src/HOL/Decision_Procs/Cooper.thy
 author hoelzl Wed, 11 Mar 2009 10:58:18 +0100 changeset 30439 57c68b3af2ea parent 30042 31039ee583fa child 31715 2eb55a82acd9 permissions -rw-r--r--
Updated paths in Decision_Procs comments and NEWS
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(*  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"
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 (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)"
(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))"
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))"
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
else if n1 \<le> n2 then CN n1 c1 (numadd (r1,Add (Mul c2 (Bound n2)) r2))
"numadd (CN n1 c1 r1, t) = CN n1 c1 (numadd (r1, t))"
"numadd (C b1, C b2) = C (b1+b2)"

(*function (sequential)
numadd :: "num \<Rightarrow> num \<Rightarrow> num"
where
(if n1 = n2 then (let c = c1 + c2
in (if c = 0 then numadd r1 r2 else
else if n1 \<le> n2 then
else
| "numadd (C b1) (C b2) = C (b1 + b2)"
apply pat_completeness apply auto*)

apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
apply (case_tac "n1 = n2")
done

lemma numadd_nb: "\<lbrakk> numbound0 t ; numbound0 s\<rbrakk> \<Longrightarrow> numbound0 (numadd (t,s))"

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)"

lemma nummul_nb: "\<And> i. numbound0 t \<Longrightarrow> numbound0 (nummul i t)"

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 (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)"
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
| "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
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
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
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
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
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
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
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"]
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'"
shows "d\<delta> p 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
from 4 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 (5 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
fix a
from 5 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e < 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "x + Inum (x#bs) e < 0" by simp
qed
thus ?case by auto
next
case (6 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
fix a
from 6 have "\<forall> x<(- Inum (a#bs) e). c*x + Inum (x#bs) e \<le> 0"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "x + Inum (x#bs) e \<le> 0" by simp
qed
thus ?case by auto
next
case (7 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
fix a
from 7 have "\<forall> x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e > 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 (8 c e) hence c1: "c=1" and nb: "numbound0 e" by simp+
fix a
from 8 have "\<forall> x<(- Inum (a#bs) e). \<not> (c*x + Inum (x#bs) e \<ge> 0)"
proof(clarsimp)
fix x assume "x < (- Inum (a#bs) e)" and"x + Inum (x#bs) e \<ge> 0"
with numbound0_I[OF nb, where bs="bs" and b="a" and b'="x"]
show "False" by simp
qed
thus ?case by auto
qed auto

lemma minusinf_repeats:
assumes d: "d\<delta> p d" and linp: "iszlfm p"
shows "Ifm bbs ((x - k*d)#bs) (minusinf p) = Ifm bbs (x #bs) (minusinf p)"
using linp d
proof(induct p rule: iszlfm.induct)
case (9 i c e) hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
hence "\<exists> k. d=i*k" by (simp add: dvd_def)
then obtain "di" where di_def: "d=i*di" by blast
show ?case
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI)
assume
"i dvd c * x - c*(k*d) + Inum (x # bs) e"
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
assume
"i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
qed
next
case (10 i c e)  hence nbe: "numbound0 e"  and id: "i dvd d" by simp+
hence "\<exists> k. d=i*k" by (simp add: dvd_def)
then obtain "di" where di_def: "d=i*di" by blast
show ?case
proof(simp add: numbound0_I[OF nbe,where bs="bs" and b="x - k * d" and b'="x"] right_diff_distrib, rule iffI)
assume
"i dvd c * x - c*(k*d) + Inum (x # bs) e"
(is "?ri dvd ?rc*?rx - ?rc*(?rk*?rd) + ?I x e" is "?ri dvd ?rt")
hence "\<exists> (l::int). ?rt = i * l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x+ ?I x e = i*l+c*(k * i*di)"
hence "\<exists> (l::int). c*x+ ?I x e = i*(l + c*k*di)"
hence "\<exists> (l::int). c*x+ ?I x e = i*l" by blast
thus "i dvd c*x + Inum (x # bs) e" by (simp add: dvd_def)
next
assume
"i dvd c*x + Inum (x # bs) e" (is "?ri dvd ?rc*?rx+?e")
hence "\<exists> (l::int). c*x+?e = i*l" by (simp add: dvd_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*d)" by simp
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*l - c*(k*i*di)" by (simp add: di_def)
hence "\<exists> (l::int). c*x - c*(k*d) +?e = i*((l - c*k*di))" by (simp add: algebra_simps)
hence "\<exists> (l::int). c*x - c * (k*d) +?e = i*l"
by blast
thus "i dvd c*x - c*(k*d) + Inum (x # bs) e" by (simp add: dvd_def)
qed
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="x - k*d" and b'="x"])

lemma mirror\<alpha>\<beta>:
assumes lp: "iszlfm p"
shows "(Inum (i#bs)) ` set (\<alpha> p) = (Inum (i#bs)) ` set (\<beta> (mirror p))"
using lp
by (induct p rule: mirror.induct, auto)

lemma mirror:
assumes lp: "iszlfm p"
shows "Ifm bbs (x#bs) (mirror p) = Ifm bbs ((- x)#bs) p"
using lp
proof(induct p rule: iszlfm.induct)
case (9 j c e) hence nb: "numbound0 e" by simp
have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
by simp
finally show ?case .
next
case (10 j c e) hence nb: "numbound0 e" by simp
have "Ifm bbs (x#bs) (mirror (Dvd j (CN 0 c e))) = (j dvd c*x - Inum (x#bs) e)" (is "_ = (j dvd c*x - ?e)") by simp
also have "\<dots> = (j dvd (- (c*x - ?e)))"
by (simp only: dvd_minus_iff)
also have "\<dots> = (j dvd (c* (- x)) + ?e)"
also have "\<dots> = Ifm bbs ((- x)#bs) (Dvd j (CN 0 c e))"
using numbound0_I[OF nb, where bs="bs" and b="x" and b'="- x"]
by simp
finally show ?case by simp
qed (auto simp add: numbound0_I[where bs="bs" and b="x" and b'="- x"] gr0_conv_Suc)

lemma mirror_l: "iszlfm p \<and> d\<beta> p 1
\<Longrightarrow> iszlfm (mirror p) \<and> d\<beta> (mirror p) 1"
by (induct p rule: mirror.induct, auto)

lemma mirror_\<delta>: "iszlfm p \<Longrightarrow> \<delta> (mirror p) = \<delta> p"
by (induct p rule: mirror.induct,auto)

lemma \<beta>_numbound0: assumes lp: "iszlfm p"
shows "\<forall> b\<in> set (\<beta> p). numbound0 b"
using lp by (induct p rule: \<beta>.induct,auto)

lemma d\<beta>_mono:
assumes linp: "iszlfm p"
and dr: "d\<beta> p l"
and d: "l dvd l'"
shows "d\<beta> p l'"
using dr linp dvd_trans[of _ "l" "l'", simplified d]
by (induct p rule: iszlfm.induct) simp_all

lemma \<alpha>_l: assumes lp: "iszlfm p"
shows "\<forall> b\<in> set (\<alpha> p). numbound0 b"
using lp
by(induct p rule: \<alpha>.induct, auto)

lemma \<zeta>:
assumes linp: "iszlfm p"
shows "\<zeta> p > 0 \<and> d\<beta> p (\<zeta> p)"
using linp
proof(induct p rule: iszlfm.induct)
case (1 p q)
from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)"  by simp
from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: zlcm_pos)
next
case (2 p q)
from prems have dl1: "\<zeta> p dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems have dl2: "\<zeta> q dvd zlcm (\<zeta> p) (\<zeta> q)" by simp
from prems d\<beta>_mono[where p = "p" and l="\<zeta> p" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
d\<beta>_mono[where p = "q" and l="\<zeta> q" and l'="zlcm (\<zeta> p) (\<zeta> q)"]
dl1 dl2 show ?case by (auto simp add: zlcm_pos)

lemma a\<beta>: assumes linp: "iszlfm p" and d: "d\<beta> p l" and lp: "l > 0"
shows "iszlfm (a\<beta> p l) \<and> d\<beta> (a\<beta> p l) 1 \<and> (Ifm bbs (l*x #bs) (a\<beta> p l) = Ifm bbs (x#bs) p)"
using linp d
proof (induct p rule: iszlfm.induct)
case (5 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l*x + (l div c) * Inum (x # bs) e < 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e < 0)"
by simp
also have "\<dots> = ((l div c) * (c*x + Inum (x # bs) e) < (l div c) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e < 0)"
using mult_less_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"] be  by simp
next
case (6 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l*x + (l div c) * Inum (x# bs) e \<le> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<le> 0)"
by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<le> ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c*x + Inum (x # bs) e \<le> 0)"
using mult_le_0_iff [where a="(l div c)" and b="c*x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"]  be by simp
next
case (7 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l*x + (l div c)* Inum (x # bs) e > 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e > 0)"
by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) > ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e > 0)"
using zero_less_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
next
case (8 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l*x + (l div c)* Inum (x # bs) e \<ge> 0) =
((c*(l div c))*x + (l div c)* Inum (x # bs) e \<ge> 0)"
by simp
also have "\<dots> = ((l div c)*(c*x + Inum (x # bs) e) \<ge> ((l div c)) * 0)"
also have "\<dots> = (c*x + Inum (x # bs) e \<ge> 0)" using ldcp
zero_le_mult_iff [where a="l div c" and b="c*x + Inum (x # bs) e"] by simp
finally show ?case using be numbound0_I[OF be,where b="l*x" and b'="x" and bs="bs"]
by simp
next
case (3 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l * x + (l div c) * Inum (x # bs) e = 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e = 0)"
by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) = ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e = 0)"
using mult_eq_0_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
next
case (4 c e) hence cp: "c>0" and be: "numbound0 e" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(l * x + (l div c) * Inum (x # bs) e \<noteq> 0) =
((c * (l div c)) * x + (l div c) * Inum (x # bs) e \<noteq> 0)"
by simp
also have "\<dots> = ((l div c) * (c * x + Inum (x # bs) e) \<noteq> ((l div c)) * 0)" by (simp add: algebra_simps)
also have "\<dots> = (c * x + Inum (x # bs) e \<noteq> 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e"] ldcp by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"]  be  by simp
next
case (9 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)"  by simp
also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be  mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
next
case (10 j c e) hence cp: "c>0" and be: "numbound0 e" and jp: "j > 0" and d': "c dvd l" by simp+
from lp cp have clel: "c\<le>l" by (simp add: zdvd_imp_le [OF d' lp])
from cp have cnz: "c \<noteq> 0" by simp
have "c div c\<le> l div c"
by (simp add: zdiv_mono1[OF clel cp])
then have ldcp:"0 < l div c"
have "c * (l div c) = c* (l div c) + l mod c" using d' dvd_eq_mod_eq_0[of "c" "l"] by simp
hence cl:"c * (l div c) =l" using zmod_zdiv_equality[where a="l" and b="c", symmetric]
by simp
hence "(\<exists> (k::int). l * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k) = (\<exists> (k::int). (c * (l div c)) * x + (l div c) * Inum (x # bs) e = ((l div c) * j) * k)"  by simp
also have "\<dots> = (\<exists> (k::int). (l div c) * (c * x + Inum (x # bs) e - j * k) = (l div c)*0)" by (simp add: algebra_simps)
also fix k have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e - j * k = 0)"
using zero_le_mult_iff [where a="(l div c)" and b="c * x + Inum (x # bs) e - j * k"] ldcp by simp
also have "\<dots> = (\<exists> (k::int). c * x + Inum (x # bs) e = j * k)" by simp
finally show ?case using numbound0_I[OF be,where b="(l * x)" and b'="x" and bs="bs"] be  mult_strict_mono[OF ldcp jp ldcp ] by (simp add: dvd_def)
qed (auto simp add: gr0_conv_Suc numbound0_I[where bs="bs" and b="(l * x)" and b'="x"])

lemma a\<beta>_ex: assumes linp: "iszlfm p" and d: "d\<beta> p l" and lp: "l>0"
shows "(\<exists> x. l dvd x \<and> Ifm bbs (x #bs) (a\<beta> p l)) = (\<exists> (x::int). Ifm bbs (x#bs) p)"
(is "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> x. ?P' x)")
proof-
have "(\<exists> x. l dvd x \<and> ?P x) = (\<exists> (x::int). ?P (l*x))"
using unity_coeff_ex[where l="l" and P="?P", simplified] by simp
also have "\<dots> = (\<exists> (x::int). ?P' x)" using a\<beta>[OF linp d lp] by simp
finally show ?thesis  .
qed

lemma \<beta>:
assumes lp: "iszlfm p"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
and nob: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). x = b + j)"
and p: "Ifm bbs (x#bs) p" (is "?P x")
shows "?P (x - d)"
using lp u d dp nob p
proof(induct p rule: iszlfm.induct)
case (5 c e) hence c1: "c=1" and  bn:"numbound0 e" by simp+
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems
show ?case by simp
next
case (6 c e)  hence c1: "c=1" and  bn:"numbound0 e" by simp+
with dp p c1 numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] prems
show ?case by simp
next
case (7 c e) hence p: "Ifm bbs (x #bs) (Gt (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e" by simp+
let ?e = "Inum (x # bs) e"
{assume "(x-d) +?e > 0" hence ?case using c1
numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] by simp}
moreover
{assume H: "\<not> (x-d) + ?e > 0"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (Gt (CN 0 c e)))" by simp
from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
have nob: "\<not> (\<exists> j\<in> {1 ..d}. x =  - ?e + j)" by auto
from H p have "x + ?e > 0 \<and> x + ?e \<le> d" by (simp add: c1)
hence "x + ?e \<ge> 1 \<and> x + ?e \<le> d"  by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. x = (- ?e + j)"
with nob have ?case by auto}
ultimately show ?case by blast
next
case (8 c e) hence p: "Ifm bbs (x #bs) (Ge (CN 0 c e))" and c1: "c=1" and bn:"numbound0 e"
by simp+
let ?e = "Inum (x # bs) e"
{assume "(x-d) +?e \<ge> 0" hence ?case using  c1
numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"]
by simp}
moreover
{assume H: "\<not> (x-d) + ?e \<ge> 0"
let ?v="Sub (C -1) e"
have vb: "?v \<in> set (\<beta> (Ge (CN 0 c e)))" by simp
from prems(11)[simplified simp_thms Inum.simps \<beta>.simps set.simps bex_simps numbound0_I[OF bn,where b="a" and b'="x" and bs="bs"]]
have nob: "\<not> (\<exists> j\<in> {1 ..d}. x =  - ?e - 1 + j)" by auto
from H p have "x + ?e \<ge> 0 \<and> x + ?e < d" by (simp add: c1)
hence "x + ?e +1 \<ge> 1 \<and> x + ?e + 1 \<le> d"  by simp
hence "\<exists> (j::int) \<in> {1 .. d}. j = x + ?e + 1" by simp
hence "\<exists> (j::int) \<in> {1 .. d}. x= - ?e - 1 + j" by (simp add: algebra_simps)
with nob have ?case by simp }
ultimately show ?case by blast
next
case (3 c e) hence p: "Ifm bbs (x #bs) (Eq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
let ?e = "Inum (x # bs) e"
let ?v="(Sub (C -1) e)"
have vb: "?v \<in> set (\<beta> (Eq (CN 0 c e)))" by simp
from p have "x= - ?e" by (simp add: c1) with prems(11) show ?case using dp
by simp (erule ballE[where x="1"],
simp_all add:algebra_simps numbound0_I[OF bn,where b="x"and b'="a"and bs="bs"])
next
case (4 c e)hence p: "Ifm bbs (x #bs) (NEq (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
let ?e = "Inum (x # bs) e"
let ?v="Neg e"
have vb: "?v \<in> set (\<beta> (NEq (CN 0 c e)))" by simp
{assume "x - d + Inum (((x -d)) # bs) e \<noteq> 0"
hence ?case by (simp add: c1)}
moreover
{assume H: "x - d + Inum (((x -d)) # bs) e = 0"
hence "x = - Inum (((x -d)) # bs) e + d" by simp
hence "x = - Inum (a # bs) e + d"
by (simp add: numbound0_I[OF bn,where b="x - d"and b'="a"and bs="bs"])
with prems(11) have ?case using dp by simp}
ultimately show ?case by blast
next
case (9 j c e) hence p: "Ifm bbs (x #bs) (Dvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
let ?e = "Inum (x # bs) e"
from prems have id: "j dvd d" by simp
from c1 have "?p x = (j dvd (x+ ?e))" by simp
also have "\<dots> = (j dvd x - d + ?e)"
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
finally show ?case
using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
next
case (10 j c e) hence p: "Ifm bbs (x #bs) (NDvd j (CN 0 c e))" (is "?p x") and c1: "c=1" and bn:"numbound0 e" by simp+
let ?e = "Inum (x # bs) e"
from prems have id: "j dvd d" by simp
from c1 have "?p x = (\<not> j dvd (x+ ?e))" by simp
also have "\<dots> = (\<not> j dvd x - d + ?e)"
using zdvd_period[OF id, where x="x" and c="-1" and t="?e"] by simp
finally show ?case using numbound0_I[OF bn,where b="(x-d)" and b'="x" and bs="bs"] c1 p by simp
qed (auto simp add: numbound0_I[where bs="bs" and b="(x - d)" and b'="x"] gr0_conv_Suc)

lemma \<beta>':
assumes lp: "iszlfm p"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
shows "\<forall> x. \<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> set(\<beta> p). Ifm bbs ((Inum (a#bs) b + j) #bs) p) \<longrightarrow> Ifm bbs (x#bs) p \<longrightarrow> Ifm bbs ((x - d)#bs) p" (is "\<forall> x. ?b \<longrightarrow> ?P x \<longrightarrow> ?P (x - d)")
proof(clarify)
fix x
assume nb:"?b" and px: "?P x"
hence nb2: "\<not>(\<exists>(j::int) \<in> {1 .. d}. \<exists> b\<in> (Inum (a#bs)) ` set(\<beta> p). x = b + j)"
by auto
from  \<beta>[OF lp u d dp nb2 px] show "?P (x -d )" .
qed
lemma cpmi_eq: "0 < D \<Longrightarrow> (EX z::int. ALL x. x < z --> (P x = P1 x))
==> ALL x.~(EX (j::int) : {1..D}. EX (b::int) : B. P(b+j)) --> P (x) --> P (x - D)
==> (ALL (x::int). ALL (k::int). ((P1 x)= (P1 (x-k*D))))
==> (EX (x::int). P(x)) = ((EX (j::int) : {1..D} . (P1(j))) | (EX (j::int) : {1..D}. EX (b::int) : B. P (b+j)))"
apply(rule iffI)
prefer 2
apply(drule minusinfinity)
apply assumption+
apply(fastsimp)
apply clarsimp
apply(subgoal_tac "!!k. 0<=k \<Longrightarrow> !x. P x \<longrightarrow> P (x - k*D)")
apply(frule_tac x = x and z=z in decr_lemma)
apply(subgoal_tac "P1(x - (\<bar>x - z\<bar> + 1) * D)")
prefer 2
apply(subgoal_tac "0 <= (\<bar>x - z\<bar> + 1)")
prefer 2 apply arith
apply fastsimp
apply(drule (1)  periodic_finite_ex)
apply blast
apply(blast dest:decr_mult_lemma)
done

theorem cp_thm:
assumes lp: "iszlfm p"
and u: "d\<beta> p 1"
and d: "d\<delta> p d"
and dp: "d > 0"
shows "(\<exists> (x::int). Ifm bbs (x #bs) p) = (\<exists> j\<in> {1.. d}. Ifm bbs (j #bs) (minusinf p) \<or> (\<exists> b \<in> set (\<beta> p). Ifm bbs ((Inum (i#bs) b + j) #bs) p))"
(is "(\<exists> (x::int). ?P (x)) = (\<exists> j\<in> ?D. ?M j \<or> (\<exists> b\<in> ?B. ?P (?I b + j)))")
proof-
from minusinf_inf[OF lp u]
have th: "\<exists>(z::int). \<forall>x<z. ?P (x) = ?M x" by blast
let ?B' = "{?I b | b. b\<in> ?B}"
have BB': "(\<exists>j\<in>?D. \<exists>b\<in> ?B. ?P (?I b +j)) = (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P (b + j))" by auto
hence th2: "\<forall> x. \<not> (\<exists> j \<in> ?D. \<exists> b \<in> ?B'. ?P ((b + j))) \<longrightarrow> ?P (x) \<longrightarrow> ?P ((x - d))"
using \<beta>'[OF lp u d dp, where a="i" and bbs = "bbs"] by blast
from minusinf_repeats[OF d lp]
have th3: "\<forall> x k. ?M x = ?M (x-k*d)" by simp
from cpmi_eq[OF dp th th2 th3] BB' show ?thesis by blast
qed

(* Implement the right hand sides of Cooper's theorem and Ferrante and Rackoff. *)
lemma mirror_ex:
assumes lp: "iszlfm p"
shows "(\<exists> x. Ifm bbs (x#bs) (mirror p)) = (\<exists> x. Ifm bbs (x#bs) p)"
(is "(\<exists> x. ?I x ?mp) = (\<exists> x. ?I x p)")
proof(auto)
fix x assume "?I x ?mp" hence "?I (- x) p" using mirror[OF lp] by blast
thus "\<exists> x. ?I x p" by blast
next
fix x assume "?I x p" hence "?I (- x) ?mp"
using mirror[OF lp, where x="- x", symmetric] by auto
thus "\<exists> x. ?I x ?mp" by blast
qed

lemma cp_thm':
assumes lp: "iszlfm p"
and up: "d\<beta> p 1" and dd: "d\<delta> p d" and dp: "d > 0"
shows "(\<exists> x. Ifm bbs (x#bs) p) = ((\<exists> j\<in> {1 .. d}. Ifm bbs (j#bs) (minusinf p)) \<or> (\<exists> j\<in> {1.. d}. \<exists> b\<in> (Inum (i#bs)) ` set (\<beta> p). Ifm bbs ((b+j)#bs) p))"
using cp_thm[OF lp up dd dp,where i="i"] by auto

constdefs unit:: "fm \<Rightarrow> fm \<times> num list \<times> int"
"unit p \<equiv> (let p' = zlfm p ; l = \<zeta> p' ; q = And (Dvd l (CN 0 1 (C 0))) (a\<beta> p' l); d = \<delta> q;
B = remdups (map simpnum (\<beta> q)) ; a = remdups (map simpnum (\<alpha> q))
in if length B \<le> length a then (q,B,d) else (mirror q, a,d))"

lemma unit: assumes qf: "qfree p"
shows "\<And> q B d. unit p = (q,B,d) \<Longrightarrow> ((\<exists> x. Ifm bbs (x#bs) p) = (\<exists> x. Ifm bbs (x#bs) q)) \<and> (Inum (i#bs)) ` set B = (Inum (i#bs)) ` set (\<beta> q) \<and> d\<beta> q 1 \<and> d\<delta> q d \<and> d >0 \<and> iszlfm q \<and> (\<forall> b\<in> set B. numbound0 b)"
proof-
fix q B d
assume qBd: "unit p = (q,B,d)"
let ?thes = "((\<exists> x. Ifm bbs (x#bs) p) = (\<exists> x. Ifm bbs (x#bs) q)) \<and>
Inum (i#bs) ` set B = Inum (i#bs) ` set (\<beta> q) \<and>
d\<beta> q 1 \<and> d\<delta> q d \<and> 0 < d \<and> iszlfm q \<and> (\<forall> b\<in> set B. numbound0 b)"
let ?I = "\<lambda> x p. Ifm bbs (x#bs) p"
let ?p' = "zlfm p"
let ?l = "\<zeta> ?p'"
let ?q = "And (Dvd ?l (CN 0 1 (C 0))) (a\<beta> ?p' ?l)"
let ?d = "\<delta> ?q"
let ?B = "set (\<beta> ?q)"
let ?B'= "remdups (map simpnum (\<beta> ?q))"
let ?A = "set (\<alpha> ?q)"
let ?A'= "remdups (map simpnum (\<alpha> ?q))"
from conjunct1[OF zlfm_I[OF qf, where bs="bs"]]
have pp': "\<forall> i. ?I i ?p' = ?I i p" by auto
from conjunct2[OF zlfm_I[OF qf, where bs="bs" and i="i"]]
have lp': "iszlfm ?p'" .
from lp' \<zeta>[where p="?p'"] have lp: "?l >0" and dl: "d\<beta> ?p' ?l" by auto
from a\<beta>_ex[where p="?p'" and l="?l" and bs="bs", OF lp' dl lp] pp'
have pq_ex:"(\<exists> (x::int). ?I x p) = (\<exists> x. ?I x ?q)" by simp
from lp' lp a\<beta>[OF lp' dl lp] have lq:"iszlfm ?q" and uq: "d\<beta> ?q 1"  by auto
from \<delta>[OF lq] have dp:"?d >0" and dd: "d\<delta> ?q ?d" by blast+
let ?N = "\<lambda> t. Inum (i#bs) t"
have "?N ` set ?B' = ((?N o simpnum) ` ?B)" by auto
also have "\<dots> = ?N ` ?B" using simpnum_ci[where bs="i#bs"] by auto
finally have BB': "?N ` set ?B' = ?N ` ?B" .
have "?N ` set ?A' = ((?N o simpnum) ` ?A)" by auto
also have "\<dots> = ?N ` ?A" using simpnum_ci[where bs="i#bs"] by auto
finally have AA': "?N ` set ?A' = ?N ` ?A" .
from \<beta>_numbound0[OF lq] have B_nb:"\<forall> b\<in> set ?B'. numbound0 b"
from \<alpha>_l[OF lq] have A_nb: "\<forall> b\<in> set ?A'. numbound0 b"
{assume "length ?B' \<le> length ?A'"
hence q:"q=?q" and "B = ?B'" and d:"d = ?d"
using qBd by (auto simp add: Let_def unit_def)
with BB' B_nb have b: "?N ` (set B) = ?N ` set (\<beta> q)"
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
with pq_ex dp uq dd lq q d have ?thes by simp}
moreover
{assume "\<not> (length ?B' \<le> length ?A')"
hence q:"q=mirror ?q" and "B = ?A'" and d:"d = ?d"
using qBd by (auto simp add: Let_def unit_def)
with AA' mirror\<alpha>\<beta>[OF lq] A_nb have b:"?N ` (set B) = ?N ` set (\<beta> q)"
and bn: "\<forall>b\<in> set B. numbound0 b" by simp+
from mirror_ex[OF lq] pq_ex q
have pqm_eq:"(\<exists> (x::int). ?I x p) = (\<exists> (x::int). ?I x q)" by simp
from lq uq q mirror_l[where p="?q"]
have lq': "iszlfm q" and uq: "d\<beta> q 1" by auto
from \<delta>[OF lq'] mirror_\<delta>[OF lq] q d have dq:"d\<delta> q d " by auto
from pqm_eq b bn uq lq' dp dq q dp d have ?thes by simp
}
ultimately show ?thes by blast
qed
(* Cooper's Algorithm *)

constdefs cooper :: "fm \<Rightarrow> fm"
"cooper p \<equiv>
(let (q,B,d) = unit p; js = iupt 1 d;
mq = simpfm (minusinf q);
md = evaldjf (\<lambda> j. simpfm (subst0 (C j) mq)) js
in if md = T then T else
(let qd = evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) q))
[(b,j). b\<leftarrow>B,j\<leftarrow>js]
in decr (disj md qd)))"
lemma cooper: assumes qf: "qfree p"
shows "((\<exists> x. Ifm bbs (x#bs) p) = (Ifm bbs bs (cooper p))) \<and> qfree (cooper p)"
(is "(?lhs = ?rhs) \<and> _")
proof-
let ?I = "\<lambda> x p. Ifm bbs (x#bs) p"
let ?q = "fst (unit p)"
let ?B = "fst (snd(unit p))"
let ?d = "snd (snd (unit p))"
let ?js = "iupt 1 ?d"
let ?mq = "minusinf ?q"
let ?smq = "simpfm ?mq"
let ?md = "evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js"
fix i
let ?N = "\<lambda> t. Inum (i#bs) t"
let ?Bjs = "[(b,j). b\<leftarrow>?B,j\<leftarrow>?js]"
let ?qd = "evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs"
have qbf:"unit p = (?q,?B,?d)" by simp
from unit[OF qf qbf] have pq_ex: "(\<exists>(x::int). ?I x p) = (\<exists> (x::int). ?I x ?q)" and
B:"?N ` set ?B = ?N ` set (\<beta> ?q)" and
uq:"d\<beta> ?q 1" and dd: "d\<delta> ?q ?d" and dp: "?d > 0" and
lq: "iszlfm ?q" and
Bn: "\<forall> b\<in> set ?B. numbound0 b" by auto
from zlin_qfree[OF lq] have qfq: "qfree ?q" .
from simpfm_qf[OF minusinf_qfree[OF qfq]] have qfmq: "qfree ?smq".
have jsnb: "\<forall> j \<in> set ?js. numbound0 (C j)" by simp
hence "\<forall> j\<in> set ?js. bound0 (subst0 (C j) ?smq)"
by (auto simp only: subst0_bound0[OF qfmq])
hence th: "\<forall> j\<in> set ?js. bound0 (simpfm (subst0 (C j) ?smq))"
from evaldjf_bound0[OF th] have mdb: "bound0 ?md" by simp
from Bn jsnb have "\<forall> (b,j) \<in> set ?Bjs. numbound0 (Add b (C j))"
by simp
hence "\<forall> (b,j) \<in> set ?Bjs. bound0 (subst0 (Add b (C j)) ?q)"
using subst0_bound0[OF qfq] by blast
hence "\<forall> (b,j) \<in> set ?Bjs. bound0 (simpfm (subst0 (Add b (C j)) ?q))"
using simpfm_bound0  by blast
hence th': "\<forall> x \<in> set ?Bjs. bound0 ((\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) x)"
by auto
from evaldjf_bound0 [OF th'] have qdb: "bound0 ?qd" by simp
from mdb qdb
have mdqdb: "bound0 (disj ?md ?qd)" by (simp only: disj_def, cases "?md=T \<or> ?qd=T", simp_all)
from trans [OF pq_ex cp_thm'[OF lq uq dd dp,where i="i"]] B
have "?lhs = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> ?N ` set ?B. Ifm bbs ((b+ j)#bs) ?q))" by auto
also have "\<dots> = (\<exists> j\<in> {1.. ?d}. ?I j ?mq \<or> (\<exists> b\<in> set ?B. Ifm bbs ((?N b+ j)#bs) ?q))" by simp
also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?mq ) \<or> (\<exists> j\<in> {1.. ?d}. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp only: Inum.simps) blast
also have "\<dots> = ((\<exists> j\<in> {1.. ?d}. ?I j ?smq ) \<or> (\<exists> j\<in> {1.. ?d}. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))" by (simp add: simpfm)
also have "\<dots> = ((\<exists> j\<in> set ?js. (\<lambda> j. ?I i (simpfm (subst0 (C j) ?smq))) j) \<or> (\<exists> j\<in> set ?js. \<exists> b\<in> set ?B. Ifm bbs ((?N (Add b (C j)))#bs) ?q))"
by (simp only: simpfm subst0_I[OF qfmq] iupt_set) auto
also have "\<dots> = (?I i (evaldjf (\<lambda> j. simpfm (subst0 (C j) ?smq)) ?js) \<or> (\<exists> j\<in> set ?js. \<exists> b\<in> set ?B. ?I i (subst0 (Add b (C j)) ?q)))"
by (simp only: evaldjf_ex subst0_I[OF qfq])
also have "\<dots>= (?I i ?md \<or> (\<exists> (b,j) \<in> set ?Bjs. (\<lambda> (b,j). ?I i (simpfm (subst0 (Add b (C j)) ?q))) (b,j)))"
by (simp only: simpfm set_concat set_map concat_map_singleton UN_simps) blast
also have "\<dots> = (?I i ?md \<or> (?I i (evaldjf (\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)) ?Bjs)))"
by (simp only: evaldjf_ex[where bs="i#bs" and f="\<lambda> (b,j). simpfm (subst0 (Add b (C j)) ?q)" and ps="?Bjs"]) (auto simp add: split_def)
finally have mdqd: "?lhs = (?I i ?md \<or> ?I i ?qd)" by simp
also have "\<dots> = (?I i (disj ?md ?qd))" by (simp add: disj)
also have "\<dots> = (Ifm bbs bs (decr (disj ?md ?qd)))" by (simp only: decr [OF mdqdb])
finally have mdqd2: "?lhs = (Ifm bbs bs (decr (disj ?md ?qd)))" .
{assume mdT: "?md = T"
hence cT:"cooper p = T"
by (simp only: cooper_def unit_def split_def Let_def if_True) simp
from mdT have lhs:"?lhs" using mdqd by simp
from mdT have "?rhs" by (simp add: cooper_def unit_def split_def)
with lhs cT have ?thesis by simp }
moreover
{assume mdT: "?md \<noteq> T" hence "cooper p = decr (disj ?md ?qd)"
by (simp only: cooper_def unit_def split_def Let_def if_False)
with mdqd2 decr_qf[OF mdqdb] have ?thesis by simp }
ultimately show ?thesis by blast
qed

definition pa :: "fm \<Rightarrow> fm" where
"pa p = qelim (prep p) cooper"

theorem mirqe: "(Ifm bbs bs (pa p) = Ifm bbs bs p) \<and> qfree (pa p)"
using qelim_ci cooper prep by (auto simp add: pa_def)

definition
cooper_test :: "unit \<Rightarrow> fm"
where
"cooper_test u = pa (E (A (Imp (Ge (Sub (Bound 0) (Bound 1)))
(E (E (Eq (Sub (Add (Mul 3 (Bound 1)) (Mul 5 (Bound 0)))
(Bound 2))))))))"

ML {* @{code cooper_test} () *}

(*
code_reserved SML oo
export_code pa in SML module_name GeneratedCooper file "~~/src/HOL/Tools/Qelim/raw_generated_cooper.ML"
*)

oracle linzqe_oracle = {*
let

fun num_of_term vs (t as Free (xn, xT)) = (case AList.lookup (op =) vs t
| SOME n => @{code Bound} n)
| num_of_term vs @{term "0::int"} = @{code C} 0
| num_of_term vs @{term "1::int"} = @{code C} 1
| num_of_term vs (@{term "number_of :: int \<Rightarrow> int"} \$ t) = @{code C} (HOLogic.dest_numeral t)
| num_of_term vs (Bound i) = @{code Bound} i
| num_of_term vs (@{term "uminus :: int \<Rightarrow> int"} \$ t') = @{code Neg} (num_of_term vs t')
| num_of_term vs (@{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} \$ t1 \$ t2) =
@{code Add} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} \$ t1 \$ t2) =
@{code Sub} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} \$ t1 \$ t2) =
(case try HOLogic.dest_number t1
of SOME (_, i) => @{code Mul} (i, num_of_term vs t2)
| NONE => (case try HOLogic.dest_number t2
of SOME (_, i) => @{code Mul} (i, num_of_term vs t1)
| NONE => error "num_of_term: unsupported multiplication"))
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);

fun fm_of_term ps vs @{term True} = @{code T}
| fm_of_term ps vs @{term False} = @{code F}
| fm_of_term ps vs (@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} \$ t1 \$ t2) =
@{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} \$ t1 \$ t2) =
@{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} \$ t1 \$ t2) =
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term ps vs (@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} \$ t1 \$ t2) =
(case try HOLogic.dest_number t1
of SOME (_, i) => @{code Dvd} (i, num_of_term vs t2)
| NONE => error "num_of_term: unsupported dvd")
| fm_of_term ps vs (@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$ t1 \$ t2) =
@{code Iff} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term "op &"} \$ t1 \$ t2) =
@{code And} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term "op |"} \$ t1 \$ t2) =
@{code Or} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term "op -->"} \$ t1 \$ t2) =
@{code Imp} (fm_of_term ps vs t1, fm_of_term ps vs t2)
| fm_of_term ps vs (@{term "Not"} \$ t') =
@{code NOT} (fm_of_term ps vs t')
| fm_of_term ps vs (Const ("Ex", _) \$ Abs (xn, xT, p)) =
let
val (xn', p') = variant_abs (xn, xT, p);
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
in @{code E} (fm_of_term ps vs' p) end
| fm_of_term ps vs (Const ("All", _) \$ Abs (xn, xT, p)) =
let
val (xn', p') = variant_abs (xn, xT, p);
val vs' = (Free (xn', xT), 0) :: map (fn (v, n) => (v, n + 1)) vs;
in @{code A} (fm_of_term ps vs' p) end
| fm_of_term ps vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);

fun term_of_num vs (@{code C} i) = HOLogic.mk_number HOLogic.intT i
| term_of_num vs (@{code Bound} n) = fst (the (find_first (fn (_, m) => n = m) vs))
| term_of_num vs (@{code Neg} t') = @{term "uminus :: int \<Rightarrow> int"} \$ term_of_num vs t'
| term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: int \<Rightarrow> int \<Rightarrow> int"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: int \<Rightarrow> int \<Rightarrow> int"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: int \<Rightarrow> int \<Rightarrow> int"} \$
term_of_num vs (@{code C} i) \$ term_of_num vs t2
| term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));

fun term_of_fm ps vs @{code T} = HOLogic.true_const
| term_of_fm ps vs @{code F} = HOLogic.false_const
| term_of_fm ps vs (@{code Lt} t) =
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} \$ term_of_num vs t \$ @{term "0::int"}
| term_of_fm ps vs (@{code Le} t) =
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} \$ term_of_num vs t \$ @{term "0::int"}
| term_of_fm ps vs (@{code Gt} t) =
@{term "op < :: int \<Rightarrow> int \<Rightarrow> bool"} \$ @{term "0::int"} \$ term_of_num vs t
| term_of_fm ps vs (@{code Ge} t) =
@{term "op \<le> :: int \<Rightarrow> int \<Rightarrow> bool"} \$ @{term "0::int"} \$ term_of_num vs t
| term_of_fm ps vs (@{code Eq} t) =
@{term "op = :: int \<Rightarrow> int \<Rightarrow> bool"} \$ term_of_num vs t \$ @{term "0::int"}
| term_of_fm ps vs (@{code NEq} t) =
term_of_fm ps vs (@{code NOT} (@{code Eq} t))
| term_of_fm ps vs (@{code Dvd} (i, t)) =
@{term "op dvd :: int \<Rightarrow> int \<Rightarrow> bool"} \$ term_of_num vs (@{code C} i) \$ term_of_num vs t
| term_of_fm ps vs (@{code NDvd} (i, t)) =
term_of_fm ps vs (@{code NOT} (@{code Dvd} (i, t)))
| term_of_fm ps vs (@{code NOT} t') =
HOLogic.Not \$ term_of_fm ps vs t'
| term_of_fm ps vs (@{code And} (t1, t2)) =
HOLogic.conj \$ term_of_fm ps vs t1 \$ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Or} (t1, t2)) =
HOLogic.disj \$ term_of_fm ps vs t1 \$ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Imp} (t1, t2)) =
HOLogic.imp \$ term_of_fm ps vs t1 \$ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Iff} (t1, t2)) =
@{term "op = :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$ term_of_fm ps vs t1 \$ term_of_fm ps vs t2
| term_of_fm ps vs (@{code Closed} n) = (fst o the) (find_first (fn (_, m) => m = n) ps)
| term_of_fm ps vs (@{code NClosed} n) = term_of_fm ps vs (@{code NOT} (@{code Closed} n));

fun term_bools acc t =
let
val is_op = member (op =) [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
@{term "op = :: int => _"}, @{term "op < :: int => _"},
@{term "op <= :: int => _"}, @{term "Not"}, @{term "All :: (int => _) => _"},
@{term "Ex :: (int => _) => _"}, @{term "True"}, @{term "False"}]
fun is_ty t = not (fastype_of t = HOLogic.boolT)
in case t
of (l as f \$ a) \$ b => if is_ty t orelse is_op t then term_bools (term_bools acc l)b
else insert (op aconv) t acc
| f \$ a => if is_ty t orelse is_op t then term_bools (term_bools acc f) a
else insert (op aconv) t acc
| Abs p => term_bools acc (snd (variant_abs p))
| _ => if is_ty t orelse is_op t then acc else insert (op aconv) t acc
end;

in fn ct =>
let
val thy = Thm.theory_of_cterm ct;
val t = Thm.term_of ct;
val fs = OldTerm.term_frees t;
val bs = term_bools [] t;
val vs = fs ~~ (0 upto (length fs - 1))
val ps = bs ~~ (0 upto (length bs - 1))
val t' = (term_of_fm ps vs o @{code pa} o fm_of_term ps vs) t;
in (Thm.cterm_of thy o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
end;
*}

use "cooper_tac.ML"
setup "Cooper_Tac.setup"

text {* Tests *}

lemma "\<exists> (j::int). \<forall> x\<ge>j. (\<exists> a b. x = 3*a+5*b)"
by cooper

lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
by cooper

theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
by cooper

theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
(\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
by cooper

theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==>  3 dvd z ==>
2 dvd (y::int) ==> (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
by cooper

theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
by cooper

lemma "ALL (x::int) >=8. EX i j. 5*i + 3*j = x"
by cooper

lemma "ALL (y::int) (z::int) (n::int). 3 dvd z --> 2 dvd (y::int) --> (EX (x::int).  2*x =  y) & (EX (k::int). 3*k = z)"
by cooper

lemma "ALL(x::int) y. x < y --> 2 * x + 1 < 2 * y"
by cooper

lemma "ALL(x::int) y. 2 * x + 1 ~= 2 * y"
by cooper

lemma "EX(x::int) y. 0 < x  & 0 <= y  & 3 * x - 5 * y = 1"
by cooper

lemma "~ (EX(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
by cooper

lemma "ALL(x::int). (2 dvd x) --> (EX(y::int). x = 2*y)"
by cooper

lemma "ALL(x::int). (2 dvd x) = (EX(y::int). x = 2*y)"
by cooper

lemma "ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y + 1))"
by cooper

lemma "~ (ALL(x::int). ((2 dvd x) = (ALL(y::int). x ~= 2*y+1) | (EX(q::int) (u::int) i. 3*i + 2*q - u < 17) --> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
by cooper

lemma "~ (ALL(i::int). 4 <= i --> (EX x y. 0 <= x & 0 <= y & 3 * x + 5 * y = i))"
by cooper

lemma "EX j. ALL (x::int) >= j. EX i j. 5*i + 3*j = x"
by cooper

theorem "(\<forall>(y::int). 3 dvd y) ==> \<forall>(x::int). b < x --> a \<le> x"
by cooper

theorem "!! (y::int) (z::int) (n::int). 3 dvd z ==> 2 dvd (y::int) ==>
(\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
by cooper

theorem "!! (y::int) (z::int) n. Suc(n::nat) < 6 ==>  3 dvd z ==>
2 dvd (y::int) ==> (\<exists>(x::int).  2*x =  y) & (\<exists>(k::int). 3*k = z)"
by cooper

theorem "\<forall>(x::nat). \<exists>(y::nat). (0::nat) \<le> 5 --> y = 5 + x "
by cooper

theorem "\<forall>(x::nat). \<exists>(y::nat). y = 5 + x | x div 6 + 1= 2"
by cooper

theorem "\<exists>(x::int). 0 < x"
by cooper

theorem "\<forall>(x::int) y. x < y --> 2 * x + 1 < 2 * y"
by cooper

theorem "\<forall>(x::int) y. 2 * x + 1 \<noteq> 2 * y"
by cooper

theorem "\<exists>(x::int) y. 0 < x  & 0 \<le> y  & 3 * x - 5 * y = 1"
by cooper

theorem "~ (\<exists>(x::int) (y::int) (z::int). 4*x + (-6::int)*y = 1)"
by cooper

theorem "~ (\<exists>(x::int). False)"
by cooper

theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
by cooper

theorem "\<forall>(x::int). (2 dvd x) --> (\<exists>(y::int). x = 2*y)"
by cooper

theorem "\<forall>(x::int). (2 dvd x) = (\<exists>(y::int). x = 2*y)"
by cooper

theorem "\<forall>(x::int). ((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y + 1))"
by cooper

theorem "~ (\<forall>(x::int).
((2 dvd x) = (\<forall>(y::int). x \<noteq> 2*y+1) |
(\<exists>(q::int) (u::int) i. 3*i + 2*q - u < 17)
--> 0 < x | ((~ 3 dvd x) &(x + 8 = 0))))"
by cooper

theorem "~ (\<forall>(i::int). 4 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
by cooper

theorem "\<forall>(i::int). 8 \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
by cooper

theorem "\<exists>(j::int). \<forall>i. j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i)"
by cooper

theorem "~ (\<forall>j (i::int). j \<le> i --> (\<exists>x y. 0 \<le> x & 0 \<le> y & 3 * x + 5 * y = i))"
by cooper

theorem "(\<exists>m::nat. n = 2 * m) --> (n + 1) div 2 = n div 2"
by cooper

end
```