src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
author wenzelm
Sat Apr 16 16:15:37 2011 +0200 (2011-04-16)
changeset 42361 23f352990944
parent 42284 326f57825e1a
child 42364 8c674b3b8e44
permissions -rw-r--r--
modernized structure Proof_Context;
     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 Dense_Linear_Order DP_Library
     9   "~~/src/HOL/Library/Efficient_Nat"
    10 begin
    11 
    12 subsection {* Terms *}
    13 
    14 datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
    15   | Neg tm | Sub tm tm | CNP nat poly tm
    16   (* A size for poly to make inductive proofs simpler*)
    17 
    18 primrec tmsize :: "tm \<Rightarrow> nat" where
    19   "tmsize (CP c) = polysize c"
    20 | "tmsize (Bound n) = 1"
    21 | "tmsize (Neg a) = 1 + tmsize a"
    22 | "tmsize (Add a b) = 1 + tmsize a + tmsize b"
    23 | "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
    24 | "tmsize (Mul c a) = 1 + polysize c + tmsize a"
    25 | "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
    26 
    27   (* Semantics of terms tm *)
    28 primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
    29   "Itm vs bs (CP c) = (Ipoly vs c)"
    30 | "Itm vs bs (Bound n) = bs!n"
    31 | "Itm vs bs (Neg a) = -(Itm vs bs a)"
    32 | "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
    33 | "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
    34 | "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
    35 | "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"   
    36 
    37 
    38 fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
    39   "allpolys P (CP c) = P c"
    40 | "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
    41 | "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
    42 | "allpolys P (Neg p) = allpolys P p"
    43 | "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
    44 | "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
    45 | "allpolys P p = True"
    46 
    47 primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
    48   "tmboundslt n (CP c) = True"
    49 | "tmboundslt n (Bound m) = (m < n)"
    50 | "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
    51 | "tmboundslt n (Neg a) = tmboundslt n a"
    52 | "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
    53 | "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
    54 | "tmboundslt n (Mul i a) = tmboundslt n a"
    55 
    56 primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
    57   "tmbound0 (CP c) = True"
    58 | "tmbound0 (Bound n) = (n>0)"
    59 | "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
    60 | "tmbound0 (Neg a) = tmbound0 a"
    61 | "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
    62 | "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
    63 | "tmbound0 (Mul i a) = tmbound0 a"
    64 lemma tmbound0_I:
    65   assumes nb: "tmbound0 a"
    66   shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
    67 using nb
    68 by (induct a rule: tm.induct,auto)
    69 
    70 primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
    71   "tmbound n (CP c) = True"
    72 | "tmbound n (Bound m) = (n \<noteq> m)"
    73 | "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
    74 | "tmbound n (Neg a) = tmbound n a"
    75 | "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
    76 | "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
    77 | "tmbound n (Mul i a) = tmbound n a"
    78 lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
    79 
    80 lemma tmbound_I: 
    81   assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
    82   shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
    83   using nb le bnd
    84   by (induct t rule: tm.induct , auto)
    85 
    86 fun decrtm0:: "tm \<Rightarrow> tm" where
    87   "decrtm0 (Bound n) = Bound (n - 1)"
    88 | "decrtm0 (Neg a) = Neg (decrtm0 a)"
    89 | "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
    90 | "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
    91 | "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
    92 | "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
    93 | "decrtm0 a = a"
    94 
    95 fun incrtm0:: "tm \<Rightarrow> tm" where
    96   "incrtm0 (Bound n) = Bound (n + 1)"
    97 | "incrtm0 (Neg a) = Neg (incrtm0 a)"
    98 | "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
    99 | "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
   100 | "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
   101 | "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
   102 | "incrtm0 a = a"
   103 
   104 lemma decrtm0: assumes nb: "tmbound0 t"
   105   shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
   106   using nb by (induct t rule: decrtm0.induct, simp_all)
   107 
   108 lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
   109   by (induct t rule: decrtm0.induct, simp_all)
   110 
   111 primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
   112   "decrtm m (CP c) = (CP c)"
   113 | "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
   114 | "decrtm m (Neg a) = Neg (decrtm m a)"
   115 | "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
   116 | "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
   117 | "decrtm m (Mul c a) = Mul c (decrtm m a)"
   118 | "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
   119 
   120 primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   121   "removen n [] = []"
   122 | "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
   123 
   124 lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
   125   by (induct xs arbitrary: n, auto)
   126 
   127 lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
   128   by (induct xs arbitrary: n, auto)
   129 
   130 lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
   131   by (induct xs arbitrary: n, auto)
   132 lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
   133   else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
   134 proof(induct xs arbitrary: n m)
   135   case Nil thus ?case by simp
   136 next
   137   case (Cons x xs n m)
   138   {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
   139   moreover
   140   {assume nxs: "\<not> (n \<ge> length (x#xs))" 
   141     {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
   142     moreover
   143     {assume mln: "\<not> (m < n)" 
   144       {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
   145       moreover
   146       {assume mxs: "\<not> (m \<le> length (x#xs))" 
   147         have th: "length (removen n (x#xs)) = length xs" 
   148           using removen_length[where n="n" and xs="x#xs"] nxs by simp
   149         with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
   150         hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
   151           using th nth_length_exceeds[OF mxs'] by auto
   152         hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
   153           by auto
   154         hence ?case using nxs mln mxs by auto }
   155       ultimately have ?case by blast
   156     }
   157     ultimately have ?case by blast
   158   } ultimately show ?case by blast
   159 qed
   160 
   161 lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
   162   and nle: "m \<le> length bs" 
   163   shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
   164   using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth)
   165 
   166 primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   167   "tmsubst0 t (CP c) = CP c"
   168 | "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
   169 | "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
   170 | "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
   171 | "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
   172 | "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
   173 | "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
   174 lemma tmsubst0:
   175   shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
   176   by (induct a rule: tm.induct) auto
   177 
   178 lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
   179   by (induct a rule: tm.induct) auto
   180 
   181 primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
   182   "tmsubst n t (CP c) = CP c"
   183 | "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
   184 | "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
   185              else CNP m c (tmsubst n t a))"
   186 | "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
   187 | "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
   188 | "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
   189 | "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
   190 
   191 lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
   192   shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
   193 using nb nlt
   194 by (induct a rule: tm.induct,auto)
   195 
   196 lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
   197 shows "tmbound0 (tmsubst 0 t a)"
   198 using tnb
   199 by (induct a rule: tm.induct, auto)
   200 
   201 lemma tmsubst_nb: assumes tnb: "tmbound m t"
   202 shows "tmbound m (tmsubst m t a)"
   203 using tnb
   204 by (induct a rule: tm.induct, auto)
   205 lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
   206   by (induct t, auto)
   207   (* Simplification *)
   208 
   209 consts
   210   tmadd:: "tm \<times> tm \<Rightarrow> tm"
   211 recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
   212   "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
   213   (if n1=n2 then 
   214   (let c = c1 +\<^sub>p c2
   215   in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
   216   else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
   217   else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
   218   "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
   219   "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
   220   "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
   221   "tmadd (a,b) = Add a b"
   222 
   223 lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
   224 apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
   225 apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
   226 apply (case_tac "n1 = n2", simp_all add: field_simps)
   227 apply (simp only: right_distrib[symmetric]) 
   228 by (auto simp del: polyadd simp add: polyadd[symmetric])
   229 
   230 lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
   231 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   232 
   233 lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
   234 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   235 lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
   236 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   237 
   238 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)
   239 
   240 fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" where
   241   "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
   242 | "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
   243 | "tmmul t = (\<lambda> i. Mul i t)"
   244 
   245 lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
   246 by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
   247 
   248 lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
   249 by (induct t arbitrary: i rule: tmmul.induct, auto )
   250 
   251 lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
   252 by (induct t arbitrary: n rule: tmmul.induct, auto )
   253 lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
   254 by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
   255 
   256 lemma tmmul_allpolys_npoly[simp]: 
   257   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   258   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)
   259 
   260 definition tmneg :: "tm \<Rightarrow> tm" where
   261   "tmneg t \<equiv> tmmul t (C (- 1,1))"
   262 
   263 definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   264   "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
   265 
   266 lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
   267 using tmneg_def[of t] 
   268 apply simp
   269 apply (subst number_of_Min)
   270 apply (simp only: of_int_minus)
   271 apply simp
   272 done
   273 
   274 lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
   275 using tmneg_def by simp
   276 
   277 lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
   278 using tmneg_def by simp
   279 lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
   280 using tmneg_def by simp
   281 lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
   282 lemma tmneg_allpolys_npoly[simp]: 
   283   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   284   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
   285   unfolding tmneg_def by auto
   286 
   287 lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
   288 using tmsub_def by simp
   289 
   290 lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
   291 using tmsub_def by simp
   292 lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
   293 using tmsub_def by simp
   294 lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
   295 using tmsub_def by simp
   296 lemma tmsub_allpolys_npoly[simp]: 
   297   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   298   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
   299   unfolding tmsub_def by (simp add: isnpoly_def)
   300 
   301 fun simptm:: "tm \<Rightarrow> tm" where
   302   "simptm (CP j) = CP (polynate j)"
   303 | "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
   304 | "simptm (Neg t) = tmneg (simptm t)"
   305 | "simptm (Add t s) = tmadd (simptm t,simptm s)"
   306 | "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
   307 | "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
   308 | "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))"
   309 
   310 lemma polynate_stupid: 
   311   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   312   shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a::{field_char_0, field_inverse_zero})" 
   313 apply (subst polynate[symmetric])
   314 apply simp
   315 done
   316 
   317 lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
   318 by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 
   319 
   320 lemma simptm_tmbound0[simp]: 
   321   "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
   322 by (induct t rule: simptm.induct, auto simp add: Let_def)
   323 
   324 lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
   325 by (induct t rule: simptm.induct, auto simp add: Let_def)
   326 lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
   327 by (induct t rule: simptm.induct, auto simp add: Let_def)
   328 
   329 lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
   330   by (simp_all add: isnpoly_def)
   331 lemma simptm_allpolys_npoly[simp]: 
   332   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   333   shows "allpolys isnpoly (simptm p)"
   334   by (induct p rule: simptm.induct, auto simp add: Let_def)
   335 
   336 declare let_cong[fundef_cong del]
   337 
   338 fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" where
   339   "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
   340 | "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
   341 | "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
   342 | "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
   343 | "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
   344 | "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
   345 | "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
   346 | "split0 t = (0\<^sub>p, t)"
   347 
   348 declare let_cong[fundef_cong]
   349 
   350 lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
   351   apply (rule exI[where x="fst (split0 p)"])
   352   apply (rule exI[where x="snd (split0 p)"])
   353   by simp
   354 
   355 lemma split0:
   356   "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
   357   apply (induct t rule: split0.induct)
   358   apply simp
   359   apply (simp add: Let_def split_def field_simps)
   360   apply (simp add: Let_def split_def field_simps)
   361   apply (simp add: Let_def split_def field_simps)
   362   apply (simp add: Let_def split_def field_simps)
   363   apply (simp add: Let_def split_def field_simps)
   364   apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
   365   apply (simp add: Let_def split_def field_simps)
   366   apply (simp add: Let_def split_def field_simps)
   367   done
   368 
   369 lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
   370 proof-
   371   fix c' t'
   372   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   373   with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
   374 qed
   375 
   376 lemma split0_nb0: 
   377   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   378   shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
   379 proof-
   380   fix c' t'
   381   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   382   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
   383 qed
   384 
   385 lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   386   shows "tmbound0 (snd (split0 t))"
   387   using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
   388 
   389 
   390 lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
   391   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   392 
   393 lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
   394   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   395 
   396 lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   397  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   398 
   399 lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
   400 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   401 
   402 lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   403  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   404 
   405 lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
   406 by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
   407 
   408 lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   409   shows 
   410   "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
   411   by (induct p rule: split0.induct, 
   412     auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
   413     Let_def split_def split0_stupid)
   414 
   415 subsection{* Formulae *}
   416 
   417 datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
   418   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
   419 
   420 
   421   (* A size for fm *)
   422 fun fmsize :: "fm \<Rightarrow> nat" where
   423   "fmsize (NOT p) = 1 + fmsize p"
   424 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
   425 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"
   426 | "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
   427 | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
   428 | "fmsize (E p) = 1 + fmsize p"
   429 | "fmsize (A p) = 4+ fmsize p"
   430 | "fmsize p = 1"
   431   (* several lemmas about fmsize *)
   432 lemma fmsize_pos[termination_simp]: "fmsize p > 0"        
   433 by (induct p rule: fmsize.induct) simp_all
   434 
   435   (* Semantics of formulae (fm) *)
   436 primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
   437   "Ifm vs bs T = True"
   438 | "Ifm vs bs F = False"
   439 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
   440 | "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
   441 | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
   442 | "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
   443 | "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
   444 | "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
   445 | "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
   446 | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
   447 | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
   448 | "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
   449 | "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
   450 
   451 fun not:: "fm \<Rightarrow> fm" where
   452   "not (NOT (NOT p)) = not p"
   453 | "not (NOT p) = p"
   454 | "not T = F"
   455 | "not F = T"
   456 | "not (Lt t) = Le (tmneg t)"
   457 | "not (Le t) = Lt (tmneg t)"
   458 | "not (Eq t) = NEq t"
   459 | "not (NEq t) = Eq t"
   460 | "not p = NOT p"
   461 lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
   462 by (induct p rule: not.induct) auto
   463 
   464 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   465   "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 
   466    if p = q then p else And p q)"
   467 lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
   468 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
   469 
   470 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   471   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
   472        else if p=q then p else Or p q)"
   473 
   474 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
   475 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
   476 
   477 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   478   "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 
   479     else Imp p q)"
   480 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
   481 by (cases "p=F \<or> q=T",simp_all add: imp_def) 
   482 
   483 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   484   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
   485        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 
   486   Iff p q)"
   487 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
   488   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
   489 
   490   (* Quantifier freeness *)
   491 fun qfree:: "fm \<Rightarrow> bool" where
   492   "qfree (E p) = False"
   493 | "qfree (A p) = False"
   494 | "qfree (NOT p) = qfree p" 
   495 | "qfree (And p q) = (qfree p \<and> qfree q)" 
   496 | "qfree (Or  p q) = (qfree p \<and> qfree q)" 
   497 | "qfree (Imp p q) = (qfree p \<and> qfree q)" 
   498 | "qfree (Iff p q) = (qfree p \<and> qfree q)"
   499 | "qfree p = True"
   500 
   501   (* Boundedness and substitution *)
   502 
   503 primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
   504   "boundslt n T = True"
   505 | "boundslt n F = True"
   506 | "boundslt n (Lt t) = (tmboundslt n t)"
   507 | "boundslt n (Le t) = (tmboundslt n t)"
   508 | "boundslt n (Eq t) = (tmboundslt n t)"
   509 | "boundslt n (NEq t) = (tmboundslt n t)"
   510 | "boundslt n (NOT p) = boundslt n p"
   511 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
   512 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
   513 | "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
   514 | "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
   515 | "boundslt n (E p) = boundslt (Suc n) p"
   516 | "boundslt n (A p) = boundslt (Suc n) p"
   517 
   518 fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
   519   "bound0 T = True"
   520 | "bound0 F = True"
   521 | "bound0 (Lt a) = tmbound0 a"
   522 | "bound0 (Le a) = tmbound0 a"
   523 | "bound0 (Eq a) = tmbound0 a"
   524 | "bound0 (NEq a) = tmbound0 a"
   525 | "bound0 (NOT p) = bound0 p"
   526 | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
   527 | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
   528 | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
   529 | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
   530 | "bound0 p = False"
   531 lemma bound0_I:
   532   assumes bp: "bound0 p"
   533   shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
   534 using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
   535 by (induct p rule: bound0.induct,auto)
   536 
   537 primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
   538   "bound m T = True"
   539 | "bound m F = True"
   540 | "bound m (Lt t) = tmbound m t"
   541 | "bound m (Le t) = tmbound m t"
   542 | "bound m (Eq t) = tmbound m t"
   543 | "bound m (NEq t) = tmbound m t"
   544 | "bound m (NOT p) = bound m p"
   545 | "bound m (And p q) = (bound m p \<and> bound m q)"
   546 | "bound m (Or p q) = (bound m p \<and> bound m q)"
   547 | "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
   548 | "bound m (Iff p q) = (bound m p \<and> bound m q)"
   549 | "bound m (E p) = bound (Suc m) p"
   550 | "bound m (A p) = bound (Suc m) p"
   551 
   552 lemma bound_I:
   553   assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
   554   shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
   555   using bnd nb le tmbound_I[where bs=bs and vs = vs]
   556 proof(induct p arbitrary: bs n rule: fm.induct)
   557   case (E p bs n) 
   558   {fix y
   559     from E have bnd: "boundslt (length (y#bs)) p" 
   560       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   561     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   562   thus ?case by simp 
   563 next
   564   case (A p bs n) {fix y
   565     from A have bnd: "boundslt (length (y#bs)) p" 
   566       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   567     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   568   thus ?case by simp 
   569 qed auto
   570 
   571 fun decr0 :: "fm \<Rightarrow> fm" where
   572   "decr0 (Lt a) = Lt (decrtm0 a)"
   573 | "decr0 (Le a) = Le (decrtm0 a)"
   574 | "decr0 (Eq a) = Eq (decrtm0 a)"
   575 | "decr0 (NEq a) = NEq (decrtm0 a)"
   576 | "decr0 (NOT p) = NOT (decr0 p)" 
   577 | "decr0 (And p q) = conj (decr0 p) (decr0 q)"
   578 | "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
   579 | "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
   580 | "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
   581 | "decr0 p = p"
   582 
   583 lemma decr0: assumes nb: "bound0 p"
   584   shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
   585   using nb 
   586   by (induct p rule: decr0.induct, simp_all add: decrtm0)
   587 
   588 primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
   589   "decr m T = T"
   590 | "decr m F = F"
   591 | "decr m (Lt t) = (Lt (decrtm m t))"
   592 | "decr m (Le t) = (Le (decrtm m t))"
   593 | "decr m (Eq t) = (Eq (decrtm m t))"
   594 | "decr m (NEq t) = (NEq (decrtm m t))"
   595 | "decr m (NOT p) = NOT (decr m p)" 
   596 | "decr m (And p q) = conj (decr m p) (decr m q)"
   597 | "decr m (Or p q) = disj (decr m p) (decr m q)"
   598 | "decr m (Imp p q) = imp (decr m p) (decr m q)"
   599 | "decr m (Iff p q) = iff (decr m p) (decr m q)"
   600 | "decr m (E p) = E (decr (Suc m) p)"
   601 | "decr m (A p) = A (decr (Suc m) p)"
   602 
   603 lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
   604   and nle: "m < length bs" 
   605   shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
   606   using bnd nb nle
   607 proof(induct p arbitrary: bs m rule: fm.induct)
   608   case (E p bs m) 
   609   {fix x
   610     from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   611   and nle: "Suc m < length (x#bs)" by auto
   612     from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   613   } thus ?case by auto 
   614 next
   615   case (A p bs m)  
   616   {fix x
   617     from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   618   and nle: "Suc m < length (x#bs)" by auto
   619     from A(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   620   } thus ?case by auto
   621 qed (auto simp add: decrtm removen_nth)
   622 
   623 primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
   624   "subst0 t T = T"
   625 | "subst0 t F = F"
   626 | "subst0 t (Lt a) = Lt (tmsubst0 t a)"
   627 | "subst0 t (Le a) = Le (tmsubst0 t a)"
   628 | "subst0 t (Eq a) = Eq (tmsubst0 t a)"
   629 | "subst0 t (NEq a) = NEq (tmsubst0 t a)"
   630 | "subst0 t (NOT p) = NOT (subst0 t p)"
   631 | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
   632 | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
   633 | "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
   634 | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
   635 | "subst0 t (E p) = E p"
   636 | "subst0 t (A p) = A p"
   637 
   638 lemma subst0: assumes qf: "qfree p"
   639   shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
   640 using qf tmsubst0[where x="x" and bs="bs" and t="t"]
   641 by (induct p rule: fm.induct, auto)
   642 
   643 lemma subst0_nb:
   644   assumes bp: "tmbound0 t" and qf: "qfree p"
   645   shows "bound0 (subst0 t p)"
   646 using qf tmsubst0_nb[OF bp] bp
   647 by (induct p rule: fm.induct, auto)
   648 
   649 primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
   650   "subst n t T = T"
   651 | "subst n t F = F"
   652 | "subst n t (Lt a) = Lt (tmsubst n t a)"
   653 | "subst n t (Le a) = Le (tmsubst n t a)"
   654 | "subst n t (Eq a) = Eq (tmsubst n t a)"
   655 | "subst n t (NEq a) = NEq (tmsubst n t a)"
   656 | "subst n t (NOT p) = NOT (subst n t p)"
   657 | "subst n t (And p q) = And (subst n t p) (subst n t q)"
   658 | "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
   659 | "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
   660 | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
   661 | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
   662 | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
   663 
   664 lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
   665   shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
   666   using nb nlm
   667 proof (induct p arbitrary: bs n t rule: fm.induct)
   668   case (E p bs n) 
   669   {fix x 
   670     from E have bn: "boundslt (length (x#bs)) p" by simp 
   671     from E have nlm: "Suc n \<le> length (x#bs)" by simp
   672     from E(1)[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 
   673     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   674     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   675 thus ?case by simp 
   676 next
   677   case (A p bs n)   
   678   {fix x 
   679     from A have bn: "boundslt (length (x#bs)) p" by simp 
   680     from A have nlm: "Suc n \<le> length (x#bs)" by simp
   681     from A(1)[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 
   682     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   683     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   684 thus ?case by simp 
   685 qed(auto simp add: tmsubst)
   686 
   687 lemma subst_nb: assumes tnb: "tmbound m t"
   688 shows "bound m (subst m t p)"
   689 using tnb tmsubst_nb incrtm0_tmbound
   690 by (induct p arbitrary: m t rule: fm.induct, auto)
   691 
   692 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
   693 by (induct p rule: not.induct, auto)
   694 lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
   695 by (induct p rule: not.induct, auto)
   696 lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
   697 by (induct p rule: not.induct, auto)
   698 lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
   699  by (induct p rule: not.induct, auto)
   700 
   701 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
   702 using conj_def by auto 
   703 lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
   704 using conj_def by auto 
   705 lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
   706 using conj_def by auto 
   707 lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
   708 using conj_def by auto 
   709 
   710 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
   711 using disj_def by auto 
   712 lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
   713 using disj_def by auto 
   714 lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
   715 using disj_def by auto 
   716 lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
   717 using disj_def by auto 
   718 
   719 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
   720 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
   721 lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
   722 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   723 lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
   724 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   725 lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
   726 using imp_def by auto 
   727 
   728 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
   729   by (unfold iff_def,cases "p=q", auto)
   730 lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
   731 using iff_def by (unfold iff_def,cases "p=q", auto)
   732 lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
   733 using iff_def by (unfold iff_def,cases "p=q", auto)
   734 lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
   735 using iff_def by auto 
   736 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
   737 by (induct p, simp_all)
   738 
   739 fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
   740   "isatom T = True"
   741 | "isatom F = True"
   742 | "isatom (Lt a) = True"
   743 | "isatom (Le a) = True"
   744 | "isatom (Eq a) = True"
   745 | "isatom (NEq a) = True"
   746 | "isatom p = False"
   747 
   748 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
   749 by (induct p, simp_all)
   750 
   751 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
   752   "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
   753   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
   754 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
   755   "evaldjf f ps \<equiv> foldr (djf f) ps F"
   756 
   757 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
   758 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
   759 (cases "f p", simp_all add: Let_def djf_def) 
   760 
   761 lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
   762   by(induct ps, simp_all add: evaldjf_def djf_Or)
   763 
   764 lemma evaldjf_bound0: 
   765   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
   766   shows "bound0 (evaldjf f xs)"
   767   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   768 
   769 lemma evaldjf_qf: 
   770   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
   771   shows "qfree (evaldjf f xs)"
   772   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   773 
   774 fun disjuncts :: "fm \<Rightarrow> fm list" where
   775   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
   776 | "disjuncts F = []"
   777 | "disjuncts p = [p]"
   778 
   779 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
   780 by(induct p rule: disjuncts.induct, auto)
   781 
   782 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
   783 proof-
   784   assume nb: "bound0 p"
   785   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
   786   thus ?thesis by (simp only: list_all_iff)
   787 qed
   788 
   789 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
   790 proof-
   791   assume qf: "qfree p"
   792   hence "list_all qfree (disjuncts p)"
   793     by (induct p rule: disjuncts.induct, auto)
   794   thus ?thesis by (simp only: list_all_iff)
   795 qed
   796 
   797 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   798   "DJ f p \<equiv> evaldjf f (disjuncts p)"
   799 
   800 lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
   801   and fF: "f F = F"
   802   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
   803 proof-
   804   have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
   805     by (simp add: DJ_def evaldjf_ex) 
   806   also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
   807   finally show ?thesis .
   808 qed
   809 
   810 lemma DJ_qf: assumes 
   811   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
   812   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
   813 proof(clarify)
   814   fix  p assume qf: "qfree p"
   815   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
   816   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
   817   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
   818   
   819   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
   820 qed
   821 
   822 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))"
   823   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
   824 proof(clarify)
   825   fix p::fm and bs
   826   assume qf: "qfree p"
   827   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
   828   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
   829   have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
   830     by (simp add: DJ_def evaldjf_ex)
   831   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
   832   also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
   833   finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
   834 qed
   835 
   836 fun conjuncts :: "fm \<Rightarrow> fm list" where
   837   "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
   838 | "conjuncts T = []"
   839 | "conjuncts p = [p]"
   840 
   841 definition list_conj :: "fm list \<Rightarrow> fm" where
   842   "list_conj ps \<equiv> foldr conj ps T"
   843 
   844 definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   845   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
   846                    in conj (decr0 (list_conj yes)) (f (list_conj no)))"
   847 
   848 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
   849 proof-
   850   assume qf: "qfree p"
   851   hence "list_all qfree (conjuncts p)"
   852     by (induct p rule: conjuncts.induct, auto)
   853   thus ?thesis by (simp only: list_all_iff)
   854 qed
   855 
   856 lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
   857 by(induct p rule: conjuncts.induct, auto)
   858 
   859 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
   860 proof-
   861   assume nb: "bound0 p"
   862   hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
   863   thus ?thesis by (simp only: list_all_iff)
   864 qed
   865 
   866 fun islin :: "fm \<Rightarrow> bool" where
   867   "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)"
   868 | "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)"
   869 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   870 | "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   871 | "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   872 | "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   873 | "islin (NOT p) = False"
   874 | "islin (Imp p q) = False"
   875 | "islin (Iff p q) = False"
   876 | "islin p = bound0 p"
   877 
   878 lemma islin_stupid: assumes nb: "tmbound0 p"
   879   shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
   880   using nb by (cases p, auto, case_tac nat, auto)+
   881 
   882 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
   883 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
   884 definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
   885 definition "neq p = not (eq p)"
   886 
   887 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
   888   apply(simp add: lt_def)
   889   apply(cases p, simp_all)
   890   apply (case_tac poly, simp_all add: isnpoly_def)
   891   done
   892 
   893 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
   894   apply(simp add: le_def)
   895   apply(cases p, simp_all)
   896   apply (case_tac poly, simp_all add: isnpoly_def)
   897   done
   898 
   899 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
   900   apply(simp add: eq_def)
   901   apply(cases p, simp_all)
   902   apply (case_tac poly, simp_all add: isnpoly_def)
   903   done
   904 
   905 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
   906   by(simp add: neq_def eq)
   907 
   908 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
   909   apply (simp add: lt_def)
   910   apply (cases p, simp_all)
   911   apply (case_tac poly, simp_all)
   912   apply (case_tac nat, simp_all)
   913   done
   914 
   915 lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
   916   apply (simp add: le_def)
   917   apply (cases p, simp_all)
   918   apply (case_tac poly, simp_all)
   919   apply (case_tac nat, simp_all)
   920   done
   921 
   922 lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
   923   apply (simp add: eq_def)
   924   apply (cases p, simp_all)
   925   apply (case_tac poly, simp_all)
   926   apply (case_tac nat, simp_all)
   927   done
   928 
   929 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
   930   apply (simp add: neq_def eq_def)
   931   apply (cases p, simp_all)
   932   apply (case_tac poly, simp_all)
   933   apply (case_tac nat, simp_all)
   934   done
   935 
   936 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
   937 definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
   938 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
   939 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
   940 
   941 lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   942   shows "islin (simplt t)"
   943   unfolding simplt_def 
   944   using split0_nb0'
   945 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])
   946   
   947 lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   948   shows "islin (simple t)"
   949   unfolding simple_def 
   950   using split0_nb0'
   951 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)
   952 lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   953   shows "islin (simpeq t)"
   954   unfolding simpeq_def 
   955   using split0_nb0'
   956 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)
   957 
   958 lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   959   shows "islin (simpneq t)"
   960   unfolding simpneq_def 
   961   using split0_nb0'
   962 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)
   963 
   964 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
   965   by (cases "split0 s", auto)
   966 lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   967   and n: "allpolys isnpoly t"
   968   shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
   969   using n
   970   by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
   971 lemma simplt[simp]:
   972   shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
   973 proof-
   974   have n: "allpolys isnpoly (simptm t)" by simp
   975   let ?t = "simptm t"
   976   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
   977       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
   978       by (simp add: simplt_def Let_def split_def lt)}
   979   moreover
   980   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
   981     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
   982   }
   983   ultimately show ?thesis by blast
   984 qed
   985 
   986 lemma simple[simp]:
   987   shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
   988 proof-
   989   have n: "allpolys isnpoly (simptm t)" by simp
   990   let ?t = "simptm t"
   991   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
   992       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
   993       by (simp add: simple_def Let_def split_def le)}
   994   moreover
   995   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
   996     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
   997   }
   998   ultimately show ?thesis by blast
   999 qed
  1000 
  1001 lemma simpeq[simp]:
  1002   shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
  1003 proof-
  1004   have n: "allpolys isnpoly (simptm t)" by simp
  1005   let ?t = "simptm t"
  1006   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1007       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
  1008       by (simp add: simpeq_def Let_def split_def)}
  1009   moreover
  1010   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1011     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
  1012   }
  1013   ultimately show ?thesis by blast
  1014 qed
  1015 
  1016 lemma simpneq[simp]:
  1017   shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
  1018 proof-
  1019   have n: "allpolys isnpoly (simptm t)" by simp
  1020   let ?t = "simptm t"
  1021   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1022       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
  1023       by (simp add: simpneq_def Let_def split_def )}
  1024   moreover
  1025   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1026     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
  1027   }
  1028   ultimately show ?thesis by blast
  1029 qed
  1030 
  1031 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
  1032   apply (simp add: lt_def)
  1033   apply (cases t, auto)
  1034   apply (case_tac poly, auto)
  1035   done
  1036 
  1037 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
  1038   apply (simp add: le_def)
  1039   apply (cases t, auto)
  1040   apply (case_tac poly, auto)
  1041   done
  1042 
  1043 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
  1044   apply (simp add: eq_def)
  1045   apply (cases t, auto)
  1046   apply (case_tac poly, auto)
  1047   done
  1048 
  1049 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
  1050   apply (simp add: neq_def eq_def)
  1051   apply (cases t, auto)
  1052   apply (case_tac poly, auto)
  1053   done
  1054 
  1055 lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1056   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
  1057   using split0 [of "simptm t" vs bs]
  1058 proof(simp add: simplt_def Let_def split_def)
  1059   assume nb: "tmbound0 t"
  1060   hence nb': "tmbound0 (simptm t)" by simp
  1061   let ?c = "fst (split0 (simptm t))"
  1062   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1063   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1064   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1065   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1066   from iffD1[OF isnpolyh_unique[OF ths] th]
  1067   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1068   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
  1069        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
  1070 qed
  1071 
  1072 lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1073   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
  1074   using split0 [of "simptm t" vs bs]
  1075 proof(simp add: simple_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 (le (snd (split0 (simptm t))))) \<and>
  1086        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
  1087 qed
  1088 
  1089 lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1090   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
  1091   using split0 [of "simptm t" vs bs]
  1092 proof(simp add: simpeq_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 (eq (snd (split0 (simptm t))))) \<and>
  1103        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
  1104 qed
  1105 
  1106 lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1107   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
  1108   using split0 [of "simptm t" vs bs]
  1109 proof(simp add: simpneq_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 (neq (snd (split0 (simptm t))))) \<and>
  1120        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
  1121 qed
  1122 
  1123 fun conjs   :: "fm \<Rightarrow> fm list" where
  1124   "conjs (And p q) = (conjs p)@(conjs q)"
  1125 | "conjs T = []"
  1126 | "conjs p = [p]"
  1127 lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
  1128 by (induct p rule: conjs.induct, auto)
  1129 definition list_disj :: "fm list \<Rightarrow> fm" where
  1130   "list_disj ps \<equiv> foldr disj ps F"
  1131 
  1132 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
  1133   by (induct ps, auto simp add: list_conj_def)
  1134 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
  1135   by (induct ps, auto simp add: list_conj_def conj_qf)
  1136 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
  1137   by (induct ps, auto simp add: list_disj_def)
  1138 
  1139 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
  1140   unfolding conj_def by auto
  1141 
  1142 lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
  1143   apply (induct p rule: conjs.induct) 
  1144   apply (unfold conjs.simps)
  1145   apply (unfold set_append)
  1146   apply (unfold ball_Un)
  1147   apply (unfold bound.simps)
  1148   apply auto
  1149   done
  1150 
  1151 lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
  1152   apply (induct p rule: conjs.induct) 
  1153   apply (unfold conjs.simps)
  1154   apply (unfold set_append)
  1155   apply (unfold ball_Un)
  1156   apply (unfold boundslt.simps)
  1157   apply blast
  1158 by simp_all
  1159 
  1160 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
  1161   unfolding list_conj_def
  1162   by (induct ps, auto simp add: conj_boundslt)
  1163 
  1164 lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
  1165   shows "bound n (list_conj ps)"
  1166   using bnd
  1167   unfolding list_conj_def
  1168   by (induct ps, auto simp add: conj_nb)
  1169 
  1170 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
  1171 unfolding list_conj_def by (induct ps , auto)
  1172 
  1173 lemma CJNB_qe: 
  1174   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1175   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
  1176 proof(clarify)
  1177   fix bs p
  1178   assume qfp: "qfree p"
  1179   let ?cjs = "conjuncts p"
  1180   let ?yes = "fst (partition bound0 ?cjs)"
  1181   let ?no = "snd (partition bound0 ?cjs)"
  1182   let ?cno = "list_conj ?no"
  1183   let ?cyes = "list_conj ?yes"
  1184   have part: "partition bound0 ?cjs = (?yes,?no)" by simp
  1185   from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
  1186   hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
  1187   hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
  1188   from conjuncts_qf[OF qfp] partition_set[OF part] 
  1189   have " \<forall>q\<in> set ?no. qfree q" by auto
  1190   hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
  1191   with qe have cno_qf:"qfree (qe ?cno )" 
  1192     and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
  1193   from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
  1194     by (simp add: CJNB_def Let_def conj_qf split_def)
  1195   {fix bs
  1196     from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
  1197     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
  1198       using partition_set[OF part] by auto
  1199     finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
  1200   hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
  1201   also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
  1202     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
  1203   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
  1204     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
  1205   also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
  1206     using qe[rule_format, OF no_qf] by auto
  1207   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" 
  1208     by (simp add: Let_def CJNB_def split_def)
  1209   with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
  1210 qed
  1211 
  1212 consts simpfm :: "fm \<Rightarrow> fm"
  1213 recdef simpfm "measure fmsize"
  1214   "simpfm (Lt t) = simplt (simptm t)"
  1215   "simpfm (Le t) = simple (simptm t)"
  1216   "simpfm (Eq t) = simpeq(simptm t)"
  1217   "simpfm (NEq t) = simpneq(simptm t)"
  1218   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
  1219   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
  1220   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
  1221   "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
  1222   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
  1223   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
  1224   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
  1225   "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
  1226   "simpfm (NOT (Eq t)) = simpneq t"
  1227   "simpfm (NOT (NEq t)) = simpeq t"
  1228   "simpfm (NOT (Le t)) = simplt (Neg t)"
  1229   "simpfm (NOT (Lt t)) = simple (Neg t)"
  1230   "simpfm (NOT (NOT p)) = simpfm p"
  1231   "simpfm (NOT T) = F"
  1232   "simpfm (NOT F) = T"
  1233   "simpfm p = p"
  1234 
  1235 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
  1236 by(induct p arbitrary: bs rule: simpfm.induct, auto)
  1237 
  1238 lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1239   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
  1240 by (induct p rule: simpfm.induct, auto)
  1241 
  1242 lemma lt_qf[simp]: "qfree (lt t)"
  1243   apply (cases t, auto simp add: lt_def)
  1244   by (case_tac poly, auto)
  1245 
  1246 lemma le_qf[simp]: "qfree (le t)"
  1247   apply (cases t, auto simp add: le_def)
  1248   by (case_tac poly, auto)
  1249 
  1250 lemma eq_qf[simp]: "qfree (eq t)"
  1251   apply (cases t, auto simp add: eq_def)
  1252   by (case_tac poly, auto)
  1253 
  1254 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
  1255 
  1256 lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
  1257 lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
  1258 lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
  1259 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
  1260 
  1261 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
  1262 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
  1263 
  1264 lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
  1265 lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
  1266 
  1267 lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1268   shows "qfree p \<Longrightarrow> islin (simpfm p)" 
  1269   apply (induct p rule: simpfm.induct)
  1270   apply (simp_all add: conj_lin disj_lin)
  1271   done
  1272 
  1273 consts prep :: "fm \<Rightarrow> fm"
  1274 recdef prep "measure fmsize"
  1275   "prep (E T) = T"
  1276   "prep (E F) = F"
  1277   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
  1278   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
  1279   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
  1280   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
  1281   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
  1282   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
  1283   "prep (E p) = E (prep p)"
  1284   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
  1285   "prep (A p) = prep (NOT (E (NOT p)))"
  1286   "prep (NOT (NOT p)) = prep p"
  1287   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
  1288   "prep (NOT (A p)) = prep (E (NOT p))"
  1289   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
  1290   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
  1291   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
  1292   "prep (NOT p) = not (prep p)"
  1293   "prep (Or p q) = disj (prep p) (prep q)"
  1294   "prep (And p q) = conj (prep p) (prep q)"
  1295   "prep (Imp p q) = prep (Or (NOT p) q)"
  1296   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
  1297   "prep p = p"
  1298 (hints simp add: fmsize_pos)
  1299 lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
  1300 by (induct p arbitrary: bs rule: prep.induct, auto)
  1301 
  1302 
  1303 
  1304   (* Generic quantifier elimination *)
  1305 function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
  1306   "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
  1307 | "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
  1308 | "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
  1309 | "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
  1310 | "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
  1311 | "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
  1312 | "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
  1313 | "qelim p = (\<lambda> y. simpfm p)"
  1314 by pat_completeness simp_all
  1315 termination by (relation "measure fmsize") auto
  1316 
  1317 lemma qelim:
  1318   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1319   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
  1320 using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
  1321 by (induct p rule: qelim.induct) auto
  1322 
  1323 subsection{* Core Procedure *}
  1324 
  1325 fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
  1326   "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
  1327 | "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
  1328 | "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1329 | "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1330 | "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
  1331 | "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
  1332 | "minusinf p = p"
  1333 
  1334 fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
  1335   "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
  1336 | "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
  1337 | "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1338 | "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1339 | "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
  1340 | "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
  1341 | "plusinf p = p"
  1342 
  1343 lemma minusinf_inf: assumes lp:"islin p"
  1344   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1345   using lp
  1346 proof (induct p rule: minusinf.induct)
  1347   case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1348 next
  1349   case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1350 next
  1351   case (3 c e) hence nbe: "tmbound0 e" by simp
  1352   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1353   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1354   let ?c = "Ipoly vs c"
  1355   let ?e = "Itm vs (y#bs) e"
  1356   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1357   moreover {assume "?c = 0" hence ?case 
  1358       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1359   moreover {assume cp: "?c > 0"
  1360     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1361         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1362       hence "?c * x + ?e < 0" by simp
  1363       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1364         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1365   moreover {assume cp: "?c < 0"
  1366     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1367         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1368       hence "?c * x + ?e > 0" by simp
  1369       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1370         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1371   ultimately show ?case by blast
  1372 next
  1373   case (4 c e)  hence nbe: "tmbound0 e" by simp
  1374   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1375   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1376   let ?c = "Ipoly vs c"
  1377   let ?e = "Itm vs (y#bs) e"
  1378   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1379   moreover {assume "?c = 0" hence ?case using eqs 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) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1385         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] 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) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1391         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1392   ultimately show ?case by blast
  1393 next
  1394   case (5 c e)  hence nbe: "tmbound0 e" by simp
  1395   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1396   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1397   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
  1398   let ?c = "Ipoly vs c"
  1399   let ?e = "Itm vs (y#bs) e"
  1400   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1401   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1402   moreover {assume cp: "?c > 0"
  1403     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1404         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1405       hence "?c * x + ?e < 0" by simp
  1406       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1407         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1408   moreover {assume cp: "?c < 0"
  1409     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1410         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1411       hence "?c * x + ?e > 0" by simp
  1412       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1413         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1414   ultimately show ?case by blast
  1415 next
  1416   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1417   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1418   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1419   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1420   let ?c = "Ipoly vs c"
  1421   let ?e = "Itm vs (y#bs) e"
  1422   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1423   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1424   moreover {assume cp: "?c > 0"
  1425     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1426         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1427       hence "?c * x + ?e < 0" by simp
  1428       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1429         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1430   moreover {assume cp: "?c < 0"
  1431     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1432         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1433       hence "?c * x + ?e > 0" by simp
  1434       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1435         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1436   ultimately show ?case by blast
  1437 qed (auto)
  1438 
  1439 lemma plusinf_inf: assumes lp:"islin p"
  1440   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1441   using lp
  1442 proof (induct p rule: plusinf.induct)
  1443   case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1444 next
  1445   case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1446 next
  1447   case (3 c e) hence nbe: "tmbound0 e" by simp
  1448   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1449   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1450   let ?c = "Ipoly vs c"
  1451   let ?e = "Itm vs (y#bs) e"
  1452   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1453   moreover {assume "?c = 0" hence ?case 
  1454       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1455   moreover {assume cp: "?c > 0"
  1456     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e" 
  1457         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1458       hence "?c * x + ?e > 0" by simp
  1459       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1460         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1461   moreover {assume cp: "?c < 0"
  1462     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1463         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1464       hence "?c * x + ?e < 0" by simp
  1465       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1466         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1467   ultimately show ?case by blast
  1468 next
  1469   case (4 c e) hence nbe: "tmbound0 e" by simp
  1470   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1471   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1472   let ?c = "Ipoly vs c"
  1473   let ?e = "Itm vs (y#bs) e"
  1474   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1475   moreover {assume "?c = 0" hence ?case using eqs 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) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1481         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] 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) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1487         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1488   ultimately show ?case by blast
  1489 next
  1490   case (5 c e) hence nbe: "tmbound0 e" by simp
  1491   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1492   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1493   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]
  1494   let ?c = "Ipoly vs c"
  1495   let ?e = "Itm vs (y#bs) e"
  1496   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1497   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1498   moreover {assume cp: "?c > 0"
  1499     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1500         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1501       hence "?c * x + ?e > 0" by simp
  1502       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1503         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1504   moreover {assume cp: "?c < 0"
  1505     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1506         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1507       hence "?c * x + ?e < 0" by simp
  1508       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1509         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1510   ultimately show ?case by blast
  1511 next
  1512   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1513   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1514   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1515   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1516   let ?c = "Ipoly vs c"
  1517   let ?e = "Itm vs (y#bs) e"
  1518   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1519   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1520   moreover {assume cp: "?c > 0"
  1521     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1522         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1523       hence "?c * x + ?e > 0" by simp
  1524       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1525         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1526   moreover {assume cp: "?c < 0"
  1527     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1528         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1529       hence "?c * x + ?e < 0" by simp
  1530       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1531         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1532   ultimately show ?case by blast
  1533 qed (auto)
  1534 
  1535 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)" 
  1536   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1537 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)" 
  1538   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1539 
  1540 lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
  1541   shows "\<exists>x. Ifm vs (x#bs) p"
  1542 proof-
  1543   from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1544   have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
  1545   from minusinf_inf[OF lp, where bs="bs"] 
  1546   obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
  1547   from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
  1548   moreover have "z - 1 < z" by simp
  1549   ultimately show ?thesis using z_def by auto
  1550 qed
  1551 
  1552 lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
  1553   shows "\<exists>x. Ifm vs (x#bs) p"
  1554 proof-
  1555   from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1556   have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
  1557   from plusinf_inf[OF lp, where bs="bs"] 
  1558   obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
  1559   from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
  1560   moreover have "z + 1 > z" by simp
  1561   ultimately show ?thesis using z_def by auto
  1562 qed
  1563 
  1564 fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
  1565   "uset (And p q) = uset p @ uset q"
  1566 | "uset (Or p q) = uset p @ uset q"
  1567 | "uset (Eq (CNP 0 a e))  = [(a,e)]"
  1568 | "uset (Le (CNP 0 a e))  = [(a,e)]"
  1569 | "uset (Lt (CNP 0 a e))  = [(a,e)]"
  1570 | "uset (NEq (CNP 0 a e)) = [(a,e)]"
  1571 | "uset p = []"
  1572 
  1573 lemma uset_l:
  1574   assumes lp: "islin p"
  1575   shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
  1576 using lp by(induct p rule: uset.induct,auto)
  1577 
  1578 lemma minusinf_uset0:
  1579   assumes lp: "islin p"
  1580   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
  1581   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1582   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c" 
  1583 proof-
  1584   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)" 
  1585     using lp nmi ex
  1586     apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
  1587     apply (auto simp add: linorder_not_less order_le_less)
  1588     done 
  1589   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
  1590   hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
  1591     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
  1592     by (auto simp add: mult_commute del: divide_minus_left)
  1593   thus ?thesis using csU by auto
  1594 qed
  1595 
  1596 lemma minusinf_uset:
  1597   assumes lp: "islin p"
  1598   and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
  1599   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1600   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c" 
  1601 proof-
  1602   from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))" 
  1603     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
  1604   from minusinf_uset0[OF lp nmi' ex] 
  1605   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
  1606   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1607   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1608 qed
  1609 
  1610 
  1611 lemma plusinf_uset0:
  1612   assumes lp: "islin p"
  1613   and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
  1614   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1615   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c" 
  1616 proof-
  1617   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)" 
  1618     using lp nmi ex
  1619     apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
  1620     apply (auto simp add: linorder_not_less order_le_less)
  1621     done 
  1622   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
  1623   hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
  1624     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
  1625     by (auto simp add: mult_commute del: divide_minus_left)
  1626   thus ?thesis using csU by auto
  1627 qed
  1628 
  1629 lemma plusinf_uset:
  1630   assumes lp: "islin p"
  1631   and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
  1632   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1633   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c" 
  1634 proof-
  1635   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))" 
  1636     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
  1637   from plusinf_uset0[OF lp nmi' ex] 
  1638   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
  1639   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1640   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1641 qed
  1642 
  1643 lemma lin_dense: 
  1644   assumes lp: "islin p"
  1645   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)" 
  1646   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
  1647   and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
  1648   and ly: "l < y" and yu: "y < u"
  1649   shows "Ifm vs (y#bs) p"
  1650 using lp px noS
  1651 proof (induct p rule: islin.induct) 
  1652   case (5 c s)
  1653   from "5.prems" 
  1654   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1655     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
  1656     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
  1657   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1658   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1659   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1660   moreover
  1661   {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"])}
  1662   moreover
  1663   {assume c: "?N c > 0"
  1664       from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1665       have px': "x < - ?Nt x s / ?N c" 
  1666         by (auto simp add: not_less field_simps) 
  1667     {assume y: "y < - ?Nt x s / ?N c" 
  1668       hence "y * ?N c < - ?Nt x s"
  1669         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1670       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1671       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1672     moreover
  1673     {assume y: "y > -?Nt x s / ?N c" 
  1674       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1675       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1676       with lx px' have "False" by simp  hence ?case by simp }
  1677     ultimately have ?case using ycs by blast
  1678   }
  1679   moreover
  1680   {assume c: "?N c < 0"
  1681       from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]  
  1682       have px': "x > - ?Nt x s / ?N c" 
  1683         by (auto simp add: not_less field_simps) 
  1684     {assume y: "y > - ?Nt x s / ?N c" 
  1685       hence "y * ?N c < - ?Nt x s"
  1686         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1687       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1688       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1689     moreover
  1690     {assume y: "y < -?Nt x s / ?N c" 
  1691       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1692       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1693       with xu px' have "False" by simp  hence ?case by simp }
  1694     ultimately have ?case using ycs by blast
  1695   }
  1696   ultimately show ?case by blast
  1697 next
  1698   case (6 c s)
  1699   from "6.prems" 
  1700   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1701     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
  1702     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
  1703   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1704   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1705   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1706   moreover
  1707   {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"])}
  1708   moreover
  1709   {assume c: "?N c > 0"
  1710       from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1711       have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps) 
  1712     {assume y: "y < - ?Nt x s / ?N c" 
  1713       hence "y * ?N c < - ?Nt x s"
  1714         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1715       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1716       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1717     moreover
  1718     {assume y: "y > -?Nt x s / ?N c" 
  1719       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1720       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1721       with lx px' have "False" by simp  hence ?case by simp }
  1722     ultimately have ?case using ycs by blast
  1723   }
  1724   moreover
  1725   {assume c: "?N c < 0"
  1726       from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]  
  1727       have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps) 
  1728     {assume y: "y > - ?Nt x s / ?N c" 
  1729       hence "y * ?N c < - ?Nt x s"
  1730         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1731       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1732       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1733     moreover
  1734     {assume y: "y < -?Nt x s / ?N c" 
  1735       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1736       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1737       with xu px' have "False" by simp  hence ?case by simp }
  1738     ultimately have ?case using ycs by blast
  1739   }
  1740   ultimately show ?case by blast
  1741 next
  1742     case (3 c s)
  1743   from "3.prems" 
  1744   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1745     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
  1746     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
  1747   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1748   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1749   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1750   moreover
  1751   {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"])}
  1752   moreover
  1753   {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
  1754     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1755     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1756     {assume y: "y < -?Nt x s / ?N c" 
  1757       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1758       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1759       with xu px' have "False" by simp  hence ?case by simp }
  1760     moreover
  1761     {assume y: "y > -?Nt x s / ?N c" 
  1762       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1763       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1764       with lx px' have "False" by simp  hence ?case by simp }
  1765     ultimately have ?case using ycs by blast
  1766   }
  1767   moreover
  1768   {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
  1769     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1770     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1771     {assume y: "y < -?Nt x s / ?N c" 
  1772       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1773       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1774       with xu px' have "False" by simp  hence ?case by simp }
  1775     moreover
  1776     {assume y: "y > -?Nt x s / ?N c" 
  1777       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1778       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1779       with lx px' have "False" by simp  hence ?case by simp }
  1780     ultimately have ?case using ycs by blast
  1781   }
  1782   ultimately show ?case by blast
  1783 next
  1784     case (4 c s)
  1785   from "4.prems" 
  1786   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1787     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
  1788     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
  1789   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1790   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1791   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
  1792   moreover
  1793   {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"])}
  1794   moreover
  1795   {assume c: "?N c \<noteq> 0"
  1796     from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
  1797       by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
  1798   ultimately show ?case by blast
  1799 qed (auto simp add: 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"])
  1800 
  1801 lemma one_plus_one_pos[simp]: "(1::'a::{linordered_field}) + 1 > 0"
  1802 proof-
  1803   have op: "(1::'a) > 0" by simp
  1804   from add_pos_pos[OF op op] show ?thesis . 
  1805 qed
  1806 
  1807 lemma one_plus_one_nonzero[simp]: "(1::'a::{linordered_field}) + 1 \<noteq> 0" 
  1808   using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le) 
  1809 
  1810 lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{linordered_field})" 
  1811 proof-
  1812   have "(u + u) = (1 + 1) * u" by (simp add: field_simps)
  1813   hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
  1814   with nonzero_mult_divide_cancel_left[OF one_plus_one_nonzero, of u] show ?thesis by simp
  1815 qed
  1816 
  1817 lemma inf_uset:
  1818   assumes lp: "islin p"
  1819   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
  1820   and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
  1821   and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
  1822   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" 
  1823 proof-
  1824   let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
  1825   let ?N = "Ipoly vs"
  1826   let ?U = "set (uset p)"
  1827   from ex obtain a where pa: "?I a p" by blast
  1828   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
  1829   have nmi': "\<not> (?I a (?M p))" by simp
  1830   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
  1831   have npi': "\<not> (?I a (?P p))" by simp
  1832   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"
  1833   proof-
  1834     let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
  1835     have fM: "finite ?M" by auto
  1836     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] 
  1837     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
  1838     then obtain "c" "t" "d" "s" where 
  1839       ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U" 
  1840       and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
  1841     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 
  1842     have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
  1843     from ctU have Mne: "?M \<noteq> {}" by auto
  1844     hence Une: "?U \<noteq> {}" by simp
  1845     let ?l = "Min ?M"
  1846     let ?u = "Max ?M"
  1847     have linM: "?l \<in> ?M" using fM Mne by simp
  1848     have uinM: "?u \<in> ?M" using fM Mne by simp
  1849     have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
  1850     have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto 
  1851     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
  1852     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
  1853     have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
  1854     have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
  1855     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
  1856     have "(\<exists> s\<in> ?M. ?I s p) \<or> 
  1857       (\<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)" .
  1858     moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
  1859       hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
  1860       then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
  1861       from half_sum_eq[of u] pu tuu 
  1862       have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / (1 + 1)) p" by simp
  1863       with tuU have ?thesis by blast}
  1864     moreover{
  1865       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"
  1866       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
  1867         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"
  1868         by blast
  1869       from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
  1870       then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
  1871       from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
  1872       then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
  1873       from t1x xt2 have t1t2: "t1 < t2" by simp
  1874       let ?u = "(t1 + t2) / (1 + 1)"
  1875       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
  1876       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
  1877       with t1uU t2uU t1u t2u have ?thesis by blast}
  1878     ultimately show ?thesis by blast
  1879   qed
  1880   then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U" 
  1881     and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / (1 + 1)) p" by blast
  1882   from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
  1883   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
  1884     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
  1885   have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / (1 + 1)) p" by simp
  1886   with lnU smU
  1887   show ?thesis by auto
  1888 qed
  1889 
  1890     (* The Ferrante - Rackoff Theorem *)
  1891 
  1892 theorem fr_eq: 
  1893   assumes lp: "islin p"
  1894   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))"
  1895   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  1896 proof
  1897   assume px: "\<exists> x. ?I x p"
  1898   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  1899   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  1900   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  1901     from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
  1902   ultimately show "?D" by blast
  1903 next
  1904   assume "?D" 
  1905   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  1906   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  1907   moreover {assume f:"?F" hence "?E" by blast}
  1908   ultimately show "?E" by blast
  1909 qed
  1910 
  1911 section{* First implementation : Naive by encoding all case splits locally *}
  1912 definition "msubsteq c t d s a r = 
  1913   evaldjf (split conj) 
  1914   [(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)))),
  1915    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1916    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1917    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
  1918 
  1919 lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  1920   shows "bound0 (msubsteq c t d s a r)"
  1921 proof-
  1922   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)))),
  1923    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1924    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1925    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
  1926     using lp by (simp add: Let_def t s )
  1927   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
  1928 qed
  1929 
  1930 lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
  1931   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")
  1932 proof-
  1933   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  1934   let ?N = "\<lambda>p. Ipoly vs p"
  1935   let ?c = "?N c"
  1936   let ?d = "?N d"
  1937   let ?t = "?Nt x t"
  1938   let ?s = "?Nt x s"
  1939   let ?a = "?N a"
  1940   let ?r = "?Nt x r"
  1941   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  1942   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  1943   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
  1944   moreover
  1945   {assume c: "?c = 0" and d: "?d=0"
  1946     hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
  1947   moreover 
  1948   {assume c: "?c = 0" and d: "?d\<noteq>0"
  1949     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  1950     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1951     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>))"])
  1952     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) = 0" 
  1953       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  1954     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r= 0"
  1955       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  1956     
  1957     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r = 0" using d by simp 
  1958     finally have ?thesis using c d 
  1959       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)
  1960       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1961       apply simp
  1962       done}
  1963   moreover
  1964   {assume c: "?c \<noteq> 0" and d: "?d=0"
  1965     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  1966     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1967     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r = 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  1968     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) = 0" 
  1969       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  1970     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r= 0"
  1971       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  1972     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r = 0" using c by simp 
  1973     finally have ?thesis using c d 
  1974       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
  1975       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1976       apply simp
  1977       done }
  1978   moreover
  1979   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  1980     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  1981     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  1982       by (simp add: field_simps)
  1983     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1984     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r = 0" 
  1985       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  1986     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) =0 "
  1987       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  1988     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r =0" 
  1989       using nonzero_mult_divide_cancel_left [OF dc] c d
  1990       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  1991     finally  have ?thesis using c d 
  1992       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
  1993       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1994       apply (simp add: field_simps)
  1995       done }
  1996   ultimately show ?thesis by blast
  1997 qed
  1998 
  1999 
  2000 definition "msubstneq c t d s a r = 
  2001   evaldjf (split conj) 
  2002   [(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)))),
  2003    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2004    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2005    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
  2006 
  2007 lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2008   shows "bound0 (msubstneq c t d s a r)"
  2009 proof-
  2010   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)))), 
  2011     (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2012     (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2013     (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
  2014     using lp by (simp add: Let_def t s )
  2015   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
  2016 qed
  2017 
  2018 lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
  2019   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")
  2020 proof-
  2021   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  2022   let ?N = "\<lambda>p. Ipoly vs p"
  2023   let ?c = "?N c"
  2024   let ?d = "?N d"
  2025   let ?t = "?Nt x t"
  2026   let ?s = "?Nt x s"
  2027   let ?a = "?N a"
  2028   let ?r = "?Nt x r"
  2029   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2030   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2031   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
  2032   moreover
  2033   {assume c: "?c = 0" and d: "?d=0"
  2034     hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
  2035   moreover 
  2036   {assume c: "?c = 0" and d: "?d\<noteq>0"
  2037     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  2038     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2039     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>))"])
  2040     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) \<noteq> 0" 
  2041       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  2042     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r\<noteq> 0"
  2043       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  2044     
  2045     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r \<noteq> 0" using d by simp 
  2046     finally have ?thesis using c d 
  2047       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)
  2048       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2049       apply simp
  2050       done}
  2051   moreover
  2052   {assume c: "?c \<noteq> 0" and d: "?d=0"
  2053     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  2054     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2055     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  2056     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) \<noteq> 0" 
  2057       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  2058     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r \<noteq> 0"
  2059       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  2060     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r \<noteq> 0" using c by simp 
  2061     finally have ?thesis using c d 
  2062       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
  2063       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2064       apply simp
  2065       done }
  2066   moreover
  2067   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  2068     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2069     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2070       by (simp add: field_simps)
  2071     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2072     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r \<noteq> 0" 
  2073       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2074     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) \<noteq> 0 "
  2075       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2076     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r \<noteq> 0" 
  2077       using nonzero_mult_divide_cancel_left[OF dc] c d
  2078       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2079     finally  have ?thesis using c d 
  2080       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
  2081       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2082       apply (simp add: field_simps)
  2083       done }
  2084   ultimately show ?thesis by blast
  2085 qed
  2086 
  2087 definition "msubstlt c t d s a r = 
  2088   evaldjf (split conj) 
  2089   [(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)))),
  2090   (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)))),
  2091    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2092    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2093    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2094    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2095    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"
  2096 
  2097 lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2098   shows "bound0 (msubstlt c t d s a r)"
  2099 proof-
  2100   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)))),
  2101   (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)))),
  2102    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2103    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2104    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2105    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2106    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
  2107     using lp by (simp add: Let_def t s lt_nb )
  2108   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
  2109 qed
  2110 
  2111 
  2112 lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))" 
  2113   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow> 
  2114   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")
  2115 proof-
  2116   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2117   let ?N = "\<lambda>p. Ipoly vs p"
  2118   let ?c = "?N c"
  2119   let ?d = "?N d"
  2120   let ?t = "?Nt x t"
  2121   let ?s = "?Nt x s"
  2122   let ?a = "?N a"
  2123   let ?r = "?Nt x r"
  2124   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2125   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2126   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
  2127   moreover
  2128   {assume c: "?c=0" and d: "?d=0"
  2129     hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
  2130   moreover
  2131   {assume dc: "?c*?d > 0" 
  2132     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2133     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2134     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2135     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2136     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2137       by (simp add: field_simps)
  2138     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2139     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2140       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2141     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) < 0"
  2142       
  2143       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
  2144     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r < 0" 
  2145       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2146       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2147     finally  have ?thesis using dc c d  nc nd dc'
  2148       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) 
  2149     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2150     by (simp add: field_simps order_less_not_sym[OF dc])}
  2151   moreover
  2152   {assume dc: "?c*?d < 0" 
  2153 
  2154     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2155       by (simp add: mult_less_0_iff field_simps) 
  2156     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2157     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2158     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2159       by (simp add: field_simps)
  2160     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2161     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2162       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2163 
  2164     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) > 0"
  2165       
  2166       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
  2167     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r < 0" 
  2168       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2169       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2170     finally  have ?thesis using dc c d  nc nd
  2171       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) 
  2172       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2173       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2174   moreover
  2175   {assume c: "?c > 0" and d: "?d=0"  
  2176     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2177     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2178     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2179     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2180     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2181     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) < 0"
  2182       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
  2183     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r < 0" 
  2184       using nonzero_mult_divide_cancel_left[OF c'] c
  2185       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2186     finally have ?thesis using c d nc nd 
  2187       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2188       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2189       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2190       by (simp add: field_simps )  }
  2191   moreover
  2192   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2193     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2194     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2195     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2196     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2197     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) > 0"
  2198       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
  2199     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r < 0" 
  2200       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2201         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2202     finally have ?thesis using c d nc nd 
  2203       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2204       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2205       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2206       by (simp add: field_simps )    }
  2207   moreover
  2208   moreover
  2209   {assume c: "?c = 0" and d: "?d>0"  
  2210     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2211     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2212     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2213     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2214     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2215     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) < 0"
  2216       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
  2217     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r < 0" 
  2218       using nonzero_mult_divide_cancel_left[OF d'] d
  2219       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2220     finally have ?thesis using c d nc nd 
  2221       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2222       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2223       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2224       by (simp add: field_simps)  }
  2225   moreover
  2226   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2227     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2228     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2229     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2230     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2231     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) > 0"
  2232       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
  2233     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r < 0" 
  2234       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2235         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2236     finally have ?thesis using c d nc nd 
  2237       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2238       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2239       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2240       by (simp add: field_simps )    }
  2241 ultimately show ?thesis by blast
  2242 qed
  2243 
  2244 definition "msubstle c t d s a r = 
  2245   evaldjf (split conj) 
  2246   [(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)))),
  2247   (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)))),
  2248    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2249    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2250    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2251    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2252    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"
  2253 
  2254 lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2255   shows "bound0 (msubstle c t d s a r)"
  2256 proof-
  2257   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)))),
  2258   (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)))),
  2259    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2260    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2261    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2262    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2263    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
  2264     using lp by (simp add: Let_def t s lt_nb )
  2265   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
  2266 qed
  2267 
  2268 lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))" 
  2269   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow> 
  2270   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")
  2271 proof-
  2272   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2273   let ?N = "\<lambda>p. Ipoly vs p"
  2274   let ?c = "?N c"
  2275   let ?d = "?N d"
  2276   let ?t = "?Nt x t"
  2277   let ?s = "?Nt x s"
  2278   let ?a = "?N a"
  2279   let ?r = "?Nt x r"
  2280   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2281   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2282   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
  2283   moreover
  2284   {assume c: "?c=0" and d: "?d=0"
  2285     hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
  2286   moreover
  2287   {assume dc: "?c*?d > 0" 
  2288     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2289     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2290     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2291     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2292     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2293       by (simp add: field_simps)
  2294     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2295     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2296       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2297     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) <= 0"
  2298       
  2299       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
  2300     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r <= 0" 
  2301       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2302       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2303     finally  have ?thesis using dc c d  nc nd dc'
  2304       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) 
  2305     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2306     by (simp add: field_simps order_less_not_sym[OF dc])}
  2307   moreover
  2308   {assume dc: "?c*?d < 0" 
  2309 
  2310     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2311       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
  2312     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2313     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2314     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2315       by (simp add: field_simps)
  2316     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2317     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2318       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2319 
  2320     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) >= 0"
  2321       
  2322       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
  2323     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r <= 0" 
  2324       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2325       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2326     finally  have ?thesis using dc c d  nc nd
  2327       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) 
  2328       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2329       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2330   moreover
  2331   {assume c: "?c > 0" and d: "?d=0"  
  2332     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2333     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2334     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2335     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2336     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2337     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) <= 0"
  2338       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
  2339     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r <= 0" 
  2340       using nonzero_mult_divide_cancel_left[OF c'] c
  2341       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2342     finally have ?thesis using c d nc nd 
  2343       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2344       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2345       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2346       by (simp add: field_simps )  }
  2347   moreover
  2348   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2349     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2350     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2351     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2352     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2353     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) >= 0"
  2354       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
  2355     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r <= 0" 
  2356       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2357         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2358     finally have ?thesis using c d nc nd 
  2359       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2360       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2361       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2362       by (simp add: field_simps )    }
  2363   moreover
  2364   moreover
  2365   {assume c: "?c = 0" and d: "?d>0"  
  2366     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2367     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2368     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2369     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2370     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2371     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) <= 0"
  2372       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
  2373     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r <= 0" 
  2374       using nonzero_mult_divide_cancel_left[OF d'] d
  2375       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2376     finally have ?thesis using c d nc nd 
  2377       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2378       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2379       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2380       by (simp add: field_simps )  }
  2381   moreover
  2382   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2383     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2384     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2385     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2386     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2387     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) >= 0"
  2388       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
  2389     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r <= 0" 
  2390       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2391         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2392     finally have ?thesis using c d nc nd 
  2393       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2394       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2395       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2396       by (simp add: field_simps )    }
  2397 ultimately show ?thesis by blast
  2398 qed
  2399 
  2400 
  2401 fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
  2402   "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
  2403 | "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
  2404 | "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
  2405 | "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
  2406 | "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
  2407 | "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
  2408 | "msubst p ((c,t),(d,s)) = p"
  2409 
  2410 lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
  2411   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"
  2412   using lp
  2413 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])
  2414 
  2415 lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
  2416   shows "bound0 (msubst p ((c,t),(d,s)))"
  2417   using lp t s
  2418   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
  2419 
  2420 lemma fr_eq_msubst: 
  2421   assumes lp: "islin p"
  2422   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)))))"
  2423   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  2424 proof-
  2425 from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2426 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2427   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"
  2428   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2429   from msubst_I[OF lp norm, of vs x bs t s] pts
  2430   have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
  2431 moreover
  2432 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2433   and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
  2434   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2435   from msubst_I[OF lp norm, of vs x bs t s] pts
  2436   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" ..}
  2437 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
  2438 from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
  2439 qed 
  2440 
  2441 lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  2442   shows "qfree p \<Longrightarrow> islin (simpfm p)"
  2443   by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
  2444 
  2445 definition 
  2446   "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2447   in if (mp = T \<or> pp = T) then T 
  2448      else (let U = alluopairs (remdups (uset  q))
  2449            in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
  2450 
  2451 lemma ferrack: 
  2452   assumes qf: "qfree p"
  2453   shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
  2454   (is "_ \<and> (?rhs = ?lhs)")
  2455 proof-
  2456   let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
  2457   let ?N = "\<lambda> t. Ipoly vs t"
  2458   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2459   let ?q = "simpfm p" 
  2460   let ?U = "remdups(uset ?q)"
  2461   let ?Up = "alluopairs ?U"
  2462   let ?mp = "minusinf ?q"
  2463   let ?pp = "plusinf ?q"
  2464   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2465   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2466   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2467   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2468   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"
  2469     by simp
  2470   {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
  2471     from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
  2472     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]
  2473     have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
  2474   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
  2475   {fix x assume xUp: "x \<in> set ?Up" 
  2476     then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U" 
  2477       and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto  
  2478     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] 
  2479     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
  2480     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] 
  2481     have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
  2482   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
  2483   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
  2484   with mp_nb pp_nb 
  2485   have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
  2486   from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
  2487   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2488   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
  2489   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_bex[OF th0] by simp
  2490   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)" 
  2491     by (simp add: evaldjf_ex)
  2492   also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
  2493   also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
  2494     apply (simp add: ferrack_def Let_def)
  2495     by (cases "?mp = T \<or> ?pp = T", auto)
  2496   finally show ?thesis using thqf by blast
  2497 qed
  2498 
  2499 definition "frpar p = simpfm (qelim p ferrack)"
  2500 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
  2501 proof-
  2502   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
  2503   from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
  2504 qed
  2505 
  2506 
  2507 section{* Second implemenation: Case splits not local *}
  2508 
  2509 lemma fr_eq2:  assumes lp: "islin p"
  2510   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow> 
  2511    ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> 
  2512     (Ifm vs (0#bs) p) \<or> 
  2513     (\<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> 
  2514     (\<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))"
  2515   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
  2516 proof
  2517   assume px: "\<exists> x. ?I x p"
  2518   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  2519   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  2520   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  2521     from inf_uset[OF lp nmi npi, OF px] 
  2522     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"
  2523       by auto
  2524     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2525     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2526     let ?s = "Itm vs (x # bs) s"
  2527     let ?t = "Itm vs (x # bs) t"
  2528     have eq2: "\<And>(x::'a). x + x = (1 + 1) * x"
  2529       by  (simp add: field_simps)
  2530     {assume "?c = 0 \<and> ?d = 0"
  2531       with ct have ?D by simp}
  2532     moreover
  2533     {assume z: "?c = 0" "?d \<noteq> 0"
  2534       from z have ?D using ct by auto}
  2535     moreover
  2536     {assume z: "?c \<noteq> 0" "?d = 0"
  2537       with ct have ?D by auto }
  2538     moreover
  2539     {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
  2540       from z have ?F using ct
  2541         apply - apply (rule bexI[where x = "(c,t)"], simp_all)
  2542         by (rule bexI[where x = "(d,s)"], simp_all)
  2543       hence ?D by blast}
  2544     ultimately have ?D by auto}
  2545   ultimately show "?D" by blast
  2546 next
  2547   assume "?D" 
  2548   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  2549   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  2550   moreover {assume f:"?F" hence "?E" by blast}
  2551   ultimately show "?E" by blast
  2552 qed
  2553 
  2554 definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
  2555 definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
  2556 definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
  2557 definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
  2558 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
  2559 
  2560 lemma msubsteq2: 
  2561   assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
  2562   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")
  2563   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2564   by (simp add: msubsteq2_def field_simps)
  2565 
  2566 lemma msubstltpos: 
  2567   assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
  2568   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")
  2569   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2570   by (simp add: msubstltpos_def field_simps)
  2571 
  2572 lemma msubstlepos: 
  2573   assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
  2574   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")
  2575   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2576   by (simp add: msubstlepos_def field_simps)
  2577 
  2578 lemma msubstltneg: 
  2579   assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
  2580   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")
  2581   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2582   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
  2583 
  2584 lemma msubstleneg: 
  2585   assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
  2586   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")
  2587   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2588   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
  2589 
  2590 fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2591   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
  2592 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
  2593 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2594 | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2595 | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
  2596 | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
  2597 | "msubstpos p c t = p"
  2598     
  2599 lemma msubstpos_I: 
  2600   assumes lp: "islin p" and pos: "Ipoly vs c > 0"
  2601   shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2602   using lp pos
  2603   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)
  2604 
  2605 fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2606   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
  2607 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
  2608 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2609 | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2610 | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
  2611 | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
  2612 | "msubstneg p c t = p"
  2613 
  2614 lemma msubstneg_I: 
  2615   assumes lp: "islin p" and pos: "Ipoly vs c < 0"
  2616   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2617   using lp pos
  2618   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)
  2619 
  2620 
  2621 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)))"
  2622 
  2623 lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
  2624   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2625 proof-
  2626   let ?c = "Ipoly vs c"
  2627   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))" 
  2628     by (simp_all add: polyneg_norm)
  2629   from nz have "?c > 0 \<or> ?c < 0" by arith
  2630   moreover
  2631   {assume c: "?c < 0"
  2632     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"]
  2633     have ?thesis by (auto simp add: msubst2_def)}
  2634   moreover
  2635   {assume c: "?c > 0"
  2636     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"]
  2637     have ?thesis by (auto simp add: msubst2_def)}
  2638   ultimately show ?thesis by blast
  2639 qed
  2640 
  2641 term msubsteq2
  2642 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
  2643   by (simp add: msubsteq2_def)
  2644 
  2645 lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
  2646   by (simp add: msubstltpos_def)
  2647 lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
  2648   by (simp add: msubstltneg_def)
  2649 
  2650 lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
  2651   by (simp add: msubstlepos_def)
  2652 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
  2653   by (simp add: msubstleneg_def)
  2654 
  2655 lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
  2656   shows "bound0 (msubstpos p c t)"
  2657 using lp tnb
  2658 by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
  2659 
  2660 lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2661   shows "bound0 (msubstneg p c t)"
  2662 using lp tnb
  2663 by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
  2664 
  2665 lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2666   shows "bound0 (msubst2 p c t)"
  2667 using lp tnb
  2668 by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
  2669     
  2670 lemma of_int2: "of_int 2 = 1 + 1"
  2671 proof-
  2672   have "(2::int) = 1 + 1" by simp
  2673   hence "of_int 2 = of_int (1 + 1)" by simp
  2674   thus ?thesis unfolding of_int_add by simp
  2675 qed
  2676 
  2677 lemma of_int_minus2: "of_int (-2) = - (1 + 1)"
  2678 proof-
  2679   have th: "(-2::int) = - 2" by simp
  2680   show ?thesis unfolding th by (simp only: of_int_minus of_int2)
  2681 qed
  2682 
  2683 
  2684 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
  2685   by (induct p rule: islin.induct, auto simp add: bound0_qf)
  2686 lemma fr_eq_msubst2: 
  2687   assumes lp: "islin p"
  2688   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)))))"
  2689   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
  2690 proof-
  2691   from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2692   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2693   have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
  2694   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
  2695   
  2696   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)"
  2697   proof-
  2698     {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
  2699       from H(1) th have "isnpoly n" by blast
  2700       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
  2701       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
  2702         by (simp add: polyneg_norm nn)
  2703       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 
  2704         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
  2705       from msubst2[OF lp nn nn2(1), of x bs t]
  2706       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"
  2707         using H(2) nn2 by (simp add: of_int_minus2 del: minus_add_distrib)}
  2708     moreover
  2709     {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"
  2710       from H(1) th have "isnpoly n" by blast
  2711       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
  2712         using H(2) by (simp_all add: polymul_norm n2)
  2713       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)}
  2714     ultimately show ?thesis by blast
  2715   qed
  2716   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).
  2717      \<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)" 
  2718   proof-
  2719     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2720      "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
  2721       from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2722       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" 
  2723         by (simp_all add: polymul_norm n2)
  2724       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)))"
  2725         by (simp_all add: polyneg_norm nn)
  2726       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"
  2727         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)
  2728       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
  2729       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" 
  2730         apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)
  2731         by (simp add: mult_commute)}
  2732     moreover
  2733     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2734       "\<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"
  2735      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2736       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"
  2737         using H(3,4) by (simp_all add: polymul_norm n2)
  2738       from msubst2[OF lp nn, of x bs ] H(3,4,5) 
  2739       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)}
  2740     ultimately show ?thesis by blast
  2741   qed
  2742   from fr_eq2[OF lp, of vs bs x] show ?thesis
  2743     unfolding eq0 eq1 eq2 by blast  
  2744 qed
  2745 
  2746 definition 
  2747 "ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2748  in if (mp = T \<or> pp = T) then T 
  2749   else (let U = remdups (uset  q)
  2750     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, 
  2751    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)]))"
  2752 
  2753 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
  2754 
  2755 lemma ferrack2: assumes qf: "qfree p"
  2756   shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
  2757   (is "_ \<and> (?rhs = ?lhs)")
  2758 proof-
  2759   let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
  2760   let ?N = "\<lambda> t. Ipoly vs t"
  2761   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2762   let ?q = "simpfm p" 
  2763   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
  2764   let ?U = "remdups(uset ?q)"
  2765   let ?Up = "alluopairs ?U"
  2766   let ?mp = "minusinf ?q"
  2767   let ?pp = "plusinf ?q"
  2768   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2769   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2770   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2771   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2772   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"
  2773     by simp
  2774   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)" 
  2775   proof-
  2776     {fix c t assume ct: "(c,t) \<in> set ?U"
  2777       hence tnb: "tmbound0 t" using U_l by blast
  2778       from msubst2_nb[OF lq tnb]
  2779       have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
  2780     thus ?thesis by auto
  2781   qed
  2782   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)" 
  2783   proof-
  2784     {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
  2785       from badc U_l alluopairs_set1[of ?U] 
  2786       have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
  2787       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}
  2788     thus ?thesis by auto
  2789   qed
  2790   have stupid: "bound0 F" by simp
  2791   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, 
  2792    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)]"
  2793   from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
  2794   have nb: "bound0 ?R "
  2795     by (simp add: list_disj_def disj_nb0 simpfm_bound0)
  2796   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))"
  2797 
  2798   {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
  2799     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))" 
  2800       by auto (simp add: isnpoly_def)
  2801     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
  2802       using norm by (simp_all add: polymul_norm)
  2803     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)))"
  2804       by (simp_all add: polyneg_norm norm2)
  2805     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")
  2806     proof
  2807       assume H: ?lhs
  2808       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" 
  2809         by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
  2810       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2811         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2812       show ?rhs by (simp add: field_simps)
  2813     next
  2814       assume H: ?rhs
  2815       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" 
  2816         by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
  2817       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2818         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2819       show ?lhs by (simp add: field_simps)
  2820     qed}
  2821   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
  2822     by clarsimp
  2823 
  2824   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2825   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))))"
  2826     using fr_eq_msubst2[OF lq, of vs bs x] by simp
  2827   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)))"
  2828     by (simp add: split_def)
  2829   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)))"
  2830     using alluopairs_bex[OF th0] by simp 
  2831   also have "\<dots> \<longleftrightarrow> ?I ?R" 
  2832     by (simp add: list_disj_def evaldjf_ex split_def)
  2833   also have "\<dots> \<longleftrightarrow> ?rhs"
  2834     unfolding ferrack2_def
  2835     apply (cases "?mp = T") 
  2836     apply (simp add: list_disj_def)
  2837     apply (cases "?pp = T") 
  2838     apply (simp add: list_disj_def)
  2839     by (simp_all add: Let_def decr0[OF nb])
  2840   finally show ?thesis using decr0_qf[OF nb]  
  2841     by (simp  add: ferrack2_def Let_def)
  2842 qed
  2843 
  2844 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
  2845 proof-
  2846   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
  2847   from qelim[OF th, of "prep p" bs] 
  2848 show ?thesis  unfolding frpar2_def by (auto simp add: prep)
  2849 qed
  2850 
  2851 ML {* 
  2852 structure ReflectedFRPar = 
  2853 struct
  2854 
  2855 val bT = HOLogic.boolT;
  2856 fun num rT x = HOLogic.mk_number rT x;
  2857 fun rrelT rT = [rT,rT] ---> rT;
  2858 fun rrT rT = [rT, rT] ---> bT;
  2859 fun divt rT = Const(@{const_name Rings.divide},rrelT rT);
  2860 fun timest rT = Const(@{const_name Groups.times},rrelT rT);
  2861 fun plust rT = Const(@{const_name Groups.plus},rrelT rT);
  2862 fun minust rT = Const(@{const_name Groups.minus},rrelT rT);
  2863 fun uminust rT = Const(@{const_name Groups.uminus}, rT --> rT);
  2864 fun powt rT = Const(@{const_name "power"}, [rT,@{typ "nat"}] ---> rT);
  2865 val brT = [bT, bT] ---> bT;
  2866 val nott = @{term "Not"};
  2867 val conjt = @{term HOL.conj};
  2868 val disjt = @{term HOL.disj};
  2869 val impt = @{term HOL.implies};
  2870 val ifft = @{term "op = :: bool => _"}
  2871 fun llt rT = Const(@{const_name Orderings.less},rrT rT);
  2872 fun lle rT = Const(@{const_name Orderings.less},rrT rT);
  2873 fun eqt rT = Const(@{const_name HOL.eq},rrT rT);
  2874 fun rz rT = Const(@{const_name Groups.zero},rT);
  2875 
  2876 fun dest_nat t = case t of
  2877   Const (@{const_name Suc}, _) $ t' => 1 + dest_nat t'
  2878 | _ => (snd o HOLogic.dest_number) t;
  2879 
  2880 fun num_of_term m t = 
  2881  case t of
  2882    Const(@{const_name Groups.uminus},_)$t => @{code poly.Neg} (num_of_term m t)
  2883  | Const(@{const_name Groups.plus},_)$a$b => @{code poly.Add} (num_of_term m a, num_of_term m b)
  2884  | Const(@{const_name Groups.minus},_)$a$b => @{code poly.Sub} (num_of_term m a, num_of_term m b)
  2885  | Const(@{const_name Groups.times},_)$a$b => @{code poly.Mul} (num_of_term m a, num_of_term m b)
  2886  | Const(@{const_name Power.power},_)$a$n => @{code poly.Pw} (num_of_term m a, dest_nat n)
  2887  | Const(@{const_name Rings.divide},_)$a$b => @{code poly.C} (HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
  2888  | _ => (@{code poly.C} (HOLogic.dest_number t |> snd,1) 
  2889          handle TERM _ => @{code poly.Bound} (AList.lookup (op aconv) m t |> the));
  2890 
  2891 fun tm_of_term m m' t = 
  2892  case t of
  2893    Const(@{const_name Groups.uminus},_)$t => @{code Neg} (tm_of_term m m' t)
  2894  | Const(@{const_name Groups.plus},_)$a$b => @{code Add} (tm_of_term m m' a, tm_of_term m m' b)
  2895  | Const(@{const_name Groups.minus},_)$a$b => @{code Sub} (tm_of_term m m' a, tm_of_term m m' b)
  2896  | Const(@{const_name Groups.times},_)$a$b => @{code Mul} (num_of_term m' a, tm_of_term m m' b)
  2897  | _ => (@{code CP} (num_of_term m' t) 
  2898          handle TERM _ => @{code Bound} (AList.lookup (op aconv) m t |> the)
  2899               | Option => @{code Bound} (AList.lookup (op aconv) m t |> the));
  2900 
  2901 fun term_of_num T m t = 
  2902  case t of
  2903   @{code poly.C} (a,b) => (if b = 1 then num T a else if b=0 then (rz T) 
  2904                                         else (divt T) $ num T a $ num T b)
  2905 | @{code poly.Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2906 | @{code poly.Add} (a,b) => (plust T)$(term_of_num T m a)$(term_of_num T m b)
  2907 | @{code poly.Mul} (a,b) => (timest T)$(term_of_num T m a)$(term_of_num T m b)
  2908 | @{code poly.Sub} (a,b) => (minust T)$(term_of_num T m a)$(term_of_num T m b)
  2909 | @{code poly.Neg} a => (uminust T)$(term_of_num T m a)
  2910 | @{code poly.Pw} (a,n) => (powt T)$(term_of_num T m t)$(HOLogic.mk_number HOLogic.natT n)
  2911 | @{code poly.CN} (c,n,p) => term_of_num T m (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)))
  2912 | _ => error "term_of_num: Unknown term";
  2913 
  2914 fun term_of_tm T m m' t = 
  2915  case t of
  2916   @{code CP} p => term_of_num T m' p
  2917 | @{code Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2918 | @{code Add} (a,b) => (plust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2919 | @{code Mul} (a,b) => (timest T)$(term_of_num T m' a)$(term_of_tm T m m' b)
  2920 | @{code Sub} (a,b) => (minust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2921 | @{code Neg} a => (uminust T)$(term_of_tm T m m' a)
  2922 | @{code CNP} (n,c,p) => term_of_tm T m m' (@{code Add}
  2923      (@{code Mul} (c, @{code Bound} n), p))
  2924 | _ => error "term_of_tm: Unknown term";
  2925 
  2926 fun fm_of_term m m' fm = 
  2927  case fm of
  2928     Const(@{const_name True},_) => @{code T}
  2929   | Const(@{const_name False},_) => @{code F}
  2930   | Const(@{const_name Not},_)$p => @{code NOT} (fm_of_term m m' p)
  2931   | Const(@{const_name HOL.conj},_)$p$q => @{code And} (fm_of_term m m' p, fm_of_term m m' q)
  2932   | Const(@{const_name HOL.disj},_)$p$q => @{code Or} (fm_of_term m m' p, fm_of_term m m' q)
  2933   | Const(@{const_name HOL.implies},_)$p$q => @{code Imp} (fm_of_term m m' p, fm_of_term m m' q)
  2934   | Const(@{const_name HOL.eq},ty)$p$q => 
  2935        if domain_type ty = bT then @{code Iff} (fm_of_term m m' p, fm_of_term m m' q)
  2936        else @{code Eq} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2937   | Const(@{const_name Orderings.less},_)$p$q => 
  2938         @{code Lt} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2939   | Const(@{const_name Orderings.less_eq},_)$p$q => 
  2940         @{code Le} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2941   | Const(@{const_name Ex},_)$Abs(xn,xT,p) => 
  2942      let val (xn', p') = Syntax_Trans.variant_abs (xn,xT,p)  (* FIXME !? *)
  2943          val x = Free(xn',xT)
  2944          fun incr i = i + 1
  2945          val m0 = (x,0):: (map (apsnd incr) m)
  2946       in @{code E} (fm_of_term m0 m' p') end
  2947   | Const(@{const_name All},_)$Abs(xn,xT,p) => 
  2948      let val (xn', p') = Syntax_Trans.variant_abs (xn,xT,p)  (* FIXME !? *)
  2949          val x = Free(xn',xT)
  2950          fun incr i = i + 1
  2951          val m0 = (x,0):: (map (apsnd incr) m)
  2952       in @{code A} (fm_of_term m0 m' p') end
  2953   | _ => error "fm_of_term";
  2954 
  2955 
  2956 fun term_of_fm T m m' t = 
  2957   case t of
  2958     @{code T} => Const(@{const_name True},bT)
  2959   | @{code F} => Const(@{const_name False},bT)
  2960   | @{code NOT} p => nott $ (term_of_fm T m m' p)
  2961   | @{code And} (p,q) => conjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  2962   | @{code Or} (p,q) => disjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  2963   | @{code Imp} (p,q) => impt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  2964   | @{code Iff} (p,q) => ifft $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  2965   | @{code Lt} p => (llt T) $ (term_of_tm T m m' p) $ (rz T)
  2966   | @{code Le} p => (lle T) $ (term_of_tm T m m' p) $ (rz T)
  2967   | @{code Eq} p => (eqt T) $ (term_of_tm T m m' p) $ (rz T)
  2968   | @{code NEq} p => nott $ ((eqt T) $ (term_of_tm T m m' p) $ (rz T))
  2969   | _ => error "term_of_fm: quantifiers!!!!???";
  2970 
  2971 fun frpar_oracle (T,m, m', fm) = 
  2972  let 
  2973    val t = HOLogic.dest_Trueprop fm
  2974    val im = 0 upto (length m - 1)
  2975    val im' = 0 upto (length m' - 1)   
  2976  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  2977                                                      (@{code frpar} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  2978  end;
  2979 
  2980 fun frpar_oracle2 (T,m, m', fm) = 
  2981  let 
  2982    val t = HOLogic.dest_Trueprop fm
  2983    val im = 0 upto (length m - 1)
  2984    val im' = 0 upto (length m' - 1)   
  2985  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  2986                                                      (@{code frpar2} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  2987  end;
  2988 
  2989 end;
  2990 
  2991 
  2992 *}
  2993 
  2994 oracle frpar_oracle = {* fn (ty, ts, ts', ct) => 
  2995  let 
  2996   val thy = Thm.theory_of_cterm ct
  2997  in cterm_of thy (ReflectedFRPar.frpar_oracle (ty,ts, ts', term_of ct))
  2998  end *}
  2999 
  3000 oracle frpar_oracle2 = {* fn (ty, ts, ts', ct) => 
  3001  let 
  3002   val thy = Thm.theory_of_cterm ct
  3003  in cterm_of thy (ReflectedFRPar.frpar_oracle2 (ty,ts, ts', term_of ct))
  3004  end *}
  3005 
  3006 ML{* 
  3007 structure FRParTac = 
  3008 struct
  3009 
  3010 fun frpar_tac T ps ctxt i = 
  3011  Object_Logic.full_atomize_tac i
  3012  THEN (fn st =>
  3013   let
  3014     val g = List.nth (cprems_of st, i - 1)
  3015     val thy = Proof_Context.theory_of ctxt
  3016     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3017     val th = frpar_oracle (T, fs,ps, (* Pattern.eta_long [] *)g)
  3018   in rtac (th RS iffD2) i st end);
  3019 
  3020 fun frpar2_tac T ps ctxt i = 
  3021  Object_Logic.full_atomize_tac i
  3022  THEN (fn st =>
  3023   let
  3024     val g = List.nth (cprems_of st, i - 1)
  3025     val thy = Proof_Context.theory_of ctxt
  3026     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3027     val th = frpar_oracle2 (T, fs,ps, (* Pattern.eta_long [] *)g)
  3028   in rtac (th RS iffD2) i st end);
  3029 
  3030 end;
  3031 
  3032 *}
  3033 
  3034 method_setup frpar = {*
  3035 let
  3036  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3037  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3038  val parsN = "pars"
  3039  val typN = "type"
  3040  val any_keyword = keyword parsN || keyword typN
  3041  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3042  val cterms = thms >> map Drule.dest_term;
  3043  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3044  val typ = Scan.unless any_keyword Args.typ
  3045 in
  3046  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3047   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar_tac T ps ctxt))
  3048 end
  3049 *} "Parametric QE for linear Arithmetic over fields, Version 1"
  3050 
  3051 method_setup frpar2 = {*
  3052 let
  3053  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3054  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3055  val parsN = "pars"
  3056  val typN = "type"
  3057  val any_keyword = keyword parsN || keyword typN
  3058  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3059  val cterms = thms >> map Drule.dest_term;
  3060  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3061  val typ = Scan.unless any_keyword Args.typ
  3062 in
  3063  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3064   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar2_tac T ps ctxt))
  3065 end
  3066 *} "Parametric QE for linear Arithmetic over fields, Version 2"
  3067 
  3068 
  3069 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3070   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3071   apply (simp add: field_simps)
  3072   apply (rule spec[where x=y])
  3073   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3074   by simp
  3075 
  3076 text{* Collins/Jones Problem *}
  3077 (*
  3078 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"
  3079 proof-
  3080   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")
  3081 by (simp add: field_simps)
  3082 have "?rhs"
  3083 
  3084   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}")
  3085   apply (simp add: field_simps)
  3086 oops
  3087 *)
  3088 (*
  3089 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"
  3090 apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3091 oops
  3092 *)
  3093 
  3094 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3095   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3096   apply (simp add: field_simps)
  3097   apply (rule spec[where x=y])
  3098   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3099   by simp
  3100 
  3101 text{* Collins/Jones Problem *}
  3102 
  3103 (*
  3104 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"
  3105 proof-
  3106   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")
  3107 by (simp add: field_simps)
  3108 have "?rhs"
  3109   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}")
  3110   apply simp
  3111 oops
  3112 *)
  3113 
  3114 (*
  3115 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"
  3116 apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3117 apply (simp add: field_simps linorder_neq_iff[symmetric])
  3118 apply ferrack
  3119 oops
  3120 *)
  3121 end