src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
author krauss
Wed Feb 23 11:15:06 2011 +0100 (2011-02-23)
changeset 41822 27afef7d6c37
parent 41821 c118ae98dfbf
child 41823 81d64ec48427
permissions -rw-r--r--
recdef -> fun
     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
     9   Reflected_Multivariate_Polynomial
    10   Dense_Linear_Order
    11   "~~/src/HOL/Library/Efficient_Nat"
    12 begin
    13 
    14 subsection {* Terms *}
    15 
    16 datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
    17   | Neg tm | Sub tm tm | CNP nat poly tm
    18   (* A size for poly to make inductive proofs simpler*)
    19 
    20 primrec tmsize :: "tm \<Rightarrow> nat" where
    21   "tmsize (CP c) = polysize c"
    22 | "tmsize (Bound n) = 1"
    23 | "tmsize (Neg a) = 1 + tmsize a"
    24 | "tmsize (Add a b) = 1 + tmsize a + tmsize b"
    25 | "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
    26 | "tmsize (Mul c a) = 1 + polysize c + tmsize a"
    27 | "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
    28 
    29   (* Semantics of terms tm *)
    30 primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
    31   "Itm vs bs (CP c) = (Ipoly vs c)"
    32 | "Itm vs bs (Bound n) = bs!n"
    33 | "Itm vs bs (Neg a) = -(Itm vs bs a)"
    34 | "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
    35 | "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
    36 | "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
    37 | "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"   
    38 
    39 
    40 fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
    41   "allpolys P (CP c) = P c"
    42 | "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
    43 | "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
    44 | "allpolys P (Neg p) = allpolys P p"
    45 | "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
    46 | "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
    47 | "allpolys P p = True"
    48 
    49 primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
    50   "tmboundslt n (CP c) = True"
    51 | "tmboundslt n (Bound m) = (m < n)"
    52 | "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
    53 | "tmboundslt n (Neg a) = tmboundslt n a"
    54 | "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
    55 | "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
    56 | "tmboundslt n (Mul i a) = tmboundslt n a"
    57 
    58 primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
    59   "tmbound0 (CP c) = True"
    60 | "tmbound0 (Bound n) = (n>0)"
    61 | "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
    62 | "tmbound0 (Neg a) = tmbound0 a"
    63 | "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
    64 | "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
    65 | "tmbound0 (Mul i a) = tmbound0 a"
    66 lemma tmbound0_I:
    67   assumes nb: "tmbound0 a"
    68   shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
    69 using nb
    70 by (induct a rule: tm.induct,auto simp add: nth_pos2)
    71 
    72 primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
    73   "tmbound n (CP c) = True"
    74 | "tmbound n (Bound m) = (n \<noteq> m)"
    75 | "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
    76 | "tmbound n (Neg a) = tmbound n a"
    77 | "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
    78 | "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
    79 | "tmbound n (Mul i a) = tmbound n a"
    80 lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
    81 
    82 lemma tmbound_I: 
    83   assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
    84   shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
    85   using nb le bnd
    86   by (induct t rule: tm.induct , auto)
    87 
    88 fun decrtm0:: "tm \<Rightarrow> tm" where
    89   "decrtm0 (Bound n) = Bound (n - 1)"
    90 | "decrtm0 (Neg a) = Neg (decrtm0 a)"
    91 | "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
    92 | "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
    93 | "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
    94 | "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
    95 | "decrtm0 a = a"
    96 
    97 fun incrtm0:: "tm \<Rightarrow> tm" where
    98   "incrtm0 (Bound n) = Bound (n + 1)"
    99 | "incrtm0 (Neg a) = Neg (incrtm0 a)"
   100 | "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
   101 | "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
   102 | "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
   103 | "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
   104 | "incrtm0 a = a"
   105 
   106 lemma decrtm0: assumes nb: "tmbound0 t"
   107   shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
   108   using nb by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)
   109 
   110 lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
   111   by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)
   112 
   113 primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
   114   "decrtm m (CP c) = (CP c)"
   115 | "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
   116 | "decrtm m (Neg a) = Neg (decrtm m a)"
   117 | "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
   118 | "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
   119 | "decrtm m (Mul c a) = Mul c (decrtm m a)"
   120 | "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
   121 
   122 primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
   123   "removen n [] = []"
   124 | "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
   125 
   126 lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
   127   by (induct xs arbitrary: n, auto)
   128 
   129 lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
   130   by (induct xs arbitrary: n, auto)
   131 
   132 lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
   133   by (induct xs arbitrary: n, auto)
   134 lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
   135   else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
   136 proof(induct xs arbitrary: n m)
   137   case Nil thus ?case by simp
   138 next
   139   case (Cons x xs n m)
   140   {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
   141   moreover
   142   {assume nxs: "\<not> (n \<ge> length (x#xs))" 
   143     {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
   144     moreover
   145     {assume mln: "\<not> (m < n)" 
   146       {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
   147       moreover
   148       {assume mxs: "\<not> (m \<le> length (x#xs))" 
   149         have th: "length (removen n (x#xs)) = length xs" 
   150           using removen_length[where n="n" and xs="x#xs"] nxs by simp
   151         with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
   152         hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
   153           using th nth_length_exceeds[OF mxs'] by auto
   154         hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
   155           by auto
   156         hence ?case using nxs mln mxs by auto }
   157       ultimately have ?case by blast
   158     }
   159     ultimately have ?case by blast
   160   } ultimately show ?case by blast
   161 qed
   162 
   163 lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
   164   and nle: "m \<le> length bs" 
   165   shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
   166   using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth)
   167 
   168 primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   169   "tmsubst0 t (CP c) = CP c"
   170 | "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
   171 | "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
   172 | "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
   173 | "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
   174 | "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
   175 | "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
   176 lemma tmsubst0:
   177   shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
   178   by (induct a rule: tm.induct) (auto simp add: nth_pos2)
   179 
   180 lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
   181   by (induct a rule: tm.induct) (auto simp add: nth_pos2)
   182 
   183 primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
   184   "tmsubst n t (CP c) = CP c"
   185 | "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
   186 | "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
   187              else CNP m c (tmsubst n t a))"
   188 | "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
   189 | "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
   190 | "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
   191 | "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
   192 
   193 lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
   194   shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
   195 using nb nlt
   196 by (induct a rule: tm.induct,auto simp add: nth_pos2)
   197 
   198 lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
   199 shows "tmbound0 (tmsubst 0 t a)"
   200 using tnb
   201 by (induct a rule: tm.induct, auto)
   202 
   203 lemma tmsubst_nb: assumes tnb: "tmbound m t"
   204 shows "tmbound m (tmsubst m t a)"
   205 using tnb
   206 by (induct a rule: tm.induct, auto)
   207 lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
   208   by (induct t, auto)
   209   (* Simplification *)
   210 
   211 consts
   212   tmadd:: "tm \<times> tm \<Rightarrow> tm"
   213 recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
   214   "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
   215   (if n1=n2 then 
   216   (let c = c1 +\<^sub>p c2
   217   in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
   218   else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
   219   else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
   220   "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
   221   "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
   222   "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
   223   "tmadd (a,b) = Add a b"
   224 
   225 lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
   226 apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
   227 apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
   228 apply (case_tac "n1 = n2", simp_all add: field_simps)
   229 apply (simp only: right_distrib[symmetric]) 
   230 by (auto simp del: polyadd simp add: polyadd[symmetric])
   231 
   232 lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
   233 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   234 
   235 lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
   236 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   237 lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
   238 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
   239 
   240 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)
   241 
   242 fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" where
   243   "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
   244 | "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
   245 | "tmmul t = (\<lambda> i. Mul i t)"
   246 
   247 lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
   248 by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
   249 
   250 lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
   251 by (induct t arbitrary: i rule: tmmul.induct, auto )
   252 
   253 lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
   254 by (induct t arbitrary: n rule: tmmul.induct, auto )
   255 lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
   256 by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
   257 
   258 lemma tmmul_allpolys_npoly[simp]: 
   259   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   260   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)
   261 
   262 definition tmneg :: "tm \<Rightarrow> tm" where
   263   "tmneg t \<equiv> tmmul t (C (- 1,1))"
   264 
   265 definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
   266   "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
   267 
   268 lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
   269 using tmneg_def[of t] 
   270 apply simp
   271 apply (subst number_of_Min)
   272 apply (simp only: of_int_minus)
   273 apply simp
   274 done
   275 
   276 lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
   277 using tmneg_def by simp
   278 
   279 lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
   280 using tmneg_def by simp
   281 lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
   282 using tmneg_def by simp
   283 lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
   284 lemma tmneg_allpolys_npoly[simp]: 
   285   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   286   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
   287   unfolding tmneg_def by auto
   288 
   289 lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
   290 using tmsub_def by simp
   291 
   292 lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
   293 using tmsub_def by simp
   294 lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
   295 using tmsub_def by simp
   296 lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
   297 using tmsub_def by simp
   298 lemma tmsub_allpolys_npoly[simp]: 
   299   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   300   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
   301   unfolding tmsub_def by (simp add: isnpoly_def)
   302 
   303 fun simptm:: "tm \<Rightarrow> tm" where
   304   "simptm (CP j) = CP (polynate j)"
   305 | "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
   306 | "simptm (Neg t) = tmneg (simptm t)"
   307 | "simptm (Add t s) = tmadd (simptm t,simptm s)"
   308 | "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
   309 | "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
   310 | "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))"
   311 
   312 lemma polynate_stupid: 
   313   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   314   shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a::{field_char_0, field_inverse_zero})" 
   315 apply (subst polynate[symmetric])
   316 apply simp
   317 done
   318 
   319 lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
   320 by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 
   321 
   322 lemma simptm_tmbound0[simp]: 
   323   "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
   324 by (induct t rule: simptm.induct, auto simp add: Let_def)
   325 
   326 lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
   327 by (induct t rule: simptm.induct, auto simp add: Let_def)
   328 lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
   329 by (induct t rule: simptm.induct, auto simp add: Let_def)
   330 
   331 lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
   332   by (simp_all add: isnpoly_def)
   333 lemma simptm_allpolys_npoly[simp]: 
   334   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   335   shows "allpolys isnpoly (simptm p)"
   336   by (induct p rule: simptm.induct, auto simp add: Let_def)
   337 
   338 declare let_cong[fundef_cong del]
   339 
   340 fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" where
   341   "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
   342 | "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
   343 | "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
   344 | "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
   345 | "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
   346 | "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
   347 | "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
   348 | "split0 t = (0\<^sub>p, t)"
   349 
   350 declare let_cong[fundef_cong]
   351 
   352 lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
   353   apply (rule exI[where x="fst (split0 p)"])
   354   apply (rule exI[where x="snd (split0 p)"])
   355   by simp
   356 
   357 lemma split0:
   358   "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
   359   apply (induct t rule: split0.induct)
   360   apply simp
   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 field_simps)
   365   apply (simp add: Let_def split_def field_simps)
   366   apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
   367   apply (simp add: Let_def split_def field_simps)
   368   apply (simp add: Let_def split_def field_simps)
   369   done
   370 
   371 lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
   372 proof-
   373   fix c' t'
   374   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   375   with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
   376 qed
   377 
   378 lemma split0_nb0: 
   379   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   380   shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
   381 proof-
   382   fix c' t'
   383   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
   384   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
   385 qed
   386 
   387 lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   388   shows "tmbound0 (snd (split0 t))"
   389   using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
   390 
   391 
   392 lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
   393   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   394 
   395 lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
   396   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   397 
   398 lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   399  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   400 
   401 lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
   402 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   403 
   404 lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
   405  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
   406 
   407 lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
   408 by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
   409 
   410 lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   411   shows 
   412   "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
   413   by (induct p rule: split0.induct, 
   414     auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
   415     Let_def split_def split0_stupid)
   416 
   417 subsection{* Formulae *}
   418 
   419 datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
   420   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
   421 
   422 
   423   (* A size for fm *)
   424 fun fmsize :: "fm \<Rightarrow> nat" where
   425   "fmsize (NOT p) = 1 + fmsize p"
   426 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
   427 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"
   428 | "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
   429 | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
   430 | "fmsize (E p) = 1 + fmsize p"
   431 | "fmsize (A p) = 4+ fmsize p"
   432 | "fmsize p = 1"
   433   (* several lemmas about fmsize *)
   434 lemma fmsize_pos[termination_simp]: "fmsize p > 0"        
   435 by (induct p rule: fmsize.induct) simp_all
   436 
   437   (* Semantics of formulae (fm) *)
   438 primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
   439   "Ifm vs bs T = True"
   440 | "Ifm vs bs F = False"
   441 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
   442 | "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
   443 | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
   444 | "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
   445 | "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
   446 | "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
   447 | "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
   448 | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
   449 | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
   450 | "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
   451 | "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
   452 
   453 fun not:: "fm \<Rightarrow> fm" where
   454   "not (NOT (NOT p)) = not p"
   455 | "not (NOT p) = p"
   456 | "not T = F"
   457 | "not F = T"
   458 | "not (Lt t) = Le (tmneg t)"
   459 | "not (Le t) = Lt (tmneg t)"
   460 | "not (Eq t) = NEq t"
   461 | "not (NEq t) = Eq t"
   462 | "not p = NOT p"
   463 lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
   464 by (induct p rule: not.induct) auto
   465 
   466 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   467   "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 
   468    if p = q then p else And p q)"
   469 lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
   470 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
   471 
   472 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   473   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
   474        else if p=q then p else Or p q)"
   475 
   476 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
   477 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
   478 
   479 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   480   "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 
   481     else Imp p q)"
   482 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
   483 by (cases "p=F \<or> q=T",simp_all add: imp_def) 
   484 
   485 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
   486   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
   487        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 
   488   Iff p q)"
   489 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
   490   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
   491 
   492   (* Quantifier freeness *)
   493 fun qfree:: "fm \<Rightarrow> bool" where
   494   "qfree (E p) = False"
   495 | "qfree (A p) = False"
   496 | "qfree (NOT p) = qfree p" 
   497 | "qfree (And p q) = (qfree p \<and> qfree q)" 
   498 | "qfree (Or  p q) = (qfree p \<and> qfree q)" 
   499 | "qfree (Imp p q) = (qfree p \<and> qfree q)" 
   500 | "qfree (Iff p q) = (qfree p \<and> qfree q)"
   501 | "qfree p = True"
   502 
   503   (* Boundedness and substitution *)
   504 
   505 primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
   506   "boundslt n T = True"
   507 | "boundslt n F = True"
   508 | "boundslt n (Lt t) = (tmboundslt n t)"
   509 | "boundslt n (Le t) = (tmboundslt n t)"
   510 | "boundslt n (Eq t) = (tmboundslt n t)"
   511 | "boundslt n (NEq t) = (tmboundslt n t)"
   512 | "boundslt n (NOT p) = boundslt n p"
   513 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
   514 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
   515 | "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
   516 | "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
   517 | "boundslt n (E p) = boundslt (Suc n) p"
   518 | "boundslt n (A p) = boundslt (Suc n) p"
   519 
   520 fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
   521   "bound0 T = True"
   522 | "bound0 F = True"
   523 | "bound0 (Lt a) = tmbound0 a"
   524 | "bound0 (Le a) = tmbound0 a"
   525 | "bound0 (Eq a) = tmbound0 a"
   526 | "bound0 (NEq a) = tmbound0 a"
   527 | "bound0 (NOT p) = bound0 p"
   528 | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
   529 | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
   530 | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
   531 | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
   532 | "bound0 p = False"
   533 lemma bound0_I:
   534   assumes bp: "bound0 p"
   535   shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
   536 using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
   537 by (induct p rule: bound0.induct,auto simp add: nth_pos2)
   538 
   539 primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
   540   "bound m T = True"
   541 | "bound m F = True"
   542 | "bound m (Lt t) = tmbound m t"
   543 | "bound m (Le t) = tmbound m t"
   544 | "bound m (Eq t) = tmbound m t"
   545 | "bound m (NEq t) = tmbound m t"
   546 | "bound m (NOT p) = bound m p"
   547 | "bound m (And p q) = (bound m p \<and> bound m q)"
   548 | "bound m (Or p q) = (bound m p \<and> bound m q)"
   549 | "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
   550 | "bound m (Iff p q) = (bound m p \<and> bound m q)"
   551 | "bound m (E p) = bound (Suc m) p"
   552 | "bound m (A p) = bound (Suc m) p"
   553 
   554 lemma bound_I:
   555   assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
   556   shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
   557   using bnd nb le tmbound_I[where bs=bs and vs = vs]
   558 proof(induct p arbitrary: bs n rule: fm.induct)
   559   case (E p bs n) 
   560   {fix y
   561     from E have bnd: "boundslt (length (y#bs)) p" 
   562       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   563     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   564   thus ?case by simp 
   565 next
   566   case (A p bs n) {fix y
   567     from A have bnd: "boundslt (length (y#bs)) p" 
   568       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
   569     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
   570   thus ?case by simp 
   571 qed auto
   572 
   573 fun decr0 :: "fm \<Rightarrow> fm" where
   574   "decr0 (Lt a) = Lt (decrtm0 a)"
   575 | "decr0 (Le a) = Le (decrtm0 a)"
   576 | "decr0 (Eq a) = Eq (decrtm0 a)"
   577 | "decr0 (NEq a) = NEq (decrtm0 a)"
   578 | "decr0 (NOT p) = NOT (decr0 p)" 
   579 | "decr0 (And p q) = conj (decr0 p) (decr0 q)"
   580 | "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
   581 | "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
   582 | "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
   583 | "decr0 p = p"
   584 
   585 lemma decr0: assumes nb: "bound0 p"
   586   shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
   587   using nb 
   588   by (induct p rule: decr0.induct, simp_all add: decrtm0)
   589 
   590 primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
   591   "decr m T = T"
   592 | "decr m F = F"
   593 | "decr m (Lt t) = (Lt (decrtm m t))"
   594 | "decr m (Le t) = (Le (decrtm m t))"
   595 | "decr m (Eq t) = (Eq (decrtm m t))"
   596 | "decr m (NEq t) = (NEq (decrtm m t))"
   597 | "decr m (NOT p) = NOT (decr m p)" 
   598 | "decr m (And p q) = conj (decr m p) (decr m q)"
   599 | "decr m (Or p q) = disj (decr m p) (decr m q)"
   600 | "decr m (Imp p q) = imp (decr m p) (decr m q)"
   601 | "decr m (Iff p q) = iff (decr m p) (decr m q)"
   602 | "decr m (E p) = E (decr (Suc m) p)"
   603 | "decr m (A p) = A (decr (Suc m) p)"
   604 
   605 lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
   606   and nle: "m < length bs" 
   607   shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
   608   using bnd nb nle
   609 proof(induct p arbitrary: bs m rule: fm.induct)
   610   case (E p bs m) 
   611   {fix x
   612     from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   613   and nle: "Suc m < length (x#bs)" by auto
   614     from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   615   } thus ?case by auto 
   616 next
   617   case (A p bs m)  
   618   {fix x
   619     from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
   620   and nle: "Suc m < length (x#bs)" by auto
   621     from A(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
   622   } thus ?case by auto
   623 qed (auto simp add: decrtm removen_nth)
   624 
   625 primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
   626   "subst0 t T = T"
   627 | "subst0 t F = F"
   628 | "subst0 t (Lt a) = Lt (tmsubst0 t a)"
   629 | "subst0 t (Le a) = Le (tmsubst0 t a)"
   630 | "subst0 t (Eq a) = Eq (tmsubst0 t a)"
   631 | "subst0 t (NEq a) = NEq (tmsubst0 t a)"
   632 | "subst0 t (NOT p) = NOT (subst0 t p)"
   633 | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
   634 | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
   635 | "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
   636 | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
   637 | "subst0 t (E p) = E p"
   638 | "subst0 t (A p) = A p"
   639 
   640 lemma subst0: assumes qf: "qfree p"
   641   shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
   642 using qf tmsubst0[where x="x" and bs="bs" and t="t"]
   643 by (induct p rule: fm.induct, auto)
   644 
   645 lemma subst0_nb:
   646   assumes bp: "tmbound0 t" and qf: "qfree p"
   647   shows "bound0 (subst0 t p)"
   648 using qf tmsubst0_nb[OF bp] bp
   649 by (induct p rule: fm.induct, auto)
   650 
   651 primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
   652   "subst n t T = T"
   653 | "subst n t F = F"
   654 | "subst n t (Lt a) = Lt (tmsubst n t a)"
   655 | "subst n t (Le a) = Le (tmsubst n t a)"
   656 | "subst n t (Eq a) = Eq (tmsubst n t a)"
   657 | "subst n t (NEq a) = NEq (tmsubst n t a)"
   658 | "subst n t (NOT p) = NOT (subst n t p)"
   659 | "subst n t (And p q) = And (subst n t p) (subst n t q)"
   660 | "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
   661 | "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
   662 | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
   663 | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
   664 | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
   665 
   666 lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
   667   shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
   668   using nb nlm
   669 proof (induct p arbitrary: bs n t rule: fm.induct)
   670   case (E p bs n) 
   671   {fix x 
   672     from E have bn: "boundslt (length (x#bs)) p" by simp 
   673     from E have nlm: "Suc n \<le> length (x#bs)" by simp
   674     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 
   675     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   676     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   677 thus ?case by simp 
   678 next
   679   case (A p bs n)   
   680   {fix x 
   681     from A have bn: "boundslt (length (x#bs)) p" by simp 
   682     from A have nlm: "Suc n \<le> length (x#bs)" by simp
   683     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 
   684     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
   685     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
   686 thus ?case by simp 
   687 qed(auto simp add: tmsubst)
   688 
   689 lemma subst_nb: assumes tnb: "tmbound m t"
   690 shows "bound m (subst m t p)"
   691 using tnb tmsubst_nb incrtm0_tmbound
   692 by (induct p arbitrary: m t rule: fm.induct, auto)
   693 
   694 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
   695 by (induct p rule: not.induct, auto)
   696 lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
   697 by (induct p rule: not.induct, auto)
   698 lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
   699 by (induct p rule: not.induct, auto)
   700 lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
   701  by (induct p rule: not.induct, auto)
   702 
   703 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
   704 using conj_def by auto 
   705 lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
   706 using conj_def by auto 
   707 lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
   708 using conj_def by auto 
   709 lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
   710 using conj_def by auto 
   711 
   712 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
   713 using disj_def by auto 
   714 lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
   715 using disj_def by auto 
   716 lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
   717 using disj_def by auto 
   718 lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
   719 using disj_def by auto 
   720 
   721 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
   722 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
   723 lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (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_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
   726 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
   727 lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
   728 using imp_def by auto 
   729 
   730 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
   731   by (unfold iff_def,cases "p=q", auto)
   732 lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
   733 using iff_def by (unfold iff_def,cases "p=q", auto)
   734 lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
   735 using iff_def by (unfold iff_def,cases "p=q", auto)
   736 lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
   737 using iff_def by auto 
   738 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
   739 by (induct p, simp_all)
   740 
   741 fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
   742   "isatom T = True"
   743 | "isatom F = True"
   744 | "isatom (Lt a) = True"
   745 | "isatom (Le a) = True"
   746 | "isatom (Eq a) = True"
   747 | "isatom (NEq a) = True"
   748 | "isatom p = False"
   749 
   750 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
   751 by (induct p, simp_all)
   752 
   753 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
   754   "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
   755   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
   756 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
   757   "evaldjf f ps \<equiv> foldr (djf f) ps F"
   758 
   759 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
   760 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
   761 (cases "f p", simp_all add: Let_def djf_def) 
   762 
   763 lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
   764   by(induct ps, simp_all add: evaldjf_def djf_Or)
   765 
   766 lemma evaldjf_bound0: 
   767   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
   768   shows "bound0 (evaldjf f xs)"
   769   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   770 
   771 lemma evaldjf_qf: 
   772   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
   773   shows "qfree (evaldjf f xs)"
   774   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 
   775 
   776 fun disjuncts :: "fm \<Rightarrow> fm list" where
   777   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
   778 | "disjuncts F = []"
   779 | "disjuncts p = [p]"
   780 
   781 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
   782 by(induct p rule: disjuncts.induct, auto)
   783 
   784 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
   785 proof-
   786   assume nb: "bound0 p"
   787   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
   788   thus ?thesis by (simp only: list_all_iff)
   789 qed
   790 
   791 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
   792 proof-
   793   assume qf: "qfree p"
   794   hence "list_all qfree (disjuncts p)"
   795     by (induct p rule: disjuncts.induct, auto)
   796   thus ?thesis by (simp only: list_all_iff)
   797 qed
   798 
   799 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   800   "DJ f p \<equiv> evaldjf f (disjuncts p)"
   801 
   802 lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
   803   and fF: "f F = F"
   804   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
   805 proof-
   806   have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
   807     by (simp add: DJ_def evaldjf_ex) 
   808   also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
   809   finally show ?thesis .
   810 qed
   811 
   812 lemma DJ_qf: assumes 
   813   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
   814   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
   815 proof(clarify)
   816   fix  p assume qf: "qfree p"
   817   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
   818   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
   819   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
   820   
   821   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
   822 qed
   823 
   824 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))"
   825   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
   826 proof(clarify)
   827   fix p::fm and bs
   828   assume qf: "qfree p"
   829   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
   830   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
   831   have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
   832     by (simp add: DJ_def evaldjf_ex)
   833   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
   834   also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
   835   finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
   836 qed
   837 
   838 fun conjuncts :: "fm \<Rightarrow> fm list" where
   839   "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
   840 | "conjuncts T = []"
   841 | "conjuncts p = [p]"
   842 
   843 definition list_conj :: "fm list \<Rightarrow> fm" where
   844   "list_conj ps \<equiv> foldr conj ps T"
   845 
   846 definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
   847   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
   848                    in conj (decr0 (list_conj yes)) (f (list_conj no)))"
   849 
   850 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
   851 proof-
   852   assume qf: "qfree p"
   853   hence "list_all qfree (conjuncts p)"
   854     by (induct p rule: conjuncts.induct, auto)
   855   thus ?thesis by (simp only: list_all_iff)
   856 qed
   857 
   858 lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
   859 by(induct p rule: conjuncts.induct, auto)
   860 
   861 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
   862 proof-
   863   assume nb: "bound0 p"
   864   hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
   865   thus ?thesis by (simp only: list_all_iff)
   866 qed
   867 
   868 fun islin :: "fm \<Rightarrow> bool" where
   869   "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)"
   870 | "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)"
   871 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   872 | "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   873 | "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   874 | "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
   875 | "islin (NOT p) = False"
   876 | "islin (Imp p q) = False"
   877 | "islin (Iff p q) = False"
   878 | "islin p = bound0 p"
   879 
   880 lemma islin_stupid: assumes nb: "tmbound0 p"
   881   shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
   882   using nb by (cases p, auto, case_tac nat, auto)+
   883 
   884 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
   885 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
   886 definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
   887 definition "neq p = not (eq p)"
   888 
   889 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
   890   apply(simp add: lt_def)
   891   apply(cases p, simp_all)
   892   apply (case_tac poly, simp_all add: isnpoly_def)
   893   done
   894 
   895 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
   896   apply(simp add: le_def)
   897   apply(cases p, simp_all)
   898   apply (case_tac poly, simp_all add: isnpoly_def)
   899   done
   900 
   901 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
   902   apply(simp add: eq_def)
   903   apply(cases p, simp_all)
   904   apply (case_tac poly, simp_all add: isnpoly_def)
   905   done
   906 
   907 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
   908   by(simp add: neq_def eq)
   909 
   910 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
   911   apply (simp add: lt_def)
   912   apply (cases p, simp_all)
   913   apply (case_tac poly, simp_all)
   914   apply (case_tac nat, simp_all)
   915   done
   916 
   917 lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
   918   apply (simp add: le_def)
   919   apply (cases p, simp_all)
   920   apply (case_tac poly, simp_all)
   921   apply (case_tac nat, simp_all)
   922   done
   923 
   924 lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
   925   apply (simp add: eq_def)
   926   apply (cases p, simp_all)
   927   apply (case_tac poly, simp_all)
   928   apply (case_tac nat, simp_all)
   929   done
   930 
   931 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
   932   apply (simp add: neq_def eq_def)
   933   apply (cases p, simp_all)
   934   apply (case_tac poly, simp_all)
   935   apply (case_tac nat, simp_all)
   936   done
   937 
   938 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
   939 definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
   940 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
   941 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
   942 
   943 lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   944   shows "islin (simplt t)"
   945   unfolding simplt_def 
   946   using split0_nb0'
   947 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])
   948   
   949 lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   950   shows "islin (simple t)"
   951   unfolding simple_def 
   952   using split0_nb0'
   953 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)
   954 lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   955   shows "islin (simpeq t)"
   956   unfolding simpeq_def 
   957   using split0_nb0'
   958 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)
   959 
   960 lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   961   shows "islin (simpneq t)"
   962   unfolding simpneq_def 
   963   using split0_nb0'
   964 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)
   965 
   966 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
   967   by (cases "split0 s", auto)
   968 lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
   969   and n: "allpolys isnpoly t"
   970   shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
   971   using n
   972   by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
   973 lemma simplt[simp]:
   974   shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
   975 proof-
   976   have n: "allpolys isnpoly (simptm t)" by simp
   977   let ?t = "simptm t"
   978   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
   979       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
   980       by (simp add: simplt_def Let_def split_def lt)}
   981   moreover
   982   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
   983     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
   984   }
   985   ultimately show ?thesis by blast
   986 qed
   987 
   988 lemma simple[simp]:
   989   shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
   990 proof-
   991   have n: "allpolys isnpoly (simptm t)" by simp
   992   let ?t = "simptm t"
   993   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
   994       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
   995       by (simp add: simple_def Let_def split_def le)}
   996   moreover
   997   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
   998     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
   999   }
  1000   ultimately show ?thesis by blast
  1001 qed
  1002 
  1003 lemma simpeq[simp]:
  1004   shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
  1005 proof-
  1006   have n: "allpolys isnpoly (simptm t)" by simp
  1007   let ?t = "simptm t"
  1008   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1009       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
  1010       by (simp add: simpeq_def Let_def split_def)}
  1011   moreover
  1012   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1013     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
  1014   }
  1015   ultimately show ?thesis by blast
  1016 qed
  1017 
  1018 lemma simpneq[simp]:
  1019   shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
  1020 proof-
  1021   have n: "allpolys isnpoly (simptm t)" by simp
  1022   let ?t = "simptm t"
  1023   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
  1024       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
  1025       by (simp add: simpneq_def Let_def split_def )}
  1026   moreover
  1027   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
  1028     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
  1029   }
  1030   ultimately show ?thesis by blast
  1031 qed
  1032 
  1033 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
  1034   apply (simp add: lt_def)
  1035   apply (cases t, auto)
  1036   apply (case_tac poly, auto)
  1037   done
  1038 
  1039 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
  1040   apply (simp add: le_def)
  1041   apply (cases t, auto)
  1042   apply (case_tac poly, auto)
  1043   done
  1044 
  1045 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
  1046   apply (simp add: eq_def)
  1047   apply (cases t, auto)
  1048   apply (case_tac poly, auto)
  1049   done
  1050 
  1051 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
  1052   apply (simp add: neq_def eq_def)
  1053   apply (cases t, auto)
  1054   apply (case_tac poly, auto)
  1055   done
  1056 
  1057 lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1058   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
  1059   using split0 [of "simptm t" vs bs]
  1060 proof(simp add: simplt_def Let_def split_def)
  1061   assume nb: "tmbound0 t"
  1062   hence nb': "tmbound0 (simptm t)" by simp
  1063   let ?c = "fst (split0 (simptm t))"
  1064   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1065   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1066   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1067   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1068   from iffD1[OF isnpolyh_unique[OF ths] th]
  1069   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1070   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
  1071        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
  1072 qed
  1073 
  1074 lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1075   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
  1076   using split0 [of "simptm t" vs bs]
  1077 proof(simp add: simple_def Let_def split_def)
  1078   assume nb: "tmbound0 t"
  1079   hence nb': "tmbound0 (simptm t)" by simp
  1080   let ?c = "fst (split0 (simptm t))"
  1081   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1082   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1083   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1084   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1085   from iffD1[OF isnpolyh_unique[OF ths] th]
  1086   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1087   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
  1088        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
  1089 qed
  1090 
  1091 lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1092   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
  1093   using split0 [of "simptm t" vs bs]
  1094 proof(simp add: simpeq_def Let_def split_def)
  1095   assume nb: "tmbound0 t"
  1096   hence nb': "tmbound0 (simptm t)" by simp
  1097   let ?c = "fst (split0 (simptm t))"
  1098   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1099   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1100   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1101   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1102   from iffD1[OF isnpolyh_unique[OF ths] th]
  1103   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1104   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
  1105        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
  1106 qed
  1107 
  1108 lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1109   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
  1110   using split0 [of "simptm t" vs bs]
  1111 proof(simp add: simpneq_def Let_def split_def)
  1112   assume nb: "tmbound0 t"
  1113   hence nb': "tmbound0 (simptm t)" by simp
  1114   let ?c = "fst (split0 (simptm t))"
  1115   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  1116   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  1117   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  1118   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  1119   from iffD1[OF isnpolyh_unique[OF ths] th]
  1120   have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  1121   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
  1122        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
  1123 qed
  1124 
  1125 fun conjs   :: "fm \<Rightarrow> fm list" where
  1126   "conjs (And p q) = (conjs p)@(conjs q)"
  1127 | "conjs T = []"
  1128 | "conjs p = [p]"
  1129 lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
  1130 by (induct p rule: conjs.induct, auto)
  1131 definition list_disj :: "fm list \<Rightarrow> fm" where
  1132   "list_disj ps \<equiv> foldr disj ps F"
  1133 
  1134 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
  1135   by (induct ps, auto simp add: list_conj_def)
  1136 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
  1137   by (induct ps, auto simp add: list_conj_def conj_qf)
  1138 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
  1139   by (induct ps, auto simp add: list_disj_def)
  1140 
  1141 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
  1142   unfolding conj_def by auto
  1143 
  1144 lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
  1145   apply (induct p rule: conjs.induct) 
  1146   apply (unfold conjs.simps)
  1147   apply (unfold set_append)
  1148   apply (unfold ball_Un)
  1149   apply (unfold bound.simps)
  1150   apply auto
  1151   done
  1152 
  1153 lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
  1154   apply (induct p rule: conjs.induct) 
  1155   apply (unfold conjs.simps)
  1156   apply (unfold set_append)
  1157   apply (unfold ball_Un)
  1158   apply (unfold boundslt.simps)
  1159   apply blast
  1160 by simp_all
  1161 
  1162 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
  1163   unfolding list_conj_def
  1164   by (induct ps, auto simp add: conj_boundslt)
  1165 
  1166 lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
  1167   shows "bound n (list_conj ps)"
  1168   using bnd
  1169   unfolding list_conj_def
  1170   by (induct ps, auto simp add: conj_nb)
  1171 
  1172 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
  1173 unfolding list_conj_def by (induct ps , auto)
  1174 
  1175 lemma CJNB_qe: 
  1176   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1177   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
  1178 proof(clarify)
  1179   fix bs p
  1180   assume qfp: "qfree p"
  1181   let ?cjs = "conjuncts p"
  1182   let ?yes = "fst (partition bound0 ?cjs)"
  1183   let ?no = "snd (partition bound0 ?cjs)"
  1184   let ?cno = "list_conj ?no"
  1185   let ?cyes = "list_conj ?yes"
  1186   have part: "partition bound0 ?cjs = (?yes,?no)" by simp
  1187   from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
  1188   hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
  1189   hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
  1190   from conjuncts_qf[OF qfp] partition_set[OF part] 
  1191   have " \<forall>q\<in> set ?no. qfree q" by auto
  1192   hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
  1193   with qe have cno_qf:"qfree (qe ?cno )" 
  1194     and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
  1195   from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
  1196     by (simp add: CJNB_def Let_def conj_qf split_def)
  1197   {fix bs
  1198     from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
  1199     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
  1200       using partition_set[OF part] by auto
  1201     finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
  1202   hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
  1203   also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
  1204     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
  1205   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
  1206     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
  1207   also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
  1208     using qe[rule_format, OF no_qf] by auto
  1209   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" 
  1210     by (simp add: Let_def CJNB_def split_def)
  1211   with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
  1212 qed
  1213 
  1214 consts simpfm :: "fm \<Rightarrow> fm"
  1215 recdef simpfm "measure fmsize"
  1216   "simpfm (Lt t) = simplt (simptm t)"
  1217   "simpfm (Le t) = simple (simptm t)"
  1218   "simpfm (Eq t) = simpeq(simptm t)"
  1219   "simpfm (NEq t) = simpneq(simptm t)"
  1220   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
  1221   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
  1222   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
  1223   "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
  1224   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
  1225   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
  1226   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
  1227   "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
  1228   "simpfm (NOT (Eq t)) = simpneq t"
  1229   "simpfm (NOT (NEq t)) = simpeq t"
  1230   "simpfm (NOT (Le t)) = simplt (Neg t)"
  1231   "simpfm (NOT (Lt t)) = simple (Neg t)"
  1232   "simpfm (NOT (NOT p)) = simpfm p"
  1233   "simpfm (NOT T) = F"
  1234   "simpfm (NOT F) = T"
  1235   "simpfm p = p"
  1236 
  1237 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
  1238 by(induct p arbitrary: bs rule: simpfm.induct, auto)
  1239 
  1240 lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1241   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
  1242 by (induct p rule: simpfm.induct, auto)
  1243 
  1244 lemma lt_qf[simp]: "qfree (lt t)"
  1245   apply (cases t, auto simp add: lt_def)
  1246   by (case_tac poly, auto)
  1247 
  1248 lemma le_qf[simp]: "qfree (le t)"
  1249   apply (cases t, auto simp add: le_def)
  1250   by (case_tac poly, auto)
  1251 
  1252 lemma eq_qf[simp]: "qfree (eq t)"
  1253   apply (cases t, auto simp add: eq_def)
  1254   by (case_tac poly, auto)
  1255 
  1256 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
  1257 
  1258 lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
  1259 lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
  1260 lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
  1261 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
  1262 
  1263 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
  1264 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
  1265 
  1266 lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
  1267 lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
  1268 
  1269 lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  1270   shows "qfree p \<Longrightarrow> islin (simpfm p)" 
  1271   apply (induct p rule: simpfm.induct)
  1272   apply (simp_all add: conj_lin disj_lin)
  1273   done
  1274 
  1275 consts prep :: "fm \<Rightarrow> fm"
  1276 recdef prep "measure fmsize"
  1277   "prep (E T) = T"
  1278   "prep (E F) = F"
  1279   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
  1280   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
  1281   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
  1282   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
  1283   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
  1284   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
  1285   "prep (E p) = E (prep p)"
  1286   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
  1287   "prep (A p) = prep (NOT (E (NOT p)))"
  1288   "prep (NOT (NOT p)) = prep p"
  1289   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
  1290   "prep (NOT (A p)) = prep (E (NOT p))"
  1291   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
  1292   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
  1293   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
  1294   "prep (NOT p) = not (prep p)"
  1295   "prep (Or p q) = disj (prep p) (prep q)"
  1296   "prep (And p q) = conj (prep p) (prep q)"
  1297   "prep (Imp p q) = prep (Or (NOT p) q)"
  1298   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
  1299   "prep p = p"
  1300 (hints simp add: fmsize_pos)
  1301 lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
  1302 by (induct p arbitrary: bs rule: prep.induct, auto)
  1303 
  1304 
  1305 
  1306   (* Generic quantifier elimination *)
  1307 function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
  1308   "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
  1309 | "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
  1310 | "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
  1311 | "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
  1312 | "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
  1313 | "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
  1314 | "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
  1315 | "qelim p = (\<lambda> y. simpfm p)"
  1316 by pat_completeness simp_all
  1317 termination by (relation "measure fmsize") auto
  1318 
  1319 lemma qelim:
  1320   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  1321   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
  1322 using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
  1323 by (induct p rule: qelim.induct) auto
  1324 
  1325 subsection{* Core Procedure *}
  1326 
  1327 fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
  1328   "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
  1329 | "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
  1330 | "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1331 | "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1332 | "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
  1333 | "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
  1334 | "minusinf p = p"
  1335 
  1336 fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
  1337   "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
  1338 | "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
  1339 | "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  1340 | "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  1341 | "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
  1342 | "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
  1343 | "plusinf p = p"
  1344 
  1345 lemma minusinf_inf: assumes lp:"islin p"
  1346   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1347   using lp
  1348 proof (induct p rule: minusinf.induct)
  1349   case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1350 next
  1351   case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
  1352 next
  1353   case (3 c e) hence nbe: "tmbound0 e" by simp
  1354   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1355   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1356   let ?c = "Ipoly vs c"
  1357   let ?e = "Itm vs (y#bs) e"
  1358   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1359   moreover {assume "?c = 0" hence ?case 
  1360       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1361   moreover {assume cp: "?c > 0"
  1362     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1363         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1364       hence "?c * x + ?e < 0" by simp
  1365       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1366         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1367   moreover {assume cp: "?c < 0"
  1368     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1369         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1370       hence "?c * x + ?e > 0" by simp
  1371       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
  1372         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1373   ultimately show ?case by blast
  1374 next
  1375   case (4 c e)  hence nbe: "tmbound0 e" by simp
  1376   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1377   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1378   let ?c = "Ipoly vs c"
  1379   let ?e = "Itm vs (y#bs) e"
  1380   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1381   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1382   moreover {assume cp: "?c > 0"
  1383     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1384         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1385       hence "?c * x + ?e < 0" by simp
  1386       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1387         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1388   moreover {assume cp: "?c < 0"
  1389     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1390         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1391       hence "?c * x + ?e > 0" by simp
  1392       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
  1393         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1394   ultimately show ?case by blast
  1395 next
  1396   case (5 c e)  hence nbe: "tmbound0 e" by simp
  1397   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1398   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1399   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
  1400   let ?c = "Ipoly vs c"
  1401   let ?e = "Itm vs (y#bs) e"
  1402   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1403   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1404   moreover {assume cp: "?c > 0"
  1405     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1406         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1407       hence "?c * x + ?e < 0" by simp
  1408       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1409         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1410   moreover {assume cp: "?c < 0"
  1411     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1412         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1413       hence "?c * x + ?e > 0" by simp
  1414       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
  1415         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1416   ultimately show ?case by blast
  1417 next
  1418   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1419   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1420   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1421   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1422   let ?c = "Ipoly vs c"
  1423   let ?e = "Itm vs (y#bs) e"
  1424   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1425   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1426   moreover {assume cp: "?c > 0"
  1427     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
  1428         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1429       hence "?c * x + ?e < 0" by simp
  1430       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1431         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1432   moreover {assume cp: "?c < 0"
  1433     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
  1434         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1435       hence "?c * x + ?e > 0" by simp
  1436       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
  1437         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1438   ultimately show ?case by blast
  1439 qed (auto)
  1440 
  1441 lemma plusinf_inf: assumes lp:"islin p"
  1442   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  1443   using lp
  1444 proof (induct p rule: plusinf.induct)
  1445   case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1446 next
  1447   case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
  1448 next
  1449   case (3 c e) hence nbe: "tmbound0 e" by simp
  1450   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1451   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1452   let ?c = "Ipoly vs c"
  1453   let ?e = "Itm vs (y#bs) e"
  1454   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1455   moreover {assume "?c = 0" hence ?case 
  1456       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  1457   moreover {assume cp: "?c > 0"
  1458     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e" 
  1459         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1460       hence "?c * x + ?e > 0" by simp
  1461       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1462         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  1463   moreover {assume cp: "?c < 0"
  1464     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1465         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1466       hence "?c * x + ?e < 0" by simp
  1467       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
  1468         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  1469   ultimately show ?case by blast
  1470 next
  1471   case (4 c e) hence nbe: "tmbound0 e" by simp
  1472   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1473   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  1474   let ?c = "Ipoly vs c"
  1475   let ?e = "Itm vs (y#bs) e"
  1476   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1477   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1478   moreover {assume cp: "?c > 0"
  1479     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1480         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1481       hence "?c * x + ?e > 0" by simp
  1482       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1483         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1484   moreover {assume cp: "?c < 0"
  1485     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1486         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1487       hence "?c * x + ?e < 0" by simp
  1488       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
  1489         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  1490   ultimately show ?case by blast
  1491 next
  1492   case (5 c e) hence nbe: "tmbound0 e" by simp
  1493   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1494   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1495   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]
  1496   let ?c = "Ipoly vs c"
  1497   let ?e = "Itm vs (y#bs) e"
  1498   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1499   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1500   moreover {assume cp: "?c > 0"
  1501     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1502         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1503       hence "?c * x + ?e > 0" by simp
  1504       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1505         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1506   moreover {assume cp: "?c < 0"
  1507     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1508         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1509       hence "?c * x + ?e < 0" by simp
  1510       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
  1511         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  1512   ultimately show ?case by blast
  1513 next
  1514   case (6 c e)  hence nbe: "tmbound0 e" by simp
  1515   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  1516   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  1517   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  1518   let ?c = "Ipoly vs c"
  1519   let ?e = "Itm vs (y#bs) e"
  1520   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  1521   moreover {assume "?c = 0" hence ?case using eqs by auto}
  1522   moreover {assume cp: "?c > 0"
  1523     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
  1524         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1525       hence "?c * x + ?e > 0" by simp
  1526       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1527         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1528   moreover {assume cp: "?c < 0"
  1529     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
  1530         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
  1531       hence "?c * x + ?e < 0" by simp
  1532       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
  1533         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  1534   ultimately show ?case by blast
  1535 qed (auto)
  1536 
  1537 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)" 
  1538   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1539 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)" 
  1540   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
  1541 
  1542 lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
  1543   shows "\<exists>x. Ifm vs (x#bs) p"
  1544 proof-
  1545   from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1546   have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
  1547   from minusinf_inf[OF lp, where bs="bs"] 
  1548   obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
  1549   from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
  1550   moreover have "z - 1 < z" by simp
  1551   ultimately show ?thesis using z_def by auto
  1552 qed
  1553 
  1554 lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
  1555   shows "\<exists>x. Ifm vs (x#bs) p"
  1556 proof-
  1557   from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  1558   have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
  1559   from plusinf_inf[OF lp, where bs="bs"] 
  1560   obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
  1561   from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
  1562   moreover have "z + 1 > z" by simp
  1563   ultimately show ?thesis using z_def by auto
  1564 qed
  1565 
  1566 fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
  1567   "uset (And p q) = uset p @ uset q"
  1568 | "uset (Or p q) = uset p @ uset q"
  1569 | "uset (Eq (CNP 0 a e))  = [(a,e)]"
  1570 | "uset (Le (CNP 0 a e))  = [(a,e)]"
  1571 | "uset (Lt (CNP 0 a e))  = [(a,e)]"
  1572 | "uset (NEq (CNP 0 a e)) = [(a,e)]"
  1573 | "uset p = []"
  1574 
  1575 lemma uset_l:
  1576   assumes lp: "islin p"
  1577   shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
  1578 using lp by(induct p rule: uset.induct,auto)
  1579 
  1580 lemma minusinf_uset0:
  1581   assumes lp: "islin p"
  1582   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
  1583   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1584   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c" 
  1585 proof-
  1586   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)" 
  1587     using lp nmi ex
  1588     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
  1589     apply (auto simp add: linorder_not_less order_le_less)
  1590     done 
  1591   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
  1592   hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
  1593     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
  1594     by (auto simp add: mult_commute del: divide_minus_left)
  1595   thus ?thesis using csU by auto
  1596 qed
  1597 
  1598 lemma minusinf_uset:
  1599   assumes lp: "islin p"
  1600   and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
  1601   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1602   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c" 
  1603 proof-
  1604   from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))" 
  1605     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
  1606   from minusinf_uset0[OF lp nmi' ex] 
  1607   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
  1608   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1609   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1610 qed
  1611 
  1612 
  1613 lemma plusinf_uset0:
  1614   assumes lp: "islin p"
  1615   and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
  1616   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1617   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c" 
  1618 proof-
  1619   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)" 
  1620     using lp nmi ex
  1621     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
  1622     apply (auto simp add: linorder_not_less order_le_less)
  1623     done 
  1624   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
  1625   hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
  1626     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
  1627     by (auto simp add: mult_commute del: divide_minus_left)
  1628   thus ?thesis using csU by auto
  1629 qed
  1630 
  1631 lemma plusinf_uset:
  1632   assumes lp: "islin p"
  1633   and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
  1634   and ex: "Ifm vs (x#bs) p" (is "?I x p")
  1635   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c" 
  1636 proof-
  1637   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))" 
  1638     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
  1639   from plusinf_uset0[OF lp nmi' ex] 
  1640   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
  1641   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  1642   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
  1643 qed
  1644 
  1645 lemma lin_dense: 
  1646   assumes lp: "islin p"
  1647   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)" 
  1648   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
  1649   and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
  1650   and ly: "l < y" and yu: "y < u"
  1651   shows "Ifm vs (y#bs) p"
  1652 using lp px noS
  1653 proof (induct p rule: islin.induct) 
  1654   case (5 c s)
  1655   from "5.prems" 
  1656   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1657     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
  1658     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
  1659   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1660   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1661   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1662   moreover
  1663   {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"])}
  1664   moreover
  1665   {assume c: "?N c > 0"
  1666       from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1667       have px': "x < - ?Nt x s / ?N c" 
  1668         by (auto simp add: not_less field_simps) 
  1669     {assume y: "y < - ?Nt x s / ?N c" 
  1670       hence "y * ?N c < - ?Nt x s"
  1671         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1672       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1673       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1674     moreover
  1675     {assume y: "y > -?Nt x s / ?N c" 
  1676       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1677       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1678       with lx px' have "False" by simp  hence ?case by simp }
  1679     ultimately have ?case using ycs by blast
  1680   }
  1681   moreover
  1682   {assume c: "?N c < 0"
  1683       from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]  
  1684       have px': "x > - ?Nt x s / ?N c" 
  1685         by (auto simp add: not_less field_simps) 
  1686     {assume y: "y > - ?Nt x s / ?N c" 
  1687       hence "y * ?N c < - ?Nt x s"
  1688         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1689       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1690       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1691     moreover
  1692     {assume y: "y < -?Nt x s / ?N c" 
  1693       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1694       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1695       with xu px' have "False" by simp  hence ?case by simp }
  1696     ultimately have ?case using ycs by blast
  1697   }
  1698   ultimately show ?case by blast
  1699 next
  1700   case (6 c s)
  1701   from "6.prems" 
  1702   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1703     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
  1704     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
  1705   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1706   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1707   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1708   moreover
  1709   {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"])}
  1710   moreover
  1711   {assume c: "?N c > 0"
  1712       from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
  1713       have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps) 
  1714     {assume y: "y < - ?Nt x s / ?N c" 
  1715       hence "y * ?N c < - ?Nt x s"
  1716         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1717       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1718       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1719     moreover
  1720     {assume y: "y > -?Nt x s / ?N c" 
  1721       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1722       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1723       with lx px' have "False" by simp  hence ?case by simp }
  1724     ultimately have ?case using ycs by blast
  1725   }
  1726   moreover
  1727   {assume c: "?N c < 0"
  1728       from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]  
  1729       have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps) 
  1730     {assume y: "y > - ?Nt x s / ?N c" 
  1731       hence "y * ?N c < - ?Nt x s"
  1732         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
  1733       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
  1734       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
  1735     moreover
  1736     {assume y: "y < -?Nt x s / ?N c" 
  1737       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1738       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1739       with xu px' have "False" by simp  hence ?case by simp }
  1740     ultimately have ?case using ycs by blast
  1741   }
  1742   ultimately show ?case by blast
  1743 next
  1744     case (3 c s)
  1745   from "3.prems" 
  1746   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1747     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
  1748     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
  1749   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1750   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1751   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  1752   moreover
  1753   {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"])}
  1754   moreover
  1755   {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
  1756     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1757     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1758     {assume y: "y < -?Nt x s / ?N c" 
  1759       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1760       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1761       with xu px' have "False" by simp  hence ?case by simp }
  1762     moreover
  1763     {assume y: "y > -?Nt x s / ?N c" 
  1764       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1765       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1766       with lx px' have "False" by simp  hence ?case by simp }
  1767     ultimately have ?case using ycs by blast
  1768   }
  1769   moreover
  1770   {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
  1771     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
  1772     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
  1773     {assume y: "y < -?Nt x s / ?N c" 
  1774       with ly have eu: "l < - ?Nt x s / ?N c" by auto
  1775       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
  1776       with xu px' have "False" by simp  hence ?case by simp }
  1777     moreover
  1778     {assume y: "y > -?Nt x s / ?N c" 
  1779       with yu have eu: "u > - ?Nt x s / ?N c" by auto
  1780       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
  1781       with lx px' have "False" by simp  hence ?case by simp }
  1782     ultimately have ?case using ycs by blast
  1783   }
  1784   ultimately show ?case by blast
  1785 next
  1786     case (4 c s)
  1787   from "4.prems" 
  1788   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
  1789     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
  1790     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
  1791   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  1792   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  1793   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
  1794   moreover
  1795   {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"])}
  1796   moreover
  1797   {assume c: "?N c \<noteq> 0"
  1798     from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
  1799       by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
  1800   ultimately show ?case by blast
  1801 qed (auto simp add: nth_pos2 tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])
  1802 
  1803 lemma one_plus_one_pos[simp]: "(1::'a::{linordered_field}) + 1 > 0"
  1804 proof-
  1805   have op: "(1::'a) > 0" by simp
  1806   from add_pos_pos[OF op op] show ?thesis . 
  1807 qed
  1808 
  1809 lemma one_plus_one_nonzero[simp]: "(1::'a::{linordered_field}) + 1 \<noteq> 0" 
  1810   using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le) 
  1811 
  1812 lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{linordered_field})" 
  1813 proof-
  1814   have "(u + u) = (1 + 1) * u" by (simp add: field_simps)
  1815   hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
  1816   with nonzero_mult_divide_cancel_left[OF one_plus_one_nonzero, of u] show ?thesis by simp
  1817 qed
  1818 
  1819 lemma inf_uset:
  1820   assumes lp: "islin p"
  1821   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
  1822   and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
  1823   and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
  1824   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" 
  1825 proof-
  1826   let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
  1827   let ?N = "Ipoly vs"
  1828   let ?U = "set (uset p)"
  1829   from ex obtain a where pa: "?I a p" by blast
  1830   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
  1831   have nmi': "\<not> (?I a (?M p))" by simp
  1832   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
  1833   have npi': "\<not> (?I a (?P p))" by simp
  1834   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"
  1835   proof-
  1836     let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
  1837     have fM: "finite ?M" by auto
  1838     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] 
  1839     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
  1840     then obtain "c" "t" "d" "s" where 
  1841       ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U" 
  1842       and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
  1843     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 
  1844     have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
  1845     from ctU have Mne: "?M \<noteq> {}" by auto
  1846     hence Une: "?U \<noteq> {}" by simp
  1847     let ?l = "Min ?M"
  1848     let ?u = "Max ?M"
  1849     have linM: "?l \<in> ?M" using fM Mne by simp
  1850     have uinM: "?u \<in> ?M" using fM Mne by simp
  1851     have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
  1852     have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto 
  1853     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
  1854     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
  1855     have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
  1856     have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
  1857     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
  1858     have "(\<exists> s\<in> ?M. ?I s p) \<or> 
  1859       (\<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)" .
  1860     moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
  1861       hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
  1862       then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
  1863       from half_sum_eq[of u] pu tuu 
  1864       have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / (1 + 1)) p" by simp
  1865       with tuU have ?thesis by blast}
  1866     moreover{
  1867       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"
  1868       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
  1869         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"
  1870         by blast
  1871       from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
  1872       then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
  1873       from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
  1874       then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
  1875       from t1x xt2 have t1t2: "t1 < t2" by simp
  1876       let ?u = "(t1 + t2) / (1 + 1)"
  1877       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
  1878       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
  1879       with t1uU t2uU t1u t2u have ?thesis by blast}
  1880     ultimately show ?thesis by blast
  1881   qed
  1882   then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U" 
  1883     and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / (1 + 1)) p" by blast
  1884   from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
  1885   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
  1886     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
  1887   have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / (1 + 1)) p" by simp
  1888   with lnU smU
  1889   show ?thesis by auto
  1890 qed
  1891 
  1892     (* The Ferrante - Rackoff Theorem *)
  1893 
  1894 theorem fr_eq: 
  1895   assumes lp: "islin p"
  1896   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))"
  1897   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  1898 proof
  1899   assume px: "\<exists> x. ?I x p"
  1900   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  1901   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  1902   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  1903     from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
  1904   ultimately show "?D" by blast
  1905 next
  1906   assume "?D" 
  1907   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  1908   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  1909   moreover {assume f:"?F" hence "?E" by blast}
  1910   ultimately show "?E" by blast
  1911 qed
  1912 
  1913 section{* First implementation : Naive by encoding all case splits locally *}
  1914 definition "msubsteq c t d s a r = 
  1915   evaldjf (split conj) 
  1916   [(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)))),
  1917    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1918    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1919    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
  1920 
  1921 lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  1922   shows "bound0 (msubsteq c t d s a r)"
  1923 proof-
  1924   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)))),
  1925    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  1926    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  1927    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
  1928     using lp by (simp add: Let_def t s )
  1929   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
  1930 qed
  1931 
  1932 lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
  1933   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")
  1934 proof-
  1935   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  1936   let ?N = "\<lambda>p. Ipoly vs p"
  1937   let ?c = "?N c"
  1938   let ?d = "?N d"
  1939   let ?t = "?Nt x t"
  1940   let ?s = "?Nt x s"
  1941   let ?a = "?N a"
  1942   let ?r = "?Nt x r"
  1943   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  1944   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  1945   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
  1946   moreover
  1947   {assume c: "?c = 0" and d: "?d=0"
  1948     hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
  1949   moreover 
  1950   {assume c: "?c = 0" and d: "?d\<noteq>0"
  1951     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  1952     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1953     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>))"])
  1954     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) = 0" 
  1955       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  1956     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r= 0"
  1957       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  1958     
  1959     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r = 0" using d by simp 
  1960     finally have ?thesis using c d 
  1961       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)
  1962       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1963       apply simp
  1964       done}
  1965   moreover
  1966   {assume c: "?c \<noteq> 0" and d: "?d=0"
  1967     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  1968     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1969     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r = 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  1970     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) = 0" 
  1971       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  1972     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r= 0"
  1973       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  1974     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r = 0" using c by simp 
  1975     finally have ?thesis using c d 
  1976       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
  1977       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1978       apply simp
  1979       done }
  1980   moreover
  1981   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  1982     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  1983     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  1984       by (simp add: field_simps)
  1985     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
  1986     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r = 0" 
  1987       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  1988     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) =0 "
  1989       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  1990     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r =0" 
  1991       using nonzero_mult_divide_cancel_left [OF dc] c d
  1992       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  1993     finally  have ?thesis using c d 
  1994       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
  1995       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  1996       apply (simp add: field_simps)
  1997       done }
  1998   ultimately show ?thesis by blast
  1999 qed
  2000 
  2001 
  2002 definition "msubstneq c t d s a r = 
  2003   evaldjf (split conj) 
  2004   [(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)))),
  2005    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2006    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2007    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
  2008 
  2009 lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2010   shows "bound0 (msubstneq c t d s a r)"
  2011 proof-
  2012   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)))), 
  2013     (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2014     (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2015     (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
  2016     using lp by (simp add: Let_def t s )
  2017   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
  2018 qed
  2019 
  2020 lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
  2021   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")
  2022 proof-
  2023   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  2024   let ?N = "\<lambda>p. Ipoly vs p"
  2025   let ?c = "?N c"
  2026   let ?d = "?N d"
  2027   let ?t = "?Nt x t"
  2028   let ?s = "?Nt x s"
  2029   let ?a = "?N a"
  2030   let ?r = "?Nt x r"
  2031   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2032   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2033   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
  2034   moreover
  2035   {assume c: "?c = 0" and d: "?d=0"
  2036     hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
  2037   moreover 
  2038   {assume c: "?c = 0" and d: "?d\<noteq>0"
  2039     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
  2040     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2041     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>))"])
  2042     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) \<noteq> 0" 
  2043       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
  2044     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r\<noteq> 0"
  2045       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
  2046     
  2047     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r \<noteq> 0" using d by simp 
  2048     finally have ?thesis using c d 
  2049       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)
  2050       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2051       apply simp
  2052       done}
  2053   moreover
  2054   {assume c: "?c \<noteq> 0" and d: "?d=0"
  2055     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
  2056     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2057     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
  2058     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) \<noteq> 0" 
  2059       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
  2060     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r \<noteq> 0"
  2061       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
  2062     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r \<noteq> 0" using c by simp 
  2063     finally have ?thesis using c d 
  2064       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
  2065       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2066       apply simp
  2067       done }
  2068   moreover
  2069   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
  2070     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2071     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2072       by (simp add: field_simps)
  2073     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
  2074     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r \<noteq> 0" 
  2075       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2076     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) \<noteq> 0 "
  2077       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
  2078     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r \<noteq> 0" 
  2079       using nonzero_mult_divide_cancel_left[OF dc] c d
  2080       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2081     finally  have ?thesis using c d 
  2082       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
  2083       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2084       apply (simp add: field_simps)
  2085       done }
  2086   ultimately show ?thesis by blast
  2087 qed
  2088 
  2089 definition "msubstlt c t d s a r = 
  2090   evaldjf (split conj) 
  2091   [(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)))),
  2092   (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)))),
  2093    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2094    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2095    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2096    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2097    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"
  2098 
  2099 lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2100   shows "bound0 (msubstlt c t d s a r)"
  2101 proof-
  2102   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)))),
  2103   (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)))),
  2104    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2105    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2106    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2107    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2108    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
  2109     using lp by (simp add: Let_def t s lt_nb )
  2110   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
  2111 qed
  2112 
  2113 
  2114 lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))" 
  2115   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow> 
  2116   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")
  2117 proof-
  2118   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2119   let ?N = "\<lambda>p. Ipoly vs p"
  2120   let ?c = "?N c"
  2121   let ?d = "?N d"
  2122   let ?t = "?Nt x t"
  2123   let ?s = "?Nt x s"
  2124   let ?a = "?N a"
  2125   let ?r = "?Nt x r"
  2126   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2127   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2128   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
  2129   moreover
  2130   {assume c: "?c=0" and d: "?d=0"
  2131     hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
  2132   moreover
  2133   {assume dc: "?c*?d > 0" 
  2134     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2135     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2136     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2137     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2138     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2139       by (simp add: field_simps)
  2140     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2141     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2142       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2143     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) < 0"
  2144       
  2145       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
  2146     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r < 0" 
  2147       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2148       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2149     finally  have ?thesis using dc c d  nc nd dc'
  2150       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) 
  2151     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2152     by (simp add: field_simps order_less_not_sym[OF dc])}
  2153   moreover
  2154   {assume dc: "?c*?d < 0" 
  2155 
  2156     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2157       by (simp add: mult_less_0_iff field_simps) 
  2158     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2159     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2160     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2161       by (simp add: field_simps)
  2162     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2163     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
  2164       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2165 
  2166     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) > 0"
  2167       
  2168       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
  2169     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r < 0" 
  2170       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2171       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2172     finally  have ?thesis using dc c d  nc nd
  2173       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) 
  2174       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2175       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2176   moreover
  2177   {assume c: "?c > 0" and d: "?d=0"  
  2178     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2179     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2180     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2181     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2182     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2183     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) < 0"
  2184       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
  2185     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r < 0" 
  2186       using nonzero_mult_divide_cancel_left[OF c'] c
  2187       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2188     finally have ?thesis using c d nc nd 
  2189       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2190       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2191       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2192       by (simp add: field_simps )  }
  2193   moreover
  2194   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2195     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2196     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2197     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2198     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2199     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) > 0"
  2200       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
  2201     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r < 0" 
  2202       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2203         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2204     finally have ?thesis using c d nc nd 
  2205       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2206       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2207       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2208       by (simp add: field_simps )    }
  2209   moreover
  2210   moreover
  2211   {assume c: "?c = 0" and d: "?d>0"  
  2212     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2213     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2214     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2215     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2216     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2217     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) < 0"
  2218       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
  2219     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r < 0" 
  2220       using nonzero_mult_divide_cancel_left[OF d'] d
  2221       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2222     finally have ?thesis using c d nc nd 
  2223       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2224       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2225       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2226       by (simp add: field_simps)  }
  2227   moreover
  2228   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2229     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2230     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2231     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
  2232     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2233     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) > 0"
  2234       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
  2235     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r < 0" 
  2236       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2237         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2238     finally have ?thesis using c d nc nd 
  2239       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2240       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2241       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2242       by (simp add: field_simps )    }
  2243 ultimately show ?thesis by blast
  2244 qed
  2245 
  2246 definition "msubstle c t d s a r = 
  2247   evaldjf (split conj) 
  2248   [(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)))),
  2249   (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)))),
  2250    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2251    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2252    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2253    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2254    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"
  2255 
  2256 lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  2257   shows "bound0 (msubstle c t d s a r)"
  2258 proof-
  2259   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)))),
  2260   (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)))),
  2261    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2262    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
  2263    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2264    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
  2265    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
  2266     using lp by (simp add: Let_def t s lt_nb )
  2267   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
  2268 qed
  2269 
  2270 lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))" 
  2271   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow> 
  2272   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")
  2273 proof-
  2274   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2275   let ?N = "\<lambda>p. Ipoly vs p"
  2276   let ?c = "?N c"
  2277   let ?d = "?N d"
  2278   let ?t = "?Nt x t"
  2279   let ?s = "?Nt x s"
  2280   let ?a = "?N a"
  2281   let ?r = "?Nt x r"
  2282   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  2283   note r= tmbound0_I[OF lin(3), of vs _ bs x]
  2284   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
  2285   moreover
  2286   {assume c: "?c=0" and d: "?d=0"
  2287     hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
  2288   moreover
  2289   {assume dc: "?c*?d > 0" 
  2290     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
  2291     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2292     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
  2293     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2294     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2295       by (simp add: field_simps)
  2296     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2297     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2298       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2299     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) <= 0"
  2300       
  2301       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
  2302     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r <= 0" 
  2303       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2304       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2305     finally  have ?thesis using dc c d  nc nd dc'
  2306       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) 
  2307     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2308     by (simp add: field_simps order_less_not_sym[OF dc])}
  2309   moreover
  2310   {assume dc: "?c*?d < 0" 
  2311 
  2312     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
  2313       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
  2314     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
  2315     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
  2316     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
  2317       by (simp add: field_simps)
  2318     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2319     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
  2320       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
  2321 
  2322     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) >= 0"
  2323       
  2324       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
  2325     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r <= 0" 
  2326       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
  2327       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
  2328     finally  have ?thesis using dc c d  nc nd
  2329       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) 
  2330       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2331       by (simp add: field_simps order_less_not_sym[OF dc]) }
  2332   moreover
  2333   {assume c: "?c > 0" and d: "?d=0"  
  2334     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
  2335     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
  2336     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2337     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2338     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2339     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) <= 0"
  2340       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
  2341     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r <= 0" 
  2342       using nonzero_mult_divide_cancel_left[OF c'] c
  2343       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2344     finally have ?thesis using c d nc nd 
  2345       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2346       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2347       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2348       by (simp add: field_simps )  }
  2349   moreover
  2350   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
  2351     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
  2352     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
  2353     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2354     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
  2355     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) >= 0"
  2356       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
  2357     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r <= 0" 
  2358       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
  2359         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2360     finally have ?thesis using c d nc nd 
  2361       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2362       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2363       using c order_less_not_sym[OF c] less_imp_neq[OF c]
  2364       by (simp add: field_simps )    }
  2365   moreover
  2366   moreover
  2367   {assume c: "?c = 0" and d: "?d>0"  
  2368     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
  2369     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
  2370     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2371     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2372     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2373     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) <= 0"
  2374       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
  2375     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r <= 0" 
  2376       using nonzero_mult_divide_cancel_left[OF d'] d
  2377       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
  2378     finally have ?thesis using c d nc nd 
  2379       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2380       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2381       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2382       by (simp add: field_simps )  }
  2383   moreover
  2384   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
  2385     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
  2386     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
  2387     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
  2388     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
  2389     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) >= 0"
  2390       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
  2391     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r <= 0" 
  2392       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
  2393         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
  2394     finally have ?thesis using c d nc nd 
  2395       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
  2396       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
  2397       using d order_less_not_sym[OF d] less_imp_neq[OF d]
  2398       by (simp add: field_simps )    }
  2399 ultimately show ?thesis by blast
  2400 qed
  2401 
  2402 
  2403 fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
  2404   "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
  2405 | "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
  2406 | "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
  2407 | "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
  2408 | "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
  2409 | "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
  2410 | "msubst p ((c,t),(d,s)) = p"
  2411 
  2412 lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
  2413   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"
  2414   using lp
  2415 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])
  2416 
  2417 lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
  2418   shows "bound0 (msubst p ((c,t),(d,s)))"
  2419   using lp t s
  2420   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
  2421 
  2422 lemma fr_eq_msubst: 
  2423   assumes lp: "islin p"
  2424   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)))))"
  2425   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
  2426 proof-
  2427 from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2428 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2429   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"
  2430   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2431   from msubst_I[OF lp norm, of vs x bs t s] pts
  2432   have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
  2433 moreover
  2434 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  2435   and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
  2436   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  2437   from msubst_I[OF lp norm, of vs x bs t s] pts
  2438   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" ..}
  2439 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
  2440 from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
  2441 qed 
  2442 
  2443 text {* Rest of the implementation *}
  2444 
  2445 primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where
  2446   "alluopairs [] = []"
  2447 | "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
  2448 
  2449 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
  2450 by (induct xs, auto)
  2451 
  2452 lemma alluopairs_set:
  2453   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
  2454 by (induct xs, auto)
  2455 
  2456 lemma alluopairs_ex:
  2457   assumes Pc: "\<forall> x \<in> set xs. \<forall>y\<in> set xs. P x y = P y x"
  2458   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
  2459 proof
  2460   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
  2461   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
  2462   from alluopairs_set[OF x y] P Pc x y show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
  2463     by auto
  2464 next
  2465   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
  2466   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
  2467   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
  2468   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
  2469 qed
  2470 
  2471 lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
  2472 using Nat.gr0_conv_Suc
  2473 by clarsimp
  2474 
  2475 lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
  2476   apply (induct xs, auto) done
  2477 
  2478 consts remdps:: "'a list \<Rightarrow> 'a list"
  2479 
  2480 recdef remdps "measure size"
  2481   "remdps [] = []"
  2482   "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
  2483 (hints simp add: filter_length[rule_format])
  2484 
  2485 lemma remdps_set[simp]: "set (remdps xs) = set xs"
  2486   by (induct xs rule: remdps.induct, auto)
  2487 
  2488 lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
  2489   shows "qfree p \<Longrightarrow> islin (simpfm p)"
  2490   by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
  2491 
  2492 definition 
  2493   "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2494   in if (mp = T \<or> pp = T) then T 
  2495      else (let U = alluopairs (remdps (uset  q))
  2496            in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
  2497 
  2498 lemma ferrack: 
  2499   assumes qf: "qfree p"
  2500   shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
  2501   (is "_ \<and> (?rhs = ?lhs)")
  2502 proof-
  2503   let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
  2504   let ?N = "\<lambda> t. Ipoly vs t"
  2505   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2506   let ?q = "simpfm p" 
  2507   let ?U = "remdps(uset ?q)"
  2508   let ?Up = "alluopairs ?U"
  2509   let ?mp = "minusinf ?q"
  2510   let ?pp = "plusinf ?q"
  2511   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2512   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2513   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2514   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2515   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"
  2516     by simp
  2517   {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
  2518     from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
  2519     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]
  2520     have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
  2521   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
  2522   {fix x assume xUp: "x \<in> set ?Up" 
  2523     then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U" 
  2524       and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto  
  2525     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] 
  2526     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
  2527     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] 
  2528     have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
  2529   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
  2530   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
  2531   with mp_nb pp_nb 
  2532   have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
  2533   from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
  2534   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2535   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
  2536   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists> (x,y) \<in> set ?Up. ?I ((simpfm o (msubst ?q)) (x,y)))" using alluopairs_ex[OF th0] by simp
  2537   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)" 
  2538     by (simp add: evaldjf_ex)
  2539   also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
  2540   also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
  2541     apply (simp add: ferrack_def Let_def)
  2542     by (cases "?mp = T \<or> ?pp = T", auto)
  2543   finally show ?thesis using thqf by blast
  2544 qed
  2545 
  2546 definition "frpar p = simpfm (qelim p ferrack)"
  2547 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
  2548 proof-
  2549   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
  2550   from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
  2551 qed
  2552 
  2553 
  2554 section{* Second implemenation: Case splits not local *}
  2555 
  2556 lemma fr_eq2:  assumes lp: "islin p"
  2557   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow> 
  2558    ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> 
  2559     (Ifm vs (0#bs) p) \<or> 
  2560     (\<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> 
  2561     (\<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))"
  2562   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
  2563 proof
  2564   assume px: "\<exists> x. ?I x p"
  2565   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  2566   moreover {assume "?M \<or> ?P" hence "?D" by blast}
  2567   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
  2568     from inf_uset[OF lp nmi npi, OF px] 
  2569     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"
  2570       by auto
  2571     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2572     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
  2573     let ?s = "Itm vs (x # bs) s"
  2574     let ?t = "Itm vs (x # bs) t"
  2575     have eq2: "\<And>(x::'a). x + x = (1 + 1) * x"
  2576       by  (simp add: field_simps)
  2577     {assume "?c = 0 \<and> ?d = 0"
  2578       with ct have ?D by simp}
  2579     moreover
  2580     {assume z: "?c = 0" "?d \<noteq> 0"
  2581       from z have ?D using ct by auto}
  2582     moreover
  2583     {assume z: "?c \<noteq> 0" "?d = 0"
  2584       with ct have ?D by auto }
  2585     moreover
  2586     {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
  2587       from z have ?F using ct
  2588         apply - apply (rule bexI[where x = "(c,t)"], simp_all)
  2589         by (rule bexI[where x = "(d,s)"], simp_all)
  2590       hence ?D by blast}
  2591     ultimately have ?D by auto}
  2592   ultimately show "?D" by blast
  2593 next
  2594   assume "?D" 
  2595   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  2596   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  2597   moreover {assume f:"?F" hence "?E" by blast}
  2598   ultimately show "?E" by blast
  2599 qed
  2600 
  2601 definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
  2602 definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
  2603 definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
  2604 definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
  2605 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
  2606 
  2607 lemma msubsteq2: 
  2608   assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
  2609   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")
  2610   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2611   by (simp add: msubsteq2_def field_simps)
  2612 
  2613 lemma msubstltpos: 
  2614   assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
  2615   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")
  2616   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2617   by (simp add: msubstltpos_def field_simps)
  2618 
  2619 lemma msubstlepos: 
  2620   assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
  2621   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")
  2622   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2623   by (simp add: msubstlepos_def field_simps)
  2624 
  2625 lemma msubstltneg: 
  2626   assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
  2627   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")
  2628   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2629   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
  2630 
  2631 lemma msubstleneg: 
  2632   assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
  2633   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")
  2634   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  2635   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
  2636 
  2637 fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2638   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
  2639 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
  2640 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2641 | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2642 | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
  2643 | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
  2644 | "msubstpos p c t = p"
  2645     
  2646 lemma msubstpos_I: 
  2647   assumes lp: "islin p" and pos: "Ipoly vs c > 0"
  2648   shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2649   using lp pos
  2650   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)
  2651 
  2652 fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  2653   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
  2654 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
  2655 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
  2656 | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
  2657 | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
  2658 | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
  2659 | "msubstneg p c t = p"
  2660 
  2661 lemma msubstneg_I: 
  2662   assumes lp: "islin p" and pos: "Ipoly vs c < 0"
  2663   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2664   using lp pos
  2665   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)
  2666 
  2667 
  2668 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)))"
  2669 
  2670 lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
  2671   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  2672 proof-
  2673   let ?c = "Ipoly vs c"
  2674   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))" 
  2675     by (simp_all add: polyneg_norm)
  2676   from nz have "?c > 0 \<or> ?c < 0" by arith
  2677   moreover
  2678   {assume c: "?c < 0"
  2679     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"]
  2680     have ?thesis by (auto simp add: msubst2_def)}
  2681   moreover
  2682   {assume c: "?c > 0"
  2683     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"]
  2684     have ?thesis by (auto simp add: msubst2_def)}
  2685   ultimately show ?thesis by blast
  2686 qed
  2687 
  2688 term msubsteq2
  2689 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
  2690   by (simp add: msubsteq2_def)
  2691 
  2692 lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
  2693   by (simp add: msubstltpos_def)
  2694 lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
  2695   by (simp add: msubstltneg_def)
  2696 
  2697 lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
  2698   by (simp add: msubstlepos_def)
  2699 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
  2700   by (simp add: msubstleneg_def)
  2701 
  2702 lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
  2703   shows "bound0 (msubstpos p c t)"
  2704 using lp tnb
  2705 by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
  2706 
  2707 lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2708   shows "bound0 (msubstneg p c t)"
  2709 using lp tnb
  2710 by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
  2711 
  2712 lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
  2713   shows "bound0 (msubst2 p c t)"
  2714 using lp tnb
  2715 by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
  2716     
  2717 lemma of_int2: "of_int 2 = 1 + 1"
  2718 proof-
  2719   have "(2::int) = 1 + 1" by simp
  2720   hence "of_int 2 = of_int (1 + 1)" by simp
  2721   thus ?thesis unfolding of_int_add by simp
  2722 qed
  2723 
  2724 lemma of_int_minus2: "of_int (-2) = - (1 + 1)"
  2725 proof-
  2726   have th: "(-2::int) = - 2" by simp
  2727   show ?thesis unfolding th by (simp only: of_int_minus of_int2)
  2728 qed
  2729 
  2730 
  2731 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
  2732   by (induct p rule: islin.induct, auto simp add: bound0_qf)
  2733 lemma fr_eq_msubst2: 
  2734   assumes lp: "islin p"
  2735   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)))))"
  2736   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
  2737 proof-
  2738   from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  2739   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2740   have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
  2741   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
  2742   
  2743   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)"
  2744   proof-
  2745     {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
  2746       from H(1) th have "isnpoly n" by blast
  2747       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
  2748       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
  2749         by (simp add: polyneg_norm nn)
  2750       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 
  2751         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
  2752       from msubst2[OF lp nn nn2(1), of x bs t]
  2753       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"
  2754         using H(2) nn2 by (simp add: of_int_minus2 del: minus_add_distrib)}
  2755     moreover
  2756     {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"
  2757       from H(1) th have "isnpoly n" by blast
  2758       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
  2759         using H(2) by (simp_all add: polymul_norm n2)
  2760       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)}
  2761     ultimately show ?thesis by blast
  2762   qed
  2763   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).
  2764      \<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)" 
  2765   proof-
  2766     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2767      "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
  2768       from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2769       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" 
  2770         by (simp_all add: polymul_norm n2)
  2771       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)))"
  2772         by (simp_all add: polyneg_norm nn)
  2773       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"
  2774         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)
  2775       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
  2776       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" 
  2777         apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)
  2778         by (simp add: mult_commute)}
  2779     moreover
  2780     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
  2781       "\<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"
  2782      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
  2783       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"
  2784         using H(3,4) by (simp_all add: polymul_norm n2)
  2785       from msubst2[OF lp nn, of x bs ] H(3,4,5) 
  2786       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)}
  2787     ultimately show ?thesis by blast
  2788   qed
  2789   from fr_eq2[OF lp, of vs bs x] show ?thesis
  2790     unfolding eq0 eq1 eq2 by blast  
  2791 qed
  2792 
  2793 definition 
  2794 "ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  2795  in if (mp = T \<or> pp = T) then T 
  2796   else (let U = remdps (uset  q)
  2797     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, 
  2798    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)]))"
  2799 
  2800 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
  2801 
  2802 lemma ferrack2: assumes qf: "qfree p"
  2803   shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
  2804   (is "_ \<and> (?rhs = ?lhs)")
  2805 proof-
  2806   let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
  2807   let ?N = "\<lambda> t. Ipoly vs t"
  2808   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  2809   let ?q = "simpfm p" 
  2810   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
  2811   let ?U = "remdps(uset ?q)"
  2812   let ?Up = "alluopairs ?U"
  2813   let ?mp = "minusinf ?q"
  2814   let ?pp = "plusinf ?q"
  2815   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  2816   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  2817   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  2818   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  2819   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"
  2820     by simp
  2821   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)" 
  2822   proof-
  2823     {fix c t assume ct: "(c,t) \<in> set ?U"
  2824       hence tnb: "tmbound0 t" using U_l by blast
  2825       from msubst2_nb[OF lq tnb]
  2826       have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
  2827     thus ?thesis by auto
  2828   qed
  2829   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)" 
  2830   proof-
  2831     {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
  2832       from badc U_l alluopairs_set1[of ?U] 
  2833       have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
  2834       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}
  2835     thus ?thesis by auto
  2836   qed
  2837   have stupid: "bound0 F" by simp
  2838   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, 
  2839    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)]"
  2840   from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
  2841   have nb: "bound0 ?R "
  2842     by (simp add: list_disj_def disj_nb0 simpfm_bound0)
  2843   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))"
  2844 
  2845   {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
  2846     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))" 
  2847       by auto (simp add: isnpoly_def)
  2848     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
  2849       using norm by (simp_all add: polymul_norm)
  2850     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)))"
  2851       by (simp_all add: polyneg_norm norm2)
  2852     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")
  2853     proof
  2854       assume H: ?lhs
  2855       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" 
  2856         by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
  2857       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2858         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2859       show ?rhs by (simp add: field_simps)
  2860     next
  2861       assume H: ?rhs
  2862       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" 
  2863         by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
  2864       from msubst2[OF lq norm2(1) z(1), of x bs] 
  2865         msubst2[OF lq norm2(2) z(2), of x bs] H 
  2866       show ?lhs by (simp add: field_simps)
  2867     qed}
  2868   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
  2869     by clarsimp
  2870 
  2871   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  2872   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))))"
  2873     using fr_eq_msubst2[OF lq, of vs bs x] by simp
  2874   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)))"
  2875     by (simp add: split_def)
  2876   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)))"
  2877     using alluopairs_ex[OF th0] by simp 
  2878   also have "\<dots> \<longleftrightarrow> ?I ?R" 
  2879     by (simp add: list_disj_def evaldjf_ex split_def)
  2880   also have "\<dots> \<longleftrightarrow> ?rhs"
  2881     unfolding ferrack2_def
  2882     apply (cases "?mp = T") 
  2883     apply (simp add: list_disj_def)
  2884     apply (cases "?pp = T") 
  2885     apply (simp add: list_disj_def)
  2886     by (simp_all add: Let_def decr0[OF nb])
  2887   finally show ?thesis using decr0_qf[OF nb]  
  2888     by (simp  add: ferrack2_def Let_def)
  2889 qed
  2890 
  2891 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
  2892 proof-
  2893   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
  2894   from qelim[OF th, of "prep p" bs] 
  2895 show ?thesis  unfolding frpar2_def by (auto simp add: prep)
  2896 qed
  2897 
  2898 ML {* 
  2899 structure ReflectedFRPar = 
  2900 struct
  2901 
  2902 val bT = HOLogic.boolT;
  2903 fun num rT x = HOLogic.mk_number rT x;
  2904 fun rrelT rT = [rT,rT] ---> rT;
  2905 fun rrT rT = [rT, rT] ---> bT;
  2906 fun divt rT = Const(@{const_name Rings.divide},rrelT rT);
  2907 fun timest rT = Const(@{const_name Groups.times},rrelT rT);
  2908 fun plust rT = Const(@{const_name Groups.plus},rrelT rT);
  2909 fun minust rT = Const(@{const_name Groups.minus},rrelT rT);
  2910 fun uminust rT = Const(@{const_name Groups.uminus}, rT --> rT);
  2911 fun powt rT = Const(@{const_name "power"}, [rT,@{typ "nat"}] ---> rT);
  2912 val brT = [bT, bT] ---> bT;
  2913 val nott = @{term "Not"};
  2914 val conjt = @{term HOL.conj};
  2915 val disjt = @{term HOL.disj};
  2916 val impt = @{term HOL.implies};
  2917 val ifft = @{term "op = :: bool => _"}
  2918 fun llt rT = Const(@{const_name Orderings.less},rrT rT);
  2919 fun lle rT = Const(@{const_name Orderings.less},rrT rT);
  2920 fun eqt rT = Const(@{const_name HOL.eq},rrT rT);
  2921 fun rz rT = Const(@{const_name Groups.zero},rT);
  2922 
  2923 fun dest_nat t = case t of
  2924   Const (@{const_name Suc}, _) $ t' => 1 + dest_nat t'
  2925 | _ => (snd o HOLogic.dest_number) t;
  2926 
  2927 fun num_of_term m t = 
  2928  case t of
  2929    Const(@{const_name Groups.uminus},_)$t => @{code poly.Neg} (num_of_term m t)
  2930  | Const(@{const_name Groups.plus},_)$a$b => @{code poly.Add} (num_of_term m a, num_of_term m b)
  2931  | Const(@{const_name Groups.minus},_)$a$b => @{code poly.Sub} (num_of_term m a, num_of_term m b)
  2932  | Const(@{const_name Groups.times},_)$a$b => @{code poly.Mul} (num_of_term m a, num_of_term m b)
  2933  | Const(@{const_name Power.power},_)$a$n => @{code poly.Pw} (num_of_term m a, dest_nat n)
  2934  | Const(@{const_name Rings.divide},_)$a$b => @{code poly.C} (HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
  2935  | _ => (@{code poly.C} (HOLogic.dest_number t |> snd,1) 
  2936          handle TERM _ => @{code poly.Bound} (AList.lookup (op aconv) m t |> the));
  2937 
  2938 fun tm_of_term m m' t = 
  2939  case t of
  2940    Const(@{const_name Groups.uminus},_)$t => @{code Neg} (tm_of_term m m' t)
  2941  | Const(@{const_name Groups.plus},_)$a$b => @{code Add} (tm_of_term m m' a, tm_of_term m m' b)
  2942  | Const(@{const_name Groups.minus},_)$a$b => @{code Sub} (tm_of_term m m' a, tm_of_term m m' b)
  2943  | Const(@{const_name Groups.times},_)$a$b => @{code Mul} (num_of_term m' a, tm_of_term m m' b)
  2944  | _ => (@{code CP} (num_of_term m' t) 
  2945          handle TERM _ => @{code Bound} (AList.lookup (op aconv) m t |> the)
  2946               | Option => @{code Bound} (AList.lookup (op aconv) m t |> the));
  2947 
  2948 fun term_of_num T m t = 
  2949  case t of
  2950   @{code poly.C} (a,b) => (if b = 1 then num T a else if b=0 then (rz T) 
  2951                                         else (divt T) $ num T a $ num T b)
  2952 | @{code poly.Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2953 | @{code poly.Add} (a,b) => (plust T)$(term_of_num T m a)$(term_of_num T m b)
  2954 | @{code poly.Mul} (a,b) => (timest T)$(term_of_num T m a)$(term_of_num T m b)
  2955 | @{code poly.Sub} (a,b) => (minust T)$(term_of_num T m a)$(term_of_num T m b)
  2956 | @{code poly.Neg} a => (uminust T)$(term_of_num T m a)
  2957 | @{code poly.Pw} (a,n) => (powt T)$(term_of_num T m t)$(HOLogic.mk_number HOLogic.natT n)
  2958 | @{code poly.CN} (c,n,p) => term_of_num T m (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)))
  2959 | _ => error "term_of_num: Unknown term";
  2960 
  2961 fun term_of_tm T m m' t = 
  2962  case t of
  2963   @{code CP} p => term_of_num T m' p
  2964 | @{code Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
  2965 | @{code Add} (a,b) => (plust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2966 | @{code Mul} (a,b) => (timest T)$(term_of_num T m' a)$(term_of_tm T m m' b)
  2967 | @{code Sub} (a,b) => (minust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
  2968 | @{code Neg} a => (uminust T)$(term_of_tm T m m' a)
  2969 | @{code CNP} (n,c,p) => term_of_tm T m m' (@{code Add}
  2970      (@{code Mul} (c, @{code Bound} n), p))
  2971 | _ => error "term_of_tm: Unknown term";
  2972 
  2973 fun fm_of_term m m' fm = 
  2974  case fm of
  2975     Const(@{const_name True},_) => @{code T}
  2976   | Const(@{const_name False},_) => @{code F}
  2977   | Const(@{const_name Not},_)$p => @{code NOT} (fm_of_term m m' p)
  2978   | Const(@{const_name HOL.conj},_)$p$q => @{code And} (fm_of_term m m' p, fm_of_term m m' q)
  2979   | Const(@{const_name HOL.disj},_)$p$q => @{code Or} (fm_of_term m m' p, fm_of_term m m' q)
  2980   | Const(@{const_name HOL.implies},_)$p$q => @{code Imp} (fm_of_term m m' p, fm_of_term m m' q)
  2981   | Const(@{const_name HOL.eq},ty)$p$q => 
  2982        if domain_type ty = bT then @{code Iff} (fm_of_term m m' p, fm_of_term m m' q)
  2983        else @{code Eq} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2984   | Const(@{const_name Orderings.less},_)$p$q => 
  2985         @{code Lt} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2986   | Const(@{const_name Orderings.less_eq},_)$p$q => 
  2987         @{code Le} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
  2988   | Const(@{const_name Ex},_)$Abs(xn,xT,p) => 
  2989      let val (xn', p') =  variant_abs (xn,xT,p)
  2990          val x = Free(xn',xT)
  2991          fun incr i = i + 1
  2992          val m0 = (x,0):: (map (apsnd incr) m)
  2993       in @{code E} (fm_of_term m0 m' p') end
  2994   | Const(@{const_name All},_)$Abs(xn,xT,p) => 
  2995      let val (xn', p') =  variant_abs (xn,xT,p)
  2996          val x = Free(xn',xT)
  2997          fun incr i = i + 1
  2998          val m0 = (x,0):: (map (apsnd incr) m)
  2999       in @{code A} (fm_of_term m0 m' p') end
  3000   | _ => error "fm_of_term";
  3001 
  3002 
  3003 fun term_of_fm T m m' t = 
  3004   case t of
  3005     @{code T} => Const(@{const_name True},bT)
  3006   | @{code F} => Const(@{const_name False},bT)
  3007   | @{code NOT} p => nott $ (term_of_fm T m m' p)
  3008   | @{code And} (p,q) => conjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3009   | @{code Or} (p,q) => disjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3010   | @{code Imp} (p,q) => impt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3011   | @{code Iff} (p,q) => ifft $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  3012   | @{code Lt} p => (llt T) $ (term_of_tm T m m' p) $ (rz T)
  3013   | @{code Le} p => (lle T) $ (term_of_tm T m m' p) $ (rz T)
  3014   | @{code Eq} p => (eqt T) $ (term_of_tm T m m' p) $ (rz T)
  3015   | @{code NEq} p => nott $ ((eqt T) $ (term_of_tm T m m' p) $ (rz T))
  3016   | _ => error "term_of_fm: quantifiers!!!!???";
  3017 
  3018 fun frpar_oracle (T,m, m', fm) = 
  3019  let 
  3020    val t = HOLogic.dest_Trueprop fm
  3021    val im = 0 upto (length m - 1)
  3022    val im' = 0 upto (length m' - 1)   
  3023  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  3024                                                      (@{code frpar} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  3025  end;
  3026 
  3027 fun frpar_oracle2 (T,m, m', fm) = 
  3028  let 
  3029    val t = HOLogic.dest_Trueprop fm
  3030    val im = 0 upto (length m - 1)
  3031    val im' = 0 upto (length m' - 1)   
  3032  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
  3033                                                      (@{code frpar2} (fm_of_term (m ~~ im) (m' ~~ im') t))))
  3034  end;
  3035 
  3036 end;
  3037 
  3038 
  3039 *}
  3040 
  3041 oracle frpar_oracle = {* fn (ty, ts, ts', ct) => 
  3042  let 
  3043   val thy = Thm.theory_of_cterm ct
  3044  in cterm_of thy (ReflectedFRPar.frpar_oracle (ty,ts, ts', term_of ct))
  3045  end *}
  3046 
  3047 oracle frpar_oracle2 = {* fn (ty, ts, ts', ct) => 
  3048  let 
  3049   val thy = Thm.theory_of_cterm ct
  3050  in cterm_of thy (ReflectedFRPar.frpar_oracle2 (ty,ts, ts', term_of ct))
  3051  end *}
  3052 
  3053 ML{* 
  3054 structure FRParTac = 
  3055 struct
  3056 
  3057 fun frpar_tac T ps ctxt i = 
  3058  Object_Logic.full_atomize_tac i
  3059  THEN (fn st =>
  3060   let
  3061     val g = List.nth (cprems_of st, i - 1)
  3062     val thy = ProofContext.theory_of ctxt
  3063     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3064     val th = frpar_oracle (T, fs,ps, (* Pattern.eta_long [] *)g)
  3065   in rtac (th RS iffD2) i st end);
  3066 
  3067 fun frpar2_tac T ps ctxt i = 
  3068  Object_Logic.full_atomize_tac i
  3069  THEN (fn st =>
  3070   let
  3071     val g = List.nth (cprems_of st, i - 1)
  3072     val thy = ProofContext.theory_of ctxt
  3073     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
  3074     val th = frpar_oracle2 (T, fs,ps, (* Pattern.eta_long [] *)g)
  3075   in rtac (th RS iffD2) i st end);
  3076 
  3077 end;
  3078 
  3079 *}
  3080 
  3081 method_setup frpar = {*
  3082 let
  3083  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3084  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3085  val parsN = "pars"
  3086  val typN = "type"
  3087  val any_keyword = keyword parsN || keyword typN
  3088  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3089  val cterms = thms >> map Drule.dest_term;
  3090  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3091  val typ = Scan.unless any_keyword Args.typ
  3092 in
  3093  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3094   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar_tac T ps ctxt))
  3095 end
  3096 *} "Parametric QE for linear Arithmetic over fields, Version 1"
  3097 
  3098 method_setup frpar2 = {*
  3099 let
  3100  fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
  3101  fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
  3102  val parsN = "pars"
  3103  val typN = "type"
  3104  val any_keyword = keyword parsN || keyword typN
  3105  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
  3106  val cterms = thms >> map Drule.dest_term;
  3107  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
  3108  val typ = Scan.unless any_keyword Args.typ
  3109 in
  3110  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  3111   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar2_tac T ps ctxt))
  3112 end
  3113 *} "Parametric QE for linear Arithmetic over fields, Version 2"
  3114 
  3115 
  3116 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3117   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3118   apply (simp add: field_simps)
  3119   apply (rule spec[where x=y])
  3120   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3121   by simp
  3122 
  3123 text{* Collins/Jones Problem *}
  3124 (*
  3125 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"
  3126 proof-
  3127   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")
  3128 by (simp add: field_simps)
  3129 have "?rhs"
  3130 
  3131   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}")
  3132   apply (simp add: field_simps)
  3133 oops
  3134 *)
  3135 (*
  3136 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"
  3137 apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3138 oops
  3139 *)
  3140 
  3141 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  3142   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
  3143   apply (simp add: field_simps)
  3144   apply (rule spec[where x=y])
  3145   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
  3146   by simp
  3147 
  3148 text{* Collins/Jones Problem *}
  3149 
  3150 (*
  3151 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"
  3152 proof-
  3153   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")
  3154 by (simp add: field_simps)
  3155 have "?rhs"
  3156   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}")
  3157   apply simp
  3158 oops
  3159 *)
  3160 
  3161 (*
  3162 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"
  3163 apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
  3164 apply (simp add: field_simps linorder_neq_iff[symmetric])
  3165 apply ferrack
  3166 oops
  3167 *)
  3168 end