src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
 author haftmann Sun Nov 11 19:24:01 2012 +0100 (2012-11-11) changeset 50045 2214bc566f88 parent 49962 a8cc904a6820 child 50282 fe4d4bb9f4c2 permissions -rw-r--r--
modernized, simplified and compacted oracle and proof method glue code;
corrected slips with poly.Pw and Orderings.less(_eq)
```     1 (*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
```
```     2     Author:     Amine Chaieb
```
```     3 *)
```
```     4
```
```     5 header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *}
```
```     6
```
```     7 theory Parametric_Ferrante_Rackoff
```
```     8 imports Reflected_Multivariate_Polynomial Dense_Linear_Order DP_Library
```
```     9   "~~/src/HOL/Library/Efficient_Nat" "~~/src/HOL/Library/Old_Recdef"
```
```    10 begin
```
```    11
```
```    12 subsection {* Terms *}
```
```    13
```
```    14 datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm
```
```    15   | Neg tm | Sub tm tm | CNP nat poly tm
```
```    16   (* A size for poly to make inductive proofs simpler*)
```
```    17
```
```    18 primrec tmsize :: "tm \<Rightarrow> nat" where
```
```    19   "tmsize (CP c) = polysize c"
```
```    20 | "tmsize (Bound n) = 1"
```
```    21 | "tmsize (Neg a) = 1 + tmsize a"
```
```    22 | "tmsize (Add a b) = 1 + tmsize a + tmsize b"
```
```    23 | "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
```
```    24 | "tmsize (Mul c a) = 1 + polysize c + tmsize a"
```
```    25 | "tmsize (CNP n c a) = 3 + polysize c + tmsize a "
```
```    26
```
```    27   (* Semantics of terms tm *)
```
```    28 primrec Itm :: "'a::{field_char_0, field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a" where
```
```    29   "Itm vs bs (CP c) = (Ipoly vs c)"
```
```    30 | "Itm vs bs (Bound n) = bs!n"
```
```    31 | "Itm vs bs (Neg a) = -(Itm vs bs a)"
```
```    32 | "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
```
```    33 | "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
```
```    34 | "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
```
```    35 | "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"
```
```    36
```
```    37
```
```    38 fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
```
```    39   "allpolys P (CP c) = P c"
```
```    40 | "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
```
```    41 | "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
```
```    42 | "allpolys P (Neg p) = allpolys P p"
```
```    43 | "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
```
```    44 | "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
```
```    45 | "allpolys P p = True"
```
```    46
```
```    47 primrec tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool" where
```
```    48   "tmboundslt n (CP c) = True"
```
```    49 | "tmboundslt n (Bound m) = (m < n)"
```
```    50 | "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
```
```    51 | "tmboundslt n (Neg a) = tmboundslt n a"
```
```    52 | "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
```
```    53 | "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)"
```
```    54 | "tmboundslt n (Mul i a) = tmboundslt n a"
```
```    55
```
```    56 primrec tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *) where
```
```    57   "tmbound0 (CP c) = True"
```
```    58 | "tmbound0 (Bound n) = (n>0)"
```
```    59 | "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
```
```    60 | "tmbound0 (Neg a) = tmbound0 a"
```
```    61 | "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
```
```    62 | "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)"
```
```    63 | "tmbound0 (Mul i a) = tmbound0 a"
```
```    64 lemma tmbound0_I:
```
```    65   assumes nb: "tmbound0 a"
```
```    66   shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
```
```    67 using nb
```
```    68 by (induct a rule: tm.induct,auto)
```
```    69
```
```    70 primrec tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *) where
```
```    71   "tmbound n (CP c) = True"
```
```    72 | "tmbound n (Bound m) = (n \<noteq> m)"
```
```    73 | "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
```
```    74 | "tmbound n (Neg a) = tmbound n a"
```
```    75 | "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
```
```    76 | "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)"
```
```    77 | "tmbound n (Mul i a) = tmbound n a"
```
```    78 lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)
```
```    79
```
```    80 lemma tmbound_I:
```
```    81   assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
```
```    82   shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
```
```    83   using nb le bnd
```
```    84   by (induct t rule: tm.induct , auto)
```
```    85
```
```    86 fun decrtm0:: "tm \<Rightarrow> tm" where
```
```    87   "decrtm0 (Bound n) = Bound (n - 1)"
```
```    88 | "decrtm0 (Neg a) = Neg (decrtm0 a)"
```
```    89 | "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
```
```    90 | "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
```
```    91 | "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
```
```    92 | "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
```
```    93 | "decrtm0 a = a"
```
```    94
```
```    95 fun incrtm0:: "tm \<Rightarrow> tm" where
```
```    96   "incrtm0 (Bound n) = Bound (n + 1)"
```
```    97 | "incrtm0 (Neg a) = Neg (incrtm0 a)"
```
```    98 | "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
```
```    99 | "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
```
```   100 | "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
```
```   101 | "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
```
```   102 | "incrtm0 a = a"
```
```   103
```
```   104 lemma decrtm0: assumes nb: "tmbound0 t"
```
```   105   shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
```
```   106   using nb by (induct t rule: decrtm0.induct, simp_all)
```
```   107
```
```   108 lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
```
```   109   by (induct t rule: decrtm0.induct, simp_all)
```
```   110
```
```   111 primrec decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" where
```
```   112   "decrtm m (CP c) = (CP c)"
```
```   113 | "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
```
```   114 | "decrtm m (Neg a) = Neg (decrtm m a)"
```
```   115 | "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
```
```   116 | "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
```
```   117 | "decrtm m (Mul c a) = Mul c (decrtm m a)"
```
```   118 | "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"
```
```   119
```
```   120 primrec removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list" where
```
```   121   "removen n [] = []"
```
```   122 | "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"
```
```   123
```
```   124 lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
```
```   125   by (induct xs arbitrary: n, auto)
```
```   126
```
```   127 lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
```
```   128   by (induct xs arbitrary: n, auto)
```
```   129
```
```   130 lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
```
```   131   by (induct xs arbitrary: n, auto)
```
```   132 lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m
```
```   133   else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
```
```   134 proof(induct xs arbitrary: n m)
```
```   135   case Nil thus ?case by simp
```
```   136 next
```
```   137   case (Cons x xs n m)
```
```   138   {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
```
```   139   moreover
```
```   140   {assume nxs: "\<not> (n \<ge> length (x#xs))"
```
```   141     {assume mln: "m < n" hence ?case using Cons by (cases m, auto)}
```
```   142     moreover
```
```   143     {assume mln: "\<not> (m < n)"
```
```   144       {assume mxs: "m \<le> length (x#xs)" hence ?case using Cons by (cases m, auto)}
```
```   145       moreover
```
```   146       {assume mxs: "\<not> (m \<le> length (x#xs))"
```
```   147         have th: "length (removen n (x#xs)) = length xs"
```
```   148           using removen_length[where n="n" and xs="x#xs"] nxs by simp
```
```   149         with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
```
```   150         hence "(removen n (x#xs))!m = [] ! (m - length xs)"
```
```   151           using th nth_length_exceeds[OF mxs'] by auto
```
```   152         hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))"
```
```   153           by auto
```
```   154         hence ?case using nxs mln mxs by auto }
```
```   155       ultimately have ?case by blast
```
```   156     }
```
```   157     ultimately have ?case by blast
```
```   158   } ultimately show ?case by blast
```
```   159 qed
```
```   160
```
```   161 lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t"
```
```   162   and nle: "m \<le> length bs"
```
```   163   shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
```
```   164   using bnd nb nle by (induct t rule: tm.induct) (auto simp add: removen_nth)
```
```   165
```
```   166 primrec tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm" where
```
```   167   "tmsubst0 t (CP c) = CP c"
```
```   168 | "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
```
```   169 | "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
```
```   170 | "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
```
```   171 | "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
```
```   172 | "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)"
```
```   173 | "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
```
```   174 lemma tmsubst0:
```
```   175   shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
```
```   176   by (induct a rule: tm.induct) auto
```
```   177
```
```   178 lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
```
```   179   by (induct a rule: tm.induct) auto
```
```   180
```
```   181 primrec tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" where
```
```   182   "tmsubst n t (CP c) = CP c"
```
```   183 | "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
```
```   184 | "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a)
```
```   185              else CNP m c (tmsubst n t a))"
```
```   186 | "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
```
```   187 | "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
```
```   188 | "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)"
```
```   189 | "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"
```
```   190
```
```   191 lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
```
```   192   shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
```
```   193 using nb nlt
```
```   194 by (induct a rule: tm.induct,auto)
```
```   195
```
```   196 lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
```
```   197 shows "tmbound0 (tmsubst 0 t a)"
```
```   198 using tnb
```
```   199 by (induct a rule: tm.induct, auto)
```
```   200
```
```   201 lemma tmsubst_nb: assumes tnb: "tmbound m t"
```
```   202 shows "tmbound m (tmsubst m t a)"
```
```   203 using tnb
```
```   204 by (induct a rule: tm.induct, auto)
```
```   205 lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
```
```   206   by (induct t, auto)
```
```   207   (* Simplification *)
```
```   208
```
```   209 consts
```
```   210   tmadd:: "tm \<times> tm \<Rightarrow> tm"
```
```   211 recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
```
```   212   "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
```
```   213   (if n1=n2 then
```
```   214   (let c = c1 +\<^sub>p c2
```
```   215   in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
```
```   216   else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2)))
```
```   217   else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
```
```   218   "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"
```
```   219   "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))"
```
```   220   "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
```
```   221   "tmadd (a,b) = Add a b"
```
```   222
```
```   223 lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
```
```   224 apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
```
```   225 apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
```
```   226 apply (case_tac "n1 = n2", simp_all add: field_simps)
```
```   227 apply (simp only: distrib_left[symmetric])
```
```   228 by (auto simp del: polyadd simp add: polyadd[symmetric])
```
```   229
```
```   230 lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
```
```   231 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
```
```   232
```
```   233 lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
```
```   234 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
```
```   235 lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
```
```   236 by (induct t s rule: tmadd.induct, auto simp add: Let_def)
```
```   237
```
```   238 lemma tmadd_allpolys_npoly[simp]: "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmadd(t,s))" by (induct t s rule: tmadd.induct, simp_all add: Let_def polyadd_norm)
```
```   239
```
```   240 fun tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm" where
```
```   241   "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
```
```   242 | "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
```
```   243 | "tmmul t = (\<lambda> i. Mul i t)"
```
```   244
```
```   245 lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
```
```   246 by (induct t arbitrary: i rule: tmmul.induct, simp_all add: field_simps)
```
```   247
```
```   248 lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
```
```   249 by (induct t arbitrary: i rule: tmmul.induct, auto )
```
```   250
```
```   251 lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
```
```   252 by (induct t arbitrary: n rule: tmmul.induct, auto )
```
```   253 lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
```
```   254 by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)
```
```   255
```
```   256 lemma tmmul_allpolys_npoly[simp]:
```
```   257   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   258   shows "allpolys isnpoly t \<Longrightarrow> isnpoly c \<Longrightarrow> allpolys isnpoly (tmmul t c)" by (induct t rule: tmmul.induct, simp_all add: Let_def polymul_norm)
```
```   259
```
```   260 definition tmneg :: "tm \<Rightarrow> tm" where
```
```   261   "tmneg t \<equiv> tmmul t (C (- 1,1))"
```
```   262
```
```   263 definition tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm" where
```
```   264   "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"
```
```   265
```
```   266 lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
```
```   267 using tmneg_def[of t]
```
```   268 apply simp
```
```   269 done
```
```   270
```
```   271 lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
```
```   272 using tmneg_def by simp
```
```   273
```
```   274 lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
```
```   275 using tmneg_def by simp
```
```   276 lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
```
```   277 using tmneg_def by simp
```
```   278 lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
```
```   279 lemma tmneg_allpolys_npoly[simp]:
```
```   280   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   281   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)"
```
```   282   unfolding tmneg_def by auto
```
```   283
```
```   284 lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
```
```   285 using tmsub_def by simp
```
```   286
```
```   287 lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
```
```   288 using tmsub_def by simp
```
```   289 lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
```
```   290 using tmsub_def by simp
```
```   291 lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
```
```   292 using tmsub_def by simp
```
```   293 lemma tmsub_allpolys_npoly[simp]:
```
```   294   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   295   shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)"
```
```   296   unfolding tmsub_def by (simp add: isnpoly_def)
```
```   297
```
```   298 fun simptm:: "tm \<Rightarrow> tm" where
```
```   299   "simptm (CP j) = CP (polynate j)"
```
```   300 | "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
```
```   301 | "simptm (Neg t) = tmneg (simptm t)"
```
```   302 | "simptm (Add t s) = tmadd (simptm t,simptm s)"
```
```   303 | "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
```
```   304 | "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
```
```   305 | "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))"
```
```   306
```
```   307 lemma polynate_stupid:
```
```   308   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   309   shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a)"
```
```   310 apply (subst polynate[symmetric])
```
```   311 apply simp
```
```   312 done
```
```   313
```
```   314 lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
```
```   315 by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid)
```
```   316
```
```   317 lemma simptm_tmbound0[simp]:
```
```   318   "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
```
```   319 by (induct t rule: simptm.induct, auto simp add: Let_def)
```
```   320
```
```   321 lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
```
```   322 by (induct t rule: simptm.induct, auto simp add: Let_def)
```
```   323 lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
```
```   324 by (induct t rule: simptm.induct, auto simp add: Let_def)
```
```   325
```
```   326 lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))"
```
```   327   by (simp_all add: isnpoly_def)
```
```   328 lemma simptm_allpolys_npoly[simp]:
```
```   329   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   330   shows "allpolys isnpoly (simptm p)"
```
```   331   by (induct p rule: simptm.induct, auto simp add: Let_def)
```
```   332
```
```   333 declare let_cong[fundef_cong del]
```
```   334
```
```   335 fun split0 :: "tm \<Rightarrow> (poly \<times> tm)" where
```
```   336   "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
```
```   337 | "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
```
```   338 | "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
```
```   339 | "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
```
```   340 | "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
```
```   341 | "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
```
```   342 | "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
```
```   343 | "split0 t = (0\<^sub>p, t)"
```
```   344
```
```   345 declare let_cong[fundef_cong]
```
```   346
```
```   347 lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
```
```   348   apply (rule exI[where x="fst (split0 p)"])
```
```   349   apply (rule exI[where x="snd (split0 p)"])
```
```   350   by simp
```
```   351
```
```   352 lemma split0:
```
```   353   "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
```
```   354   apply (induct t rule: split0.induct)
```
```   355   apply simp
```
```   356   apply (simp add: Let_def split_def field_simps)
```
```   357   apply (simp add: Let_def split_def field_simps)
```
```   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 mult_assoc distrib_left[symmetric])
```
```   362   apply (simp add: Let_def split_def field_simps)
```
```   363   apply (simp add: Let_def split_def field_simps)
```
```   364   done
```
```   365
```
```   366 lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
```
```   367 proof-
```
```   368   fix c' t'
```
```   369   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
```
```   370   with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
```
```   371 qed
```
```   372
```
```   373 lemma split0_nb0:
```
```   374   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   375   shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
```
```   376 proof-
```
```   377   fix c' t'
```
```   378   assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
```
```   379   with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
```
```   380 qed
```
```   381
```
```   382 lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   383   shows "tmbound0 (snd (split0 t))"
```
```   384   using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)
```
```   385
```
```   386
```
```   387 lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
```
```   388   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   389
```
```   390 lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
```
```   391   using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   392
```
```   393 lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
```
```   394  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   395
```
```   396 lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
```
```   397 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   398
```
```   399 lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
```
```   400  by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)
```
```   401
```
```   402 lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
```
```   403 by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)
```
```   404
```
```   405 lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   406   shows
```
```   407   "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
```
```   408   by (induct p rule: split0.induct,
```
```   409     auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm
```
```   410     Let_def split_def split0_stupid)
```
```   411
```
```   412 subsection{* Formulae *}
```
```   413
```
```   414 datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
```
```   415   NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm
```
```   416
```
```   417
```
```   418   (* A size for fm *)
```
```   419 fun fmsize :: "fm \<Rightarrow> nat" where
```
```   420   "fmsize (NOT p) = 1 + fmsize p"
```
```   421 | "fmsize (And p q) = 1 + fmsize p + fmsize q"
```
```   422 | "fmsize (Or p q) = 1 + fmsize p + fmsize q"
```
```   423 | "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
```
```   424 | "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
```
```   425 | "fmsize (E p) = 1 + fmsize p"
```
```   426 | "fmsize (A p) = 4+ fmsize p"
```
```   427 | "fmsize p = 1"
```
```   428   (* several lemmas about fmsize *)
```
```   429 lemma fmsize_pos[termination_simp]: "fmsize p > 0"
```
```   430 by (induct p rule: fmsize.induct) simp_all
```
```   431
```
```   432   (* Semantics of formulae (fm) *)
```
```   433 primrec Ifm ::"'a::{linordered_field_inverse_zero} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool" where
```
```   434   "Ifm vs bs T = True"
```
```   435 | "Ifm vs bs F = False"
```
```   436 | "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
```
```   437 | "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
```
```   438 | "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
```
```   439 | "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
```
```   440 | "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
```
```   441 | "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
```
```   442 | "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
```
```   443 | "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
```
```   444 | "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
```
```   445 | "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
```
```   446 | "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"
```
```   447
```
```   448 fun not:: "fm \<Rightarrow> fm" where
```
```   449   "not (NOT (NOT p)) = not p"
```
```   450 | "not (NOT p) = p"
```
```   451 | "not T = F"
```
```   452 | "not F = T"
```
```   453 | "not (Lt t) = Le (tmneg t)"
```
```   454 | "not (Le t) = Lt (tmneg t)"
```
```   455 | "not (Eq t) = NEq t"
```
```   456 | "not (NEq t) = Eq t"
```
```   457 | "not p = NOT p"
```
```   458 lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
```
```   459 by (induct p rule: not.induct) auto
```
```   460
```
```   461 definition conj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   462   "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
```
```   463    if p = q then p else And p q)"
```
```   464 lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
```
```   465 by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)
```
```   466
```
```   467 definition disj :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   468   "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p
```
```   469        else if p=q then p else Or p q)"
```
```   470
```
```   471 lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
```
```   472 by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
```
```   473
```
```   474 definition imp :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   475   "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
```
```   476     else Imp p q)"
```
```   477 lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
```
```   478 by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   479
```
```   480 definition iff :: "fm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   481   "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
```
```   482        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
```
```   483   Iff p q)"
```
```   484 lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
```
```   485   by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
```
```   486
```
```   487   (* Quantifier freeness *)
```
```   488 fun qfree:: "fm \<Rightarrow> bool" where
```
```   489   "qfree (E p) = False"
```
```   490 | "qfree (A p) = False"
```
```   491 | "qfree (NOT p) = qfree p"
```
```   492 | "qfree (And p q) = (qfree p \<and> qfree q)"
```
```   493 | "qfree (Or  p q) = (qfree p \<and> qfree q)"
```
```   494 | "qfree (Imp p q) = (qfree p \<and> qfree q)"
```
```   495 | "qfree (Iff p q) = (qfree p \<and> qfree q)"
```
```   496 | "qfree p = True"
```
```   497
```
```   498   (* Boundedness and substitution *)
```
```   499
```
```   500 primrec boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool" where
```
```   501   "boundslt n T = True"
```
```   502 | "boundslt n F = True"
```
```   503 | "boundslt n (Lt t) = (tmboundslt n t)"
```
```   504 | "boundslt n (Le t) = (tmboundslt n t)"
```
```   505 | "boundslt n (Eq t) = (tmboundslt n t)"
```
```   506 | "boundslt n (NEq t) = (tmboundslt n t)"
```
```   507 | "boundslt n (NOT p) = boundslt n p"
```
```   508 | "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
```
```   509 | "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
```
```   510 | "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
```
```   511 | "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
```
```   512 | "boundslt n (E p) = boundslt (Suc n) p"
```
```   513 | "boundslt n (A p) = boundslt (Suc n) p"
```
```   514
```
```   515 fun bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *) where
```
```   516   "bound0 T = True"
```
```   517 | "bound0 F = True"
```
```   518 | "bound0 (Lt a) = tmbound0 a"
```
```   519 | "bound0 (Le a) = tmbound0 a"
```
```   520 | "bound0 (Eq a) = tmbound0 a"
```
```   521 | "bound0 (NEq a) = tmbound0 a"
```
```   522 | "bound0 (NOT p) = bound0 p"
```
```   523 | "bound0 (And p q) = (bound0 p \<and> bound0 q)"
```
```   524 | "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
```
```   525 | "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
```
```   526 | "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
```
```   527 | "bound0 p = False"
```
```   528 lemma bound0_I:
```
```   529   assumes bp: "bound0 p"
```
```   530   shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
```
```   531 using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
```
```   532 by (induct p rule: bound0.induct,auto)
```
```   533
```
```   534 primrec bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *) where
```
```   535   "bound m T = True"
```
```   536 | "bound m F = True"
```
```   537 | "bound m (Lt t) = tmbound m t"
```
```   538 | "bound m (Le t) = tmbound m t"
```
```   539 | "bound m (Eq t) = tmbound m t"
```
```   540 | "bound m (NEq t) = tmbound m t"
```
```   541 | "bound m (NOT p) = bound m p"
```
```   542 | "bound m (And p q) = (bound m p \<and> bound m q)"
```
```   543 | "bound m (Or p q) = (bound m p \<and> bound m q)"
```
```   544 | "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
```
```   545 | "bound m (Iff p q) = (bound m p \<and> bound m q)"
```
```   546 | "bound m (E p) = bound (Suc m) p"
```
```   547 | "bound m (A p) = bound (Suc m) p"
```
```   548
```
```   549 lemma bound_I:
```
```   550   assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
```
```   551   shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
```
```   552   using bnd nb le tmbound_I[where bs=bs and vs = vs]
```
```   553 proof(induct p arbitrary: bs n rule: fm.induct)
```
```   554   case (E p bs n)
```
```   555   {fix y
```
```   556     from E have bnd: "boundslt (length (y#bs)) p"
```
```   557       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
```
```   558     from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
```
```   559   thus ?case by simp
```
```   560 next
```
```   561   case (A p bs n) {fix y
```
```   562     from A have bnd: "boundslt (length (y#bs)) p"
```
```   563       and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
```
```   564     from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
```
```   565   thus ?case by simp
```
```   566 qed auto
```
```   567
```
```   568 fun decr0 :: "fm \<Rightarrow> fm" where
```
```   569   "decr0 (Lt a) = Lt (decrtm0 a)"
```
```   570 | "decr0 (Le a) = Le (decrtm0 a)"
```
```   571 | "decr0 (Eq a) = Eq (decrtm0 a)"
```
```   572 | "decr0 (NEq a) = NEq (decrtm0 a)"
```
```   573 | "decr0 (NOT p) = NOT (decr0 p)"
```
```   574 | "decr0 (And p q) = conj (decr0 p) (decr0 q)"
```
```   575 | "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
```
```   576 | "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
```
```   577 | "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
```
```   578 | "decr0 p = p"
```
```   579
```
```   580 lemma decr0: assumes nb: "bound0 p"
```
```   581   shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
```
```   582   using nb
```
```   583   by (induct p rule: decr0.induct, simp_all add: decrtm0)
```
```   584
```
```   585 primrec decr :: "nat \<Rightarrow> fm \<Rightarrow> fm" where
```
```   586   "decr m T = T"
```
```   587 | "decr m F = F"
```
```   588 | "decr m (Lt t) = (Lt (decrtm m t))"
```
```   589 | "decr m (Le t) = (Le (decrtm m t))"
```
```   590 | "decr m (Eq t) = (Eq (decrtm m t))"
```
```   591 | "decr m (NEq t) = (NEq (decrtm m t))"
```
```   592 | "decr m (NOT p) = NOT (decr m p)"
```
```   593 | "decr m (And p q) = conj (decr m p) (decr m q)"
```
```   594 | "decr m (Or p q) = disj (decr m p) (decr m q)"
```
```   595 | "decr m (Imp p q) = imp (decr m p) (decr m q)"
```
```   596 | "decr m (Iff p q) = iff (decr m p) (decr m q)"
```
```   597 | "decr m (E p) = E (decr (Suc m) p)"
```
```   598 | "decr m (A p) = A (decr (Suc m) p)"
```
```   599
```
```   600 lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p"
```
```   601   and nle: "m < length bs"
```
```   602   shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
```
```   603   using bnd nb nle
```
```   604 proof(induct p arbitrary: bs m rule: fm.induct)
```
```   605   case (E p bs m)
```
```   606   {fix x
```
```   607     from E have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p"
```
```   608   and nle: "Suc m < length (x#bs)" by auto
```
```   609     from E(1)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
```
```   610   } thus ?case by auto
```
```   611 next
```
```   612   case (A p bs m)
```
```   613   {fix x
```
```   614     from A 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 A(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 qed (auto simp add: decrtm removen_nth)
```
```   619
```
```   620 primrec subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   621   "subst0 t T = T"
```
```   622 | "subst0 t F = F"
```
```   623 | "subst0 t (Lt a) = Lt (tmsubst0 t a)"
```
```   624 | "subst0 t (Le a) = Le (tmsubst0 t a)"
```
```   625 | "subst0 t (Eq a) = Eq (tmsubst0 t a)"
```
```   626 | "subst0 t (NEq a) = NEq (tmsubst0 t a)"
```
```   627 | "subst0 t (NOT p) = NOT (subst0 t p)"
```
```   628 | "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
```
```   629 | "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
```
```   630 | "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
```
```   631 | "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
```
```   632 | "subst0 t (E p) = E p"
```
```   633 | "subst0 t (A p) = A p"
```
```   634
```
```   635 lemma subst0: assumes qf: "qfree p"
```
```   636   shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
```
```   637 using qf tmsubst0[where x="x" and bs="bs" and t="t"]
```
```   638 by (induct p rule: fm.induct, auto)
```
```   639
```
```   640 lemma subst0_nb:
```
```   641   assumes bp: "tmbound0 t" and qf: "qfree p"
```
```   642   shows "bound0 (subst0 t p)"
```
```   643 using qf tmsubst0_nb[OF bp] bp
```
```   644 by (induct p rule: fm.induct, auto)
```
```   645
```
```   646 primrec subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" where
```
```   647   "subst n t T = T"
```
```   648 | "subst n t F = F"
```
```   649 | "subst n t (Lt a) = Lt (tmsubst n t a)"
```
```   650 | "subst n t (Le a) = Le (tmsubst n t a)"
```
```   651 | "subst n t (Eq a) = Eq (tmsubst n t a)"
```
```   652 | "subst n t (NEq a) = NEq (tmsubst n t a)"
```
```   653 | "subst n t (NOT p) = NOT (subst n t p)"
```
```   654 | "subst n t (And p q) = And (subst n t p) (subst n t q)"
```
```   655 | "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
```
```   656 | "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
```
```   657 | "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
```
```   658 | "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
```
```   659 | "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"
```
```   660
```
```   661 lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
```
```   662   shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
```
```   663   using nb nlm
```
```   664 proof (induct p arbitrary: bs n t rule: fm.induct)
```
```   665   case (E p bs n)
```
```   666   {fix x
```
```   667     from E have bn: "boundslt (length (x#bs)) p" by simp
```
```   668     from E have nlm: "Suc n \<le> length (x#bs)" by simp
```
```   669     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
```
```   670     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
```
```   671     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }
```
```   672 thus ?case by simp
```
```   673 next
```
```   674   case (A p bs n)
```
```   675   {fix x
```
```   676     from A have bn: "boundslt (length (x#bs)) p" by simp
```
```   677     from A have nlm: "Suc n \<le> length (x#bs)" by simp
```
```   678     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
```
```   679     hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
```
```   680     by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }
```
```   681 thus ?case by simp
```
```   682 qed(auto simp add: tmsubst)
```
```   683
```
```   684 lemma subst_nb: assumes tnb: "tmbound m t"
```
```   685 shows "bound m (subst m t p)"
```
```   686 using tnb tmsubst_nb incrtm0_tmbound
```
```   687 by (induct p arbitrary: m t rule: fm.induct, auto)
```
```   688
```
```   689 lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
```
```   690 by (induct p rule: not.induct, auto)
```
```   691 lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
```
```   692 by (induct p rule: not.induct, auto)
```
```   693 lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
```
```   694 by (induct p rule: not.induct, auto)
```
```   695 lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
```
```   696  by (induct p rule: not.induct, auto)
```
```   697
```
```   698 lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
```
```   699 using conj_def by auto
```
```   700 lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
```
```   701 using conj_def by auto
```
```   702 lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
```
```   703 using conj_def by auto
```
```   704 lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
```
```   705 using conj_def by auto
```
```   706
```
```   707 lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
```
```   708 using disj_def by auto
```
```   709 lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
```
```   710 using disj_def by auto
```
```   711 lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
```
```   712 using disj_def by auto
```
```   713 lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
```
```   714 using disj_def by auto
```
```   715
```
```   716 lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
```
```   717 using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
```
```   718 lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
```
```   719 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
```
```   720 lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
```
```   721 using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
```
```   722 lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
```
```   723 using imp_def by auto
```
```   724
```
```   725 lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
```
```   726   by (unfold iff_def,cases "p=q", auto)
```
```   727 lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
```
```   728 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   729 lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
```
```   730 using iff_def by (unfold iff_def,cases "p=q", auto)
```
```   731 lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
```
```   732 using iff_def by auto
```
```   733 lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
```
```   734 by (induct p, simp_all)
```
```   735
```
```   736 fun isatom :: "fm \<Rightarrow> bool" (* test for atomicity *) where
```
```   737   "isatom T = True"
```
```   738 | "isatom F = True"
```
```   739 | "isatom (Lt a) = True"
```
```   740 | "isatom (Le a) = True"
```
```   741 | "isatom (Eq a) = True"
```
```   742 | "isatom (NEq a) = True"
```
```   743 | "isatom p = False"
```
```   744
```
```   745 lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
```
```   746 by (induct p, simp_all)
```
```   747
```
```   748 definition djf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm" where
```
```   749   "djf f p q \<equiv> (if q=T then T else if q=F then f p else
```
```   750   (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
```
```   751 definition evaldjf :: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm" where
```
```   752   "evaldjf f ps \<equiv> foldr (djf f) ps F"
```
```   753
```
```   754 lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
```
```   755 by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def)
```
```   756 (cases "f p", simp_all add: Let_def djf_def)
```
```   757
```
```   758 lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
```
```   759   by(induct ps, simp_all add: evaldjf_def djf_Or)
```
```   760
```
```   761 lemma evaldjf_bound0:
```
```   762   assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
```
```   763   shows "bound0 (evaldjf f xs)"
```
```   764   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   765
```
```   766 lemma evaldjf_qf:
```
```   767   assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
```
```   768   shows "qfree (evaldjf f xs)"
```
```   769   using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto)
```
```   770
```
```   771 fun disjuncts :: "fm \<Rightarrow> fm list" where
```
```   772   "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
```
```   773 | "disjuncts F = []"
```
```   774 | "disjuncts p = [p]"
```
```   775
```
```   776 lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
```
```   777 by(induct p rule: disjuncts.induct, auto)
```
```   778
```
```   779 lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
```
```   780 proof-
```
```   781   assume nb: "bound0 p"
```
```   782   hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
```
```   783   thus ?thesis by (simp only: list_all_iff)
```
```   784 qed
```
```   785
```
```   786 lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
```
```   787 proof-
```
```   788   assume qf: "qfree p"
```
```   789   hence "list_all qfree (disjuncts p)"
```
```   790     by (induct p rule: disjuncts.induct, auto)
```
```   791   thus ?thesis by (simp only: list_all_iff)
```
```   792 qed
```
```   793
```
```   794 definition DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
```
```   795   "DJ f p \<equiv> evaldjf f (disjuncts p)"
```
```   796
```
```   797 lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
```
```   798   and fF: "f F = F"
```
```   799   shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
```
```   800 proof-
```
```   801   have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
```
```   802     by (simp add: DJ_def evaldjf_ex)
```
```   803   also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
```
```   804   finally show ?thesis .
```
```   805 qed
```
```   806
```
```   807 lemma DJ_qf: assumes
```
```   808   fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
```
```   809   shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
```
```   810 proof(clarify)
```
```   811   fix  p assume qf: "qfree p"
```
```   812   have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
```
```   813   from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
```
```   814   with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
```
```   815
```
```   816   from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
```
```   817 qed
```
```   818
```
```   819 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))"
```
```   820   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
```
```   821 proof(clarify)
```
```   822   fix p::fm and bs
```
```   823   assume qf: "qfree p"
```
```   824   from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
```
```   825   from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
```
```   826   have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
```
```   827     by (simp add: DJ_def evaldjf_ex)
```
```   828   also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
```
```   829   also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
```
```   830   finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
```
```   831 qed
```
```   832
```
```   833 fun conjuncts :: "fm \<Rightarrow> fm list" where
```
```   834   "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
```
```   835 | "conjuncts T = []"
```
```   836 | "conjuncts p = [p]"
```
```   837
```
```   838 definition list_conj :: "fm list \<Rightarrow> fm" where
```
```   839   "list_conj ps \<equiv> foldr conj ps T"
```
```   840
```
```   841 definition CJNB :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm" where
```
```   842   "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
```
```   843                    in conj (decr0 (list_conj yes)) (f (list_conj no)))"
```
```   844
```
```   845 lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
```
```   846 proof-
```
```   847   assume qf: "qfree p"
```
```   848   hence "list_all qfree (conjuncts p)"
```
```   849     by (induct p rule: conjuncts.induct, auto)
```
```   850   thus ?thesis by (simp only: list_all_iff)
```
```   851 qed
```
```   852
```
```   853 lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
```
```   854 by(induct p rule: conjuncts.induct, auto)
```
```   855
```
```   856 lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
```
```   857 proof-
```
```   858   assume nb: "bound0 p"
```
```   859   hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
```
```   860   thus ?thesis by (simp only: list_all_iff)
```
```   861 qed
```
```   862
```
```   863 fun islin :: "fm \<Rightarrow> bool" where
```
```   864   "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)"
```
```   865 | "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)"
```
```   866 | "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   867 | "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   868 | "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   869 | "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
```
```   870 | "islin (NOT p) = False"
```
```   871 | "islin (Imp p q) = False"
```
```   872 | "islin (Iff p q) = False"
```
```   873 | "islin p = bound0 p"
```
```   874
```
```   875 lemma islin_stupid: assumes nb: "tmbound0 p"
```
```   876   shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
```
```   877   using nb by (cases p, auto, case_tac nat, auto)+
```
```   878
```
```   879 definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
```
```   880 definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
```
```   881 definition eq where "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
```
```   882 definition "neq p = not (eq p)"
```
```   883
```
```   884 lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
```
```   885   apply(simp add: lt_def)
```
```   886   apply(cases p, simp_all)
```
```   887   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   888   done
```
```   889
```
```   890 lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
```
```   891   apply(simp add: le_def)
```
```   892   apply(cases p, simp_all)
```
```   893   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   894   done
```
```   895
```
```   896 lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
```
```   897   apply(simp add: eq_def)
```
```   898   apply(cases p, simp_all)
```
```   899   apply (case_tac poly, simp_all add: isnpoly_def)
```
```   900   done
```
```   901
```
```   902 lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
```
```   903   by(simp add: neq_def eq)
```
```   904
```
```   905 lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
```
```   906   apply (simp add: lt_def)
```
```   907   apply (cases p, simp_all)
```
```   908   apply (case_tac poly, simp_all)
```
```   909   apply (case_tac nat, simp_all)
```
```   910   done
```
```   911
```
```   912 lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
```
```   913   apply (simp add: le_def)
```
```   914   apply (cases p, simp_all)
```
```   915   apply (case_tac poly, simp_all)
```
```   916   apply (case_tac nat, simp_all)
```
```   917   done
```
```   918
```
```   919 lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
```
```   920   apply (simp add: eq_def)
```
```   921   apply (cases p, simp_all)
```
```   922   apply (case_tac poly, simp_all)
```
```   923   apply (case_tac nat, simp_all)
```
```   924   done
```
```   925
```
```   926 lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
```
```   927   apply (simp add: neq_def eq_def)
```
```   928   apply (cases p, simp_all)
```
```   929   apply (case_tac poly, simp_all)
```
```   930   apply (case_tac nat, simp_all)
```
```   931   done
```
```   932
```
```   933 definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
```
```   934 definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
```
```   935 definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
```
```   936 definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"
```
```   937
```
```   938 lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   939   shows "islin (simplt t)"
```
```   940   unfolding simplt_def
```
```   941   using split0_nb0'
```
```   942 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])
```
```   943
```
```   944 lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   945   shows "islin (simple t)"
```
```   946   unfolding simple_def
```
```   947   using split0_nb0'
```
```   948 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)
```
```   949 lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   950   shows "islin (simpeq t)"
```
```   951   unfolding simpeq_def
```
```   952   using split0_nb0'
```
```   953 by (auto simp add: Let_def split_def isnpoly_fst_split0[OF simptm_allpolys_npoly] islin_stupid allpolys_split0[OF simptm_allpolys_npoly] eq_lin)
```
```   954
```
```   955 lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   956   shows "islin (simpneq t)"
```
```   957   unfolding simpneq_def
```
```   958   using split0_nb0'
```
```   959 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)
```
```   960
```
```   961 lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
```
```   962   by (cases "split0 s", auto)
```
```   963 lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```   964   and n: "allpolys isnpoly t"
```
```   965   shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
```
```   966   using n
```
```   967   by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
```
```   968 lemma simplt[simp]:
```
```   969   shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
```
```   970 proof-
```
```   971   have n: "allpolys isnpoly (simptm t)" by simp
```
```   972   let ?t = "simptm t"
```
```   973   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```   974       using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
```
```   975       by (simp add: simplt_def Let_def split_def lt)}
```
```   976   moreover
```
```   977   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```   978     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
```
```   979   }
```
```   980   ultimately show ?thesis by blast
```
```   981 qed
```
```   982
```
```   983 lemma simple[simp]:
```
```   984   shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
```
```   985 proof-
```
```   986   have n: "allpolys isnpoly (simptm t)" by simp
```
```   987   let ?t = "simptm t"
```
```   988   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```   989       using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
```
```   990       by (simp add: simple_def Let_def split_def le)}
```
```   991   moreover
```
```   992   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```   993     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
```
```   994   }
```
```   995   ultimately show ?thesis by blast
```
```   996 qed
```
```   997
```
```   998 lemma simpeq[simp]:
```
```   999   shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
```
```  1000 proof-
```
```  1001   have n: "allpolys isnpoly (simptm t)" by simp
```
```  1002   let ?t = "simptm t"
```
```  1003   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```  1004       using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
```
```  1005       by (simp add: simpeq_def Let_def split_def)}
```
```  1006   moreover
```
```  1007   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```  1008     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
```
```  1009   }
```
```  1010   ultimately show ?thesis by blast
```
```  1011 qed
```
```  1012
```
```  1013 lemma simpneq[simp]:
```
```  1014   shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
```
```  1015 proof-
```
```  1016   have n: "allpolys isnpoly (simptm t)" by simp
```
```  1017   let ?t = "simptm t"
```
```  1018   {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
```
```  1019       using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
```
```  1020       by (simp add: simpneq_def Let_def split_def )}
```
```  1021   moreover
```
```  1022   {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
```
```  1023     hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
```
```  1024   }
```
```  1025   ultimately show ?thesis by blast
```
```  1026 qed
```
```  1027
```
```  1028 lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
```
```  1029   apply (simp add: lt_def)
```
```  1030   apply (cases t, auto)
```
```  1031   apply (case_tac poly, auto)
```
```  1032   done
```
```  1033
```
```  1034 lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
```
```  1035   apply (simp add: le_def)
```
```  1036   apply (cases t, auto)
```
```  1037   apply (case_tac poly, auto)
```
```  1038   done
```
```  1039
```
```  1040 lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
```
```  1041   apply (simp add: eq_def)
```
```  1042   apply (cases t, auto)
```
```  1043   apply (case_tac poly, auto)
```
```  1044   done
```
```  1045
```
```  1046 lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
```
```  1047   apply (simp add: neq_def eq_def)
```
```  1048   apply (cases t, auto)
```
```  1049   apply (case_tac poly, auto)
```
```  1050   done
```
```  1051
```
```  1052 lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1053   shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
```
```  1054   using split0 [of "simptm t" "vs::'a list" bs]
```
```  1055 proof(simp add: simplt_def Let_def split_def)
```
```  1056   assume nb: "tmbound0 t"
```
```  1057   hence nb': "tmbound0 (simptm t)" by simp
```
```  1058   let ?c = "fst (split0 (simptm t))"
```
```  1059   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1060   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1061   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1062   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1063   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1064   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1065   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
```
```  1066        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
```
```  1067 qed
```
```  1068
```
```  1069 lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1070   shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
```
```  1071   using split0 [of "simptm t" "vs::'a list" bs]
```
```  1072 proof(simp add: simple_def Let_def split_def)
```
```  1073   assume nb: "tmbound0 t"
```
```  1074   hence nb': "tmbound0 (simptm t)" by simp
```
```  1075   let ?c = "fst (split0 (simptm t))"
```
```  1076   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1077   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1078   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1079   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1080   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1081   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1082   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
```
```  1083        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
```
```  1084 qed
```
```  1085
```
```  1086 lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1087   shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
```
```  1088   using split0 [of "simptm t" "vs::'a list" bs]
```
```  1089 proof(simp add: simpeq_def Let_def split_def)
```
```  1090   assume nb: "tmbound0 t"
```
```  1091   hence nb': "tmbound0 (simptm t)" by simp
```
```  1092   let ?c = "fst (split0 (simptm t))"
```
```  1093   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1094   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1095   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1096   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1097   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1098   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1099   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
```
```  1100        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
```
```  1101 qed
```
```  1102
```
```  1103 lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1104   shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
```
```  1105   using split0 [of "simptm t" "vs::'a list" bs]
```
```  1106 proof(simp add: simpneq_def Let_def split_def)
```
```  1107   assume nb: "tmbound0 t"
```
```  1108   hence nb': "tmbound0 (simptm t)" by simp
```
```  1109   let ?c = "fst (split0 (simptm t))"
```
```  1110   from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
```
```  1111   have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
```
```  1112   from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
```
```  1113   have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
```
```  1114   from iffD1[OF isnpolyh_unique[OF ths] th]
```
```  1115   have "fst (split0 (simptm t)) = 0\<^sub>p" .
```
```  1116   thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
```
```  1117        fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
```
```  1118 qed
```
```  1119
```
```  1120 fun conjs   :: "fm \<Rightarrow> fm list" where
```
```  1121   "conjs (And p q) = (conjs p)@(conjs q)"
```
```  1122 | "conjs T = []"
```
```  1123 | "conjs p = [p]"
```
```  1124 lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
```
```  1125 by (induct p rule: conjs.induct, auto)
```
```  1126 definition list_disj :: "fm list \<Rightarrow> fm" where
```
```  1127   "list_disj ps \<equiv> foldr disj ps F"
```
```  1128
```
```  1129 lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
```
```  1130   by (induct ps, auto simp add: list_conj_def)
```
```  1131 lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
```
```  1132   by (induct ps, auto simp add: list_conj_def)
```
```  1133 lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
```
```  1134   by (induct ps, auto simp add: list_disj_def)
```
```  1135
```
```  1136 lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
```
```  1137   unfolding conj_def by auto
```
```  1138
```
```  1139 lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
```
```  1140   apply (induct p rule: conjs.induct)
```
```  1141   apply (unfold conjs.simps)
```
```  1142   apply (unfold set_append)
```
```  1143   apply (unfold ball_Un)
```
```  1144   apply (unfold bound.simps)
```
```  1145   apply auto
```
```  1146   done
```
```  1147
```
```  1148 lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
```
```  1149   apply (induct p rule: conjs.induct)
```
```  1150   apply (unfold conjs.simps)
```
```  1151   apply (unfold set_append)
```
```  1152   apply (unfold ball_Un)
```
```  1153   apply (unfold boundslt.simps)
```
```  1154   apply blast
```
```  1155 by simp_all
```
```  1156
```
```  1157 lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
```
```  1158   unfolding list_conj_def
```
```  1159   by (induct ps, auto simp add: conj_boundslt)
```
```  1160
```
```  1161 lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
```
```  1162   shows "bound n (list_conj ps)"
```
```  1163   using bnd
```
```  1164   unfolding list_conj_def
```
```  1165   by (induct ps, auto simp add: conj_nb)
```
```  1166
```
```  1167 lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
```
```  1168 unfolding list_conj_def by (induct ps , auto)
```
```  1169
```
```  1170 lemma CJNB_qe:
```
```  1171   assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
```
```  1172   shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
```
```  1173 proof(clarify)
```
```  1174   fix bs p
```
```  1175   assume qfp: "qfree p"
```
```  1176   let ?cjs = "conjuncts p"
```
```  1177   let ?yes = "fst (partition bound0 ?cjs)"
```
```  1178   let ?no = "snd (partition bound0 ?cjs)"
```
```  1179   let ?cno = "list_conj ?no"
```
```  1180   let ?cyes = "list_conj ?yes"
```
```  1181   have part: "partition bound0 ?cjs = (?yes,?no)" by simp
```
```  1182   from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast
```
```  1183   hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb')
```
```  1184   hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
```
```  1185   from conjuncts_qf[OF qfp] partition_set[OF part]
```
```  1186   have " \<forall>q\<in> set ?no. qfree q" by auto
```
```  1187   hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
```
```  1188   with qe have cno_qf:"qfree (qe ?cno )"
```
```  1189     and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
```
```  1190   from cno_qf yes_qf have qf: "qfree (CJNB qe p)"
```
```  1191     by (simp add: CJNB_def Let_def conj_qf split_def)
```
```  1192   {fix bs
```
```  1193     from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
```
```  1194     also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
```
```  1195       using partition_set[OF part] by auto
```
```  1196     finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
```
```  1197   hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
```
```  1198   also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
```
```  1199     using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
```
```  1200   also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
```
```  1201     by (auto simp add: decr0[OF yes_nb] simp del: partition_filter_conv)
```
```  1202   also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
```
```  1203     using qe[rule_format, OF no_qf] by auto
```
```  1204   finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)"
```
```  1205     by (simp add: Let_def CJNB_def split_def)
```
```  1206   with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
```
```  1207 qed
```
```  1208
```
```  1209 consts simpfm :: "fm \<Rightarrow> fm"
```
```  1210 recdef simpfm "measure fmsize"
```
```  1211   "simpfm (Lt t) = simplt (simptm t)"
```
```  1212   "simpfm (Le t) = simple (simptm t)"
```
```  1213   "simpfm (Eq t) = simpeq(simptm t)"
```
```  1214   "simpfm (NEq t) = simpneq(simptm t)"
```
```  1215   "simpfm (And p q) = conj (simpfm p) (simpfm q)"
```
```  1216   "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
```
```  1217   "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
```
```  1218   "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
```
```  1219   "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
```
```  1220   "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
```
```  1221   "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
```
```  1222   "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
```
```  1223   "simpfm (NOT (Eq t)) = simpneq t"
```
```  1224   "simpfm (NOT (NEq t)) = simpeq t"
```
```  1225   "simpfm (NOT (Le t)) = simplt (Neg t)"
```
```  1226   "simpfm (NOT (Lt t)) = simple (Neg t)"
```
```  1227   "simpfm (NOT (NOT p)) = simpfm p"
```
```  1228   "simpfm (NOT T) = F"
```
```  1229   "simpfm (NOT F) = T"
```
```  1230   "simpfm p = p"
```
```  1231
```
```  1232 lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
```
```  1233 by(induct p arbitrary: bs rule: simpfm.induct, auto)
```
```  1234
```
```  1235 lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1236   shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
```
```  1237 by (induct p rule: simpfm.induct, auto)
```
```  1238
```
```  1239 lemma lt_qf[simp]: "qfree (lt t)"
```
```  1240   apply (cases t, auto simp add: lt_def)
```
```  1241   by (case_tac poly, auto)
```
```  1242
```
```  1243 lemma le_qf[simp]: "qfree (le t)"
```
```  1244   apply (cases t, auto simp add: le_def)
```
```  1245   by (case_tac poly, auto)
```
```  1246
```
```  1247 lemma eq_qf[simp]: "qfree (eq t)"
```
```  1248   apply (cases t, auto simp add: eq_def)
```
```  1249   by (case_tac poly, auto)
```
```  1250
```
```  1251 lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)
```
```  1252
```
```  1253 lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
```
```  1254 lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
```
```  1255 lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
```
```  1256 lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)
```
```  1257
```
```  1258 lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
```
```  1259 by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)
```
```  1260
```
```  1261 lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
```
```  1262 lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)
```
```  1263
```
```  1264 lemma   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  1265   shows "qfree p \<Longrightarrow> islin (simpfm p)"
```
```  1266   apply (induct p rule: simpfm.induct)
```
```  1267   apply (simp_all add: conj_lin disj_lin)
```
```  1268   done
```
```  1269
```
```  1270 consts prep :: "fm \<Rightarrow> fm"
```
```  1271 recdef prep "measure fmsize"
```
```  1272   "prep (E T) = T"
```
```  1273   "prep (E F) = F"
```
```  1274   "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
```
```  1275   "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
```
```  1276   "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))"
```
```  1277   "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
```
```  1278   "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
```
```  1279   "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
```
```  1280   "prep (E p) = E (prep p)"
```
```  1281   "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
```
```  1282   "prep (A p) = prep (NOT (E (NOT p)))"
```
```  1283   "prep (NOT (NOT p)) = prep p"
```
```  1284   "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
```
```  1285   "prep (NOT (A p)) = prep (E (NOT p))"
```
```  1286   "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
```
```  1287   "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
```
```  1288   "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
```
```  1289   "prep (NOT p) = not (prep p)"
```
```  1290   "prep (Or p q) = disj (prep p) (prep q)"
```
```  1291   "prep (And p q) = conj (prep p) (prep q)"
```
```  1292   "prep (Imp p q) = prep (Or (NOT p) q)"
```
```  1293   "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
```
```  1294   "prep p = p"
```
```  1295 (hints simp add: fmsize_pos)
```
```  1296 lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
```
```  1297 by (induct p arbitrary: bs rule: prep.induct, auto)
```
```  1298
```
```  1299
```
```  1300
```
```  1301   (* Generic quantifier elimination *)
```
```  1302 function (sequential) qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm" where
```
```  1303   "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
```
```  1304 | "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
```
```  1305 | "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
```
```  1306 | "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))"
```
```  1307 | "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))"
```
```  1308 | "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
```
```  1309 | "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
```
```  1310 | "qelim p = (\<lambda> y. simpfm p)"
```
```  1311 by pat_completeness simp_all
```
```  1312 termination by (relation "measure fmsize") auto
```
```  1313
```
```  1314 lemma qelim:
```
```  1315   assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
```
```  1316   shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
```
```  1317 using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
```
```  1318 by (induct p rule: qelim.induct) auto
```
```  1319
```
```  1320 subsection{* Core Procedure *}
```
```  1321
```
```  1322 fun minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*) where
```
```  1323   "minusinf (And p q) = conj (minusinf p) (minusinf q)"
```
```  1324 | "minusinf (Or p q) = disj (minusinf p) (minusinf q)"
```
```  1325 | "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
```
```  1326 | "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
```
```  1327 | "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
```
```  1328 | "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
```
```  1329 | "minusinf p = p"
```
```  1330
```
```  1331 fun plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*) where
```
```  1332   "plusinf (And p q) = conj (plusinf p) (plusinf q)"
```
```  1333 | "plusinf (Or p q) = disj (plusinf p) (plusinf q)"
```
```  1334 | "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
```
```  1335 | "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
```
```  1336 | "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
```
```  1337 | "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
```
```  1338 | "plusinf p = p"
```
```  1339
```
```  1340 lemma minusinf_inf: assumes lp:"islin p"
```
```  1341   shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
```
```  1342   using lp
```
```  1343 proof (induct p rule: minusinf.induct)
```
```  1344   case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
```
```  1345 next
```
```  1346   case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
```
```  1347 next
```
```  1348   case (3 c e) hence nbe: "tmbound0 e" by simp
```
```  1349   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1350   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1351   let ?c = "Ipoly vs c"
```
```  1352   let ?e = "Itm vs (y#bs) e"
```
```  1353   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1354   moreover {assume "?c = 0" hence ?case
```
```  1355       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
```
```  1356   moreover {assume cp: "?c > 0"
```
```  1357     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1358         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1359       hence "?c * x + ?e < 0" by simp
```
```  1360       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
```
```  1361         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
```
```  1362   moreover {assume cp: "?c < 0"
```
```  1363     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1364         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1365       hence "?c * x + ?e > 0" by simp
```
```  1366       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
```
```  1367         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
```
```  1368   ultimately show ?case by blast
```
```  1369 next
```
```  1370   case (4 c e)  hence nbe: "tmbound0 e" by simp
```
```  1371   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1372   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1373   let ?c = "Ipoly vs c"
```
```  1374   let ?e = "Itm vs (y#bs) e"
```
```  1375   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1376   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1377   moreover {assume cp: "?c > 0"
```
```  1378     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1379         using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1380       hence "?c * x + ?e < 0" by simp
```
```  1381       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
```
```  1382         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1383   moreover {assume cp: "?c < 0"
```
```  1384     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1385         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1386       hence "?c * x + ?e > 0" by simp
```
```  1387       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
```
```  1388         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1389   ultimately show ?case by blast
```
```  1390 next
```
```  1391   case (5 c e)  hence nbe: "tmbound0 e" by simp
```
```  1392   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1393   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1394   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
```
```  1395   let ?c = "Ipoly vs c"
```
```  1396   let ?e = "Itm vs (y#bs) e"
```
```  1397   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1398   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1399   moreover {assume cp: "?c > 0"
```
```  1400     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1401         using pos_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) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
```
```  1404         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1405   moreover {assume cp: "?c < 0"
```
```  1406     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1407         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1408       hence "?c * x + ?e > 0" by simp
```
```  1409       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
```
```  1410         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
```
```  1411   ultimately show ?case by blast
```
```  1412 next
```
```  1413   case (6 c e)  hence nbe: "tmbound0 e" by simp
```
```  1414   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1415   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1416   note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
```
```  1417   let ?c = "Ipoly vs c"
```
```  1418   let ?e = "Itm vs (y#bs) e"
```
```  1419   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1420   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1421   moreover {assume cp: "?c > 0"
```
```  1422     {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
```
```  1423         using pos_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) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
```
```  1426         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1427   moreover {assume cp: "?c < 0"
```
```  1428     {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
```
```  1429         using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1430       hence "?c * x + ?e > 0" by simp
```
```  1431       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
```
```  1432         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1433   ultimately show ?case by blast
```
```  1434 qed (auto)
```
```  1435
```
```  1436 lemma plusinf_inf: assumes lp:"islin p"
```
```  1437   shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
```
```  1438   using lp
```
```  1439 proof (induct p rule: plusinf.induct)
```
```  1440   case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
```
```  1441 next
```
```  1442   case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
```
```  1443 next
```
```  1444   case (3 c e) hence nbe: "tmbound0 e" by simp
```
```  1445   from 3 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1446   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1447   let ?c = "Ipoly vs c"
```
```  1448   let ?e = "Itm vs (y#bs) e"
```
```  1449   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1450   moreover {assume "?c = 0" hence ?case
```
```  1451       using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
```
```  1452   moreover {assume cp: "?c > 0"
```
```  1453     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1454         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1455       hence "?c * x + ?e > 0" by simp
```
```  1456       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
```
```  1457         using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
```
```  1458   moreover {assume cp: "?c < 0"
```
```  1459     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1460         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1461       hence "?c * x + ?e < 0" by simp
```
```  1462       hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
```
```  1463         using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
```
```  1464   ultimately show ?case by blast
```
```  1465 next
```
```  1466   case (4 c e) hence nbe: "tmbound0 e" by simp
```
```  1467   from 4 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1468   note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
```
```  1469   let ?c = "Ipoly vs c"
```
```  1470   let ?e = "Itm vs (y#bs) e"
```
```  1471   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1472   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1473   moreover {assume cp: "?c > 0"
```
```  1474     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1475         using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1476       hence "?c * x + ?e > 0" by simp
```
```  1477       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
```
```  1478         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1479   moreover {assume cp: "?c < 0"
```
```  1480     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1481         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1482       hence "?c * x + ?e < 0" by simp
```
```  1483       hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
```
```  1484         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
```
```  1485   ultimately show ?case by blast
```
```  1486 next
```
```  1487   case (5 c e) hence nbe: "tmbound0 e" by simp
```
```  1488   from 5 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1489   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1490   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]
```
```  1491   let ?c = "Ipoly vs c"
```
```  1492   let ?e = "Itm vs (y#bs) e"
```
```  1493   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1494   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1495   moreover {assume cp: "?c > 0"
```
```  1496     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1497         using pos_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) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
```
```  1500         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1501   moreover {assume cp: "?c < 0"
```
```  1502     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1503         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1504       hence "?c * x + ?e < 0" by simp
```
```  1505       hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
```
```  1506         using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
```
```  1507   ultimately show ?case by blast
```
```  1508 next
```
```  1509   case (6 c e)  hence nbe: "tmbound0 e" by simp
```
```  1510   from 6 have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
```
```  1511   hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
```
```  1512   note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
```
```  1513   let ?c = "Ipoly vs c"
```
```  1514   let ?e = "Itm vs (y#bs) e"
```
```  1515   have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
```
```  1516   moreover {assume "?c = 0" hence ?case using eqs by auto}
```
```  1517   moreover {assume cp: "?c > 0"
```
```  1518     {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
```
```  1519         using pos_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) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
```
```  1522         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1523   moreover {assume cp: "?c < 0"
```
```  1524     {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
```
```  1525         using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
```
```  1526       hence "?c * x + ?e < 0" by simp
```
```  1527       hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
```
```  1528         using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
```
```  1529   ultimately show ?case by blast
```
```  1530 qed (auto)
```
```  1531
```
```  1532 lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)"
```
```  1533   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
```
```  1534 lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)"
```
```  1535   by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
```
```  1536
```
```  1537 lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
```
```  1538   shows "\<exists>x. Ifm vs (x#bs) p"
```
```  1539 proof-
```
```  1540   from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
```
```  1541   have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
```
```  1542   from minusinf_inf[OF lp, where bs="bs"]
```
```  1543   obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
```
```  1544   from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
```
```  1545   moreover have "z - 1 < z" by simp
```
```  1546   ultimately show ?thesis using z_def by auto
```
```  1547 qed
```
```  1548
```
```  1549 lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
```
```  1550   shows "\<exists>x. Ifm vs (x#bs) p"
```
```  1551 proof-
```
```  1552   from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
```
```  1553   have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
```
```  1554   from plusinf_inf[OF lp, where bs="bs"]
```
```  1555   obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
```
```  1556   from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
```
```  1557   moreover have "z + 1 > z" by simp
```
```  1558   ultimately show ?thesis using z_def by auto
```
```  1559 qed
```
```  1560
```
```  1561 fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
```
```  1562   "uset (And p q) = uset p @ uset q"
```
```  1563 | "uset (Or p q) = uset p @ uset q"
```
```  1564 | "uset (Eq (CNP 0 a e))  = [(a,e)]"
```
```  1565 | "uset (Le (CNP 0 a e))  = [(a,e)]"
```
```  1566 | "uset (Lt (CNP 0 a e))  = [(a,e)]"
```
```  1567 | "uset (NEq (CNP 0 a e)) = [(a,e)]"
```
```  1568 | "uset p = []"
```
```  1569
```
```  1570 lemma uset_l:
```
```  1571   assumes lp: "islin p"
```
```  1572   shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
```
```  1573 using lp by(induct p rule: uset.induct,auto)
```
```  1574
```
```  1575 lemma minusinf_uset0:
```
```  1576   assumes lp: "islin p"
```
```  1577   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
```
```  1578   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1579   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c"
```
```  1580 proof-
```
```  1581   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)"
```
```  1582     using lp nmi ex
```
```  1583     apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
```
```  1584     apply (auto simp add: linorder_not_less order_le_less)
```
```  1585     done
```
```  1586   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
```
```  1587   hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
```
```  1588     using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
```
```  1589       by (auto simp add: mult_commute)
```
```  1590   thus ?thesis using csU by auto
```
```  1591 qed
```
```  1592
```
```  1593 lemma minusinf_uset:
```
```  1594   assumes lp: "islin p"
```
```  1595   and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
```
```  1596   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1597   shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c"
```
```  1598 proof-
```
```  1599   from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))"
```
```  1600     by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
```
```  1601   from minusinf_uset0[OF lp nmi' ex]
```
```  1602   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
```
```  1603   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
```
```  1604   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
```
```  1605 qed
```
```  1606
```
```  1607
```
```  1608 lemma plusinf_uset0:
```
```  1609   assumes lp: "islin p"
```
```  1610   and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
```
```  1611   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1612   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c"
```
```  1613 proof-
```
```  1614   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)"
```
```  1615     using lp nmi ex
```
```  1616     apply (induct p rule: minusinf.induct, auto simp add: eq le lt polyneg_norm)
```
```  1617     apply (auto simp add: linorder_not_less order_le_less)
```
```  1618     done
```
```  1619   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
```
```  1620   hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
```
```  1621     using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
```
```  1622     by (auto simp add: mult_commute del: divide_minus_left)
```
```  1623   thus ?thesis using csU by auto
```
```  1624 qed
```
```  1625
```
```  1626 lemma plusinf_uset:
```
```  1627   assumes lp: "islin p"
```
```  1628   and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
```
```  1629   and ex: "Ifm vs (x#bs) p" (is "?I x p")
```
```  1630   shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c"
```
```  1631 proof-
```
```  1632   from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))"
```
```  1633     by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
```
```  1634   from plusinf_uset0[OF lp nmi' ex]
```
```  1635   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
```
```  1636   from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
```
```  1637   from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
```
```  1638 qed
```
```  1639
```
```  1640 lemma lin_dense:
```
```  1641   assumes lp: "islin p"
```
```  1642   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)"
```
```  1643   (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
```
```  1644   and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
```
```  1645   and ly: "l < y" and yu: "y < u"
```
```  1646   shows "Ifm vs (y#bs) p"
```
```  1647 using lp px noS
```
```  1648 proof (induct p rule: islin.induct)
```
```  1649   case (5 c s)
```
```  1650   from "5.prems"
```
```  1651   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1652     and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
```
```  1653     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
```
```  1654   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1655   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1656   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1657   moreover
```
```  1658   {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"])}
```
```  1659   moreover
```
```  1660   {assume c: "?N c > 0"
```
```  1661       from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1662       have px': "x < - ?Nt x s / ?N c"
```
```  1663         by (auto simp add: not_less field_simps)
```
```  1664     {assume y: "y < - ?Nt x s / ?N c"
```
```  1665       hence "y * ?N c < - ?Nt x s"
```
```  1666         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1667       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1668       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1669     moreover
```
```  1670     {assume y: "y > -?Nt x s / ?N c"
```
```  1671       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1672       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1673       with lx px' have "False" by simp  hence ?case by simp }
```
```  1674     ultimately have ?case using ycs by blast
```
```  1675   }
```
```  1676   moreover
```
```  1677   {assume c: "?N c < 0"
```
```  1678       from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1679       have px': "x > - ?Nt x s / ?N c"
```
```  1680         by (auto simp add: not_less field_simps)
```
```  1681     {assume y: "y > - ?Nt x s / ?N c"
```
```  1682       hence "y * ?N c < - ?Nt x s"
```
```  1683         by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1684       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1685       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1686     moreover
```
```  1687     {assume y: "y < -?Nt x s / ?N c"
```
```  1688       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1689       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1690       with xu px' have "False" by simp  hence ?case by simp }
```
```  1691     ultimately have ?case using ycs by blast
```
```  1692   }
```
```  1693   ultimately show ?case by blast
```
```  1694 next
```
```  1695   case (6 c s)
```
```  1696   from "6.prems"
```
```  1697   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1698     and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
```
```  1699     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
```
```  1700   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1701   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1702   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1703   moreover
```
```  1704   {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"])}
```
```  1705   moreover
```
```  1706   {assume c: "?N c > 0"
```
```  1707       from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1708       have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less field_simps)
```
```  1709     {assume y: "y < - ?Nt x s / ?N c"
```
```  1710       hence "y * ?N c < - ?Nt x s"
```
```  1711         by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
```
```  1712       hence "?N c * y + ?Nt x s < 0" by (simp add: field_simps)
```
```  1713       hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
```
```  1714     moreover
```
```  1715     {assume y: "y > -?Nt x s / ?N c"
```
```  1716       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1717       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1718       with lx px' have "False" by simp  hence ?case by simp }
```
```  1719     ultimately have ?case using ycs by blast
```
```  1720   }
```
```  1721   moreover
```
```  1722   {assume c: "?N c < 0"
```
```  1723       from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]
```
```  1724       have px': "x >= - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1725     {assume y: "y > - ?Nt x s / ?N c"
```
```  1726       hence "y * ?N c < - ?Nt x s"
```
```  1727         by (simp add: neg_divide_less_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 ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1733       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1734       with xu px' have "False" by simp  hence ?case by simp }
```
```  1735     ultimately have ?case using ycs by blast
```
```  1736   }
```
```  1737   ultimately show ?case by blast
```
```  1738 next
```
```  1739     case (3 c s)
```
```  1740   from "3.prems"
```
```  1741   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1742     and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
```
```  1743     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
```
```  1744   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1745   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1746   have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
```
```  1747   moreover
```
```  1748   {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"])}
```
```  1749   moreover
```
```  1750   {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
```
```  1751     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
```
```  1752     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1753     {assume y: "y < -?Nt x s / ?N c"
```
```  1754       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1755       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1756       with xu px' have "False" by simp  hence ?case by simp }
```
```  1757     moreover
```
```  1758     {assume y: "y > -?Nt x s / ?N c"
```
```  1759       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1760       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1761       with lx px' have "False" by simp  hence ?case by simp }
```
```  1762     ultimately have ?case using ycs by blast
```
```  1763   }
```
```  1764   moreover
```
```  1765   {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
```
```  1766     from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
```
```  1767     have px': "x = - ?Nt x s / ?N c" by (simp add: field_simps)
```
```  1768     {assume y: "y < -?Nt x s / ?N c"
```
```  1769       with ly have eu: "l < - ?Nt x s / ?N c" by auto
```
```  1770       with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
```
```  1771       with xu px' have "False" by simp  hence ?case by simp }
```
```  1772     moreover
```
```  1773     {assume y: "y > -?Nt x s / ?N c"
```
```  1774       with yu have eu: "u > - ?Nt x s / ?N c" by auto
```
```  1775       with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
```
```  1776       with lx px' have "False" by simp  hence ?case by simp }
```
```  1777     ultimately have ?case using ycs by blast
```
```  1778   }
```
```  1779   ultimately show ?case by blast
```
```  1780 next
```
```  1781     case (4 c s)
```
```  1782   from "4.prems"
```
```  1783   have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
```
```  1784     and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
```
```  1785     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
```
```  1786   from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
```
```  1787   hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
```
```  1788   have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
```
```  1789   moreover
```
```  1790   {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"])}
```
```  1791   moreover
```
```  1792   {assume c: "?N c \<noteq> 0"
```
```  1793     from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
```
```  1794       by (simp add: field_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
```
```  1795   ultimately show ?case by blast
```
```  1796 qed (auto simp add: tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])
```
```  1797
```
```  1798 lemma inf_uset:
```
```  1799   assumes lp: "islin p"
```
```  1800   and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
```
```  1801   and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
```
```  1802   and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
```
```  1803   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) / 2) p"
```
```  1804 proof-
```
```  1805   let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
```
```  1806   let ?N = "Ipoly vs"
```
```  1807   let ?U = "set (uset p)"
```
```  1808   from ex obtain a where pa: "?I a p" by blast
```
```  1809   from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
```
```  1810   have nmi': "\<not> (?I a (?M p))" by simp
```
```  1811   from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
```
```  1812   have npi': "\<not> (?I a (?P p))" by simp
```
```  1813   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) / 2) p"
```
```  1814   proof-
```
```  1815     let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
```
```  1816     have fM: "finite ?M" by auto
```
```  1817     from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa]
```
```  1818     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
```
```  1819     then obtain "c" "t" "d" "s" where
```
```  1820       ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U"
```
```  1821       and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
```
```  1822     from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1
```
```  1823     have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
```
```  1824     from ctU have Mne: "?M \<noteq> {}" by auto
```
```  1825     hence Une: "?U \<noteq> {}" by simp
```
```  1826     let ?l = "Min ?M"
```
```  1827     let ?u = "Max ?M"
```
```  1828     have linM: "?l \<in> ?M" using fM Mne by simp
```
```  1829     have uinM: "?u \<in> ?M" using fM Mne by simp
```
```  1830     have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
```
```  1831     have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto
```
```  1832     have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
```
```  1833     have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
```
```  1834     have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
```
```  1835     have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
```
```  1836     from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
```
```  1837     have "(\<exists> s\<in> ?M. ?I s p) \<or>
```
```  1838       (\<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)" .
```
```  1839     moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
```
```  1840       hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
```
```  1841       then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
```
```  1842       from pu tuu
```
```  1843       have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / 2) p" by simp
```
```  1844       with tuU have ?thesis by blast}
```
```  1845     moreover{
```
```  1846       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"
```
```  1847       then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
```
```  1848         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"
```
```  1849         by blast
```
```  1850       from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
```
```  1851       then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
```
```  1852       from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
```
```  1853       then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
```
```  1854       from t1x xt2 have t1t2: "t1 < t2" by simp
```
```  1855       let ?u = "(t1 + t2) / 2"
```
```  1856       from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
```
```  1857       from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
```
```  1858       with t1uU t2uU t1u t2u have ?thesis by blast}
```
```  1859     ultimately show ?thesis by blast
```
```  1860   qed
```
```  1861   then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U"
```
```  1862     and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / 2) p" by blast
```
```  1863   from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
```
```  1864   from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
```
```  1865     tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
```
```  1866   have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / 2) p" by simp
```
```  1867   with lnU smU
```
```  1868   show ?thesis by auto
```
```  1869 qed
```
```  1870
```
```  1871     (* The Ferrante - Rackoff Theorem *)
```
```  1872
```
```  1873 theorem fr_eq:
```
```  1874   assumes lp: "islin p"
```
```  1875   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) / 2)#bs) p))"
```
```  1876   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  1877 proof
```
```  1878   assume px: "\<exists> x. ?I x p"
```
```  1879   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  1880   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  1881   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  1882     from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
```
```  1883   ultimately show "?D" by blast
```
```  1884 next
```
```  1885   assume "?D"
```
```  1886   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
```
```  1887   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
```
```  1888   moreover {assume f:"?F" hence "?E" by blast}
```
```  1889   ultimately show "?E" by blast
```
```  1890 qed
```
```  1891
```
```  1892 section{* First implementation : Naive by encoding all case splits locally *}
```
```  1893 definition "msubsteq c t d s a r =
```
```  1894   evaldjf (split conj)
```
```  1895   [(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)))),
```
```  1896    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1897    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1898    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"
```
```  1899
```
```  1900 lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  1901   shows "bound0 (msubsteq c t d s a r)"
```
```  1902 proof-
```
```  1903   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)))),
```
```  1904    (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1905    (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1906    (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
```
```  1907     using lp by (simp add: Let_def t s )
```
```  1908   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
```
```  1909 qed
```
```  1910
```
```  1911 lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
```
```  1912   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) / 2)#bs) (Eq (CNP 0 a r))" (is "?lhs = ?rhs")
```
```  1913 proof-
```
```  1914   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
```
```  1915   let ?N = "\<lambda>p. Ipoly vs p"
```
```  1916   let ?c = "?N c"
```
```  1917   let ?d = "?N d"
```
```  1918   let ?t = "?Nt x t"
```
```  1919   let ?s = "?Nt x s"
```
```  1920   let ?a = "?N a"
```
```  1921   let ?r = "?Nt x r"
```
```  1922   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  1923   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  1924   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
```
```  1925   moreover
```
```  1926   {assume c: "?c = 0" and d: "?d=0"
```
```  1927     hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
```
```  1928   moreover
```
```  1929   {assume c: "?c = 0" and d: "?d\<noteq>0"
```
```  1930     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" by simp
```
```  1931     have "?rhs = Ifm vs (-?s / (2*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1932     also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r = 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
```
```  1933     also have "\<dots> \<longleftrightarrow> 2*?d * (?a * (-?s / (2*?d)) + ?r) = 0"
```
```  1934       using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp
```
```  1935     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r= 0"
```
```  1936       by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left)
```
```  1937
```
```  1938     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r = 0" using d by simp
```
```  1939     finally have ?thesis using c d
```
```  1940       by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubsteq_def Let_def evaldjf_ex)
```
```  1941   }
```
```  1942   moreover
```
```  1943   {assume c: "?c \<noteq> 0" and d: "?d=0"
```
```  1944     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" by simp
```
```  1945     have "?rhs = Ifm vs (-?t / (2*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1946     also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r = 0" by (simp add: r[of "- (?t/ (2 * ?c))"])
```
```  1947     also have "\<dots> \<longleftrightarrow> 2*?c * (?a * (-?t / (2*?c)) + ?r) = 0"
```
```  1948       using c mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp
```
```  1949     also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r= 0"
```
```  1950       by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left)
```
```  1951     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r = 0" using c by simp
```
```  1952     finally have ?thesis using c d
```
```  1953       by (simp add: r[of "- (?t/ (2*?c))"] msubsteq_def Let_def evaldjf_ex)
```
```  1954   }
```
```  1955   moreover
```
```  1956   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *2 \<noteq> 0" by simp
```
```  1957     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  1958     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  1959       by (simp add: field_simps)
```
```  1960     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
```
```  1961     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r = 0"
```
```  1962       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  1963     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) =0 "
```
```  1964       using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
```
```  1965     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r =0"
```
```  1966       using nonzero_mult_divide_cancel_left [OF dc] c d
```
```  1967       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  1968     finally  have ?thesis using c d
```
```  1969       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex field_simps)
```
```  1970   }
```
```  1971   ultimately show ?thesis by blast
```
```  1972 qed
```
```  1973
```
```  1974
```
```  1975 definition "msubstneq c t d s a r =
```
```  1976   evaldjf (split conj)
```
```  1977   [(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)))),
```
```  1978    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1979    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1980    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"
```
```  1981
```
```  1982 lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  1983   shows "bound0 (msubstneq c t d s a r)"
```
```  1984 proof-
```
```  1985   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)))),
```
```  1986     (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  1987     (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  1988     (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
```
```  1989     using lp by (simp add: Let_def t s )
```
```  1990   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
```
```  1991 qed
```
```  1992
```
```  1993 lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
```
```  1994   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) /2)#bs) (NEq (CNP 0 a r))" (is "?lhs = ?rhs")
```
```  1995 proof-
```
```  1996   let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
```
```  1997   let ?N = "\<lambda>p. Ipoly vs p"
```
```  1998   let ?c = "?N c"
```
```  1999   let ?d = "?N d"
```
```  2000   let ?t = "?Nt x t"
```
```  2001   let ?s = "?Nt x s"
```
```  2002   let ?a = "?N a"
```
```  2003   let ?r = "?Nt x r"
```
```  2004   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2005   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2006   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
```
```  2007   moreover
```
```  2008   {assume c: "?c = 0" and d: "?d=0"
```
```  2009     hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
```
```  2010   moreover
```
```  2011   {assume c: "?c = 0" and d: "?d\<noteq>0"
```
```  2012     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = -?s / (2*?d)" by simp
```
```  2013     have "?rhs = Ifm vs (-?s / (2*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2014     also have "\<dots> \<longleftrightarrow> ?a * (-?s / (2*?d)) + ?r \<noteq> 0" by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
```
```  2015     also have "\<dots> \<longleftrightarrow> 2*?d * (?a * (-?s / (2*?d)) + ?r) \<noteq> 0"
```
```  2016       using d mult_cancel_left[of "2*?d" "(?a * (-?s / (2*?d)) + ?r)" 0] by simp
```
```  2017     also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * (2*?d / (2*?d)) + 2*?d*?r\<noteq> 0"
```
```  2018       by (simp add: field_simps distrib_left[of "2*?d"] del: distrib_left)
```
```  2019
```
```  2020     also have "\<dots> \<longleftrightarrow> - (?a * ?s) + 2*?d*?r \<noteq> 0" using d by simp
```
```  2021     finally have ?thesis using c d
```
```  2022       by (simp add: r[of "- (Itm vs (x # bs) s / (2 * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubstneq_def Let_def evaldjf_ex)
```
```  2023   }
```
```  2024   moreover
```
```  2025   {assume c: "?c \<noteq> 0" and d: "?d=0"
```
```  2026     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = -?t / (2*?c)" by simp
```
```  2027     have "?rhs = Ifm vs (-?t / (2*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2028     also have "\<dots> \<longleftrightarrow> ?a * (-?t / (2*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ (2 * ?c))"])
```
```  2029     also have "\<dots> \<longleftrightarrow> 2*?c * (?a * (-?t / (2*?c)) + ?r) \<noteq> 0"
```
```  2030       using c mult_cancel_left[of "2*?c" "(?a * (-?t / (2*?c)) + ?r)" 0] by simp
```
```  2031     also have "\<dots> \<longleftrightarrow> (?a * -?t)* (2*?c) / (2*?c) + 2*?c*?r \<noteq> 0"
```
```  2032       by (simp add: field_simps distrib_left[of "2*?c"] del: distrib_left)
```
```  2033     also have "\<dots> \<longleftrightarrow> - (?a * ?t) + 2*?c*?r \<noteq> 0" using c by simp
```
```  2034     finally have ?thesis using c d
```
```  2035       by (simp add: r[of "- (?t/ (2*?c))"] msubstneq_def Let_def evaldjf_ex)
```
```  2036   }
```
```  2037   moreover
```
```  2038   {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *2 \<noteq> 0" by simp
```
```  2039     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2040     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  2041       by (simp add: field_simps)
```
```  2042     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
```
```  2043     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r \<noteq> 0"
```
```  2044       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  2045     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) \<noteq> 0 "
```
```  2046       using c d mult_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
```
```  2047     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r \<noteq> 0"
```
```  2048       using nonzero_mult_divide_cancel_left[OF dc] c d
```
```  2049       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  2050     finally  have ?thesis using c d
```
```  2051       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex field_simps)
```
```  2052   }
```
```  2053   ultimately show ?thesis by blast
```
```  2054 qed
```
```  2055
```
```  2056 definition "msubstlt c t d s a r =
```
```  2057   evaldjf (split conj)
```
```  2058   [(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)))),
```
```  2059   (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)))),
```
```  2060    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2061    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2062    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2063    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2064    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"
```
```  2065
```
```  2066 lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2067   shows "bound0 (msubstlt c t d s a r)"
```
```  2068 proof-
```
```  2069   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)))),
```
```  2070   (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)))),
```
```  2071    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2072    (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2073    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2074    (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2075    (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
```
```  2076     using lp by (simp add: Let_def t s lt_nb )
```
```  2077   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
```
```  2078 qed
```
```  2079
```
```  2080
```
```  2081 lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))"
```
```  2082   shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow>
```
```  2083   Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Lt (CNP 0 a r))" (is "?lhs = ?rhs")
```
```  2084 proof-
```
```  2085   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2086   let ?N = "\<lambda>p. Ipoly vs p"
```
```  2087   let ?c = "?N c"
```
```  2088   let ?d = "?N d"
```
```  2089   let ?t = "?Nt x t"
```
```  2090   let ?s = "?Nt x s"
```
```  2091   let ?a = "?N a"
```
```  2092   let ?r = "?Nt x r"
```
```  2093   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2094   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2095   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
```
```  2096   moreover
```
```  2097   {assume c: "?c=0" and d: "?d=0"
```
```  2098     hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
```
```  2099   moreover
```
```  2100   {assume dc: "?c*?d > 0"
```
```  2101     from dc have dc': "2*?c *?d > 0" by simp
```
```  2102     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2103     from dc' have dc'': "\<not> 2*?c *?d < 0" by simp
```
```  2104     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2105     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  2106       by (simp add: field_simps)
```
```  2107     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2108     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0"
```
```  2109       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  2110     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) < 0"
```
```  2111
```
```  2112       using dc' dc'' mult_less_cancel_left_disj[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
```
```  2113     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r < 0"
```
```  2114       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
```
```  2115       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  2116     finally  have ?thesis using dc c d  nc nd dc'
```
```  2117       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2118   }
```
```  2119   moreover
```
```  2120   {assume dc: "?c*?d < 0"
```
```  2121
```
```  2122     from dc have dc': "2*?c *?d < 0"
```
```  2123       by (simp add: mult_less_0_iff field_simps)
```
```  2124     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2125     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2126     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  2127       by (simp add: field_simps)
```
```  2128     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2129     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r < 0"
```
```  2130       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  2131
```
```  2132     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) > 0"
```
```  2133
```
```  2134       using dc' order_less_not_sym[OF dc'] mult_less_cancel_left_disj[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp
```
```  2135     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2*?c*?d*?r < 0"
```
```  2136       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
```
```  2137       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  2138     finally  have ?thesis using dc c d  nc nd
```
```  2139       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2140   }
```
```  2141   moreover
```
```  2142   {assume c: "?c > 0" and d: "?d=0"
```
```  2143     from c have c'': "2*?c > 0" by (simp add: zero_less_mult_iff)
```
```  2144     from c have c': "2*?c \<noteq> 0" by simp
```
```  2145     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
```
```  2146     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2147     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r < 0" by (simp add: r[of "- (?t / (2*?c))"])
```
```  2148     also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) < 0"
```
```  2149       using c mult_less_cancel_left_disj[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
```
```  2150     also have "\<dots> \<longleftrightarrow> - ?a*?t+  2*?c *?r < 0"
```
```  2151       using nonzero_mult_divide_cancel_left[OF c'] c
```
```  2152       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
```
```  2153     finally have ?thesis using c d nc nd
```
```  2154       by(simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2155   }
```
```  2156   moreover
```
```  2157   {assume c: "?c < 0" and d: "?d=0"  hence c': "2*?c \<noteq> 0" by simp
```
```  2158     from c have c'': "2*?c < 0" by (simp add: mult_less_0_iff)
```
```  2159     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
```
```  2160     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2161     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r < 0" by (simp add: r[of "- (?t / (2*?c))"])
```
```  2162     also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) > 0"
```
```  2163       using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_less_cancel_left_disj[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp
```
```  2164     also have "\<dots> \<longleftrightarrow> ?a*?t -  2*?c *?r < 0"
```
```  2165       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
```
```  2166         by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
```
```  2167     finally have ?thesis using c d nc nd
```
```  2168       by(simp add: r[of "- (?t / (2*?c))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2169   }
```
```  2170   moreover
```
```  2171   moreover
```
```  2172   {assume c: "?c = 0" and d: "?d>0"
```
```  2173     from d have d'': "2*?d > 0" by (simp add: zero_less_mult_iff)
```
```  2174     from d have d': "2*?d \<noteq> 0" by simp
```
```  2175     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
```
```  2176     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2177     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r < 0" by (simp add: r[of "- (?s / (2*?d))"])
```
```  2178     also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) < 0"
```
```  2179       using d mult_less_cancel_left_disj[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
```
```  2180     also have "\<dots> \<longleftrightarrow> - ?a*?s+  2*?d *?r < 0"
```
```  2181       using nonzero_mult_divide_cancel_left[OF d'] d
```
```  2182       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
```
```  2183     finally have ?thesis using c d nc nd
```
```  2184       by(simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2185   }
```
```  2186   moreover
```
```  2187   {assume c: "?c = 0" and d: "?d<0"  hence d': "2*?d \<noteq> 0" by simp
```
```  2188     from d have d'': "2*?d < 0" by (simp add: mult_less_0_iff)
```
```  2189     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
```
```  2190     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
```
```  2191     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r < 0" by (simp add: r[of "- (?s / (2*?d))"])
```
```  2192     also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) > 0"
```
```  2193       using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_less_cancel_left_disj[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp
```
```  2194     also have "\<dots> \<longleftrightarrow> ?a*?s -  2*?d *?r < 0"
```
```  2195       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
```
```  2196         by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
```
```  2197     finally have ?thesis using c d nc nd
```
```  2198       by(simp add: r[of "- (?s / (2*?d))"] msubstlt_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2199   }
```
```  2200 ultimately show ?thesis by blast
```
```  2201 qed
```
```  2202
```
```  2203 definition "msubstle c t d s a r =
```
```  2204   evaldjf (split conj)
```
```  2205   [(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)))),
```
```  2206   (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)))),
```
```  2207    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2208    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2209    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2210    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2211    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"
```
```  2212
```
```  2213 lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2214   shows "bound0 (msubstle c t d s a r)"
```
```  2215 proof-
```
```  2216   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)))),
```
```  2217   (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)))),
```
```  2218    (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2219    (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
```
```  2220    (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2221    (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
```
```  2222    (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
```
```  2223     using lp by (simp add: Let_def t s lt_nb )
```
```  2224   from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
```
```  2225 qed
```
```  2226
```
```  2227 lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))"
```
```  2228   shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow>
```
```  2229   Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /2)#bs) (Le (CNP 0 a r))" (is "?lhs = ?rhs")
```
```  2230 proof-
```
```  2231   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2232   let ?N = "\<lambda>p. Ipoly vs p"
```
```  2233   let ?c = "?N c"
```
```  2234   let ?d = "?N d"
```
```  2235   let ?t = "?Nt x t"
```
```  2236   let ?s = "?Nt x s"
```
```  2237   let ?a = "?N a"
```
```  2238   let ?r = "?Nt x r"
```
```  2239   from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
```
```  2240   note r= tmbound0_I[OF lin(3), of vs _ bs x]
```
```  2241   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
```
```  2242   moreover
```
```  2243   {assume c: "?c=0" and d: "?d=0"
```
```  2244     hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
```
```  2245   moreover
```
```  2246   {assume dc: "?c*?d > 0"
```
```  2247     from dc have dc': "2*?c *?d > 0" by simp
```
```  2248     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2249     from dc' have dc'': "\<not> 2*?c *?d < 0" by simp
```
```  2250     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2251     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  2252       by (simp add: field_simps)
```
```  2253     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2254     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r <= 0"
```
```  2255       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  2256     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) <= 0"
```
```  2257
```
```  2258       using dc' dc'' mult_le_cancel_left[of "2 * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r" 0] by simp
```
```  2259     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + 2*?c*?d*?r <= 0"
```
```  2260       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
```
```  2261       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  2262     finally  have ?thesis using dc c d  nc nd dc'
```
```  2263       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2264   }
```
```  2265   moreover
```
```  2266   {assume dc: "?c*?d < 0"
```
```  2267
```
```  2268     from dc have dc': "2*?c *?d < 0"
```
```  2269       by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
```
```  2270     hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
```
```  2271     from add_frac_eq[OF c d, of "- ?t" "- ?s"]
```
```  2272     have th: "(- ?t / ?c + - ?s / ?d)/2 = - (?d * ?t + ?c* ?s )/ (2*?c*?d)"
```
```  2273       by (simp add: field_simps)
```
```  2274     have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ (2*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2275     also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r <= 0"
```
```  2276       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"])
```
```  2277
```
```  2278     also have "\<dots> \<longleftrightarrow> (2 * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ (2*?c*?d)) + ?r) >= 0"
```
```  2279
```
```  2280       using dc' order_less_not_sym[OF dc'] mult_le_cancel_left[of "2 * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ (2*?c*?d)) + ?r"] by simp
```
```  2281     also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - 2*?c*?d*?r <= 0"
```
```  2282       using nonzero_mult_divide_cancel_left[of "2*?c*?d"] c d
```
```  2283       by (simp add: algebra_simps diff_divide_distrib del: distrib_right)
```
```  2284     finally  have ?thesis using dc c d  nc nd
```
```  2285       by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / (2 * ?c * ?d)"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2286   }
```
```  2287   moreover
```
```  2288   {assume c: "?c > 0" and d: "?d=0"
```
```  2289     from c have c'': "2*?c > 0" by (simp add: zero_less_mult_iff)
```
```  2290     from c have c': "2*?c \<noteq> 0" by simp
```
```  2291     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
```
```  2292     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2293     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r <= 0" by (simp add: r[of "- (?t / (2*?c))"])
```
```  2294     also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) <= 0"
```
```  2295       using c mult_le_cancel_left[of "2 * ?c" "?a* (- ?t / (2*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
```
```  2296     also have "\<dots> \<longleftrightarrow> - ?a*?t+  2*?c *?r <= 0"
```
```  2297       using nonzero_mult_divide_cancel_left[OF c'] c
```
```  2298       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
```
```  2299     finally have ?thesis using c d nc nd
```
```  2300       by(simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2301   }
```
```  2302   moreover
```
```  2303   {assume c: "?c < 0" and d: "?d=0"  hence c': "2*?c \<noteq> 0" by simp
```
```  2304     from c have c'': "2*?c < 0" by (simp add: mult_less_0_iff)
```
```  2305     from d have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?t / (2*?c)"  by (simp add: field_simps)
```
```  2306     have "?rhs \<longleftrightarrow> Ifm vs (- ?t / (2*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2307     also have "\<dots> \<longleftrightarrow> ?a* (- ?t / (2*?c))+ ?r <= 0" by (simp add: r[of "- (?t / (2*?c))"])
```
```  2308     also have "\<dots> \<longleftrightarrow> 2*?c * (?a* (- ?t / (2*?c))+ ?r) >= 0"
```
```  2309       using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_le_cancel_left[of "2 * ?c" 0 "?a* (- ?t / (2*?c))+ ?r"] by simp
```
```  2310     also have "\<dots> \<longleftrightarrow> ?a*?t -  2*?c *?r <= 0"
```
```  2311       using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
```
```  2312         by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
```
```  2313     finally have ?thesis using c d nc nd
```
```  2314       by(simp add: r[of "- (?t / (2*?c))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2315   }
```
```  2316   moreover
```
```  2317   moreover
```
```  2318   {assume c: "?c = 0" and d: "?d>0"
```
```  2319     from d have d'': "2*?d > 0" by (simp add: zero_less_mult_iff)
```
```  2320     from d have d': "2*?d \<noteq> 0" by simp
```
```  2321     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
```
```  2322     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2323     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r <= 0" by (simp add: r[of "- (?s / (2*?d))"])
```
```  2324     also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) <= 0"
```
```  2325       using d mult_le_cancel_left[of "2 * ?d" "?a* (- ?s / (2*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
```
```  2326     also have "\<dots> \<longleftrightarrow> - ?a*?s+  2*?d *?r <= 0"
```
```  2327       using nonzero_mult_divide_cancel_left[OF d'] d
```
```  2328       by (simp add: algebra_simps diff_divide_distrib less_le del: distrib_right)
```
```  2329     finally have ?thesis using c d nc nd
```
```  2330       by(simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2331   }
```
```  2332   moreover
```
```  2333   {assume c: "?c = 0" and d: "?d<0"  hence d': "2*?d \<noteq> 0" by simp
```
```  2334     from d have d'': "2*?d < 0" by (simp add: mult_less_0_iff)
```
```  2335     from c have th: "(- ?t / ?c + - ?s / ?d)/2 = - ?s / (2*?d)"  by (simp add: field_simps)
```
```  2336     have "?rhs \<longleftrightarrow> Ifm vs (- ?s / (2*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
```
```  2337     also have "\<dots> \<longleftrightarrow> ?a* (- ?s / (2*?d))+ ?r <= 0" by (simp add: r[of "- (?s / (2*?d))"])
```
```  2338     also have "\<dots> \<longleftrightarrow> 2*?d * (?a* (- ?s / (2*?d))+ ?r) >= 0"
```
```  2339       using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_le_cancel_left[of "2 * ?d" 0 "?a* (- ?s / (2*?d))+ ?r"] by simp
```
```  2340     also have "\<dots> \<longleftrightarrow> ?a*?s -  2*?d *?r <= 0"
```
```  2341       using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
```
```  2342         by (simp add: algebra_simps diff_divide_distrib del:  distrib_right)
```
```  2343     finally have ?thesis using c d nc nd
```
```  2344       by(simp add: r[of "- (?s / (2*?d))"] msubstle_def Let_def evaldjf_ex field_simps lt polyneg_norm polymul_norm)
```
```  2345   }
```
```  2346 ultimately show ?thesis by blast
```
```  2347 qed
```
```  2348
```
```  2349
```
```  2350 fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
```
```  2351   "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
```
```  2352 | "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
```
```  2353 | "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
```
```  2354 | "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
```
```  2355 | "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
```
```  2356 | "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
```
```  2357 | "msubst p ((c,t),(d,s)) = p"
```
```  2358
```
```  2359 lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
```
```  2360   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) /2)#bs) p"
```
```  2361   using lp
```
```  2362 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>)) / 2" 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>)) / 2" and b'=x and bs = bs and vs=vs] msubsteq msubstneq msubstlt[OF nc nd] msubstle[OF nc nd])
```
```  2363
```
```  2364 lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
```
```  2365   shows "bound0 (msubst p ((c,t),(d,s)))"
```
```  2366   using lp t s
```
```  2367   by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)
```
```  2368
```
```  2369 lemma fr_eq_msubst:
```
```  2370   assumes lp: "islin p"
```
```  2371   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)))))"
```
```  2372   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
```
```  2373 proof-
```
```  2374 from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
```
```  2375 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
```
```  2376   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>) / 2 # bs) p"
```
```  2377   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
```
```  2378   from msubst_I[OF lp norm, of vs x bs t s] pts
```
```  2379   have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
```
```  2380 moreover
```
```  2381 {fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)"
```
```  2382   and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
```
```  2383   from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
```
```  2384   from msubst_I[OF lp norm, of vs x bs t s] pts
```
```  2385   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>) / 2 # bs) p" ..}
```
```  2386 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>) / 2 # bs) p) \<longleftrightarrow> ?F" by blast
```
```  2387 from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
```
```  2388 qed
```
```  2389
```
```  2390 lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})"
```
```  2391   shows "qfree p \<Longrightarrow> islin (simpfm p)"
```
```  2392   by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)
```
```  2393
```
```  2394 definition
```
```  2395   "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
```
```  2396   in if (mp = T \<or> pp = T) then T
```
```  2397      else (let U = alluopairs (remdups (uset  q))
```
```  2398            in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"
```
```  2399
```
```  2400 lemma ferrack:
```
```  2401   assumes qf: "qfree p"
```
```  2402   shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
```
```  2403   (is "_ \<and> (?rhs = ?lhs)")
```
```  2404 proof-
```
```  2405   let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
```
```  2406   let ?N = "\<lambda> t. Ipoly vs t"
```
```  2407   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2408   let ?q = "simpfm p"
```
```  2409   let ?U = "remdups(uset ?q)"
```
```  2410   let ?Up = "alluopairs ?U"
```
```  2411   let ?mp = "minusinf ?q"
```
```  2412   let ?pp = "plusinf ?q"
```
```  2413   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2414   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
```
```  2415   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
```
```  2416   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
```
```  2417   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"
```
```  2418     by simp
```
```  2419   {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
```
```  2420     from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
```
```  2421     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]
```
```  2422     have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: field_simps)}
```
```  2423   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
```
```  2424   {fix x assume xUp: "x \<in> set ?Up"
```
```  2425     then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
```
```  2426       and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto
```
```  2427     from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU]
```
```  2428     have nbs: "tmbound0 t" "tmbound0 s" by simp_all
```
```  2429     from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]]
```
```  2430     have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
```
```  2431   with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
```
```  2432   have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
```
```  2433   with mp_nb pp_nb
```
```  2434   have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
```
```  2435   from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
```
```  2436   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
```
```  2437   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
```
```  2438   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists> (x,y) \<in> set ?Up. ?I ((simpfm o (msubst ?q)) (x,y)))" using alluopairs_bex[OF th0] by simp
```
```  2439   also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)"
```
```  2440     by (simp add: evaldjf_ex)
```
```  2441   also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
```
```  2442   also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
```
```  2443     apply (simp add: ferrack_def Let_def)
```
```  2444     by (cases "?mp = T \<or> ?pp = T", auto)
```
```  2445   finally show ?thesis using thqf by blast
```
```  2446 qed
```
```  2447
```
```  2448 definition "frpar p = simpfm (qelim p ferrack)"
```
```  2449 lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
```
```  2450 proof-
```
```  2451   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
```
```  2452   from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
```
```  2453 qed
```
```  2454
```
```  2455
```
```  2456 section{* Second implemenation: Case splits not local *}
```
```  2457
```
```  2458 lemma fr_eq2:  assumes lp: "islin p"
```
```  2459   shows "(\<exists> x. Ifm vs (x#bs) p) \<longleftrightarrow>
```
```  2460    ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or>
```
```  2461     (Ifm vs (0#bs) p) \<or>
```
```  2462     (\<exists> (n,t) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t /  (Ipoly vs n * 2))#bs) p) \<or>
```
```  2463     (\<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) /2)#bs) p))"
```
```  2464   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
```
```  2465 proof
```
```  2466   assume px: "\<exists> x. ?I x p"
```
```  2467   have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
```
```  2468   moreover {assume "?M \<or> ?P" hence "?D" by blast}
```
```  2469   moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
```
```  2470     from inf_uset[OF lp nmi npi, OF px]
```
```  2471     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"
```
```  2472       by auto
```
```  2473     let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
```
```  2474     let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
```
```  2475     let ?s = "Itm vs (x # bs) s"
```
```  2476     let ?t = "Itm vs (x # bs) t"
```
```  2477     have eq2: "\<And>(x::'a). x + x = 2 * x"
```
```  2478       by  (simp add: field_simps)
```
```  2479     {assume "?c = 0 \<and> ?d = 0"
```
```  2480       with ct have ?D by simp}
```
```  2481     moreover
```
```  2482     {assume z: "?c = 0" "?d \<noteq> 0"
```
```  2483       from z have ?D using ct by auto}
```
```  2484     moreover
```
```  2485     {assume z: "?c \<noteq> 0" "?d = 0"
```
```  2486       with ct have ?D by auto }
```
```  2487     moreover
```
```  2488     {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
```
```  2489       from z have ?F using ct
```
```  2490         apply - apply (rule bexI[where x = "(c,t)"], simp_all)
```
```  2491         by (rule bexI[where x = "(d,s)"], simp_all)
```
```  2492       hence ?D by blast}
```
```  2493     ultimately have ?D by auto}
```
```  2494   ultimately show "?D" by blast
```
```  2495 next
```
```  2496   assume "?D"
```
```  2497   moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
```
```  2498   moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
```
```  2499   moreover {assume f:"?F" hence "?E" by blast}
```
```  2500   ultimately show "?E" by blast
```
```  2501 qed
```
```  2502
```
```  2503 definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
```
```  2504 definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
```
```  2505 definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
```
```  2506 definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
```
```  2507 definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"
```
```  2508
```
```  2509 lemma msubsteq2:
```
```  2510   assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
```
```  2511   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")
```
```  2512   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2513   by (simp add: msubsteq2_def field_simps)
```
```  2514
```
```  2515 lemma msubstltpos:
```
```  2516   assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
```
```  2517   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")
```
```  2518   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2519   by (simp add: msubstltpos_def field_simps)
```
```  2520
```
```  2521 lemma msubstlepos:
```
```  2522   assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
```
```  2523   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")
```
```  2524   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2525   by (simp add: msubstlepos_def field_simps)
```
```  2526
```
```  2527 lemma msubstltneg:
```
```  2528   assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
```
```  2529   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")
```
```  2530   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2531   by (simp add: msubstltneg_def field_simps del: minus_add_distrib)
```
```  2532
```
```  2533 lemma msubstleneg:
```
```  2534   assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
```
```  2535   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")
```
```  2536   using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
```
```  2537   by (simp add: msubstleneg_def field_simps del: minus_add_distrib)
```
```  2538
```
```  2539 fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
```
```  2540   "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
```
```  2541 | "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
```
```  2542 | "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
```
```  2543 | "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
```
```  2544 | "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
```
```  2545 | "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
```
```  2546 | "msubstpos p c t = p"
```
```  2547
```
```  2548 lemma msubstpos_I:
```
```  2549   assumes lp: "islin p" and pos: "Ipoly vs c > 0"
```
```  2550   shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2551   using lp pos
```
```  2552   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)
```
```  2553
```
```  2554 fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
```
```  2555   "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
```
```  2556 | "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
```
```  2557 | "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
```
```  2558 | "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
```
```  2559 | "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
```
```  2560 | "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
```
```  2561 | "msubstneg p c t = p"
```
```  2562
```
```  2563 lemma msubstneg_I:
```
```  2564   assumes lp: "islin p" and pos: "Ipoly vs c < 0"
```
```  2565   shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2566   using lp pos
```
```  2567   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)
```
```  2568
```
```  2569
```
```  2570 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)))"
```
```  2571
```
```  2572 lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
```
```  2573   shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
```
```  2574 proof-
```
```  2575   let ?c = "Ipoly vs c"
```
```  2576   from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))"
```
```  2577     by (simp_all add: polyneg_norm)
```
```  2578   from nz have "?c > 0 \<or> ?c < 0" by arith
```
```  2579   moreover
```
```  2580   {assume c: "?c < 0"
```
```  2581     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"]
```
```  2582     have ?thesis by (auto simp add: msubst2_def)}
```
```  2583   moreover
```
```  2584   {assume c: "?c > 0"
```
```  2585     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"]
```
```  2586     have ?thesis by (auto simp add: msubst2_def)}
```
```  2587   ultimately show ?thesis by blast
```
```  2588 qed
```
```  2589
```
```  2590 term msubsteq2
```
```  2591 lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
```
```  2592   by (simp add: msubsteq2_def)
```
```  2593
```
```  2594 lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
```
```  2595   by (simp add: msubstltpos_def)
```
```  2596 lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
```
```  2597   by (simp add: msubstltneg_def)
```
```  2598
```
```  2599 lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
```
```  2600   by (simp add: msubstlepos_def)
```
```  2601 lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
```
```  2602   by (simp add: msubstleneg_def)
```
```  2603
```
```  2604 lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
```
```  2605   shows "bound0 (msubstpos p c t)"
```
```  2606 using lp tnb
```
```  2607 by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)
```
```  2608
```
```  2609 lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
```
```  2610   shows "bound0 (msubstneg p c t)"
```
```  2611 using lp tnb
```
```  2612 by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)
```
```  2613
```
```  2614 lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{field_char_0, field_inverse_zero})" and lp: "islin p" and tnb: "tmbound0 t"
```
```  2615   shows "bound0 (msubst2 p c t)"
```
```  2616 using lp tnb
```
```  2617 by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
```
```  2618
```
```  2619 lemma mult_minus2_left: "-2 * (x::'a::comm_ring_1) = - (2 * x)"
```
```  2620   by simp
```
```  2621
```
```  2622 lemma mult_minus2_right: "(x::'a::comm_ring_1) * -2 = - (x * 2)"
```
```  2623   by simp
```
```  2624
```
```  2625 lemma islin_qf: "islin p \<Longrightarrow> qfree p"
```
```  2626   by (induct p rule: islin.induct, auto simp add: bound0_qf)
```
```  2627 lemma fr_eq_msubst2:
```
```  2628   assumes lp: "islin p"
```
```  2629   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)))))"
```
```  2630   (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
```
```  2631 proof-
```
```  2632   from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
```
```  2633   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2634   have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
```
```  2635   note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
```
```  2636
```
```  2637   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> * 2) # bs) p)"
```
```  2638   proof-
```
```  2639     {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
```
```  2640       from H(1) th have "isnpoly n" by blast
```
```  2641       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
```
```  2642       have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
```
```  2643         by (simp add: polyneg_norm nn)
```
```  2644       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
```
```  2645         by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
```
```  2646       from msubst2[OF lp nn nn2(1), of x bs t]
```
```  2647       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> * 2) # bs) p"
```
```  2648         using H(2) nn2 by (simp add: mult_minus2_right)}
```
```  2649     moreover
```
```  2650     {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> * 2) # bs) p"
```
```  2651       from H(1) th have "isnpoly n" by blast
```
```  2652       hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
```
```  2653         using H(2) by (simp_all add: polymul_norm n2)
```
```  2654       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: mult_minus2_right)}
```
```  2655     ultimately show ?thesis by blast
```
```  2656   qed
```
```  2657   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).
```
```  2658      \<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>) / 2 # bs) p)"
```
```  2659   proof-
```
```  2660     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
```
```  2661      "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))"
```
```  2662       from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
```
```  2663       hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)"
```
```  2664         by (simp_all add: polymul_norm n2)
```
```  2665       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)))"
```
```  2666         by (simp_all add: polyneg_norm nn)
```
```  2667       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"
```
```  2668         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)
```
```  2669       from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
```
```  2670       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>) / 2 # bs) p"
```
```  2671         apply (simp add: add_divide_distrib mult_minus2_left)
```
```  2672         by (simp add: mult_commute)}
```
```  2673     moreover
```
```  2674     {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)"
```
```  2675       "\<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>) / 2 # bs) p"
```
```  2676      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
```
```  2677       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"
```
```  2678         using H(3,4) by (simp_all add: polymul_norm n2)
```
```  2679       from msubst2[OF lp nn, of x bs ] H(3,4,5)
```
```  2680       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 mult_minus2_left) by (simp add: mult_commute)}
```
```  2681     ultimately show ?thesis by blast
```
```  2682   qed
```
```  2683   from fr_eq2[OF lp, of vs bs x] show ?thesis
```
```  2684     unfolding eq0 eq1 eq2 by blast
```
```  2685 qed
```
```  2686
```
```  2687 definition
```
```  2688 "ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
```
```  2689  in if (mp = T \<or> pp = T) then T
```
```  2690   else (let U = remdups (uset  q)
```
```  2691     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,
```
```  2692    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)]))"
```
```  2693
```
```  2694 definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"
```
```  2695
```
```  2696 lemma ferrack2: assumes qf: "qfree p"
```
```  2697   shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
```
```  2698   (is "_ \<and> (?rhs = ?lhs)")
```
```  2699 proof-
```
```  2700   let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
```
```  2701   let ?N = "\<lambda> t. Ipoly vs t"
```
```  2702   let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
```
```  2703   let ?q = "simpfm p"
```
```  2704   let ?qz = "subst0 (CP 0\<^sub>p) ?q"
```
```  2705   let ?U = "remdups(uset ?q)"
```
```  2706   let ?Up = "alluopairs ?U"
```
```  2707   let ?mp = "minusinf ?q"
```
```  2708   let ?pp = "plusinf ?q"
```
```  2709   let ?I = "\<lambda>p. Ifm vs (x#bs) p"
```
```  2710   from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
```
```  2711   from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
```
```  2712   from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
```
```  2713   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"
```
```  2714     by simp
```
```  2715   have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)"
```
```  2716   proof-
```
```  2717     {fix c t assume ct: "(c,t) \<in> set ?U"
```
```  2718       hence tnb: "tmbound0 t" using U_l by blast
```
```  2719       from msubst2_nb[OF lq tnb]
```
```  2720       have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
```
```  2721     thus ?thesis by auto
```
```  2722   qed
```
```  2723   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)"
```
```  2724   proof-
```
```  2725     {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
```
```  2726       from badc U_l alluopairs_set1[of ?U]
```
```  2727       have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
```
```  2728       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}
```
```  2729     thus ?thesis by auto
```
```  2730   qed
```
```  2731   have stupid: "bound0 F" by simp
```
```  2732   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,
```
```  2733    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)]"
```
```  2734   from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
```
```  2735   have nb: "bound0 ?R "
```
```  2736     by (simp add: list_disj_def simpfm_bound0)
```
```  2737   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))"
```
```  2738
```
```  2739   {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
```
```  2740     from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))"
```
```  2741       by auto (simp add: isnpoly_def)
```
```  2742     have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
```
```  2743       using norm by (simp_all add: polymul_norm)
```
```  2744     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)))"
```
```  2745       by (simp_all add: polyneg_norm norm2)
```
```  2746     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")
```
```  2747     proof
```
```  2748       assume H: ?lhs
```
```  2749       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"
```
```  2750         by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
```
```  2751       from msubst2[OF lq norm2(1) z(1), of x bs]
```
```  2752         msubst2[OF lq norm2(2) z(2), of x bs] H
```
```  2753       show ?rhs by (simp add: field_simps)
```
```  2754     next
```
```  2755       assume H: ?rhs
```
```  2756       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"
```
```  2757         by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
```
```  2758       from msubst2[OF lq norm2(1) z(1), of x bs]
```
```  2759         msubst2[OF lq norm2(2) z(2), of x bs] H
```
```  2760       show ?lhs by (simp add: field_simps)
```
```  2761     qed}
```
```  2762   hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
```
```  2763     by clarsimp
```
```  2764
```
```  2765   have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
```
```  2766   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))))"
```
```  2767     using fr_eq_msubst2[OF lq, of vs bs x] by simp
```
```  2768   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)))"
```
```  2769     by (simp add: split_def)
```
```  2770   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)))"
```
```  2771     using alluopairs_bex[OF th0] by simp
```
```  2772   also have "\<dots> \<longleftrightarrow> ?I ?R"
```
```  2773     by (simp add: list_disj_def evaldjf_ex split_def)
```
```  2774   also have "\<dots> \<longleftrightarrow> ?rhs"
```
```  2775     unfolding ferrack2_def
```
```  2776     apply (cases "?mp = T")
```
```  2777     apply (simp add: list_disj_def)
```
```  2778     apply (cases "?pp = T")
```
```  2779     apply (simp add: list_disj_def)
```
```  2780     by (simp_all add: Let_def decr0[OF nb])
```
```  2781   finally show ?thesis using decr0_qf[OF nb]
```
```  2782     by (simp  add: ferrack2_def Let_def)
```
```  2783 qed
```
```  2784
```
```  2785 lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
```
```  2786 proof-
```
```  2787   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
```
```  2788   from qelim[OF th, of "prep p" bs]
```
```  2789 show ?thesis  unfolding frpar2_def by (auto simp add: prep)
```
```  2790 qed
```
```  2791
```
```  2792 oracle frpar_oracle = {*
```
```  2793 let
```
```  2794
```
```  2795 fun binopT T = T --> T --> T;
```
```  2796 fun relT T = T --> T --> @{typ bool};
```
```  2797
```
```  2798 val dest_num = snd o HOLogic.dest_number;
```
```  2799
```
```  2800 fun try_dest_num t = SOME ((snd o HOLogic.dest_number) t)
```
```  2801   handle TERM _ => NONE;
```
```  2802
```
```  2803 fun dest_nat (t as Const (@{const_name Suc}, _)) = HOLogic.dest_nat t
```
```  2804   | dest_nat t = dest_num t;
```
```  2805
```
```  2806 fun the_index ts t =
```
```  2807   let
```
```  2808     val k = find_index (fn t' => t aconv t') ts;
```
```  2809   in if k < 0 then raise General.Subscript else k end;
```
```  2810
```
```  2811 fun num_of_term ps (Const (@{const_name Groups.uminus}, _) \$ t) = @{code poly.Neg} (num_of_term ps t)
```
```  2812   | num_of_term ps (Const (@{const_name Groups.plus}, _) \$ a \$ b) = @{code poly.Add} (num_of_term ps a, num_of_term ps b)
```
```  2813   | num_of_term ps (Const (@{const_name Groups.minus}, _) \$ a \$ b) = @{code poly.Sub} (num_of_term ps a, num_of_term ps b)
```
```  2814   | num_of_term ps (Const (@{const_name Groups.times}, _) \$ a \$ b) = @{code poly.Mul} (num_of_term ps a, num_of_term ps b)
```
```  2815   | num_of_term ps (Const (@{const_name Power.power}, _) \$ a \$ n) = @{code poly.Pw} (num_of_term ps a, dest_nat n)
```
```  2816   | num_of_term ps (Const (@{const_name Fields.divide}, _) \$ a \$ b) = @{code poly.C} (dest_num a, dest_num b)
```
```  2817   | num_of_term ps t = (case try_dest_num t
```
```  2818      of SOME k => @{code poly.C} (k, 1)
```
```  2819       | NONE => @{code poly.Bound} (the_index ps t));
```
```  2820
```
```  2821 fun tm_of_term fs ps (Const(@{const_name Groups.uminus}, _) \$ t) = @{code Neg} (tm_of_term fs ps t)
```
```  2822   | tm_of_term fs ps (Const(@{const_name Groups.plus}, _) \$ a \$ b) = @{code Add} (tm_of_term fs ps a, tm_of_term fs ps b)
```
```  2823   | tm_of_term fs ps (Const(@{const_name Groups.minus}, _) \$ a \$ b) = @{code Sub} (tm_of_term fs ps a, tm_of_term fs ps b)
```
```  2824   | tm_of_term fs ps (Const(@{const_name Groups.times}, _) \$ a \$ b) = @{code Mul} (num_of_term ps a, tm_of_term fs ps b)
```
```  2825   | tm_of_term fs ps t = (@{code CP} (num_of_term ps t)
```
```  2826       handle TERM _ => @{code Bound} (the_index fs t)
```
```  2827            | General.Subscript => @{code Bound} (the_index fs t));
```
```  2828
```
```  2829 fun fm_of_term fs ps @{term True} = @{code T}
```
```  2830   | fm_of_term fs ps @{term False} = @{code F}
```
```  2831   | fm_of_term fs ps (Const (@{const_name Not}, _) \$ p) = @{code NOT} (fm_of_term fs ps p)
```
```  2832   | fm_of_term fs ps (Const (@{const_name HOL.conj}, _) \$ p \$ q) = @{code And} (fm_of_term fs ps p, fm_of_term fs ps q)
```
```  2833   | fm_of_term fs ps (Const (@{const_name HOL.disj}, _) \$ p \$ q) = @{code Or} (fm_of_term fs ps p, fm_of_term fs ps q)
```
```  2834   | fm_of_term fs ps (Const (@{const_name HOL.implies}, _) \$ p \$ q) = @{code Imp} (fm_of_term fs ps p, fm_of_term fs ps q)
```
```  2835   | fm_of_term fs ps (@{term HOL.iff} \$ p \$ q) = @{code Iff} (fm_of_term fs ps p, fm_of_term fs ps q)
```
```  2836   | fm_of_term fs ps (Const (@{const_name HOL.eq}, T) \$ p \$ q) =
```
```  2837       @{code Eq} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q))
```
```  2838   | fm_of_term fs ps (Const (@{const_name Orderings.less}, _) \$ p \$ q) =
```
```  2839       @{code Lt} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q))
```
```  2840   | fm_of_term fs ps (Const (@{const_name Orderings.less_eq}, _) \$ p \$ q) =
```
```  2841       @{code Le} (@{code Sub} (tm_of_term fs ps p, tm_of_term fs ps q))
```
```  2842   | fm_of_term fs ps (Const (@{const_name Ex}, _) \$ Abs (abs as (_, xT, _))) =
```
```  2843       let
```
```  2844         val (xn', p') = Syntax_Trans.variant_abs abs;  (* FIXME !? *)
```
```  2845       in @{code E} (fm_of_term (Free (xn', xT) :: fs) ps p') end
```
```  2846   | fm_of_term fs ps (Const (@{const_name All},_) \$ Abs (abs as (_, xT, _))) =
```
```  2847       let
```
```  2848         val (xn', p') = Syntax_Trans.variant_abs abs;  (* FIXME !? *)
```
```  2849       in @{code A} (fm_of_term (Free (xn', xT) :: fs) ps p') end
```
```  2850   | fm_of_term fs ps _ = error "fm_of_term";
```
```  2851
```
```  2852 fun term_of_num T ps (@{code poly.C} (a, b)) =
```
```  2853     (if b = 1 then HOLogic.mk_number T a
```
```  2854      else if b = 0 then Const (@{const_name Groups.zero}, T)
```
```  2855      else Const (@{const_name Fields.divide}, binopT T) \$ HOLogic.mk_number T a \$ HOLogic.mk_number T b)
```
```  2856   | term_of_num T ps (@{code poly.Bound} i) = nth ps i
```
```  2857   | term_of_num T ps (@{code poly.Add} (a, b)) = Const (@{const_name Groups.plus}, binopT T) \$ term_of_num T ps a \$ term_of_num T ps b
```
```  2858   | term_of_num T ps (@{code poly.Mul} (a, b)) = Const (@{const_name Groups.times}, binopT T) \$ term_of_num T ps a \$ term_of_num T ps b
```
```  2859   | term_of_num T ps (@{code poly.Sub} (a, b)) = Const (@{const_name Groups.minus}, binopT T) \$ term_of_num T ps a \$ term_of_num T ps b
```
```  2860   | term_of_num T ps (@{code poly.Neg} a) = Const (@{const_name Groups.uminus}, T --> T) \$ term_of_num T ps a
```
```  2861   | term_of_num T ps (@{code poly.Pw} (a, n)) =
```
```  2862       Const (@{const_name Power.power}, T --> @{typ nat} --> T) \$ term_of_num T ps a \$ HOLogic.mk_number HOLogic.natT n
```
```  2863   | term_of_num T ps (@{code poly.CN} (c, n, p)) =
```
```  2864       term_of_num T ps (@{code poly.Add} (c, @{code poly.Mul} (@{code poly.Bound} n, p)));
```
```  2865
```
```  2866 fun term_of_tm T fs ps (@{code CP} p) = term_of_num T ps p
```
```  2867   | term_of_tm T fs ps (@{code Bound} i) = nth fs i
```
```  2868   | term_of_tm T fs ps (@{code Add} (a, b)) = Const (@{const_name Groups.plus}, binopT T) \$ term_of_tm T fs ps a \$ term_of_tm T fs ps b
```
```  2869   | term_of_tm T fs ps (@{code Mul} (a, b)) = Const (@{const_name Groups.times}, binopT T) \$ term_of_num T ps a \$ term_of_tm T fs ps b
```
```  2870   | term_of_tm T fs ps (@{code Sub} (a, b)) = Const (@{const_name Groups.minus}, binopT T) \$ term_of_tm T fs ps a \$ term_of_tm T fs ps b
```
```  2871   | term_of_tm T fs ps (@{code Neg} a) = Const (@{const_name Groups.uminus}, T --> T) \$ term_of_tm T fs ps a
```
```  2872   | term_of_tm T fs ps (@{code CNP} (n, c, p)) = term_of_tm T fs ps
```
```  2873      (@{code Add} (@{code Mul} (c, @{code Bound} n), p));
```
```  2874
```
```  2875 fun term_of_fm T fs ps @{code T} = @{term True}
```
```  2876   | term_of_fm T fs ps @{code F} = @{term False}
```
```  2877   | term_of_fm T fs ps (@{code NOT} p) = @{term Not} \$ term_of_fm T fs ps p
```
```  2878   | term_of_fm T fs ps (@{code And} (p, q)) = @{term HOL.conj} \$ term_of_fm T fs ps p \$ term_of_fm T fs ps q
```
```  2879   | term_of_fm T fs ps (@{code Or} (p, q)) = @{term HOL.disj} \$ term_of_fm T fs ps p \$ term_of_fm T fs ps q
```
```  2880   | term_of_fm T fs ps (@{code Imp} (p, q)) = @{term HOL.implies} \$ term_of_fm T fs ps p \$ term_of_fm T fs ps q
```
```  2881   | term_of_fm T fs ps (@{code Iff} (p, q)) = @{term HOL.iff} \$ term_of_fm T fs ps p \$ term_of_fm T fs ps q
```
```  2882   | term_of_fm T fs ps (@{code Lt} p) = Const (@{const_name Orderings.less}, relT T) \$ term_of_tm T fs ps p \$ Const (@{const_name Groups.zero}, T)
```
```  2883   | term_of_fm T fs ps (@{code Le} p) = Const (@{const_name Orderings.less_eq}, relT T) \$ term_of_tm T fs ps p \$ Const (@{const_name Groups.zero}, T)
```
```  2884   | term_of_fm T fs ps (@{code Eq} p) = Const (@{const_name HOL.eq}, relT T) \$ term_of_tm T fs ps p \$ Const (@{const_name Groups.zero}, T)
```
```  2885   | term_of_fm T fs ps (@{code NEq} p) = @{term Not} \$ (Const (@{const_name HOL.eq}, relT T) \$ term_of_tm T fs ps p \$ Const (@{const_name Groups.zero}, T))
```
```  2886   | term_of_fm T fs ps _ = error "term_of_fm: quantifiers";
```
```  2887
```
```  2888 fun frpar_procedure alternative T ps fm =
```
```  2889   let
```
```  2890     val frpar = if alternative then @{code frpar2} else @{code frpar};
```
```  2891     val fs = subtract (op aconv) (map Free (Term.add_frees fm [])) ps;
```
```  2892     val eval = term_of_fm T fs ps o frpar o fm_of_term fs ps;
```
```  2893     val t = HOLogic.dest_Trueprop fm;
```
```  2894   in
```
```  2895     (HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, eval t)
```
```  2896   end;
```
```  2897
```
```  2898 in
```
```  2899
```
```  2900   fn (ctxt, alternative, ty, ps, ct) =>
```
```  2901     cterm_of (Proof_Context.theory_of ctxt)
```
```  2902       (frpar_procedure alternative ty ps (term_of ct))
```
```  2903
```
```  2904 end
```
```  2905 *}
```
```  2906
```
```  2907 ML {*
```
```  2908 structure Parametric_Ferrante_Rackoff =
```
```  2909 struct
```
```  2910
```
```  2911 fun tactic ctxt alternative T ps =
```
```  2912   Object_Logic.full_atomize_tac
```
```  2913   THEN' CSUBGOAL (fn (g, i) =>
```
```  2914     let
```
```  2915       val th = frpar_oracle (ctxt, alternative, T, ps, g)
```
```  2916     in rtac (th RS iffD2) i end);
```
```  2917
```
```  2918 fun method alternative =
```
```  2919   let
```
```  2920     fun keyword k = Scan.lift (Args.\$\$\$ k -- Args.colon) >> K ()
```
```  2921     val parsN = "pars"
```
```  2922     val typN = "type"
```
```  2923     val any_keyword = keyword parsN || keyword typN
```
```  2924     val terms = Scan.repeat (Scan.unless any_keyword Args.term)
```
```  2925     val typ = Scan.unless any_keyword Args.typ
```
```  2926   in
```
```  2927     (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
```
```  2928       (fn (T, ps) => fn ctxt => SIMPLE_METHOD' (tactic ctxt alternative T ps))
```
```  2929   end
```
```  2930
```
```  2931 end;
```
```  2932 *}
```
```  2933
```
```  2934 method_setup frpar = {*
```
```  2935   Parametric_Ferrante_Rackoff.method false
```
```  2936 *} "parametric QE for linear Arithmetic over fields"
```
```  2937
```
```  2938 method_setup frpar2 = {*
```
```  2939   Parametric_Ferrante_Rackoff.method true
```
```  2940 *} "parametric QE for linear Arithmetic over fields, Version 2"
```
```  2941
```
```  2942 lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
```
```  2943   apply (frpar type: "'a::{linordered_field_inverse_zero}" pars: "y::'a::{linordered_field_inverse_zero}")
```
```  2944   apply (simp add: field_simps)
```
```  2945   apply (rule spec[where x=y])
```
```  2946   apply (frpar type: "'a::{linordered_field_inverse_zero}" pars: "z::'a::{linordered_field_inverse_zero}")
```
```  2947   by simp
```
```  2948
```
```  2949 lemma "\<exists>(x::'a::{linordered_field_inverse_zero}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
```
```  2950   apply (frpar2 type: "'a::{linordered_field_inverse_zero}" pars: "y::'a::{linordered_field_inverse_zero}")
```
```  2951   apply (simp add: field_simps)
```
```  2952   apply (rule spec[where x=y])
```
```  2953   apply (frpar2 type: "'a::{linordered_field_inverse_zero}" pars: "z::'a::{linordered_field_inverse_zero}")
```
```  2954   by simp
```
```  2955
```
```  2956 text{* Collins/Jones Problem *}
```
```  2957 (*
```
```  2958 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"
```
```  2959 proof-
```
```  2960   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")
```
```  2961 by (simp add: field_simps)
```
```  2962 have "?rhs"
```
```  2963
```
```  2964   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}")
```
```  2965   apply (simp add: field_simps)
```
```  2966 oops
```
```  2967 *)
```
```  2968 (*
```
```  2969 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"
```
```  2970 apply (frpar type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
```
```  2971 oops
```
```  2972 *)
```
```  2973
```
```  2974 text{* Collins/Jones Problem *}
```
```  2975
```
```  2976 (*
```
```  2977 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"
```
```  2978 proof-
```
```  2979   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")
```
```  2980 by (simp add: field_simps)
```
```  2981 have "?rhs"
```
```  2982   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}")
```
```  2983   apply simp
```
```  2984 oops
```
```  2985 *)
```
```  2986
```
```  2987 (*
```
```  2988 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"
```
```  2989 apply (frpar2 type: "'a::{linordered_field_inverse_zero, number_ring}" pars: "t::'a::{linordered_field_inverse_zero, number_ring}")
```
```  2990 apply (simp add: field_simps linorder_neq_iff[symmetric])
```
```  2991 apply ferrack
```
```  2992 oops
```
```  2993 *)
```
```  2994 end
```
```  2995
```