreover
  2453 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2454   and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
  2455   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2456   from msubst_I[OF lp norm, of vs x bs t s] pts
  2457   have "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p" ..}
  2458 ultimately have th': "(\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p) \<longleftrightarrow> ?F" by blast
  2459 from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
  2460 qed 
  2461 
  2462 text {* Rest of the implementation *}
  2463 
  2464 primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where
  2465   "alluopairs [] = []"
  2466 | "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
  2467 
  2468 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
  2469 by (induct xs, auto)
  2470 
  2471 lemma alluopairs_set:
  2472   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
  2473 by (induct xs, auto)
  2474 
  2475 lemma alluopairs_ex:
  2476   assumes Pc: "\<forall> x \<in> set xs. \<forall>y\<in> set xs. P x y = P y x"
  2477   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
  2478 proof
  2479   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
  2480   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
  2481   from alluopairs_set[OF x y] P Pc x y show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
  2482     by auto
  2483 next
  2484   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
  2485   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
  2486   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
  2487   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
  2488 qed
  2489 
  2490 lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  2491 using Nat.gr0_conv_Suc
  2492 by clarsimp
  2493 
  2494 lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
  2495   apply (induct xs, auto) done
  2496 
  2497 consts remdps:: "'a list \<Rightarrow> 'a list"
  2498 
  2499 recdef remdps "measure size"
  2500   "remdps [] = []"
  2501   "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
  2502 (hints simp add: filter_length[rule_format])
  2503 
  2504 lemma remdps_set[simp]: "set (remdps xs) = set xs"
  2505   by (induct xs rule: remdps.induct, auto)
  2506 
  2507 lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  2508   shows "qfree p \<Longrightarrow> islin (simpfm p)"
  2509   by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
  2510 
  2511 definition 
  2512   "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2513   in if (mp = T \<or> pp = T) then T 
  2514      else (let U = alluopairs (remdps (uset  q))
  2515            in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
  2516 
  2517 lemma ferrack: 
  2518   assumes qf: "qfree p"
  2519   shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
  2520   (is "_ \<and> (?rhs = ?lhs)")
  2521 proof-
  2522   let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
  2523   let ?N = "\<lambda> t. Ipoly vs t"
  2524   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2525   let ?q = "simpfm p" 
  2526   let ?U = "remdps(uset ?q)"
  2527   let ?Up = "alluopairs ?U"
  2528   let ?mp = "minusinf ?q"
  2529   let ?pp = "plusinf ?q"
  2530   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2531   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2532   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2533   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2534   from uset_l[OF lq] have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
  2535     by simp
  2536   {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
  2537     from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
  2538     from msubst_I[OF lq norm, of vs x bs t s] msubst_I[OF lq norm(2,1), of vs x bs s t]
  2539     have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
  2540   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (msubst ?q (x, y)) \<longleftrightarrow> ?I (msubst ?q (y, x))" by clarsimp
  2541   {fix x assume xUp: "x \<in> set ?Up" 
  2542     then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U" 
  2543       and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto  
  2544     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] 
  2545     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
  2546     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] 
  2547     have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
  2548   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
  2549   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
  2550   with mp_nb pp_nb 
  2551   have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
  2552   from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
  2553   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2554   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists>(c, t)\<in>set ?U. \<exists>(d, s)\<in>set ?U. ?I (msubst (simpfm p) ((c, t), d, s)))" using fr_eq_msubst[OF lq, of vs bs x] by simp
  2555   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists> (x,y) \<in> set ?Up. ?I ((simpfm o (msubst ?q)) (x,y)))" using alluopairs_ex[OF th0] by simp
  2556   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)" 
  2557     by (simp add: evaldjf_ex)
  2558   also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
  2559   also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
  2560     apply (simp add: ferrack_def Let_def)
  2561     by (cases "?mp = T \<or> ?pp = T", auto)
  2562   finally show ?thesis using thqf by blast
  2563 qed
  2564 
  2565 definition "frpar p = simpfm (qelim p ferrack)"
  2566 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
  2567 proof-
  2568   from ferrack have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack p) \<and> Ifm vs bs (ferrack p) = Ifm vs bs (E p)" by blast
  2569   from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
  2570 qed
  2571 
  2572 declare polyadd.simps[code]
  2573 lemma [simp,code]: "polyadd (CN c n p, CN c' n' p') = 
  2574     (if n < n' then CN (polyadd(c,CN c' n' p')) n p
  2575      else if n'<n then CN (polyadd(CN c n p, c')) n' p'
  2576      else (let cc' = polyadd (c,c') ; 
  2577                pp' = polyadd (p,p')
  2578            in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
  2579   by (simp add: Let_def stupid)
  2580 
  2581 
  2582 section{* Second implemenation: Case splits not local *}
  2583 
  2584 lemma fr_eq2:  assumes lp: "islin p"
  2585   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow> 
  2586    ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> 
  2587     (Ifm vs (0#bs) p) \<or> 
  2588     (\<exists> (n,t) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t /  (Ipoly vs n * (1 + 1)))#bs) p) \<or> 
  2589     (\<exists> (n,t) \<in> set (uset p). \<exists> (m,s) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ipoly vs m \<noteq> 0 \<and> Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /(1 + 1))#bs) p))"
  2590   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
  2591 proof
  2592   assume px: "\<exists> x. ?I x p"
  2593   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  2594   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  2595   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  2596     from inf_uset[OF lp nmi npi, OF px] 
  2597     obtain c t d s where ct: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" "?I ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / ((1\<Colon>'a) + (1\<Colon>'a))) p"
  2598       by auto
  2599     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2600     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2601     let ?s = "Itm vs (x # bs) s"
  2602     let ?t = "Itm vs (x # bs) t"
  2603     have eq2: "\<And>(x::'a). x + x = (1 + 1) * x"
  2604       by  (simp add: field_simps)
  2605     {assume "?c = 0 \<and> ?d = 0"
  2606       with ct have ?D by simp}
  2607     moreover
  2608     {assume z: "?c = 0" "?d \<noteq> 0"
  2609       from z have ?D using ct by auto}
  2610     moreover
  2611     {assume z: "?c \<noteq> 0" "?d = 0"
  2612       with ct have ?D by auto }
  2613     moreover
  2614     {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
  2615       from z have ?F using ct
  2616         apply - apply (rule bexI[where x = "(c,t)"], simp_all)
  2617         by (rule bexI[where x = "(d,s)"], simp_all)
  2618       hence ?D by blast}
  2619     ultimately have ?D by auto}
  2620   ultimately show "?D" by blast
  2621 next
  2622   assume "?D" 
  2623   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  2624   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  2625   moreover {assume f:"?F" hence "?E" by blast}
  2626   ultimately show "?E" by blast
  2627 qed
  2628 
  2629 definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
  2630 definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
  2631 definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
  2632 definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
  2633 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
  2634 
  2635 lemma msubsteq2: 
  2636   assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
  2637   shows "Ifm vs (x#bs) (msubsteq2 c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Eq (CNP 0 a b))" (is "?lhs = ?rhs")
  2638   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2639   by (simp add: msubsteq2_def field_simps)
  2640 
  2641 lemma msubstltpos: 
  2642   assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
  2643   shows "Ifm vs (x#bs) (msubstltpos c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Lt (CNP 0 a b))" (is "?lhs = ?rhs")
  2644   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2645   by (simp add: msubstltpos_def field_simps)
  2646 
  2647 lemma msubstlepos: 
  2648   assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
  2649   shows "Ifm vs (x#bs) (msubstlepos c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))" (is "?lhs = ?rhs")
  2650   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2651   by (simp add: msubstlepos_def field_simps)
  2652 
  2653 lemma msubstltneg: 
  2654   assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
  2655   shows "Ifm vs (x#bs) (msubstltneg c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Lt (CNP 0 a b))" (is "?lhs = ?rhs")
  2656   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2657   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
  2658 
  2659 lemma msubstleneg: 
  2660   assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
  2661   shows "Ifm vs (x#bs) (msubstleneg c t a b) = Ifm vs (((Itm vs (x#bs) t /  Ipoly vs c ))#bs) (Le (CNP 0 a b))" (is "?lhs = ?rhs")
  2662   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2663   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
  2664 
  2665 fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2666   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
  2667 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
  2668 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2669 | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2670 | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
  2671 | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
  2672 | "msubstpos p c t = p"
  2673     
  2674 lemma msubstpos_I: 
  2675   assumes lp: "islin p" and pos: "Ipoly vs c > 0"
  2676   shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2677   using lp pos
  2678   by (induct p rule: islin.induct, auto simp add: msubsteq2 msubstltpos[OF pos] msubstlepos[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
  2679 
  2680 fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2681   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
  2682 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
  2683 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2684 | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2685 | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
  2686 | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
  2687 | "msubstneg p c t = p"
  2688 
  2689 lemma msubstneg_I: 
  2690   assumes lp: "islin p" and pos: "Ipoly vs c < 0"
  2691   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2692   using lp pos
  2693   by (induct p rule: islin.induct, auto simp add: msubsteq2 msubstltneg[OF pos] msubstleneg[OF pos] tmbound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] bound0_I[of _ vs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" bs x] field_simps)
  2694 
  2695 
  2696 definition "msubst2 p c t = disj (conj (lt (CP (polyneg c))) (simpfm (msubstpos p c t))) (conj (lt (CP c)) (simpfm (msubstneg p c t)))"
  2697 
  2698 lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
  2699   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2700 proof-
  2701   let ?c = "Ipoly vs c"
  2702   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))" 
  2703     by (simp_all add: polyneg_norm)
  2704   from nz have "?c > 0 \<or> ?c < 0" by arith
  2705   moreover
  2706   {assume c: "?c < 0"
  2707     from c msubstneg_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"]
  2708     have ?thesis by (auto simp add: msubst2_def)}
  2709   moreover
  2710   {assume c: "?c > 0"
  2711     from c msubstpos_I[OF lp c, of x bs t] lt[OF anc(1), of vs "x#bs"] lt[OF anc(2), of vs "x#bs"]
  2712     have ?thesis by (auto simp add: msubst2_def)}
  2713   ultimately show ?thesis by blast
  2714 qed
  2715 
  2716 term msubsteq2
  2717 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
  2718   by (simp add: msubsteq2_def)
  2719 
  2720 lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
  2721   by (simp add: msubstltpos_def)
  2722 lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
  2723   by (simp add: msubstltneg_def)
  2724 
  2725 lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
  2726   by (simp add: msubstlepos_def)
  2727 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
  2728   by (simp add: msubstleneg_def)
  2729 
  2730 lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
  2731   shows "bound0 (msubstpos p c t)"
  2732 using lp tnb
  2733 by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
  2734 
  2735 lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2736   shows "bound0 (msubstneg p c t)"
  2737 using lp tnb
  2738 by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
  2739 
  2740 lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2741   shows "bound0 (msubst2 p c t)"
  2742 using lp tnb
  2743 by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
  2744     
  2745 lemma of_int2: "of_int 2 = 1 + 1"
  2746 proof-
  2747   have "(2::int) = 1 + 1" by simp
  2748   hence "of_int 2 = of_int (1 + 1)" by simp
  2749   thus ?thesis unfolding of_int_add by simp
  2750 qed
  2751 
  2752 lemma of_int_minus2: "of_int (-2) = - (1 + 1)"
  2753 proof-
  2754   have th: "(-2::int) = - 2" by simp
  2755   show ?thesis unfolding th by (simp only: of_int_minus of_int2)
  2756 qed
  2757 
  2758 
  2759 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
  2760   by (induct p rule: islin.induct, auto simp add: bound0_qf)
  2761 lemma fr_eq_msubst2: 
  2762   assumes lp: "islin p"
  2763   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow> ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> Ifm vs (x#bs) (subst0 (CP 0\<^sub>p) p) \<or> (\<exists>(n, t)\<in>set (uset p). Ifm vs (x# bs) (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<or> (\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))))"
  2764   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
  2765 proof-
  2766   from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2767   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2768   have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
  2769   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
  2770   
  2771   have eq1: "(\<exists>(n, t)\<in>set (uset p). ?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<longleftrightarrow> (\<exists>(n, t)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p)"
  2772   proof-
  2773     {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
  2774       from H(1) th have "isnpoly n" by blast
  2775       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
  2776       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
  2777         by (simp add: polyneg_norm nn)
  2778       hence nn2: "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>n \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" using H(2) nn' nn 
  2779         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
  2780       from msubst2[OF lp nn nn2(1), of x bs t]
  2781       have "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p"
  2782         using H(2) nn2 by (simp add: of_int_minus2 del: minus_add_distrib)}
  2783     moreover
  2784     {fix n t assume H: "(n, t)\<in>set (uset p)" "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p"
  2785       from H(1) th have "isnpoly n" by blast
  2786       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
  2787         using H(2) by (simp_all add: polymul_norm n2)
  2788       from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)" using H(2,3) by (simp add: of_int_minus2 del: minus_add_distrib)}
  2789     ultimately show ?thesis by blast
  2790   qed
  2791   have eq2: "(\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))) \<longleftrightarrow> (\<exists>(n, t)\<in>set (uset p).
  2792      \<exists>(m, s)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p)" 
  2793   proof-
  2794     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2795      "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
  2796       from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2797       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" 
  2798         by (simp_all add: polymul_norm n2)
  2799       have stupid: "allpolys isnpoly (CP (~\<^sub>p (C (-2, 1) *\<^sub>p c *\<^sub>p d)))" "allpolys isnpoly (CP ((C (-2, 1) *\<^sub>p c *\<^sub>p d)))"
  2800         by (simp_all add: polyneg_norm nn)
  2801       have nn': "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
  2802         using H(3) by (auto simp add: msubst2_def lt[OF stupid(1)]  lt[OF stupid(2)] zero_less_mult_iff mult_less_0_iff)
  2803       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
  2804       have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p" 
  2805         apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)
  2806         by (simp add: mult_commute)}
  2807     moreover
  2808     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2809       "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p"
  2810      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2811       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" "\<lparr>(C (-2, 1) *\<^sub>p c*\<^sub>p d)\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
  2812         using H(3,4) by (simp_all add: polymul_norm n2)
  2813       from msubst2[OF lp nn, of x bs ] H(3,4,5) 
  2814       have "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))" apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)by (simp add: mult_commute)}
  2815     ultimately show ?thesis by blast
  2816   qed
  2817   from fr_eq2[OF lp, of vs bs x] show ?thesis
  2818     unfolding eq0 eq1 eq2 by blast  
  2819 qed
  2820 
  2821 definition 
  2822 "ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2823  in if (mp = T \<or> pp = T) then T 
  2824   else (let U = remdps (uset  q)
  2825     in decr0 (list_disj [mp, pp, simpfm (subst0 (CP 0\<^sub>p) q), evaldjf (\<lambda>(c,t). msubst2 q (c *\<^sub>p C (-2, 1)) t) U, 
  2826    evaldjf (\<lambda>((b,a),(d,c)). msubst2 q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs U)]))"
  2827 
  2828 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
  2829 
  2830 lemma ferrack2: assumes qf: "qfree p"
  2831   shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
  2832   (is "_ \<and> (?rhs = ?lhs)")
  2833 proof-
  2834   let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
  2835   let ?N = "\<lambda> t. Ipoly vs t"
  2836   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2837   let ?q = "simpfm p" 
  2838   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
  2839   let ?U = "remdps(uset ?q)"
  2840   let ?Up = "alluopairs ?U"
  2841   let ?mp = "minusinf ?q"
  2842   let ?pp = "plusinf ?q"
  2843   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2844   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2845   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2846   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2847   from uset_l[OF lq] have U_l: "\<forall>(c, s)\<in>set ?U. isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
  2848     by simp
  2849   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)" 
  2850   proof-
  2851     {fix c t assume ct: "(c,t) \<in> set ?U"
  2852       hence tnb: "tmbound0 t" using U_l by blast
  2853       from msubst2_nb[OF lq tnb]
  2854       have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
  2855     thus ?thesis by auto
  2856   qed
  2857   have bnd1: "\<forall>x \<in> set ?Up. bound0 ((\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) x)" 
  2858   proof-
  2859     {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
  2860       from badc U_l alluopairs_set1[of ?U] 
  2861       have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
  2862       from msubst2_nb[OF lq nb] have "bound0 ((\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) ((b,a),(d,c)))" by simp}
  2863     thus ?thesis by auto
  2864   qed
  2865   have stupid: "bound0 F" by simp
  2866   let ?R = "list_disj [?mp, ?pp, simpfm (subst0 (CP 0\<^sub>p) ?q), evaldjf (\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) ?U, 
  2867    evaldjf (\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) (alluopairs ?U)]"
  2868   from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
  2869   have nb: "bound0 ?R "
  2870     by (simp add: list_disj_def disj_nb0 simpfm_bound0)
  2871   let ?s = "\<lambda>((b,a),(d,c)). msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))"
  2872 
  2873   {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
  2874     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))" 
  2875       by auto (simp add: isnpoly_def)
  2876     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
  2877       using norm by (simp_all add: polymul_norm)
  2878     have stupid: "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p b*\<^sub>p d))" "allpolys isnpoly (CP (C (-2, 1) *\<^sub>p d*\<^sub>p b))" "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p b*\<^sub>p d)))" "allpolys isnpoly (CP (~\<^sub>p(C (-2, 1) *\<^sub>p d*\<^sub>p b)))"
  2879       by (simp_all add: polyneg_norm norm2)
  2880     have "?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))) = ?I (msubst2 ?q (C (-2, 1) *\<^sub>p d*\<^sub>p b) (Add (Mul b c) (Mul d a)))" (is "?lhs \<longleftrightarrow> ?rhs")
  2881     proof
  2882       assume H: ?lhs
  2883       hence z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" 
  2884         by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
  2885       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2886         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2887       show ?rhs by (simp add: field_simps)
  2888     next
  2889       assume H: ?rhs
  2890       hence z: "\<lparr>C (-2, 1) *\<^sub>p b *\<^sub>p d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>C (-2, 1) *\<^sub>p d *\<^sub>p b\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" 
  2891         by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
  2892       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2893         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2894       show ?lhs by (simp add: field_simps)
  2895     qed}
  2896   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
  2897     by clarsimp
  2898 
  2899   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2900   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists>(b, a)\<in>set ?U. \<exists>(d, c)\<in>set ?U. ?I (msubst2 ?q (C (-2, 1) *\<^sub>p b*\<^sub>p d) (Add (Mul d a) (Mul b c))))"
  2901     using fr_eq_msubst2[OF lq, of vs bs x] by simp
  2902   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists> x\<in>set ?U. \<exists> y \<in>set ?U. ?I (?s (x,y)))"
  2903     by (simp add: split_def)
  2904   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (subst0 (CP 0\<^sub>p) ?q) \<or> (\<exists>(n,t) \<in> set ?U. ?I (msubst2 ?q (n *\<^sub>p C (-2, 1)) t)) \<or> (\<exists> (x,y) \<in> set ?Up. ?I (?s (x,y)))"
  2905     using alluopairs_ex[OF th0] by simp 
  2906   also have "\<dots> \<longleftrightarrow> ?I ?R" 
  2907     by (simp add: list_disj_def evaldjf_ex split_def)
  2908   also have "\<dots> \<longleftrightarrow> ?rhs"
  2909     unfolding ferrack2_def
  2910     apply (cases "?mp = T") 
  2911     apply (simp add: list_disj_def)
  2912     apply (cases "?pp = T") 
  2913     apply (simp add: list_disj_def)
  2914     by (simp_all add: Let_def decr0[OF nb])
  2915   finally show ?thesis using decr0_qf[OF nb]  
  2916     by (simp  add: ferrack2_def Let_def)
  2917 qed
  2918 
  2919 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
  2920 proof-
  2921   from ferrack2 have th: "\<forall>bs p. qfree p \<longrightarrow> qfree (ferrack2 p) \<and> Ifm vs bs (ferrack2 p) = Ifm vs bs (E p)" by blast
  2922   from qelim[OF th, of "prep p" bs] 
  2923 show ?thesis  unfolding frpar2_def by (auto simp add: prep)
  2924 qed
  2925 
  2926 ML {* 
  2927 structure ReflectedFRPar = 
  2928 struct
  2929 
  2930 val bT = HOLogic.boolT;
  2931 fun num rT x = HOLogic.mk_number rT x;
  2932 fun rrelT rT = [rT,rT] ---> rT;
  2933 fun rrT rT = [rT, rT] ---> bT;
  2934 fun divt rT = Const(@{const_name Rings.divide},rrelT rT);
  2935 fun timest rT = Const(@{const_name Groups.times},rrelT rT);
  2936 fun plust rT = Const(@{const_name Groups.plus},rrelT rT);
  2937 fun minust rT = Const(@{const_name Groups.minus},rrelT rT);
  2938 fun uminust rT = Const(@{const_name Groups.uminus}, rT --> rT);
  2939 fun powt rT = Const(@{const_name "power"}, [rT,@{typ "nat"}] ---> rT);
  2940 val brT = [bT, bT] ---> bT;
  2941 val nott = @{term "Not"};
  2942 val conjt = @{term HOL.conj};
  2943 val disjt = @{term HOL.disj};
  2944 val impt = @{term HOL.implies};
  2945 val ifft = @{term "op = :: bool => _"}
  2946 fun llt rT = Const(@{const_name Orderings.less},rrT rT);
  2947 fun lle rT = Const(@{const_name Orderings.less},rrT rT);
  2948 fun eqt rT = Const(@{const_name HOL.eq},rrT rT);
  2949 fun rz rT = Const(@{const_name Groups.zero},rT);
  2950 
  2951 fun dest_nat t = case t of
  2952   Const (@{const_name Suc}, _) $ t' => 1 + dest_nat t'
  2953 | _ => (snd o HOLogic.dest_number) t;
  2954 
  2955 fun num_of_term m t = 
  2956  case t of
  2957    Const(@{const_name Groups.uminus},_)$t => @{code poly.Neg} (num_of_term m t)
  2958  | Const(@{const_name Groups.plus},_)$a$b => @{code poly.Add} (num_of_term m a, num_of_term m b)
  2959  | Const(@{const_name Groups.minus},_)$a$b => @{code poly.Sub} (num_of_term m a, num_of_term m b)
  2960  | Const(@{const_name Groups.times},_)$a$b => @{code poly.Mul} (num_of_term m a, num_of_term m b)
  2961  | Const(@{const_name Power.power},_)$a$n => @{code poly.Pw} (num_of_term m a, dest_nat n)
  2962  | Const(@{const_name Rings.divide},_)$a$b => @{code poly.C} (HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
  2963  | _ => (@{code poly.C} (HOLogic.dest_number t |> snd,1) 
  2964          handle TERM _ => @{code poly.Bound} (AList.lookup (op aconv) m t |> the));
  2965 
  2966 fun tm_of_term m m' t = 
  2967  case t of
  2968    Const(@{const_name Groups.uminus},_)$t => @{code Neg} (tm_of_term m m' t)
  2969  | Const(@{const_name Groups.plus},_)$a$b => @{code Add} (tm_of_term m m' a, tm_of_term m m' b)
  2970  | Const(@{const_name Groups.minus},_)$a$b => @{code Sub} (tm_of_term m m' a, tm_of_term m m' b)
  2971  | Const(@{const_name Groups.times},_)$a$b => @{code Mul} (num_of_term m' a, tm_of_term m m' b)
  2972  | _ => (@{code CP} (num_of_term m' t) 
  2973          handle TERM _ => @{code Bound} (AList.lookup (op aconv) m t |> the)
  2974               | Option => @{code Bound} (AList.lookup (op aconv) m t |> the));
  2975 
  2976 fun term_of_num T m t = 
  2977  case t of
  2978   @{code poly.C} (a,b) => (if b = 1 then num T a else if b=0 then (rz T) 
  2979                                         else (divt T) $ num T a $ num T b)
  2980 | @{code poly.Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2981 | @{code poly.Add} (a,b) => (plust T)$(term_of_num T m a)$(term_of_num T m b)
  2982 | @{code poly.Mul} (a,b) => (timest T)$(term_of_num T m a)$(term_of_num T m b)
  2983 | @{code poly.Sub} (a,b) => (minust T)$(term_of_num T m a)$(term_of_num T m b)
  2984 | @{code poly.Neg} a => (uminust T)$(term_of_num T m a)
  2985 | @{code poly.Pw} (a,n) => (powt T)$(term_of_num T m t)$(HOLogic.mk_number HOLogic.natT n)
  2986 | @{code poly.CN} (c,n,p) => term_of_num T m (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)))
  2987 | _ => error "term_of_num: Unknown term";
  2988 
  2989 fun term_of_tm T m m' t = 
  2990  case t of
  2991   @{code CP} p => term_of_num T m' p
  2992 | @{code Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2993 | @{code Add} (a,b) => (plust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2994 | @{code Mul} (a,b) => (timest T)$(term_of_num T m' a)$(term_of_tm T m m' b)
  2995 | @{code Sub} (a,b) => (minust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2996 | @{code Neg} a => (uminust T)$(term_of_tm T m m' a)
  2997 | @{code CNP} (n,c,p) => term_of_tm T m m' (@{code Add}
  2998      (@{code Mul} (c, @{code Bound} n), p))
  2999 | _ => error "term_of_tm: Unknown term";
  3000 
  3001 fun fm_of_term m m' fm = 
  3002  case fm of
  3003     Const(@{const_name True},_) => @{code T}
  3004   | Const(@{const_name False},_) => @{code F}
  3005   | Const(@{const_name Not},_)$p => @{code NOT} (fm_of_term m m' p)
  3006   | Const(@{const_name HOL.conj},_)$p$q => @{code And} (fm_of_term m m' p, fm_of_term m m' q)
  3007   | Const(@{const_name HOL.disj},_)$p$q => @{code Or} (fm_of_term m m' p, fm_of_term m m' q)
  3008   | Const(@{const_name HOL.implies},_)$p$q => @{code Imp} (fm_of_term m m' p, fm_of_term m m' q)
  3009   | Const(@{const_name HOL.eq},ty)$p$q => 
  3010        if domain_type ty = bT then @{code Iff} (fm_of_term m m' p, fm_of_term m m' q)
  3011        else @{code Eq} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  3012   | Const(@{const_name Orderings.less},_)$p$q => 
  3013         @{code Lt} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  3014   | Const(@{const_name Orderings.less_eq},_)$p$q => 
  3015         @{code Le} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  3016   | Const(@{const_name Ex},_)$Abs(xn,xT,p) => 
  3017      let val (xn', p') =  variant_abs (xn,xT,p)
  3018          val x = Free(xn',xT)
  3019          fun incr i = i + 1
  3020          val m0 = (x,0):: (map (apsnd incr) m)
  3021       in @{code E} (fm_of_term m0 m' p') end
  3022   | Const(@{const_name All},_)$Abs(xn,xT,p) => 
  3023      let val (xn', p') =  variant_abs (xn,xT,p)
  3024          val x = Free(xn',xT)
  3025          fun incr i = i + 1
  3026          val m0 = (x,0):: (map (apsnd incr) m)
  3027       in @{code A} (fm_of_term m0 m' p') end
  3028   | _ => error "fm_of_term";
  3029 
  3030 
  3031 fun term_of_fm T m m' t = 
  3032   case t of
  3033     @{code T} => Const(@{const_name True},bT)
  3034   | @{code F} => Const(@{const_name False},bT)
  3035   | @{code NOT} p => nott $ (term_of_fm T m m' p)
  3036   | @{code And} (p,q) => conjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3037   | @{code Or} (p,q) => disjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3038   | @{code Imp} (p,q) => impt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3039   | @{code Iff} (p,q) => ifft $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3040   | @{code Lt} p => (llt T) $ (term_of_tm T m m' p) $ (rz T)
  3041   | @{code Le} p => (lle T) $ (term_of_tm T m m' p) $ (rz T)
  3042   | @{code Eq} p => (eqt T) $ (term_of_tm T m m' p) $ (rz T)
  3043   | @{code NEq} p => nott $ ((eqt T) $ (term_of_tm T m m' p) $ (rz T))
  3044   | _ => error "term_of_fm: quantifiers!!!!???";
  3045 
  3046 fun frpar_oracle (T,m, m', fm) = 
  3047  let 
  3048    val t = HOLogic.dest_Trueprop fm
  3049    val im = 0 upto (length m - 1)
  3050    val im' = 0 upto (length m' - 1)   
  3051  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  3052                                                      (@{code frpar} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  3053  end;
  3054 
  3055 fun frpar_oracle2 (T,m, m', fm) = 
  3056  let 
  3057    val t = HOLogic.dest_Trueprop fm
  3058    val im = 0 upto (length m - 1)
  3059    val im' = 0 upto (length m' - 1)   
  3060  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  3061                                                      (@{code frpar2} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  3062  end;
  3063 
  3064 end;
  3065 
  3066 
  3067 *}
  3068 
  3069 oracle frpar_oracle = {* fn (ty, ts, ts', ct) => 
  3070  let 
  3071   val thy = Thm.theory_of_cterm ct
  3072  in cterm_of thy (ReflectedFRPar.frpar_oracle (ty,ts, ts', term_of ct))
  3073  end *}
  3074 
  3075 oracle frpar_oracle2 = {* fn (ty, ts, ts', ct) => 
  3076  let 
  3077   val thy = Thm.theory_of_cterm ct
  3078  in cterm_of thy (ReflectedFRPar.frpar_oracle2 (ty,ts, ts', term_of ct))
  3079  end *}
  3080 
  3081 ML{* 
  3082 structure FRParTac = 
  3083 struct
  3084 
  3085 fun frpar_tac T ps ctxt i = 
  3086  Object_Logic.full_atomize_tac i
  3087  THEN (fn st =>
  3088   let
  3089     val g = List.nth (cprems_of st, i - 1)
  3090     val thy = ProofContext.theory_of ctxt
  3091     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3092     val th = frpar_oracle (T, fs,ps, (* Pattern.eta_long [] *)g)
  3093   in rtac (th RS iffD2) i st end);
  3094 
  3095 fun frpar2_tac T ps ctxt i = 
  3096  Object_Logic.full_atomize_tac i
  3097  THEN (fn st =>
  3098   let
  3099     val g = List.nth (cprems_of st, i - 1)
  3100     val thy = ProofContext.theory_of ctxt
  3101     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3102     val th = frpar_oracle2 (T, fs,ps, (* Pattern.eta_long [] *)g)
  3103   in rtac (th RS iffD2) i st end);
  3104 
  3105 end;
  3106 
  3107 *}
  3108 
  3109 method_setup frpar = {*
  3110 let
  3111  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3112  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3113  val parsN = "pars"
  3114  val typN = "type"
  3115  val any_keyword = keyword parsN || keyword typN
  3116  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3117  val cterms = thms >> map Drule.dest_term;
  3118  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3119  val typ = Scan.unless any_keyword Args.typ
  3120 in
  3121  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3122   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar_tac T ps ctxt))
  3123 end
  3124 *} "Parametric QE for linear Arithmetic over fields, Version 1"
  3125 
  3126 method_setup frpar2 = {*
  3127 let
  3128  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3129  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3130  val parsN = "pars"
  3131  val typN = "type"
  3132  val any_keyword = keyword parsN || keyword typN
  3133  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3134  val cterms = thms >> map Drule.dest_term;
  3135  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3136  val typ = Scan.unless any_keyword Args.typ
  3137 in
  3138  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3139   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar2_tac T ps ctxt))
  3140 end
  3141 *} "Parametric QE for linear Arithmetic over fields, Version 2"
  3142 
  3143 
  3144 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3145   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3146   apply (simp add: field_simps)
  3147   apply (rule spec[where x=y])
  3148   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3149   by simp
  3150 
  3151 text{* Collins/Jones Problem *}
  3152 (*
  3153 lemma "\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
  3154 proof-
  3155   have "(\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
  3156 by (simp add: field_simps)
  3157 have "?rhs"
  3158 
  3159   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "a::'a::{linordered_field_inverse_zero, number_ring}" "b::'a::{linordered_field_inverse_zero, number_ring}")
  3160   apply (simp add: field_simps)
  3161 oops
  3162 *)
  3163 (*
  3164 lemma "ALL (x::'a::{linordered_field_inverse_zero, number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
  3165 apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3166 oops
  3167 *)
  3168 
  3169 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3170   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3171   apply (simp add: field_simps)
  3172   apply (rule spec[where x=y])
  3173   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3174   by simp
  3175 
  3176 text{* Collins/Jones Problem *}
  3177 
  3178 (*
  3179 lemma "\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0"
  3180 proof-
  3181   have "(\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < (2 - 3*r) *(a^2 + b^2) + (2*a)*r \<and> (2 - 3*r) *(a^2 + b^2) + 4*a*r - 2*a - r < 0) \<longleftrightarrow> (\<exists>(r::'a::{linordered_field_inverse_zero, number_ring}). 0 < r \<and> r < 1 \<and> 0 < 2 *(a^2 + b^2) - (3*(a^2 + b^2)) * r + (2*a)*r \<and> 2*(a^2 + b^2) - (3*(a^2 + b^2) - 4*a + 1)*r - 2*a < 0)" (is "?lhs \<longleftrightarrow> ?rhs")
  3182 by (simp add: field_simps)
  3183 have "?rhs"
  3184   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "a::'a::{linordered_field_inverse_zero, number_ring}" "b::'a::{linordered_field_inverse_zero, number_ring}")
  3185   apply simp
  3186 oops
  3187 *)
  3188 
  3189 (*
  3190 lemma "ALL (x::'a::{linordered_field_inverse_zero, number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
  3191 apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3192 apply (simp add: field_simps linorder_neq_iff[symmetric])
  3193 apply ferrack
  3194 oops
  3195 *)
  3196 end