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Datei: how_to_build_eliza_chatterbot.htm Sprache: Isabelle

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

theory Ferrack
imports Complex_Main Dense_Linear_Order DP_Library
  "HOL-Library.Code_Target_Numeral"
begin

section \<open>Quantifier elimination for \<open>\<real> (0, 1, +, <)\<close>\<close>

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

datatype (plugins del: size) num = C int | Bound nat | CN nat int num | Neg num | Add num num| Sub num num
  | Mul int num

instantiation num :: size
begin

primrec size_num :: "num \ nat"
where
  "size_num (C c) = 1"
| "size_num (Bound n) = 1"
| "size_num (Neg a) = 1 + size_num a"
| "size_num (Add a b) = 1 + size_num a + size_num b"
| "size_num (Sub a b) = 3 + size_num a + size_num b"
| "size_num (Mul c a) = 1 + size_num a"
| "size_num (CN n c a) = 3 + size_num a "

instance ..

end

  (* Semantics of numeral terms (num) *)
primrec Inum :: "real list \ num \ real"
where
  "Inum bs (C c) = (real_of_int c)"
| "Inum bs (Bound n) = bs!n"
| "Inum bs (CN n c a) = (real_of_int 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_of_int c) * Inum bs a"
    (* FORMULAE *)
datatype (plugins del: size) 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

instantiation fm :: size
begin

primrec size_fm :: "fm \ nat"
where
  "size_fm (NOT p) = 1 + size_fm p"
| "size_fm (And p q) = 1 + size_fm p + size_fm q"
| "size_fm (Or p q) = 1 + size_fm p + size_fm q"
| "size_fm (Imp p q) = 3 + size_fm p + size_fm q"
| "size_fm (Iff p q) = 3 + 2 * (size_fm p + size_fm q)"
| "size_fm (E p) = 1 + size_fm p"
| "size_fm (A p) = 4 + size_fm p"
| "size_fm T = 1"
| "size_fm F = 1"
| "size_fm (Lt _) = 1"
| "size_fm (Le _) = 1"
| "size_fm (Gt _) = 1"
| "size_fm (Ge _) = 1"
| "size_fm (Eq _) = 1"
| "size_fm (NEq _) = 1"

instance ..

end

lemma size_fm_pos [simp]: "size p > 0" for p :: fm
  by (induct p) simp_all

  (* Semantics of formulae (fm) *)
primrec Ifm ::"real list \ fm \ bool"
where
  "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 \ 0)"
| "Ifm bs (Ge a) = (Inum bs a \ 0)"
| "Ifm bs (Eq a) = (Inum bs a = 0)"
| "Ifm bs (NEq a) = (Inum bs a \ 0)"
| "Ifm bs (NOT p) = (\ (Ifm bs p))"
| "Ifm bs (And p q) = (Ifm bs p \ Ifm bs q)"
| "Ifm bs (Or p q) = (Ifm bs p \ Ifm bs q)"
| "Ifm bs (Imp p q) = ((Ifm bs p) \ (Ifm bs q))"
| "Ifm bs (Iff p q) = (Ifm bs p = Ifm bs q)"
| "Ifm bs (E p) = (\x. Ifm (x#bs) p)"
| "Ifm bs (A p) = (\x. Ifm (x#bs) p)"

lemma IfmLeSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Le (Sub s t)) = (s' \ t')"
  by simp

lemma IfmLtSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Lt (Sub s t)) = (s' < t')"
  by simp

lemma IfmEqSub: "\ Inum bs s = s' ; Inum bs t = t' \ \ Ifm bs (Eq (Sub s t)) = (s' = t')"
  by simp

lemma IfmNOT: " (Ifm bs p = P) \ (Ifm bs (NOT p) = (\P))"
  by simp

lemma IfmAnd: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (And p q) = (P \ Q))"
  by simp

lemma IfmOr: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Or p q) = (P \ Q))"
  by simp

lemma IfmImp: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Imp p q) = (P \ Q))"
  by simp

lemma IfmIff: " \ Ifm bs p = P ; Ifm bs q = Q\ \ (Ifm bs (Iff p q) = (P = Q))"
  by simp

lemma IfmE: " (!! x. Ifm (x#bs) p = P x) \ (Ifm bs (E p) = (\x. P x))"
  by simp

lemma IfmA: " (!! x. Ifm (x#bs) p = P x) \ (Ifm bs (A p) = (\x. P x))"
  by simp

fun not:: "fm \ fm"
where
  "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

definition conj :: "fm \ fm \ fm"
where
  "conj p q =
   (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 \ q = F", simp_all add: conj_def) (cases p, simp_all)

definition disj :: "fm \ fm \ fm"
where
  "disj p q =
   (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 \ q = T", simp_all add: disj_def) (cases p, simp_all)

definition imp :: "fm \ fm \ fm"
where
  "imp p q =
   (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 \ q = T") (simp_all add: imp_def)

definition iff :: "fm \ fm \ fm"
where
  "iff p q =
   (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 \ T \ P \ F \ Q \ T \ Q \ F \ P \ Q \ conj P Q = And P Q"
  by (simp_all add: conj_def)

lemma disj_simps:
  "disj T Q = T"
  "disj P T = T"
  "disj F Q = Q"
  "disj P F = P"
  "disj P P = P"
  "P \ T \ P \ F \ Q \ T \ Q \ F \ P \ Q \ disj P Q = Or P Q"
  by (simp_all add: disj_def)

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 \ T \ P \ F \ P \ Q \ Q \ T \ Q \ F \ imp P Q = Imp P Q"
  by (simp_all add: imp_def)

lemma trivNOT: "p \ NOT p" "NOT p \ p"
  by (induct p) auto

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 \ NOT T \ iff T p = p"
  "p\ NOT T \ iff p T = p"
  "p\q \ p\ NOT q \ q\ NOT p \ p\ F \ q\ F \ p \ T \ q \ T \ iff p q = Iff p q"
  using trivNOT
  by (simp_all add: iff_def, cases p, auto)

  (* Quantifier freeness *)
fun qfree:: "fm \ bool"
where
  "qfree (E p) = False"
| "qfree (A p) = False"
| "qfree (NOT p) = qfree p"
| "qfree (And p q) = (qfree p \ qfree q)"
| "qfree (Or p q) = (qfree p \ qfree q)"
| "qfree (Imp p q) = (qfree p \ qfree q)"
| "qfree (Iff p q) = (qfree p \ qfree q)"
| "qfree p = True"

  (* Boundedness and substitution *)
primrec numbound0:: "num \ bool" (* a num is INDEPENDENT of Bound 0 *)
where
  "numbound0 (C c) = True"
| "numbound0 (Bound n) = (n > 0)"
| "numbound0 (CN n c a) = (n \ 0 \ numbound0 a)"
| "numbound0 (Neg a) = numbound0 a"
| "numbound0 (Add a b) = (numbound0 a \ numbound0 b)"
| "numbound0 (Sub a b) = (numbound0 a \ 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) simp_all

primrec bound0:: "fm \ bool" (* A Formula is independent of Bound 0 *)
where
  "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 \ bound0 q)"
| "bound0 (Or p q) = (bound0 p \ bound0 q)"
| "bound0 (Imp p q) = ((bound0 p) \ (bound0 q))"
| "bound0 (Iff p q) = (bound0 p \ 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) auto

lemma not_qf[simp]: "qfree p \ qfree (not p)"
  by (cases p) auto

lemma not_bn[simp]: "bound0 p \ bound0 (not p)"
  by (cases p) auto

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

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

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

lemma iff_qf[simp]: "\qfree p ; qfree q\ \ qfree (iff p q)"
  unfolding iff_def by (cases "p = q") auto
lemma iff_nb[simp]: "\bound0 p ; bound0 q\ \ bound0 (iff p q)"
  using iff_def unfolding iff_def by (cases "p = q") auto

fun decrnum:: "num \ num"
where
  "decrnum (Bound n) = Bound (n - 1)"
| "decrnum (Neg a) = Neg (decrnum a)"
| "decrnum (Add a b) = Add (decrnum a) (decrnum b)"
| "decrnum (Sub a b) = Sub (decrnum a) (decrnum b)"
| "decrnum (Mul c a) = Mul c (decrnum a)"
| "decrnum (CN n c a) = CN (n - 1) c (decrnum a)"
| "decrnum a = a"

fun decr :: "fm \ fm"
where
  "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

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: decrnum)

lemma decr_qf: "bound0 p \ qfree (decr p)"
  by (induct p) simp_all

fun isatom :: "fm \ bool" (* test for atomicity *)
where
  "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 \ qfree p"
  by (induct p) simp_all

definition djf :: "('a \ fm) \ 'a \ fm \ fm"
where
  "djf f p q =
   (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))"

definition evaldjf :: "('a \ fm) \ 'a list \ fm"
  where "evaldjf f ps = foldr (djf f) ps F"

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

lemma djf_simps:
  "djf f p T = T"
  "djf f p F = f p"
  "q \ T \ q \ F \ djf f p q = (let fp = f p in case fp of T \ T | F \ q | _ \ Or (f p) q)"
  by (simp_all add: djf_def)

lemma evaldjf_ex: "Ifm bs (evaldjf f ps) \ (\p \ set ps. Ifm bs (f p))"
  by (induct ps) (simp_all add: evaldjf_def djf_Or)

lemma evaldjf_bound0:
  assumes nb: "\x\ 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: "\x\ 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)

fun disjuncts :: "fm \ fm list"
where
  "disjuncts (Or p q) = disjuncts p @ disjuncts q"
| "disjuncts F = []"
| "disjuncts p = [p]"

lemma disjuncts: "(\q\ set (disjuncts p). Ifm bs q) = Ifm bs p"
  by (induct p rule: disjuncts.induct) auto

lemma disjuncts_nb: "bound0 p \ \q\ set (disjuncts p). bound0 q"
proof -
  assume nb: "bound0 p"
  then have "list_all bound0 (disjuncts p)"
    by (induct p rule: disjuncts.induct) auto
  then show ?thesis
    by (simp only: list_all_iff)
qed

lemma disjuncts_qf: "qfree p \ \q\ set (disjuncts p). qfree q"
proof -
  assume qf: "qfree p"
  then have "list_all qfree (disjuncts p)"
    by (induct p rule: disjuncts.induct) auto
  then show ?thesis
    by (simp only: list_all_iff)
qed

definition DJ :: "(fm \ fm) \ fm \ fm"
  where "DJ f p = evaldjf f (disjuncts p)"

lemma DJ:
  assumes fdj: "\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) = (\q \ set (disjuncts p). Ifm bs (f q))"
    by (simp add: DJ_def evaldjf_ex)
  also have "\ = Ifm bs (f p)"
    using fdj fF by (induct p rule: disjuncts.induct) auto
  finally show ?thesis .
qed

lemma DJ_qf:
  assumes fqf: "\p. qfree p \ qfree (f p)"
  shows "\p. qfree p \ 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 "\q\ set (disjuncts p). qfree q" .
  with fqf have th':"\q\ 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: "\bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))"
  shows "\bs p. qfree p \ qfree (DJ qe p) \ (Ifm bs ((DJ qe p)) = Ifm bs (E p))"
proof clarify
  fix p :: fm
  fix bs
  assume qf: "qfree p"
  from qe have qth: "\p. qfree p \ 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) \ (\q\ set (disjuncts p). Ifm bs (qe q))"
    by (simp add: DJ_def evaldjf_ex)
  also have "\ \ (\q \ set(disjuncts p). Ifm bs (E q))"
    using qe disjuncts_qf[OF qf] by auto
  also have "\ = Ifm bs (E p)"
    by (induct p rule: disjuncts.induct) auto
  finally show "qfree (DJ qe p) \ Ifm bs (DJ qe p) = Ifm bs (E p)"
    using qfth by blast
qed

  (* Simplification *)

fun maxcoeff:: "num \ int"
where
  "maxcoeff (C i) = \i\"
| "maxcoeff (CN n c t) = max \c\ (maxcoeff t)"
| "maxcoeff t = 1"

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

fun numgcdh:: "num \ int \ int"
where
  "numgcdh (C i) = (\g. gcd i g)"
| "numgcdh (CN n c t) = (\g. gcd c (numgcdh t g))"
| "numgcdh t = (\g. 1)"

definition numgcd :: "num \ int"
  where "numgcd t = numgcdh t (maxcoeff t)"

fun reducecoeffh:: "num \ int \ num"
where
  "reducecoeffh (C i) = (\g. C (i div g))"
| "reducecoeffh (CN n c t) = (\g. CN n (c div g) (reducecoeffh t g))"
| "reducecoeffh t = (\g. t)"

definition reducecoeff :: "num \ num"
where
  "reducecoeff t =
   (let g = numgcd t
    in if g = 0 then C 0 else if g = 1 then t else reducecoeffh t g)"

fun dvdnumcoeff:: "num \ int \ bool"
where
  "dvdnumcoeff (C i) = (\g. g dvd i)"
| "dvdnumcoeff (CN n c t) = (\g. g dvd c \ dvdnumcoeff t g)"
| "dvdnumcoeff t = (\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])

declare dvd_trans [trans add]

lemma natabs0: "nat \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: natabs0 maxcoeff_pos max.absorb2)

lemma numgcdh_pos:
  assumes gp: "g \ 0"
  shows "numgcdh t g \ 0"
  using gp by (induct t rule: numgcdh.induct) auto

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

lemma reducecoeffh:
  assumes gt: "dvdnumcoeff t g"
    and gp: "g > 0"
  shows "real_of_int g *(Inum bs (reducecoeffh t g)) = Inum bs t"
  using gt
proof (induct t rule: reducecoeffh.induct)
  case (1 i)
  then have gd: "g dvd i"
    by simp
  with assms show ?case
    by (simp add: real_of_int_div[OF gd])
next
  case (2 n c t)
  then have gd: "g dvd c"
    by simp
  from assms 2 show ?case
    by (simp add: real_of_int_div[OF gd] algebra_simps)
qed (auto simp add: numgcd_def gp)

fun ismaxcoeff:: "num \ int \ bool"
where
  "ismaxcoeff (C i) = (\x. \i\ \ x)"
| "ismaxcoeff (CN n c t) = (\x. \c\ \ x \ ismaxcoeff t x)"
| "ismaxcoeff t = (\x. True)"

lemma ismaxcoeff_mono: "ismaxcoeff t c \ c \ c' \ 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)
  then have H:"ismaxcoeff t (maxcoeff t)" .
  have thh: "maxcoeff t \ max \c\ (maxcoeff t)"
    by simp
  from ismaxcoeff_mono[OF H thh] show ?case
    by simp
qed simp_all

lemma zgcd_gt1:
  "\i\ > 1 \ \j\ > 1 \ \i\ = 0 \ \j\ > 1 \ \i\ > 1 \ \j\ = 0"
  if "gcd i j > 1" for i j :: int
proof -
  have "\k\ \ 1 \ k = - 1 \ k = 0 \ k = 1" for k :: int
    by auto
  with that show ?thesis
    by (auto simp add: not_less)
qed

lemma numgcdh0:"numgcdh t m = 0 \ m =0"
  by (induct t rule: numgcdh.induct) auto

lemma dvdnumcoeff_aux:
  assumes "ismaxcoeff t m"
    and mp: "m \ 0"
    and "numgcdh t m > 1"
  shows "dvdnumcoeff t (numgcdh t m)"
  using assms
proof (induct t rule: numgcdh.induct)
  case (2 n c t)
  let ?g = "numgcdh t m"
  from 2 have th: "gcd c ?g > 1"
    by simp
  from zgcd_gt1[OF th] numgcdh_pos[OF mp, where t="t"]
  consider "\c\ > 1" "?g > 1" | "\c\ = 0" "?g > 1" | "?g = 0"
    by auto
  then show ?case
  proof cases
    case 1
    with 2 have th: "dvdnumcoeff t ?g"
      by simp
    have th': "gcd c ?g dvd ?g"
      by simp
    from dvdnumcoeff_trans[OF th' th] show ?thesis
      by simp
  next
    case "2'": 2
    with 2 have th: "dvdnumcoeff t ?g"
      by simp
    have th': "gcd c ?g dvd ?g"
      by simp
    from dvdnumcoeff_trans[OF th' th] show ?thesis
      by simp
  next
    case 3
    then have "m = 0" by (rule numgcdh0)
    with 2 3 show ?thesis by simp
  qed
qed auto

lemma dvdnumcoeff_aux2:
  assumes "numgcd t > 1"
  shows "dvdnumcoeff t (numgcd t) \ numgcd t > 0"
  using assms
proof (simp add: numgcd_def)
  let ?mc = "maxcoeff t"
  let ?g = "numgcdh t ?mc"
  have th1: "ismaxcoeff t ?mc"
    by (rule maxcoeff_ismaxcoeff)
  have th2: "?mc \ 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_of_int (numgcd t) * (Inum bs (reducecoeff t)) = Inum bs t"
proof -
  let ?g = "numgcd t"
  have "?g \ 0"
    by (simp add: numgcd_pos)
  then consider "?g = 0" | "?g = 1" | "?g > 1" by atomize_elim auto
  then show ?thesis
  proof cases
    case 1
    then show ?thesis by (simp add: numgcd0)
  next
    case 2
    then show ?thesis by (simp add: reducecoeff_def)
  next
    case g1: 3
    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 show ?thesis
      by (simp add: reducecoeff_def Let_def)
  qed
qed

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

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

fun numadd:: "num \ num \ num"
where
  "numadd (CN n1 c1 r1) (CN n2 c2 r2) =
   (if n1 = n2 then
    (let c = c1 + c2
     in (if c = 0 then numadd r1 r2 else CN n1 c (numadd r1 r2)))
    else if n1 \<le> n2 then (CN n1 c1 (numadd r1 (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 t (CN n2 c2 r2) = CN n2 c2 (numadd t r2)"
| "numadd (C b1) (C b2) = C (b1 + b2)"
| "numadd a b = Add a b"

lemma numadd [simp]: "Inum bs (numadd t s) = Inum bs (Add t s)"
  by (induct t s rule: numadd.induct) (simp_all add: Let_def algebra_simps add_eq_0_iff)

lemma numadd_nb [simp]: "numbound0 t \ numbound0 s \ numbound0 (numadd t s)"
  by (induct t s rule: numadd.induct) (simp_all add: Let_def)

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

lemma nummul[simp]: "\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]: "\i. numbound0 t \ numbound0 (nummul t i)"
  by (induct t rule: nummul.induct) auto

definition numneg :: "num \ num"
  where "numneg t = nummul t (- 1)"

definition numsub :: "num \ num \ num"
  where "numsub s t = (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 \ 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]: "\ numbound0 t ; numbound0 s\ \ numbound0 (numsub t s)"
  using numsub_def by simp

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

lemma simpnum_numbound0[simp]: "numbound0 t \ numbound0 (simpnum t)"
  by (induct t) simp_all

fun nozerocoeff:: "num \ bool"
where
  "nozerocoeff (C c) = True"
| "nozerocoeff (CN n c t) = (c \ 0 \ nozerocoeff t)"
| "nozerocoeff t = True"

lemma numadd_nz : "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numadd a b)"
  by (induct a b rule: numadd.induct) (simp_all add: Let_def)

lemma nummul_nz : "\i. i\0 \ nozerocoeff a \ nozerocoeff (nummul a i)"
  by (induct a rule: nummul.induct) (auto simp add: Let_def numadd_nz)

lemma numneg_nz : "nozerocoeff a \ nozerocoeff (numneg a)"
  by (simp add: numneg_def nummul_nz)

lemma numsub_nz: "nozerocoeff a \ nozerocoeff b \ nozerocoeff (numsub a b)"
  by (simp add: numsub_def numneg_nz numadd_nz)

lemma simpnum_nz: "nozerocoeff (simpnum t)"
  by (induct t) (simp_all add: numadd_nz numneg_nz numsub_nz nummul_nz)

lemma maxcoeff_nz: "nozerocoeff t \ maxcoeff t = 0 \ t = C 0"
proof (induct t rule: maxcoeff.induct)
  case (2 n c t)
  then have cnz: "c \ 0" and mx: "max \c\ (maxcoeff t) = 0"
    by simp_all
  have "max \c\ (maxcoeff t) \ \c\"
    by simp
  with cnz have "max \c\ (maxcoeff t) > 0"
    by arith
  with 2 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

definition simp_num_pair :: "(num \ int) \ num \ int"
where
  "simp_num_pair =
    (\<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' = gcd 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 "((\(t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) =
    ((\<lambda>(t,n). Inum bs t / real_of_int n) (t, n))"
  (is "?lhs = ?rhs")
proof -
  let ?t' = "simpnum t"
  let ?g = "numgcd ?t'"
  let ?g' = "gcd n ?g"
  show ?thesis
  proof (cases "n = 0")
    case True
    then show ?thesis
      by (simp add: Let_def simp_num_pair_def)
  next
    case nnz: False
    show ?thesis
    proof (cases "?g > 1")
      case False
      then show ?thesis by (simp add: Let_def simp_num_pair_def)
    next
      case g1: True
      then have g0: "?g > 0"
        by simp
      from g1 nnz have gp0: "?g' \ 0"
        by simp
      then have g'p: "?g' > 0"
        using gcd_ge_0_int[where x="n" and y="numgcd ?t'"] by arith
      then consider "?g' = 1" | "?g' > 1" by arith
      then show ?thesis
      proof cases
        case 1
        then show ?thesis
          by (simp add: Let_def simp_num_pair_def)
      next
        case g'1: 2
        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
        have gpdd: "?g' dvd n" by simp
        have gpdgp: "?g' dvd ?g'" by simp
        from reducecoeffh[OF dvdnumcoeff_trans[OF gpdg th1] g'p]
        have th2:"real_of_int ?g' * ?t = Inum bs ?t'"
          by simp
        from g1 g'1 have "?lhs = ?t / real_of_int (n div ?g')"
          by (simp add: simp_num_pair_def Let_def)
        also have "\ = (real_of_int ?g' * ?t) / (real_of_int ?g' * (real_of_int (n div ?g')))"
          by simp
        also have "\ = (Inum bs ?t' / real_of_int n)"
          using real_of_int_div[OF gpdd] th2 gp0 by simp
        finally have "?lhs = Inum bs t / real_of_int n"
          by simp
        then show ?thesis
          by (simp add: simp_num_pair_def)
      qed
    qed
  qed
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' \ n' > 0"
proof -
  let ?t' = "simpnum t"
  let ?g = "numgcd ?t'"
  let ?g' = "gcd n ?g"
  show ?thesis
  proof (cases "n = 0")
    case True
    then show ?thesis
      using assms by (simp add: Let_def simp_num_pair_def)
  next
    case nnz: False
    show ?thesis
    proof (cases "?g > 1")
      case False
      then show ?thesis
        using assms by (auto simp add: Let_def simp_num_pair_def)
    next
      case g1: True
      then have g0: "?g > 0" by simp
      from g1 nnz have gp0: "?g' \ 0" by simp
      then have g'p: "?g' > 0" using gcd_ge_0_int[where x="n" and y="numgcd ?t'"]
        by arith
      then consider "?g'= 1" | "?g' > 1" by arith
      then show ?thesis
      proof cases
        case 1
        then show ?thesis
          using assms g1 by (auto simp add: Let_def simp_num_pair_def)
      next
        case g'1: 2
        have gpdg: "?g' dvd ?g" by simp
        have gpdd: "?g' dvd n" by simp
        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 div_self[OF gp0]] have "n div ?g' > 0"
          by simp
        then show ?thesis
          using assms g1 g'1
          by (auto simp add: simp_num_pair_def Let_def reducecoeffh_numbound0)
      qed
    qed
  qed
qed

fun simpfm :: "fm \ 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 \ if (v < 0) then T else F | _ \ Lt a')"
| "simpfm (Le a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Le a')"
| "simpfm (Gt a) = (let a' = simpnum a in case a' of C v \ if (v > 0) then T else F | _ \ Gt a')"
| "simpfm (Ge a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ Ge a')"
| "simpfm (Eq a) = (let a' = simpnum a in case a' of C v \ if (v = 0) then T else F | _ \ Eq a')"
| "simpfm (NEq a) = (let a' = simpnum a in case a' of C v \ if (v \ 0) then T else F | _ \ 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
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
next
  case (7 a)
  let ?sa = "simpnum a"
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
    by simp
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
next
  case (8 a)
  let ?sa = "simpnum a"
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
    by simp
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
next
  case (9 a)
  let ?sa = "simpnum a"
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
    by simp
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
next
  case (10 a)
  let ?sa = "simpnum a"
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
    by simp
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
next
  case (11 a)
  let ?sa = "simpnum a"
  from simpnum_ci have sa: "Inum bs ?sa = Inum bs a"
    by simp
  consider v where "?sa = C v" | "\ (\v. ?sa = C v)" by blast
  then show ?case
  proof cases
    case 1
    then show ?thesis using sa by simp
  next
    case 2
    then show ?thesis using sa by (cases ?sa) (simp_all add: Let_def)
  qed
qed (induct p rule: simpfm.induct, simp_all)

lemma simpfm_bound0: "bound0 p \ bound0 (simpfm p)"
proof (induct p rule: simpfm.induct)
  case (6 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
next
  case (7 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
next
  case (8 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
next
  case (9 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
next
  case (10 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
next
  case (11 a)
  then have nb: "numbound0 a" by simp
  then have "numbound0 (simpnum a)" by (simp only: simpnum_numbound0[OF nb])
  then show ?case by (cases "simpnum a") (auto simp add: Let_def)
qed (auto simp add: disj_def imp_def iff_def conj_def)

lemma simpfm_qf: "qfree p \ qfree (simpfm p)"
  apply (induct p rule: simpfm.induct)
  apply (auto simp add: Let_def)
  apply (case_tac "simpnum a", auto)+
  done

fun prep :: "fm \ fm"
where
  "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: "\bs. Ifm bs (prep p) = Ifm bs p"
  by (induct p rule: prep.induct) auto

  (* Generic quantifier elimination *)
fun qelim :: "fm \ (fm \ fm) \ fm"
where
  "qelim (E p) = (\qe. DJ qe (qelim p qe))"
| "qelim (A p) = (\qe. not (qe ((qelim (NOT p) qe))))"
| "qelim (NOT p) = (\qe. not (qelim p qe))"
| "qelim (And p q) = (\qe. conj (qelim p qe) (qelim q qe))"
| "qelim (Or p q) = (\qe. disj (qelim p qe) (qelim q qe))"
| "qelim (Imp p q) = (\qe. imp (qelim p qe) (qelim q qe))"
| "qelim (Iff p q) = (\qe. iff (qelim p qe) (qelim q qe))"
| "qelim p = (\y. simpfm p)"

lemma qelim_ci:
  assumes qe_inv: "\bs p. qfree p \ qfree (qe p) \ (Ifm bs (qe p) = Ifm bs (E p))"
  shows "\bs. qfree (qelim p qe) \ (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: simpfm simpfm_qf simp del: simpfm.simps)

fun minusinf:: "fm \ fm" (* Virtual substitution of -\*)
where
  "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"

fun plusinf:: "fm \ fm" (* Virtual substitution of +\*)
where
  "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"

fun isrlfm :: "fm \ bool" (* Linearity test for fm *)
where
  "isrlfm (And p q) = (isrlfm p \ isrlfm q)"
| "isrlfm (Or p q) = (isrlfm p \ isrlfm q)"
| "isrlfm (Eq (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm (NEq (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm (Lt (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm (Le (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm (Gt (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm (Ge (CN 0 c e)) = (c>0 \ numbound0 e)"
| "isrlfm p = (isatom p \ (bound0 p))"

  (* splits the bounded from the unbounded part*)
fun rsplit0 :: "num \ int \ num"
where
  "rsplit0 (Bound 0) = (1,C 0)"
| "rsplit0 (Add a b) = (let (ca,ta) = rsplit0 a; (cb,tb) = rsplit0 b in (ca + cb, Add ta tb))"
| "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: "Inum bs ((case_prod (CN 0)) (rsplit0 t)) = Inum bs t \ 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 2 have nb: "numbound0 (snd(rsplit0 (Add a b)))"
    by (cases "rsplit0 a") (auto simp add: Let_def split_def)
  have "Inum bs ((case_prod (CN 0)) (rsplit0 (Add a b))) =
    Inum bs ((case_prod (CN 0)) ?sa)+Inum bs ((case_prod (CN 0)) ?sb)"
    by (simp add: Let_def split_def algebra_simps)
  also have "\ = Inum bs a + Inum bs b"
    using 2 by (cases "rsplit0 a") auto
  finally show ?case
    using nb by simp
qed (auto simp add: Let_def split_def algebra_simps, simp add: distrib_left[symmetric])

    (* Linearize a formula*)
definition lt :: "int \ num \ 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 \ num \ 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 \ num \ 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 \ num \ 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 \ num \ 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 \ num \ 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 \ Ifm bs (case_prod lt (rsplit0 t)) =
  Ifm bs (Lt t) \<and> isrlfm (case_prod lt (rsplit0 t))"
  using rsplit0[where bs = "bs" and t="t"]
  by (auto simp add: lt_def split_def, cases "snd(rsplit0 t)", auto,
    rename_tac nat a b, case_tac "nat", auto)

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

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

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

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

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

lemma conj_lin: "isrlfm p \ isrlfm q \ isrlfm (conj p q)"
  by (auto simp add: conj_def)

lemma disj_lin: "isrlfm p \ isrlfm q \ isrlfm (disj p q)"
  by (auto simp add: disj_def)

fun rlfm :: "fm \ fm"
where
  "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) = case_prod lt (rsplit0 a)"
| "rlfm (Le a) = case_prod le (rsplit0 a)"
| "rlfm (Gt a) = case_prod gt (rsplit0 a)"
| "rlfm (Ge a) = case_prod ge (rsplit0 a)"
| "rlfm (Eq a) = case_prod eq (rsplit0 a)"
| "rlfm (NEq a) = case_prod 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"

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

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

lemma rplusinf_inf:
  assumes lp: "isrlfm p"
  shows "\z. \x > z. Ifm (x#bs) (plusinf p) = Ifm (x#bs) p" (is "\z. \x. ?P z x p")
using lp
proof (induct p rule: isrlfm.induct)
  case (1 p q)
  then show ?case
    apply auto
    apply (rule_tac x= "max z za" in exI)
    apply auto
    done
next
  case (2 p q)
  then show ?case
    apply auto
    apply (rule_tac x= "max z za" in exI)
    apply auto
    done
next
  case (3 c e)
  from 3 have nb: "numbound0 e" by simp
  from 3 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e = "Inum (a # bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int c * x + ?e > 0" by arith
    then have "real_of_int c * x + ?e \ 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
  }
  then have "\x > ?z. ?P ?z x (Eq (CN 0 c e))" by simp
  then show ?case by blast
next
  case (4 c e)
  from 4 have nb: "numbound0 e" by simp
  from 4 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e = "Inum (a # bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int c * x + ?e > 0" by arith
    then have "real_of_int c * x + ?e \ 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
  }
  then have "\x > ?z. ?P ?z x (NEq (CN 0 c e))" by simp
  then show ?case by blast
next
  case (5 c e)
  from 5 have nb: "numbound0 e" by simp
  from 5 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e = "Inum (a # bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int 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
  }
  then have "\x > ?z. ?P ?z x (Lt (CN 0 c e))" by simp
  then show ?case by blast
next
  case (6 c e)
  from 6 have nb: "numbound0 e" by simp
  from 6 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e = "Inum (a # bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int 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
  }
  then have "\x > ?z. ?P ?z x (Le (CN 0 c e))" by simp
  then show ?case by blast
next
  case (7 c e)
  from 7 have nb: "numbound0 e" by simp
  from 7 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e = "Inum (a # bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int 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
  }
  then have "\x > ?z. ?P ?z x (Gt (CN 0 c e))" by simp
  then show ?case by blast
next
  case (8 c e)
  from 8 have nb: "numbound0 e" by simp
  from 8 have cp: "real_of_int c > 0" by simp
  fix a
  let ?e="Inum (a#bs) e"
  let ?z = "(- ?e) / real_of_int c"
  {
    fix x
    assume xz: "x > ?z"
    with mult_strict_right_mono [OF xz cp] cp
    have "(real_of_int c * x > - ?e)" by (simp add: ac_simps)
    then have "real_of_int 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
  }
  then have "\x > ?z. ?P ?z x (Ge (CN 0 c e))" by simp
  then show ?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 "\x. Ifm (x#bs) p"
proof -
  from bound0_I [OF rminusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  have th: "\x. Ifm (x#bs) (minusinf p)" by auto
  from rminusinf_inf[OF lp, where bs="bs"]
  obtain z where z_def: "\x
  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 "\x. Ifm (x # bs) p"
proof -
  from bound0_I [OF rplusinf_bound0[OF lp], where b="a" and bs ="bs"] ex
  have th: "\x. Ifm (x # bs) (plusinf p)" by auto
  from rplusinf_inf[OF lp, where bs="bs"]
  obtain z where z_def: "\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

fun uset :: "fm \ (num \ int) list"
where
  "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 = []"

fun usubst :: "fm \ num \ int \ fm"
where
  "usubst (And p q) = (\(t,n). And (usubst p (t,n)) (usubst q (t,n)))"
| "usubst (Or p q) = (\(t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
| "usubst (Eq (CN 0 c e)) = (\(t,n). Eq (Add (Mul c t) (Mul n e)))"
| "usubst (NEq (CN 0 c e)) = (\(t,n). NEq (Add (Mul c t) (Mul n e)))"
| "usubst (Lt (CN 0 c e)) = (\(t,n). Lt (Add (Mul c t) (Mul n e)))"
| "usubst (Le (CN 0 c e)) = (\(t,n). Le (Add (Mul c t) (Mul n e)))"
| "usubst (Gt (CN 0 c e)) = (\(t,n). Gt (Add (Mul c t) (Mul n e)))"
| "usubst (Ge (CN 0 c e)) = (\(t,n). Ge (Add (Mul c t) (Mul n e)))"
| "usubst p = (\(t, n). p)"

lemma usubst_I:
  assumes lp: "isrlfm p"
    and np: "real_of_int n > 0"
    and nbt: "numbound0 t"
  shows "(Ifm (x # bs) (usubst p (t,n)) =
    Ifm (((Inum (x # bs) t) / (real_of_int n)) # bs) p) \<and> bound0 (usubst p (t, n))"
  (is "(?I x (usubst p (t, n)) = ?I ?u p) \ ?B p"
   is "(_ = ?I (?t/?n) p) \ _"
   is "(_ = ?I (?N x t /_) p) \ _")
  using lp
proof (induct p rule: usubst.induct)
  case (5 c e)
  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
  have "?I ?u (Lt (CN 0 c e)) \ real_of_int 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 "\ \ ?n * (real_of_int c * (?t / ?n)) + ?n*(?N x e) < 0"
    by (simp only: pos_less_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ \ real_of_int 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)
  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
  have "?I ?u (Le (CN 0 c e)) \ real_of_int 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 "\ = (?n*(real_of_int c *(?t/?n)) + ?n*(?N x e) \ 0)"
    by (simp only: pos_le_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ = (real_of_int c *?t + ?n* (?N x e) \ 0)" using np by simp
  finally show ?case using nbt nb by (simp add: algebra_simps)
next
  case (7 c e)
  with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
  have "?I ?u (Gt (CN 0 c e)) \ real_of_int 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 "\ \ ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e > 0"
    by (simp only: pos_divide_less_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ \ real_of_int 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)
  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
  have "?I ?u (Ge (CN 0 c e)) \ real_of_int 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 "\ \ ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e \ 0"
    by (simp only: pos_divide_le_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ \ real_of_int c * ?t + ?n * ?N x e \ 0" using np by simp
  finally show ?case using nbt nb by (simp add: algebra_simps)
next
  case (3 c e)
  with assms have cp: "c > 0" and nb: "numbound0 e" by simp_all
  from np have np: "real_of_int n \ 0" by simp
  have "?I ?u (Eq (CN 0 c e)) \ real_of_int 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 "\ \ ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e = 0"
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ \ real_of_int 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) with assms have cp: "c >0" and nb: "numbound0 e" by simp_all
  from np have np: "real_of_int n \ 0" by simp
  have "?I ?u (NEq (CN 0 c e)) \ real_of_int 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 "\ \ ?n * (real_of_int c * (?t / ?n)) + ?n * ?N x e \ 0"
    by (simp only: nonzero_eq_divide_eq[OF np, where a="real_of_int c *(?t/?n) + (?N x e)"
      and b="0", simplified div_0]) (simp only: algebra_simps)
  also have "\ \ real_of_int c * ?t + ?n * ?N x e \ 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_of_int n" and b'="x"])

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

lemma rminusinf_uset:
  assumes lp: "isrlfm p"
    and nmi: "\ (Ifm (a # bs) (minusinf p))" (is "\ (Ifm (a # bs) (?M p))")
    and ex: "Ifm (x#bs) p" (is "?I x p")
  shows "\(s,m) \ set (uset p). x \ Inum (a#bs) s / real_of_int m"
    (is "\(s,m) \ ?U p. x \ ?N a s / real_of_int m")
proof -
  have "\(s,m) \ set (uset p). real_of_int m * x \ Inum (a#bs) s"
    (is "\(s,m) \ ?U p. real_of_int m *x \ ?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"])
  then obtain s m where smU: "(s,m) \ set (uset p)" and mx: "real_of_int m * x \ ?N a s"
    by blast
  from uset_l[OF lp] smU have mp: "real_of_int m > 0"
    by auto
  from pos_divide_le_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real_of_int m"
    by (auto simp add: mult.commute)
  then show ?thesis
    using smU by auto
qed

lemma rplusinf_uset:
  assumes lp: "isrlfm p"
    and nmi: "\ (Ifm (a # bs) (plusinf p))" (is "\ (Ifm (a # bs) (?M p))")
    and ex: "Ifm (x # bs) p" (is "?I x p")
  shows "\(s,m) \ set (uset p). x \ Inum (a#bs) s / real_of_int m"
    (is "\(s,m) \ ?U p. x \ ?N a s / real_of_int m")
proof -
  have "\(s,m) \ set (uset p). real_of_int m * x \ Inum (a#bs) s"
    (is "\(s,m) \ ?U p. real_of_int m *x \ ?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"])
  then obtain s m where smU: "(s,m) \ set (uset p)" and mx: "real_of_int m * x \ ?N a s"
    by blast
  from uset_l[OF lp] smU have mp: "real_of_int m > 0"
    by auto
  from pos_le_divide_eq[OF mp, where a="x" and b="?N a s", symmetric] mx have "x \ ?N a s / real_of_int m"
    by (auto simp add: mult.commute)
  then show ?thesis
    using smU by auto
qed

lemma lin_dense:
  assumes lp: "isrlfm p"
    and noS: "\t. l < t \ t< u \ t \ (\(t,n). Inum (x#bs) t / real_of_int n) ` set (uset p)"
      (is "\t. _ \ _ \ t \ (\(t,n). ?N x t / real_of_int 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)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from 5 have "x * real_of_int c + ?N x e < 0"
    by (simp add: algebra_simps)
  then have pxc: "x < (- ?N x e) / real_of_int c"
    by (simp only: pos_less_divide_eq[OF cp, where a="x" and b="-?N x e"])
  from 5 have noSc:"\t. l < t \ t < u \ t \ (- ?N x e) / real_of_int c"
    by auto
  with ly yu have yne: "y \ - ?N x e / real_of_int c"
    by auto
  then consider "y < (-?N x e)/ real_of_int c" | "y > (- ?N x e) / real_of_int c"
    by atomize_elim auto
  then show ?case
  proof cases
    case 1
    then have "y * real_of_int c < - ?N x e"
      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
    then have "real_of_int c * y + ?N x e < 0"
      by (simp add: algebra_simps)
    then show ?thesis
      using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
  next
    case 2
    with yu have eu: "u > (- ?N x e) / real_of_int c"
      by auto
    with noSc ly yu have "(- ?N x e) / real_of_int c \ l"
      by (cases "(- ?N x e) / real_of_int c > l") auto
    with lx pxc have False
      by auto
    then show ?thesis ..
  qed
next
  case (6 c e)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from 6 have "x * real_of_int c + ?N x e \ 0"
    by (simp add: algebra_simps)
  then have pxc: "x \ (- ?N x e) / real_of_int c"
    by (simp only: pos_le_divide_eq[OF cp, where a="x" and b="-?N x e"])
  from 6 have noSc:"\t. l < t \ t < u \ t \ (- ?N x e) / real_of_int c"
    by auto
  with ly yu have yne: "y \ - ?N x e / real_of_int c"
    by auto
  then consider "y < (- ?N x e) / real_of_int c" | "y > (-?N x e) / real_of_int c"
    by atomize_elim auto
  then show ?case
  proof cases
    case 1
    then have "y * real_of_int c < - ?N x e"
      by (simp add: pos_less_divide_eq[OF cp, where a="y" and b="-?N x e", symmetric])
    then have "real_of_int c * y + ?N x e < 0"
      by (simp add: algebra_simps)
    then show ?thesis
      using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
  next
    case 2
    with yu have eu: "u > (- ?N x e) / real_of_int c"
      by auto
    with noSc ly yu have "(- ?N x e) / real_of_int c \ l"
      by (cases "(- ?N x e) / real_of_int c > l") auto
    with lx pxc have False
      by auto
    then show ?thesis ..
  qed
next
  case (7 c e)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from 7 have "x * real_of_int c + ?N x e > 0"
    by (simp add: algebra_simps)
  then have pxc: "x > (- ?N x e) / real_of_int c"
    by (simp only: pos_divide_less_eq[OF cp, where a="x" and b="-?N x e"])
  from 7 have noSc: "\t. l < t \ t < u \ t \ (- ?N x e) / real_of_int c"
    by auto
  with ly yu have yne: "y \ - ?N x e / real_of_int c"
    by auto
  then consider "y > (- ?N x e) / real_of_int c" | "y < (-?N x e) / real_of_int c"
    by atomize_elim auto
  then show ?case
  proof cases
    case 1
    then have "y * real_of_int c > - ?N x e"
      by (simp add: pos_divide_less_eq[OF cp, where a="y" and b="-?N x e", symmetric])
    then have "real_of_int c * y + ?N x e > 0"
      by (simp add: algebra_simps)
    then show ?thesis
      using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"] by simp
  next
    case 2
    with ly have eu: "l < (- ?N x e) / real_of_int c"
      by auto
    with noSc ly yu have "(- ?N x e) / real_of_int c \ u"
      by (cases "(- ?N x e) / real_of_int c > l") auto
    with xu pxc have False by auto
    then show ?thesis ..
  qed
next
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