src/HOL/Decision_Procs/Cooper.thy

Updated paths in Decision_Procs comments and NEWS

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

theory Cooper
imports Complex_Main Efficient_Nat
uses ("cooper_tac.ML")
begin

function iupt :: "int \<Rightarrow> int \<Rightarrow> int list" where
  "iupt i j = (if j < i then [] else i # iupt (i+1) j)"
by pat_completeness auto
termination by (relation "measure (\<lambda> (i, j). nat (j-i+1))") auto

lemma iupt_set: "set (iupt i j) = {i..j}"
  by (induct rule: iupt.induct) (simp add: simp_from_to)

(* Periodicity of dvd *)

  (*********************************************************************************)
  (****                            SHADOW SYNTAX AND SEMANTICS                  ****)
  (*********************************************************************************)

datatype num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num 
  | Mul int num

  (* A size for num to make inductive proofs simpler*)
primrec num_size :: "num \<Rightarrow> nat" where
  "num_size (C c) = 1"
| "num_size (Bound n) = 1"
| "num_size (Neg a) = 1 + num_size a"
| "num_size (Add a b) = 1 + num_size a + num_size b"
| "num_size (Sub a b) = 3 + num_size a + num_size b"
| "num_size (CN n c a) = 4 + num_size a"
| "num_size (Mul c a) = 1 + num_size a"

primrec Inum :: "int list \<Rightarrow> num \<Rightarrow> int" where
  "Inum bs (C c) = c"
| "Inum bs (Bound n) = bs!n"
| "Inum bs (CN n c a) = c * (bs!n) + (Inum bs a)"
| "Inum bs (Neg a) = -(Inum bs a)"
| "Inum bs (Add a b) = Inum bs a + Inum bs b"
| "Inum bs (Sub a b) = Inum bs a - Inum bs b"
| "Inum bs (Mul c a) = c* Inum bs a"

datatype fm  = 
  T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num| Dvd int num| NDvd int num|
  NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm 
  | Closed nat | NClosed nat


  (* 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 (Dvd i t) = 2"
  "fmsize (NDvd i t) = 2"
  "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 ::"bool list \<Rightarrow> int list \<Rightarrow> fm \<Rightarrow> bool"
primrec
  "Ifm bbs bs T = True"
  "Ifm bbs bs F = False"
  "Ifm bbs bs (Lt a) = (Inum bs a < 0)"
  "Ifm bbs bs (Gt a) = (Inum bs a > 0)"
  "Ifm bbs bs (Le a) = (Inum bs a \<le> 0)"
  "Ifm bbs bs (Ge a) = (Inum bs a \<ge> 0)"
  "Ifm bbs bs (Eq a) = (Inum bs a = 0)"
  "Ifm bbs bs (NEq a) = (Inum bs a \<noteq> 0)"
  "Ifm bbs bs (Dvd i b) = (i dvd Inum bs b)"
  "Ifm bbs bs (NDvd i b) = (\<not>(i dvd Inum bs b))"
  "Ifm bbs bs (NOT p) = (\<not> (Ifm bbs bs p))"
  "Ifm bbs bs (And p q) = (Ifm bbs bs p \<and> Ifm bbs bs q)"
  "Ifm bbs bs (Or p q) = (Ifm bbs bs p \<or> Ifm bbs bs q)"
  "Ifm bbs bs (Imp p q) = ((Ifm bbs bs p) \<longrightarrow> (Ifm bbs bs q))"
  "Ifm bbs bs (Iff p q) = (Ifm bbs bs p = Ifm bbs bs q)"
  "Ifm bbs bs (E p) = (\<exists> x. Ifm bbs (x#bs) p)"
  "Ifm bbs bs (A p) = (\<forall> x. Ifm bbs (x#bs) p)"
  "Ifm bbs bs (Closed n) = bbs!n"
  "Ifm bbs bs (NClosed n) = (\<not> bbs!n)"

consts prep :: "fm \<Rightarrow> fm"
recdef prep "measure fmsize"
  "prep (E T) = T"
  "prep (E F) = F"
  "prep (E (Or p q)) = Or (prep (E p)) (prep (E q))"
  "prep (E (Imp p q)) = Or (prep (E (NOT p))) (prep (E q))"
  "prep (E (Iff p q)) = Or (prep (E (And p q))) (prep (E (And (NOT p) (NOT q))))" 
  "prep (E (NOT (And p q))) = Or (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))) = Or (prep (E (And p (NOT q)))) (prep (E(And (NOT p) q)))"
  "prep (E p) = E (prep p)"
  "prep (A (And p q)) = And (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)) = Or (prep (NOT p)) (prep (NOT q))"
  "prep (NOT (A p)) = prep (E (NOT p))"
  "prep (NOT (Or p q)) = And (prep (NOT p)) (prep (NOT q))"
  "prep (NOT (Imp p q)) = And (prep p) (prep (NOT q))"
  "prep (NOT (Iff p q)) = Or (prep (And p (NOT q))) (prep (And (NOT p) q))"
  "prep (NOT p) = NOT (prep p)"
  "prep (Or p q) = Or (prep p) (prep q)"
  "prep (And p q) = And (prep p) (prep q)"
  "prep (Imp p q) = prep (Or (NOT p) q)"
  "prep (Iff p q) = Or (prep (And p q)) (prep (And (NOT p) (NOT q)))"
  "prep p = p"
(hints simp add: fmsize_pos)
lemma prep: "Ifm bbs bs (prep p) = Ifm bbs bs p"
by (induct p arbitrary: bs rule: prep.induct, 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 
  numbound0:: "num \<Rightarrow> bool" (* a num is INDEPENDENT of Bound 0 *)
  bound0:: "fm \<Rightarrow> bool" (* A Formula is independent of Bound 0 *)
  subst0:: "num \<Rightarrow> fm \<Rightarrow> fm" (* substitue a num into a formula for Bound 0 *)
primrec
  "numbound0 (C c) = True"
  "numbound0 (Bound n) = (n>0)"
  "numbound0 (CN n i a) = (n >0 \<and> numbound0 a)"
  "numbound0 (Neg a) = numbound0 a"
  "numbound0 (Add a b) = (numbound0 a \<and> numbound0 b)"
  "numbound0 (Sub a b) = (numbound0 a \<and> numbound0 b)" 
  "numbound0 (Mul i a) = numbound0 a"

lemma numbound0_I:
  assumes nb: "numbound0 a"
  shows "Inum (b#bs) a = Inum (b'#bs) a"
using nb
by (induct a rule: numbound0.induct) (auto simp add: gr0_conv_Suc)

primrec
  "bound0 T = True"
  "bound0 F = True"
  "bound0 (Lt a) = numbound0 a"
  "bound0 (Le a) = numbound0 a"
  "bound0 (Gt a) = numbound0 a"
  "bound0 (Ge a) = numbound0 a"
  "bound0 (Eq a) = numbound0 a"
  "bound0 (NEq a) = numbound0 a"
  "bound0 (Dvd i a) = numbound0 a"
  "bound0 (NDvd i a) = numbound0 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 (E p) = False"
  "bound0 (A p) = False"
  "bound0 (Closed P) = True"
  "bound0 (NClosed P) = True"
lemma bound0_I:
  assumes bp: "bound0 p"
  shows "Ifm bbs (b#bs) p = Ifm bbs (b'#bs) p"
using bp numbound0_I[where b="b" and bs="bs" and b'="b'"]
by (induct p rule: bound0.induct) (auto simp add: gr0_conv_Suc)

fun numsubst0:: "num \<Rightarrow> num \<Rightarrow> num" where
  "numsubst0 t (C c) = (C c)"
| "numsubst0 t (Bound n) = (if n=0 then t else Bound n)"
| "numsubst0 t (CN 0 i a) = Add (Mul i t) (numsubst0 t a)"
| "numsubst0 t (CN n i a) = CN n i (numsubst0 t a)"
| "numsubst0 t (Neg a) = Neg (numsubst0 t a)"
| "numsubst0 t (Add a b) = Add (numsubst0 t a) (numsubst0 t b)"
| "numsubst0 t (Sub a b) = Sub (numsubst0 t a) (numsubst0 t b)" 
| "numsubst0 t (Mul i a) = Mul i (numsubst0 t a)"

lemma numsubst0_I:
  "Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b#bs) a)#bs) t"
by (induct t rule: numsubst0.induct,auto simp:nth_Cons')

lemma numsubst0_I':
  "numbound0 a \<Longrightarrow> Inum (b#bs) (numsubst0 a t) = Inum ((Inum (b'#bs) a)#bs) t"
by (induct t rule: numsubst0.induct, auto simp: nth_Cons' numbound0_I[where b="b" and b'="b'"])

primrec
  "subst0 t T = T"
  "subst0 t F = F"
  "subst0 t (Lt a) = Lt (numsubst0 t a)"
  "subst0 t (Le a) = Le (numsubst0 t a)"
  "subst0 t (Gt a) = Gt (numsubst0 t a)"
  "subst0 t (Ge a) = Ge (numsubst0 t a)"
  "subst0 t (Eq a) = Eq (numsubst0 t a)"
  "subst0 t (NEq a) = NEq (numsubst0 t a)"
  "subst0 t (Dvd i a) = Dvd i (numsubst0 t a)"
  "subst0 t (NDvd i a) = NDvd i (numsubst0 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 (Closed P) = (Closed P)"
  "subst0 t (NClosed P) = (NClosed P)"

lemma subst0_I: assumes qfp: "qfree p"
  shows "Ifm bbs (b#bs) (subst0 a p) = Ifm bbs ((Inum (b#bs) a)#bs) p"
  using qfp numsubst0_I[where b="b" and bs="bs" and a="a"]
  by (induct p) (simp_all add: gr0_conv_Suc)


consts 
  decrnum:: "num \<Rightarrow> num" 
  decr :: "fm \<Rightarrow> fm"

recdef decrnum "measure size"
  "decrnum (Bound n) = Bound (n - 1)"
  "decrnum (Neg a) = Neg (decrnum a)"
  "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
  "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
  "decrnum (Mul c a) = Mul c (decrnum a)"
  "decrnum (CN n i a) = (CN (n - 1) i (decrnum a))"
  "decrnum a = a"

recdef decr "measure size"
  "decr (Lt a) = Lt (decrnum a)"
  "decr (Le a) = Le (decrnum a)"
  "decr (Gt a) = Gt (decrnum a)"
  "decr (Ge a) = Ge (decrnum a)"
  "decr (Eq a) = Eq (decrnum a)"
  "decr (NEq a) = NEq (decrnum a)"
  "decr (Dvd i a) = Dvd i (decrnum a)"
  "decr (NDvd i a) = NDvd i (decrnum a)"
  "decr (NOT p) = NOT (decr p)" 
  "decr (And p q) = And (decr p) (decr q)"
  "decr (Or p q) = Or (decr p) (decr q)"
  "decr (Imp p q) = Imp (decr p) (decr q)"
  "decr (Iff p q) = Iff (decr p) (decr q)"
  "decr p = p"

lemma decrnum: assumes nb: "numbound0 t"
  shows "Inum (x#bs) t = Inum bs (decrnum t)"
  using nb by (induct t rule: decrnum.induct, auto simp add: gr0_conv_Suc)

lemma decr: assumes nb: "bound0 p"
  shows "Ifm bbs (x#bs) p = Ifm bbs bs (decr p)"
  using nb 
  by (induct p rule: decr.induct, simp_all add: gr0_conv_Suc decrnum)

lemma decr_qf: "bound0 p \<Longrightarrow> qfree (decr 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 (Gt a) = True"
  "isatom (Ge a) = True"
  "isatom (Eq a) = True"
  "isatom (NEq a) = True"
  "isatom (Dvd i b) = True"
  "isatom (NDvd i b) = True"
  "isatom (Closed P) = True"
  "isatom (NClosed P) = True"
  "isatom p = False"

lemma numsubst0_numbound0: assumes nb: "numbound0 t"
  shows "numbound0 (numsubst0 t a)"
using nb apply (induct a rule: numbound0.induct)
apply simp_all
apply (case_tac n, simp_all)
done

lemma subst0_bound0: assumes qf: "qfree p" and nb: "numbound0 t"
  shows "bound0 (subst0 t p)"
using qf numsubst0_numbound0[OF nb] by (induct p  rule: subst0.induct, auto)

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 bbs bs (djf f p q) = Ifm bbs 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 bbs bs (evaldjf f ps) = (\<exists> p \<in> set ps. Ifm bbs 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 bbs bs q) = Ifm bbs 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. f (Or p q) = Or (f p) (f q)"
  and fF: "f F = F"
  shows "Ifm bbs bs (DJ f p) = Ifm bbs bs (f p)"
proof-
  have "Ifm bbs bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bbs bs (f q))"
    by (simp add: DJ_def evaldjf_ex) 
  also have "\<dots> = Ifm bbs 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 bbs bs (qe p) = Ifm bbs bs (E p))"
  shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bbs bs ((DJ qe p)) = Ifm bbs 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 bbs bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bbs bs (qe q))"
    by (simp add: DJ_def evaldjf_ex)
  also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bbs bs (E q))" using qe disjuncts_qf[OF qf] by auto
  also have "\<dots> = Ifm bbs bs (E p)" by (induct p rule: disjuncts.induct, auto)
  finally show "qfree (DJ qe p) \<and> Ifm bbs bs (DJ qe p) = Ifm bbs bs (E p)" using qfth by blast
qed
  (* Simplification *)

  (* Algebraic simplifications for nums *)
consts bnds:: "num \<Rightarrow> nat list"
  lex_ns:: "nat list \<times> nat list \<Rightarrow> bool"
recdef bnds "measure size"
  "bnds (Bound n) = [n]"
  "bnds (CN n c a) = n#(bnds a)"
  "bnds (Neg a) = bnds a"
  "bnds (Add a b) = (bnds a)@(bnds b)"
  "bnds (Sub a b) = (bnds a)@(bnds b)"
  "bnds (Mul i a) = bnds a"
  "bnds a = []"
recdef lex_ns "measure (\<lambda> (xs,ys). length xs + length ys)"
  "lex_ns ([], ms) = True"
  "lex_ns (ns, []) = False"
  "lex_ns (n#ns, m#ms) = (n<m \<or> ((n = m) \<and> lex_ns (ns,ms))) "
constdefs lex_bnd :: "num \<Rightarrow> num \<Rightarrow> bool"
  "lex_bnd t s \<equiv> lex_ns (bnds t, bnds s)"

consts
  numadd:: "num \<times> num \<Rightarrow> num"
recdef numadd "measure (\<lambda> (t,s). num_size t + num_size s)"
  "numadd (CN n1 c1 r1 ,CN n2 c2 r2) =
  (if n1=n2 then 
  (let c = c1 + c2
  in (if c=0 then numadd(r1,r2) else CN n1 c (numadd (r1,r2))))
  else if n1 \<le> n2 then CN n1 c1 (numadd (r1,Add (Mul c2 (Bound n2)) r2))
  else CN n2 c2 (numadd (Add (Mul c1 (Bound n1)) r1,r2)))"
  "numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))"  
  "numadd (t,CN n2 c2 r2) = CN n2 c2 (numadd (t,r2))" 
  "numadd (C b1, C b2) = C (b1+b2)"
  "numadd (a,b) = Add a b"

(*function (sequential)
  numadd :: "num \<Rightarrow> num \<Rightarrow> num"
where
  "numadd (Add (Mul c1 (Bound n1)) r1) (Add (Mul c2 (Bound n2)) r2) =
      (if n1 = n2 then (let c = c1 + c2
      in (if c = 0 then numadd r1 r2 else
        Add (Mul c (Bound n1)) (numadd r1 r2)))
      else if n1 \<le> n2 then
        Add (Mul c1 (Bound n1)) (numadd r1 (Add (Mul c2 (Bound n2)) r2))
      else
        Add (Mul c2 (Bound n2)) (numadd (Add (Mul c1 (Bound n1)) r1) r2))"
  | "numadd (Add (Mul c1 (Bound n1)) r1) t =
      Add (Mul c1 (Bound n1)) (numadd r1 t)"  
  | "numadd t (Add (Mul c2 (Bound n2)) r2) =
      Add (Mul c2 (Bound n2)) (numadd t r2)" 
  | "numadd (C b1) (C b2) = C (b1 + b2)"
  | "numadd a b = Add a b"
apply pat_completeness apply auto*)
  
lemma numadd: "Inum bs (numadd (t,s)) = Inum bs (Add t s)"
apply (induct t s rule: numadd.induct, simp_all add: Let_def)
apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
 apply (case_tac "n1 = n2")
  apply(simp_all add: algebra_simps)
apply(simp add: left_distrib[symmetric])
done

lemma numadd_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"
by (induct t s rule: numadd.induct, auto simp add: Let_def)

fun
  nummul :: "int \<Rightarrow> num \<Rightarrow> num"
where
  "nummul i (C j) = C (i * j)"
  | "nummul i (CN n c t) = CN n (c*i) (nummul i t)"
  | "nummul i t = Mul i t"

lemma nummul: "\<And> i. Inum bs (nummul i t) = Inum bs (Mul i t)"
by (induct t rule: nummul.induct, auto simp add: algebra_simps numadd)

lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"
by (induct t rule: nummul.induct, auto simp add: numadd_nb)

constdefs numneg :: "num \<Rightarrow> num"
  "numneg t \<equiv> nummul (- 1) t"

constdefs numsub :: "num \<Rightarrow> num \<Rightarrow> num"
  "numsub s t \<equiv> (if s = t then C 0 else numadd (s, numneg t))"

lemma numneg: "Inum bs (numneg t) = Inum bs (Neg t)"
using numneg_def nummul by simp

lemma numneg_nb: "numbound0 t \<Longrightarrow> numbound0 (numneg t)"
using numneg_def nummul_nb by simp

lemma numsub: "Inum bs (numsub a b) = Inum bs (Sub a b)"
using numneg numadd numsub_def by simp

lemma numsub_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numsub t s)"
using numsub_def numadd_nb numneg_nb by simp

fun
  simpnum :: "num \<Rightarrow> num"
where
  "simpnum (C j) = C j"
  | "simpnum (Bound n) = CN n 1 (C 0)"
  | "simpnum (Neg t) = numneg (simpnum t)"
  | "simpnum (Add t s) = numadd (simpnum t, simpnum s)"
  | "simpnum (Sub t s) = numsub (simpnum t) (simpnum s)"
  | "simpnum (Mul i t) = (if i = 0 then C 0 else nummul i (simpnum t))"
  | "simpnum t = t"

lemma simpnum_ci: "Inum bs (simpnum t) = Inum bs t"
by (induct t rule: simpnum.induct, auto simp add: numneg numadd numsub nummul)

lemma simpnum_numbound0: 
  "numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
by (induct t rule: simpnum.induct, auto simp add: numadd_nb numsub_nb nummul_nb numneg_nb)

fun
  not :: "fm \<Rightarrow> fm"
where
  "not (NOT p) = p"
  | "not T = F"
  | "not F = T"
  | "not p = NOT p"
lemma not: "Ifm bbs bs (not p) = Ifm bbs bs (NOT p)"
by (cases p) auto
lemma not_qf: "qfree p \<Longrightarrow> qfree (not p)"
by (cases p, auto)
lemma not_bn: "bound0 p \<Longrightarrow> bound0 (not p)"
by (cases p, 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 And p q)"
lemma conj: "Ifm bbs bs (conj p q) = Ifm bbs bs (And p q)"
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)

lemma conj_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (conj p q)"
using conj_def by auto 
lemma conj_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (conj p q)"
using conj_def by auto 

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 Or p q)"

lemma disj: "Ifm bbs bs (disj p q) = Ifm bbs bs (Or p q)"
by (cases "p=T \<or> q=T",simp_all add: disj_def) (cases p,simp_all)
lemma disj_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (disj p q)"
using disj_def by auto 
lemma disj_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (disj p q)"
using disj_def by auto 

constdefs   imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
  "imp p q \<equiv> (if (p = F \<or> q=T) then T else if p=T then q else if q=F then not p else Imp p q)"
lemma imp: "Ifm bbs bs (imp p q) = Ifm bbs bs (Imp p q)"
by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) (simp_all add: not)
lemma imp_qf: "\<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,cases p) (simp_all add: not_qf) 
lemma imp_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (imp p q)"
using imp_def by (cases "p=F \<or> q=T",simp_all add: imp_def,cases p) simp_all

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: "Ifm bbs bs (iff p q) = Ifm bbs bs (Iff p q)"
  by (unfold iff_def,cases "p=q", simp,cases "p=not q", simp add:not) 
(cases "not p= q", auto simp add:not)
lemma iff_qf: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
  by (unfold iff_def,cases "p=q", auto simp add: not_qf)
lemma iff_nb: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto simp add: not_bn)

function (sequential)
  simpfm :: "fm \<Rightarrow> fm"
where
  "simpfm (And p q) = conj (simpfm p) (simpfm q)"
  | "simpfm (Or p q) = disj (simpfm p) (simpfm q)"
  | "simpfm (Imp p q) = imp (simpfm p) (simpfm q)"
  | "simpfm (Iff p q) = iff (simpfm p) (simpfm q)"
  | "simpfm (NOT p) = not (simpfm p)"
  | "simpfm (Lt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v < 0) then T else F 
      | _ \<Rightarrow> Lt a')"
  | "simpfm (Le a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<le> 0)  then T else F | _ \<Rightarrow> Le a')"
  | "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v > 0)  then T else F | _ \<Rightarrow> Gt a')"
  | "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<ge> 0)  then T else F | _ \<Rightarrow> Ge a')"
  | "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v = 0)  then T else F | _ \<Rightarrow> Eq a')"
  | "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \<Rightarrow> if (v \<noteq> 0)  then T else F | _ \<Rightarrow> NEq a')"
  | "simpfm (Dvd i a) = (if i=0 then simpfm (Eq a)
             else if (abs i = 1) then T
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (i dvd v)  then T else F | _ \<Rightarrow> Dvd i a')"
  | "simpfm (NDvd i a) = (if i=0 then simpfm (NEq a) 
             else if (abs i = 1) then F
             else let a' = simpnum a in case a' of C v \<Rightarrow> if (\<not>(i dvd v)) then T else F | _ \<Rightarrow> NDvd i a')"
  | "simpfm p = p"
by pat_completeness auto
termination by (relation "measure fmsize") auto

lemma simpfm: "Ifm bbs bs (simpfm p) = Ifm bbs bs p"
proof(induct p rule: simpfm.induct)
  case (6 a) let ?sa = "simpnum a" from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (7 a)  let ?sa = "simpnum a" 
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (8 a)  let ?sa = "simpnum a" 
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (9 a)  let ?sa = "simpnum a" 
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (10 a)  let ?sa = "simpnum a" 
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (11 a)  let ?sa = "simpnum a" 
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a" by simp
  {fix v assume "?sa = C v" hence ?case using sa by simp }
  moreover {assume "\<not> (\<exists> v. ?sa = C v)" hence ?case using sa 
      by (cases ?sa, simp_all add: Let_def)}
  ultimately show ?case by blast
next
  case (12 i a)  let ?sa = "simpnum a" from simpnum_ci 
  have sa: "Inum bs ?sa = Inum bs a" by simp
  have "i=0 \<or> abs i = 1 \<or> (i\<noteq>0 \<and> (abs i \<noteq> 1))" by auto
  {assume "i=0" hence ?case using "12.hyps" by (simp add: dvd_def Let_def)}
  moreover 
  {assume i1: "abs i = 1"
      from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
      have ?case using i1 apply (cases "i=0", simp_all add: Let_def) 
	by (cases "i > 0", simp_all)}
  moreover   
  {assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
	by (cases "abs i = 1", auto) }
    moreover {assume "\<not> (\<exists> v. ?sa = C v)" 
      hence "simpfm (Dvd i a) = Dvd i ?sa" using inz cond 
	by (cases ?sa, auto simp add: Let_def)
      hence ?case using sa by simp}
    ultimately have ?case by blast}
  ultimately show ?case by blast
next
  case (13 i a)  let ?sa = "simpnum a" from simpnum_ci 
  have sa: "Inum bs ?sa = Inum bs a" by simp
  have "i=0 \<or> abs i = 1 \<or> (i\<noteq>0 \<and> (abs i \<noteq> 1))" by auto
  {assume "i=0" hence ?case using "13.hyps" by (simp add: dvd_def Let_def)}
  moreover 
  {assume i1: "abs i = 1"
      from one_dvd[of "Inum bs a"] uminus_dvd_conv[where d="1" and t="Inum bs a"]
      have ?case using i1 apply (cases "i=0", simp_all add: Let_def)
      apply (cases "i > 0", simp_all) done}
  moreover   
  {assume inz: "i\<noteq>0" and cond: "abs i \<noteq> 1"
    {fix v assume "?sa = C v" hence ?case using sa[symmetric] inz cond
	by (cases "abs i = 1", auto) }
    moreover {assume "\<not> (\<exists> v. ?sa = C v)" 
      hence "simpfm (NDvd i a) = NDvd i ?sa" using inz cond 
	by (cases ?sa, auto simp add: Let_def)
      hence ?case using sa by simp}
    ultimately have ?case by blast}
  ultimately show ?case by blast
qed (induct p rule: simpfm.induct, simp_all add: conj disj imp iff not)

lemma simpfm_bound0: "bound0 p \<Longrightarrow> bound0 (simpfm p)"
proof(induct p rule: simpfm.induct)
  case (6 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (7 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (8 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (9 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (10 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (11 a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (12 i a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
next
  case (13 i a) hence nb: "numbound0 a" by simp
  hence "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  thus ?case by (cases "simpnum a", auto simp add: Let_def)
qed(auto simp add: disj_def imp_def iff_def conj_def not_bn)

lemma simpfm_qf: "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)
 (case_tac "simpnum a",auto)+

  (* Generic quantifier elimination *)
consts qelim :: "fm \<Rightarrow> (fm \<Rightarrow> fm) \<Rightarrow> fm"
recdef qelim "measure fmsize"
  "qelim (E p) = (\<lambda> qe. DJ 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)"

(*function (sequential)
  qelim :: "(fm \<Rightarrow> fm) \<Rightarrow> fm \<Rightarrow> fm"
where
  "qelim qe (E p) = DJ qe (qelim qe p)"
  | "qelim qe (A p) = not (qe ((qelim qe (NOT p))))"
  | "qelim qe (NOT p) = not (qelim qe p)"
  | "qelim qe (And p q) = conj (qelim qe p) (qelim qe q)" 
  | "qelim qe (Or  p q) = disj (qelim qe p) (qelim qe q)" 
  | "qelim qe (Imp p q) = imp (qelim qe p) (qelim qe q)"
  | "qelim qe (Iff p q) = iff (qelim qe p) (qelim qe q)"
  | "qelim qe p = simpfm p"
by pat_completeness auto
termination by (relation "measure (fmsize o snd)") auto*)

lemma qelim_ci:
  assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bbs bs (qe p) = Ifm bbs bs (E p))"
  shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bbs bs (qelim p qe) = Ifm bbs bs p)"
using qe_inv DJ_qe[OF qe_inv] 
by(induct p rule: qelim.induct) 
(auto simp add: not disj conj iff imp not_qf disj_qf conj_qf imp_qf iff_qf 
  simpfm simpfm_qf simp del: simpfm.simps)
  (* Linearity for fm where Bound 0 ranges over \<int> *)

fun
  zsplit0 :: "num \<Rightarrow> int \<times> num" (* splits the bounded from the unbounded part*)
where
  "zsplit0 (C c) = (0,C c)"
  | "zsplit0 (Bound n) = (if n=0 then (1, C 0) else (0,Bound n))"
  | "zsplit0 (CN n i a) = 
      (let (i',a') =  zsplit0 a 
       in if n=0 then (i+i', a') else (i',CN n i a'))"
  | "zsplit0 (Neg a) = (let (i',a') =  zsplit0 a in (-i', Neg a'))"
  | "zsplit0 (Add a b) = (let (ia,a') =  zsplit0 a ; 
                            (ib,b') =  zsplit0 b 
                            in (ia+ib, Add a' b'))"
  | "zsplit0 (Sub a b) = (let (ia,a') =  zsplit0 a ; 
                            (ib,b') =  zsplit0 b 
                            in (ia-ib, Sub a' b'))"
  | "zsplit0 (Mul i a) = (let (i',a') =  zsplit0 a in (i*i', Mul i a'))"

lemma zsplit0_I:
  shows "\<And> n a. zsplit0 t = (n,a) \<Longrightarrow> (Inum ((x::int) #bs) (CN 0 n a) = Inum (x #bs) t) \<and> numbound0 a"
  (is "\<And> n a. ?S t = (n,a) \<Longrightarrow> (?I x (CN 0 n a) = ?I x t) \<and> ?N a")
proof(induct t rule: zsplit0.induct)
  case (1 c n a) thus ?case by auto 
next
  case (2 m n a) thus ?case by (cases "m=0") auto
next
  case (3 m i a n a')
  let ?j = "fst (zsplit0 a)"
  let ?b = "snd (zsplit0 a)"
  have abj: "zsplit0 a = (?j,?b)" by simp 
  {assume "m\<noteq>0" 
    with prems(1)[OF abj] prems(2) have ?case by (auto simp add: Let_def split_def)}
  moreover
  {assume m0: "m =0"
    from abj have th: "a'=?b \<and> n=i+?j" using prems 
      by (simp add: Let_def split_def)
    from abj prems  have th2: "(?I x (CN 0 ?j ?b) = ?I x a) \<and> ?N ?b" by blast
    from th have "?I x (CN 0 n a') = ?I x (CN 0 (i+?j) ?b)" by simp
    also from th2 have "\<dots> = ?I x (CN 0 i (CN 0 ?j ?b))" by (simp add: left_distrib)
  finally have "?I x (CN 0 n a') = ?I  x (CN 0 i a)" using th2 by simp
  with th2 th have ?case using m0 by blast} 
ultimately show ?case by blast
next
  case (4 t n a)
  let ?nt = "fst (zsplit0 t)"
  let ?at = "snd (zsplit0 t)"
  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Neg ?at \<and> n=-?nt" using prems 
    by (simp add: Let_def split_def)
  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
  from th2[simplified] th[simplified] show ?case by simp
next
  case (5 s t n a)
  let ?ns = "fst (zsplit0 s)"
  let ?as = "snd (zsplit0 s)"
  let ?nt = "fst (zsplit0 t)"
  let ?at = "snd (zsplit0 t)"
  have abjs: "zsplit0 s = (?ns,?as)" by simp 
  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
  ultimately have th: "a=Add ?as ?at \<and> n=?ns + ?nt" using prems 
    by (simp add: Let_def split_def)
  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
  from th3[simplified] th2[simplified] th[simplified] show ?case 
    by (simp add: left_distrib)
next
  case (6 s t n a)
  let ?ns = "fst (zsplit0 s)"
  let ?as = "snd (zsplit0 s)"
  let ?nt = "fst (zsplit0 t)"
  let ?at = "snd (zsplit0 t)"
  have abjs: "zsplit0 s = (?ns,?as)" by simp 
  moreover have abjt:  "zsplit0 t = (?nt,?at)" by simp 
  ultimately have th: "a=Sub ?as ?at \<and> n=?ns - ?nt" using prems 
    by (simp add: Let_def split_def)
  from abjs[symmetric] have bluddy: "\<exists> x y. (x,y) = zsplit0 s" by blast
  from prems have "(\<exists> x y. (x,y) = zsplit0 s) \<longrightarrow> (\<forall>xa xb. zsplit0 t = (xa, xb) \<longrightarrow> Inum (x # bs) (CN 0 xa xb) = Inum (x # bs) t \<and> numbound0 xb)" by auto
  with bluddy abjt have th3: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
  from abjs prems  have th2: "(?I x (CN 0 ?ns ?as) = ?I x s) \<and> ?N ?as" by blast
  from th3[simplified] th2[simplified] th[simplified] show ?case 
    by (simp add: left_diff_distrib)
next
  case (7 i t n a)
  let ?nt = "fst (zsplit0 t)"
  let ?at = "snd (zsplit0 t)"
  have abj: "zsplit0 t = (?nt,?at)" by simp hence th: "a=Mul i ?at \<and> n=i*?nt" using prems 
    by (simp add: Let_def split_def)
  from abj prems  have th2: "(?I x (CN 0 ?nt ?at) = ?I x t) \<and> ?N ?at" by blast
  hence " ?I x (Mul i t) = i * ?I x (CN 0 ?nt ?at)" by simp
  also have "\<dots> = ?I x (CN 0 (i*?nt) (Mul i ?at))" by (simp add: right_distrib)
  finally show ?case using th th2 by simp
qed

consts
  iszlfm :: "fm \<Rightarrow> bool"   (* Linearity test for fm *)
recdef iszlfm "measure size"
  "iszlfm (And p q) = (iszlfm p \<and> iszlfm q)" 
  "iszlfm (Or p q) = (iszlfm p \<and> iszlfm q)" 
  "iszlfm (Eq  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
  "iszlfm (NEq (CN 0 c e)) = (c>0 \<and> numbound0 e)"
  "iszlfm (Lt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
  "iszlfm (Le  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
  "iszlfm (Gt  (CN 0 c e)) = (c>0 \<and> numbound0 e)"
  "iszlfm (Ge  (CN 0 c e)) = ( c>0 \<and> numbound0 e)"
  "iszlfm (Dvd i (CN 0 c e)) = 
                 (c>0 \<and> i>0 \<and> numbound0 e)"
  "iszlfm (NDvd i (CN 0 c e))= 
                 (c>0 \<and> i>0 \<and> numbound0 e)"
  "iszlfm p = (isatom p \<and> (bound0 p))"

lemma zlin_qfree: "iszlfm p \<Longrightarrow> qfree p"
  by (induct p rule: iszlfm.induct) auto

consts
  zlfm :: "fm \<Rightarrow> fm"       (* Linearity transformation for fm *)
recdef zlfm "measure fmsize"
  "zlfm (And p q) = And (zlfm p) (zlfm q)"
  "zlfm (Or p q) = Or (zlfm p) (zlfm q)"
  "zlfm (Imp p q) = Or (zlfm (NOT p)) (zlfm q)"
  "zlfm (Iff p q) = Or (And (zlfm p) (zlfm q)) (And (zlfm (NOT p)) (zlfm (NOT q)))"
  "zlfm (Lt a) = (let (c,r) = zsplit0 a in 
     if c=0 then Lt r else 
     if c>0 then (Lt (CN 0 c r)) else (Gt (CN 0 (- c) (Neg r))))"
  "zlfm (Le a) = (let (c,r) = zsplit0 a in 
     if c=0 then Le r else 
     if c>0 then (Le (CN 0 c r)) else (Ge (CN 0 (- c) (Neg r))))"
  "zlfm (Gt a) = (let (c,r) = zsplit0 a in 
     if c=0 then Gt r else 
     if c>0 then (Gt (CN 0 c r)) else (Lt (CN 0 (- c) (Neg r))))"
  "zlfm (Ge a) = (let (c,r) = zsplit0 a in 
     if c=0 then Ge r else 
     if c>0 then (Ge (CN 0 c r)) else (Le (CN 0 (- c) (Neg r))))"
  "zlfm (Eq a) = (let (c,r) = zsplit0 a in 
     if c=0 then Eq r else 
     if c>0 then (Eq (CN 0 c r)) else (Eq (CN 0 (- c) (Neg r))))"
  "zlfm (NEq a) = (let (c,r) = zsplit0 a in 
     if c=0 then NEq r else 
     if c>0 then (NEq (CN 0 c r)) else (NEq (CN 0 (- c) (Neg r))))"
  "zlfm (Dvd i a) = (if i=0 then zlfm (Eq a) 
        else (let (c,r) = zsplit0 a in 
              if c=0 then (Dvd (abs i) r) else 
      if c>0 then (Dvd (abs i) (CN 0 c r))
      else (Dvd (abs i) (CN 0 (- c) (Neg r)))))"
  "zlfm (NDvd i a) = (if i=0 then zlfm (NEq a) 
        else (let (c,r) = zsplit0 a in 
              if c=0 then (NDvd (abs i) r) else 
      if c>0 then (NDvd (abs i) (CN 0 c r))
      else (NDvd (abs i) (CN 0 (- c) (Neg r)))))"
  "zlfm (NOT (And p q)) = Or (zlfm (NOT p)) (zlfm (NOT q))"
  "zlfm (NOT (Or p q)) = And (zlfm (NOT p)) (zlfm (NOT q))"
  "zlfm (NOT (Imp p q)) = And (zlfm p) (zlfm (NOT q))"
  "zlfm (NOT (Iff p q)) = Or (And(zlfm p) (zlfm(NOT q))) (And (zlfm(NOT p)) (zlfm q))"
  "zlfm (NOT (NOT p)) = zlfm p"
  "zlfm (NOT T) = F"
  "zlfm (NOT F) = T"
  "zlfm (NOT (Lt a)) = zlfm (Ge a)"
  "zlfm (NOT (Le a)) = zlfm (Gt a)"
  "zlfm (NOT (Gt a)) = zlfm (Le a)"
  "zlfm (NOT (Ge a)) = zlfm (Lt a)"
  "zlfm (NOT (Eq a)) = zlfm (NEq a)"
  "zlfm (NOT (NEq a)) = zlfm (Eq a)"
  "zlfm (NOT (Dvd i a)) = zlfm (NDvd i a)"
  "zlfm (NOT (NDvd i a)) = zlfm (Dvd i a)"
  "zlfm (NOT (Closed P)) = NClosed P"
  "zlfm (NOT (NClosed P)) = Closed P"
  "zlfm p = p" (hints simp add: fmsize_pos)

lemma zlfm_I:
  assumes qfp: "qfree p"
  shows "(Ifm bbs (i#bs) (zlfm p) = Ifm bbs (i# bs) p) \<and> iszlfm (zlfm p)"
  (is "(?I (?l p) = ?I p) \<and> ?L (?l p)")
using qfp
proof(induct p rule: zlfm.induct)
  case (5 a) 
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (6 a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (7 a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (8 a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (9 a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (10 a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  from prems Ia nb  show ?case 
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done
next
  case (11 j a)  
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
  moreover
  {assume "j=0" hence z: "zlfm (Dvd j a) = (zlfm (Eq a))" by (simp add: Let_def) 
    hence ?case using prems by (simp del: zlfm.simps)}
  moreover
  {assume "?c=0" and "j\<noteq>0" hence ?case 
      using zsplit0_I[OF spl, where x="i" and bs="bs"]
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done}
  moreover
  {assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
      by (simp add: nb Let_def split_def)
    hence ?case using Ia cp jnz by (simp add: Let_def split_def)}
  moreover
  {assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
      by (simp add: nb Let_def split_def)
    hence ?case using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r" ]
      by (simp add: Let_def split_def) }
  ultimately show ?case by blast
next
  case (12 j a) 
  let ?c = "fst (zsplit0 a)"
  let ?r = "snd (zsplit0 a)"
  have spl: "zsplit0 a = (?c,?r)" by simp
  from zsplit0_I[OF spl, where x="i" and bs="bs"] 
  have Ia:"Inum (i # bs) a = Inum (i #bs) (CN 0 ?c ?r)" and nb: "numbound0 ?r" by auto 
  let ?N = "\<lambda> t. Inum (i#bs) t"
  have "j=0 \<or> (j\<noteq>0 \<and> ?c = 0) \<or> (j\<noteq>0 \<and> ?c >0) \<or> (j\<noteq> 0 \<and> ?c<0)" by arith
  moreover
  {assume "j=0" hence z: "zlfm (NDvd j a) = (zlfm (NEq a))" by (simp add: Let_def) 
    hence ?case using prems by (simp del: zlfm.simps)}
  moreover
  {assume "?c=0" and "j\<noteq>0" hence ?case 
      using zsplit0_I[OF spl, where x="i" and bs="bs"]
    apply (auto simp add: Let_def split_def algebra_simps) 
    apply (cases "?r",auto)
    apply (case_tac nat, auto)
    done}
  moreover
  {assume cp: "?c > 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
      by (simp add: nb Let_def split_def)
    hence ?case using Ia cp jnz by (simp add: Let_def split_def) }
  moreover
  {assume cn: "?c < 0" and jnz: "j\<noteq>0" hence l: "?L (?l (Dvd j a))" 
      by (simp add: nb Let_def split_def)
    hence ?case using Ia cn jnz dvd_minus_iff[of "abs j" "?c*i + ?N ?r"]
      by (simp add: Let_def split_def)}
  ultimately show ?case by blast
qed auto

consts 
  plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
  minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
  \<delta> :: "fm \<Rightarrow> int" (* Compute lcm {d| N\<^isup>?\<^isup> Dvd c*x+t \<in> p}*)
  d\<delta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* checks if a given l divides all the ds above*)

recdef minusinf "measure size"
  "minusinf (And p q) = And (minusinf p) (minusinf q)" 
  "minusinf (Or p q) = Or (minusinf p) (minusinf q)" 
  "minusinf (Eq  (CN 0 c e)) = F"
  "minusinf (NEq (CN 0 c e)) = T"
  "minusinf (Lt  (CN 0 c e)) = T"
  "minusinf (Le  (CN 0 c e)) = T"
  "minusinf (Gt  (CN 0 c e)) = F"
  "minusinf (Ge  (CN 0 c e)) = F"
  "minusinf p = p"

lemma minusinf_qfree: "qfree p \<Longrightarrow> qfree (minusinf p)"
  by (induct p rule: minusinf.induct, auto)

recdef plusinf "measure size"
  "plusinf (And p q) = And (plusinf p) (plusinf q)" 
  "plusinf (Or p q) = Or (plusinf p) (plusinf q)" 
  "plusinf (Eq  (CN 0 c e)) = F"
  "plusinf (NEq (CN 0 c e)) = T"
  "plusinf (Lt  (CN 0 c e)) = F"
  "plusinf (Le  (CN 0 c e)) = F"
  "plusinf (Gt  (CN 0 c e)) = T"
  "plusinf (Ge  (CN 0 c e)) = T"
  "plusinf p = p"

recdef \<delta> "measure size"
  "\<delta> (And p q) = zlcm (\<delta> p) (\<delta> q)" 
  "\<delta> (Or p q) = zlcm (\<delta> p) (\<delta> q)" 
  "\<delta> (Dvd i (CN 0 c e)) = i"
  "\<delta> (NDvd i (CN 0 c e)) = i"
  "\<delta> p = 1"

recdef d\<delta> "measure size"
  "d\<delta> (And p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
  "d\<delta> (Or p q) = (\<lambda> d. d\<delta> p d \<and> d\<delta> q d)" 
  "d\<delta> (Dvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
  "d\<delta> (NDvd i (CN 0 c e)) = (\<lambda> d. i dvd d)"
  "d\<delta> p = (\<lambda> d. True)"

lemma delta_mono: 
  assumes lin: "iszlfm p"
  and d: "d dvd d'"
  and ad: "d\<delta> p d"
  shows "d\<delta> p d'"
  using lin ad d
proof(induct p rule: iszlfm.induct)
  case (9 i c e)  thus ?case using d
    by (simp add: dvd_trans[of "i" "d" "d'"])
next
  case (10 i c e) thus ?case using d
    by (simp add: dvd_trans[of "i" "d" "d'"])
qed simp_all

lemma \<delta> : assumes lin:"iszlfm p"
  shows "d\<delta> p (\<delta> p) \<and> \<delta> p >0"
using lin
proof (induct p rule: iszlfm.induct)
  case (1 p q) 
  let ?d = "\<delta> (And p q)"
  from prems zlcm_pos have dp: "?d >0" by simp
  have d1: "\<delta> p dvd \<delta> (And p q)" using prems by simp
  hence th: "d\<delta> p ?d" using delta_mono prems(3-4) by(simp del:dvd_zlcm_self1)
  have "\<delta> q dvd \<delta> (And p q)" using prems by simp
  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2)
  from th th' dp show ?case by simp
next
  case (2 p q)  
  let ?d = "\<delta> (And p q)"
  from prems zlcm_pos have dp: "?d >0" by simp
  have "\<delta> p dvd \<delta> (And p q)" using prems by simp
  hence th: "d\<delta> p ?d" using delta_mono prems by(simp del:dvd_zlcm_self1)
  have "\<delta> q dvd \<delta> (And p q)" using prems by simp
  hence th': "d\<delta> q ?d" using delta_mono prems by(simp del:dvd_zlcm_self2)
  from th th' dp show ?case by simp
qed simp_all


consts 
  a\<beta> :: "fm \<Rightarrow> int \<Rightarrow> fm" (* adjusts the coeffitients of a formula *)
  d\<beta> :: "fm \<Rightarrow> int \<Rightarrow> bool" (* tests if all coeffs c of c divide a given l*)
  \<zeta>  :: "fm \<Rightarrow> int" (* computes the lcm of all coefficients of x*)
  \<beta> :: "fm \<Rightarrow> num list"
  \<alpha> :: "fm \<Rightarrow> num list"

recdef a\<beta> "measure size"
  "a\<beta> (And p q) = (\<lambda> k. And (a\<beta> p k) (a\<beta> q k))" 
  "a\<beta> (Or p q) = (\<lambda> k. Or (a\<beta> p k) (a\<beta> q k))" 
  "a\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. Eq (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (NEq (CN 0 c e)) = (\<lambda> k. NEq (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. Lt (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (Le  (CN 0 c e)) = (\<lambda> k. Le (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. Gt (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. Ge (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. Dvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. NDvd ((k div c)*i) (CN 0 1 (Mul (k div c) e)))"
  "a\<beta> p = (\<lambda> k. p)"

recdef d\<beta> "measure size"
  "d\<beta> (And p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
  "d\<beta> (Or p q) = (\<lambda> k. (d\<beta> p k) \<and> (d\<beta> q k))" 
  "d\<beta> (Eq  (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (NEq (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (Lt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (Le  (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (Gt  (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (Ge  (CN 0 c e)) = (\<lambda> k. c dvd k)"
  "d\<beta> (Dvd i (CN 0 c e)) =(\<lambda> k. c dvd k)"
  "d\<beta> (NDvd i (CN 0 c e))=(\<lambda> k. c dvd k)"
  "d\<beta> p = (\<lambda> k. True)"

recdef \<zeta> "measure size"
  "\<zeta> (And p q) = zlcm (\<zeta> p) (\<zeta> q)" 
  "\<zeta> (Or p q) = zlcm (\<zeta> p) (\<zeta> q)" 
  "\<zeta> (Eq  (CN 0 c e)) = c"
  "\<zeta> (NEq (CN 0 c e)) = c"
  "\<zeta> (Lt  (CN 0 c e)) = c"
  "\<zeta> (Le  (CN 0 c e)) = c"
  "\<zeta> (Gt  (CN 0 c e)) = c"
  "\<zeta> (Ge  (CN 0 c e)) = c"
  "\<zeta> (Dvd i (CN 0 c e)) = c"
  "\<zeta> (NDvd i (CN 0 c e))= c"
  "\<zeta> p = 1"

recdef \<beta> "measure size"
  "\<beta> (And p q) = (\<beta> p @ \<beta> q)" 
  "\<beta> (Or p q) = (\<beta> p @ \<beta> q)" 
  "\<beta> (Eq  (CN 0 c e)) = [Sub (C -1) e]"
  "\<beta> (NEq (CN 0 c e)) = [Neg e]"
  "\<beta> (Lt  (CN 0 c e)) = []"
  "\<beta> (Le  (CN 0 c e)) = []"
  "\<beta> (Gt  (CN 0 c e)) = [Neg e]"
  "\<beta> (Ge  (CN 0 c e)) = [Sub (C -1) e]"
  "\<beta> p = []"

recdef \<alpha> "measure size"
  "\<alpha> (And p q) = (\<alpha> p @ \<alpha> q)" 
  "\<alpha> (Or p q) = (\<alpha> p @ \<alpha> q)" 
  "\<alpha> (Eq  (CN 0 c e)) = [Add (C -1) e]"
  "\<alpha> (NEq (CN 0 c e)) = [e]"
  "\<alpha> (Lt  (CN 0 c e)) = [e]"
  "\<alpha> (Le  (CN 0 c e)) = [Add (C -1) e]"
  "\<alpha> (Gt  (CN 0 c e)) = []"
  "\<alpha> (Ge  (CN 0 c e)) = []"
  "\<alpha> p = []"
consts mirror :: "fm \<Rightarrow> fm"
recdef mirror "measure size"
  "mirror (And p q) = And (mirror p) (mirror q)" 
  "mirror (Or p q) = Or (mirror p) (mirror q)" 
  "mirror (Eq  (CN 0 c e)) = Eq (CN 0 c (Neg e))"
  "mirror (NEq (CN 0 c e)) = NEq (CN 0 c (Neg e))"
  "mirror (Lt  (CN 0 c e)) = Gt (CN 0 c (Neg e))"
  "mirror (Le  (CN 0 c e)) = Ge (CN 0 c (Neg e))"
  "mirror (Gt  (CN 0 c e)) = Lt (CN 0 c (Neg e))"
  "mirror (Ge  (CN 0 c e)) = Le (CN 0 c (Neg e))"
  "mirror (Dvd i (CN 0 c e)) = Dvd i (CN 0 c (Neg e))"
  "mirror (NDvd i (CN 0 c e)) = NDvd i (CN 0 c (Neg e))"
  "mirror p = p"
    (* Lemmas for the correctness of \<sigma>\<rho> *)
lemma dvd1_eq1: "x >0 \<Longrightarrow> (x::int) dvd 1 = (x = 1)"
by simp

lemma minusinf_inf:
  assumes linp: "iszlfm p"
  and u: "d\<beta> p 1"
  shows "\<exists> (z::int). \<forall> x < z. Ifm bbs (x#bs) (minusinf p) = Ifm bbs (x#bs) p"
  (is "?P p" is "\<exists> (z::int). \<forall> x < z. ?I x (?M p) = ?I x p")
using linp u
proof (induct p rule: minusinf.induct)
  case (1 p q) thus ?case 
    by auto (rule_tac x="min z za" in exI,simp)
next
  case (2 p q) thus ?case 
    by auto (rule_tac x="min z za" in exI,simp)
next
  case (3 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
  fix a
  from 3 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<noteq> 0"
  proof(clarsimp)
    fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e = 0"
    with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
    show "False" by simp
  qed
  thus ?case by auto
next
  case (4 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
  fix a
  1234</