src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
 author krauss Wed Feb 23 10:48:57 2011 +0100 (2011-02-23) changeset 41821 c118ae98dfbf parent 41816 7a55699805dc child 41822 27afef7d6c37 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 consts split0 :: "tm \<Rightarrow> (poly \<times> tm)"
```
```   339 recdef split0 "measure tmsize"
```
```   340   "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
```
```   341   "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
```
```   342   "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
```
```   343   "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
```
```   344   "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
```
```   345   "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
```
```   346   "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
```
```   347   "split0 t = (0\<^sub>p, t)"
```
```   348
```
```   349 lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
```
```   350   apply (rule exI[where x="fst (split0 p)"])
```
```   351   apply (rule exI[where x="snd (split0 p)"])
```
```   352   by simp
```
```   353
```
```   354 lemma split0:
```
```   355   "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
```
```   356   apply (induct t rule: split0.induct)
```
```   357   apply simp
```
```   358   apply (simp add: Let_def split_def field_simps)
```
```   359   apply (simp add: Let_def split_def field_simps)
```
```   360   apply (simp add: Let_def split_def field_simps)
```
```   361   apply (simp add: Let_def split_def field_simps)
```
```   362   apply (simp add: Let_def split_def field_simps)
```
```   363   apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
```
```   364   apply (simp add: Let_def split_def field_simps)
```
```   365   apply (simp add: Let_def split_def field_simps)
```
```   366   done
```
```   367
```
```   368 lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
```
```   369 proof-
```
```   370   fix c' t'
```
```   371   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
```
```   372   with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
```
```   373 qed
```
```   374
```
```   375 lemma split0_nb0:
```
```   376   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   377   shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
```
```   378 proof-
```
```   379   fix c' t'
```
```   380   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
```
```   381   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
```
```   382 qed
```
```   383
```
```   384 lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   385   shows "tmbound0 (snd (split0 t))"
```
```   386   using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
```
```   387
```
```   388
```
```   389 lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
```
```   390   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   391
```
```   392 lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt 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 tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
```
```   396  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   397
```
```   398 lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
```
```   399 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   400
```
```   401 lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
```
```   402  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   403
```
```   404 lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
```
```   405 by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
```
```   406
```
```   407 lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   408   shows
```
```   409   "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
```
```   410   by (induct p rule: split0.induct,
```
```   411     auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm
```
```   412     Let_def split_def split0_stupid)
```
```   413
```
```   414 subsection{* Formulae *}
```
```   415
```
```   416 datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
```
```   417   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
```
```   418
```
```   419
```
```   420   (* A size for fm *)
```
```   421 consts fmsize :: "fm \<Rightarrow> nat"
```
```   422 recdef fmsize "measure size"
```
```   423   "fmsize (NOT p) = 1 + fmsize p"
```
```   424   "fmsize (And p q) = 1 + fmsize p + fmsize q"
```
```   425   "fmsize (Or p q) = 1 + fmsize p + fmsize q"
```
```   426   "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
```
```   427   "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
```
```   428   "fmsize (E p) = 1 + fmsize p"
```
```   429   "fmsize (A p) = 4+ fmsize p"
```
```   430   "fmsize p = 1"
```
```   431   (* several lemmas about fmsize *)
```
```   432 lemma fmsize_pos: "fmsize p > 0"
```
```   433 by (induct p rule: fmsize.induct) simp_all
```
```   434
```
```   435   (* Semantics of formulae (fm) *)
```
```   436 primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
```
```   437   "Ifm vs bs T = True"
```
```   438 | "Ifm vs bs F = False"
```
```   439 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
```
```   440 | "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
```
```   441 | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
```
```   442 | "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
```
```   443 | "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
```
```   444 | "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
```
```   445 | "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
```
```   446 | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
```
```   447 | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
```
```   448 | "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
```
```   449 | "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
```
```   450
```
```   451 consts not:: "fm \<Rightarrow> fm"
```
```   452 recdef not "measure size"
```
```   453   "not (NOT (NOT p)) = not p"
```
```   454   "not (NOT p) = p"
```
```   455   "not T = F"
```
```   456   "not F = T"
```
```   457   "not (Lt t) = Le (tmneg t)"
```
```   458   "not (Le t) = Lt (tmneg t)"
```
```   459   "not (Eq t) = NEq t"
```
```   460   "not (NEq t) = Eq t"
```
```   461   "not p = NOT p"
```
```   462 lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
```
```   463 by (induct p rule: not.induct) auto
```
```   464
```
```   465 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   466   "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
```
```   467    if p = q then p else And p q)"
```
```   468 lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
```
```   469 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
```
```   470
```
```   471 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   472   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
```
```   473        else if p=q then p else Or p q)"
```
```   474
```
```   475 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
```
```   476 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
```
```   477
```
```   478 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   479   "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
```
```   480     else Imp p q)"
```
```   481 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
```
```   482 by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   483
```
```   484 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   485   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
```
```   486        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
```
```   487   Iff p q)"
```
```   488 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
```
```   489   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
```
```   490   (* Quantifier freeness *)
```
```   491 consts qfree:: "fm \<Rightarrow> bool"
```
```   492 recdef qfree "measure size"
```
```   493   "qfree (E p) = False"
```
```   494   "qfree (A p) = False"
```
```   495   "qfree (NOT p) = qfree p"
```
```   496   "qfree (And p q) = (qfree p \<and> qfree q)"
```
```   497   "qfree (Or  p q) = (qfree p \<and> qfree q)"
```
```   498   "qfree (Imp p q) = (qfree p \<and> qfree q)"
```
```   499   "qfree (Iff p q) = (qfree p \<and> qfree q)"
```
```   500   "qfree p = True"
```
```   501
```
```   502   (* Boundedness and substitution *)
```
```   503
```
```   504 primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
```
```   505   "boundslt n T = True"
```
```   506 | "boundslt n F = True"
```
```   507 | "boundslt n (Lt t) = (tmboundslt n t)"
```
```   508 | "boundslt n (Le t) = (tmboundslt n t)"
```
```   509 | "boundslt n (Eq t) = (tmboundslt n t)"
```
```   510 | "boundslt n (NEq t) = (tmboundslt n t)"
```
```   511 | "boundslt n (NOT p) = boundslt n p"
```
```   512 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
```
```   513 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
```
```   514 | "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
```
```   515 | "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
```
```   516 | "boundslt n (E p) = boundslt (Suc n) p"
```
```   517 | "boundslt n (A p) = boundslt (Suc n) p"
```
```   518
```
```   519 consts
```
```   520   bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
```
```   521   decr0 :: "fm \<Rightarrow> fm"
```
```   522 recdef bound0 "measure size"
```
```   523   "bound0 T = True"
```
```   524   "bound0 F = True"
```
```   525   "bound0 (Lt a) = tmbound0 a"
```
```   526   "bound0 (Le a) = tmbound0 a"
```
```   527   "bound0 (Eq a) = tmbound0 a"
```
```   528   "bound0 (NEq a) = tmbound0 a"
```
```   529   "bound0 (NOT p) = bound0 p"
```
```   530   "bound0 (And p q) = (bound0 p \<and> bound0 q)"
```
```   531   "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
```
```   532   "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
```
```   533   "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
```
```   534   "bound0 p = False"
```
```   535 lemma bound0_I:
```
```   536   assumes bp: "bound0 p"
```
```   537   shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
```
```   538 using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
```
```   539 by (induct p rule: bound0.induct,auto simp add: nth_pos2)
```
```   540
```
```   541 primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
```
```   542   "bound m T = True"
```
```   543 | "bound m F = True"
```
```   544 | "bound m (Lt t) = tmbound m t"
```
```   545 | "bound m (Le t) = tmbound m t"
```
```   546 | "bound m (Eq t) = tmbound m t"
```
```   547 | "bound m (NEq t) = tmbound m t"
```
```   548 | "bound m (NOT p) = bound m p"
```
```   549 | "bound m (And p q) = (bound m p \<and> bound m q)"
```
```   550 | "bound m (Or p q) = (bound m p \<and> bound m q)"
```
```   551 | "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
```
```   552 | "bound m (Iff p q) = (bound m p \<and> bound m q)"
```
```   553 | "bound m (E p) = bound (Suc m) p"
```
```   554 | "bound m (A p) = bound (Suc m) p"
```
```   555
```
```   556 lemma bound_I:
```
```   557   assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
```
```   558   shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
```
```   559   using bnd nb le tmbound_I[where bs=bs and vs = vs]
```
```   560 proof(induct p arbitrary: bs n rule: fm.induct)
```
```   561   case (E p bs n)
```
```   562   {fix y
```
```   563     from E have bnd: "boundslt (length (y#bs)) p"
```
```   564       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
```
```   565     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
```
```   566   thus ?case by simp
```
```   567 next
```
```   568   case (A p bs n) {fix y
```
```   569     from A have bnd: "boundslt (length (y#bs)) p"
```
```   570       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
```
```   571     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
```
```   572   thus ?case by simp
```
```   573 qed auto
```
```   574
```
```   575 recdef decr0 "measure size"
```
```   576   "decr0 (Lt a) = Lt (decrtm0 a)"
```
```   577   "decr0 (Le a) = Le (decrtm0 a)"
```
```   578   "decr0 (Eq a) = Eq (decrtm0 a)"
```
```   579   "decr0 (NEq a) = NEq (decrtm0 a)"
```
```   580   "decr0 (NOT p) = NOT (decr0 p)"
```
```   581   "decr0 (And p q) = conj (decr0 p) (decr0 q)"
```
```   582   "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
```
```   583   "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
```
```   584   "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
```
```   585   "decr0 p = p"
```
```   586
```
```   587 lemma decr0: assumes nb: "bound0 p"
```
```   588   shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
```
```   589   using nb
```
```   590   by (induct p rule: decr0.induct, simp_all add: decrtm0)
```
```   591
```
```   592 primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
```
```   593   "decr m T = T"
```
```   594 | "decr m F = F"
```
```   595 | "decr m (Lt t) = (Lt (decrtm m t))"
```
```   596 | "decr m (Le t) = (Le (decrtm m t))"
```
```   597 | "decr m (Eq t) = (Eq (decrtm m t))"
```
```   598 | "decr m (NEq t) = (NEq (decrtm m t))"
```
```   599 | "decr m (NOT p) = NOT (decr m p)"
```
```   600 | "decr m (And p q) = conj (decr m p) (decr m q)"
```
```   601 | "decr m (Or p q) = disj (decr m p) (decr m q)"
```
```   602 | "decr m (Imp p q) = imp (decr m p) (decr m q)"
```
```   603 | "decr m (Iff p q) = iff (decr m p) (decr m q)"
```
```   604 | "decr m (E p) = E (decr (Suc m) p)"
```
```   605 | "decr m (A p) = A (decr (Suc m) p)"
```
```   606
```
```   607 lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p"
```
```   608   and nle: "m < length bs"
```
```   609   shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
```
```   610   using bnd nb nle
```
```   611 proof(induct p arbitrary: bs m rule: fm.induct)
```
```   612   case (E p bs m)
```
```   613   {fix x
```
```   614     from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p"
```
```   615   and nle: "Suc m < length (x#bs)" by auto
```
```   616     from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
```
```   617   } thus ?case by auto
```
```   618 next
```
```   619   case (A p bs m)
```
```   620   {fix x
```
```   621     from A have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p"
```
```   622   and nle: "Suc m < length (x#bs)" by auto
```
```   623     from A(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
```
```   624   } thus ?case by auto
```
```   625 qed (auto simp add: decrtm removen_nth)
```
```   626
```
```   627 primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   628   "subst0 t T = T"
```
```   629 | "subst0 t F = F"
```
```   630 | "subst0 t (Lt a) = Lt (tmsubst0 t a)"
```
```   631 | "subst0 t (Le a) = Le (tmsubst0 t a)"
```
```   632 | "subst0 t (Eq a) = Eq (tmsubst0 t a)"
```
```   633 | "subst0 t (NEq a) = NEq (tmsubst0 t a)"
```
```   634 | "subst0 t (NOT p) = NOT (subst0 t p)"
```
```   635 | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
```
```   636 | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
```
```   637 | "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
```
```   638 | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
```
```   639 | "subst0 t (E p) = E p"
```
```   640 | "subst0 t (A p) = A p"
```
```   641
```
```   642 lemma subst0: assumes qf: "qfree p"
```
```   643   shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
```
```   644 using qf tmsubst0[where x="x" and bs="bs" and t="t"]
```
```   645 by (induct p rule: fm.induct, auto)
```
```   646
```
```   647 lemma subst0_nb:
```
```   648   assumes bp: "tmbound0 t" and qf: "qfree p"
```
```   649   shows "bound0 (subst0 t p)"
```
```   650 using qf tmsubst0_nb[OF bp] bp
```
```   651 by (induct p rule: fm.induct, auto)
```
```   652
```
```   653 primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   654   "subst n t T = T"
```
```   655 | "subst n t F = F"
```
```   656 | "subst n t (Lt a) = Lt (tmsubst n t a)"
```
```   657 | "subst n t (Le a) = Le (tmsubst n t a)"
```
```   658 | "subst n t (Eq a) = Eq (tmsubst n t a)"
```
```   659 | "subst n t (NEq a) = NEq (tmsubst n t a)"
```
```   660 | "subst n t (NOT p) = NOT (subst n t p)"
```
```   661 | "subst n t (And p q) = And (subst n t p) (subst n t q)"
```
```   662 | "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
```
```   663 | "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
```
```   664 | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
```
```   665 | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
```
```   666 | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
```
```   667
```
```   668 lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
```
```   669   shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
```
```   670   using nb nlm
```
```   671 proof (induct p arbitrary: bs n t rule: fm.induct)
```
```   672   case (E p bs n)
```
```   673   {fix x
```
```   674     from E have bn: "boundslt (length (x#bs)) p" by simp
```
```   675     from E have nlm: "Suc n \<le> length (x#bs)" by simp
```
```   676     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
```
```   677     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
```
```   678     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }
```
```   679 thus ?case by simp
```
```   680 next
```
```   681   case (A p bs n)
```
```   682   {fix x
```
```   683     from A have bn: "boundslt (length (x#bs)) p" by simp
```
```   684     from A have nlm: "Suc n \<le> length (x#bs)" by simp
```
```   685     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
```
```   686     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
```
```   687     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }
```
```   688 thus ?case by simp
```
```   689 qed(auto simp add: tmsubst)
```
```   690
```
```   691 lemma subst_nb: assumes tnb: "tmbound m t"
```
```   692 shows "bound m (subst m t p)"
```
```   693 using tnb tmsubst_nb incrtm0_tmbound
```
```   694 by (induct p arbitrary: m t rule: fm.induct, auto)
```
```   695
```
```   696 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
```
```   697 by (induct p rule: not.induct, auto)
```
```   698 lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
```
```   699 by (induct p rule: not.induct, auto)
```
```   700 lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
```
```   701 by (induct p rule: not.induct, auto)
```
```   702 lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
```
```   703  by (induct p rule: not.induct, auto)
```
```   704
```
```   705 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
```
```   706 using conj_def by auto
```
```   707 lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
```
```   708 using conj_def by auto
```
```   709 lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
```
```   710 using conj_def by auto
```
```   711 lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
```
```   712 using conj_def by auto
```
```   713
```
```   714 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
```
```   715 using disj_def by auto
```
```   716 lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
```
```   717 using disj_def by auto
```
```   718 lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
```
```   719 using disj_def by auto
```
```   720 lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
```
```   721 using disj_def by auto
```
```   722
```
```   723 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
```
```   724 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   725 lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (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_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
```
```   728 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
```
```   729 lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
```
```   730 using imp_def by auto
```
```   731
```
```   732 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
```
```   733   by (unfold iff_def,cases "p=q", auto)
```
```   734 lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
```
```   735 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   736 lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
```
```   737 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   738 lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
```
```   739 using iff_def by auto
```
```   740 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
```
```   741 by (induct p, simp_all)
```
```   742
```
```   743 consts
```
```   744   isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
```
```   745 recdef isatom "measure size"
```
```   746   "isatom T = True"
```
```   747   "isatom F = True"
```
```   748   "isatom (Lt a) = True"
```
```   749   "isatom (Le a) = True"
```
```   750   "isatom (Eq a) = True"
```
```   751   "isatom (NEq a) = True"
```
```   752   "isatom p = False"
```
```   753
```
```   754 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
```
```   755 by (induct p, simp_all)
```
```   756
```
```   757 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
```
```   758   "djf f p q \<equiv> (if q=T then T else if q=F then f p else
```
```   759   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
```
```   760 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
```
```   761   "evaldjf f ps \<equiv> foldr (djf f) ps F"
```
```   762
```
```   763 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
```
```   764 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
```
```   765 (cases "f p", simp_all add: Let_def djf_def)
```
```   766
```
```   767 lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
```
```   768   by(induct ps, simp_all add: evaldjf_def djf_Or)
```
```   769
```
```   770 lemma evaldjf_bound0:
```
```   771   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
```
```   772   shows "bound0 (evaldjf f xs)"
```
```   773   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   774
```
```   775 lemma evaldjf_qf:
```
```   776   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
```
```   777   shows "qfree (evaldjf f xs)"
```
```   778   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   779
```
```   780 consts disjuncts :: "fm \<Rightarrow> fm list"
```
```   781 recdef disjuncts "measure size"
```
```   782   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
```
```   783   "disjuncts F = []"
```
```   784   "disjuncts p = [p]"
```
```   785
```
```   786 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
```
```   787 by(induct p rule: disjuncts.induct, auto)
```
```   788
```
```   789 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
```
```   790 proof-
```
```   791   assume nb: "bound0 p"
```
```   792   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
```
```   793   thus ?thesis by (simp only: list_all_iff)
```
```   794 qed
```
```   795
```
```   796 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
```
```   797 proof-
```
```   798   assume qf: "qfree p"
```
```   799   hence "list_all qfree (disjuncts p)"
```
```   800     by (induct p rule: disjuncts.induct, auto)
```
```   801   thus ?thesis by (simp only: list_all_iff)
```
```   802 qed
```
```   803
```
```   804 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
```
```   805   "DJ f p \<equiv> evaldjf f (disjuncts p)"
```
```   806
```
```   807 lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
```
```   808   and fF: "f F = F"
```
```   809   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
```
```   810 proof-
```
```   811   have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
```
```   812     by (simp add: DJ_def evaldjf_ex)
```
```   813   also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
```
```   814   finally show ?thesis .
```
```   815 qed
```
```   816
```
```   817 lemma DJ_qf: assumes
```
```   818   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
```
```   819   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
```
```   820 proof(clarify)
```
```   821   fix  p assume qf: "qfree p"
```
```   822   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
```
```   823   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
```
```   824   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
```
```   825
```
```   826   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
```
```   827 qed
```
```   828
```
```   829 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))"
```
```   830   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
```
```   831 proof(clarify)
```
```   832   fix p::fm and bs
```
```   833   assume qf: "qfree p"
```
```   834   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
```
```   835   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
```
```   836   have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
```
```   837     by (simp add: DJ_def evaldjf_ex)
```
```   838   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
```
```   839   also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
```
```   840   finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
```
```   841 qed
```
```   842
```
```   843 consts conjuncts :: "fm \<Rightarrow> fm list"
```
```   844
```
```   845 recdef conjuncts "measure size"
```
```   846   "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
```
```   847   "conjuncts T = []"
```
```   848   "conjuncts p = [p]"
```
```   849
```
```   850 definition list_conj :: "fm list \<Rightarrow> fm" where
```
```   851   "list_conj ps \<equiv> foldr conj ps T"
```
```   852
```
```   853 definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
```
```   854   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
```
```   855                    in conj (decr0 (list_conj yes)) (f (list_conj no)))"
```
```   856
```
```   857 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
```
```   858 proof-
```
```   859   assume qf: "qfree p"
```
```   860   hence "list_all qfree (conjuncts p)"
```
```   861     by (induct p rule: conjuncts.induct, auto)
```
```   862   thus ?thesis by (simp only: list_all_iff)
```
```   863 qed
```
```   864
```
```   865 lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
```
```   866 by(induct p rule: conjuncts.induct, auto)
```
```   867
```
```   868 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
```
```   869 proof-
```
```   870   assume nb: "bound0 p"
```
```   871   hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
```
```   872   thus ?thesis by (simp only: list_all_iff)
```
```   873 qed
```
```   874
```
```   875 fun islin :: "fm \<Rightarrow> bool" where
```
```   876   "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)"
```
```   877 | "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)"
```
```   878 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   879 | "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   880 | "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   881 | "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   882 | "islin (NOT p) = False"
```
```   883 | "islin (Imp p q) = False"
```
```   884 | "islin (Iff p q) = False"
```
```   885 | "islin p = bound0 p"
```
```   886
```
```   887 lemma islin_stupid: assumes nb: "tmbound0 p"
```
```   888   shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
```
```   889   using nb by (cases p, auto, case_tac nat, auto)+
```
```   890
```
```   891 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
```
```   892 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
```
```   893 definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
```
```   894 definition "neq p = not (eq p)"
```
```   895
```
```   896 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
```
```   897   apply(simp add: lt_def)
```
```   898   apply(cases p, simp_all)
```
```   899   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   900   done
```
```   901
```
```   902 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
```
```   903   apply(simp add: le_def)
```
```   904   apply(cases p, simp_all)
```
```   905   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   906   done
```
```   907
```
```   908 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
```
```   909   apply(simp add: eq_def)
```
```   910   apply(cases p, simp_all)
```
```   911   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   912   done
```
```   913
```
```   914 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
```
```   915   by(simp add: neq_def eq)
```
```   916
```
```   917 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
```
```   918   apply (simp add: lt_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 le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
```
```   925   apply (simp add: le_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 eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
```
```   932   apply (simp add: 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 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
```
```   939   apply (simp add: neq_def eq_def)
```
```   940   apply (cases p, simp_all)
```
```   941   apply (case_tac poly, simp_all)
```
```   942   apply (case_tac nat, simp_all)
```
```   943   done
```
```   944
```
```   945 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
```
```   946 definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
```
```   947 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
```
```   948 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
```
```   949
```
```   950 lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   951   shows "islin (simplt t)"
```
```   952   unfolding simplt_def
```
```   953   using split0_nb0'
```
```   954 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])
```
```   955
```
```   956 lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   957   shows "islin (simple t)"
```
```   958   unfolding simple_def
```
```   959   using split0_nb0'
```
```   960 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)
```
```   961 lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   962   shows "islin (simpeq t)"
```
```   963   unfolding simpeq_def
```
```   964   using split0_nb0'
```
```   965 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)
```
```   966
```
```   967 lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   968   shows "islin (simpneq t)"
```
```   969   unfolding simpneq_def
```
```   970   using split0_nb0'
```
```   971 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)
```
```   972
```
```   973 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
```
```   974   by (cases "split0 s", auto)
```
```   975 lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   976   and n: "allpolys isnpoly t"
```
```   977   shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
```
```   978   using n
```
```   979   by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
```
```   980 lemma simplt[simp]:
```
```   981   shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
```
```   982 proof-
```
```   983   have n: "allpolys isnpoly (simptm t)" by simp
```
```   984   let ?t = "simptm t"
```
```   985   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```   986       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
```
```   987       by (simp add: simplt_def Let_def split_def lt)}
```
```   988   moreover
```
```   989   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```   990     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
```
```   991   }
```
```   992   ultimately show ?thesis by blast
```
```   993 qed
```
```   994
```
```   995 lemma simple[simp]:
```
```   996   shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
```
```   997 proof-
```
```   998   have n: "allpolys isnpoly (simptm t)" by simp
```
```   999   let ?t = "simptm t"
```
```  1000   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```  1001       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
```
```  1002       by (simp add: simple_def Let_def split_def le)}
```
```  1003   moreover
```
```  1004   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```  1005     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
```
```  1006   }
```
```  1007   ultimately show ?thesis by blast
```
```  1008 qed
```
```  1009
```
```  1010 lemma simpeq[simp]:
```
```  1011   shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
```
```  1012 proof-
```
```  1013   have n: "allpolys isnpoly (simptm t)" by simp
```
```  1014   let ?t = "simptm t"
```
```  1015   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```  1016       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
```
```  1017       by (simp add: simpeq_def Let_def split_def)}
```
```  1018   moreover
```
```  1019   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```  1020     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
```
```  1021   }
```
```  1022   ultimately show ?thesis by blast
```
```  1023 qed
```
```  1024
```
```  1025 lemma simpneq[simp]:
```
```  1026   shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
```
```  1027 proof-
```
```  1028   have n: "allpolys isnpoly (simptm t)" by simp
```
```  1029   let ?t = "simptm t"
```
```  1030   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```  1031       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
```
```  1032       by (simp add: simpneq_def Let_def split_def )}
```
```  1033   moreover
```
```  1034   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```  1035     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
```
```  1036   }
```
```  1037   ultimately show ?thesis by blast
```
```  1038 qed
```
```  1039
```
```  1040 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
```
```  1041   apply (simp add: lt_def)
```
```  1042   apply (cases t, auto)
```
```  1043   apply (case_tac poly, auto)
```
```  1044   done
```
```  1045
```
```  1046 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
```
```  1047   apply (simp add: le_def)
```
```  1048   apply (cases t, auto)
```
```  1049   apply (case_tac poly, auto)
```
```  1050   done
```
```  1051
```
```  1052 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
```
```  1053   apply (simp add: eq_def)
```
```  1054   apply (cases t, auto)
```
```  1055   apply (case_tac poly, auto)
```
```  1056   done
```
```  1057
```
```  1058 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
```
```  1059   apply (simp add: neq_def eq_def)
```
```  1060   apply (cases t, auto)
```
```  1061   apply (case_tac poly, auto)
```
```  1062   done
```
```  1063
```
```  1064 lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1065   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
```
```  1066   using split0 [of "simptm t" vs bs]
```
```  1067 proof(simp add: simplt_def Let_def split_def)
```
```  1068   assume nb: "tmbound0 t"
```
```  1069   hence nb': "tmbound0 (simptm t)" by simp
```
```  1070   let ?c = "fst (split0 (simptm t))"
```
```  1071   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1072   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1073   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1074   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1075   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1076   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1077   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
```
```  1078        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
```
```  1079 qed
```
```  1080
```
```  1081 lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1082   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
```
```  1083   using split0 [of "simptm t" vs bs]
```
```  1084 proof(simp add: simple_def Let_def split_def)
```
```  1085   assume nb: "tmbound0 t"
```
```  1086   hence nb': "tmbound0 (simptm t)" by simp
```
```  1087   let ?c = "fst (split0 (simptm t))"
```
```  1088   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1089   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1090   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1091   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1092   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1093   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1094   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
```
```  1095        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
```
```  1096 qed
```
```  1097
```
```  1098 lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1099   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
```
```  1100   using split0 [of "simptm t" vs bs]
```
```  1101 proof(simp add: simpeq_def Let_def split_def)
```
```  1102   assume nb: "tmbound0 t"
```
```  1103   hence nb': "tmbound0 (simptm t)" by simp
```
```  1104   let ?c = "fst (split0 (simptm t))"
```
```  1105   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1106   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1107   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1108   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1109   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1110   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1111   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
```
```  1112        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
```
```  1113 qed
```
```  1114
```
```  1115 lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1116   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
```
```  1117   using split0 [of "simptm t" vs bs]
```
```  1118 proof(simp add: simpneq_def Let_def split_def)
```
```  1119   assume nb: "tmbound0 t"
```
```  1120   hence nb': "tmbound0 (simptm t)" by simp
```
```  1121   let ?c = "fst (split0 (simptm t))"
```
```  1122   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1123   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1124   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1125   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1126   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1127   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1128   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
```
```  1129        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
```
```  1130 qed
```
```  1131
```
```  1132 consts conjs   :: "fm \<Rightarrow> fm list"
```
```  1133 recdef conjs "measure size"
```
```  1134   "conjs (And p q) = (conjs p)@(conjs q)"
```
```  1135   "conjs T = []"
```
```  1136   "conjs p = [p]"
```
```  1137 lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
```
```  1138 by (induct p rule: conjs.induct, auto)
```
```  1139 definition list_disj :: "fm list \<Rightarrow> fm" where
```
```  1140   "list_disj ps \<equiv> foldr disj ps F"
```
```  1141
```
```  1142 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
```
```  1143   by (induct ps, auto simp add: list_conj_def)
```
```  1144 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
```
```  1145   by (induct ps, auto simp add: list_conj_def conj_qf)
```
```  1146 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
```
```  1147   by (induct ps, auto simp add: list_disj_def)
```
```  1148
```
```  1149 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
```
```  1150   unfolding conj_def by auto
```
```  1151
```
```  1152 lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
```
```  1153   apply (induct p rule: conjs.induct)
```
```  1154   apply (unfold conjs.simps)
```
```  1155   apply (unfold set_append)
```
```  1156   apply (unfold ball_Un)
```
```  1157   apply (unfold bound.simps)
```
```  1158   apply auto
```
```  1159   done
```
```  1160
```
```  1161 lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
```
```  1162   apply (induct p rule: conjs.induct)
```
```  1163   apply (unfold conjs.simps)
```
```  1164   apply (unfold set_append)
```
```  1165   apply (unfold ball_Un)
```
```  1166   apply (unfold boundslt.simps)
```
```  1167   apply blast
```
```  1168 by simp_all
```
```  1169
```
```  1170 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
```
```  1171   unfolding list_conj_def
```
```  1172   by (induct ps, auto simp add: conj_boundslt)
```
```  1173
```
```  1174 lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
```
```  1175   shows "bound n (list_conj ps)"
```
```  1176   using bnd
```
```  1177   unfolding list_conj_def
```
```  1178   by (induct ps, auto simp add: conj_nb)
```
```  1179
```
```  1180 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
```
```  1181 unfolding list_conj_def by (induct ps , auto)
```
```  1182
```
```  1183 lemma CJNB_qe:
```
```  1184   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
```
```  1185   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
```
```  1186 proof(clarify)
```
```  1187   fix bs p
```
```  1188   assume qfp: "qfree p"
```
```  1189   let ?cjs = "conjuncts p"
```
```  1190   let ?yes = "fst (partition bound0 ?cjs)"
```
```  1191   let ?no = "snd (partition bound0 ?cjs)"
```
```  1192   let ?cno = "list_conj ?no"
```
```  1193   let ?cyes = "list_conj ?yes"
```
```  1194   have part: "partition bound0 ?cjs = (?yes,?no)" by simp
```
```  1195   from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast
```
```  1196   hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb')
```
```  1197   hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
```
```  1198   from conjuncts_qf[OF qfp] partition_set[OF part]
```
```  1199   have " \<forall>q\<in> set ?no. qfree q" by auto
```
```  1200   hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
```
```  1201   with qe have cno_qf:"qfree (qe ?cno )"
```
```  1202     and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
```
```  1203   from cno_qf yes_qf have qf: "qfree (CJNB qe p)"
```
```  1204     by (simp add: CJNB_def Let_def conj_qf split_def)
```
```  1205   {fix bs
```
```  1206     from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
```
```  1207     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
```
```  1208       using partition_set[OF part] by auto
```
```  1209     finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
```
```  1210   hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
```
```  1211   also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
```
```  1212     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
```
```  1213   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
```
```  1214     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
```
```  1215   also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
```
```  1216     using qe[rule_format, OF no_qf] by auto
```
```  1217   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)"
```
```  1218     by (simp add: Let_def CJNB_def split_def)
```
```  1219   with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
```
```  1220 qed
```
```  1221
```
```  1222 consts simpfm :: "fm \<Rightarrow> fm"
```
```  1223 recdef simpfm "measure fmsize"
```
```  1224   "simpfm (Lt t) = simplt (simptm t)"
```
```  1225   "simpfm (Le t) = simple (simptm t)"
```
```  1226   "simpfm (Eq t) = simpeq(simptm t)"
```
```  1227   "simpfm (NEq t) = simpneq(simptm t)"
```
```  1228   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
```
```  1229   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
```
```  1230   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
```
```  1231   "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
```
```  1232   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
```
```  1233   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
```
```  1234   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
```
```  1235   "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
```
```  1236   "simpfm (NOT (Eq t)) = simpneq t"
```
```  1237   "simpfm (NOT (NEq t)) = simpeq t"
```
```  1238   "simpfm (NOT (Le t)) = simplt (Neg t)"
```
```  1239   "simpfm (NOT (Lt t)) = simple (Neg t)"
```
```  1240   "simpfm (NOT (NOT p)) = simpfm p"
```
```  1241   "simpfm (NOT T) = F"
```
```  1242   "simpfm (NOT F) = T"
```
```  1243   "simpfm p = p"
```
```  1244
```
```  1245 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
```
```  1246 by(induct p arbitrary: bs rule: simpfm.induct, auto)
```
```  1247
```
```  1248 lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1249   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
```
```  1250 by (induct p rule: simpfm.induct, auto)
```
```  1251
```
```  1252 lemma lt_qf[simp]: "qfree (lt t)"
```
```  1253   apply (cases t, auto simp add: lt_def)
```
```  1254   by (case_tac poly, auto)
```
```  1255
```
```  1256 lemma le_qf[simp]: "qfree (le t)"
```
```  1257   apply (cases t, auto simp add: le_def)
```
```  1258   by (case_tac poly, auto)
```
```  1259
```
```  1260 lemma eq_qf[simp]: "qfree (eq t)"
```
```  1261   apply (cases t, auto simp add: eq_def)
```
```  1262   by (case_tac poly, auto)
```
```  1263
```
```  1264 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
```
```  1265
```
```  1266 lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
```
```  1267 lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
```
```  1268 lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
```
```  1269 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
```
```  1270
```
```  1271 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
```
```  1272 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
```
```  1273
```
```  1274 lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
```
```  1275 lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
```
```  1276
```
```  1277 lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1278   shows "qfree p \<Longrightarrow> islin (simpfm p)"
```
```  1279   apply (induct p rule: simpfm.induct)
```
```  1280   apply (simp_all add: conj_lin disj_lin)
```
```  1281   done
```
```  1282
```
```  1283 consts prep :: "fm \<Rightarrow> fm"
```
```  1284 recdef prep "measure fmsize"
```
```  1285   "prep (E T) = T"
```
```  1286   "prep (E F) = F"
```
```  1287   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
```
```  1288   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
```
```  1289   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
```
```  1290   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
```
```  1291   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
```
```  1292   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
```
```  1293   "prep (E p) = E (prep p)"
```
```  1294   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
```
```  1295   "prep (A p) = prep (NOT (E (NOT p)))"
```
```  1296   "prep (NOT (NOT p)) = prep p"
```
```  1297   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
```
```  1298   "prep (NOT (A p)) = prep (E (NOT p))"
```
```  1299   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
```
```  1300   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
```
```  1301   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
```
```  1302   "prep (NOT p) = not (prep p)"
```
```  1303   "prep (Or p q) = disj (prep p) (prep q)"
```
```  1304   "prep (And p q) = conj (prep p) (prep q)"
```
```  1305   "prep (Imp p q) = prep (Or (NOT p) q)"
```
```  1306   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
```
```  1307   "prep p = p"
```
```  1308 (hints simp add: fmsize_pos)
```
```  1309 lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
```
```  1310 by (induct p arbitrary: bs rule: prep.induct, auto)
```
```  1311
```
```  1312
```
```  1313
```
```  1314   (* Generic quantifier elimination *)
```
```  1315 consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
```
```  1316 recdef qelim "measure fmsize"
```
```  1317   "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
```
```  1318   "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
```
```  1319   "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
```
```  1320   "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))"
```
```  1321   "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))"
```
```  1322   "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
```
```  1323   "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
```
```  1324   "qelim p = (\<lambda> y. simpfm p)"
```
```  1325
```
```  1326
```
```  1327 lemma qelim:
```
```  1328   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
```
```  1329   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
```
```  1330 using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
```
```  1331 by (induct p rule: qelim.induct) auto
```
```  1332
```
```  1333 subsection{* Core Procedure *}
```
```  1334
```
```  1335 consts
```
```  1336   plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
```
```  1337   minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
```
```  1338 recdef minusinf "measure size"
```
```  1339   "minusinf (And p q) = conj (minusinf p) (minusinf q)"
```
```  1340   "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
```
```  1341   "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
```
```  1342   "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
```
```  1343   "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
```
```  1344   "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
```
```  1345   "minusinf p = p"
```
```  1346
```
```  1347 recdef plusinf "measure size"
```
```  1348   "plusinf (And p q) = conj (plusinf p) (plusinf q)"
```
```  1349   "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
```
```  1350   "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
```
```  1351   "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
```
```  1352   "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
```
```  1353   "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
```
```  1354   "plusinf p = p"
```
```  1355
```
```  1356 lemma minusinf_inf: assumes lp:"islin p"
```
```  1357   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
```
```  1358   using lp
```
```  1359 proof (induct p rule: minusinf.induct)
```
```  1360   case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
```
```  1361 next
```
```  1362   case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
```
```  1363 next
```
```  1364   case (3 c e) hence nbe: "tmbound0 e" by simp
```
```  1365   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1366   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1367   let ?c = "Ipoly vs c"
```
```  1368   let ?e = "Itm vs (y#bs) e"
```
```  1369   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1370   moreover {assume "?c = 0" hence ?case
```
```  1371       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
```
```  1372   moreover {assume cp: "?c > 0"
```
```  1373     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1374         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1375       hence "?c * x + ?e < 0" by simp
```
```  1376       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
```
```  1377         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
```
```  1378   moreover {assume cp: "?c < 0"
```
```  1379     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1380         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1381       hence "?c * x + ?e > 0" by simp
```
```  1382       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
```
```  1383         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
```
```  1384   ultimately show ?case by blast
```
```  1385 next
```
```  1386   case (4 c e)  hence nbe: "tmbound0 e" by simp
```
```  1387   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1388   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1389   let ?c = "Ipoly vs c"
```
```  1390   let ?e = "Itm vs (y#bs) e"
```
```  1391   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1392   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1393   moreover {assume cp: "?c > 0"
```
```  1394     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1395         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1396       hence "?c * x + ?e < 0" by simp
```
```  1397       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
```
```  1398         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1399   moreover {assume cp: "?c < 0"
```
```  1400     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1401         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1402       hence "?c * x + ?e > 0" by simp
```
```  1403       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
```
```  1404         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1405   ultimately show ?case by blast
```
```  1406 next
```
```  1407   case (5 c e)  hence nbe: "tmbound0 e" by simp
```
```  1408   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1409   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1410   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
```
```  1411   let ?c = "Ipoly vs c"
```
```  1412   let ?e = "Itm vs (y#bs) e"
```
```  1413   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1414   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1415   moreover {assume cp: "?c > 0"
```
```  1416     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1417         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1418       hence "?c * x + ?e < 0" by simp
```
```  1419       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
```
```  1420         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1421   moreover {assume cp: "?c < 0"
```
```  1422     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1423         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1424       hence "?c * x + ?e > 0" by simp
```
```  1425       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
```
```  1426         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
```
```  1427   ultimately show ?case by blast
```
```  1428 next
```
```  1429   case (6 c e)  hence nbe: "tmbound0 e" by simp
```
```  1430   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1431   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1432   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
```
```  1433   let ?c = "Ipoly vs c"
```
```  1434   let ?e = "Itm vs (y#bs) e"
```
```  1435   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1436   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1437   moreover {assume cp: "?c > 0"
```
```  1438     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1439         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1440       hence "?c * x + ?e < 0" by simp
```
```  1441       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
```
```  1442         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1443   moreover {assume cp: "?c < 0"
```
```  1444     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1445         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1446       hence "?c * x + ?e > 0" by simp
```
```  1447       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
```
```  1448         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1449   ultimately show ?case by blast
```
```  1450 qed (auto)
```
```  1451
```
```  1452 lemma plusinf_inf: assumes lp:"islin p"
```
```  1453   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
```
```  1454   using lp
```
```  1455 proof (induct p rule: plusinf.induct)
```
```  1456   case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
```
```  1457 next
```
```  1458   case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
```
```  1459 next
```
```  1460   case (3 c e) hence nbe: "tmbound0 e" by simp
```
```  1461   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1462   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1463   let ?c = "Ipoly vs c"
```
```  1464   let ?e = "Itm vs (y#bs) e"
```
```  1465   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1466   moreover {assume "?c = 0" hence ?case
```
```  1467       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
```
```  1468   moreover {assume cp: "?c > 0"
```
```  1469     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1470         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1471       hence "?c * x + ?e > 0" by simp
```
```  1472       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
```
```  1473         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
```
```  1474   moreover {assume cp: "?c < 0"
```
```  1475     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1476         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1477       hence "?c * x + ?e < 0" by simp
```
```  1478       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
```
```  1479         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
```
```  1480   ultimately show ?case by blast
```
```  1481 next
```
```  1482   case (4 c e) hence nbe: "tmbound0 e" by simp
```
```  1483   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1484   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1485   let ?c = "Ipoly vs c"
```
```  1486   let ?e = "Itm vs (y#bs) e"
```
```  1487   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1488   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1489   moreover {assume cp: "?c > 0"
```
```  1490     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1491         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1492       hence "?c * x + ?e > 0" by simp
```
```  1493       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
```
```  1494         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1495   moreover {assume cp: "?c < 0"
```
```  1496     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1497         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1498       hence "?c * x + ?e < 0" by simp
```
```  1499       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
```
```  1500         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1501   ultimately show ?case by blast
```
```  1502 next
```
```  1503   case (5 c e) hence nbe: "tmbound0 e" by simp
```
```  1504   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1505   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1506   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]
```
```  1507   let ?c = "Ipoly vs c"
```
```  1508   let ?e = "Itm vs (y#bs) e"
```
```  1509   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1510   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1511   moreover {assume cp: "?c > 0"
```
```  1512     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1513         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1514       hence "?c * x + ?e > 0" by simp
```
```  1515       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
```
```  1516         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1517   moreover {assume cp: "?c < 0"
```
```  1518     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1519         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1520       hence "?c * x + ?e < 0" by simp
```
```  1521       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
```
```  1522         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
```
```  1523   ultimately show ?case by blast
```
```  1524 next
```
```  1525   case (6 c e)  hence nbe: "tmbound0 e" by simp
```
```  1526   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1527   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1528   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
```
```  1529   let ?c = "Ipoly vs c"
```
```  1530   let ?e = "Itm vs (y#bs) e"
```
```  1531   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1532   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1533   moreover {assume cp: "?c > 0"
```
```  1534     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1535         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1536       hence "?c * x + ?e > 0" by simp
```
```  1537       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
```
```  1538         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1539   moreover {assume cp: "?c < 0"
```
```  1540     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1541         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1542       hence "?c * x + ?e < 0" by simp
```
```  1543       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
```
```  1544         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1545   ultimately show ?case by blast
```
```  1546 qed (auto)
```
```  1547
```
```  1548 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)"
```
```  1549   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
```
```  1550 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)"
```
```  1551   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
```
```  1552
```
```  1553 lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
```
```  1554   shows "\<exists>x. Ifm vs (x#bs) p"
```
```  1555 proof-
```
```  1556   from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
```
```  1557   have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
```
```  1558   from minusinf_inf[OF lp, where bs="bs"]
```
```  1559   obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
```
```  1560   from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
```
```  1561   moreover have "z - 1 < z" by simp
```
```  1562   ultimately show ?thesis using z_def by auto
```
```  1563 qed
```
```  1564
```
```  1565 lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
```
```  1566   shows "\<exists>x. Ifm vs (x#bs) p"
```
```  1567 proof-
```
```  1568   from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
```
```  1569   have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
```
```  1570   from plusinf_inf[OF lp, where bs="bs"]
```
```  1571   obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
```
```  1572   from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
```
```  1573   moreover have "z + 1 > z" by simp
```
```  1574   ultimately show ?thesis using z_def by auto
```
```  1575 qed
```
```  1576
```
```  1577 fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
```
```  1578   "uset (And p q) = uset p @ uset q"
```
```  1579 | "uset (Or p q) = uset p @ uset q"
```
```  1580 | "uset (Eq (CNP 0 a e))  = [(a,e)]"
```
```  1581 | "uset (Le (CNP 0 a e))  = [(a,e)]"
```
```  1582 | "uset (Lt (CNP 0 a e))  = [(a,e)]"
```
```  1583 | "uset (NEq (CNP 0 a e)) = [(a,e)]"
```
```  1584 | "uset p = []"
```
```  1585
```
```  1586 lemma uset_l:
```
```  1587   assumes lp: "islin p"
```
```  1588   shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
```
```  1589 using lp by(induct p rule: uset.induct,auto)
```
```  1590
```
```  1591 lemma minusinf_uset0:
```
```  1592   assumes lp: "islin p"
```
```  1593   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
```
```  1594   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1595   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c"
```
```  1596 proof-
```
```  1597   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)"
```
```  1598     using lp nmi ex
```
```  1599     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
```
```  1600     apply (auto simp add: linorder_not_less order_le_less)
```
```  1601     done
```
```  1602   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
```
```  1603   hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
```
```  1604     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
```
```  1605     by (auto simp add: mult_commute del: divide_minus_left)
```
```  1606   thus ?thesis using csU by auto
```
```  1607 qed
```
```  1608
```
```  1609 lemma minusinf_uset:
```
```  1610   assumes lp: "islin p"
```
```  1611   and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
```
```  1612   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1613   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c"
```
```  1614 proof-
```
```  1615   from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))"
```
```  1616     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
```
```  1617   from minusinf_uset0[OF lp nmi' ex]
```
```  1618   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
```
```  1619   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
```
```  1620   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
```
```  1621 qed
```
```  1622
```
```  1623
```
```  1624 lemma plusinf_uset0:
```
```  1625   assumes lp: "islin p"
```
```  1626   and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
```
```  1627   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1628   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c"
```
```  1629 proof-
```
```  1630   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)"
```
```  1631     using lp nmi ex
```
```  1632     apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
```
```  1633     apply (auto simp add: linorder_not_less order_le_less)
```
```  1634     done
```
```  1635   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
```
```  1636   hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
```
```  1637     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
```
```  1638     by (auto simp add: mult_commute del: divide_minus_left)
```
```  1639   thus ?thesis using csU by auto
```
```  1640 qed
```
```  1641
```
```  1642 lemma plusinf_uset:
```
```  1643   assumes lp: "islin p"
```
```  1644   and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
```
```  1645   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1646   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c"
```
```  1647 proof-
```
```  1648   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))"
```
```  1649     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
```
```  1650   from plusinf_uset0[OF lp nmi' ex]
```
```  1651   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
```
```  1652   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
```
```  1653   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
```
```  1654 qed
```
```  1655
```
```  1656 lemma lin_dense:
```
```  1657   assumes lp: "islin p"
```
```  1658   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)"
```
```  1659   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
```
```  1660   and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
```
```  1661   and ly: "l < y" and yu: "y < u"
```
```  1662   shows "Ifm vs (y#bs) p"
```
```  1663 using lp px noS
```
```  1664 proof (induct p rule: islin.induct)
```
```  1665   case (5 c s)
```
```  1666   from "5.prems"
```
```  1667   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1668     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
```
```  1669     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
```
```  1670   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1671   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1672   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1673   moreover
```
```  1674   {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"])}
```
```  1675   moreover
```
```  1676   {assume c: "?N c > 0"
```
```  1677       from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1678       have px': "x < - ?Nt x s / ?N c"
```
```  1679         by (auto simp add: not_less field_simps)
```
```  1680     {assume y: "y < - ?Nt x s / ?N c"
```
```  1681       hence "y * ?N c < - ?Nt x s"
```
```  1682         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1683       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1684       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1685     moreover
```
```  1686     {assume y: "y > -?Nt x s / ?N c"
```
```  1687       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1688       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1689       with lx px' have "False" by simp  hence ?case by simp }
```
```  1690     ultimately have ?case using ycs by blast
```
```  1691   }
```
```  1692   moreover
```
```  1693   {assume c: "?N c < 0"
```
```  1694       from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1695       have px': "x > - ?Nt x s / ?N c"
```
```  1696         by (auto simp add: not_less field_simps)
```
```  1697     {assume y: "y > - ?Nt x s / ?N c"
```
```  1698       hence "y * ?N c < - ?Nt x s"
```
```  1699         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1700       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1701       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1702     moreover
```
```  1703     {assume y: "y < -?Nt x s / ?N c"
```
```  1704       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1705       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1706       with xu px' have "False" by simp  hence ?case by simp }
```
```  1707     ultimately have ?case using ycs by blast
```
```  1708   }
```
```  1709   ultimately show ?case by blast
```
```  1710 next
```
```  1711   case (6 c s)
```
```  1712   from "6.prems"
```
```  1713   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1714     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
```
```  1715     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
```
```  1716   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1717   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1718   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1719   moreover
```
```  1720   {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"])}
```
```  1721   moreover
```
```  1722   {assume c: "?N c > 0"
```
```  1723       from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1724       have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps)
```
```  1725     {assume y: "y < - ?Nt x s / ?N c"
```
```  1726       hence "y * ?N c < - ?Nt x s"
```
```  1727         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1728       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1729       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1730     moreover
```
```  1731     {assume y: "y > -?Nt x s / ?N c"
```
```  1732       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1733       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1734       with lx px' have "False" by simp  hence ?case by simp }
```
```  1735     ultimately have ?case using ycs by blast
```
```  1736   }
```
```  1737   moreover
```
```  1738   {assume c: "?N c < 0"
```
```  1739       from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1740       have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1741     {assume y: "y > - ?Nt x s / ?N c"
```
```  1742       hence "y * ?N c < - ?Nt x s"
```
```  1743         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1744       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1745       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1746     moreover
```
```  1747     {assume y: "y < -?Nt x s / ?N c"
```
```  1748       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1749       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1750       with xu px' have "False" by simp  hence ?case by simp }
```
```  1751     ultimately have ?case using ycs by blast
```
```  1752   }
```
```  1753   ultimately show ?case by blast
```
```  1754 next
```
```  1755     case (3 c s)
```
```  1756   from "3.prems"
```
```  1757   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1758     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
```
```  1759     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
```
```  1760   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1761   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1762   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1763   moreover
```
```  1764   {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"])}
```
```  1765   moreover
```
```  1766   {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
```
```  1767     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
```
```  1768     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1769     {assume y: "y < -?Nt x s / ?N c"
```
```  1770       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1771       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1772       with xu px' have "False" by simp  hence ?case by simp }
```
```  1773     moreover
```
```  1774     {assume y: "y > -?Nt x s / ?N c"
```
```  1775       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1776       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1777       with lx px' have "False" by simp  hence ?case by simp }
```
```  1778     ultimately have ?case using ycs by blast
```
```  1779   }
```
```  1780   moreover
```
```  1781   {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
```
```  1782     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
```
```  1783     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1784     {assume y: "y < -?Nt x s / ?N c"
```
```  1785       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1786       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1787       with xu px' have "False" by simp  hence ?case by simp }
```
```  1788     moreover
```
```  1789     {assume y: "y > -?Nt x s / ?N c"
```
```  1790       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1791       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1792       with lx px' have "False" by simp  hence ?case by simp }
```
```  1793     ultimately have ?case using ycs by blast
```
```  1794   }
```
```  1795   ultimately show ?case by blast
```
```  1796 next
```
```  1797     case (4 c s)
```
```  1798   from "4.prems"
```
```  1799   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1800     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
```
```  1801     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
```
```  1802   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1803   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1804   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
```
```  1805   moreover
```
```  1806   {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"])}
```
```  1807   moreover
```
```  1808   {assume c: "?N c \<noteq> 0"
```
```  1809     from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
```
```  1810       by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
```
```  1811   ultimately show ?case by blast
```
```  1812 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"])
```
```  1813
```
```  1814 lemma one_plus_one_pos[simp]: "(1::'a::{linordered_field}) + 1 > 0"
```
```  1815 proof-
```
```  1816   have op: "(1::'a) > 0" by simp
```
```  1817   from add_pos_pos[OF op op] show ?thesis .
```
```  1818 qed
```
```  1819
```
```  1820 lemma one_plus_one_nonzero[simp]: "(1::'a::{linordered_field}) + 1 \<noteq> 0"
```
```  1821   using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le)
```
```  1822
```
```  1823 lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{linordered_field})"
```
```  1824 proof-
```
```  1825   have "(u + u) = (1 + 1) * u" by (simp add: field_simps)
```
```  1826   hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
```
```  1827   with nonzero_mult_divide_cancel_left[OF one_plus_one_nonzero, of u] show ?thesis by simp
```
```  1828 qed
```
```  1829
```
```  1830 lemma inf_uset:
```
```  1831   assumes lp: "islin p"
```
```  1832   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
```
```  1833   and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
```
```  1834   and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
```
```  1835   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"
```
```  1836 proof-
```
```  1837   let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
```
```  1838   let ?N = "Ipoly vs"
```
```  1839   let ?U = "set (uset p)"
```
```  1840   from ex obtain a where pa: "?I a p" by blast
```
```  1841   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
```
```  1842   have nmi': "\<not> (?I a (?M p))" by simp
```
```  1843   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
```
```  1844   have npi': "\<not> (?I a (?P p))" by simp
```
```  1845   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"
```
```  1846   proof-
```
```  1847     let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
```
```  1848     have fM: "finite ?M" by auto
```
```  1849     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa]
```
```  1850     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
```
```  1851     then obtain "c" "t" "d" "s" where
```
```  1852       ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U"
```
```  1853       and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
```
```  1854     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1
```
```  1855     have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
```
```  1856     from ctU have Mne: "?M \<noteq> {}" by auto
```
```  1857     hence Une: "?U \<noteq> {}" by simp
```
```  1858     let ?l = "Min ?M"
```
```  1859     let ?u = "Max ?M"
```
```  1860     have linM: "?l \<in> ?M" using fM Mne by simp
```
```  1861     have uinM: "?u \<in> ?M" using fM Mne by simp
```
```  1862     have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
```
```  1863     have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto
```
```  1864     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
```
```  1865     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
```
```  1866     have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
```
```  1867     have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
```
```  1868     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
```
```  1869     have "(\<exists> s\<in> ?M. ?I s p) \<or>
```
```  1870       (\<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)" .
```
```  1871     moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
```
```  1872       hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
```
```  1873       then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
```
```  1874       from half_sum_eq[of u] pu tuu
```
```  1875       have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / (1 + 1)) p" by simp
```
```  1876       with tuU have ?thesis by blast}
```
```  1877     moreover{
```
```  1878       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"
```
```  1879       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
```
```  1880         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"
```
```  1881         by blast
```
```  1882       from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
```
```  1883       then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
```
```  1884       from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
```
```  1885       then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
```
```  1886       from t1x xt2 have t1t2: "t1 < t2" by simp
```
```  1887       let ?u = "(t1 + t2) / (1 + 1)"
```
```  1888       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
```
```  1889       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
```
```  1890       with t1uU t2uU t1u t2u have ?thesis by blast}
```
```  1891     ultimately show ?thesis by blast
```
```  1892   qed
```
```  1893   then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U"
```
```  1894     and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / (1 + 1)) p" by blast
```
```  1895   from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
```
```  1896   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
```
```  1897     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
```
```  1898   have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / (1 + 1)) p" by simp
```
```  1899   with lnU smU
```
```  1900   show ?thesis by auto
```
```  1901 qed
```
```  1902
```
```  1903     (* The Ferrante - Rackoff Theorem *)
```
```  1904
```
```  1905 theorem fr_eq:
```
```  1906   assumes lp: "islin p"
```
```  1907   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))"
```
```  1908   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1909 proof
```
```  1910   assume px: "\<exists> x. ?I x p"
```
```  1911   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1912   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1913   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1914     from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
```
```  1915   ultimately show "?D" by blast
```
```  1916 next
```
```  1917   assume "?D"
```
```  1918   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
```
```  1919   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
```
```  1920   moreover {assume f:"?F" hence "?E" by blast}
```
```  1921   ultimately show "?E" by blast
```
```  1922 qed
```
```  1923
```
```  1924 section{* First implementation : Naive by encoding all case splits locally *}
```
```  1925 definition "msubsteq c t d s a r =
```
```  1926   evaldjf (split conj)
```
```  1927   [(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)))),
```
```  1928    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1929    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1930    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
```
```  1931
```
```  1932 lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  1933   shows "bound0 (msubsteq c t d s a r)"
```
```  1934 proof-
```
```  1935   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)))),
```
```  1936    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1937    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1938    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
```
```  1939     using lp by (simp add: Let_def t s )
```
```  1940   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
```
```  1941 qed
```
```  1942
```
```  1943 lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
```
```  1944   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")
```
```  1945 proof-
```
```  1946   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
```
```  1947   let ?N = "\<lambda>p. Ipoly vs p"
```
```  1948   let ?c = "?N c"
```
```  1949   let ?d = "?N d"
```
```  1950   let ?t = "?Nt x t"
```
```  1951   let ?s = "?Nt x s"
```
```  1952   let ?a = "?N a"
```
```  1953   let ?r = "?Nt x r"
```
```  1954   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  1955   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  1956   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
```
```  1957   moreover
```
```  1958   {assume c: "?c = 0" and d: "?d=0"
```
```  1959     hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
```
```  1960   moreover
```
```  1961   {assume c: "?c = 0" and d: "?d\<noteq>0"
```
```  1962     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
```
```  1963     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1964     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>))"])
```
```  1965     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) = 0"
```
```  1966       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
```
```  1967     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r= 0"
```
```  1968       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
```
```  1969
```
```  1970     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r = 0" using d by simp
```
```  1971     finally have ?thesis using c d
```
```  1972       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)
```
```  1973       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  1974       apply simp
```
```  1975       done}
```
```  1976   moreover
```
```  1977   {assume c: "?c \<noteq> 0" and d: "?d=0"
```
```  1978     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
```
```  1979     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1980     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r = 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
```
```  1981     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) = 0"
```
```  1982       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
```
```  1983     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r= 0"
```
```  1984       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
```
```  1985     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r = 0" using c by simp
```
```  1986     finally have ?thesis using c d
```
```  1987       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
```
```  1988       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  1989       apply simp
```
```  1990       done }
```
```  1991   moreover
```
```  1992   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
```
```  1993     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  1994     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  1995       by (simp add: field_simps)
```
```  1996     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1997     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r = 0"
```
```  1998       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  1999     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) =0 "
```
```  2000       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
```
```  2001     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r =0"
```
```  2002       using nonzero_mult_divide_cancel_left [OF dc] c d
```
```  2003       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2004     finally  have ?thesis using c d
```
```  2005       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
```
```  2006       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2007       apply (simp add: field_simps)
```
```  2008       done }
```
```  2009   ultimately show ?thesis by blast
```
```  2010 qed
```
```  2011
```
```  2012
```
```  2013 definition "msubstneq c t d s a r =
```
```  2014   evaldjf (split conj)
```
```  2015   [(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)))),
```
```  2016    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2017    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2018    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
```
```  2019
```
```  2020 lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2021   shows "bound0 (msubstneq c t d s a r)"
```
```  2022 proof-
```
```  2023   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)))),
```
```  2024     (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2025     (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2026     (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
```
```  2027     using lp by (simp add: Let_def t s )
```
```  2028   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
```
```  2029 qed
```
```  2030
```
```  2031 lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
```
```  2032   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")
```
```  2033 proof-
```
```  2034   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
```
```  2035   let ?N = "\<lambda>p. Ipoly vs p"
```
```  2036   let ?c = "?N c"
```
```  2037   let ?d = "?N d"
```
```  2038   let ?t = "?Nt x t"
```
```  2039   let ?s = "?Nt x s"
```
```  2040   let ?a = "?N a"
```
```  2041   let ?r = "?Nt x r"
```
```  2042   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2043   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2044   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
```
```  2045   moreover
```
```  2046   {assume c: "?c = 0" and d: "?d=0"
```
```  2047     hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
```
```  2048   moreover
```
```  2049   {assume c: "?c = 0" and d: "?d\<noteq>0"
```
```  2050     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
```
```  2051     have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2052     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>))"])
```
```  2053     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) \<noteq> 0"
```
```  2054       using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
```
```  2055     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r\<noteq> 0"
```
```  2056       by (simp add: field_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
```
```  2057
```
```  2058     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r \<noteq> 0" using d by simp
```
```  2059     finally have ?thesis using c d
```
```  2060       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)
```
```  2061       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2062       apply simp
```
```  2063       done}
```
```  2064   moreover
```
```  2065   {assume c: "?c \<noteq> 0" and d: "?d=0"
```
```  2066     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
```
```  2067     have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2068     also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
```
```  2069     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) \<noteq> 0"
```
```  2070       using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
```
```  2071     also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r \<noteq> 0"
```
```  2072       by (simp add: field_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
```
```  2073     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r \<noteq> 0" using c by simp
```
```  2074     finally have ?thesis using c d
```
```  2075       apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
```
```  2076       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2077       apply simp
```
```  2078       done }
```
```  2079   moreover
```
```  2080   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
```
```  2081     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2082     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  2083       by (simp add: field_simps)
```
```  2084     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2085     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r \<noteq> 0"
```
```  2086       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  2087     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) \<noteq> 0 "
```
```  2088       using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
```
```  2089     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r \<noteq> 0"
```
```  2090       using nonzero_mult_divide_cancel_left[OF dc] c d
```
```  2091       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2092     finally  have ?thesis using c d
```
```  2093       apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
```
```  2094       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2095       apply (simp add: field_simps)
```
```  2096       done }
```
```  2097   ultimately show ?thesis by blast
```
```  2098 qed
```
```  2099
```
```  2100 definition "msubstlt c t d s a r =
```
```  2101   evaldjf (split conj)
```
```  2102   [(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)]"
```
```  2109
```
```  2110 lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2111   shows "bound0 (msubstlt c t d s a r)"
```
```  2112 proof-
```
```  2113   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)))),
```
```  2114   (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)))),
```
```  2115    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2116    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2117    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2118    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2119    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
```
```  2120     using lp by (simp add: Let_def t s lt_nb )
```
```  2121   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
```
```  2122 qed
```
```  2123
```
```  2124
```
```  2125 lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))"
```
```  2126   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow>
```
```  2127   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")
```
```  2128 proof-
```
```  2129   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2130   let ?N = "\<lambda>p. Ipoly vs p"
```
```  2131   let ?c = "?N c"
```
```  2132   let ?d = "?N d"
```
```  2133   let ?t = "?Nt x t"
```
```  2134   let ?s = "?Nt x s"
```
```  2135   let ?a = "?N a"
```
```  2136   let ?r = "?Nt x r"
```
```  2137   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2138   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2139   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
```
```  2140   moreover
```
```  2141   {assume c: "?c=0" and d: "?d=0"
```
```  2142     hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
```
```  2143   moreover
```
```  2144   {assume dc: "?c*?d > 0"
```
```  2145     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
```
```  2146     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2147     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
```
```  2148     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2149     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  2150       by (simp add: field_simps)
```
```  2151     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2152     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0"
```
```  2153       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  2154     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) < 0"
```
```  2155
```
```  2156       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
```
```  2157     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r < 0"
```
```  2158       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
```
```  2159       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2160     finally  have ?thesis using dc c d  nc nd dc'
```
```  2161       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)
```
```  2162     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2163     by (simp add: field_simps order_less_not_sym[OF dc])}
```
```  2164   moreover
```
```  2165   {assume dc: "?c*?d < 0"
```
```  2166
```
```  2167     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
```
```  2168       by (simp add: mult_less_0_iff field_simps)
```
```  2169     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2170     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2171     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  2172       by (simp add: field_simps)
```
```  2173     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2174     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0"
```
```  2175       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  2176
```
```  2177     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) > 0"
```
```  2178
```
```  2179       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
```
```  2180     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r < 0"
```
```  2181       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
```
```  2182       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2183     finally  have ?thesis using dc c d  nc nd
```
```  2184       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)
```
```  2185       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2186       by (simp add: field_simps order_less_not_sym[OF dc]) }
```
```  2187   moreover
```
```  2188   {assume c: "?c > 0" and d: "?d=0"
```
```  2189     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
```
```  2190     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
```
```  2191     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
```
```  2192     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2193     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
```
```  2194     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) < 0"
```
```  2195       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
```
```  2196     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r < 0"
```
```  2197       using nonzero_mult_divide_cancel_left[OF c'] c
```
```  2198       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
```
```  2199     finally have ?thesis using c d nc nd
```
```  2200       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2201       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2202       using c order_less_not_sym[OF c] less_imp_neq[OF c]
```
```  2203       by (simp add: field_simps )  }
```
```  2204   moreover
```
```  2205   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
```
```  2206     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
```
```  2207     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
```
```  2208     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2209     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
```
```  2210     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) > 0"
```
```  2211       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
```
```  2212     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r < 0"
```
```  2213       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
```
```  2214         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
```
```  2215     finally have ?thesis using c d nc nd
```
```  2216       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2217       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2218       using c order_less_not_sym[OF c] less_imp_neq[OF c]
```
```  2219       by (simp add: field_simps )    }
```
```  2220   moreover
```
```  2221   moreover
```
```  2222   {assume c: "?c = 0" and d: "?d>0"
```
```  2223     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
```
```  2224     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
```
```  2225     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
```
```  2226     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2227     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
```
```  2228     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) < 0"
```
```  2229       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
```
```  2230     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r < 0"
```
```  2231       using nonzero_mult_divide_cancel_left[OF d'] d
```
```  2232       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
```
```  2233     finally have ?thesis using c d nc nd
```
```  2234       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2235       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2236       using d order_less_not_sym[OF d] less_imp_neq[OF d]
```
```  2237       by (simp add: field_simps)  }
```
```  2238   moreover
```
```  2239   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
```
```  2240     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
```
```  2241     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
```
```  2242     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2243     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
```
```  2244     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) > 0"
```
```  2245       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
```
```  2246     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r < 0"
```
```  2247       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
```
```  2248         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
```
```  2249     finally have ?thesis using c d nc nd
```
```  2250       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2251       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2252       using d order_less_not_sym[OF d] less_imp_neq[OF d]
```
```  2253       by (simp add: field_simps )    }
```
```  2254 ultimately show ?thesis by blast
```
```  2255 qed
```
```  2256
```
```  2257 definition "msubstle c t d s a r =
```
```  2258   evaldjf (split conj)
```
```  2259   [(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)]"
```
```  2266
```
```  2267 lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2268   shows "bound0 (msubstle c t d s a r)"
```
```  2269 proof-
```
```  2270   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)))),
```
```  2271   (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)))),
```
```  2272    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2273    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2274    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2275    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2276    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
```
```  2277     using lp by (simp add: Let_def t s lt_nb )
```
```  2278   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
```
```  2279 qed
```
```  2280
```
```  2281 lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))"
```
```  2282   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow>
```
```  2283   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")
```
```  2284 proof-
```
```  2285   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2286   let ?N = "\<lambda>p. Ipoly vs p"
```
```  2287   let ?c = "?N c"
```
```  2288   let ?d = "?N d"
```
```  2289   let ?t = "?Nt x t"
```
```  2290   let ?s = "?Nt x s"
```
```  2291   let ?a = "?N a"
```
```  2292   let ?r = "?Nt x r"
```
```  2293   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2294   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2295   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
```
```  2296   moreover
```
```  2297   {assume c: "?c=0" and d: "?d=0"
```
```  2298     hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
```
```  2299   moreover
```
```  2300   {assume dc: "?c*?d > 0"
```
```  2301     from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
```
```  2302     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2303     from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
```
```  2304     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2305     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  2306       by (simp add: field_simps)
```
```  2307     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2308     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0"
```
```  2309       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  2310     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) <= 0"
```
```  2311
```
```  2312       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
```
```  2313     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r <= 0"
```
```  2314       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
```
```  2315       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2316     finally  have ?thesis using dc c d  nc nd dc'
```
```  2317       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)
```
```  2318     apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2319     by (simp add: field_simps order_less_not_sym[OF dc])}
```
```  2320   moreover
```
```  2321   {assume dc: "?c*?d < 0"
```
```  2322
```
```  2323     from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
```
```  2324       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
```
```  2325     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2326     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2327     have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)"
```
```  2328       by (simp add: field_simps)
```
```  2329     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2330     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0"
```
```  2331       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
```
```  2332
```
```  2333     also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) >= 0"
```
```  2334
```
```  2335       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
```
```  2336     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r <= 0"
```
```  2337       using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
```
```  2338       by (simp add: algebra_simps diff_divide_distrib del: left_distrib)
```
```  2339     finally  have ?thesis using dc c d  nc nd
```
```  2340       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)
```
```  2341       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2342       by (simp add: field_simps order_less_not_sym[OF dc]) }
```
```  2343   moreover
```
```  2344   {assume c: "?c > 0" and d: "?d=0"
```
```  2345     from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
```
```  2346     from c have c': "(1 + 1)*?c \<noteq> 0" by simp
```
```  2347     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
```
```  2348     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2349     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
```
```  2350     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) <= 0"
```
```  2351       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
```
```  2352     also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r <= 0"
```
```  2353       using nonzero_mult_divide_cancel_left[OF c'] c
```
```  2354       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
```
```  2355     finally have ?thesis using c d nc nd
```
```  2356       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2357       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2358       using c order_less_not_sym[OF c] less_imp_neq[OF c]
```
```  2359       by (simp add: field_simps )  }
```
```  2360   moreover
```
```  2361   {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
```
```  2362     from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
```
```  2363     from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: field_simps)
```
```  2364     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2365     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
```
```  2366     also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) >= 0"
```
```  2367       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
```
```  2368     also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r <= 0"
```
```  2369       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
```
```  2370         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
```
```  2371     finally have ?thesis using c d nc nd
```
```  2372       apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2373       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2374       using c order_less_not_sym[OF c] less_imp_neq[OF c]
```
```  2375       by (simp add: field_simps )    }
```
```  2376   moreover
```
```  2377   moreover
```
```  2378   {assume c: "?c = 0" and d: "?d>0"
```
```  2379     from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
```
```  2380     from d have d': "(1 + 1)*?d \<noteq> 0" by simp
```
```  2381     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
```
```  2382     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2383     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
```
```  2384     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) <= 0"
```
```  2385       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
```
```  2386     also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r <= 0"
```
```  2387       using nonzero_mult_divide_cancel_left[OF d'] d
```
```  2388       by (simp add: algebra_simps diff_divide_distrib less_le del: left_distrib)
```
```  2389     finally have ?thesis using c d nc nd
```
```  2390       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2391       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2392       using d order_less_not_sym[OF d] less_imp_neq[OF d]
```
```  2393       by (simp add: field_simps )  }
```
```  2394   moreover
```
```  2395   {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
```
```  2396     from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
```
```  2397     from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: field_simps)
```
```  2398     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2399     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
```
```  2400     also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) >= 0"
```
```  2401       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
```
```  2402     also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r <= 0"
```
```  2403       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
```
```  2404         by (simp add: algebra_simps diff_divide_distrib del:  left_distrib)
```
```  2405     finally have ?thesis using c d nc nd
```
```  2406       apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2407       apply (simp only: one_add_one_is_two[symmetric] of_int_add)
```
```  2408       using d order_less_not_sym[OF d] less_imp_neq[OF d]
```
```  2409       by (simp add: field_simps )    }
```
```  2410 ultimately show ?thesis by blast
```
```  2411 qed
```
```  2412
```
```  2413
```
```  2414 fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
```
```  2415   "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
```
```  2416 | "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
```
```  2417 | "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
```
```  2418 | "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
```
```  2419 | "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
```
```  2420 | "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
```
```  2421 | "msubst p ((c,t),(d,s)) = p"
```
```  2422
```
```  2423 lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
```
```  2424   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"
```
```  2425   using lp
```
```  2426 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])
```
```  2427
```
```  2428 lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2429   shows "bound0 (msubst p ((c,t),(d,s)))"
```
```  2430   using lp t s
```
```  2431   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
```
```  2432
```
```  2433 lemma fr_eq_msubst:
```
```  2434   assumes lp: "islin p"
```
```  2435   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)))))"
```
```  2436   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  2437 proof-
```
```  2438 from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
```
```  2439 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
```
```  2440   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"
```
```  2441   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
```
```  2442   from msubst_I[OF lp norm, of vs x bs t s] pts
```
```  2443   have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
```
```  2444 moreover
```
```  2445 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
```
```  2446   and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
```
```  2447   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
```
```  2448   from msubst_I[OF lp norm, of vs x bs t s] pts
```
```  2449   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" ..}
```
```  2450 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
```
```  2451 from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
```
```  2452 qed
```
```  2453
```
```  2454 text {* Rest of the implementation *}
```
```  2455
```
```  2456 primrec alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list" where
```
```  2457   "alluopairs [] = []"
```
```  2458 | "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"
```
```  2459
```
```  2460 lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
```
```  2461 by (induct xs, auto)
```
```  2462
```
```  2463 lemma alluopairs_set:
```
```  2464   "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
```
```  2465 by (induct xs, auto)
```
```  2466
```
```  2467 lemma alluopairs_ex:
```
```  2468   assumes Pc: "\<forall> x \<in> set xs. \<forall>y\<in> set xs. P x y = P y x"
```
```  2469   shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
```
```  2470 proof
```
```  2471   assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
```
```  2472   then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
```
```  2473   from alluopairs_set[OF x y] P Pc x y show"\<exists>(x, y)\<in>set (alluopairs xs). P x y"
```
```  2474     by auto
```
```  2475 next
```
```  2476   assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
```
```  2477   then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
```
```  2478   from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
```
```  2479   with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
```
```  2480 qed
```
```  2481
```
```  2482 lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
```
```  2483 using Nat.gr0_conv_Suc
```
```  2484 by clarsimp
```
```  2485
```
```  2486 lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
```
```  2487   apply (induct xs, auto) done
```
```  2488
```
```  2489 consts remdps:: "'a list \<Rightarrow> 'a list"
```
```  2490
```
```  2491 recdef remdps "measure size"
```
```  2492   "remdps [] = []"
```
```  2493   "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
```
```  2494 (hints simp add: filter_length[rule_format])
```
```  2495
```
```  2496 lemma remdps_set[simp]: "set (remdps xs) = set xs"
```
```  2497   by (induct xs rule: remdps.induct, auto)
```
```  2498
```
```  2499 lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  2500   shows "qfree p \<Longrightarrow> islin (simpfm p)"
```
```  2501   by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
```
```  2502
```
```  2503 definition
```
```  2504   "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
```
```  2505   in if (mp = T \<or> pp = T) then T
```
```  2506      else (let U = alluopairs (remdps (uset  q))
```
```  2507            in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
```
```  2508
```
```  2509 lemma ferrack:
```
```  2510   assumes qf: "qfree p"
```
```  2511   shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
```
```  2512   (is "_ \<and> (?rhs = ?lhs)")
```
```  2513 proof-
```
```  2514   let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
```
```  2515   let ?N = "\<lambda> t. Ipoly vs t"
```
```  2516   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2517   let ?q = "simpfm p"
```
```  2518   let ?U = "remdps(uset ?q)"
```
```  2519   let ?Up = "alluopairs ?U"
```
```  2520   let ?mp = "minusinf ?q"
```
```  2521   let ?pp = "plusinf ?q"
```
```  2522   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2523   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
```
```  2524   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
```
```  2525   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
```
```  2526   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"
```
```  2527     by simp
```
```  2528   {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
```
```  2529     from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
```
```  2530     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]
```
```  2531     have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
```
```  2532   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
```
```  2533   {fix x assume xUp: "x \<in> set ?Up"
```
```  2534     then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
```
```  2535       and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto
```
```  2536     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU]
```
```  2537     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
```
```  2538     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]]
```
```  2539     have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
```
```  2540   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
```
```  2541   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
```
```  2542   with mp_nb pp_nb
```
```  2543   have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
```
```  2544   from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
```
```  2545   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
```
```  2546   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
```
```  2547   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
```
```  2548   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)"
```
```  2549     by (simp add: evaldjf_ex)
```
```  2550   also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
```
```  2551   also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
```
```  2552     apply (simp add: ferrack_def Let_def)
```
```  2553     by (cases "?mp = T \<or> ?pp = T", auto)
```
```  2554   finally show ?thesis using thqf by blast
```
```  2555 qed
```
```  2556
```
```  2557 definition "frpar p = simpfm (qelim p ferrack)"
```
```  2558 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
```
```  2559 proof-
```
```  2560   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
```
```  2561   from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
```
```  2562 qed
```
```  2563
```
```  2564
```
```  2565 section{* Second implemenation: Case splits not local *}
```
```  2566
```
```  2567 lemma fr_eq2:  assumes lp: "islin p"
```
```  2568   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow>
```
```  2569    ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or>
```
```  2570     (Ifm vs (0#bs) p) \<or>
```
```  2571     (\<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>
```
```  2572     (\<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))"
```
```  2573   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
```
```  2574 proof
```
```  2575   assume px: "\<exists> x. ?I x p"
```
```  2576   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  2577   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  2578   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  2579     from inf_uset[OF lp nmi npi, OF px]
```
```  2580     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"
```
```  2581       by auto
```
```  2582     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
```
```  2583     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
```
```  2584     let ?s = "Itm vs (x # bs) s"
```
```  2585     let ?t = "Itm vs (x # bs) t"
```
```  2586     have eq2: "\<And>(x::'a). x + x = (1 + 1) * x"
```
```  2587       by  (simp add: field_simps)
```
```  2588     {assume "?c = 0 \<and> ?d = 0"
```
```  2589       with ct have ?D by simp}
```
```  2590     moreover
```
```  2591     {assume z: "?c = 0" "?d \<noteq> 0"
```
```  2592       from z have ?D using ct by auto}
```
```  2593     moreover
```
```  2594     {assume z: "?c \<noteq> 0" "?d = 0"
```
```  2595       with ct have ?D by auto }
```
```  2596     moreover
```
```  2597     {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
```
```  2598       from z have ?F using ct
```
```  2599         apply - apply (rule bexI[where x = "(c,t)"], simp_all)
```
```  2600         by (rule bexI[where x = "(d,s)"], simp_all)
```
```  2601       hence ?D by blast}
```
```  2602     ultimately have ?D by auto}
```
```  2603   ultimately show "?D" by blast
```
```  2604 next
```
```  2605   assume "?D"
```
```  2606   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
```
```  2607   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
```
```  2608   moreover {assume f:"?F" hence "?E" by blast}
```
```  2609   ultimately show "?E" by blast
```
```  2610 qed
```
```  2611
```
```  2612 definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
```
```  2613 definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
```
```  2614 definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
```
```  2615 definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
```
```  2616 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
```
```  2617
```
```  2618 lemma msubsteq2:
```
```  2619   assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
```
```  2620   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")
```
```  2621   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2622   by (simp add: msubsteq2_def field_simps)
```
```  2623
```
```  2624 lemma msubstltpos:
```
```  2625   assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
```
```  2626   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")
```
```  2627   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2628   by (simp add: msubstltpos_def field_simps)
```
```  2629
```
```  2630 lemma msubstlepos:
```
```  2631   assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
```
```  2632   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")
```
```  2633   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2634   by (simp add: msubstlepos_def field_simps)
```
```  2635
```
```  2636 lemma msubstltneg:
```
```  2637   assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
```
```  2638   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")
```
```  2639   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2640   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
```
```  2641
```
```  2642 lemma msubstleneg:
```
```  2643   assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
```
```  2644   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")
```
```  2645   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2646   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
```
```  2647
```
```  2648 fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
```
```  2649   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
```
```  2650 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
```
```  2651 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
```
```  2652 | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
```
```  2653 | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
```
```  2654 | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
```
```  2655 | "msubstpos p c t = p"
```
```  2656
```
```  2657 lemma msubstpos_I:
```
```  2658   assumes lp: "islin p" and pos: "Ipoly vs c > 0"
```
```  2659   shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2660   using lp pos
```
```  2661   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)
```
```  2662
```
```  2663 fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
```
```  2664   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
```
```  2665 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
```
```  2666 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
```
```  2667 | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
```
```  2668 | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
```
```  2669 | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
```
```  2670 | "msubstneg p c t = p"
```
```  2671
```
```  2672 lemma msubstneg_I:
```
```  2673   assumes lp: "islin p" and pos: "Ipoly vs c < 0"
```
```  2674   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2675   using lp pos
```
```  2676   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)
```
```  2677
```
```  2678
```
```  2679 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)))"
```
```  2680
```
```  2681 lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
```
```  2682   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2683 proof-
```
```  2684   let ?c = "Ipoly vs c"
```
```  2685   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))"
```
```  2686     by (simp_all add: polyneg_norm)
```
```  2687   from nz have "?c > 0 \<or> ?c < 0" by arith
```
```  2688   moreover
```
```  2689   {assume c: "?c < 0"
```
```  2690     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"]
```
```  2691     have ?thesis by (auto simp add: msubst2_def)}
```
```  2692   moreover
```
```  2693   {assume c: "?c > 0"
```
```  2694     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"]
```
```  2695     have ?thesis by (auto simp add: msubst2_def)}
```
```  2696   ultimately show ?thesis by blast
```
```  2697 qed
```
```  2698
```
```  2699 term msubsteq2
```
```  2700 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
```
```  2701   by (simp add: msubsteq2_def)
```
```  2702
```
```  2703 lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
```
```  2704   by (simp add: msubstltpos_def)
```
```  2705 lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
```
```  2706   by (simp add: msubstltneg_def)
```
```  2707
```
```  2708 lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
```
```  2709   by (simp add: msubstlepos_def)
```
```  2710 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
```
```  2711   by (simp add: msubstleneg_def)
```
```  2712
```
```  2713 lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
```
```  2714   shows "bound0 (msubstpos p c t)"
```
```  2715 using lp tnb
```
```  2716 by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
```
```  2717
```
```  2718 lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
```
```  2719   shows "bound0 (msubstneg p c t)"
```
```  2720 using lp tnb
```
```  2721 by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
```
```  2722
```
```  2723 lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
```
```  2724   shows "bound0 (msubst2 p c t)"
```
```  2725 using lp tnb
```
```  2726 by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
```
```  2727
```
```  2728 lemma of_int2: "of_int 2 = 1 + 1"
```
```  2729 proof-
```
```  2730   have "(2::int) = 1 + 1" by simp
```
```  2731   hence "of_int 2 = of_int (1 + 1)" by simp
```
```  2732   thus ?thesis unfolding of_int_add by simp
```
```  2733 qed
```
```  2734
```
```  2735 lemma of_int_minus2: "of_int (-2) = - (1 + 1)"
```
```  2736 proof-
```
```  2737   have th: "(-2::int) = - 2" by simp
```
```  2738   show ?thesis unfolding th by (simp only: of_int_minus of_int2)
```
```  2739 qed
```
```  2740
```
```  2741
```
```  2742 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
```
```  2743   by (induct p rule: islin.induct, auto simp add: bound0_qf)
```
```  2744 lemma fr_eq_msubst2:
```
```  2745   assumes lp: "islin p"
```
```  2746   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)))))"
```
```  2747   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
```
```  2748 proof-
```
```  2749   from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
```
```  2750   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2751   have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
```
```  2752   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
```
```  2753
```
```  2754   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)"
```
```  2755   proof-
```
```  2756     {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
```
```  2757       from H(1) th have "isnpoly n" by blast
```
```  2758       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
```
```  2759       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
```
```  2760         by (simp add: polyneg_norm nn)
```
```  2761       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
```
```  2762         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
```
```  2763       from msubst2[OF lp nn nn2(1), of x bs t]
```
```  2764       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"
```
```  2765         using H(2) nn2 by (simp add: of_int_minus2 del: minus_add_distrib)}
```
```  2766     moreover
```
```  2767     {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"
```
```  2768       from H(1) th have "isnpoly n" by blast
```
```  2769       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
```
```  2770         using H(2) by (simp_all add: polymul_norm n2)
```
```  2771       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)}
```
```  2772     ultimately show ?thesis by blast
```
```  2773   qed
```
```  2774   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).
```
```  2775      \<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)"
```
```  2776   proof-
```
```  2777     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
```
```  2778      "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
```
```  2779       from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
```
```  2780       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)"
```
```  2781         by (simp_all add: polymul_norm n2)
```
```  2782       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)))"
```
```  2783         by (simp_all add: polyneg_norm nn)
```
```  2784       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"
```
```  2785         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)
```
```  2786       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
```
```  2787       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"
```
```  2788         apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)
```
```  2789         by (simp add: mult_commute)}
```
```  2790     moreover
```
```  2791     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
```
```  2792       "\<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"
```
```  2793      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
```
```  2794       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"
```
```  2795         using H(3,4) by (simp_all add: polymul_norm n2)
```
```  2796       from msubst2[OF lp nn, of x bs ] H(3,4,5)
```
```  2797       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)}
```
```  2798     ultimately show ?thesis by blast
```
```  2799   qed
```
```  2800   from fr_eq2[OF lp, of vs bs x] show ?thesis
```
```  2801     unfolding eq0 eq1 eq2 by blast
```
```  2802 qed
```
```  2803
```
```  2804 definition
```
```  2805 "ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
```
```  2806  in if (mp = T \<or> pp = T) then T
```
```  2807   else (let U = remdps (uset  q)
```
```  2808     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,
```
```  2809    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)]))"
```
```  2810
```
```  2811 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
```
```  2812
```
```  2813 lemma ferrack2: assumes qf: "qfree p"
```
```  2814   shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
```
```  2815   (is "_ \<and> (?rhs = ?lhs)")
```
```  2816 proof-
```
```  2817   let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
```
```  2818   let ?N = "\<lambda> t. Ipoly vs t"
```
```  2819   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2820   let ?q = "simpfm p"
```
```  2821   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
```
```  2822   let ?U = "remdps(uset ?q)"
```
```  2823   let ?Up = "alluopairs ?U"
```
```  2824   let ?mp = "minusinf ?q"
```
```  2825   let ?pp = "plusinf ?q"
```
```  2826   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2827   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
```
```  2828   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
```
```  2829   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
```
```  2830   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"
```
```  2831     by simp
```
```  2832   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)"
```
```  2833   proof-
```
```  2834     {fix c t assume ct: "(c,t) \<in> set ?U"
```
```  2835       hence tnb: "tmbound0 t" using U_l by blast
```
```  2836       from msubst2_nb[OF lq tnb]
```
```  2837       have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
```
```  2838     thus ?thesis by auto
```
```  2839   qed
```
```  2840   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)"
```
```  2841   proof-
```
```  2842     {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
```
```  2843       from badc U_l alluopairs_set1[of ?U]
```
```  2844       have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
```
```  2845       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}
```
```  2846     thus ?thesis by auto
```
```  2847   qed
```
```  2848   have stupid: "bound0 F" by simp
```
```  2849   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,
```
```  2850    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)]"
```
```  2851   from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
```
```  2852   have nb: "bound0 ?R "
```
```  2853     by (simp add: list_disj_def disj_nb0 simpfm_bound0)
```
```  2854   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))"
```
```  2855
```
```  2856   {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
```
```  2857     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))"
```
```  2858       by auto (simp add: isnpoly_def)
```
```  2859     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
```
```  2860       using norm by (simp_all add: polymul_norm)
```
```  2861     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)))"
```
```  2862       by (simp_all add: polyneg_norm norm2)
```
```  2863     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")
```
```  2864     proof
```
```  2865       assume H: ?lhs
```
```  2866       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"
```
```  2867         by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
```
```  2868       from msubst2[OF lq norm2(1) z(1), of x bs]
```
```  2869         msubst2[OF lq norm2(2) z(2), of x bs] H
```
```  2870       show ?rhs by (simp add: field_simps)
```
```  2871     next
```
```  2872       assume H: ?rhs
```
```  2873       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"
```
```  2874         by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
```
```  2875       from msubst2[OF lq norm2(1) z(1), of x bs]
```
```  2876         msubst2[OF lq norm2(2) z(2), of x bs] H
```
```  2877       show ?lhs by (simp add: field_simps)
```
```  2878     qed}
```
```  2879   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
```
```  2880     by clarsimp
```
```  2881
```
```  2882   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
```
```  2883   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))))"
```
```  2884     using fr_eq_msubst2[OF lq, of vs bs x] by simp
```
```  2885   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)))"
```
```  2886     by (simp add: split_def)
```
```  2887   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)))"
```
```  2888     using alluopairs_ex[OF th0] by simp
```
```  2889   also have "\<dots> \<longleftrightarrow> ?I ?R"
```
```  2890     by (simp add: list_disj_def evaldjf_ex split_def)
```
```  2891   also have "\<dots> \<longleftrightarrow> ?rhs"
```
```  2892     unfolding ferrack2_def
```
```  2893     apply (cases "?mp = T")
```
```  2894     apply (simp add: list_disj_def)
```
```  2895     apply (cases "?pp = T")
```
```  2896     apply (simp add: list_disj_def)
```
```  2897     by (simp_all add: Let_def decr0[OF nb])
```
```  2898   finally show ?thesis using decr0_qf[OF nb]
```
```  2899     by (simp  add: ferrack2_def Let_def)
```
```  2900 qed
```
```  2901
```
```  2902 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
```
```  2903 proof-
```
```  2904   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
```
```  2905   from qelim[OF th, of "prep p" bs]
```
```  2906 show ?thesis  unfolding frpar2_def by (auto simp add: prep)
```
```  2907 qed
```
```  2908
```
```  2909 ML {*
```
```  2910 structure ReflectedFRPar =
```
```  2911 struct
```
```  2912
```
```  2913 val bT = HOLogic.boolT;
```
```  2914 fun num rT x = HOLogic.mk_number rT x;
```
```  2915 fun rrelT rT = [rT,rT] ---> rT;
```
```  2916 fun rrT rT = [rT, rT] ---> bT;
```
```  2917 fun divt rT = Const(@{const_name Rings.divide},rrelT rT);
```
```  2918 fun timest rT = Const(@{const_name Groups.times},rrelT rT);
```
```  2919 fun plust rT = Const(@{const_name Groups.plus},rrelT rT);
```
```  2920 fun minust rT = Const(@{const_name Groups.minus},rrelT rT);
```
```  2921 fun uminust rT = Const(@{const_name Groups.uminus}, rT --> rT);
```
```  2922 fun powt rT = Const(@{const_name "power"}, [rT,@{typ "nat"}] ---> rT);
```
```  2923 val brT = [bT, bT] ---> bT;
```
```  2924 val nott = @{term "Not"};
```
```  2925 val conjt = @{term HOL.conj};
```
```  2926 val disjt = @{term HOL.disj};
```
```  2927 val impt = @{term HOL.implies};
```
```  2928 val ifft = @{term "op = :: bool => _"}
```
```  2929 fun llt rT = Const(@{const_name Orderings.less},rrT rT);
```
```  2930 fun lle rT = Const(@{const_name Orderings.less},rrT rT);
```
```  2931 fun eqt rT = Const(@{const_name HOL.eq},rrT rT);
```
```  2932 fun rz rT = Const(@{const_name Groups.zero},rT);
```
```  2933
```
```  2934 fun dest_nat t = case t of
```
```  2935   Const (@{const_name Suc}, _) \$ t' => 1 + dest_nat t'
```
```  2936 | _ => (snd o HOLogic.dest_number) t;
```
```  2937
```
```  2938 fun num_of_term m t =
```
```  2939  case t of
```
```  2940    Const(@{const_name Groups.uminus},_)\$t => @{code poly.Neg} (num_of_term m t)
```
```  2941  | Const(@{const_name Groups.plus},_)\$a\$b => @{code poly.Add} (num_of_term m a, num_of_term m b)
```
```  2942  | Const(@{const_name Groups.minus},_)\$a\$b => @{code poly.Sub} (num_of_term m a, num_of_term m b)
```
```  2943  | Const(@{const_name Groups.times},_)\$a\$b => @{code poly.Mul} (num_of_term m a, num_of_term m b)
```
```  2944  | Const(@{const_name Power.power},_)\$a\$n => @{code poly.Pw} (num_of_term m a, dest_nat n)
```
```  2945  | Const(@{const_name Rings.divide},_)\$a\$b => @{code poly.C} (HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
```
```  2946  | _ => (@{code poly.C} (HOLogic.dest_number t |> snd,1)
```
```  2947          handle TERM _ => @{code poly.Bound} (AList.lookup (op aconv) m t |> the));
```
```  2948
```
```  2949 fun tm_of_term m m' t =
```
```  2950  case t of
```
```  2951    Const(@{const_name Groups.uminus},_)\$t => @{code Neg} (tm_of_term m m' t)
```
```  2952  | Const(@{const_name Groups.plus},_)\$a\$b => @{code Add} (tm_of_term m m' a, tm_of_term m m' b)
```
```  2953  | Const(@{const_name Groups.minus},_)\$a\$b => @{code Sub} (tm_of_term m m' a, tm_of_term m m' b)
```
```  2954  | Const(@{const_name Groups.times},_)\$a\$b => @{code Mul} (num_of_term m' a, tm_of_term m m' b)
```
```  2955  | _ => (@{code CP} (num_of_term m' t)
```
```  2956          handle TERM _ => @{code Bound} (AList.lookup (op aconv) m t |> the)
```
```  2957               | Option => @{code Bound} (AList.lookup (op aconv) m t |> the));
```
```  2958
```
```  2959 fun term_of_num T m t =
```
```  2960  case t of
```
```  2961   @{code poly.C} (a,b) => (if b = 1 then num T a else if b=0 then (rz T)
```
```  2962                                         else (divt T) \$ num T a \$ num T b)
```
```  2963 | @{code poly.Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
```
```  2964 | @{code poly.Add} (a,b) => (plust T)\$(term_of_num T m a)\$(term_of_num T m b)
```
```  2965 | @{code poly.Mul} (a,b) => (timest T)\$(term_of_num T m a)\$(term_of_num T m b)
```
```  2966 | @{code poly.Sub} (a,b) => (minust T)\$(term_of_num T m a)\$(term_of_num T m b)
```
```  2967 | @{code poly.Neg} a => (uminust T)\$(term_of_num T m a)
```
```  2968 | @{code poly.Pw} (a,n) => (powt T)\$(term_of_num T m t)\$(HOLogic.mk_number HOLogic.natT n)
```
```  2969 | @{code poly.CN} (c,n,p) => term_of_num T m (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)))
```
```  2970 | _ => error "term_of_num: Unknown term";
```
```  2971
```
```  2972 fun term_of_tm T m m' t =
```
```  2973  case t of
```
```  2974   @{code CP} p => term_of_num T m' p
```
```  2975 | @{code Bound} i => AList.lookup (op = : int*int -> bool) m i |> the
```
```  2976 | @{code Add} (a,b) => (plust T)\$(term_of_tm T m m' a)\$(term_of_tm T m m' b)
```
```  2977 | @{code Mul} (a,b) => (timest T)\$(term_of_num T m' a)\$(term_of_tm T m m' b)
```
```  2978 | @{code Sub} (a,b) => (minust T)\$(term_of_tm T m m' a)\$(term_of_tm T m m' b)
```
```  2979 | @{code Neg} a => (uminust T)\$(term_of_tm T m m' a)
```
```  2980 | @{code CNP} (n,c,p) => term_of_tm T m m' (@{code Add}
```
```  2981      (@{code Mul} (c, @{code Bound} n), p))
```
```  2982 | _ => error "term_of_tm: Unknown term";
```
```  2983
```
```  2984 fun fm_of_term m m' fm =
```
```  2985  case fm of
```
```  2986     Const(@{const_name True},_) => @{code T}
```
```  2987   | Const(@{const_name False},_) => @{code F}
```
```  2988   | Const(@{const_name Not},_)\$p => @{code NOT} (fm_of_term m m' p)
```
```  2989   | Const(@{const_name HOL.conj},_)\$p\$q => @{code And} (fm_of_term m m' p, fm_of_term m m' q)
```
```  2990   | Const(@{const_name HOL.disj},_)\$p\$q => @{code Or} (fm_of_term m m' p, fm_of_term m m' q)
```
```  2991   | Const(@{const_name HOL.implies},_)\$p\$q => @{code Imp} (fm_of_term m m' p, fm_of_term m m' q)
```
```  2992   | Const(@{const_name HOL.eq},ty)\$p\$q =>
```
```  2993        if domain_type ty = bT then @{code Iff} (fm_of_term m m' p, fm_of_term m m' q)
```
```  2994        else @{code Eq} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
```
```  2995   | Const(@{const_name Orderings.less},_)\$p\$q =>
```
```  2996         @{code Lt} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
```
```  2997   | Const(@{const_name Orderings.less_eq},_)\$p\$q =>
```
```  2998         @{code Le} (@{code Sub} (tm_of_term m m' p, tm_of_term m m' q))
```
```  2999   | Const(@{const_name Ex},_)\$Abs(xn,xT,p) =>
```
```  3000      let val (xn', p') =  variant_abs (xn,xT,p)
```
```  3001          val x = Free(xn',xT)
```
```  3002          fun incr i = i + 1
```
```  3003          val m0 = (x,0):: (map (apsnd incr) m)
```
```  3004       in @{code E} (fm_of_term m0 m' p') end
```
```  3005   | Const(@{const_name All},_)\$Abs(xn,xT,p) =>
```
```  3006      let val (xn', p') =  variant_abs (xn,xT,p)
```
```  3007          val x = Free(xn',xT)
```
```  3008          fun incr i = i + 1
```
```  3009          val m0 = (x,0):: (map (apsnd incr) m)
```
```  3010       in @{code A} (fm_of_term m0 m' p') end
```
```  3011   | _ => error "fm_of_term";
```
```  3012
```
```  3013
```
```  3014 fun term_of_fm T m m' t =
```
```  3015   case t of
```
```  3016     @{code T} => Const(@{const_name True},bT)
```
```  3017   | @{code F} => Const(@{const_name False},bT)
```
```  3018   | @{code NOT} p => nott \$ (term_of_fm T m m' p)
```
```  3019   | @{code And} (p,q) => conjt \$ (term_of_fm T m m' p) \$ (term_of_fm T m m' q)
```
```  3020   | @{code Or} (p,q) => disjt \$ (term_of_fm T m m' p) \$ (term_of_fm T m m' q)
```
```  3021   | @{code Imp} (p,q) => impt \$ (term_of_fm T m m' p) \$ (term_of_fm T m m' q)
```
```  3022   | @{code Iff} (p,q) => ifft \$ (term_of_fm T m m' p) \$ (term_of_fm T m m' q)
```
```  3023   | @{code Lt} p => (llt T) \$ (term_of_tm T m m' p) \$ (rz T)
```
```  3024   | @{code Le} p => (lle T) \$ (term_of_tm T m m' p) \$ (rz T)
```
```  3025   | @{code Eq} p => (eqt T) \$ (term_of_tm T m m' p) \$ (rz T)
```
```  3026   | @{code NEq} p => nott \$ ((eqt T) \$ (term_of_tm T m m' p) \$ (rz T))
```
```  3027   | _ => error "term_of_fm: quantifiers!!!!???";
```
```  3028
```
```  3029 fun frpar_oracle (T,m, m', fm) =
```
```  3030  let
```
```  3031    val t = HOLogic.dest_Trueprop fm
```
```  3032    val im = 0 upto (length m - 1)
```
```  3033    val im' = 0 upto (length m' - 1)
```
```  3034  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')
```
```  3035                                                      (@{code frpar} (fm_of_term (m ~~ im) (m' ~~ im') t))))
```
```  3036  end;
```
```  3037
```
```  3038 fun frpar_oracle2 (T,m, m', fm) =
```
```  3039  let
```
```  3040    val t = HOLogic.dest_Trueprop fm
```
```  3041    val im = 0 upto (length m - 1)
```
```  3042    val im' = 0 upto (length m' - 1)
```
```  3043  in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')
```
```  3044                                                      (@{code frpar2} (fm_of_term (m ~~ im) (m' ~~ im') t))))
```
```  3045  end;
```
```  3046
```
```  3047 end;
```
```  3048
```
```  3049
```
```  3050 *}
```
```  3051
```
```  3052 oracle frpar_oracle = {* fn (ty, ts, ts', ct) =>
```
```  3053  let
```
```  3054   val thy = Thm.theory_of_cterm ct
```
```  3055  in cterm_of thy (ReflectedFRPar.frpar_oracle (ty,ts, ts', term_of ct))
```
```  3056  end *}
```
```  3057
```
```  3058 oracle frpar_oracle2 = {* fn (ty, ts, ts', ct) =>
```
```  3059  let
```
```  3060   val thy = Thm.theory_of_cterm ct
```
```  3061  in cterm_of thy (ReflectedFRPar.frpar_oracle2 (ty,ts, ts', term_of ct))
```
```  3062  end *}
```
```  3063
```
```  3064 ML{*
```
```  3065 structure FRParTac =
```
```  3066 struct
```
```  3067
```
```  3068 fun frpar_tac T ps ctxt i =
```
```  3069  Object_Logic.full_atomize_tac i
```
```  3070  THEN (fn st =>
```
```  3071   let
```
```  3072     val g = List.nth (cprems_of st, i - 1)
```
```  3073     val thy = ProofContext.theory_of ctxt
```
```  3074     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
```
```  3075     val th = frpar_oracle (T, fs,ps, (* Pattern.eta_long [] *)g)
```
```  3076   in rtac (th RS iffD2) i st end);
```
```  3077
```
```  3078 fun frpar2_tac T ps ctxt i =
```
```  3079  Object_Logic.full_atomize_tac i
```
```  3080  THEN (fn st =>
```
```  3081   let
```
```  3082     val g = List.nth (cprems_of st, i - 1)
```
```  3083     val thy = ProofContext.theory_of ctxt
```
```  3084     val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
```
```  3085     val th = frpar_oracle2 (T, fs,ps, (* Pattern.eta_long [] *)g)
```
```  3086   in rtac (th RS iffD2) i st end);
```
```  3087
```
```  3088 end;
```
```  3089
```
```  3090 *}
```
```  3091
```
```  3092 method_setup frpar = {*
```
```  3093 let
```
```  3094  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```  3095  fun simple_keyword k = Scan.lift (Args.\$\$\$ k) >> K ()
```
```  3096  val parsN = "pars"
```
```  3097  val typN = "type"
```
```  3098  val any_keyword = keyword parsN || keyword typN
```
```  3099  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
```
```  3100  val cterms = thms >> map Drule.dest_term;
```
```  3101  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
```
```  3102  val typ = Scan.unless any_keyword Args.typ
```
```  3103 in
```
```  3104  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
```
```  3105   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar_tac T ps ctxt))
```
```  3106 end
```
```  3107 *} "Parametric QE for linear Arithmetic over fields, Version 1"
```
```  3108
```
```  3109 method_setup frpar2 = {*
```
```  3110 let
```
```  3111  fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```  3112  fun simple_keyword k = Scan.lift (Args.\$\$\$ k) >> K ()
```
```  3113  val parsN = "pars"
```
```  3114  val typN = "type"
```
```  3115  val any_keyword = keyword parsN || keyword typN
```
```  3116  val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
```
```  3117  val cterms = thms >> map Drule.dest_term;
```
```  3118  val terms = Scan.repeat (Scan.unless any_keyword Args.term)
```
```  3119  val typ = Scan.unless any_keyword Args.typ
```
```  3120 in
```
```  3121  (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
```
```  3122   (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar2_tac T ps ctxt))
```
```  3123 end
```
```  3124 *} "Parametric QE for linear Arithmetic over fields, Version 2"
```
```  3125
```
```  3126
```
```  3127 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
```
```  3128   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3129   apply (simp add: field_simps)
```
```  3130   apply (rule spec[where x=y])
```
```  3131   apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3132   by simp
```
```  3133
```
```  3134 text{* Collins/Jones Problem *}
```
```  3135 (*
```
```  3136 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"
```
```  3137 proof-
```
```  3138   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")
```
```  3139 by (simp add: field_simps)
```
```  3140 have "?rhs"
```
```  3141
```
```  3142   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}")
```
```  3143   apply (simp add: field_simps)
```
```  3144 oops
```
```  3145 *)
```
```  3146 (*
```
```  3147 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"
```
```  3148 apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3149 oops
```
```  3150 *)
```
```  3151
```
```  3152 lemma "\<exists>(x::'a::{linordered_field_inverse_zero, number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
```
```  3153   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "y::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3154   apply (simp add: field_simps)
```
```  3155   apply (rule spec[where x=y])
```
```  3156   apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "z::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3157   by simp
```
```  3158
```
```  3159 text{* Collins/Jones Problem *}
```
```  3160
```
```  3161 (*
```
```  3162 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"
```
```  3163 proof-
```
```  3164   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")
```
```  3165 by (simp add: field_simps)
```
```  3166 have "?rhs"
```
```  3167   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}")
```
```  3168   apply simp
```
```  3169 oops
```
```  3170 *)
```
```  3171
```
```  3172 (*
```
```  3173 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"
```
```  3174 apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
```
```  3175 apply (simp add: field_simps linorder_neq_iff[symmetric])
```
```  3176 apply ferrack
```
```  3177 oops
```
```  3178 *)
```
`  3179 end`