src/HOL/Decision_Procs/Ferrack.thy
 author hoelzl Wed, 11 Mar 2009 10:58:18 +0100 changeset 30439 57c68b3af2ea parent 30042 31039ee583fa child 30684 c98a64746c69 permissions -rw-r--r--
Updated paths in Decision_Procs comments and NEWS
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(*  Title:      HOL/Decision_Procs/Ferrack.thy
Author:     Amine Chaieb
*)

theory Ferrack
imports Complex_Main Dense_Linear_Order Efficient_Nat
uses ("ferrack_tac.ML")
begin

section {* Quantifier elimination for @{text "\<real> (0, 1, +, <)"} *}

(*********************************************************************************)
(*          SOME GENERAL STUFF< HAS TO BE MOVED IN SOME LIB                      *)
(*********************************************************************************)

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

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

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

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

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

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

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

recdef remdps "measure size"
"remdps [] = []"
"remdps (x#xs) = (x#(remdps (List.filter (\<lambda> y. y \<noteq> x) xs)))"

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

(*********************************************************************************)
(****                            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*)
consts num_size :: "num \<Rightarrow> nat"
primrec
"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 (Mul c a) = 1 + num_size a"
"num_size (CN n c a) = 3 + num_size a "

(* Semantics of numeral terms (num) *)
consts Inum :: "real list \<Rightarrow> num \<Rightarrow> real"
primrec
"Inum bs (C c) = (real c)"
"Inum bs (Bound n) = bs!n"
"Inum bs (CN n c a) = (real 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) = (real c) * Inum bs a"
(* FORMULAE *)
datatype fm  =
T| F| Lt num| Le num| Gt num| Ge num| Eq num| NEq num|
NOT fm| And fm fm|  Or fm fm| Imp fm fm| Iff fm fm| E fm| A fm

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

(* Semantics of formulae (fm) *)
consts Ifm ::"real list \<Rightarrow> fm \<Rightarrow> bool"
primrec
"Ifm bs T = True"
"Ifm bs F = False"
"Ifm bs (Lt a) = (Inum bs a < 0)"
"Ifm bs (Gt a) = (Inum bs a > 0)"
"Ifm bs (Le a) = (Inum bs a \<le> 0)"
"Ifm bs (Ge a) = (Inum bs a \<ge> 0)"
"Ifm bs (Eq a) = (Inum bs a = 0)"
"Ifm bs (NEq a) = (Inum bs a \<noteq> 0)"
"Ifm bs (NOT p) = (\<not> (Ifm bs p))"
"Ifm bs (And p q) = (Ifm bs p \<and> Ifm bs q)"
"Ifm bs (Or p q) = (Ifm bs p \<or> Ifm bs q)"
"Ifm bs (Imp p q) = ((Ifm bs p) \<longrightarrow> (Ifm bs q))"
"Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
"Ifm bs (E p) = (\<exists> x. Ifm (x#bs) p)"
"Ifm bs (A p) = (\<forall> x. Ifm (x#bs) p)"

lemma IfmLeSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Le (Sub s t)) = (s' \<le> t')"
apply simp
done

lemma IfmLtSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Lt (Sub s t)) = (s' < t')"
apply simp
done
lemma IfmEqSub: "\<lbrakk> Inum bs s = s' ; Inum bs t = t' \<rbrakk> \<Longrightarrow> Ifm bs (Eq (Sub s t)) = (s' = t')"
apply simp
done
lemma IfmNOT: " (Ifm bs p = P) \<Longrightarrow> (Ifm bs (NOT p) = (\<not>P))"
apply simp
done
lemma IfmAnd: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (And p q) = (P \<and> Q))"
apply simp
done
lemma IfmOr: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Or p q) = (P \<or> Q))"
apply simp
done
lemma IfmImp: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Imp p q) = (P \<longrightarrow> Q))"
apply simp
done
lemma IfmIff: " \<lbrakk> Ifm bs p = P ; Ifm bs q = Q\<rbrakk> \<Longrightarrow> (Ifm bs (Iff p q) = (P = Q))"
apply simp
done

lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (E p) = (\<exists>x. P x))"
apply simp
done
lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \<Longrightarrow> (Ifm bs (A p) = (\<forall>x. P x))"
apply simp
done

consts not:: "fm \<Rightarrow> fm"
recdef not "measure size"
"not (NOT p) = p"
"not T = F"
"not F = T"
"not p = NOT p"
lemma not[simp]: "Ifm bs (not p) = Ifm bs (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
if p = q then p else And p q)"
lemma conj[simp]: "Ifm bs (conj p q) = Ifm bs (And p q)"
by (cases "p=F \<or> q=F",simp_all add: conj_def) (cases p,simp_all)

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

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

constdefs  imp :: "fm \<Rightarrow> fm \<Rightarrow> fm"
"imp p q \<equiv> (if (p = F \<or> q=T \<or> p=q) then T else if p=T then q else if q=F then not p
else Imp p q)"
lemma imp[simp]: "Ifm bs (imp p q) = Ifm bs (Imp p q)"
by (cases "p=F \<or> q=T",simp_all add: imp_def)

constdefs   iff :: "fm \<Rightarrow> fm \<Rightarrow> fm"
"iff p q \<equiv> (if (p = q) then T else if (p = NOT q \<or> NOT p = q) then F else
if p=F then not q else if q=F then not p else if p=T then q else if q=T then p else
Iff p q)"
lemma iff[simp]: "Ifm bs (iff p q) = Ifm bs (Iff p q)"
by (unfold iff_def,cases "p=q", simp,cases "p=NOT q", simp) (cases "NOT p= q", auto)

lemma conj_simps:
"conj F Q = F"
"conj P F = F"
"conj T Q = Q"
"conj P T = P"
"conj P P = P"
"P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> conj P Q = And P Q"

lemma disj_simps:
"disj T Q = T"
"disj P T = T"
"disj F Q = Q"
"disj P F = P"
"disj P P = P"
"P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> disj P Q = Or P Q"
lemma imp_simps:
"imp F Q = T"
"imp P T = T"
"imp T Q = Q"
"imp P F = not P"
"imp P P = T"
"P \<noteq> T \<Longrightarrow> P \<noteq> F \<Longrightarrow> P \<noteq> Q \<Longrightarrow> Q \<noteq> T \<Longrightarrow> Q \<noteq> F \<Longrightarrow> imp P Q = Imp P Q"
lemma trivNOT: "p \<noteq> NOT p" "NOT p \<noteq> p"
apply (induct p, auto)
done

lemma iff_simps:
"iff p p = T"
"iff p (NOT p) = F"
"iff (NOT p) p = F"
"iff p F = not p"
"iff F p = not p"
"p \<noteq> NOT T \<Longrightarrow> iff T p = p"
"p\<noteq> NOT T \<Longrightarrow> iff p T = p"
"p\<noteq>q \<Longrightarrow> p\<noteq> NOT q \<Longrightarrow> q\<noteq> NOT p \<Longrightarrow> p\<noteq> F \<Longrightarrow> q\<noteq> F \<Longrightarrow> p \<noteq> T \<Longrightarrow> q \<noteq> T \<Longrightarrow> iff p q = Iff p q"
using trivNOT
by (simp_all add: iff_def, cases p, 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 *)
primrec
"numbound0 (C c) = True"
"numbound0 (Bound n) = (n>0)"
"numbound0 (CN n c a) = (n\<noteq>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: nth_pos2)

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 (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"

lemma bound0_I:
assumes bp: "bound0 p"
shows "Ifm (b#bs) p = Ifm (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: nth_pos2)

lemma not_qf[simp]: "qfree p \<Longrightarrow> qfree (not p)"
by (cases p, auto)
lemma not_bn[simp]: "bound0 p \<Longrightarrow> bound0 (not p)"
by (cases p, auto)

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

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

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

lemma iff_qf[simp]: "\<lbrakk>qfree p ; qfree q\<rbrakk> \<Longrightarrow> qfree (iff p q)"
by (unfold iff_def,cases "p=q", auto)
lemma iff_nb[simp]: "\<lbrakk>bound0 p ; bound0 q\<rbrakk> \<Longrightarrow> bound0 (iff p q)"
using iff_def by (unfold iff_def,cases "p=q", auto)

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 c a) = CN (n - 1) c (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 (NOT p) = NOT (decr p)"
"decr (And p q) = conj (decr p) (decr q)"
"decr (Or p q) = disj (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, simp_all add: nth_pos2)

lemma decr: assumes nb: "bound0 p"
shows "Ifm (x#bs) p = Ifm bs (decr p)"
using nb
by (induct p rule: decr.induct, simp_all add: nth_pos2 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 p = False"

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

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

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

lemma djf_simps:
"djf f p T = T"
"djf f p F = f p"
"q\<noteq>T \<Longrightarrow> q\<noteq>F \<Longrightarrow> djf f p q = (let fp = f p in case fp of T \<Rightarrow> T | F \<Rightarrow> q | _ \<Rightarrow> Or (f p) q)"

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

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

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

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

lemma DJ: assumes fdj: "\<forall> p q. Ifm bs (f (Or p q)) = Ifm bs (Or (f p) (f q))"
and fF: "f F = F"
shows "Ifm bs (DJ f p) = Ifm bs (f p)"
proof-
have "Ifm bs (DJ f p) = (\<exists> q \<in> set (disjuncts p). Ifm bs (f q))"
also have "\<dots> = Ifm 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 bs (qe p) = Ifm bs (E p))"
shows "\<forall> bs p. qfree p \<longrightarrow> qfree (DJ qe p) \<and> (Ifm bs ((DJ qe p)) = Ifm 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 bs (DJ qe p) = (\<exists> q\<in> set (disjuncts p). Ifm bs (qe q))"
also have "\<dots> = (\<exists> q \<in> set(disjuncts p). Ifm bs (E q))" using qe disjuncts_qf[OF qf] by auto
also have "\<dots> = Ifm bs (E p)" by (induct p rule: disjuncts.induct, auto)
finally show "qfree (DJ qe p) \<and> Ifm bs (DJ qe p) = Ifm bs (E p)" using qfth by blast
qed
(* Simplification *)
consts
numgcd :: "num \<Rightarrow> int"
numgcdh:: "num \<Rightarrow> int \<Rightarrow> int"
reducecoeffh:: "num \<Rightarrow> int \<Rightarrow> num"
reducecoeff :: "num \<Rightarrow> num"
dvdnumcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
consts maxcoeff:: "num \<Rightarrow> int"
recdef maxcoeff "measure size"
"maxcoeff (C i) = abs i"
"maxcoeff (CN n c t) = max (abs c) (maxcoeff t)"
"maxcoeff t = 1"

lemma maxcoeff_pos: "maxcoeff t \<ge> 0"
by (induct t rule: maxcoeff.induct, auto)

recdef numgcdh "measure size"
"numgcdh (C i) = (\<lambda>g. zgcd i g)"
"numgcdh (CN n c t) = (\<lambda>g. zgcd c (numgcdh t g))"
"numgcdh t = (\<lambda>g. 1)"
defs numgcd_def [code]: "numgcd t \<equiv> numgcdh t (maxcoeff t)"

recdef reducecoeffh "measure size"
"reducecoeffh (C i) = (\<lambda> g. C (i div g))"
"reducecoeffh (CN n c t) = (\<lambda> g. CN n (c div g) (reducecoeffh t g))"
"reducecoeffh t = (\<lambda>g. t)"

defs reducecoeff_def: "reducecoeff t \<equiv>
(let g = numgcd t in
if g = 0 then C 0 else if g=1 then t else reducecoeffh t g)"

recdef dvdnumcoeff "measure size"
"dvdnumcoeff (C i) = (\<lambda> g. g dvd i)"
"dvdnumcoeff (CN n c t) = (\<lambda> g. g dvd c \<and> (dvdnumcoeff t g))"
"dvdnumcoeff t = (\<lambda>g. False)"

lemma dvdnumcoeff_trans:
assumes gdg: "g dvd g'" and dgt':"dvdnumcoeff t g'"
shows "dvdnumcoeff t g"
using dgt' gdg
by (induct t rule: dvdnumcoeff.induct, simp_all add: gdg dvd_trans[OF gdg])

lemma natabs0: "(nat (abs x) = 0) = (x = 0)"
by arith

lemma numgcd0:
assumes g0: "numgcd t = 0"
shows "Inum bs t = 0"
using g0[simplified numgcd_def]
by (induct t rule: numgcdh.induct, auto simp add: zgcd_def gcd_zero natabs0 max_def maxcoeff_pos)

lemma numgcdh_pos: assumes gp: "g \<ge> 0" shows "numgcdh t g \<ge> 0"
using gp
by (induct t rule: numgcdh.induct, auto simp add: zgcd_def)

lemma numgcd_pos: "numgcd t \<ge>0"
by (simp add: numgcd_def numgcdh_pos maxcoeff_pos)

lemma reducecoeffh:
assumes gt: "dvdnumcoeff t g" and gp: "g > 0"
shows "real g *(Inum bs (reducecoeffh t g)) = Inum bs t"
using gt
proof(induct t rule: reducecoeffh.induct)
case (1 i) hence gd: "g dvd i" by simp
from gp have gnz: "g \<noteq> 0" by simp
from prems show ?case by (simp add: real_of_int_div[OF gnz gd])
next
case (2 n c t)  hence gd: "g dvd c" by simp
from gp have gnz: "g \<noteq> 0" by simp
from prems show ?case by (simp add: real_of_int_div[OF gnz gd] algebra_simps)
qed (auto simp add: numgcd_def gp)
consts ismaxcoeff:: "num \<Rightarrow> int \<Rightarrow> bool"
recdef ismaxcoeff "measure size"
"ismaxcoeff (C i) = (\<lambda> x. abs i \<le> x)"
"ismaxcoeff (CN n c t) = (\<lambda>x. abs c \<le> x \<and> (ismaxcoeff t x))"
"ismaxcoeff t = (\<lambda>x. True)"

lemma ismaxcoeff_mono: "ismaxcoeff t c \<Longrightarrow> c \<le> c' \<Longrightarrow> ismaxcoeff t c'"
by (induct t rule: ismaxcoeff.induct, auto)

lemma maxcoeff_ismaxcoeff: "ismaxcoeff t (maxcoeff t)"
proof (induct t rule: maxcoeff.induct)
case (2 n c t)
hence H:"ismaxcoeff t (maxcoeff t)" .
have thh: "maxcoeff t \<le> max (abs c) (maxcoeff t)" by (simp add: le_maxI2)
from ismaxcoeff_mono[OF H thh] show ?case by (simp add: le_maxI1)
qed simp_all

lemma zgcd_gt1: "zgcd i j > 1 \<Longrightarrow> ((abs i > 1 \<and> abs j > 1) \<or> (abs i = 0 \<and> abs j > 1) \<or> (abs i > 1 \<and> abs j = 0))"
apply (cases "abs i = 0", simp_all add: zgcd_def)
apply (cases "abs j = 0", simp_all)
apply (cases "abs i = 1", simp_all)
apply (cases "abs j = 1", simp_all)
apply auto
done
lemma numgcdh0:"numgcdh t m = 0 \<Longrightarrow>  m =0"
by (induct t rule: numgcdh.induct, auto simp add:zgcd0)

lemma dvdnumcoeff_aux:
assumes "ismaxcoeff t m" and mp:"m \<ge> 0" and "numgcdh t m > 1"
shows "dvdnumcoeff t (numgcdh t m)"
using prems
proof(induct t rule: numgcdh.induct)
case (2 n c t)
let ?g = "numgcdh t m"
from prems have th:"zgcd c ?g > 1" by simp
from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
have "(abs c > 1 \<and> ?g > 1) \<or> (abs c = 0 \<and> ?g > 1) \<or> (abs c > 1 \<and> ?g = 0)" by simp
moreover {assume "abs c > 1" and gp: "?g > 1" with prems
have th: "dvdnumcoeff t ?g" by simp
have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)}
moreover {assume "abs c = 0 \<and> ?g > 1"
with prems have th: "dvdnumcoeff t ?g" by simp
have th': "zgcd c ?g dvd ?g" by (simp add:zgcd_zdvd2)
from dvdnumcoeff_trans[OF th' th] have ?case by (simp add: zgcd_zdvd1)
hence ?case by simp }
moreover {assume "abs c > 1" and g0:"?g = 0"
from numgcdh0[OF g0] have "m=0". with prems   have ?case by simp }
ultimately show ?case by blast

lemma dvdnumcoeff_aux2:
assumes "numgcd t > 1" shows "dvdnumcoeff t (numgcd t) \<and> numgcd t > 0"
using prems
let ?mc = "maxcoeff t"
let ?g = "numgcdh t ?mc"
have th1: "ismaxcoeff t ?mc" by (rule maxcoeff_ismaxcoeff)
have th2: "?mc \<ge> 0" by (rule maxcoeff_pos)
assume H: "numgcdh t ?mc > 1"
from dvdnumcoeff_aux[OF th1 th2 H]  show "dvdnumcoeff t ?g" .
qed

lemma reducecoeff: "real (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
proof-
let ?g = "numgcd t"
have "?g \<ge> 0"  by (simp add: numgcd_pos)
hence	"?g = 0 \<or> ?g = 1 \<or> ?g > 1" by auto
moreover {assume "?g = 0" hence ?thesis by (simp add: numgcd0)}
moreover {assume "?g = 1" hence ?thesis by (simp add: reducecoeff_def)}
moreover { assume g1:"?g > 1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff t ?g" and g0: "?g > 0" by blast+
from reducecoeffh[OF th1 g0, where bs="bs"] g1 have ?thesis
ultimately show ?thesis by blast
qed

lemma reducecoeffh_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeffh t g)"
by (induct t rule: reducecoeffh.induct, auto)

lemma reducecoeff_numbound0: "numbound0 t \<Longrightarrow> numbound0 (reducecoeff t)"
using reducecoeffh_numbound0 by (simp add: reducecoeff_def Let_def)

consts
simpnum:: "num \<Rightarrow> num"
numadd:: "num \<times> num \<Rightarrow> num"
nummul:: "num \<Rightarrow> int \<Rightarrow> num"
recdef numadd "measure (\<lambda> (t,s). size t + 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,CN n2 c2 r2)))
else (CN n2 c2 (numadd (CN n1 c1 r1,r2))))"
"numadd (CN n1 c1 r1,t) = CN n1 c1 (numadd (r1, t))"
"numadd (C b1, C b2) = C (b1+b2)"

apply (case_tac "c1+c2 = 0",case_tac "n1 \<le> n2", simp_all)
apply (case_tac "n1 = n2", simp_all add: algebra_simps)
by (simp only: left_distrib[symmetric],simp)

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

recdef nummul "measure size"
"nummul (C j) = (\<lambda> i. C (i*j))"
"nummul (CN n c a) = (\<lambda> i. CN n (i*c) (nummul a i))"
"nummul t = (\<lambda> i. Mul i t)"

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

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

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

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

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

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

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

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

recdef simpnum "measure size"
"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 (simpnum t) i)"
"simpnum (CN n c t) = (if c = 0 then simpnum t else numadd (CN n c (C 0),simpnum t))"

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

lemma simpnum_numbound0[simp]:
"numbound0 t \<Longrightarrow> numbound0 (simpnum t)"
by (induct t rule: simpnum.induct, auto)

consts nozerocoeff:: "num \<Rightarrow> bool"
recdef nozerocoeff "measure size"
"nozerocoeff (C c) = True"
"nozerocoeff (CN n c t) = (c\<noteq>0 \<and> nozerocoeff t)"
"nozerocoeff t = True"

lemma numadd_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numadd (a,b))"

lemma nummul_nz : "\<And> i. i\<noteq>0 \<Longrightarrow> nozerocoeff a \<Longrightarrow> nozerocoeff (nummul a i)"

lemma numneg_nz : "nozerocoeff a \<Longrightarrow> nozerocoeff (numneg a)"

lemma numsub_nz: "nozerocoeff a \<Longrightarrow> nozerocoeff b \<Longrightarrow> nozerocoeff (numsub a b)"

lemma simpnum_nz: "nozerocoeff (simpnum t)"

lemma maxcoeff_nz: "nozerocoeff t \<Longrightarrow> maxcoeff t = 0 \<Longrightarrow> t = C 0"
proof (induct t rule: maxcoeff.induct)
case (2 n c t)
hence cnz: "c \<noteq>0" and mx: "max (abs c) (maxcoeff t) = 0" by simp+
have "max (abs c) (maxcoeff t) \<ge> abs c" by (simp add: le_maxI1)
with cnz have "max (abs c) (maxcoeff t) > 0" by arith
with prems show ?case by simp
qed auto

lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0"
proof-
from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def)
from numgcdh0[OF th]  have th:"maxcoeff t = 0" .
from maxcoeff_nz[OF nz th] show ?thesis .
qed

constdefs simp_num_pair:: "(num \<times> int) \<Rightarrow> num \<times> int"
"simp_num_pair \<equiv> (\<lambda> (t,n). (if n = 0 then (C 0, 0) else
(let t' = simpnum t ; g = numgcd t' in
if g > 1 then (let g' = zgcd n g in
if g' = 1 then (t',n)
else (reducecoeffh t' g', n div g'))
else (t',n))))"

lemma simp_num_pair_ci:
shows "((\<lambda> (t,n). Inum bs t / real n) (simp_num_pair (t,n))) = ((\<lambda> (t,n). Inum bs t / real n) (t,n))"
(is "?lhs = ?rhs")
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
let ?g' = "zgcd n ?g"
{assume nz: "n = 0" hence ?thesis by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis by (simp add: Let_def simp_num_pair_def simpnum_ci)}
moreover {assume g'1:"?g'>1"
from dvdnumcoeff_aux2[OF g1] have th1:"dvdnumcoeff ?t' ?g" ..
let ?tt = "reducecoeffh ?t' ?g'"
let ?t = "Inum bs ?tt"
have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
have gpdgp: "?g' dvd ?g'" by simp
from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
have th2:"real ?g' * ?t = Inum bs ?t'" by simp
from prems have "?lhs = ?t / real (n div ?g')" by (simp add: simp_num_pair_def Let_def)
also have "\<dots> = (real ?g' * ?t) / (real ?g' * (real (n div ?g')))" by simp
also have "\<dots> = (Inum bs ?t' / real n)"
using real_of_int_div[OF gp0 gpdd] th2 gp0 by simp
finally have "?lhs = Inum bs t / real n" by (simp add: simpnum_ci)
then have ?thesis using prems by (simp add: simp_num_pair_def)}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed

lemma simp_num_pair_l: assumes tnb: "numbound0 t" and np: "n >0" and tn: "simp_num_pair (t,n) = (t',n')"
shows "numbound0 t' \<and> n' >0"
proof-
let ?t' = "simpnum t"
let ?g = "numgcd ?t'"
let ?g' = "zgcd n ?g"
{assume nz: "n = 0" hence ?thesis using prems by (simp add: Let_def simp_num_pair_def)}
moreover
{ assume nnz: "n \<noteq> 0"
{assume "\<not> ?g > 1" hence ?thesis  using prems by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
moreover
{assume g1:"?g>1" hence g0: "?g > 0" by simp
from zgcd0 g1 nnz have gp0: "?g' \<noteq> 0" by simp
hence g'p: "?g' > 0" using zgcd_pos[where i="n" and j="numgcd ?t'"] by arith
hence "?g'= 1 \<or> ?g' > 1" by arith
moreover {assume "?g'=1" hence ?thesis using prems
by (auto simp add: Let_def simp_num_pair_def simpnum_numbound0)}
moreover {assume g'1:"?g'>1"
have gpdg: "?g' dvd ?g" by (simp add: zgcd_zdvd2)
have gpdd: "?g' dvd n" by (simp add: zgcd_zdvd1)
have gpdgp: "?g' dvd ?g'" by simp
from zdvd_imp_le[OF gpdd np] have g'n: "?g' \<le> n" .
from zdiv_mono1[OF g'n g'p, simplified zdiv_self[OF gp0]]
have "n div ?g' >0" by simp
hence ?thesis using prems
by(auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0 simpnum_numbound0)}
ultimately have ?thesis by blast}
ultimately have ?thesis by blast}
ultimately show ?thesis by blast
qed

consts simpfm :: "fm \<Rightarrow> fm"
recdef simpfm "measure fmsize"
"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 p = p"
lemma simpfm: "Ifm bs (simpfm p) = Ifm 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
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)
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)+

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

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

lemma qelim_ci:
assumes qe_inv: "\<forall> bs p. qfree p \<longrightarrow> qfree (qe p) \<and> (Ifm bs (qe p) = Ifm bs (E p))"
shows "\<And> bs. qfree (qelim p qe) \<and> (Ifm bs (qelim p qe) = Ifm 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)

consts
plusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of +\<infinity>*)
minusinf:: "fm \<Rightarrow> fm" (* Virtual substitution of -\<infinity>*)
recdef minusinf "measure size"
"minusinf (And p q) = conj (minusinf p) (minusinf q)"
"minusinf (Or p q) = disj (minusinf p) (minusinf q)"
"minusinf (Eq  (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"

recdef plusinf "measure size"
"plusinf (And p q) = conj (plusinf p) (plusinf q)"
"plusinf (Or p q) = disj (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"

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

(* splits the bounded from the unbounded part*)
consts rsplit0 :: "num \<Rightarrow> int \<times> num"
recdef rsplit0 "measure num_size"
"rsplit0 (Bound 0) = (1,C 0)"
"rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a ; (cb,tb) = rsplit0 b
"rsplit0 (Sub a b) = rsplit0 (Add a (Neg b))"
"rsplit0 (Neg a) = (let (c,t) = rsplit0 a in (-c,Neg t))"
"rsplit0 (Mul c a) = (let (ca,ta) = rsplit0 a in (c*ca,Mul c ta))"
"rsplit0 (CN 0 c a) = (let (ca,ta) = rsplit0 a in (c+ca,ta))"
"rsplit0 (CN n c a) = (let (ca,ta) = rsplit0 a in (ca,CN n c ta))"
"rsplit0 t = (0,t)"
lemma rsplit0:
shows "Inum bs ((split (CN 0)) (rsplit0 t)) = Inum bs t \<and> numbound0 (snd (rsplit0 t))"
proof (induct t rule: rsplit0.induct)
case (2 a b)
let ?sa = "rsplit0 a" let ?sb = "rsplit0 b"
let ?ca = "fst ?sa" let ?cb = "fst ?sb"
let ?ta = "snd ?sa" let ?tb = "snd ?sb"
from prems have nb: "numbound0 (snd(rsplit0 (Add a b)))"
by(cases "rsplit0 a",auto simp add: Let_def split_def)
have "Inum bs ((split (CN 0)) (rsplit0 (Add a b))) =
Inum bs ((split (CN 0)) ?sa)+Inum bs ((split (CN 0)) ?sb)"
by (simp add: Let_def split_def algebra_simps)
also have "\<dots> = Inum bs a + Inum bs b" using prems by (cases "rsplit0 a", simp_all)
finally show ?case using nb by simp

(* Linearize a formula*)
definition
lt :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"lt c t = (if c = 0 then (Lt t) else if c > 0 then (Lt (CN 0 c t))
else (Gt (CN 0 (-c) (Neg t))))"

definition
le :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"le c t = (if c = 0 then (Le t) else if c > 0 then (Le (CN 0 c t))
else (Ge (CN 0 (-c) (Neg t))))"

definition
gt :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"gt c t = (if c = 0 then (Gt t) else if c > 0 then (Gt (CN 0 c t))
else (Lt (CN 0 (-c) (Neg t))))"

definition
ge :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"ge c t = (if c = 0 then (Ge t) else if c > 0 then (Ge (CN 0 c t))
else (Le (CN 0 (-c) (Neg t))))"

definition
eq :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"eq c t = (if c = 0 then (Eq t) else if c > 0 then (Eq (CN 0 c t))
else (Eq (CN 0 (-c) (Neg t))))"

definition
neq :: "int \<Rightarrow> num \<Rightarrow> fm"
where
"neq c t = (if c = 0 then (NEq t) else if c > 0 then (NEq (CN 0 c t))
else (NEq (CN 0 (-c) (Neg t))))"

lemma lt: "numnoabs t \<Longrightarrow> Ifm bs (split lt (rsplit0 t)) = Ifm bs (Lt t) \<and> isrlfm (split lt (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: lt_def split_def,cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma le: "numnoabs t \<Longrightarrow> Ifm bs (split le (rsplit0 t)) = Ifm bs (Le t) \<and> isrlfm (split le (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: le_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma gt: "numnoabs t \<Longrightarrow> Ifm bs (split gt (rsplit0 t)) = Ifm bs (Gt t) \<and> isrlfm (split gt (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: gt_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma ge: "numnoabs t \<Longrightarrow> Ifm bs (split ge (rsplit0 t)) = Ifm bs (Ge t) \<and> isrlfm (split ge (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: ge_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma eq: "numnoabs t \<Longrightarrow> Ifm bs (split eq (rsplit0 t)) = Ifm bs (Eq t) \<and> isrlfm (split eq (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: eq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma neq: "numnoabs t \<Longrightarrow> Ifm bs (split neq (rsplit0 t)) = Ifm bs (NEq t) \<and> isrlfm (split neq (rsplit0 t))"
using rsplit0[where bs = "bs" and t="t"]
by (auto simp add: neq_def split_def) (cases "snd(rsplit0 t)",auto,case_tac "nat",auto)

lemma conj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (conj p q)"
lemma disj_lin: "isrlfm p \<Longrightarrow> isrlfm q \<Longrightarrow> isrlfm (disj p q)"

consts rlfm :: "fm \<Rightarrow> fm"
recdef rlfm "measure fmsize"
"rlfm (And p q) = conj (rlfm p) (rlfm q)"
"rlfm (Or p q) = disj (rlfm p) (rlfm q)"
"rlfm (Imp p q) = disj (rlfm (NOT p)) (rlfm q)"
"rlfm (Iff p q) = disj (conj (rlfm p) (rlfm q)) (conj (rlfm (NOT p)) (rlfm (NOT q)))"
"rlfm (Lt a) = split lt (rsplit0 a)"
"rlfm (Le a) = split le (rsplit0 a)"
"rlfm (Gt a) = split gt (rsplit0 a)"
"rlfm (Ge a) = split ge (rsplit0 a)"
"rlfm (Eq a) = split eq (rsplit0 a)"
"rlfm (NEq a) = split neq (rsplit0 a)"
"rlfm (NOT (And p q)) = disj (rlfm (NOT p)) (rlfm (NOT q))"
"rlfm (NOT (Or p q)) = conj (rlfm (NOT p)) (rlfm (NOT q))"
"rlfm (NOT (Imp p q)) = conj (rlfm p) (rlfm (NOT q))"
"rlfm (NOT (Iff p q)) = disj (conj(rlfm p) (rlfm(NOT q))) (conj(rlfm(NOT p)) (rlfm q))"
"rlfm (NOT (NOT p)) = rlfm p"
"rlfm (NOT T) = F"
"rlfm (NOT F) = T"
"rlfm (NOT (Lt a)) = rlfm (Ge a)"
"rlfm (NOT (Le a)) = rlfm (Gt a)"
"rlfm (NOT (Gt a)) = rlfm (Le a)"
"rlfm (NOT (Ge a)) = rlfm (Lt a)"
"rlfm (NOT (Eq a)) = rlfm (NEq a)"
"rlfm (NOT (NEq a)) = rlfm (Eq a)"
"rlfm p = p" (hints simp add: fmsize_pos)

lemma rlfm_I:
assumes qfp: "qfree p"
shows "(Ifm bs (rlfm p) = Ifm bs p) \<and> isrlfm (rlfm p)"
using qfp
by (induct p rule: rlfm.induct, auto simp add: lt le gt ge eq neq conj disj conj_lin disj_lin)

(* Operations needed for Ferrante and Rackoff *)
lemma rminusinf_inf:
assumes lp: "isrlfm p"
shows "\<exists> z. \<forall> x < z. Ifm (x#bs) (minusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
using lp
proof (induct p rule: minusinf.induct)
case (1 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
next
case (2 p q) thus ?case by (auto,rule_tac x= "min z za" in exI) auto
next
case (3 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
hence "real c * x + ?e \<noteq> 0" by simp
with xz have "?P ?z x (Eq (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp  }
hence "\<forall> x < ?z. ?P ?z x (Eq (CN 0 c e))" by simp
thus ?case by blast
next
case (4 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
hence "real c * x + ?e \<noteq> 0" by simp
with xz have "?P ?z x (NEq (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x < ?z. ?P ?z x (NEq (CN 0 c e))" by simp
thus ?case by blast
next
case (5 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
with xz have "?P ?z x (Lt (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]  by simp }
hence "\<forall> x < ?z. ?P ?z x (Lt (CN 0 c e))" by simp
thus ?case by blast
next
case (6 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
with xz have "?P ?z x (Le (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x < ?z. ?P ?z x (Le (CN 0 c e))" by simp
thus ?case by blast
next
case (7 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
with xz have "?P ?z x (Gt (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x < ?z. ?P ?z x (Gt (CN 0 c e))" by simp
thus ?case by blast
next
case (8 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x < ?z"
hence "(real c * x < - ?e)"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="- ?e"] mult_ac)
hence "real c * x + ?e < 0" by arith
with xz have "?P ?z x (Ge (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x < ?z. ?P ?z x (Ge (CN 0 c e))" by simp
thus ?case by blast
qed simp_all

lemma rplusinf_inf:
assumes lp: "isrlfm p"
shows "\<exists> z. \<forall> x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\<exists> z. \<forall> x. ?P z x p")
using lp
proof (induct p rule: isrlfm.induct)
case (1 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
next
case (2 p q) thus ?case by (auto,rule_tac x= "max z za" in exI) auto
next
case (3 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
hence "real c * x + ?e \<noteq> 0" by simp
with xz have "?P ?z x (Eq (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
thus ?case by blast
next
case (4 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
hence "real c * x + ?e \<noteq> 0" by simp
with xz have "?P ?z x (NEq (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
thus ?case by blast
next
case (5 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
with xz have "?P ?z x (Lt (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
thus ?case by blast
next
case (6 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
with xz have "?P ?z x (Le (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
thus ?case by blast
next
case (7 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
with xz have "?P ?z x (Gt (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"] by simp }
hence "\<forall> x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
thus ?case by blast
next
case (8 c e)
from prems have nb: "numbound0 e" by simp
from prems have cp: "real c > 0" by simp
fix a
let ?e="Inum (a#bs) e"
let ?z = "(- ?e) / real c"
{fix x
assume xz: "x > ?z"
with mult_strict_right_mono [OF xz cp] cp
have "(real c * x > - ?e)" by (simp add: mult_ac)
hence "real c * x + ?e > 0" by arith
with xz have "?P ?z x (Ge (CN 0 c e))"
using numbound0_I[OF nb, where b="x" and bs="bs" and b'="a"]   by simp }
hence "\<forall> x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
thus ?case by blast
qed simp_all

lemma rminusinf_bound0:
assumes lp: "isrlfm p"
shows "bound0 (minusinf p)"
using lp
by (induct p rule: minusinf.induct) simp_all

lemma rplusinf_bound0:
assumes lp: "isrlfm p"
shows "bound0 (plusinf p)"
using lp
by (induct p rule: plusinf.induct) simp_all

lemma rminusinf_ex:
assumes lp: "isrlfm p"
and ex: "Ifm (a#bs) (minusinf p)"
shows "\<exists> x. Ifm (x#bs) p"
proof-
from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
have th: "\<forall> x. Ifm (x#bs) (minusinf p)" by auto
from rminusinf_inf[OF lp, where bs="bs"]
obtain z where z_def: "\<forall>x<z. Ifm (x # bs) (minusinf p) = Ifm (x # bs) p" by blast
from th have "Ifm ((z - 1)#bs) (minusinf p)" by simp
moreover have "z - 1 < z" by simp
ultimately show ?thesis using z_def by auto
qed

lemma rplusinf_ex:
assumes lp: "isrlfm p"
and ex: "Ifm (a#bs) (plusinf p)"
shows "\<exists> x. Ifm (x#bs) p"
proof-
from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
have th: "\<forall> x. Ifm (x#bs) (plusinf p)" by auto
from rplusinf_inf[OF lp, where bs="bs"]
obtain z where z_def: "\<forall>x>z. Ifm (x # bs) (plusinf p) = Ifm (x # bs) p" by blast
from th have "Ifm ((z + 1)#bs) (plusinf p)" by simp
moreover have "z + 1 > z" by simp
ultimately show ?thesis using z_def by auto
qed

consts
uset:: "fm \<Rightarrow> (num \<times> int) list"
usubst :: "fm \<Rightarrow> (num \<times> int) \<Rightarrow> fm "
recdef uset "measure size"
"uset (And p q) = (uset p @ uset q)"
"uset (Or p q) = (uset p @ uset q)"
"uset (Eq  (CN 0 c e)) = [(Neg e,c)]"
"uset (NEq (CN 0 c e)) = [(Neg e,c)]"
"uset (Lt  (CN 0 c e)) = [(Neg e,c)]"
"uset (Le  (CN 0 c e)) = [(Neg e,c)]"
"uset (Gt  (CN 0 c e)) = [(Neg e,c)]"
"uset (Ge  (CN 0 c e)) = [(Neg e,c)]"
"uset p = []"
recdef usubst "measure size"
"usubst (And p q) = (\<lambda> (t,n). And (usubst p (t,n)) (usubst q (t,n)))"
"usubst (Or p q) = (\<lambda> (t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
"usubst (Eq (CN 0 c e)) = (\<lambda> (t,n). Eq (Add (Mul c t) (Mul n e)))"
"usubst (NEq (CN 0 c e)) = (\<lambda> (t,n). NEq (Add (Mul c t) (Mul n e)))"
"usubst (Lt (CN 0 c e)) = (\<lambda> (t,n). Lt (Add (Mul c t) (Mul n e)))"
"usubst (Le (CN 0 c e)) = (\<lambda> (t,n). Le (Add (Mul c t) (Mul n e)))"
"usubst (Gt (CN 0 c e)) = (\<lambda> (t,n). Gt (Add (Mul c t) (Mul n e)))"
"usubst (Ge (CN 0 c e)) = (\<lambda> (t,n). Ge (Add (Mul c t) (Mul n e)))"
"usubst p = (\<lambda> (t,n). p)"

lemma usubst_I: assumes lp: "isrlfm p"
and np: "real n > 0" and nbt: "numbound0 t"
shows "(Ifm (x#bs) (usubst p (t,n)) = Ifm (((Inum (x#bs) t)/(real n))#bs) p) \<and> bound0 (usubst p (t,n))" (is "(?I x (usubst p (t,n)) = ?I ?u p) \<and> ?B p" is "(_ = ?I (?t/?n) p) \<and> _" is "(_ = ?I (?N x t /_) p) \<and> _")
using lp
proof(induct p rule: usubst.induct)
case (5 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Lt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) < 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) < 0)"
by (simp only: pos_less_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) < 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (6 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Le (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<le> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<le> 0)"
by (simp only: pos_le_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<le> 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (7 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Gt (CN 0 c e)) = (real c *(?t/?n) + (?N x e) > 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) > 0)"
by (simp only: pos_divide_less_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) > 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (8 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
have "?I ?u (Ge (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<ge> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<ge> 0)"
by (simp only: pos_divide_le_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<ge> 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (3 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
have "?I ?u (Eq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) = 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) = 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) = 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
next
case (4 c e) from prems have cp: "c >0" and nb: "numbound0 e" by simp+
from np have np: "real n \<noteq> 0" by simp
have "?I ?u (NEq (CN 0 c e)) = (real c *(?t/?n) + (?N x e) \<noteq> 0)"
using numbound0_I[OF nb, where bs="bs" and b="?u" and b'="x"] by simp
also have "\<dots> = (?n*(real c *(?t/?n)) + ?n*(?N x e) \<noteq> 0)"
by (simp only: nonzero_eq_divide_eq[OF np, where a="real c *(?t/?n) + (?N x e)"
and b="0", simplified divide_zero_left]) (simp only: algebra_simps)
also have "\<dots> = (real c *?t + ?n* (?N x e) \<noteq> 0)"
using np by simp
finally show ?case using nbt nb by (simp add: algebra_simps)
qed(simp_all add: nbt numbound0_I[where bs ="bs" and b="(Inum (x#bs) t)/ real n" and b'="x"] nth_pos2)

lemma uset_l:
assumes lp: "isrlfm p"
shows "\<forall> (t,k) \<in> set (uset p). numbound0 t \<and> k >0"
using lp
by(induct p rule: uset.induct,auto)

lemma rminusinf_uset:
assumes lp: "isrlfm p"
and nmi: "\<not> (Ifm (a#bs) (minusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
and ex: "Ifm (x#bs) p" (is "?I x p")
shows "\<exists> (s,m) \<in> set (uset p). x \<ge> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<ge> ?N a s / real m")
proof-
have "\<exists> (s,m) \<in> set (uset p). real m * x \<ge> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<ge> ?N a s")
using lp nmi ex
by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<ge> ?N a s" by blast
from uset_l[OF lp] smU have mp: "real m > 0" by auto
from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<ge> ?N a s / real m"
thus ?thesis using smU by auto
qed

lemma rplusinf_uset:
assumes lp: "isrlfm p"
and nmi: "\<not> (Ifm (a#bs) (plusinf p))" (is "\<not> (Ifm (a#bs) (?M p))")
and ex: "Ifm (x#bs) p" (is "?I x p")
shows "\<exists> (s,m) \<in> set (uset p). x \<le> Inum (a#bs) s / real m" (is "\<exists> (s,m) \<in> ?U p. x \<le> ?N a s / real m")
proof-
have "\<exists> (s,m) \<in> set (uset p). real m * x \<le> Inum (a#bs) s " (is "\<exists> (s,m) \<in> ?U p. real m *x \<le> ?N a s")
using lp nmi ex
by (induct p rule: minusinf.induct, auto simp add:numbound0_I[where bs="bs" and b="a" and b'="x"] nth_pos2)
then obtain s m where smU: "(s,m) \<in> set (uset p)" and mx: "real m * x \<le> ?N a s" by blast
from uset_l[OF lp] smU have mp: "real m > 0" by auto
from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \<le> ?N a s / real m"
thus ?thesis using smU by auto
qed

lemma lin_dense:
assumes lp: "isrlfm p"
and noS: "\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> (\<lambda> (t,n). Inum (x#bs) t / real n) ` set (uset p)"
(is "\<forall> t. _ \<and> _ \<longrightarrow> t \<notin> (\<lambda> (t,n). ?N x t / real n ) ` (?U p)")
and lx: "l < x" and xu:"x < u" and px:" Ifm (x#bs) p"
and ly: "l < y" and yu: "y < u"
shows "Ifm (y#bs) p"
using lp px noS
proof (induct p rule: isrlfm.induct)
case (5 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from prems have "x * real c + ?N x e < 0" by (simp add: algebra_simps)
hence pxc: "x < (- ?N x e) / real c"
by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
with lx pxc have "False" by auto
hence ?case by simp }
ultimately show ?case by blast
next
case (6 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp +
from prems have "x * real c + ?N x e \<le> 0" by (simp add: algebra_simps)
hence pxc: "x \<le> (- ?N x e) / real c"
by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
moreover {assume y: "y < (-?N x e)/ real c"
hence "y * real c < - ?N x e"
by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
hence "real c * y + ?N x e < 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y > (- ?N x e) / real c"
with yu have eu: "u > (- ?N x e) / real c" by auto
with noSc ly yu have "(- ?N x e) / real c \<le> l" by (cases "(- ?N x e) / real c > l", auto)
with lx pxc have "False" by auto
hence ?case by simp }
ultimately show ?case by blast
next
case (7 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from prems have "x * real c + ?N x e > 0" by (simp add: algebra_simps)
hence pxc: "x > (- ?N x e) / real c"
by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
with xu pxc have "False" by auto
hence ?case by simp }
ultimately show ?case by blast
next
case (8 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from prems have "x * real c + ?N x e \<ge> 0" by (simp add: algebra_simps)
hence pxc: "x \<ge> (- ?N x e) / real c"
by (simp only: pos_divide_le_eq[OF cp, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y < (- ?N x e) / real c \<or> y > (-?N x e) / real c" by auto
moreover {assume y: "y > (-?N x e)/ real c"
hence "y * real c > - ?N x e"
by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
hence "real c * y + ?N x e > 0" by (simp add: algebra_simps)
hence ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp}
moreover {assume y: "y < (- ?N x e) / real c"
with ly have eu: "l < (- ?N x e) / real c" by auto
with noSc ly yu have "(- ?N x e) / real c \<ge> u" by (cases "(- ?N x e) / real c > l", auto)
with xu pxc have "False" by auto
hence ?case by simp }
ultimately show ?case by blast
next
case (3 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from cp have cnz: "real c \<noteq> 0" by simp
from prems have "x * real c + ?N x e = 0" by (simp add: algebra_simps)
hence pxc: "x = (- ?N x e) / real c"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with lx xu have yne: "x \<noteq> - ?N x e / real c" by auto
with pxc show ?case by simp
next
case (4 c e) hence cp: "real c > 0" and nb: "numbound0 e" by simp+
from cp have cnz: "real c \<noteq> 0" by simp
from prems have noSc:"\<forall> t. l < t \<and> t < u \<longrightarrow> t \<noteq> (- ?N x e) / real c" by auto
with ly yu have yne: "y \<noteq> - ?N x e / real c" by auto
hence "y* real c \<noteq> -?N x e"
by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
hence "y* real c + ?N x e \<noteq> 0" by (simp add: algebra_simps)
thus ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
qed (auto simp add: nth_pos2 numbound0_I[where bs="bs" and b="y" and b'="x"])

lemma finite_set_intervals:
assumes px: "P (x::real)"
and lx: "l \<le> x" and xu: "x \<le> u"
and linS: "l\<in> S" and uinS: "u \<in> S"
and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
shows "\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
proof-
let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
let ?xM = "{y. y\<in> S \<and> x \<le> y}"
let ?a = "Max ?Mx"
let ?b = "Min ?xM"
have MxS: "?Mx \<subseteq> S" by blast
hence fMx: "finite ?Mx" using fS finite_subset by auto
from lx linS have linMx: "l \<in> ?Mx" by blast
hence Mxne: "?Mx \<noteq> {}" by blast
have xMS: "?xM \<subseteq> S" by blast
hence fxM: "finite ?xM" using fS finite_subset by auto
from xu uinS have linxM: "u \<in> ?xM" by blast
hence xMne: "?xM \<noteq> {}" by blast
have ax:"?a \<le> x" using Mxne fMx by auto
have xb:"x \<le> ?b" using xMne fxM by auto
have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
proof(clarsimp)
fix y
assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by auto
moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by simp}
moreover {assume "y \<in> ?xM" hence "y \<ge> ?b" using xMne fxM by auto with yb have "False" by simp}
ultimately show "False" by blast
qed
from ainS binS noy ax xb px show ?thesis by blast
qed

lemma finite_set_intervals2:
assumes px: "P (x::real)"
and lx: "l \<le> x" and xu: "x \<le> u"
and linS: "l\<in> S" and uinS: "u \<in> S"
and fS:"finite S" and lS: "\<forall> x\<in> S. l \<le> x" and Su: "\<forall> x\<in> S. x \<le> u"
shows "(\<exists> s\<in> S. P s) \<or> (\<exists> a \<in> S. \<exists> b \<in> S. (\<forall> y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> P x)"
proof-
from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
obtain a and b where
as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by auto
thus ?thesis using px as bs noS by blast
qed

lemma rinf_uset:
assumes lp: "isrlfm p"
and nmi: "\<not> (Ifm (x#bs) (minusinf p))" (is "\<not> (Ifm (x#bs) (?M p))")
and npi: "\<not> (Ifm (x#bs) (plusinf p))" (is "\<not> (Ifm (x#bs) (?P p))")
and ex: "\<exists> x.  Ifm (x#bs) p" (is "\<exists> x. ?I x p")
shows "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((Inum (x#bs) l / real n + Inum (x#bs) s / real m) / 2) p"
proof-
let ?N = "\<lambda> x t. Inum (x#bs) t"
let ?U = "set (uset p)"
from ex obtain a where pa: "?I a p" by blast
from bound0_I[OF rminusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] nmi
have nmi': "\<not> (?I a (?M p))" by simp
from bound0_I[OF rplusinf_bound0[OF lp], where bs="bs" and b="x" and b'="a"] npi
have npi': "\<not> (?I a (?P p))" by simp
have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). ?I ((?N a l/real n + ?N a s /real m) / 2) p"
proof-
let ?M = "(\<lambda> (t,c). ?N a t / real c) ` ?U"
have fM: "finite ?M" by auto
from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
have "\<exists> (l,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). a \<le> ?N x l / real n \<and> a \<ge> ?N x s / real m" by blast
then obtain "t" "n" "s" "m" where
tnU: "(t,n) \<in> ?U" and smU: "(s,m) \<in> ?U"
and xs1: "a \<le> ?N x s / real m" and tx1: "a \<ge> ?N x t / real n" by blast
from uset_l[OF lp] tnU smU numbound0_I[where bs="bs" and b="x" and b'="a"] xs1 tx1 have xs: "a \<le> ?N a s / real m" and tx: "a \<ge> ?N a t / real n" by auto
from tnU have Mne: "?M \<noteq> {}" by auto
hence Une: "?U \<noteq> {}" by simp
let ?l = "Min ?M"
let ?u = "Max ?M"
have linM: "?l \<in> ?M" using fM Mne by simp
have uinM: "?u \<in> ?M" using fM Mne by simp
have tnM: "?N a t / real n \<in> ?M" using tnU by auto
have smM: "?N a s / real m \<in> ?M" using smU by auto
have lM: "\<forall> t\<in> ?M. ?l \<le> t" using Mne fM by auto
have Mu: "\<forall> t\<in> ?M. t \<le> ?u" using Mne fM by auto
have "?l \<le> ?N a t / real n" using tnM Mne by simp hence lx: "?l \<le> a" using tx by simp
have "?N a s / real m \<le> ?u" using smM Mne by simp hence xu: "a \<le> ?u" using xs by simp
from finite_set_intervals2[where P="\<lambda> x. ?I x p",OF pa lx xu linM uinM fM lM Mu]
have "(\<exists> s\<in> ?M. ?I s p) \<or>
(\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p)" .
moreover { fix u assume um: "u\<in> ?M" and pu: "?I u p"
hence "\<exists> (tu,nu) \<in> ?U. u = ?N a tu / real nu" by auto
then obtain "tu" "nu" where tuU: "(tu,nu) \<in> ?U" and tuu:"u= ?N a tu / real nu" by blast
have "(u + u) / 2 = u" by auto with pu tuu
have "?I (((?N a tu / real nu) + (?N a tu / real nu)) / 2) p" by simp
with tuU have ?thesis by blast}
moreover{
assume "\<exists> t1\<in> ?M. \<exists> t2 \<in> ?M. (\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M) \<and> t1 < a \<and> a < t2 \<and> ?I a p"
then obtain t1 and t2 where t1M: "t1 \<in> ?M" and t2M: "t2\<in> ?M"
and noM: "\<forall> y. t1 < y \<and> y < t2 \<longrightarrow> y \<notin> ?M" and t1x: "t1 < a" and xt2: "a < t2" and px: "?I a p"
by blast
from t1M have "\<exists> (t1u,t1n) \<in> ?U. t1 = ?N a t1u / real t1n" by auto
then obtain "t1u" "t1n" where t1uU: "(t1u,t1n) \<in> ?U" and t1u: "t1 = ?N a t1u / real t1n" by blast
from t2M have "\<exists> (t2u,t2n) \<in> ?U. t2 = ?N a t2u / real t2n" by auto
then obtain "t2u" "t2n" where t2uU: "(t2u,t2n) \<in> ?U" and t2u: "t2 = ?N a t2u / real t2n" by blast
from t1x xt2 have t1t2: "t1 < t2" by simp
let ?u = "(t1 + t2) / 2"
from less_half_sum[OF t1t2] gt_half_sum[OF t1t2] have t1lu: "t1 < ?u" and ut2: "?u < t2" by auto
from lin_dense[OF lp noM t1x xt2 px t1lu ut2] have "?I ?u p" .
with t1uU t2uU t1u t2u have ?thesis by blast}
ultimately show ?thesis by blast
qed
then obtain "l" "n" "s"  "m" where lnU: "(l,n) \<in> ?U" and smU:"(s,m) \<in> ?U"
and pu: "?I ((?N a l / real n + ?N a s / real m) / 2) p" by blast
from lnU smU uset_l[OF lp] have nbl: "numbound0 l" and nbs: "numbound0 s" by auto
from numbound0_I[OF nbl, where bs="bs" and b="a" and b'="x"]
numbound0_I[OF nbs, where bs="bs" and b="a" and b'="x"] pu
have "?I ((?N x l / real n + ?N x s / real m) / 2) p" by simp
with lnU smU
show ?thesis by auto
qed
(* The Ferrante - Rackoff Theorem *)

theorem fr_eq:
assumes lp: "isrlfm p"
shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,n) \<in> set (uset p). \<exists> (s,m) \<in> set (uset p). Ifm ((((Inum (x#bs) t)/  real n + (Inum (x#bs) s) / real m) /2)#bs) p))"
(is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
proof
assume px: "\<exists> x. ?I x p"
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
moreover {assume "?M \<or> ?P" hence "?D" by blast}
moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
from rinf_uset[OF lp nmi npi] have "?F" using px by blast hence "?D" by blast}
ultimately show "?D" by blast
next
assume "?D"
moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
moreover {assume f:"?F" hence "?E" by blast}
ultimately show "?E" by blast
qed

lemma fr_equsubst:
assumes lp: "isrlfm p"
shows "(\<exists> x. Ifm (x#bs) p) = ((Ifm (x#bs) (minusinf p)) \<or> (Ifm (x#bs) (plusinf p)) \<or> (\<exists> (t,k) \<in> set (uset p). \<exists> (s,l) \<in> set (uset p). Ifm (x#bs) (usubst p (Add(Mul l t) (Mul k s) , 2*k*l))))"
(is "(\<exists> x. ?I x p) = (?M \<or> ?P \<or> ?F)" is "?E = ?D")
proof
assume px: "\<exists> x. ?I x p"
have "?M \<or> ?P \<or> (\<not> ?M \<and> \<not> ?P)" by blast
moreover {assume "?M \<or> ?P" hence "?D" by blast}
moreover {assume nmi: "\<not> ?M" and npi: "\<not> ?P"
let ?f ="\<lambda> (t,n). Inum (x#bs) t / real n"
let ?N = "\<lambda> t. Inum (x#bs) t"
{fix t n s m assume "(t,n)\<in> set (uset p)" and "(s,m) \<in> set (uset p)"
with uset_l[OF lp] have tnb: "numbound0 t" and np:"real n > 0" and snb: "numbound0 s" and mp:"real m > 0"
by auto
let ?st = "Add (Mul m t) (Mul n s)"
from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
from tnb snb have st_nb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
from usubst_I[OF lp mnp st_nb, where x="x" and bs="bs"]
have "?I x (usubst p (?st,2*n*m)) = ?I ((?N t / real n + ?N s / real m) /2) p" by (simp only: st[symmetric])}
with rinf_uset[OF lp nmi npi px] have "?F" by blast hence "?D" by blast}
ultimately show "?D" by blast
next
assume "?D"
moreover {assume m:"?M" from rminusinf_ex[OF lp m] have "?E" .}
moreover {assume p: "?P" from rplusinf_ex[OF lp p] have "?E" . }
moreover {fix t k s l assume "(t,k) \<in> set (uset p)" and "(s,l) \<in> set (uset p)"
and px:"?I x (usubst p (Add (Mul l t) (Mul k s), 2*k*l))"
with uset_l[OF lp] have tnb: "numbound0 t" and np:"real k > 0" and snb: "numbound0 s" and mp:"real l > 0" by auto
let ?st = "Add (Mul l t) (Mul k s)"
from mult_pos_pos[OF np mp] have mnp: "real (2*k*l) > 0"
from tnb snb have st_nb: "numbound0 ?st" by simp
from usubst_I[OF lp mnp st_nb, where bs="bs"] px have "?E" by auto}
ultimately show "?E" by blast
qed

(* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
constdefs ferrack:: "fm \<Rightarrow> fm"
"ferrack p \<equiv> (let p' = rlfm (simpfm p); mp = minusinf p'; pp = plusinf p'
in if (mp = T \<or> pp = T) then T else
(let U = remdps(map simp_num_pair
(map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m))
(alluopairs (uset p'))))
in decr (disj mp (disj pp (evaldjf (simpfm o (usubst p')) U)))))"

lemma uset_cong_aux:
assumes Ul: "\<forall> (t,n) \<in> set U. numbound0 t \<and> n >0"
shows "((\<lambda> (t,n). Inum (x#bs) t /real n) ` (set (map (\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)) (alluopairs U)))) = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (set U \<times> set U))"
(is "?lhs = ?rhs")
proof(auto)
fix t n s m
assume "((t,n),(s,m)) \<in> set (alluopairs U)"
hence th: "((t,n),(s,m)) \<in> (set U \<times> set U)"
using alluopairs_set1[where xs="U"] by blast
let ?N = "\<lambda> t. Inum (x#bs) t"
let ?st= "Add (Mul m t) (Mul n s)"
from Ul th have mnz: "m \<noteq> 0" by auto
from Ul th have  nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"

thus "(real m *  Inum (x # bs) t + real n * Inum (x # bs) s) /
(2 * real n * real m)
\<in> (\<lambda>((t, n), s, m).
(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2) `
(set U \<times> set U)"using mnz nnz th
by (rule_tac x="(s,m)" in bexI,simp_all)
(rule_tac x="(t,n)" in bexI,simp_all)
next
fix t n s m
assume tnU: "(t,n) \<in> set U" and smU:"(s,m) \<in> set U"
let ?N = "\<lambda> t. Inum (x#bs) t"
let ?st= "Add (Mul m t) (Mul n s)"
from Ul smU have mnz: "m \<noteq> 0" by auto
from Ul tnU have  nnz: "n \<noteq> 0" by auto
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
let ?P = "\<lambda> (t',n') (s',m'). (Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2"
have Pc:"\<forall> a b. ?P a b = ?P b a"
by auto
from Ul alluopairs_set1 have Up:"\<forall> ((t,n),(s,m)) \<in> set (alluopairs U). n \<noteq> 0 \<and> m \<noteq> 0" by blast
from alluopairs_ex[OF Pc, where xs="U"] tnU smU
have th':"\<exists> ((t',n'),(s',m')) \<in> set (alluopairs U). ?P (t',n') (s',m')"
by blast
then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) \<in> set (alluopairs U)"
and Pts': "?P (t',n') (s',m')" by blast
from ts'_U Up have mnz': "m' \<noteq> 0" and nnz': "n'\<noteq> 0" by auto
let ?st' = "Add (Mul m' t') (Mul n' s')"
have st': "(?N t' / real n' + ?N s' / real m')/2 = ?N ?st' / real (2*n'*m')"
from Pts' have
"(Inum (x # bs) t / real n + Inum (x # bs) s / real m)/2 = (Inum (x # bs) t' / real n' + Inum (x # bs) s' / real m')/2" by simp
also have "\<dots> = ((\<lambda>(t, n). Inum (x # bs) t / real n) ((\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) ((t',n'),(s',m'))))" by (simp add: st')
finally show "(Inum (x # bs) t / real n + Inum (x # bs) s / real m) / 2
\<in> (\<lambda>(t, n). Inum (x # bs) t / real n) `
(\<lambda>((t, n), s, m). (Add (Mul m t) (Mul n s), 2 * n * m)) `
set (alluopairs U)"
using ts'_U by blast
qed

lemma uset_cong:
assumes lp: "isrlfm p"
and UU': "((\<lambda> (t,n). Inum (x#bs) t /real n) ` U') = ((\<lambda> ((t,n),(s,m)). (Inum (x#bs) t /real n + Inum (x#bs) s /real m)/2) ` (U \<times> U))" (is "?f ` U' = ?g ` (U\<times>U)")
and U: "\<forall> (t,n) \<in> U. numbound0 t \<and> n > 0"
and U': "\<forall> (t,n) \<in> U'. numbound0 t \<and> n > 0"
shows "(\<exists> (t,n) \<in> U. \<exists> (s,m) \<in> U. Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))) = (\<exists> (t,n) \<in> U'. Ifm (x#bs) (usubst p (t,n)))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
Pst: "Ifm (x#bs) (usubst p (Add (Mul m t) (Mul n s),2*n*m))" by blast
let ?N = "\<lambda> t. Inum (x#bs) t"
from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
and snb: "numbound0 s" and mp:"m > 0"  by auto
let ?st= "Add (Mul m t) (Mul n s)"
from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
from tnU smU UU' have "?g ((t,n),(s,m)) \<in> ?f ` U'" by blast
hence "\<exists> (t',n') \<in> U'. ?g ((t,n),(s,m)) = ?f (t',n')"
by auto (rule_tac x="(a,b)" in bexI, auto)
then obtain t' n' where tnU': "(t',n') \<in> U'" and th: "?g ((t,n),(s,m)) = ?f (t',n')" by blast
from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
from usubst_I[OF lp mnp stnb, where bs="bs" and x="x"] Pst
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
from conjunct1[OF usubst_I[OF lp np' tnb', where bs="bs" and x="x"], symmetric] th[simplified split_def fst_conv snd_conv,symmetric] Pst2[simplified st[symmetric]]
have "Ifm (x # bs) (usubst p (t', n')) " by (simp only: st)
then show ?rhs using tnU' by auto
next
assume ?rhs
then obtain t' n' where tnU': "(t',n') \<in> U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))"
by blast
from tnU' UU' have "?f (t',n') \<in> ?g ` (U\<times>U)" by blast
hence "\<exists> ((t,n),(s,m)) \<in> (U\<times>U). ?f (t',n') = ?g ((t,n),(s,m))"
by auto (rule_tac x="(a,b)" in bexI, auto)
then obtain t n s m where tnU: "(t,n) \<in> U" and smU:"(s,m) \<in> U" and
th: "?f (t',n') = ?g((t,n),(s,m)) "by blast
let ?N = "\<lambda> t. Inum (x#bs) t"
from tnU smU U have tnb: "numbound0 t" and np: "n > 0"
and snb: "numbound0 s" and mp:"m > 0"  by auto
let ?st= "Add (Mul m t) (Mul n s)"
from mult_pos_pos[OF np mp] have mnp: "real (2*n*m) > 0"
by (simp add: mult_commute real_of_int_mult[symmetric] del: real_of_int_mult)
from tnb snb have stnb: "numbound0 ?st" by simp
have st: "(?N t / real n + ?N s / real m)/2 = ?N ?st / real (2*n*m)"
from U' tnU' have tnb': "numbound0 t'" and np': "real n' > 0" by auto
from usubst_I[OF lp np' tnb', where bs="bs" and x="x",simplified th[simplified split_def fst_conv snd_conv] st] Pt'
have Pst2: "Ifm (Inum (x # bs) (Add (Mul m t) (Mul n s)) / real (2 * n * m) # bs) p" by simp
with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU show ?lhs by blast
qed

lemma ferrack:
assumes qf: "qfree p"
shows "qfree (ferrack p) \<and> ((Ifm bs (ferrack p)) = (\<exists> x. Ifm (x#bs) p))"
(is "_ \<and> (?rhs = ?lhs)")
proof-
let ?I = "\<lambda> x p. Ifm (x#bs) p"
fix x
let ?N = "\<lambda> t. Inum (x#bs) t"
let ?q = "rlfm (simpfm p)"
let ?U = "uset ?q"
let ?Up = "alluopairs ?U"
let ?g = "\<lambda> ((t,n),(s,m)). (Add (Mul m t) (Mul n s) , 2*n*m)"
let ?S = "map ?g ?Up"
let ?SS = "map simp_num_pair ?S"
let ?Y = "remdps ?SS"
let ?f= "(\<lambda> (t,n). ?N t / real n)"
let ?h = "\<lambda> ((t,n),(s,m)). (?N t/real n + ?N s/ real m) /2"
let ?F = "\<lambda> p. \<exists> a \<in> set (uset p). \<exists> b \<in> set (uset p). ?I x (usubst p (?g(a,b)))"
let ?ep = "evaldjf (simpfm o (usubst ?q)) ?Y"
from rlfm_I[OF simpfm_qf[OF qf]] have lq: "isrlfm ?q" by blast
from alluopairs_set1[where xs="?U"] have UpU: "set ?Up \<le> (set ?U \<times> set ?U)" by simp
from uset_l[OF lq] have U_l: "\<forall> (t,n) \<in> set ?U. numbound0 t \<and> n > 0" .
from U_l UpU
have "\<forall> ((t,n),(s,m)) \<in> set ?Up. numbound0 t \<and> n> 0 \<and> numbound0 s \<and> m > 0" by auto
hence Snb: "\<forall> (t,n) \<in> set ?S. numbound0 t \<and> n > 0 "
have Y_l: "\<forall> (t,n) \<in> set ?Y. numbound0 t \<and> n > 0"
proof-
{ fix t n assume tnY: "(t,n) \<in> set ?Y"
hence "(t,n) \<in> set ?SS" by simp
hence "\<exists> (t',n') \<in> set ?S. simp_num_pair (t',n') = (t,n)"
by (auto simp add: split_def) (rule_tac x="((aa,ba),(ab,bb))" in bexI, simp_all)
then obtain t' n' where tn'S: "(t',n') \<in> set ?S" and tns: "simp_num_pair (t',n') = (t,n)" by blast
from tn'S Snb have tnb: "numbound0 t'" and np: "n' > 0" by auto
from simp_num_pair_l[OF tnb np tns]
have "numbound0 t \<and> n > 0" . }
thus ?thesis by blast
qed

have YU: "(?f ` set ?Y) = (?h ` (set ?U \<times> set ?U))"
proof-
from simp_num_pair_ci[where bs="x#bs"] have
"\<forall>x. (?f o simp_num_pair) x = ?f x" by auto
hence th: "?f o simp_num_pair = ?f" using ext by blast
have "(?f ` set ?Y) = ((?f o simp_num_pair) ` set ?S)" by (simp add: image_compose)
also have "\<dots> = (?f ` set ?S)" by (simp add: th)
also have "\<dots> = ((?f o ?g) ` set ?Up)"
by (simp only: set_map o_def image_compose[symmetric])
also have "\<dots> = (?h ` (set ?U \<times> set ?U))"
using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_compose[symmetric]] by blast
finally show ?thesis .
qed
have "\<forall> (t,n) \<in> set ?Y. bound0 (simpfm (usubst ?q (t,n)))"
proof-
{ fix t n assume tnY: "(t,n) \<in> set ?Y"
with Y_l have tnb: "numbound0 t" and np: "real n > 0" by auto
from usubst_I[OF lq np tnb]
have "bound0 (usubst ?q (t,n))"  by simp hence "bound0 (simpfm (usubst ?q (t,n)))"
using simpfm_bound0 by simp}
thus ?thesis by blast
qed
hence ep_nb: "bound0 ?ep"  using evaldjf_bound0[where xs="?Y" and f="simpfm o (usubst ?q)"] by auto
let ?mp = "minusinf ?q"
let ?pp = "plusinf ?q"
let ?M = "?I x ?mp"
let ?P = "?I x ?pp"
let ?res = "disj ?mp (disj ?pp ?ep)"
from rminusinf_bound0[OF lq] rplusinf_bound0[OF lq] ep_nb
have nbth: "bound0 ?res" by auto

from conjunct1[OF rlfm_I[OF simpfm_qf[OF qf]]] simpfm

have th: "?lhs = (\<exists> x. ?I x ?q)" by auto
from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M \<or> ?P \<or> ?F ?q)"
by (simp only: split_def fst_conv snd_conv)
also have "\<dots> = (?M \<or> ?P \<or> (\<exists> (t,n) \<in> set ?Y. ?I x (simpfm (usubst ?q (t,n)))))"
using uset_cong[OF lq YU U_l Y_l]  by (simp only: split_def fst_conv snd_conv simpfm)
also have "\<dots> = (Ifm (x#bs) ?res)"
using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm o (usubst ?q)",symmetric]
finally have lheq: "?lhs =  (Ifm bs (decr ?res))" using decr[OF nbth] by blast
hence lr: "?lhs = ?rhs" apply (unfold ferrack_def Let_def)
by (cases "?mp = T \<or> ?pp = T", auto) (simp add: disj_def)+
from decr_qf[OF nbth] have "qfree (ferrack p)" by (auto simp add: Let_def ferrack_def)
with lr show ?thesis by blast
qed

definition linrqe:: "fm \<Rightarrow> fm" where
"linrqe p = qelim (prep p) ferrack"

theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p \<and> qfree (linrqe p)"
using ferrack qelim_ci prep
unfolding linrqe_def by auto

definition ferrack_test :: "unit \<Rightarrow> fm" where
"ferrack_test u = linrqe (A (A (Imp (Lt (Sub (Bound 1) (Bound 0)))
(E (Eq (Sub (Add (Bound 0) (Bound 2)) (Bound 1)))))))"

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

oracle linr_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 "real (0::int)"} = @{code C} 0
| num_of_term vs @{term "real (1::int)"} = @{code C} 1
| num_of_term vs @{term "0::real"} = @{code C} 0
| num_of_term vs @{term "1::real"} = @{code C} 1
| num_of_term vs (Bound i) = @{code Bound} i
| num_of_term vs (@{term "uminus :: real \<Rightarrow> real"} \$ t') = @{code Neg} (num_of_term vs t')
| num_of_term vs (@{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = @{code Add} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = @{code Sub} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$ t1 \$ t2) = (case (num_of_term vs t1)
of @{code C} i => @{code Mul} (i, num_of_term vs t2)
| _ => error "num_of_term: unsupported Multiplication")
| num_of_term vs (@{term "real :: int \<Rightarrow> real"} \$ (@{term "number_of :: int \<Rightarrow> int"} \$ t')) = @{code C} (HOLogic.dest_numeral t')
| num_of_term vs (@{term "number_of :: int \<Rightarrow> real"} \$ t') = @{code C} (HOLogic.dest_numeral t')
| num_of_term vs t = error ("num_of_term: unknown term " ^ Syntax.string_of_term @{context} t);

fun fm_of_term vs @{term True} = @{code T}
| fm_of_term vs @{term False} = @{code F}
| fm_of_term vs (@{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$ t1 \$ t2) = @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term "op &"} \$ t1 \$ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term "op |"} \$ t1 \$ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term "op -->"} \$ t1 \$ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term "Not"} \$ t') = @{code NOT} (fm_of_term vs t')
| fm_of_term vs (Const ("Ex", _) \$ Abs (xn, xT, p)) =
@{code E} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
| fm_of_term vs (Const ("All", _) \$ Abs (xn, xT, p)) =
@{code A} (fm_of_term (map (fn (v, n) => (v, n + 1)) vs) p)
| fm_of_term vs t = error ("fm_of_term : unknown term " ^ Syntax.string_of_term @{context} t);

fun term_of_num vs (@{code C} i) = @{term "real :: int \<Rightarrow> real"} \$ 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 :: real \<Rightarrow> real"} \$ term_of_num vs t'
| term_of_num vs (@{code Add} (t1, t2)) = @{term "op + :: real \<Rightarrow> real \<Rightarrow> real"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "op - :: real \<Rightarrow> real \<Rightarrow> real"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Mul} (i, t2)) = @{term "op * :: real \<Rightarrow> real \<Rightarrow> real"} \$
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 vs @{code T} = HOLogic.true_const
| term_of_fm vs @{code F} = HOLogic.false_const
| term_of_fm vs (@{code Lt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code Le} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code Gt} t) = @{term "op < :: real \<Rightarrow> real \<Rightarrow> bool"} \$
@{term "0::real"} \$ term_of_num vs t
| term_of_fm vs (@{code Ge} t) = @{term "op \<le> :: real \<Rightarrow> real \<Rightarrow> bool"} \$
@{term "0::real"} \$ term_of_num vs t
| term_of_fm vs (@{code Eq} t) = @{term "op = :: real \<Rightarrow> real \<Rightarrow> bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
| term_of_fm vs (@{code NOT} t') = HOLogic.Not \$ term_of_fm vs t'
| term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Iff} (t1, t2)) = @{term "op \<longleftrightarrow> :: bool \<Rightarrow> bool \<Rightarrow> bool"} \$
term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs _ = error "If this is raised, Isabelle/HOL or generate_code is inconsistent.";

in fn ct =>
let
val thy = Thm.theory_of_cterm ct;
val t = Thm.term_of ct;
val fs = OldTerm.term_frees t;
val vs = fs ~~ (0 upto (length fs - 1));
val res = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, term_of_fm vs (@{code linrqe} (fm_of_term vs t))));
in Thm.cterm_of thy res end
end;
*}

use "ferrack_tac.ML"
setup Ferrack_Tac.setup

lemma
fixes x :: real
shows "2 * x \<le> 2 * x \<and> 2 * x \<le> 2 * x + 1"
apply rferrack
done

lemma
fixes x :: real
shows "\<exists>y \<le> x. x = y + 1"
apply rferrack
done

lemma
fixes x :: real
shows "\<not> (\<exists>z. x + z = x + z + 1)"
apply rferrack
done

end
```