src/HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy

tuned;

(*  Title:      HOL/Decision_Procs/Parametric_Ferrante_Rackoff.thy
    Author:     Amine Chaieb
*)

header{* A formalization of Ferrante and Rackoff's procedure with polynomial parameters, see Paper in CALCULEMUS 2008 *}

theory Parametric_Ferrante_Rackoff
imports Reflected_Multivariate_Polynomial 
  "~~/src/HOL/Decision_Procs/Dense_Linear_Order"
begin


subsection {* Terms *}

datatype tm = CP poly | Bound nat | Add tm tm | Mul poly tm 
  | Neg tm | Sub tm tm | CNP nat poly tm
  (* A size for poly to make inductive proofs simpler*)

consts tmsize :: "tm \<Rightarrow> nat"
primrec 
  "tmsize (CP c) = polysize c"
  "tmsize (Bound n) = 1"
  "tmsize (Neg a) = 1 + tmsize a"
  "tmsize (Add a b) = 1 + tmsize a + tmsize b"
  "tmsize (Sub a b) = 3 + tmsize a + tmsize b"
  "tmsize (Mul c a) = 1 + polysize c + tmsize a"
  "tmsize (CNP n c a) = 3 + polysize c + tmsize a "

  (* Semantics of terms tm *)
consts Itm :: "'a::{ring_char_0,division_by_zero,field} list \<Rightarrow> 'a list \<Rightarrow> tm \<Rightarrow> 'a"
primrec
  "Itm vs bs (CP c) = (Ipoly vs c)"
  "Itm vs bs (Bound n) = bs!n"
  "Itm vs bs (Neg a) = -(Itm vs bs a)"
  "Itm vs bs (Add a b) = Itm vs bs a + Itm vs bs b"
  "Itm vs bs (Sub a b) = Itm vs bs a - Itm vs bs b"
  "Itm vs bs (Mul c a) = (Ipoly vs c) * Itm vs bs a"
  "Itm vs bs (CNP n c t) = (Ipoly vs c)*(bs!n) + Itm vs bs t"	


fun allpolys:: "(poly \<Rightarrow> bool) \<Rightarrow> tm \<Rightarrow> bool"  where
  "allpolys P (CP c) = P c"
| "allpolys P (CNP n c p) = (P c \<and> allpolys P p)"
| "allpolys P (Mul c p) = (P c \<and> allpolys P p)"
| "allpolys P (Neg p) = allpolys P p"
| "allpolys P (Add p q) = (allpolys P p \<and> allpolys P q)"
| "allpolys P (Sub p q) = (allpolys P p \<and> allpolys P q)"
| "allpolys P p = True"

consts 
  tmboundslt:: "nat \<Rightarrow> tm \<Rightarrow> bool"
  tmbound0:: "tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound 0 *)
  tmbound:: "nat \<Rightarrow> tm \<Rightarrow> bool" (* a tm is INDEPENDENT of Bound n *)
  incrtm0:: "tm \<Rightarrow> tm"
  incrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm"
  decrtm0:: "tm \<Rightarrow> tm" 
  decrtm:: "nat \<Rightarrow> tm \<Rightarrow> tm" 
primrec
  "tmboundslt n (CP c) = True"
  "tmboundslt n (Bound m) = (m < n)"
  "tmboundslt n (CNP m c a) = (m < n \<and> tmboundslt n a)"
  "tmboundslt n (Neg a) = tmboundslt n a"
  "tmboundslt n (Add a b) = (tmboundslt n a \<and> tmboundslt n b)"
  "tmboundslt n (Sub a b) = (tmboundslt n a \<and> tmboundslt n b)" 
  "tmboundslt n (Mul i a) = tmboundslt n a"
primrec
  "tmbound0 (CP c) = True"
  "tmbound0 (Bound n) = (n>0)"
  "tmbound0 (CNP n c a) = (n\<noteq>0 \<and> tmbound0 a)"
  "tmbound0 (Neg a) = tmbound0 a"
  "tmbound0 (Add a b) = (tmbound0 a \<and> tmbound0 b)"
  "tmbound0 (Sub a b) = (tmbound0 a \<and> tmbound0 b)" 
  "tmbound0 (Mul i a) = tmbound0 a"
lemma tmbound0_I:
  assumes nb: "tmbound0 a"
  shows "Itm vs (b#bs) a = Itm vs (b'#bs) a"
using nb
by (induct a rule: tmbound0.induct,auto simp add: nth_pos2)

primrec
  "tmbound n (CP c) = True"
  "tmbound n (Bound m) = (n \<noteq> m)"
  "tmbound n (CNP m c a) = (n\<noteq>m \<and> tmbound n a)"
  "tmbound n (Neg a) = tmbound n a"
  "tmbound n (Add a b) = (tmbound n a \<and> tmbound n b)"
  "tmbound n (Sub a b) = (tmbound n a \<and> tmbound n b)" 
  "tmbound n (Mul i a) = tmbound n a"
lemma tmbound0_tmbound_iff: "tmbound 0 t = tmbound0 t" by (induct t, auto)

lemma tmbound_I: 
  assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound n t" and le: "n \<le> length bs"
  shows "Itm vs (bs[n:=x]) t = Itm vs bs t"
  using nb le bnd
  by (induct t rule: tmbound.induct , auto)

recdef decrtm0 "measure size"
  "decrtm0 (Bound n) = Bound (n - 1)"
  "decrtm0 (Neg a) = Neg (decrtm0 a)"
  "decrtm0 (Add a b) = Add (decrtm0 a) (decrtm0 b)"
  "decrtm0 (Sub a b) = Sub (decrtm0 a) (decrtm0 b)"
  "decrtm0 (Mul c a) = Mul c (decrtm0 a)"
  "decrtm0 (CNP n c a) = CNP (n - 1) c (decrtm0 a)"
  "decrtm0 a = a"
recdef incrtm0 "measure size"
  "incrtm0 (Bound n) = Bound (n + 1)"
  "incrtm0 (Neg a) = Neg (incrtm0 a)"
  "incrtm0 (Add a b) = Add (incrtm0 a) (incrtm0 b)"
  "incrtm0 (Sub a b) = Sub (incrtm0 a) (incrtm0 b)"
  "incrtm0 (Mul c a) = Mul c (incrtm0 a)"
  "incrtm0 (CNP n c a) = CNP (n + 1) c (incrtm0 a)"
  "incrtm0 a = a"
lemma decrtm0: assumes nb: "tmbound0 t"
  shows "Itm vs (x#bs) t = Itm vs bs (decrtm0 t)"
  using nb by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)
lemma incrtm0: "Itm vs (x#bs) (incrtm0 t) = Itm vs bs t"
  by (induct t rule: decrtm0.induct, simp_all add: nth_pos2)

primrec
  "decrtm m (CP c) = (CP c)"
  "decrtm m (Bound n) = (if n < m then Bound n else Bound (n - 1))"
  "decrtm m (Neg a) = Neg (decrtm m a)"
  "decrtm m (Add a b) = Add (decrtm m a) (decrtm m b)"
  "decrtm m (Sub a b) = Sub (decrtm m a) (decrtm m b)"
  "decrtm m (Mul c a) = Mul c (decrtm m a)"
  "decrtm m (CNP n c a) = (if n < m then CNP n c (decrtm m a) else CNP (n - 1) c (decrtm m a))"

consts removen:: "nat \<Rightarrow> 'a list \<Rightarrow> 'a list"
primrec
  "removen n [] = []"
  "removen n (x#xs) = (if n=0 then xs else (x#(removen (n - 1) xs)))"

lemma removen_same: "n \<ge> length xs \<Longrightarrow> removen n xs = xs"
  by (induct xs arbitrary: n, auto)

lemma nth_length_exceeds: "n \<ge> length xs \<Longrightarrow> xs!n = []!(n - length xs)"
  by (induct xs arbitrary: n, auto)

lemma removen_length: "length (removen n xs) = (if n \<ge> length xs then length xs else length xs - 1)"
  by (induct xs arbitrary: n, auto)
lemma removen_nth: "(removen n xs)!m = (if n \<ge> length xs then xs!m 
  else if m < n then xs!m else if m \<le> length xs then xs!(Suc m) else []!(m - (length xs - 1)))"
proof(induct xs arbitrary: n m)
  case Nil thus ?case by simp
next
  case (Cons x xs n m)
  {assume nxs: "n \<ge> length (x#xs)" hence ?case using removen_same[OF nxs] by simp}
  moreover
  {assume nxs: "\<not> (n \<ge> length (x#xs))" 
    {assume mln: "m < n" hence ?case using prems by (cases m, auto)}
    moreover
    {assume mln: "\<not> (m < n)" 
      
      {assume mxs: "m \<le> length (x#xs)" hence ?case using prems by (cases m, auto)}
      moreover
      {assume mxs: "\<not> (m \<le> length (x#xs))" 
	have th: "length (removen n (x#xs)) = length xs" 
	  using removen_length[where n="n" and xs="x#xs"] nxs by simp
	with mxs have mxs':"m \<ge> length (removen n (x#xs))" by auto
	hence "(removen n (x#xs))!m = [] ! (m - length xs)" 
	  using th nth_length_exceeds[OF mxs'] by auto
	hence th: "(removen n (x#xs))!m = [] ! (m - (length (x#xs) - 1))" 
	  by auto
	hence ?case using nxs mln mxs by auto }
      ultimately have ?case by blast
    }
    ultimately have ?case by blast
    
  }      ultimately show ?case by blast
qed

lemma decrtm: assumes bnd: "tmboundslt (length bs) t" and nb: "tmbound m t" 
  and nle: "m \<le> length bs" 
  shows "Itm vs (removen m bs) (decrtm m t) = Itm vs bs t"
  using bnd nb nle
  by (induct t rule: decrtm.induct, auto simp add: removen_nth)

consts tmsubst0:: "tm \<Rightarrow> tm \<Rightarrow> tm"
primrec
  "tmsubst0 t (CP c) = CP c"
  "tmsubst0 t (Bound n) = (if n=0 then t else Bound n)"
  "tmsubst0 t (CNP n c a) = (if n=0 then Add (Mul c t) (tmsubst0 t a) else CNP n c (tmsubst0 t a))"
  "tmsubst0 t (Neg a) = Neg (tmsubst0 t a)"
  "tmsubst0 t (Add a b) = Add (tmsubst0 t a) (tmsubst0 t b)"
  "tmsubst0 t (Sub a b) = Sub (tmsubst0 t a) (tmsubst0 t b)" 
  "tmsubst0 t (Mul i a) = Mul i (tmsubst0 t a)"
lemma tmsubst0:
  shows "Itm vs (x#bs) (tmsubst0 t a) = Itm vs ((Itm vs (x#bs) t)#bs) a"
by (induct a rule: tmsubst0.induct,auto simp add: nth_pos2)

lemma tmsubst0_nb: "tmbound0 t \<Longrightarrow> tmbound0 (tmsubst0 t a)"
by (induct a rule: tmsubst0.induct,auto simp add: nth_pos2)

consts tmsubst:: "nat \<Rightarrow> tm \<Rightarrow> tm \<Rightarrow> tm" 

primrec
  "tmsubst n t (CP c) = CP c"
  "tmsubst n t (Bound m) = (if n=m then t else Bound m)"
  "tmsubst n t (CNP m c a) = (if n=m then Add (Mul c t) (tmsubst n t a) 
             else CNP m c (tmsubst n t a))"
  "tmsubst n t (Neg a) = Neg (tmsubst n t a)"
  "tmsubst n t (Add a b) = Add (tmsubst n t a) (tmsubst n t b)"
  "tmsubst n t (Sub a b) = Sub (tmsubst n t a) (tmsubst n t b)" 
  "tmsubst n t (Mul i a) = Mul i (tmsubst n t a)"

lemma tmsubst: assumes nb: "tmboundslt (length bs) a" and nlt: "n \<le> length bs"
  shows "Itm vs bs (tmsubst n t a) = Itm vs (bs[n:= Itm vs bs t]) a"
using nb nlt
by (induct a rule: tmsubst0.induct,auto simp add: nth_pos2)

lemma tmsubst_nb0: assumes tnb: "tmbound0 t"
shows "tmbound0 (tmsubst 0 t a)"
using tnb
by (induct a rule: tmsubst.induct, auto)

lemma tmsubst_nb: assumes tnb: "tmbound m t"
shows "tmbound m (tmsubst m t a)"
using tnb
by (induct a rule: tmsubst.induct, auto)
lemma incrtm0_tmbound: "tmbound n t \<Longrightarrow> tmbound (Suc n) (incrtm0 t)"
  by (induct t, auto)
  (* Simplification *)

consts
  simptm:: "tm \<Rightarrow> tm"
  tmadd:: "tm \<times> tm \<Rightarrow> tm"
  tmmul:: "tm \<Rightarrow> poly \<Rightarrow> tm"
recdef tmadd "measure (\<lambda> (t,s). size t + size s)"
  "tmadd (CNP n1 c1 r1,CNP n2 c2 r2) =
  (if n1=n2 then 
  (let c = c1 +\<^sub>p c2
  in if c = 0\<^sub>p then tmadd(r1,r2) else CNP n1 c (tmadd (r1,r2)))
  else if n1 \<le> n2 then (CNP n1 c1 (tmadd (r1,CNP n2 c2 r2))) 
  else (CNP n2 c2 (tmadd (CNP n1 c1 r1,r2))))"
  "tmadd (CNP n1 c1 r1,t) = CNP n1 c1 (tmadd (r1, t))"  
  "tmadd (t,CNP n2 c2 r2) = CNP n2 c2 (tmadd (t,r2))" 
  "tmadd (CP b1, CP b2) = CP (b1 +\<^sub>p b2)"
  "tmadd (a,b) = Add a b"

lemma tmadd[simp]: "Itm vs bs (tmadd (t,s)) = Itm vs bs (Add t s)"
apply (induct t s rule: tmadd.induct, simp_all add: Let_def)
apply (case_tac "c1 +\<^sub>p c2 = 0\<^sub>p",case_tac "n1 \<le> n2", simp_all)
apply (case_tac "n1 = n2", simp_all add: ring_simps)
apply (simp only: right_distrib[symmetric]) 
by (auto simp del: polyadd simp add: polyadd[symmetric])

lemma tmadd_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmadd (t,s))"
by (induct t s rule: tmadd.induct, auto simp add: Let_def)

lemma tmadd_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmadd (t,s))"
by (induct t s rule: tmadd.induct, auto simp add: Let_def)
lemma tmadd_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmadd (t,s))"
by (induct t s rule: tmadd.induct, auto simp add: Let_def)

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)

recdef tmmul "measure size"
  "tmmul (CP j) = (\<lambda> i. CP (i *\<^sub>p j))"
  "tmmul (CNP n c a) = (\<lambda> i. CNP n (i *\<^sub>p c) (tmmul a i))"
  "tmmul t = (\<lambda> i. Mul i t)"

lemma tmmul[simp]: "Itm vs bs (tmmul t i) = Itm vs bs (Mul i t)"
by (induct t arbitrary: i rule: tmmul.induct, simp_all add: ring_simps)

lemma tmmul_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmmul t i)"
by (induct t arbitrary: i rule: tmmul.induct, auto )

lemma tmmul_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmmul t i)"
by (induct t arbitrary: n rule: tmmul.induct, auto )
lemma tmmul_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmmul t i)"
by (induct t arbitrary: i rule: tmmul.induct, auto simp add: Let_def)

lemma tmmul_allpolys_npoly[simp]: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  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)

constdefs tmneg :: "tm \<Rightarrow> tm"
  "tmneg t \<equiv> tmmul t (C (- 1,1))"

constdefs tmsub :: "tm \<Rightarrow> tm \<Rightarrow> tm"
  "tmsub s t \<equiv> (if s = t then CP 0\<^sub>p else tmadd (s,tmneg t))"

lemma tmneg[simp]: "Itm vs bs (tmneg t) = Itm vs bs (Neg t)"
using tmneg_def[of t] 
apply simp
apply (subst number_of_Min)
apply (simp only: of_int_minus)
apply simp
done

lemma tmneg_nb0[simp]: "tmbound0 t \<Longrightarrow> tmbound0 (tmneg t)"
using tmneg_def by simp

lemma tmneg_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (tmneg t)"
using tmneg_def by simp
lemma tmneg_blt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (tmneg t)"
using tmneg_def by simp
lemma [simp]: "isnpoly (C (-1,1))" unfolding isnpoly_def by simp
lemma tmneg_allpolys_npoly[simp]: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly (tmneg t)" 
  unfolding tmneg_def by auto

lemma tmsub[simp]: "Itm vs bs (tmsub a b) = Itm vs bs (Sub a b)"
using tmsub_def by simp

lemma tmsub_nb0[simp]: "\<lbrakk> tmbound0 t ; tmbound0 s\<rbrakk> \<Longrightarrow> tmbound0 (tmsub t s)"
using tmsub_def by simp
lemma tmsub_nb[simp]: "\<lbrakk> tmbound n t ; tmbound n s\<rbrakk> \<Longrightarrow> tmbound n (tmsub t s)"
using tmsub_def by simp
lemma tmsub_blt[simp]: "\<lbrakk>tmboundslt n t ; tmboundslt n s\<rbrakk> \<Longrightarrow> tmboundslt n (tmsub t s )"
using tmsub_def by simp
lemma tmsub_allpolys_npoly[simp]: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "allpolys isnpoly t \<Longrightarrow> allpolys isnpoly s \<Longrightarrow> allpolys isnpoly (tmsub t s)" 
  unfolding tmsub_def by (simp add: isnpoly_def)

recdef simptm "measure size"
  "simptm (CP j) = CP (polynate j)"
  "simptm (Bound n) = CNP n 1\<^sub>p (CP 0\<^sub>p)"
  "simptm (Neg t) = tmneg (simptm t)"
  "simptm (Add t s) = tmadd (simptm t,simptm s)"
  "simptm (Sub t s) = tmsub (simptm t) (simptm s)"
  "simptm (Mul i t) = (let i' = polynate i in if i' = 0\<^sub>p then CP 0\<^sub>p else tmmul (simptm t) i')"
  "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))"

lemma polynate_stupid: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "polynate t = 0\<^sub>p \<Longrightarrow> Ipoly bs t = (0::'a::{ring_char_0,division_by_zero, field})" 
apply (subst polynate[symmetric])
apply simp
done

lemma simptm_ci[simp]: "Itm vs bs (simptm t) = Itm vs bs t"
by (induct t rule: simptm.induct, auto simp add: tmneg tmadd tmsub tmmul Let_def polynate_stupid) 

lemma simptm_tmbound0[simp]: 
  "tmbound0 t \<Longrightarrow> tmbound0 (simptm t)"
by (induct t rule: simptm.induct, auto simp add: Let_def)

lemma simptm_nb[simp]: "tmbound n t \<Longrightarrow> tmbound n (simptm t)"
by (induct t rule: simptm.induct, auto simp add: Let_def)
lemma simptm_nlt[simp]: "tmboundslt n t \<Longrightarrow> tmboundslt n (simptm t)"
by (induct t rule: simptm.induct, auto simp add: Let_def)

lemma [simp]: "isnpoly 0\<^sub>p" and [simp]: "isnpoly (C(1,1))" 
  by (simp_all add: isnpoly_def)
lemma simptm_allpolys_npoly[simp]: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "allpolys isnpoly (simptm p)"
  by (induct p rule: simptm.induct, auto simp add: Let_def)

consts split0 :: "tm \<Rightarrow> (poly \<times> tm)"
recdef split0 "measure tmsize"
  "split0 (Bound 0) = (1\<^sub>p, CP 0\<^sub>p)"
  "split0 (CNP 0 c t) = (let (c',t') = split0 t in (c +\<^sub>p c',t'))"
  "split0 (Neg t) = (let (c,t') = split0 t in (~\<^sub>p c,Neg t'))"
  "split0 (CNP n c t) = (let (c',t') = split0 t in (c',CNP n c t'))"
  "split0 (Add s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 +\<^sub>p c2, Add s' t'))"
  "split0 (Sub s t) = (let (c1,s') = split0 s ; (c2,t') = split0 t in (c1 -\<^sub>p c2, Sub s' t'))"
  "split0 (Mul c t) = (let (c',t') = split0 t in (c *\<^sub>p c', Mul c t'))"
  "split0 t = (0\<^sub>p, t)"

lemma split0_stupid[simp]: "\<exists>x y. (x,y) = split0 p"
  apply (rule exI[where x="fst (split0 p)"])
  apply (rule exI[where x="snd (split0 p)"])
  by simp

lemma split0:
  "tmbound 0 (snd (split0 t)) \<and> (Itm vs bs (CNP 0 (fst (split0 t)) (snd (split0 t))) = Itm vs bs t)"
  apply (induct t rule: split0.induct)
  apply simp
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def mult_assoc right_distrib[symmetric])
  apply (simp add: Let_def split_def ring_simps)
  apply (simp add: Let_def split_def ring_simps)
  done

lemma split0_ci: "split0 t = (c',t') \<Longrightarrow> Itm vs bs t = Itm vs bs (CNP 0 c' t')"
proof-
  fix c' t'
  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
  with split0[where t="t" and bs="bs"] show "Itm vs bs t = Itm vs bs (CNP 0 c' t')" by simp
qed

lemma split0_nb0: 
  assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "split0 t = (c',t') \<Longrightarrow>  tmbound 0 t'"
proof-
  fix c' t'
  assume "split0 t = (c', t')" hence "c' = fst (split0 t)" and "t' = snd (split0 t)" by auto
  with conjunct1[OF split0[where t="t"]] show "tmbound 0 t'" by simp
qed

lemma split0_nb0'[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows "tmbound0 (snd (split0 t))"
  using split0_nb0[of t "fst (split0 t)" "snd (split0 t)"] by (simp add: tmbound0_tmbound_iff)


lemma split0_nb: assumes nb:"tmbound n t" shows "tmbound n (snd (split0 t))"
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)

lemma split0_blt: assumes nb:"tmboundslt n t" shows "tmboundslt n (snd (split0 t))"
  using nb by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)

lemma tmbound_split0: "tmbound 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)

lemma tmboundslt_split0: "tmboundslt n t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0 \<or> n > 0"
by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)

lemma tmboundslt0_split0: "tmboundslt 0 t \<Longrightarrow> Ipoly vs (fst(split0 t)) = 0"
 by (induct t rule: split0.induct, auto simp add: Let_def split_def split0_stupid)

lemma allpolys_split0: "allpolys isnpoly p \<Longrightarrow> allpolys isnpoly (snd (split0 p))"
by (induct p rule: split0.induct, auto simp  add: isnpoly_def Let_def split_def split0_stupid)

lemma isnpoly_fst_split0:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero, field})"
  shows 
  "allpolys isnpoly p \<Longrightarrow> isnpoly (fst (split0 p))"
  by (induct p rule: split0.induct, 
    auto simp  add: polyadd_norm polysub_norm polyneg_norm polymul_norm 
    Let_def split_def split0_stupid)

subsection{* Formulae *}

datatype fm  =  T| F| Le tm | Lt tm | Eq tm | NEq tm|
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm


  (* A size for fm *)
consts fmsize :: "fm \<Rightarrow> nat"
recdef fmsize "measure size"
  "fmsize (NOT p) = 1 + fmsize p"
  "fmsize (And p q) = 1 + fmsize p + fmsize q"
  "fmsize (Or p q) = 1 + fmsize p + fmsize q"
  "fmsize (Imp p q) = 3 + fmsize p + fmsize q"
  "fmsize (Iff p q) = 3 + 2*(fmsize p + fmsize q)"
  "fmsize (E p) = 1 + fmsize p"
  "fmsize (A p) = 4+ fmsize p"
  "fmsize p = 1"
  (* several lemmas about fmsize *)
lemma fmsize_pos: "fmsize p > 0"	
by (induct p rule: fmsize.induct) simp_all

  (* Semantics of formulae (fm) *)
consts Ifm ::"'a::{division_by_zero,ordered_field} list \<Rightarrow> 'a list \<Rightarrow> fm \<Rightarrow> bool"
primrec
  "Ifm vs bs T = True"
  "Ifm vs bs F = False"
  "Ifm vs bs (Lt a) = (Itm vs bs a < 0)"
  "Ifm vs bs (Le a) = (Itm vs bs a \<le> 0)"
  "Ifm vs bs (Eq a) = (Itm vs bs a = 0)"
  "Ifm vs bs (NEq a) = (Itm vs bs a \<noteq> 0)"
  "Ifm vs bs (NOT p) = (\<not> (Ifm vs bs p))"
  "Ifm vs bs (And p q) = (Ifm vs bs p \<and> Ifm vs bs q)"
  "Ifm vs bs (Or p q) = (Ifm vs bs p \<or> Ifm vs bs q)"
  "Ifm vs bs (Imp p q) = ((Ifm vs bs p) \<longrightarrow> (Ifm vs bs q))"
  "Ifm vs bs (Iff p q) = (Ifm vs bs p = Ifm vs bs q)"
  "Ifm vs bs (E p) = (\<exists> x. Ifm vs (x#bs) p)"
  "Ifm vs bs (A p) = (\<forall> x. Ifm vs (x#bs) p)"

consts not:: "fm \<Rightarrow> fm"
recdef not "measure size"
  "not (NOT (NOT p)) = not p"
  "not (NOT p) = p"
  "not T = F"
  "not F = T"
  "not (Lt t) = Le (tmneg t)"
  "not (Le t) = Lt (tmneg t)"
  "not (Eq t) = NEq t"
  "not (NEq t) = Eq t"
  "not p = NOT p"
lemma not[simp]: "Ifm vs bs (not p) = Ifm vs bs (NOT p)"
by (induct p rule: not.induct) auto

constdefs conj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "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 
   if p = q then p else And p q)"
lemma conj[simp]: "Ifm vs bs (conj p q) = Ifm vs bs (And p q)"
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)

constdefs disj :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "disj p q \<equiv> (if (p = T \<or> q=T) then T else if p=F then q else if q=F then p 
       else if p=q then p else Or p q)"

lemma disj[simp]: "Ifm vs bs (disj p q) = Ifm vs bs (Or p q)"
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)

constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "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 
    else Imp p q)"
lemma imp[simp]: "Ifm vs bs (imp p q) = Ifm vs bs (Imp p q)"
by (cases "p=F \<or> q=T",simp_all add: imp_def) 

constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else 
       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 
  Iff p q)"
lemma iff[simp]: "Ifm vs bs (iff p q) = Ifm vs bs (Iff p q)"
  by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)
  (* Quantifier freeness *)
consts qfree:: "fm \<Rightarrow> bool"
recdef qfree "measure size"
  "qfree (E p) = False"
  "qfree (A p) = False"
  "qfree (NOT p) = qfree p" 
  "qfree (And p q) = (qfree p \<and> qfree q)" 
  "qfree (Or  p q) = (qfree p \<and> qfree q)" 
  "qfree (Imp p q) = (qfree p \<and> qfree q)" 
  "qfree (Iff p q) = (qfree p \<and> qfree q)"
  "qfree p = True"

  (* Boundedness and substitution *)

consts boundslt :: "nat \<Rightarrow> fm \<Rightarrow> bool"
primrec
  "boundslt n T = True"
  "boundslt n F = True"
  "boundslt n (Lt t) = (tmboundslt n t)"
  "boundslt n (Le t) = (tmboundslt n t)"
  "boundslt n (Eq t) = (tmboundslt n t)"
  "boundslt n (NEq t) = (tmboundslt n t)"
  "boundslt n (NOT p) = boundslt n p"
  "boundslt n (And p q) = (boundslt n p \<and> boundslt n q)"
  "boundslt n (Or p q) = (boundslt n p \<and> boundslt n q)"
  "boundslt n (Imp p q) = ((boundslt n p) \<and> (boundslt n q))"
  "boundslt n (Iff p q) = (boundslt n p \<and> boundslt n q)"
  "boundslt n (E p) = boundslt (Suc n) p"
  "boundslt n (A p) = boundslt (Suc n) p"

consts 
  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
  bound:: "nat \<Rightarrow> fm \<Rightarrow> bool" (* A Formula is independent of Bound n *)
  decr0 :: "fm \<Rightarrow> fm"
  decr :: "nat \<Rightarrow> fm \<Rightarrow> fm"
recdef bound0 "measure size"
  "bound0 T = True"
  "bound0 F = True"
  "bound0 (Lt a) = tmbound0 a"
  "bound0 (Le a) = tmbound0 a"
  "bound0 (Eq a) = tmbound0 a"
  "bound0 (NEq a) = tmbound0 a"
  "bound0 (NOT p) = bound0 p"
  "bound0 (And p q) = (bound0 p \<and> bound0 q)"
  "bound0 (Or p q) = (bound0 p \<and> bound0 q)"
  "bound0 (Imp p q) = ((bound0 p) \<and> (bound0 q))"
  "bound0 (Iff p q) = (bound0 p \<and> bound0 q)"
  "bound0 p = False"
lemma bound0_I:
  assumes bp: "bound0 p"
  shows "Ifm vs (b#bs) p = Ifm vs (b'#bs) p"
using bp tmbound0_I[where b="b" and bs="bs" and b'="b'"]
by (induct p rule: bound0.induct,auto simp add: nth_pos2)

primrec
  "bound m T = True"
  "bound m F = True"
  "bound m (Lt t) = tmbound m t"
  "bound m (Le t) = tmbound m t"
  "bound m (Eq t) = tmbound m t"
  "bound m (NEq t) = tmbound m t"
  "bound m (NOT p) = bound m p"
  "bound m (And p q) = (bound m p \<and> bound m q)"
  "bound m (Or p q) = (bound m p \<and> bound m q)"
  "bound m (Imp p q) = ((bound m p) \<and> (bound m q))"
  "bound m (Iff p q) = (bound m p \<and> bound m q)"
  "bound m (E p) = bound (Suc m) p"
  "bound m (A p) = bound (Suc m) p"

lemma bound_I:
  assumes bnd: "boundslt (length bs) p" and nb: "bound n p" and le: "n \<le> length bs"
  shows "Ifm vs (bs[n:=x]) p = Ifm vs bs p"
  using bnd nb le tmbound_I[where bs=bs and vs = vs]
proof(induct p arbitrary: bs n rule: bound.induct)
  case (E p bs n) 
  {fix y
    from prems have bnd: "boundslt (length (y#bs)) p" 
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
    from E.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
  thus ?case by simp 
next
  case (A p bs n) {fix y
    from prems have bnd: "boundslt (length (y#bs)) p" 
      and nb: "bound (Suc n) p" and le: "Suc n \<le> length (y#bs)" by simp+
    from A.hyps[OF bnd nb le tmbound_I] have "Ifm vs ((y#bs)[Suc n:=x]) p = Ifm vs (y#bs) p" .   }
  thus ?case by simp 
qed auto

recdef decr0 "measure size"
  "decr0 (Lt a) = Lt (decrtm0 a)"
  "decr0 (Le a) = Le (decrtm0 a)"
  "decr0 (Eq a) = Eq (decrtm0 a)"
  "decr0 (NEq a) = NEq (decrtm0 a)"
  "decr0 (NOT p) = NOT (decr0 p)" 
  "decr0 (And p q) = conj (decr0 p) (decr0 q)"
  "decr0 (Or p q) = disj (decr0 p) (decr0 q)"
  "decr0 (Imp p q) = imp (decr0 p) (decr0 q)"
  "decr0 (Iff p q) = iff (decr0 p) (decr0 q)"
  "decr0 p = p"

lemma decr0: assumes nb: "bound0 p"
  shows "Ifm vs (x#bs) p = Ifm vs bs (decr0 p)"
  using nb 
  by (induct p rule: decr0.induct, simp_all add: decrtm0)

primrec
  "decr m T = T"
  "decr m F = F"
  "decr m (Lt t) = (Lt (decrtm m t))"
  "decr m (Le t) = (Le (decrtm m t))"
  "decr m (Eq t) = (Eq (decrtm m t))"
  "decr m (NEq t) = (NEq (decrtm m t))"
  "decr m (NOT p) = NOT (decr m p)" 
  "decr m (And p q) = conj (decr m p) (decr m q)"
  "decr m (Or p q) = disj (decr m p) (decr m q)"
  "decr m (Imp p q) = imp (decr m p) (decr m q)"
  "decr m (Iff p q) = iff (decr m p) (decr m q)"
  "decr m (E p) = E (decr (Suc m) p)"
  "decr m (A p) = A (decr (Suc m) p)"

lemma decr: assumes  bnd: "boundslt (length bs) p" and nb: "bound m p" 
  and nle: "m < length bs" 
  shows "Ifm vs (removen m bs) (decr m p) = Ifm vs bs p"
  using bnd nb nle
proof(induct p arbitrary: bs m rule: decr.induct)
  case (E p bs m) 
  {fix x
    from prems have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
  and nle: "Suc m < length (x#bs)" by auto
    from prems(4)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
  } thus ?case by auto 
next
  case (A p bs m)  
  {fix x
    from prems have bnd: "boundslt (length (x#bs)) p" and nb: "bound (Suc m) p" 
  and nle: "Suc m < length (x#bs)" by auto
    from prems(4)[OF bnd nb nle] have "Ifm vs (removen (Suc m) (x#bs)) (decr (Suc m) p) = Ifm vs (x#bs) p".
  } thus ?case by auto
qed (auto simp add: decrtm removen_nth)

consts
  subst0:: "tm \<Rightarrow> fm \<Rightarrow> fm"

primrec
  "subst0 t T = T"
  "subst0 t F = F"
  "subst0 t (Lt a) = Lt (tmsubst0 t a)"
  "subst0 t (Le a) = Le (tmsubst0 t a)"
  "subst0 t (Eq a) = Eq (tmsubst0 t a)"
  "subst0 t (NEq a) = NEq (tmsubst0 t a)"
  "subst0 t (NOT p) = NOT (subst0 t p)"
  "subst0 t (And p q) = And (subst0 t p) (subst0 t q)"
  "subst0 t (Or p q) = Or (subst0 t p) (subst0 t q)"
  "subst0 t (Imp p q) = Imp (subst0 t p)  (subst0 t q)"
  "subst0 t (Iff p q) = Iff (subst0 t p) (subst0 t q)"
  "subst0 t (E p) = E p"
  "subst0 t (A p) = A p"

lemma subst0: assumes qf: "qfree p"
  shows "Ifm vs (x#bs) (subst0 t p) = Ifm vs ((Itm vs (x#bs) t)#bs) p"
using qf tmsubst0[where x="x" and bs="bs" and t="t"]
by (induct p rule: subst0.induct, auto)

lemma subst0_nb:
  assumes bp: "tmbound0 t" and qf: "qfree p"
  shows "bound0 (subst0 t p)"
using qf tmsubst0_nb[OF bp] bp
by (induct p rule: subst0.induct, auto)

consts   subst:: "nat \<Rightarrow> tm \<Rightarrow> fm \<Rightarrow> fm" 
primrec
  "subst n t T = T"
  "subst n t F = F"
  "subst n t (Lt a) = Lt (tmsubst n t a)"
  "subst n t (Le a) = Le (tmsubst n t a)"
  "subst n t (Eq a) = Eq (tmsubst n t a)"
  "subst n t (NEq a) = NEq (tmsubst n t a)"
  "subst n t (NOT p) = NOT (subst n t p)"
  "subst n t (And p q) = And (subst n t p) (subst n t q)"
  "subst n t (Or p q) = Or (subst n t p) (subst n t q)"
  "subst n t (Imp p q) = Imp (subst n t p)  (subst n t q)"
  "subst n t (Iff p q) = Iff (subst n t p) (subst n t q)"
  "subst n t (E p) = E (subst (Suc n) (incrtm0 t) p)"
  "subst n t (A p) = A (subst (Suc n) (incrtm0 t) p)"

lemma subst: assumes nb: "boundslt (length bs) p" and nlm: "n \<le> length bs"
  shows "Ifm vs bs (subst n t p) = Ifm vs (bs[n:= Itm vs bs t]) p"
  using nb nlm
proof (induct p arbitrary: bs n t rule: subst0.induct)
  case (E p bs n) 
  {fix x 
    from prems have bn: "boundslt (length (x#bs)) p" by simp 
      from prems have nlm: "Suc n \<le> length (x#bs)" by simp
    from prems(3)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
thus ?case by simp 
next
  case (A p bs n)   
  {fix x 
    from prems have bn: "boundslt (length (x#bs)) p" by simp 
      from prems have nlm: "Suc n \<le> length (x#bs)" by simp
    from prems(3)[OF bn nlm] have "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs ((x#bs)[Suc n:= Itm vs (x#bs) (incrtm0 t)]) p" by simp 
    hence "Ifm vs (x#bs) (subst (Suc n) (incrtm0 t) p) = Ifm vs (x#bs[n:= Itm vs bs t]) p"
    by (simp add: incrtm0[where x="x" and bs="bs" and t="t"]) }  
thus ?case by simp 
qed(auto simp add: tmsubst)

lemma subst_nb: assumes tnb: "tmbound m t"
shows "bound m (subst m t p)"
using tnb tmsubst_nb incrtm0_tmbound
by (induct p arbitrary: m t rule: subst.induct, auto)

lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
by (induct p rule: not.induct, auto)
lemma not_bn0[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
by (induct p rule: not.induct, auto)
lemma not_nb[simp]: "bound n p \<Longrightarrow> bound n (not p)"
by (induct p rule: not.induct, auto)
lemma not_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n (not p)"
 by (induct p rule: not.induct, auto)

lemma conj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
using conj_def by auto 
lemma conj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
using conj_def by auto 
lemma conj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (conj p q)"
using conj_def by auto 
lemma conj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
using conj_def by auto 

lemma disj_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
using disj_def by auto 
lemma disj_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
using disj_def by auto 
lemma disj_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (disj p q)"
using disj_def by auto 
lemma disj_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (disj p q)"
using disj_def by auto 

lemma imp_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (imp p q)"
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def)
lemma imp_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
lemma imp_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (imp p q)"
using imp_def by (cases "p=F \<or> q=T \<or> p=q",simp_all add: imp_def)
lemma imp_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (imp p q)"
using imp_def by auto 

lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
  by (unfold iff_def,cases "p=q", auto)
lemma iff_nb0[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto)
lemma iff_nb[simp]: "\<lbrakk>bound n p ; bound n q\<rbrakk> \<Longrightarrow> bound n (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto)
lemma iff_blt[simp]: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (iff p q)"
using iff_def by auto 
lemma decr0_qf: "bound0 p \<Longrightarrow> qfree (decr0 p)"
by (induct p, simp_all)

consts 
  isatom :: "fm \<Rightarrow> bool" (* test for atomicity *)
recdef isatom "measure size"
  "isatom T = True"
  "isatom F = True"
  "isatom (Lt a) = True"
  "isatom (Le a) = True"
  "isatom (Eq a) = True"
  "isatom (NEq a) = True"
  "isatom p = False"

lemma bound0_qf: "bound0 p \<Longrightarrow> qfree p"
by (induct p, simp_all)

constdefs djf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a \<Rightarrow> fm \<Rightarrow> fm"
  "djf f p q \<equiv> (if q=T then T else if q=F then f p else 
  (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q))"
constdefs evaldjf:: "('a \<Rightarrow> fm) \<Rightarrow> 'a list \<Rightarrow> fm"
  "evaldjf f ps \<equiv> foldr (djf f) ps F"

lemma djf_Or: "Ifm vs bs (djf f p q) = Ifm vs bs (Or (f p) q)"
by (cases "q=T", simp add: djf_def,cases "q=F",simp add: djf_def) 
(cases "f p", simp_all add: Let_def djf_def) 

lemma evaldjf_ex: "Ifm vs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm vs bs (f p))"
  by(induct ps, simp_all add: evaldjf_def djf_Or)

lemma evaldjf_bound0: 
  assumes nb: "\<forall> x\<in> set xs. bound0 (f x)"
  shows "bound0 (evaldjf f xs)"
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 

lemma evaldjf_qf: 
  assumes nb: "\<forall> x\<in> set xs. qfree (f x)"
  shows "qfree (evaldjf f xs)"
  using nb by (induct xs, auto simp add: evaldjf_def djf_def Let_def) (case_tac "f a", auto) 

consts disjuncts :: "fm \<Rightarrow> fm list"
recdef disjuncts "measure size"
  "disjuncts (Or p q) = (disjuncts p) @ (disjuncts q)"
  "disjuncts F = []"
  "disjuncts p = [p]"

lemma disjuncts: "(\<exists> q\<in> set (disjuncts p). Ifm vs bs q) = Ifm vs bs p"
by(induct p rule: disjuncts.induct, auto)

lemma disjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). bound0 q"
proof-
  assume nb: "bound0 p"
  hence "list_all bound0 (disjuncts p)" by (induct p rule:disjuncts.induct,auto)
  thus ?thesis by (simp only: list_all_iff)
qed

lemma disjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (disjuncts p). qfree q"
proof-
  assume qf: "qfree p"
  hence "list_all qfree (disjuncts p)"
    by (induct p rule: disjuncts.induct, auto)
  thus ?thesis by (simp only: list_all_iff)
qed

constdefs DJ :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
  "DJ f p \<equiv> evaldjf f (disjuncts p)"

lemma DJ: assumes fdj: "\<forall> p q. Ifm vs bs (f (Or p q)) = Ifm vs bs (Or (f p) (f q))"
  and fF: "f F = F"
  shows "Ifm vs bs (DJ f p) = Ifm vs bs (f p)"
proof-
  have "Ifm vs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm vs bs (f q))"
    by (simp add: DJ_def evaldjf_ex) 
  also have "\<dots> = Ifm vs bs (f p)" using fdj fF by (induct p rule: disjuncts.induct, auto)
  finally show ?thesis .
qed

lemma DJ_qf: assumes 
  fqf: "\<forall> p. qfree p \<longrightarrow> qfree (f p)"
  shows "\<forall>p. qfree p \<longrightarrow> qfree (DJ f p) "
proof(clarify)
  fix  p assume qf: "qfree p"
  have th: "DJ f p = evaldjf f (disjuncts p)" by (simp add: DJ_def)
  from disjuncts_qf[OF qf] have "\<forall> q\<in> set (disjuncts p). qfree q" .
  with fqf have th':"\<forall> q\<in> set (disjuncts p). qfree (f q)" by blast
  
  from evaldjf_qf[OF th'] th show "qfree (DJ f p)" by simp
qed

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))"
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm vs bs ((DJ qe p)) = Ifm vs bs (E p))"
proof(clarify)
  fix p::fm and bs
  assume qf: "qfree p"
  from qe have qth: "\<forall> p. qfree p \<longrightarrow> qfree (qe p)" by blast
  from DJ_qf[OF qth] qf have qfth:"qfree (DJ qe p)" by auto
  have "Ifm vs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm vs bs (qe q))"
    by (simp add: DJ_def evaldjf_ex)
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm vs bs (E q))" using qe disjuncts_qf[OF qf] by auto
  also have "\<dots> = Ifm vs bs (E p)" by (induct p rule: disjuncts.induct, auto)
  finally show "qfree (DJ qe p) \<and> Ifm vs bs (DJ qe p) = Ifm vs bs (E p)" using qfth by blast
qed

consts conjuncts :: "fm \<Rightarrow> fm list"

recdef conjuncts "measure size"
  "conjuncts (And p q) = (conjuncts p) @ (conjuncts q)"
  "conjuncts T = []"
  "conjuncts p = [p]"

constdefs list_conj :: "fm list \<Rightarrow> fm"
  "list_conj ps \<equiv> foldr conj ps T"

constdefs CJNB:: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
  "CJNB f p \<equiv> (let cjs = conjuncts p ; (yes,no) = partition bound0 cjs
                   in conj (decr0 (list_conj yes)) (f (list_conj no)))"

lemma conjuncts_qf: "qfree p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). qfree q"
proof-
  assume qf: "qfree p"
  hence "list_all qfree (conjuncts p)"
    by (induct p rule: conjuncts.induct, auto)
  thus ?thesis by (simp only: list_all_iff)
qed

lemma conjuncts: "(\<forall> q\<in> set (conjuncts p). Ifm vs bs q) = Ifm vs bs p"
by(induct p rule: conjuncts.induct, auto)

lemma conjuncts_nb: "bound0 p \<Longrightarrow> \<forall> q\<in> set (conjuncts p). bound0 q"
proof-
  assume nb: "bound0 p"
  hence "list_all bound0 (conjuncts p)" by (induct p rule:conjuncts.induct,auto)
  thus ?thesis by (simp only: list_all_iff)
qed

fun islin :: "fm \<Rightarrow> bool" where
  "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)"
| "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)"
| "islin (Eq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
| "islin (NEq (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
| "islin (Lt (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
| "islin (Le (CNP 0 c s)) = (isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s)"
| "islin (NOT p) = False"
| "islin (Imp p q) = False"
| "islin (Iff p q) = False"
| "islin p = bound0 p"

lemma islin_stupid: assumes nb: "tmbound0 p"
  shows "islin (Lt p)" and "islin (Le p)" and "islin (Eq p)" and "islin (NEq p)"
  using nb by (cases p, auto, case_tac nat, auto)+

definition "lt p = (case p of CP (C c) \<Rightarrow> if 0>\<^sub>N c then T else F| _ \<Rightarrow> Lt p)"
definition "le p = (case p of CP (C c) \<Rightarrow> if 0\<ge>\<^sub>N c then T else F | _ \<Rightarrow> Le p)"
definition "eq p = (case p of CP (C c) \<Rightarrow> if c = 0\<^sub>N then T else F | _ \<Rightarrow> Eq p)"
definition "neq p = not (eq p)"

lemma lt: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (lt p) = Ifm vs bs (Lt p)"
  apply(simp add: lt_def)
  apply(cases p, simp_all)
  apply (case_tac poly, simp_all add: isnpoly_def)
  done

lemma le: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (le p) = Ifm vs bs (Le p)"
  apply(simp add: le_def)
  apply(cases p, simp_all)
  apply (case_tac poly, simp_all add: isnpoly_def)
  done

lemma eq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (eq p) = Ifm vs bs (Eq p)"
  apply(simp add: eq_def)
  apply(cases p, simp_all)
  apply (case_tac poly, simp_all add: isnpoly_def)
  done

lemma neq: "allpolys isnpoly p \<Longrightarrow> Ifm vs bs (neq p) = Ifm vs bs (NEq p)"
  by(simp add: neq_def eq)

lemma lt_lin: "tmbound0 p \<Longrightarrow> islin (lt p)"
  apply (simp add: lt_def)
  apply (cases p, simp_all)
  apply (case_tac poly, simp_all)
  apply (case_tac nat, simp_all)
  done

lemma le_lin: "tmbound0 p \<Longrightarrow> islin (le p)"
  apply (simp add: le_def)
  apply (cases p, simp_all)
  apply (case_tac poly, simp_all)
  apply (case_tac nat, simp_all)
  done

lemma eq_lin: "tmbound0 p \<Longrightarrow> islin (eq p)"
  apply (simp add: eq_def)
  apply (cases p, simp_all)
  apply (case_tac poly, simp_all)
  apply (case_tac nat, simp_all)
  done

lemma neq_lin: "tmbound0 p \<Longrightarrow> islin (neq p)"
  apply (simp add: neq_def eq_def)
  apply (cases p, simp_all)
  apply (case_tac poly, simp_all)
  apply (case_tac nat, simp_all)
  done

definition "simplt t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then lt s else Lt (CNP 0 c s))"
definition "simple t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then le s else Le (CNP 0 c s))"
definition "simpeq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then eq s else Eq (CNP 0 c s))"
definition "simpneq t = (let (c,s) = split0 (simptm t) in if c= 0\<^sub>p then neq s else NEq (CNP 0 c s))"

lemma simplt_islin[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "islin (simplt t)"
  unfolding simplt_def 
  using split0_nb0'
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])
  
lemma simple_islin[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "islin (simple t)"
  unfolding simple_def 
  using split0_nb0'
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)
lemma simpeq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "islin (simpeq t)"
  unfolding simpeq_def 
  using split0_nb0'
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)

lemma simpneq_islin[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "islin (simpneq t)"
  unfolding simpneq_def 
  using split0_nb0'
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)

lemma really_stupid: "\<not> (\<forall>c1 s'. (c1, s') \<noteq> split0 s)"
  by (cases "split0 s", auto)
lemma split0_npoly:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  and n: "allpolys isnpoly t"
  shows "isnpoly (fst (split0 t))" and "allpolys isnpoly (snd (split0 t))"
  using n
  by (induct t rule: split0.induct, auto simp add: Let_def split_def polyadd_norm polymul_norm polyneg_norm polysub_norm really_stupid)
lemma simplt[simp]:
  shows "Ifm vs bs (simplt t) = Ifm vs bs (Lt t)"
proof-
  have n: "allpolys isnpoly (simptm t)" by simp
  let ?t = "simptm t"
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
      using split0[of "simptm t" vs bs] lt[OF split0_npoly(2)[OF n], of vs bs]
      by (simp add: simplt_def Let_def split_def lt)}
  moreover
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simplt_def Let_def split_def)
  }
  ultimately show ?thesis by blast
qed

lemma simple[simp]:
  shows "Ifm vs bs (simple t) = Ifm vs bs (Le t)"
proof-
  have n: "allpolys isnpoly (simptm t)" by simp
  let ?t = "simptm t"
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
      using split0[of "simptm t" vs bs] le[OF split0_npoly(2)[OF n], of vs bs]
      by (simp add: simple_def Let_def split_def le)}
  moreover
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simple_def Let_def split_def)
  }
  ultimately show ?thesis by blast
qed

lemma simpeq[simp]:
  shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
proof-
  have n: "allpolys isnpoly (simptm t)" by simp
  let ?t = "simptm t"
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
      using split0[of "simptm t" vs bs] eq[OF split0_npoly(2)[OF n], of vs bs]
      by (simp add: simpeq_def Let_def split_def)}
  moreover
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpeq_def Let_def split_def)
  }
  ultimately show ?thesis by blast
qed

lemma simpneq[simp]:
  shows "Ifm vs bs (simpneq t) = Ifm vs bs (NEq t)"
proof-
  have n: "allpolys isnpoly (simptm t)" by simp
  let ?t = "simptm t"
  {assume "fst (split0 ?t) = 0\<^sub>p" hence ?thesis
      using split0[of "simptm t" vs bs] neq[OF split0_npoly(2)[OF n], of vs bs]
      by (simp add: simpneq_def Let_def split_def )}
  moreover
  {assume "fst (split0 ?t) \<noteq> 0\<^sub>p"
    hence ?thesis using  split0[of "simptm t" vs bs] by (simp add: simpneq_def Let_def split_def)
  }
  ultimately show ?thesis by blast
qed

lemma lt_nb: "tmbound0 t \<Longrightarrow> bound0 (lt t)"
  apply (simp add: lt_def)
  apply (cases t, auto)
  apply (case_tac poly, auto)
  done

lemma le_nb: "tmbound0 t \<Longrightarrow> bound0 (le t)"
  apply (simp add: le_def)
  apply (cases t, auto)
  apply (case_tac poly, auto)
  done

lemma eq_nb: "tmbound0 t \<Longrightarrow> bound0 (eq t)"
  apply (simp add: eq_def)
  apply (cases t, auto)
  apply (case_tac poly, auto)
  done

lemma neq_nb: "tmbound0 t \<Longrightarrow> bound0 (neq t)"
  apply (simp add: neq_def eq_def)
  apply (cases t, auto)
  apply (case_tac poly, auto)
  done

lemma simplt_nb[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "tmbound0 t \<Longrightarrow> bound0 (simplt t)"
  using split0 [of "simptm t" vs bs]
proof(simp add: simplt_def Let_def split_def)
  assume nb: "tmbound0 t"
  hence nb': "tmbound0 (simptm t)" by simp
  let ?c = "fst (split0 (simptm t))"
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  from iffD1[OF isnpolyh_unique[OF ths] th]
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (lt (snd (split0 (simptm t))))) \<and>
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def lt_nb)
qed

lemma simple_nb[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "tmbound0 t \<Longrightarrow> bound0 (simple t)"
  using split0 [of "simptm t" vs bs]
proof(simp add: simple_def Let_def split_def)
  assume nb: "tmbound0 t"
  hence nb': "tmbound0 (simptm t)" by simp
  let ?c = "fst (split0 (simptm t))"
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  from iffD1[OF isnpolyh_unique[OF ths] th]
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (le (snd (split0 (simptm t))))) \<and>
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simplt_def Let_def split_def le_nb)
qed

lemma simpeq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "tmbound0 t \<Longrightarrow> bound0 (simpeq t)"
  using split0 [of "simptm t" vs bs]
proof(simp add: simpeq_def Let_def split_def)
  assume nb: "tmbound0 t"
  hence nb': "tmbound0 (simptm t)" by simp
  let ?c = "fst (split0 (simptm t))"
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  from iffD1[OF isnpolyh_unique[OF ths] th]
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (eq (snd (split0 (simptm t))))) \<and>
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpeq_def Let_def split_def eq_nb)
qed

lemma simpneq_nb[simp]:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "tmbound0 t \<Longrightarrow> bound0 (simpneq t)"
  using split0 [of "simptm t" vs bs]
proof(simp add: simpneq_def Let_def split_def)
  assume nb: "tmbound0 t"
  hence nb': "tmbound0 (simptm t)" by simp
  let ?c = "fst (split0 (simptm t))"
  from tmbound_split0[OF nb'[unfolded tmbound0_tmbound_iff[symmetric]]]
  have th: "\<forall>bs. Ipoly bs ?c = Ipoly bs 0\<^sub>p" by auto
  from isnpoly_fst_split0[OF simptm_allpolys_npoly[of t]]
  have ths: "isnpolyh ?c 0" "isnpolyh 0\<^sub>p 0" by (simp_all add: isnpoly_def)
  from iffD1[OF isnpolyh_unique[OF ths] th]
  have "fst (split0 (simptm t)) = 0\<^sub>p" . 
  thus "(fst (split0 (simptm t)) = 0\<^sub>p \<longrightarrow> bound0 (neq (snd (split0 (simptm t))))) \<and>
       fst (split0 (simptm t)) = 0\<^sub>p" by (simp add: simpneq_def Let_def split_def neq_nb)
qed

consts conjs   :: "fm \<Rightarrow> fm list"
recdef conjs "measure size"
  "conjs (And p q) = (conjs p)@(conjs q)"
  "conjs T = []"
  "conjs p = [p]"
lemma conjs_ci: "(\<forall> q \<in> set (conjs p). Ifm vs bs q) = Ifm vs bs p"
by (induct p rule: conjs.induct, auto)
constdefs list_disj :: "fm list \<Rightarrow> fm"
  "list_disj ps \<equiv> foldr disj ps F"

lemma list_conj: "Ifm vs bs (list_conj ps) = (\<forall>p\<in> set ps. Ifm vs bs p)"
  by (induct ps, auto simp add: list_conj_def)
lemma list_conj_qf: " \<forall>p\<in> set ps. qfree p \<Longrightarrow> qfree (list_conj ps)"
  by (induct ps, auto simp add: list_conj_def conj_qf)
lemma list_disj: "Ifm vs bs (list_disj ps) = (\<exists>p\<in> set ps. Ifm vs bs p)"
  by (induct ps, auto simp add: list_disj_def)

lemma conj_boundslt: "boundslt n p \<Longrightarrow> boundslt n q \<Longrightarrow> boundslt n (conj p q)"
  unfolding conj_def by auto

lemma conjs_nb: "bound n p \<Longrightarrow> \<forall>q\<in> set (conjs p). bound n q"
  apply (induct p rule: conjs.induct) 
  apply (unfold conjs.simps)
  apply (unfold set_append)
  apply (unfold ball_Un)
  apply (unfold bound.simps)
  apply auto
  done

lemma conjs_boundslt: "boundslt n p \<Longrightarrow> \<forall>q\<in> set (conjs p). boundslt n q"
  apply (induct p rule: conjs.induct) 
  apply (unfold conjs.simps)
  apply (unfold set_append)
  apply (unfold ball_Un)
  apply (unfold boundslt.simps)
  apply blast
by simp_all

lemma list_conj_boundslt: " \<forall>p\<in> set ps. boundslt n p \<Longrightarrow> boundslt n (list_conj ps)"
  unfolding list_conj_def
  by (induct ps, auto simp add: conj_boundslt)

lemma list_conj_nb: assumes bnd: "\<forall>p\<in> set ps. bound n p"
  shows "bound n (list_conj ps)"
  using bnd
  unfolding list_conj_def
  by (induct ps, auto simp add: conj_nb)

lemma list_conj_nb': "\<forall>p\<in>set ps. bound0 p \<Longrightarrow> bound0 (list_conj ps)"
unfolding list_conj_def by (induct ps , auto)

lemma CJNB_qe: 
  assumes qe: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (CJNB qe p) \<and> (Ifm vs bs ((CJNB qe p)) = Ifm vs bs (E p))"
proof(clarify)
  fix bs p
  assume qfp: "qfree p"
  let ?cjs = "conjuncts p"
  let ?yes = "fst (partition bound0 ?cjs)"
  let ?no = "snd (partition bound0 ?cjs)"
  let ?cno = "list_conj ?no"
  let ?cyes = "list_conj ?yes"
  have part: "partition bound0 ?cjs = (?yes,?no)" by simp
  from partition_P[OF part] have "\<forall> q\<in> set ?yes. bound0 q" by blast 
  hence yes_nb: "bound0 ?cyes" by (simp add: list_conj_nb') 
  hence yes_qf: "qfree (decr0 ?cyes )" by (simp add: decr0_qf)
  from conjuncts_qf[OF qfp] partition_set[OF part] 
  have " \<forall>q\<in> set ?no. qfree q" by auto
  hence no_qf: "qfree ?cno"by (simp add: list_conj_qf)
  with qe have cno_qf:"qfree (qe ?cno )" 
    and noE: "Ifm vs bs (qe ?cno) = Ifm vs bs (E ?cno)" by blast+
  from cno_qf yes_qf have qf: "qfree (CJNB qe p)" 
    by (simp add: CJNB_def Let_def conj_qf split_def)
  {fix bs
    from conjuncts have "Ifm vs bs p = (\<forall>q\<in> set ?cjs. Ifm vs bs q)" by blast
    also have "\<dots> = ((\<forall>q\<in> set ?yes. Ifm vs bs q) \<and> (\<forall>q\<in> set ?no. Ifm vs bs q))"
      using partition_set[OF part] by auto
    finally have "Ifm vs bs p = ((Ifm vs bs ?cyes) \<and> (Ifm vs bs ?cno))" using list_conj[of vs bs] by simp}
  hence "Ifm vs bs (E p) = (\<exists>x. (Ifm vs (x#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))" by simp
  also have "\<dots> = (\<exists>x. (Ifm vs (y#bs) ?cyes) \<and> (Ifm vs (x#bs) ?cno))"
    using bound0_I[OF yes_nb, where bs="bs" and b'="y"] by blast
  also have "\<dots> = (Ifm vs bs (decr0 ?cyes) \<and> Ifm vs bs (E ?cno))"
    by (auto simp add: decr0[OF yes_nb])
  also have "\<dots> = (Ifm vs bs (conj (decr0 ?cyes) (qe ?cno)))"
    using qe[rule_format, OF no_qf] by auto
  finally have "Ifm vs bs (E p) = Ifm vs bs (CJNB qe p)" 
    by (simp add: Let_def CJNB_def split_def)
  with qf show "qfree (CJNB qe p) \<and> Ifm vs bs (CJNB qe p) = Ifm vs bs (E p)" by blast
qed

consts simpfm :: "fm \<Rightarrow> fm"
recdef simpfm "measure fmsize"
  "simpfm (Lt t) = simplt (simptm t)"
  "simpfm (Le t) = simple (simptm t)"
  "simpfm (Eq t) = simpeq(simptm t)"
  "simpfm (NEq t) = simpneq(simptm t)"
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
  "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
  "simpfm (Imp p q) = disj (simpfm (NOT p)) (simpfm q)"
  "simpfm (Iff p q) = disj (conj (simpfm p) (simpfm q)) (conj (simpfm (NOT p)) (simpfm (NOT q)))"
  "simpfm (NOT (And p q)) = disj (simpfm (NOT p)) (simpfm (NOT q))"
  "simpfm (NOT (Or p q)) = conj (simpfm (NOT p)) (simpfm (NOT q))"
  "simpfm (NOT (Imp p q)) = conj (simpfm p) (simpfm (NOT q))"
  "simpfm (NOT (Iff p q)) = disj (conj (simpfm p) (simpfm (NOT q))) (conj (simpfm (NOT p)) (simpfm q))"
  "simpfm (NOT (Eq t)) = simpneq t"
  "simpfm (NOT (NEq t)) = simpeq t"
  "simpfm (NOT (Le t)) = simplt (Neg t)"
  "simpfm (NOT (Lt t)) = simple (Neg t)"
  "simpfm (NOT (NOT p)) = simpfm p"
  "simpfm (NOT T) = F"
  "simpfm (NOT F) = T"
  "simpfm p = p"

lemma simpfm[simp]: "Ifm vs bs (simpfm p) = Ifm vs bs p"
by(induct p arbitrary: bs rule: simpfm.induct, auto)

lemma simpfm_bound0:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "bound0 p \<Longrightarrow> bound0 (simpfm p)"
by (induct p rule: simpfm.induct, auto)

lemma lt_qf[simp]: "qfree (lt t)"
  apply (cases t, auto simp add: lt_def)
  by (case_tac poly, auto)

lemma le_qf[simp]: "qfree (le t)"
  apply (cases t, auto simp add: le_def)
  by (case_tac poly, auto)

lemma eq_qf[simp]: "qfree (eq t)"
  apply (cases t, auto simp add: eq_def)
  by (case_tac poly, auto)

lemma neq_qf[simp]: "qfree (neq t)" by (simp add: neq_def)

lemma simplt_qf[simp]: "qfree (simplt t)" by (simp add: simplt_def Let_def split_def)
lemma simple_qf[simp]: "qfree (simple t)" by (simp add: simple_def Let_def split_def)
lemma simpeq_qf[simp]: "qfree (simpeq t)" by (simp add: simpeq_def Let_def split_def)
lemma simpneq_qf[simp]: "qfree (simpneq t)" by (simp add: simpneq_def Let_def split_def)

lemma simpfm_qf[simp]: "qfree p \<Longrightarrow> qfree (simpfm p)"
by (induct p rule: simpfm.induct, auto simp add: disj_qf imp_qf iff_qf conj_qf not_qf Let_def)

lemma disj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (disj p q)" by (simp add: disj_def)
lemma conj_lin: "islin p \<Longrightarrow> islin q \<Longrightarrow> islin (conj p q)" by (simp add: conj_def)

lemma   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "qfree p \<Longrightarrow> islin (simpfm p)" 
  apply (induct p rule: simpfm.induct)
  apply (simp_all add: conj_lin disj_lin)
  done

consts prep :: "fm \<Rightarrow> fm"
recdef prep "measure fmsize"
  "prep (E T) = T"
  "prep (E F) = F"
  "prep (E (Or p q)) = disj (prep (E p)) (prep (E q))"
  "prep (E (Imp p q)) = disj (prep (E (NOT p))) (prep (E q))"
  "prep (E (Iff p q)) = disj (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
  "prep (E (NOT (And p q))) = disj (prep (E (NOT p))) (prep (E(NOT q)))"
  "prep (E (NOT (Imp p q))) = prep (E (And p (NOT q)))"
  "prep (E (NOT (Iff p q))) = disj (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
  "prep (E p) = E (prep p)"
  "prep (A (And p q)) = conj (prep (A p)) (prep (A q))"
  "prep (A p) = prep (NOT (E (NOT p)))"
  "prep (NOT (NOT p)) = prep p"
  "prep (NOT (And p q)) = disj (prep (NOT p)) (prep (NOT q))"
  "prep (NOT (A p)) = prep (E (NOT p))"
  "prep (NOT (Or p q)) = conj (prep (NOT p)) (prep (NOT q))"
  "prep (NOT (Imp p q)) = conj (prep p) (prep (NOT q))"
  "prep (NOT (Iff p q)) = disj (prep (And p (NOT q))) (prep (And (NOT p) q))"
  "prep (NOT p) = not (prep p)"
  "prep (Or p q) = disj (prep p) (prep q)"
  "prep (And p q) = conj (prep p) (prep q)"
  "prep (Imp p q) = prep (Or (NOT p) q)"
  "prep (Iff p q) = disj (prep (And p q)) (prep (And (NOT p) (NOT q)))"
  "prep p = p"
(hints simp add: fmsize_pos)
lemma prep: "Ifm vs bs (prep p) = Ifm vs bs p"
by (induct p arbitrary: bs rule: prep.induct, auto)



  (* Generic quantifier elimination *)
consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
recdef qelim "measure fmsize"
  "qelim (E p) = (\<lambda> qe. DJ (CJNB qe) (qelim p qe))"
  "qelim (A p) = (\<lambda> qe. not (qe ((qelim (NOT p) qe))))"
  "qelim (NOT p) = (\<lambda> qe. not (qelim p qe))"
  "qelim (And p q) = (\<lambda> qe. conj (qelim p qe) (qelim q qe))" 
  "qelim (Or  p q) = (\<lambda> qe. disj (qelim p qe) (qelim q qe))" 
  "qelim (Imp p q) = (\<lambda> qe. imp (qelim p qe) (qelim q qe))"
  "qelim (Iff p q) = (\<lambda> qe. iff (qelim p qe) (qelim q qe))"
  "qelim p = (\<lambda> y. simpfm p)"


lemma qelim:
  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm vs bs (qe p) = Ifm vs bs (E p))"
  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm vs bs (qelim p qe) = Ifm vs bs p)"
using qe_inv DJ_qe[OF CJNB_qe[OF qe_inv]]
by (induct p rule: qelim.induct) auto

subsection{* Core Procedure *}

consts 
  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
recdef minusinf "measure size"
  "minusinf (And p q) = conj (minusinf p) (minusinf q)" 
  "minusinf (Or p q) = disj (minusinf p) (minusinf q)" 
  "minusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  "minusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  "minusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP (~\<^sub>p c)))"
  "minusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP (~\<^sub>p c)))"
  "minusinf p = p"

recdef plusinf "measure size"
  "plusinf (And p q) = conj (plusinf p) (plusinf q)" 
  "plusinf (Or p q) = disj (plusinf p) (plusinf q)" 
  "plusinf (Eq  (CNP 0 c e)) = conj (eq (CP c)) (eq e)"
  "plusinf (NEq (CNP 0 c e)) = disj (not (eq e)) (not (eq (CP c)))"
  "plusinf (Lt  (CNP 0 c e)) = disj (conj (eq (CP c)) (lt e)) (lt (CP c))"
  "plusinf (Le  (CNP 0 c e)) = disj (conj (eq (CP c)) (le e)) (lt (CP c))"
  "plusinf p = p"

lemma minusinf_inf: assumes lp:"islin p"
  shows "\<exists>z. \<forall>x < z. Ifm vs (x#bs) (minusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  using lp
proof (induct p rule: minusinf.induct)
  case 1 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
next
  case 2 thus ?case by (auto,rule_tac x="min z za" in exI, auto)
next
  case (3 c e) hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case 
      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
	using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
	using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Eq (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (4 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
	using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
	using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (NEq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (5 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] lt[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
	using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
	using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Lt (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (6 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  note eqs = lt[OF nc', where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x < - ?e"
	using pos_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x < -?e / ?c" hence "?c * x > - ?e"
	using neg_less_divide_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (minusinf (Le (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  ultimately show ?case by blast
qed (auto)

lemma plusinf_inf: assumes lp:"islin p"
  shows "\<exists>z. \<forall>x > z. Ifm vs (x#bs) (plusinf p) \<longleftrightarrow> Ifm vs (x#bs) p"
  using lp
proof (induct p rule: plusinf.induct)
  case 1 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
next
  case 2 thus ?case by (auto,rule_tac x="max z za" in exI, auto)
next
  case (3 c e) hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case 
      using eq[OF nc(2), of vs] eq[OF nc(1), of vs] by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e" 
	using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x" and vs=vs and bs=bs] by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
	using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Eq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Eq (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] eqs by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (4 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  note eqs = eq[OF nc(1), where ?'a = 'a] eq[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
	using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
	using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (NEq (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (NEq (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (5 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  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]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
	using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
	using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Lt (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Lt (CNP 0 c e)))"
	using eqs tmbound0_I[OF nbe, where b="y" and b'="x"] cp by auto} hence ?case by auto}
  ultimately show ?case by blast
next
  case (6 c e)  hence nbe: "tmbound0 e" by simp
  from prems have nc: "allpolys isnpoly (CP c)" "allpolys isnpoly e" by simp_all
  hence nc': "allpolys isnpoly (CP (~\<^sub>p c))" by (simp add: polyneg_norm)
  note eqs = lt[OF nc(1), where ?'a = 'a] eq [OF nc(1), where ?'a = 'a] le[OF nc(2), where ?'a = 'a]
  let ?c = "Ipoly vs c"
  let ?e = "Itm vs (y#bs) e"
  have "?c=0 \<or> ?c > 0 \<or> ?c < 0" by arith
  moreover {assume "?c = 0" hence ?case using eqs by auto}
  moreover {assume cp: "?c > 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x > - ?e"
	using pos_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e > 0" by simp
      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  moreover {assume cp: "?c < 0"
    {fix x assume xz: "x > -?e / ?c" hence "?c * x < - ?e"
	using neg_divide_less_eq[OF cp, where a="x" and b="-?e"] by (simp add: mult_commute)
      hence "?c * x + ?e < 0" by simp
      hence "Ifm vs (x#bs) (Le (CNP 0 c e)) = Ifm vs (x#bs) (plusinf (Le (CNP 0 c e)))"
	using tmbound0_I[OF nbe, where b="y" and b'="x"] cp eqs by auto} hence ?case by auto}
  ultimately show ?case by blast
qed (auto)

lemma minusinf_nb: "islin p \<Longrightarrow> bound0 (minusinf p)" 
  by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)
lemma plusinf_nb: "islin p \<Longrightarrow> bound0 (plusinf p)" 
  by (induct p rule: minusinf.induct, auto simp add: eq_nb lt_nb le_nb)

lemma minusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (minusinf p)"
  shows "\<exists>x. Ifm vs (x#bs) p"
proof-
  from bound0_I [OF minusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  have th: "\<forall> x. Ifm vs (x#bs) (minusinf p)" by auto
  from minusinf_inf[OF lp, where bs="bs"] 
  obtain z where z_def: "\<forall>x<z. Ifm vs (x # bs) (minusinf p) = Ifm vs (x # bs) p" by blast
  from th have "Ifm vs ((z - 1)#bs) (minusinf p)" by simp
  moreover have "z - 1 < z" by simp
  ultimately show ?thesis using z_def by auto
qed

lemma plusinf_ex: assumes lp: "islin p" and ex: "Ifm vs (x#bs) (plusinf p)"
  shows "\<exists>x. Ifm vs (x#bs) p"
proof-
  from bound0_I [OF plusinf_nb[OF lp], where b="a" and bs ="bs"] ex
  have th: "\<forall> x. Ifm vs (x#bs) (plusinf p)" by auto
  from plusinf_inf[OF lp, where bs="bs"] 
  obtain z where z_def: "\<forall>x>z. Ifm vs (x # bs) (plusinf p) = Ifm vs (x # bs) p" by blast
  from th have "Ifm vs ((z + 1)#bs) (plusinf p)" by simp
  moreover have "z + 1 > z" by simp
  ultimately show ?thesis using z_def by auto
qed

fun uset :: "fm \<Rightarrow> (poly \<times> tm) list" where
  "uset (And p q) = uset p @ uset q"
| "uset (Or p q) = uset p @ uset q"
| "uset (Eq (CNP 0 a e))  = [(a,e)]"
| "uset (Le (CNP 0 a e))  = [(a,e)]"
| "uset (Lt (CNP 0 a e))  = [(a,e)]"
| "uset (NEq (CNP 0 a e)) = [(a,e)]"
| "uset p = []"

lemma uset_l:
  assumes lp: "islin p"
  shows "\<forall> (c,s) \<in> set (uset p). isnpoly c \<and> c \<noteq> 0\<^sub>p \<and> tmbound0 s \<and> allpolys isnpoly s"
using lp by(induct p rule: uset.induct,auto)

lemma minusinf_uset0:
  assumes lp: "islin p"
  and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))"
  and ex: "Ifm vs (x#bs) p" (is "?I x p")
  shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (x#bs) s / Ipoly vs c" 
proof-
  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)" 
    using lp nmi ex
    apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
    apply (auto simp add: linorder_not_less order_le_less)
    done 
  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
  hence "x \<ge> (- Itm vs (x#bs) s) / Ipoly vs c"
    using divide_le_eq[of "- Itm vs (x#bs) s" "Ipoly vs c" x]
    by (auto simp add: mult_commute del: divide_minus_left)
  thus ?thesis using csU by auto
qed

lemma minusinf_uset:
  assumes lp: "islin p"
  and nmi: "\<not> (Ifm vs (a#bs) (minusinf p))"
  and ex: "Ifm vs (x#bs) p" (is "?I x p")
  shows "\<exists> (c,s) \<in> set (uset p). x \<ge> - Itm vs (a#bs) s / Ipoly vs c" 
proof-
  from nmi have nmi': "\<not> (Ifm vs (x#bs) (minusinf p))" 
    by (simp add: bound0_I[OF minusinf_nb[OF lp], where b=x and b'=a])
  from minusinf_uset0[OF lp nmi' ex] 
  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
  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
qed


lemma plusinf_uset0:
  assumes lp: "islin p"
  and nmi: "\<not> (Ifm vs (x#bs) (plusinf p))"
  and ex: "Ifm vs (x#bs) p" (is "?I x p")
  shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (x#bs) s / Ipoly vs c" 
proof-
  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)" 
    using lp nmi ex
    apply (induct p rule: minusinf.induct, auto simp add: eq le lt nth_pos2 polyneg_norm)
    apply (auto simp add: linorder_not_less order_le_less)
    done 
  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
  hence "x \<le> (- Itm vs (x#bs) s) / Ipoly vs c"
    using le_divide_eq[of x "- Itm vs (x#bs) s" "Ipoly vs c"]
    by (auto simp add: mult_commute del: divide_minus_left)
  thus ?thesis using csU by auto
qed

lemma plusinf_uset:
  assumes lp: "islin p"
  and nmi: "\<not> (Ifm vs (a#bs) (plusinf p))"
  and ex: "Ifm vs (x#bs) p" (is "?I x p")
  shows "\<exists> (c,s) \<in> set (uset p). x \<le> - Itm vs (a#bs) s / Ipoly vs c" 
proof-
  from nmi have nmi': "\<not> (Ifm vs (x#bs) (plusinf p))" 
    by (simp add: bound0_I[OF plusinf_nb[OF lp], where b=x and b'=a])
  from plusinf_uset0[OF lp nmi' ex] 
  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
  from uset_l[OF lp, rule_format, OF csU] have nb: "tmbound0 s" by simp
  from th tmbound0_I[OF nb, of vs x bs a] csU show ?thesis by auto
qed

lemma lin_dense: 
  assumes lp: "islin p"
  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)" 
  (is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (c,t). - ?Nt x t / ?N c) ` ?U p")
  and lx: "l < x" and xu:"x < u" and px:" Ifm vs (x#bs) p"
  and ly: "l < y" and yu: "y < u"
  shows "Ifm vs (y#bs) p"
using lp px noS
proof (induct p rule: islin.induct) 
  case (5 c s)
  from "5.prems" 
  have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
    and px: "Ifm vs (x # bs) (Lt (CNP 0 c s))"
    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
  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  moreover
  {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"])}
  moreover
  {assume c: "?N c > 0"
      from px pos_less_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
      have px': "x < - ?Nt x s / ?N c" 
	by (auto simp add: not_less ring_simps) 
    {assume y: "y < - ?Nt x s / ?N c" 
      hence "y * ?N c < - ?Nt x s"
	by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
      hence "?N c * y + ?Nt x s < 0" by (simp add: ring_simps)
      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
    moreover
    {assume y: "y > -?Nt x s / ?N c" 
      with yu have eu: "u > - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
      with lx px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  moreover
  {assume c: "?N c < 0"
      from px neg_divide_less_eq[OF c, where a="x" and b="-?Nt x s"]  
      have px': "x > - ?Nt x s / ?N c" 
	by (auto simp add: not_less ring_simps) 
    {assume y: "y > - ?Nt x s / ?N c" 
      hence "y * ?N c < - ?Nt x s"
	by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
      hence "?N c * y + ?Nt x s < 0" by (simp add: ring_simps)
      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
    moreover
    {assume y: "y < -?Nt x s / ?N c" 
      with ly have eu: "l < - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
      with xu px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  ultimately show ?case by blast
next
  case (6 c s)
  from "6.prems" 
  have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
    and px: "Ifm vs (x # bs) (Le (CNP 0 c s))"
    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
  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  moreover
  {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"])}
  moreover
  {assume c: "?N c > 0"
      from px pos_le_divide_eq[OF c, where a="x" and b="-?Nt x s"]  
      have px': "x <= - ?Nt x s / ?N c" by (simp add: not_less ring_simps) 
    {assume y: "y < - ?Nt x s / ?N c" 
      hence "y * ?N c < - ?Nt x s"
	by (simp add: pos_less_divide_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
      hence "?N c * y + ?Nt x s < 0" by (simp add: ring_simps)
      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
    moreover
    {assume y: "y > -?Nt x s / ?N c" 
      with yu have eu: "u > - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
      with lx px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  moreover
  {assume c: "?N c < 0"
      from px neg_divide_le_eq[OF c, where a="x" and b="-?Nt x s"]  
      have px': "x >= - ?Nt x s / ?N c" by (simp add: ring_simps) 
    {assume y: "y > - ?Nt x s / ?N c" 
      hence "y * ?N c < - ?Nt x s"
	by (simp add: neg_divide_less_eq[OF c, where a="y" and b="-?Nt x s", symmetric])
      hence "?N c * y + ?Nt x s < 0" by (simp add: ring_simps)
      hence ?case using tmbound0_I[OF lin(3), where bs="bs" and b="x" and b'="y"] by simp}
    moreover
    {assume y: "y < -?Nt x s / ?N c" 
      with ly have eu: "l < - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
      with xu px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  ultimately show ?case by blast
next
    case (3 c s)
  from "3.prems" 
  have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
    and px: "Ifm vs (x # bs) (Eq (CNP 0 c s))"
    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
  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  have ccs: "?N c = 0 \<or> ?N c < 0 \<or> ?N c > 0" by dlo
  moreover
  {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"])}
  moreover
  {assume c: "?N c > 0" hence cnz: "?N c \<noteq> 0" by simp
    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
    have px': "x = - ?Nt x s / ?N c" by (simp add: ring_simps)
    {assume y: "y < -?Nt x s / ?N c" 
      with ly have eu: "l < - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
      with xu px' have "False" by simp  hence ?case by simp }
    moreover
    {assume y: "y > -?Nt x s / ?N c" 
      with yu have eu: "u > - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
      with lx px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  moreover
  {assume c: "?N c < 0" hence cnz: "?N c \<noteq> 0" by simp
    from px eq_divide_eq[of "x" "-?Nt x s" "?N c"]  cnz
    have px': "x = - ?Nt x s / ?N c" by (simp add: ring_simps)
    {assume y: "y < -?Nt x s / ?N c" 
      with ly have eu: "l < - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<ge> u" by (cases "- ?Nt x s / ?N c < u", auto)
      with xu px' have "False" by simp  hence ?case by simp }
    moreover
    {assume y: "y > -?Nt x s / ?N c" 
      with yu have eu: "u > - ?Nt x s / ?N c" by auto
      with noS ly yu have th: "- ?Nt x s / ?N c \<le> l" by (cases "- ?Nt x s / ?N c > l", auto)
      with lx px' have "False" by simp  hence ?case by simp }
    ultimately have ?case using ycs by blast
  }
  ultimately show ?case by blast
next
    case (4 c s)
  from "4.prems" 
  have lin: "isnpoly c" "c \<noteq> 0\<^sub>p" "tmbound0 s" "allpolys isnpoly s"
    and px: "Ifm vs (x # bs) (NEq (CNP 0 c s))"
    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
  from ly yu noS have yne: "y \<noteq> - ?Nt x s / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" by simp
  hence ycs: "y < - ?Nt x s / ?N c \<or> y > -?Nt x s / ?N c" by auto
  have ccs: "?N c = 0 \<or> ?N c \<noteq> 0" by dlo
  moreover
  {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"])}
  moreover
  {assume c: "?N c \<noteq> 0"
    from yne c eq_divide_eq[of "y" "- ?Nt x s" "?N c"] have ?case
      by (simp add: ring_simps tmbound0_I[OF lin(3), of vs x bs y] sum_eq[symmetric]) }
  ultimately show ?case by blast
qed (auto simp add: nth_pos2 tmbound0_I[where vs=vs and bs="bs" and b="y" and b'="x"] bound0_I[where vs=vs and bs="bs" and b="y" and b'="x"])

lemma one_plus_one_pos[simp]: "(1::'a::{ordered_field}) + 1 > 0"
proof-
  have op: "(1::'a) > 0" by simp
  from add_pos_pos[OF op op] show ?thesis . 
qed

lemma one_plus_one_nonzero[simp]: "(1::'a::{ordered_field}) + 1 \<noteq> 0" 
  using one_plus_one_pos[where ?'a = 'a] by (simp add: less_le) 

lemma half_sum_eq: "(u + u) / (1+1) = (u::'a::{ordered_field})" 
proof-
  have "(u + u) = (1 + 1) * u" by (simp add: ring_simps)
  hence "(u + u) / (1+1) = (1 + 1)*u / (1 + 1)" by simp
  with nonzero_mult_divide_cancel_left[OF one_plus_one_nonzero, of u] show ?thesis by simp
qed

lemma inf_uset:
  assumes lp: "islin p"
  and nmi: "\<not> (Ifm vs (x#bs) (minusinf p))" (is "\<not> (Ifm vs (x#bs) (?M p))")
  and npi: "\<not> (Ifm vs (x#bs) (plusinf p))" (is "\<not> (Ifm vs (x#bs) (?P p))")
  and ex: "\<exists> x.  Ifm vs (x#bs) p" (is "\<exists> x. ?I x p")
  shows "\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). ?I ((- Itm vs (x#bs) t / Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) / (1 + 1)) p" 
proof-
  let ?Nt = "\<lambda> x t. Itm vs (x#bs) t"
  let ?N = "Ipoly vs"
  let ?U = "set (uset p)"
  from ex obtain a where pa: "?I a p" by blast
  from bound0_I[OF minusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] nmi
  have nmi': "\<not> (?I a (?M p))" by simp
  from bound0_I[OF plusinf_nb[OF lp], where bs="bs" and b="x" and b'="a"] npi
  have npi': "\<not> (?I a (?P p))" by simp
  have "\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). ?I ((- ?Nt a t/?N c + - ?Nt a s /?N d) / (1 + 1)) p"
  proof-
    let ?M = "(\<lambda> (c,t). - ?Nt a t / ?N c) ` ?U"
    have fM: "finite ?M" by auto
    from minusinf_uset[OF lp nmi pa] plusinf_uset[OF lp npi pa] 
    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
    then obtain "c" "t" "d" "s" where 
      ctU: "(c,t) \<in> ?U" and dsU: "(d,s) \<in> ?U" 
      and xs1: "a \<le> - ?Nt x s / ?N d" and tx1: "a \<ge> - ?Nt x t / ?N c" by blast
    from uset_l[OF lp] ctU dsU tmbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 
    have xs: "a \<le> - ?Nt a s / ?N d" and tx: "a \<ge> - ?Nt a t / ?N c" by auto
    from ctU have Mne: "?M \<noteq> {}" by auto
    hence Une: "?U \<noteq> {}" by simp
    let ?l = "Min ?M"
    let ?u = "Max ?M"
    have linM: "?l \<in> ?M" using fM Mne by simp
    have uinM: "?u \<in> ?M" using fM Mne by simp
    have ctM: "- ?Nt a t / ?N c \<in> ?M" using ctU by auto
    have dsM: "- ?Nt a s / ?N d \<in> ?M" using dsU by auto 
    have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
    have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
    have "?l \<le> - ?Nt a t / ?N c" using ctM Mne by simp hence lx: "?l \<le> a" using tx by simp
    have "- ?Nt a s / ?N d \<le> ?u" using dsM Mne by simp hence xu: "a \<le> ?u" using xs by simp
    from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
    have "(\<exists> s\<in> ?M. ?I s p) \<or> 
      (\<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)" .
    moreover {fix u assume um: "u\<in> ?M" and pu: "?I u p"
      hence "\<exists> (nu,tu) \<in> ?U. u = - ?Nt a tu / ?N nu" by auto
      then obtain "tu" "nu" where tuU: "(nu,tu) \<in> ?U" and tuu:"u= - ?Nt a tu / ?N nu" by blast
      from half_sum_eq[of u] pu tuu 
      have "?I (((- ?Nt a tu / ?N nu) + (- ?Nt a tu / ?N nu)) / (1 + 1)) p" by simp
      with tuU have ?thesis by blast}
    moreover{
      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"
      then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M" 
	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"
	by blast
      from t1M have "\<exists> (t1n,t1u) \<in> ?U. t1 = - ?Nt a t1u / ?N t1n" by auto
      then obtain "t1u" "t1n" where t1uU: "(t1n,t1u) \<in> ?U" and t1u: "t1 = - ?Nt a t1u / ?N t1n" by blast
      from t2M have "\<exists> (t2n,t2u) \<in> ?U. t2 = - ?Nt a t2u / ?N t2n" by auto
      then obtain "t2u" "t2n" where t2uU: "(t2n,t2u) \<in> ?U" and t2u: "t2 = - ?Nt a t2u / ?N t2n" by blast
      from t1x xt2 have t1t2: "t1 < t2" by simp
      let ?u = "(t1 + t2) / (1 + 1)"
      from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
      from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
      with t1uU t2uU t1u t2u have ?thesis by blast}
    ultimately show ?thesis by blast
  qed
  then obtain "l" "n" "s"  "m" where lnU: "(n,l) \<in> ?U" and smU:"(m,s) \<in> ?U" 
    and pu: "?I ((- ?Nt a l / ?N n + - ?Nt a s / ?N m) / (1 + 1)) p" by blast
  from lnU smU uset_l[OF lp] have nbl: "tmbound0 l" and nbs: "tmbound0 s" by auto
  from tmbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"] 
    tmbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
  have "?I ((- ?Nt x l / ?N n + - ?Nt x s / ?N m) / (1 + 1)) p" by simp
  with lnU smU
  show ?thesis by auto
qed

    (* The Ferrante - Rackoff Theorem *)

theorem fr_eq: 
  assumes lp: "islin p"
  shows "(\<exists> x. Ifm vs (x#bs) p) = ((Ifm vs (x#bs) (minusinf p)) \<or> (Ifm vs (x#bs) (plusinf p)) \<or> (\<exists> (n,t) \<in> set (uset p). \<exists> (m,s) \<in> set (uset p). Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /(1 + 1))#bs) p))"
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
proof
  assume px: "\<exists> x. ?I x p"
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
    from inf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
  ultimately show "?D" by blast
next
  assume "?D" 
  moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  moreover {assume f:"?F" hence "?E" by blast}
  ultimately show "?E" by blast
qed

section{* First implementation : Naive by encoding all case splits locally *}
definition "msubsteq c t d s a r = 
  evaldjf (split conj) 
  [(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)))),
   (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]"

lemma msubsteq_nb: assumes lp: "islin (Eq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  shows "bound0 (msubsteq c t d s a r)"
proof-
  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)))),
   (conj (Eq (CP c)) (NEq (CP d)) , Eq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (NEq (CP c)) (Eq (CP d)) , Eq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Eq r)]. bound0 (split conj x)"
    using lp by (simp add: Let_def t s )
  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubsteq_def)
qed

lemma msubsteq: assumes lp: "islin (Eq (CNP 0 a r))"
  shows "Ifm vs (x#bs) (msubsteq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Eq (CNP 0 a r))" (is "?lhs = ?rhs")
proof-
  let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  let ?N = "\<lambda>p. Ipoly vs p"
  let ?c = "?N c"
  let ?d = "?N d"
  let ?t = "?Nt x t"
  let ?s = "?Nt x s"
  let ?a = "?N a"
  let ?r = "?Nt x r"
  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  note r= tmbound0_I[OF lin(3), of vs _ bs x]
  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
  moreover
  {assume c: "?c = 0" and d: "?d=0"
    hence ?thesis  by (simp add: r[of 0] msubsteq_def Let_def evaldjf_ex)}
  moreover 
  {assume c: "?c = 0" and d: "?d\<noteq>0"
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
    have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (-?s / ((1 + 1)*?d)) + ?r = 0" by (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) = 0" 
      using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
    also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r= 0"
      by (simp add: ring_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
    
    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r = 0" using d by simp 
    finally have ?thesis using c d 
      apply (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply simp
      done}
  moreover
  {assume c: "?c \<noteq> 0" and d: "?d=0"
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
    have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (Eq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r = 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) = 0" 
      using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
    also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r= 0"
      by (simp add: ring_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r = 0" using c by simp 
    finally have ?thesis using c d 
      apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubsteq_def Let_def evaldjf_ex del: one_add_one_is_two)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply simp
      done }
  moreover
  {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Eq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r = 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) =0 "
      using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r =0" 
      using nonzero_mult_divide_cancel_left[OF dc] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using c d 
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubsteq_def Let_def evaldjf_ex ring_simps)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply (simp add: ring_simps)
      done }
  ultimately show ?thesis by blast
qed


definition "msubstneq c t d s a r = 
  evaldjf (split conj) 
  [(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)))),
   (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]"

lemma msubstneq_nb: assumes lp: "islin (NEq (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  shows "bound0 (msubstneq c t d s a r)"
proof-
  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)))), 
    (conj (Eq (CP c)) (NEq (CP d)) , NEq (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
    (conj (NEq (CP c)) (Eq (CP d)) , NEq (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
    (conj (Eq (CP c)) (Eq (CP d)) , NEq r)]. bound0 (split conj x)"
    using lp by (simp add: Let_def t s )
  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstneq_def)
qed

lemma msubstneq: assumes lp: "islin (Eq (CNP 0 a r))"
  shows "Ifm vs (x#bs) (msubstneq c t d s a r) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (NEq (CNP 0 a r))" (is "?lhs = ?rhs")
proof-
  let ?Nt = "\<lambda>(x::'a) t. Itm vs (x#bs) t"
  let ?N = "\<lambda>p. Ipoly vs p"
  let ?c = "?N c"
  let ?d = "?N d"
  let ?t = "?Nt x t"
  let ?s = "?Nt x s"
  let ?a = "?N a"
  let ?r = "?Nt x r"
  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  note r= tmbound0_I[OF lin(3), of vs _ bs x]
  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
  moreover
  {assume c: "?c = 0" and d: "?d=0"
    hence ?thesis  by (simp add: r[of 0] msubstneq_def Let_def evaldjf_ex)}
  moreover 
  {assume c: "?c = 0" and d: "?d\<noteq>0"
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?s / ((1 + 1)*?d)" by simp
    have "?rhs = Ifm vs (-?s / ((1 + 1)*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (-?s / ((1 + 1)*?d)) + ?r \<noteq> 0" by (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a * (-?s / ((1 + 1)*?d)) + ?r) \<noteq> 0" 
      using d mult_cancel_left[of "(1 + 1)*?d" "(?a * (-?s / ((1 + 1)*?d)) + ?r)" 0] by simp
    also have "\<dots> \<longleftrightarrow> (- ?a * ?s) * ((1 + 1)*?d / ((1 + 1)*?d)) + (1 + 1)*?d*?r\<noteq> 0"
      by (simp add: ring_simps right_distrib[of "(1 + 1)*?d"] del: right_distrib)
    
    also have "\<dots> \<longleftrightarrow> - (?a * ?s) + (1 + 1)*?d*?r \<noteq> 0" using d by simp 
    finally have ?thesis using c d 
      apply (simp add: r[of "- (Itm vs (x # bs) s / ((1 + 1) * \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply simp
      done}
  moreover
  {assume c: "?c \<noteq> 0" and d: "?d=0"
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = -?t / ((1 + 1)*?c)" by simp
    have "?rhs = Ifm vs (-?t / ((1 + 1)*?c) # bs) (NEq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (-?t / ((1 + 1)*?c)) + ?r \<noteq> 0" by (simp add: r[of "- (?t/ ((1 + 1)* ?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a * (-?t / ((1 + 1)*?c)) + ?r) \<noteq> 0" 
      using c mult_cancel_left[of "(1 + 1)*?c" "(?a * (-?t / ((1 + 1)*?c)) + ?r)" 0] by simp
    also have "\<dots> \<longleftrightarrow> (?a * -?t)* ((1 + 1)*?c) / ((1 + 1)*?c) + (1 + 1)*?c*?r \<noteq> 0"
      by (simp add: ring_simps right_distrib[of "(1 + 1)*?c"] del: right_distrib)
    also have "\<dots> \<longleftrightarrow> - (?a * ?t) + (1 + 1)*?c*?r \<noteq> 0" using c by simp 
    finally have ?thesis using c d 
      apply (simp add: r[of "- (?t/ ((1 + 1)*?c))"] msubstneq_def Let_def evaldjf_ex del: one_add_one_is_two)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply simp
      done }
  moreover
  {assume c: "?c \<noteq> 0" and d: "?d\<noteq>0" hence dc: "?c * ?d *(1 + 1) \<noteq> 0" by simp
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (NEq (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r \<noteq> 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) \<noteq> 0 "
      using c d mult_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r \<noteq> 0" 
      using nonzero_mult_divide_cancel_left[OF dc] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using c d 
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstneq_def Let_def evaldjf_ex ring_simps)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      apply (simp add: ring_simps)
      done }
  ultimately show ?thesis by blast
qed

definition "msubstlt c t d s a r = 
  evaldjf (split conj) 
  [(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)))),
  (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)))),
   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]"

lemma msubstlt_nb: assumes lp: "islin (Lt (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  shows "bound0 (msubstlt c t d s a r)"
proof-
  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)))),
  (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)))),
   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Lt (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP c)) (Eq (CP d)) , Lt (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Lt (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (lt (CP d)) (Eq (CP c)) , Lt (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Lt r)]. bound0 (split conj x)"
    using lp by (simp add: Let_def t s lt_nb )
  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstlt_def)
qed


lemma msubstlt: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Lt (CNP 0 a r))" 
  shows "Ifm vs (x#bs) (msubstlt c t d s a r) \<longleftrightarrow> 
  Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Lt (CNP 0 a r))" (is "?lhs = ?rhs")
proof-
  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  let ?N = "\<lambda>p. Ipoly vs p"
  let ?c = "?N c"
  let ?d = "?N d"
  let ?t = "?Nt x t"
  let ?s = "?Nt x s"
  let ?a = "?N a"
  let ?r = "?Nt x r"
  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  note r= tmbound0_I[OF lin(3), of vs _ bs x]
  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
  moreover
  {assume c: "?c=0" and d: "?d=0"
    hence ?thesis  using nc nd by (simp add: polyneg_norm lt r[of 0] msubstlt_def Let_def evaldjf_ex)}
  moreover
  {assume dc: "?c*?d > 0" 
    from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
    from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) < 0"
      
      using dc' dc'' mult_less_cancel_left_disj[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r < 0" 
      using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using dc c d  nc nd dc'
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm) 
    apply (simp only: one_add_one_is_two[symmetric] of_int_add)
    by (simp add: ring_simps order_less_not_sym[OF dc])}
  moreover
  {assume dc: "?c*?d < 0" 

    from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
      apply (simp add: mult_less_0_iff field_simps) 
      apply (rule add_neg_neg)
      apply (simp_all add: mult_less_0_iff)
      done
    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r < 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])

    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) > 0"
      
      using dc' order_less_not_sym[OF dc'] mult_less_cancel_left_disj[of "(1 + 1) * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r < 0" 
      using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using dc c d  nc nd
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm) 
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      by (simp add: ring_simps order_less_not_sym[OF dc]) }
  moreover
  {assume c: "?c > 0" and d: "?d=0"  
    from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
    from c have c': "(1 + 1)*?c \<noteq> 0" by simp
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) < 0"
      using c mult_less_cancel_left_disj[of "(1 + 1) * ?c" "?a* (- ?t / ((1 + 1)*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
    also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r < 0" 
      using nonzero_mult_divide_cancel_left[OF c'] c
      by (simp add: ring_simps diff_divide_distrib less_le del: left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using c order_less_not_sym[OF c] less_imp_neq[OF c]
      by (simp add: ring_simps )  }
  moreover
  {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
    from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r < 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) > 0"
      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_less_cancel_left_disj[of "(1 + 1) * ?c" 0 "?a* (- ?t / ((1 + 1)*?c))+ ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r < 0" 
      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
	by (simp add: ring_simps diff_divide_distrib del:  left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using c order_less_not_sym[OF c] less_imp_neq[OF c]
      by (simp add: ring_simps )    }
  moreover
  moreover
  {assume c: "?c = 0" and d: "?d>0"  
    from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
    from d have d': "(1 + 1)*?d \<noteq> 0" by simp
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) < 0"
      using d mult_less_cancel_left_disj[of "(1 + 1) * ?d" "?a* (- ?s / ((1 + 1)*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
    also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r < 0" 
      using nonzero_mult_divide_cancel_left[OF d'] d
      by (simp add: ring_simps diff_divide_distrib less_le del: left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using d order_less_not_sym[OF d] less_imp_neq[OF d]
      by (simp add: ring_simps )  }
  moreover
  {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
    from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Lt (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r < 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) > 0"
      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_less_cancel_left_disj[of "(1 + 1) * ?d" 0 "?a* (- ?s / ((1 + 1)*?d))+ ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r < 0" 
      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
	by (simp add: ring_simps diff_divide_distrib del:  left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstlt_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using d order_less_not_sym[OF d] less_imp_neq[OF d]
      by (simp add: ring_simps )    }
ultimately show ?thesis by blast
qed

definition "msubstle c t d s a r = 
  evaldjf (split conj) 
  [(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)))),
  (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)))),
   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Le r)]"

lemma msubstle_nb: assumes lp: "islin (Le (CNP 0 a r))" and t: "tmbound0 t" and s: "tmbound0 s"
  shows "bound0 (msubstle c t d s a r)"
proof-
  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)))),
  (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)))),
   (conj (lt (CP (~\<^sub>p c))) (Eq (CP d)) , Le (Add (Mul (~\<^sub>p a) t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP c)) (Eq (CP d)) , Le (Sub (Mul a t) (Mul (2\<^sub>p *\<^sub>p c) r))),
   (conj (lt (CP (~\<^sub>p d))) (Eq (CP c)) , Le (Add (Mul (~\<^sub>p a) s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (lt (CP d)) (Eq (CP c)) , Le (Sub (Mul a s) (Mul (2\<^sub>p *\<^sub>p d) r))),
   (conj (Eq (CP c)) (Eq (CP d)) , Le r)]. bound0 (split conj x)"
    using lp by (simp add: Let_def t s lt_nb )
  from evaldjf_bound0[OF th] show ?thesis by (simp add: msubstle_def)
qed

lemma msubstle: assumes nc: "isnpoly c" and nd: "isnpoly d" and lp: "islin (Le (CNP 0 a r))" 
  shows "Ifm vs (x#bs) (msubstle c t d s a r) \<longleftrightarrow> 
  Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) (Le (CNP 0 a r))" (is "?lhs = ?rhs")
proof-
  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  let ?N = "\<lambda>p. Ipoly vs p"
  let ?c = "?N c"
  let ?d = "?N d"
  let ?t = "?Nt x t"
  let ?s = "?Nt x s"
  let ?a = "?N a"
  let ?r = "?Nt x r"
  from lp have lin:"isnpoly a" "a \<noteq> 0\<^sub>p" "tmbound0 r" "allpolys isnpoly r" by simp_all
  note r= tmbound0_I[OF lin(3), of vs _ bs x]
  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
  moreover
  {assume c: "?c=0" and d: "?d=0"
    hence ?thesis  using nc nd by (simp add: polyneg_norm polymul_norm lt r[of 0] msubstle_def Let_def evaldjf_ex)}
  moreover
  {assume dc: "?c*?d > 0" 
    from mult_pos_pos[OF one_plus_one_pos dc] have dc': "(1 + 1)*?c *?d > 0" by simp
    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
    from dc' have dc'': "\<not> (1 + 1)*?c *?d < 0" by simp
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])
    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) <= 0"
      
      using dc' dc'' mult_le_cancel_left[of "(1 + 1) * ?c * ?d" "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r" 0] by simp
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )) + (1 + 1)*?c*?d*?r <= 0" 
      using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using dc c d  nc nd dc'
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm) 
    apply (simp only: one_add_one_is_two[symmetric] of_int_add)
    by (simp add: ring_simps order_less_not_sym[OF dc])}
  moreover
  {assume dc: "?c*?d < 0" 

    from dc one_plus_one_pos[where ?'a='a] have dc': "(1 + 1)*?c *?d < 0"
      by (simp add: mult_less_0_iff field_simps add_neg_neg add_pos_pos)
    hence c:"?c \<noteq> 0" and d: "?d\<noteq> 0" by auto
    from add_frac_eq[OF c d, of "- ?t" "- ?s"]
    have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)" 
      by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r <= 0" 
      by (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"])

    also have "\<dots> \<longleftrightarrow> ((1 + 1) * ?c * ?d) * (?a * (- (?d * ?t + ?c* ?s )/ ((1 + 1)*?c*?d)) + ?r) >= 0"
      
      using dc' order_less_not_sym[OF dc'] mult_le_cancel_left[of "(1 + 1) * ?c * ?d" 0 "?a * (- (?d * ?t + ?c* ?s)/ ((1 + 1)*?c*?d)) + ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a * ((?d * ?t + ?c* ?s )) - (1 + 1)*?c*?d*?r <= 0" 
      using nonzero_mult_divide_cancel_left[of "(1 + 1)*?c*?d"] c d
      by (simp add: ring_simps diff_divide_distrib del: left_distrib)
    finally  have ?thesis using dc c d  nc nd
      apply (simp add: r[of "(- (?d * ?t) + - (?c *?s)) / ((1 + 1) * ?c * ?d)"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm) 
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      by (simp add: ring_simps order_less_not_sym[OF dc]) }
  moreover
  {assume c: "?c > 0" and d: "?d=0"  
    from c have c'': "(1 + 1)*?c > 0" by (simp add: zero_less_mult_iff)
    from c have c': "(1 + 1)*?c \<noteq> 0" by simp
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) <= 0"
      using c mult_le_cancel_left[of "(1 + 1) * ?c" "?a* (- ?t / ((1 + 1)*?c))+ ?r" 0] c' c'' order_less_not_sym[OF c''] by simp
    also have "\<dots> \<longleftrightarrow> - ?a*?t+  (1 + 1)*?c *?r <= 0" 
      using nonzero_mult_divide_cancel_left[OF c'] c
      by (simp add: ring_simps diff_divide_distrib less_le del: left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using c order_less_not_sym[OF c] less_imp_neq[OF c]
      by (simp add: ring_simps )  }
  moreover
  {assume c: "?c < 0" and d: "?d=0"  hence c': "(1 + 1)*?c \<noteq> 0" by simp
    from c have c'': "(1 + 1)*?c < 0" by (simp add: mult_less_0_iff)
    from d have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?t / ((1 + 1)*?c)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?t / ((1 + 1)*?c) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?t / ((1 + 1)*?c))+ ?r <= 0" by (simp add: r[of "- (?t / ((1 + 1)*?c))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?c * (?a* (- ?t / ((1 + 1)*?c))+ ?r) >= 0"
      using c order_less_not_sym[OF c''] less_imp_neq[OF c''] c'' mult_le_cancel_left[of "(1 + 1) * ?c" 0 "?a* (- ?t / ((1 + 1)*?c))+ ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a*?t -  (1 + 1)*?c *?r <= 0" 
      using nonzero_mult_divide_cancel_left[OF c'] c order_less_not_sym[OF c''] less_imp_neq[OF c''] c''
	by (simp add: ring_simps diff_divide_distrib del:  left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?t / ((1 + 1)*?c))"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using c order_less_not_sym[OF c] less_imp_neq[OF c]
      by (simp add: ring_simps )    }
  moreover
  moreover
  {assume c: "?c = 0" and d: "?d>0"  
    from d have d'': "(1 + 1)*?d > 0" by (simp add: zero_less_mult_iff)
    from d have d': "(1 + 1)*?d \<noteq> 0" by simp
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) <= 0"
      using d mult_le_cancel_left[of "(1 + 1) * ?d" "?a* (- ?s / ((1 + 1)*?d))+ ?r" 0] d' d'' order_less_not_sym[OF d''] by simp
    also have "\<dots> \<longleftrightarrow> - ?a*?s+  (1 + 1)*?d *?r <= 0" 
      using nonzero_mult_divide_cancel_left[OF d'] d
      by (simp add: ring_simps diff_divide_distrib less_le del: left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using d order_less_not_sym[OF d] less_imp_neq[OF d]
      by (simp add: ring_simps )  }
  moreover
  {assume c: "?c = 0" and d: "?d<0"  hence d': "(1 + 1)*?d \<noteq> 0" by simp
    from d have d'': "(1 + 1)*?d < 0" by (simp add: mult_less_0_iff)
    from c have th: "(- ?t / ?c + - ?s / ?d)/(1 + 1) = - ?s / ((1 + 1)*?d)"  by (simp add: ring_simps)
    have "?rhs \<longleftrightarrow> Ifm vs (- ?s / ((1 + 1)*?d) # bs) (Le (CNP 0 a r))" by (simp only: th)
    also have "\<dots> \<longleftrightarrow> ?a* (- ?s / ((1 + 1)*?d))+ ?r <= 0" by (simp add: r[of "- (?s / ((1 + 1)*?d))"])
    also have "\<dots> \<longleftrightarrow> (1 + 1)*?d * (?a* (- ?s / ((1 + 1)*?d))+ ?r) >= 0"
      using d order_less_not_sym[OF d''] less_imp_neq[OF d''] d'' mult_le_cancel_left[of "(1 + 1) * ?d" 0 "?a* (- ?s / ((1 + 1)*?d))+ ?r"] by simp
    also have "\<dots> \<longleftrightarrow> ?a*?s -  (1 + 1)*?d *?r <= 0" 
      using nonzero_mult_divide_cancel_left[OF d'] d order_less_not_sym[OF d''] less_imp_neq[OF d''] d''
	by (simp add: ring_simps diff_divide_distrib del:  left_distrib)
    finally have ?thesis using c d nc nd 
      apply(simp add: r[of "- (?s / ((1 + 1)*?d))"] msubstle_def Let_def evaldjf_ex ring_simps lt polyneg_norm polymul_norm)
      apply (simp only: one_add_one_is_two[symmetric] of_int_add)
      using d order_less_not_sym[OF d] less_imp_neq[OF d]
      by (simp add: ring_simps )    }
ultimately show ?thesis by blast
qed


fun msubst :: "fm \<Rightarrow> (poly \<times> tm) \<times> (poly \<times> tm) \<Rightarrow> fm" where
  "msubst (And p q) ((c,t), (d,s)) = conj (msubst p ((c,t),(d,s))) (msubst q ((c,t),(d,s)))"
| "msubst (Or p q) ((c,t), (d,s)) = disj (msubst p ((c,t),(d,s))) (msubst q ((c,t), (d,s)))"
| "msubst (Eq (CNP 0 a r)) ((c,t),(d,s)) = msubsteq c t d s a r"
| "msubst (NEq (CNP 0 a r)) ((c,t),(d,s)) = msubstneq c t d s a r"
| "msubst (Lt (CNP 0 a r)) ((c,t),(d,s)) = msubstlt c t d s a r"
| "msubst (Le (CNP 0 a r)) ((c,t),(d,s)) = msubstle c t d s a r"
| "msubst p ((c,t),(d,s)) = p"

lemma msubst_I: assumes lp: "islin p" and nc: "isnpoly c" and nd: "isnpoly d"
  shows "Ifm vs (x#bs) (msubst p ((c,t),(d,s))) = Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs c + - Itm vs (x#bs) s / Ipoly vs d) /(1 + 1))#bs) p"
  using lp
by (induct p rule: islin.induct, auto simp add: tmbound0_I[where b="(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) + - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) /(1 + 1)" and b'=x and bs = bs and vs=vs] bound0_I[where b="(- (Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>) + - (Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>)) /(1 + 1)" and b'=x and bs = bs and vs=vs] msubsteq msubstneq msubstlt[OF nc nd] msubstle[OF nc nd])

lemma msubst_nb: assumes lp: "islin p" and t: "tmbound0 t" and s: "tmbound0 s"
  shows "bound0 (msubst p ((c,t),(d,s)))"
  using lp t s
  by (induct p rule: islin.induct, auto simp add: msubsteq_nb msubstneq_nb msubstlt_nb msubstle_nb)

lemma fr_eq_msubst: 
  assumes lp: "islin p"
  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)))))"
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
proof-
from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
{fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  and pts: "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p"
  from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  from msubst_I[OF lp norm, of vs x bs t s] pts
  have "Ifm vs (x # bs) (msubst p ((c, t), d, s))" ..}
moreover
{fix c t d s assume ctU: "(c,t) \<in>set (uset p)" and dsU: "(d,s) \<in>set (uset p)" 
  and pts: "Ifm vs (x # bs) (msubst p ((c, t), d, s))"
  from th[rule_format, OF ctU] th[rule_format, OF dsU] have norm:"isnpoly c" "isnpoly d" by simp_all
  from msubst_I[OF lp norm, of vs x bs t s] pts
  have "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p" ..}
ultimately have th': "(\<exists> (c,t) \<in> set (uset p). \<exists> (d,s) \<in> set (uset p). Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1+1) # bs) p) \<longleftrightarrow> ?F" by blast
from fr_eq[OF lp, of vs bs x, simplified th'] show ?thesis .
qed 

text {* Rest of the implementation *}

consts alluopairs:: "'a list \<Rightarrow> ('a \<times> 'a) list"
primrec
  "alluopairs [] = []"
  "alluopairs (x#xs) = (map (Pair x) (x#xs))@(alluopairs xs)"

lemma alluopairs_set1: "set (alluopairs xs) \<le> {(x,y). x\<in> set xs \<and> y\<in> set xs}"
by (induct xs, auto)

lemma alluopairs_set:
  "\<lbrakk>x\<in> set xs ; y \<in> set xs\<rbrakk> \<Longrightarrow> (x,y) \<in> set (alluopairs xs) \<or> (y,x) \<in> set (alluopairs xs) "
by (induct xs, auto)

lemma alluopairs_ex:
  assumes Pc: "\<forall> x \<in> set xs. \<forall>y\<in> set xs. P x y = P y x"
  shows "(\<exists> x \<in> set xs. \<exists> y \<in> set xs. P x y) = (\<exists> (x,y) \<in> set (alluopairs xs). P x y)"
proof
  assume "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y"
  then obtain x y where x: "x \<in> set xs" and y:"y \<in> set xs" and P: "P x y"  by blast
  from alluopairs_set[OF x y] P Pc x y show"\<exists>(x, y)\<in>set (alluopairs xs). P x y" 
    by auto
next
  assume "\<exists>(x, y)\<in>set (alluopairs xs). P x y"
  then obtain "x" and "y"  where xy:"(x,y) \<in> set (alluopairs xs)" and P: "P x y" by blast+
  from xy have "x \<in> set xs \<and> y\<in> set xs" using alluopairs_set1 by blast
  with P show "\<exists>x\<in>set xs. \<exists>y\<in>set xs. P x y" by blast
qed

lemma nth_pos2: "0 < n \<Longrightarrow> (x#xs) ! n = xs ! (n - 1)"
using Nat.gr0_conv_Suc
by clarsimp

lemma filter_length: "length (List.filter P xs) < Suc (length xs)"
  apply (induct xs, auto) done

consts remdps:: "'a list \<Rightarrow> 'a list"

recdef remdps "measure size"
  "remdps [] = []"
  "remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"
(hints simp add: filter_length[rule_format])

lemma remdps_set[simp]: "set (remdps xs) = set xs"
  by (induct xs rule: remdps.induct, auto)

lemma simpfm_lin:   assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})"
  shows "qfree p \<Longrightarrow> islin (simpfm p)"
  by (induct p rule: simpfm.induct, auto simp add: conj_lin disj_lin)

definition 
  "ferrack p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
  in if (mp = T \<or> pp = T) then T 
     else (let U = alluopairs (remdps (uset  q))
           in decr0 (disj mp (disj pp (evaldjf (simpfm o (msubst q)) U ))))"

lemma ferrack: 
  assumes qf: "qfree p"
  shows "qfree (ferrack p) \<and> ((Ifm vs bs (ferrack p)) = (Ifm vs bs (E p)))"
  (is "_ \<and> (?rhs = ?lhs)")
proof-
  let ?I = "\<lambda> x p. Ifm vs (x#bs) p"
  let ?N = "\<lambda> t. Ipoly vs t"
  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  let ?q = "simpfm p" 
  let ?U = "remdps(uset ?q)"
  let ?Up = "alluopairs ?U"
  let ?mp = "minusinf ?q"
  let ?pp = "plusinf ?q"
  let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  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"
    by simp
  {fix c t d s assume ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U"
    from U_l ctU dsU have norm: "isnpoly c" "isnpoly d" by auto
    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]
    have "?I (msubst ?q ((c,t),(d,s))) = ?I (msubst ?q ((d,s),(c,t)))" by (simp add: ring_simps)}
  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
  {fix x assume xUp: "x \<in> set ?Up" 
    then  obtain c t d s where ctU: "(c,t) \<in> set ?U" and dsU: "(d,s) \<in> set ?U" 
      and x: "x = ((c,t),(d,s))" using alluopairs_set1[of ?U] by auto  
    from U_l[rule_format, OF ctU] U_l[rule_format, OF dsU] 
    have nbs: "tmbound0 t" "tmbound0 s" by simp_all
    from simpfm_bound0[OF msubst_nb[OF lq nbs, of c d]] 
    have "bound0 ((simpfm o (msubst (simpfm p))) x)" using x by simp}
  with evaldjf_bound0[of ?Up "(simpfm o (msubst (simpfm p)))"]
  have "bound0 (evaldjf (simpfm o (msubst (simpfm p))) ?Up)" by blast
  with mp_nb pp_nb 
  have th1: "bound0 (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up )))" by (simp add: disj_nb)
  from decr0_qf[OF th1] have thqf: "qfree (ferrack p)" by (simp add: ferrack_def Let_def)
  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  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
  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> (\<exists> (x,y) \<in> set ?Up. ?I ((simpfm o (msubst ?q)) (x,y)))" using alluopairs_ex[OF th0] by simp
  also have "\<dots> \<longleftrightarrow> ?I ?mp \<or> ?I ?pp \<or> ?I (evaldjf (simpfm o (msubst ?q)) ?Up)" 
    by (simp add: evaldjf_ex)
  also have "\<dots> \<longleftrightarrow> ?I (disj ?mp (disj ?pp (evaldjf (simpfm o (msubst ?q)) ?Up)))" by simp
  also have "\<dots> \<longleftrightarrow> ?rhs" using decr0[OF th1, of vs x bs]
    apply (simp add: ferrack_def Let_def)
    by (cases "?mp = T \<or> ?pp = T", auto)
  finally show ?thesis using thqf by blast
qed

definition "frpar p = simpfm (qelim p ferrack)"
lemma frpar: "qfree (frpar p) \<and> (Ifm vs bs (frpar p) \<longleftrightarrow> Ifm vs bs p)"
proof-
  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
  from qelim[OF th, of p bs] show ?thesis  unfolding frpar_def by auto
qed

declare polyadd.simps[code]
lemma [simp,code]: "polyadd (CN c n p, CN c' n' p') = 
    (if n < n' then CN (polyadd(c,CN c' n' p')) n p
     else if n'<n then CN (polyadd(CN c n p, c')) n' p'
     else (let cc' = polyadd (c,c') ; 
               pp' = polyadd (p,p')
           in (if pp' = 0\<^sub>p then cc' else CN cc' n pp')))"
  by (simp add: Let_def stupid)



(*
lemmas [code func] = polysub_def
lemmas [code func del] = Zero_nat_def
code_gen  "frpar" in SML to FRParTest
*)

section{* Second implemenation: Case splits not local *}

lemma fr_eq2:  assumes lp: "islin p"
  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 (0#bs) p) \<or> 
    (\<exists> (n,t) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ifm vs ((- Itm vs (x#bs) t /  (Ipoly vs n * (1 + 1)))#bs) p) \<or> 
    (\<exists> (n,t) \<in> set (uset p). \<exists> (m,s) \<in> set (uset p). Ipoly vs n \<noteq> 0 \<and> Ipoly vs m \<noteq> 0 \<and> Ifm vs (((- Itm vs (x#bs) t /  Ipoly vs n + - Itm vs (x#bs) s / Ipoly vs m) /(1 + 1))#bs) p))"
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Z \<or> ?U \<or> ?F)" is "?E = ?D")
proof
  assume px: "\<exists> x. ?I x p"
  have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
  moreover {assume "?M \<or> ?P" hence "?D" by blast}
  moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
    from inf_uset[OF lp nmi npi, OF px] 
    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"
      by auto
    let ?c = "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
    let ?d = "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>"
    let ?s = "Itm vs (x # bs) s"
    let ?t = "Itm vs (x # bs) t"
    have eq2: "\<And>(x::'a). x + x = (1 + 1) * x"
      by  (simp add: ring_simps)
    {assume "?c = 0 \<and> ?d = 0"
      with ct have ?D by simp}
    moreover
    {assume z: "?c = 0" "?d \<noteq> 0"
      from z have ?D using ct by auto}
    moreover
    {assume z: "?c \<noteq> 0" "?d = 0"
      with ct have ?D by auto }
    moreover
    {assume z: "?c \<noteq> 0" "?d \<noteq> 0"
      from z have ?F using ct
	apply - apply (rule bexI[where x = "(c,t)"], simp_all)
	by (rule bexI[where x = "(d,s)"], simp_all)
      hence ?D by blast}
    ultimately have ?D by auto}
  ultimately show "?D" by blast
next
  assume "?D" 
  moreover {assume m:"?M" from minusinf_ex[OF lp m] have "?E" .}
  moreover {assume p: "?P" from plusinf_ex[OF lp p] have "?E" . }
  moreover {assume f:"?F" hence "?E" by blast}
  ultimately show "?E" by blast
qed

definition "msubsteq2 c t a b = Eq (Add (Mul a t) (Mul c b))"
definition "msubstltpos c t a b = Lt (Add (Mul a t) (Mul c b))"
definition "msubstlepos c t a b = Le (Add (Mul a t) (Mul c b))"
definition "msubstltneg c t a b = Lt (Neg (Add (Mul a t) (Mul c b)))"
definition "msubstleneg c t a b = Le (Neg (Add (Mul a t) (Mul c b)))"

lemma msubsteq2: 
  assumes nz: "Ipoly vs c \<noteq> 0" and l: "islin (Eq (CNP 0 a b))"
  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")
  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  by (simp add: msubsteq2_def field_simps)

lemma msubstltpos: 
  assumes nz: "Ipoly vs c > 0" and l: "islin (Lt (CNP 0 a b))"
  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")
  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  by (simp add: msubstltpos_def field_simps)

lemma msubstlepos: 
  assumes nz: "Ipoly vs c > 0" and l: "islin (Le (CNP 0 a b))"
  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")
  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  by (simp add: msubstlepos_def field_simps)

lemma msubstltneg: 
  assumes nz: "Ipoly vs c < 0" and l: "islin (Lt (CNP 0 a b))"
  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")
  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  by (simp add: msubstltneg_def field_simps del: minus_add_distrib)

lemma msubstleneg: 
  assumes nz: "Ipoly vs c < 0" and l: "islin (Le (CNP 0 a b))"
  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")
  using nz l tmbound0_I[of b vs x bs "Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup>" , symmetric]
  by (simp add: msubstleneg_def field_simps del: minus_add_distrib)

fun msubstpos :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  "msubstpos (And p q) c t = And (msubstpos p c t) (msubstpos q c t)"
| "msubstpos (Or p q) c t = Or (msubstpos p c t) (msubstpos q c t)"
| "msubstpos (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
| "msubstpos (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
| "msubstpos (Lt (CNP 0 a r)) c t = msubstltpos c t a r"
| "msubstpos (Le (CNP 0 a r)) c t = msubstlepos c t a r"
| "msubstpos p c t = p"
    
lemma msubstpos_I: 
  assumes lp: "islin p" and pos: "Ipoly vs c > 0"
  shows "Ifm vs (x#bs) (msubstpos p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  using lp pos
  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)

fun msubstneg :: "fm \<Rightarrow> poly \<Rightarrow> tm \<Rightarrow> fm" where
  "msubstneg (And p q) c t = And (msubstneg p c t) (msubstneg q c t)"
| "msubstneg (Or p q) c t = Or (msubstneg p c t) (msubstneg q c t)"
| "msubstneg (Eq (CNP 0 a r)) c t = msubsteq2 c t a r"
| "msubstneg (NEq (CNP 0 a r)) c t = NOT (msubsteq2 c t a r)"
| "msubstneg (Lt (CNP 0 a r)) c t = msubstltneg c t a r"
| "msubstneg (Le (CNP 0 a r)) c t = msubstleneg c t a r"
| "msubstneg p c t = p"

lemma msubstneg_I: 
  assumes lp: "islin p" and pos: "Ipoly vs c < 0"
  shows "Ifm vs (x#bs) (msubstneg p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
  using lp pos
  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)


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)))"

lemma msubst2: assumes lp: "islin p" and nc: "isnpoly c" and nz: "Ipoly vs c \<noteq> 0"
  shows "Ifm vs (x#bs) (msubst2 p c t) = Ifm vs (Itm vs (x#bs) t /  Ipoly vs c #bs) p"
proof-
  let ?c = "Ipoly vs c"
  from nc have anc: "allpolys isnpoly (CP c)" "allpolys isnpoly (CP (~\<^sub>p c))" 
    by (simp_all add: polyneg_norm)
  from nz have "?c > 0 \<or> ?c < 0" by arith
  moreover
  {assume c: "?c < 0"
    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"]
    have ?thesis by (auto simp add: msubst2_def)}
  moreover
  {assume c: "?c > 0"
    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"]
    have ?thesis by (auto simp add: msubst2_def)}
  ultimately show ?thesis by blast
qed

term msubsteq2
lemma msubsteq2_nb: "tmbound0 t \<Longrightarrow> islin (Eq (CNP 0 a r)) \<Longrightarrow> bound0 (msubsteq2 c t a r)"
  by (simp add: msubsteq2_def)

lemma msubstltpos_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltpos c t a r)"
  by (simp add: msubstltpos_def)
lemma msubstltneg_nb: "tmbound0 t \<Longrightarrow> islin (Lt (CNP 0 a r)) \<Longrightarrow> bound0 (msubstltneg c t a r)"
  by (simp add: msubstltneg_def)

lemma msubstlepos_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstlepos c t a r)"
  by (simp add: msubstlepos_def)
lemma msubstleneg_nb: "tmbound0 t \<Longrightarrow> islin (Le (CNP 0 a r)) \<Longrightarrow> bound0 (msubstleneg c t a r)"
  by (simp add: msubstleneg_def)

lemma msubstpos_nb: assumes lp: "islin p" and tnb: "tmbound0 t"
  shows "bound0 (msubstpos p c t)"
using lp tnb
by (induct p c t rule: msubstpos.induct, auto simp add: msubsteq2_nb msubstltpos_nb msubstlepos_nb)

lemma msubstneg_nb: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" and lp: "islin p" and tnb: "tmbound0 t"
  shows "bound0 (msubstneg p c t)"
using lp tnb
by (induct p c t rule: msubstneg.induct, auto simp add: msubsteq2_nb msubstltneg_nb msubstleneg_nb)

lemma msubst2_nb: assumes "SORT_CONSTRAINT('a::{ring_char_0,division_by_zero,field})" and lp: "islin p" and tnb: "tmbound0 t"
  shows "bound0 (msubst2 p c t)"
using lp tnb
by (simp add: msubst2_def msubstneg_nb msubstpos_nb conj_nb disj_nb lt_nb simpfm_bound0)
    
lemma of_int2: "of_int 2 = 1 + 1"
proof-
  have "(2::int) = 1 + 1" by simp
  hence "of_int 2 = of_int (1 + 1)" by simp
  thus ?thesis unfolding of_int_add by simp
qed

lemma of_int_minus2: "of_int (-2) = - (1 + 1)"
proof-
  have th: "(-2::int) = - 2" by simp
  show ?thesis unfolding th by (simp only: of_int_minus of_int2)
qed


lemma islin_qf: "islin p \<Longrightarrow> qfree p"
  by (induct p rule: islin.induct, auto simp add: bound0_qf)
lemma fr_eq_msubst2: 
  assumes lp: "islin p"
  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)))))"
  (is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?Pz \<or> ?PU \<or> ?F)" is "?E = ?D")
proof-
  from uset_l[OF lp] have th: "\<forall>(c, s)\<in>set (uset p). isnpoly c \<and> tmbound0 s" by blast
  let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  have n2: "isnpoly (C (-2,1))" by (simp add: isnpoly_def)
  note eq0 = subst0[OF islin_qf[OF lp], of vs x bs "CP 0\<^sub>p", simplified]
  
  have eq1: "(\<exists>(n, t)\<in>set (uset p). ?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)) \<longleftrightarrow> (\<exists>(n, t)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p)"
  proof-
    {fix n t assume H: "(n, t)\<in>set (uset p)" "?I(msubst2 p (n *\<^sub>p C (-2, 1)) t)"
      from H(1) th have "isnpoly n" by blast
      hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" by (simp_all add: polymul_norm n2)
      have nn': "allpolys isnpoly (CP (~\<^sub>p (n *\<^sub>p C (-2, 1))))"
	by (simp add: polyneg_norm nn)
      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 
	by (auto simp add: msubst2_def lt zero_less_mult_iff mult_less_0_iff)
      from msubst2[OF lp nn nn2(1), of x bs t]
      have "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p"
	using H(2) nn2 by (simp add: of_int_minus2 del: minus_add_distrib)}
    moreover
    {fix n t assume H: "(n, t)\<in>set (uset p)" "\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs (- Itm vs (x # bs) t / (\<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> * (1 + 1)) # bs) p"
      from H(1) th have "isnpoly n" by blast
      hence nn: "isnpoly (n *\<^sub>p (C (-2,1)))" "\<lparr>n *\<^sub>p(C (-2,1)) \<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0"
	using H(2) by (simp_all add: polymul_norm n2)
      from msubst2[OF lp nn, of x bs t] have "?I (msubst2 p (n *\<^sub>p (C (-2,1))) t)" using H(2,3) by (simp add: of_int_minus2 del: minus_add_distrib)}
    ultimately show ?thesis by blast
  qed
  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).
     \<exists>(m, s)\<in>set (uset p). \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>n\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>m\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p)" 
  proof-
    {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(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)))"
      from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
      hence nn: "isnpoly (C (-2, 1) *\<^sub>p c*\<^sub>p d)" 
	by (simp_all add: polymul_norm n2)
      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)))"
	by (simp_all add: polyneg_norm nn)
      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"
	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)
      from msubst2[OF lp nn nn'(1), of x bs ] H(3) nn'
      have "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0 \<and> Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p" 
	apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)
	by (simp add: mult_commute)}
    moreover
    {fix c t d s assume H: "(c,t) \<in> set (uset p)" "(d,s) \<in> set (uset p)" 
      "\<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "\<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup> \<noteq> 0" "Ifm vs ((- Itm vs (x # bs) t / \<lparr>c\<rparr>\<^sub>p\<^bsup>vs\<^esup> + - Itm vs (x # bs) s / \<lparr>d\<rparr>\<^sub>p\<^bsup>vs\<^esup>) / (1 + 1) # bs) p"
     from H(1,2) th have "isnpoly c" "isnpoly d" by blast+
      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"
	using H(3,4) by (simp_all add: polymul_norm n2)
      from msubst2[OF lp nn, of x bs ] H(3,4,5) 
      have "Ifm vs (x#bs) (msubst2 p (C (-2, 1) *\<^sub>p c*\<^sub>p d) (Add (Mul d t) (Mul c s)))" apply (simp add: add_divide_distrib of_int_minus2 del: minus_add_distrib)by (simp add: mult_commute)}
    ultimately show ?thesis by blast
  qed
  from fr_eq2[OF lp, of vs bs x] show ?thesis
    unfolding eq0 eq1 eq2 by blast  
qed

definition 
"ferrack2 p \<equiv> let q = simpfm p ; mp = minusinf q ; pp = plusinf q
 in if (mp = T \<or> pp = T) then T 
  else (let U = remdps (uset  q)
    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, 
   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)]))"

definition "frpar2 p = simpfm (qelim (prep p) ferrack2)"

lemma ferrack2: assumes qf: "qfree p"
  shows "qfree (ferrack2 p) \<and> ((Ifm vs bs (ferrack2 p)) = (Ifm vs bs (E p)))"
  (is "_ \<and> (?rhs = ?lhs)")
proof-
  let ?J = "\<lambda> x p. Ifm vs (x#bs) p"
  let ?N = "\<lambda> t. Ipoly vs t"
  let ?Nt = "\<lambda>x t. Itm vs (x#bs) t"
  let ?q = "simpfm p" 
  let ?qz = "subst0 (CP 0\<^sub>p) ?q"
  let ?U = "remdps(uset ?q)"
  let ?Up = "alluopairs ?U"
  let ?mp = "minusinf ?q"
  let ?pp = "plusinf ?q"
  let ?I = "\<lambda>p. Ifm vs (x#bs) p"
  from simpfm_lin[OF qf] simpfm_qf[OF qf] have lq: "islin ?q" and q_qf: "qfree ?q" .
  from minusinf_nb[OF lq] plusinf_nb[OF lq] have mp_nb: "bound0 ?mp" and pp_nb: "bound0 ?pp" .
  from bound0_qf[OF mp_nb] bound0_qf[OF pp_nb] have mp_qf: "qfree ?mp" and pp_qf: "qfree ?pp" .
  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"
    by simp
  have bnd0: "\<forall>x \<in> set ?U. bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) x)" 
  proof-
    {fix c t assume ct: "(c,t) \<in> set ?U"
      hence tnb: "tmbound0 t" using U_l by blast
      from msubst2_nb[OF lq tnb]
      have "bound0 ((\<lambda>(c,t). msubst2 ?q (c *\<^sub>p C (-2, 1)) t) (c,t))" by simp}
    thus ?thesis by auto
  qed
  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)" 
  proof-
    {fix b a d c assume badc: "((b,a),(d,c)) \<in> set ?Up"
      from badc U_l alluopairs_set1[of ?U] 
      have nb: "tmbound0 (Add (Mul d a) (Mul b c))" by auto
      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}
    thus ?thesis by auto
  qed
  have stupid: "bound0 F" by simp
  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, 
   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)]"
  from subst0_nb[of "CP 0\<^sub>p" ?q] q_qf evaldjf_bound0[OF bnd1] evaldjf_bound0[OF bnd0] mp_nb pp_nb stupid
  have nb: "bound0 ?R "
    by (simp add: list_disj_def disj_nb0 simpfm_bound0)
  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))"

  {fix b a d c assume baU: "(b,a) \<in> set ?U" and dcU: "(d,c) \<in> set ?U"
    from U_l baU dcU have norm: "isnpoly b" "isnpoly d" "isnpoly (C (-2, 1))" 
      by auto (simp add: isnpoly_def)
    have norm2: "isnpoly (C (-2, 1) *\<^sub>p b*\<^sub>p d)" "isnpoly (C (-2, 1) *\<^sub>p d*\<^sub>p b)"
      using norm by (simp_all add: polymul_norm)
    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)))"
      by (simp_all add: polyneg_norm norm2)
    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")
    proof
      assume H: ?lhs
      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" 
	by (auto simp add: msubst2_def lt[OF stupid(3)] lt[OF stupid(1)] mult_less_0_iff zero_less_mult_iff)
      from msubst2[OF lq norm2(1) z(1), of x bs] 
	msubst2[OF lq norm2(2) z(2), of x bs] H 
      show ?rhs by (simp add: ring_simps)
    next
      assume H: ?rhs
      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" 
	by (auto simp add: msubst2_def lt[OF stupid(4)] lt[OF stupid(2)] mult_less_0_iff zero_less_mult_iff)
      from msubst2[OF lq norm2(1) z(1), of x bs] 
	msubst2[OF lq norm2(2) z(2), of x bs] H 
      show ?lhs by (simp add: ring_simps)
    qed}
  hence th0: "\<forall>x \<in> set ?U. \<forall>y \<in> set ?U. ?I (?s (x, y)) \<longleftrightarrow> ?I (?s (y, x))"
    by clarsimp

  have "?lhs \<longleftrightarrow> (\<exists>x. Ifm vs (x#bs) ?q)" by simp
  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))))"
    using fr_eq_msubst2[OF lq, of vs bs x] by simp
  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)))"
    by (simp add: split_def)
  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)))"
    using alluopairs_ex[OF th0] by simp 
  also have "\<dots> \<longleftrightarrow> ?I ?R" 
    by (simp add: list_disj_def evaldjf_ex split_def)
  also have "\<dots> \<longleftrightarrow> ?rhs"
    unfolding ferrack2_def
    apply (cases "?mp = T") 
    apply (simp add: list_disj_def)
    apply (cases "?pp = T") 
    apply (simp add: list_disj_def)
    by (simp_all add: Let_def decr0[OF nb])
  finally show ?thesis using decr0_qf[OF nb]  
    by (simp  add: ferrack2_def Let_def)
qed

lemma frpar2: "qfree (frpar2 p) \<and> (Ifm vs bs (frpar2 p) \<longleftrightarrow> Ifm vs bs p)"
proof-
  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
  from qelim[OF th, of "prep p" bs] 
show ?thesis  unfolding frpar2_def by (auto simp add: prep)
qed

code_module FRPar
  contains 
  frpar = "frpar"
  frpar2 = "frpar2"
  test = "%x . frpar (E(Lt (Mul 1\<^sub>p (Bound 0))))"

ML{* 

structure ReflectedFRPar = 
struct

val bT = HOLogic.boolT;
fun num rT x = HOLogic.mk_number rT x;
fun rrelT rT = [rT,rT] ---> rT;
fun rrT rT = [rT, rT] ---> bT;
fun divt rT = Const(@{const_name "HOL.divide"},rrelT rT);
fun timest rT = Const(@{const_name "HOL.times"},rrelT rT);
fun plust rT = Const(@{const_name "HOL.plus"},rrelT rT);
fun minust rT = Const(@{const_name "HOL.minus"},rrelT rT);
fun uminust rT = Const(@{const_name "HOL.uminus"}, rT --> rT);
fun powt rT = Const(@{const_name "power"}, [rT,@{typ "nat"}] ---> rT);
val brT = [bT, bT] ---> bT;
val nott = @{term "Not"};
val conjt = @{term "op &"};
val disjt = @{term "op |"};
val impt = @{term "op -->"};
val ifft = @{term "op = :: bool => _"}
fun llt rT = Const(@{const_name "HOL.less"},rrT rT);
fun lle rT = Const(@{const_name "HOL.less"},rrT rT);
fun eqt rT = Const("op =",rrT rT);
fun rz rT = Const(@{const_name "HOL.zero"},rT);

fun dest_nat t = case t of
  Const ("Suc",_)$t' => 1 + dest_nat t'
| _ => (snd o HOLogic.dest_number) t;

fun num_of_term m t = 
 case t of
   Const(@{const_name "uminus"},_)$t => FRPar.Neg (num_of_term m t)
 | Const(@{const_name "HOL.plus"},_)$a$b => FRPar.Add (num_of_term m a, num_of_term m b)
 | Const(@{const_name "HOL.minus"},_)$a$b => FRPar.Sub (num_of_term m a, num_of_term m b)
 | Const(@{const_name "HOL.times"},_)$a$b => FRPar.Mul (num_of_term m a, num_of_term m b)
 | Const(@{const_name "power"},_)$a$n => FRPar.Pw (num_of_term m a, dest_nat n)
 | Const(@{const_name "HOL.divide"},_)$a$b => FRPar.C (HOLogic.dest_number a |> snd, HOLogic.dest_number b |> snd)
 | _ => (FRPar.C (HOLogic.dest_number t |> snd,1) 
         handle TERM _ => FRPar.Bound (AList.lookup (op aconv) m t |> the));

fun tm_of_term m m' t = 
 case t of
   Const(@{const_name "uminus"},_)$t => FRPar.tm_Neg (tm_of_term m m' t)
 | Const(@{const_name "HOL.plus"},_)$a$b => FRPar.tm_Add (tm_of_term m m' a, tm_of_term m m' b)
 | Const(@{const_name "HOL.minus"},_)$a$b => FRPar.tm_Sub (tm_of_term m m' a, tm_of_term m m' b)
 | Const(@{const_name "HOL.times"},_)$a$b => FRPar.tm_Mul (num_of_term m' a, tm_of_term m m' b)
 | _ => (FRPar.CP (num_of_term m' t) 
         handle TERM _ => FRPar.tm_Bound (AList.lookup (op aconv) m t |> the)
              | Option => FRPar.tm_Bound (AList.lookup (op aconv) m t |> the));

fun term_of_num T m t = 
 case t of
  FRPar.C (a,b) => (if b = 1 then num T a else if b=0 then (rz T) 
                                        else (divt T) $ num T a $ num T b)
| FRPar.Bound i => AList.lookup (op = : int*int -> bool) m i |> the
| FRPar.Add(a,b) => (plust T)$(term_of_num T m a)$(term_of_num T m b)
| FRPar.Mul(a,b) => (timest T)$(term_of_num T m a)$(term_of_num T m b)
| FRPar.Sub(a,b) => (minust T)$(term_of_num T m a)$(term_of_num T m b)
| FRPar.Neg a => (uminust T)$(term_of_num T m a)
| FRPar.Pw(a,n) => (powt T)$(term_of_num T m t)$(HOLogic.mk_number HOLogic.natT n)
| FRPar.CN(c,n,p) => term_of_num T m (FRPar.Add(c,FRPar.Mul(FRPar.Bound n, p)))
| _ => error "term_of_num: Unknown term";

fun term_of_tm T m m' t = 
 case t of
  FRPar.CP p => term_of_num T m' p
| FRPar.tm_Bound i => AList.lookup (op = : int*int -> bool) m i |> the
| FRPar.tm_Add(a,b) => (plust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
| FRPar.tm_Mul(a,b) => (timest T)$(term_of_num T m' a)$(term_of_tm T m m' b)
| FRPar.tm_Sub(a,b) => (minust T)$(term_of_tm T m m' a)$(term_of_tm T m m' b)
| FRPar.tm_Neg a => (uminust T)$(term_of_tm T m m' a)
| FRPar.CNP(n,c,p) => term_of_tm T m m' (FRPar.tm_Add(FRPar.tm_Mul(c, FRPar.tm_Bound n), p))
| _ => error "term_of_tm: Unknown term";

fun fm_of_term m m' fm = 
 case fm of
    Const("True",_) => FRPar.T
  | Const("False",_) => FRPar.F
  | Const("Not",_)$p => FRPar.NOT (fm_of_term m m' p)
  | Const("op &",_)$p$q => FRPar.And(fm_of_term m m' p, fm_of_term m m' q)
  | Const("op |",_)$p$q => FRPar.Or(fm_of_term m m' p, fm_of_term m m' q)
  | Const("op -->",_)$p$q => FRPar.Imp(fm_of_term m m' p, fm_of_term m m' q)
  | Const("op =",ty)$p$q => 
       if domain_type ty = bT then FRPar.Iff(fm_of_term m m' p, fm_of_term m m' q)
       else FRPar.Eq (FRPar.tm_Sub(tm_of_term m m' p, tm_of_term m m' q))
  | Const(@{const_name "HOL.less"},_)$p$q => 
        FRPar.Lt (FRPar.tm_Sub(tm_of_term m m' p, tm_of_term m m' q))
  | Const(@{const_name "HOL.less_eq"},_)$p$q => 
        FRPar.Le (FRPar.tm_Sub(tm_of_term m m' p, tm_of_term m m' q))
  | Const("Ex",_)$Abs(xn,xT,p) => 
     let val (xn', p') =  variant_abs (xn,xT,p)
         val x = Free(xn',xT)
         fun incr i = i + 1
         val m0 = (x,0):: (map (apsnd incr) m)
      in FRPar.E (fm_of_term m0 m' p') end
  | Const("All",_)$Abs(xn,xT,p) => 
     let val (xn', p') =  variant_abs (xn,xT,p)
         val x = Free(xn',xT)
         fun incr i = i + 1
         val m0 = (x,0):: (map (apsnd incr) m)
      in FRPar.A (fm_of_term m0 m' p') end
  | _ => error "fm_of_term";


fun term_of_fm T m m' t = 
  case t of
    FRPar.T => Const("True",bT)
  | FRPar.F => Const("False",bT)
  | FRPar.NOT p => nott $ (term_of_fm T m m' p)
  | FRPar.And (p,q) => conjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  | FRPar.Or (p,q) => disjt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  | FRPar.Imp (p,q) => impt $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  | FRPar.Iff (p,q) => ifft $ (term_of_fm T m m' p) $ (term_of_fm T m m' q)
  | FRPar.Lt p => (llt T) $ (term_of_tm T m m' p) $ (rz T)
  | FRPar.Le p => (lle T) $ (term_of_tm T m m' p) $ (rz T)
  | FRPar.Eq p => (eqt T) $ (term_of_tm T m m' p) $ (rz T)
  | FRPar.NEq p => nott $ ((eqt T) $ (term_of_tm T m m' p) $ (rz T))
  | _ => error "term_of_fm: quantifiers!!!!???";

fun frpar_oracle (T,m, m', fm) = 
 let 
   val t = HOLogic.dest_Trueprop fm
   val im = 0 upto (length m - 1)
   val im' = 0 upto (length m' - 1)   
 in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
                                                     (FRPar.frpar (fm_of_term (m ~~ im) (m' ~~ im') t))))
 end;

fun frpar_oracle2 (T,m, m', fm) = 
 let 
   val t = HOLogic.dest_Trueprop fm
   val im = 0 upto (length m - 1)
   val im' = 0 upto (length m' - 1)   
 in HOLogic.mk_Trueprop (HOLogic.mk_eq(t, term_of_fm T (im ~~ m) (im' ~~ m')  
                                                     (FRPar.frpar2 (fm_of_term (m ~~ im) (m' ~~ im') t))))
 end;

end;


*}

oracle frpar_oracle = {* fn (ty, ts, ts', ct) => 
 let 
  val thy = Thm.theory_of_cterm ct
 in cterm_of thy (ReflectedFRPar.frpar_oracle (ty,ts, ts', term_of ct))
 end *}

oracle frpar_oracle2 = {* fn (ty, ts, ts', ct) => 
 let 
  val thy = Thm.theory_of_cterm ct
 in cterm_of thy (ReflectedFRPar.frpar_oracle2 (ty,ts, ts', term_of ct))
 end *}

ML{* 
structure FRParTac = 
struct

fun frpar_tac T ps ctxt i = 
 (ObjectLogic.full_atomize_tac i) 
 THEN (fn st =>
  let
    val g = List.nth (cprems_of st, i - 1)
    val thy = ProofContext.theory_of ctxt
    val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
    val th = frpar_oracle (T, fs,ps, (* Pattern.eta_long [] *)g)
  in rtac (th RS iffD2) i st end);

fun frpar2_tac T ps ctxt i = 
 (ObjectLogic.full_atomize_tac i) 
 THEN (fn st =>
  let
    val g = List.nth (cprems_of st, i - 1)
    val thy = ProofContext.theory_of ctxt
    val fs = subtract (op aconv) (map Free (Term.add_frees (term_of g) [])) ps
    val th = frpar_oracle2 (T, fs,ps, (* Pattern.eta_long [] *)g)
  in rtac (th RS iffD2) i st end);

end;

*}

method_setup frpar = {*
let
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
 fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
 val parsN = "pars"
 val typN = "type"
 val any_keyword = keyword parsN || keyword typN
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
 val cterms = thms >> map Drule.dest_term;
 val terms = Scan.repeat (Scan.unless any_keyword Args.term)
 val typ = Scan.unless any_keyword Args.typ
in
 (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar_tac T ps ctxt))
end
*} "Parametric QE for linear Arithmetic over fields, Version 1"

method_setup frpar2 = {*
let
 fun keyword k = Scan.lift (Args.$$$ k -- Args.colon) >> K ()
 fun simple_keyword k = Scan.lift (Args.$$$ k) >> K ()
 val parsN = "pars"
 val typN = "type"
 val any_keyword = keyword parsN || keyword typN
 val thms = Scan.repeat (Scan.unless any_keyword Attrib.multi_thm) >> flat
 val cterms = thms >> map Drule.dest_term;
 val terms = Scan.repeat (Scan.unless any_keyword Args.term)
 val typ = Scan.unless any_keyword Args.typ
in
 (keyword typN |-- typ) -- (keyword parsN |-- terms) >>
  (fn (T,ps) => fn ctxt => SIMPLE_METHOD' (FRParTac.frpar2_tac T ps ctxt))
end
*} "Parametric QE for linear Arithmetic over fields, Version 2"


lemma "\<exists>(x::'a::{division_by_zero,ordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "y::'a::{division_by_zero,ordered_field,number_ring}")
  apply (simp add: ring_simps)
  apply (rule spec[where x=y])
  apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "z::'a::{division_by_zero,ordered_field,number_ring}")
  by simp

text{* Collins/Jones Problem *}
(*
lemma "\<exists>(r::'a::{division_by_zero,ordered_field,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"
proof-
  have "(\<exists>(r::'a::{division_by_zero,ordered_field,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::{division_by_zero,ordered_field,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")
by (simp add: ring_simps)
have "?rhs"

  apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "a::'a::{division_by_zero,ordered_field,number_ring}" "b::'a::{division_by_zero,ordered_field,number_ring}")
  apply (simp add: ring_simps)
oops
*)
(*
lemma "ALL (x::'a::{division_by_zero,ordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
apply (frpar type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "t::'a::{division_by_zero,ordered_field,number_ring}")
oops
*)

lemma "\<exists>(x::'a::{division_by_zero,ordered_field,number_ring}). y \<noteq> -1 \<longrightarrow> (y + 1)*x < 0"
  apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "y::'a::{division_by_zero,ordered_field,number_ring}")
  apply (simp add: ring_simps)
  apply (rule spec[where x=y])
  apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "z::'a::{division_by_zero,ordered_field,number_ring}")
  by simp

text{* Collins/Jones Problem *}

(*
lemma "\<exists>(r::'a::{division_by_zero,ordered_field,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"
proof-
  have "(\<exists>(r::'a::{division_by_zero,ordered_field,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::{division_by_zero,ordered_field,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")
by (simp add: ring_simps)
have "?rhs"
  apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "a::'a::{division_by_zero,ordered_field,number_ring}" "b::'a::{division_by_zero,ordered_field,number_ring}")
  apply simp
oops
*)

(*
lemma "ALL (x::'a::{division_by_zero,ordered_field,number_ring}) y. (1 - t)*x \<le> (1+t)*y \<and> (1 - t)*y \<le> (1+t)*x --> 0 \<le> y"
apply (frpar2 type: "'a::{division_by_zero,ordered_field,number_ring}" pars: "t::'a::{division_by_zero,ordered_field,number_ring}")
apply (simp add: field_simps linorder_neq_iff[symmetric])
apply ferrack
oops
*)
end