Theory Ferrack

(*  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 ‹Quantifier elimination for ‹ℝ (0, 1, +, <)››

  (*********************************************************************************)
  (****                            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 ∨ 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 ∨ 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 ∨ q = T ∨ 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 ∨ 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 ⇒ T | F ⇒ q | _ ⇒ 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) = (λg. gcd i g)"
| "numgcdh (CN n c t) = (λg. gcd c (numgcdh t g))"
| "numgcdh t = (λg. 1)"

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

fun reducecoeffh:: "num ⇒ int ⇒ num"
where
  "reducecoeffh (C i) = (λg. C (i div g))"
| "reducecoeffh (CN n c t) = (λg. CN n (c div g) (reducecoeffh t g))"
| "reducecoeffh t = (λg. 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) = (λg. g dvd i)"
| "dvdnumcoeff (CN n c t) = (λg. g dvd c ∧ dvdnumcoeff t g)"
| "dvdnumcoeff t = (λg. 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) = (λx. ¦i¦ ≤ x)"
| "ismaxcoeff (CN n c t) = (λx. ¦c¦ ≤ x ∧ ismaxcoeff t x)"
| "ismaxcoeff t = (λx. 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 ≤ 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) = (λi. C (i * j))"
| "nummul (CN n c a) = (λi. CN n (i * c) (nummul a i))"
| "nummul t = (λi. 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"
  by (induction t rule: maxcoeff.induct) 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 =
    (λ(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 "((λ(t,n). Inum bs t / real_of_int n) (simp_num_pair (t,n))) =
    ((λ(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' ≤ 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) = (λqe. DJ qe (qelim p qe))"
| "qelim (A p) = (λqe. not (qe ((qelim (Not p) qe))))"
| "qelim (Not p) = (λqe. not (qelim p qe))"
| "qelim (And p q) = (λqe. conj (qelim p qe) (qelim q qe))"
| "qelim (Or  p q) = (λqe. disj (qelim p qe) (qelim q qe))"
| "qelim (Imp p q) = (λqe. imp (qelim p qe) (qelim q qe))"
| "qelim (Iff p q) = (λqe. iff (qelim p qe) (qelim q qe))"
| "qelim p = (λy. 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) ∧ 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) ∧ 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) ∧ 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) ∧ 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) ∧ 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) ∧ 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<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 "∃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) = (λ(t,n). And (usubst p (t,n)) (usubst q (t,n)))"
| "usubst (Or p q) = (λ(t,n). Or (usubst p (t,n)) (usubst q (t,n)))"
| "usubst (Eq (CN 0 c e)) = (λ(t,n). Eq (Add (Mul c t) (Mul n e)))"
| "usubst (NEq (CN 0 c e)) = (λ(t,n). NEq (Add (Mul c t) (Mul n e)))"
| "usubst (Lt (CN 0 c e)) = (λ(t,n). Lt (Add (Mul c t) (Mul n e)))"
| "usubst (Le (CN 0 c e)) = (λ(t,n). Le (Add (Mul c t) (Mul n e)))"
| "usubst (Gt (CN 0 c e)) = (λ(t,n). Gt (Add (Mul c t) (Mul n e)))"
| "usubst (Ge (CN 0 c e)) = (λ(t,n). Ge (Add (Mul c t) (Mul n e)))"
| "usubst p = (λ(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) ∧ 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 ∉ (λ(t,n). Inum (x#bs) t / real_of_int n) ` set (uset p)"
      (is "∀t. _ ∧ _ ⟶ t ∉ (λ(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
  case (8 c e)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from 8 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_le_eq[OF cp, where a="x" and b="-?N x e"])
  from 8 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
  case (3 c e)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from cp have cnz: "real_of_int c ≠ 0"
    by simp
  from 3 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: nonzero_eq_divide_eq[OF cnz, where a="x" and b="-?N x e"])
  from 3 have noSc:"∀t. l < t ∧ t < u ⟶ t ≠ (- ?N x e) / real_of_int c"
    by auto
  with lx xu have yne: "x ≠ - ?N x e / real_of_int c"
    by auto
  with pxc show ?case
    by simp
next
  case (4 c e)
  then have cp: "real_of_int c > 0" and nb: "numbound0 e"
    by simp_all
  from cp have cnz: "real_of_int c ≠ 0"
    by simp
  from 4 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 have "y* real_of_int c ≠ -?N x e"
    by (simp only: nonzero_eq_divide_eq[OF cnz, where a="y" and b="-?N x e"]) simp
  then have "y* real_of_int c + ?N x e ≠ 0"
    by (simp add: algebra_simps)
  then show ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
    by (simp add: algebra_simps)
qed (auto simp add: numbound0_I[where bs="bs" and b="y" and b'="x"])

lemma finite_set_intervals:
  fixes x :: real
  assumes px: "P x"
    and lx: "l ≤ x"
    and xu: "x ≤ u"
    and linS: "l∈ S"
    and uinS: "u ∈ S"
    and fS: "finite S"
    and lS: "∀x∈ S. l ≤ x"
    and Su: "∀x∈ S. x ≤ u"
  shows "∃a ∈ S. ∃b ∈ S. (∀y. a < y ∧ y < b ⟶ y ∉ S) ∧ a ≤ x ∧ x ≤ b ∧ P x"
proof -
  let ?Mx = "{y. y∈ S ∧ y ≤ x}"
  let ?xM = "{y. y∈ S ∧ x ≤ y}"
  let ?a = "Max ?Mx"
  let ?b = "Min ?xM"
  have MxS: "?Mx ⊆ S"
    by blast
  then have fMx: "finite ?Mx"
    using fS finite_subset by auto
  from lx linS have linMx: "l ∈ ?Mx"
    by blast
  then have Mxne: "?Mx ≠ {}"
    by blast
  have xMS: "?xM ⊆ S"
    by blast
  then have fxM: "finite ?xM"
    using fS finite_subset by auto
  from xu uinS have linxM: "u ∈ ?xM"
    by blast
  then have xMne: "?xM ≠ {}"
    by blast
  have ax:"?a ≤ x"
    using Mxne fMx by auto
  have xb:"x ≤ ?b"
    using xMne fxM by auto
  have "?a ∈ ?Mx"
    using Max_in[OF fMx Mxne] by simp
  then have ainS: "?a ∈ S"
    using MxS by blast
  have "?b ∈ ?xM"
    using Min_in[OF fxM xMne] by simp
  then have binS: "?b ∈ S"
    using xMS by blast
  have noy: "∀y. ?a < y ∧ y < ?b ⟶ y ∉ S"
  proof clarsimp
    fix y
    assume ay: "?a < y" and yb: "y < ?b" and yS: "y ∈ S"
    from yS consider "y ∈ ?Mx" | "y ∈ ?xM"
      by atomize_elim auto
    then show False
    proof cases
      case 1
      then have "y ≤ ?a"
        using Mxne fMx by auto
      with ay show ?thesis by simp
    next
      case 2
      then have "y ≥ ?b"
        using xMne fxM by auto
      with yb show ?thesis by simp
    qed
  qed
  from ainS binS noy ax xb px show ?thesis
    by blast
qed

lemma rinf_uset:
  assumes lp: "isrlfm p"
    and nmi: "¬ (Ifm (x # bs) (minusinf p))"  (is "¬ (Ifm (x # bs) (?M p))")
    and npi: "¬ (Ifm (x # bs) (plusinf p))"  (is "¬ (Ifm (x # bs) (?P p))")
    and ex: "∃x. Ifm (x # bs) p"  (is "∃x. ?I x p")
  shows "∃(l,n) ∈ set (uset p). ∃(s,m) ∈ set (uset p).
    ?I ((Inum (x#bs) l / real_of_int n + Inum (x#bs) s / real_of_int m) / 2) p"
proof -
  let ?N = "λx 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': "¬ (?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': "¬ (?I a (?P p))"
    by simp
  have "∃(l,n) ∈ set (uset p). ∃(s,m) ∈ set (uset p). ?I ((?N a l/real_of_int n + ?N a s /real_of_int m) / 2) p"
  proof -
    let ?M = "(λ(t,c). ?N a t / real_of_int c) ` ?U"
    have fM: "finite ?M"
      by auto
    from rminusinf_uset[OF lp nmi pa] rplusinf_uset[OF lp npi pa]
    have "∃(l,n) ∈ set (uset p). ∃(s,m) ∈ set (uset p). a ≤ ?N x l / real_of_int n ∧ a ≥ ?N x s / real_of_int m"
      by blast
    then obtain "t" "n" "s" "m"
      where tnU: "(t,n) ∈ ?U"
        and smU: "(s,m) ∈ ?U"
        and xs1: "a ≤ ?N x s / real_of_int m"
        and tx1: "a ≥ ?N x t / real_of_int 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 ≤ ?N a s / real_of_int m" and tx: "a ≥ ?N a t / real_of_int n"
      by auto
    from tnU have Mne: "?M ≠ {}"
      by auto
    then have Une: "?U ≠ {}"
      by simp
    let ?l = "Min ?M"
    let ?u = "Max ?M"
    have linM: "?l ∈ ?M"
      using fM Mne by simp
    have uinM: "?u ∈ ?M"
      using fM Mne by simp
    have tnM: "?N a t / real_of_int n ∈ ?M"
      using tnU by auto
    have smM: "?N a s / real_of_int m ∈ ?M"
      using smU by auto
    have lM: "∀t∈ ?M. ?l ≤ t"
      using Mne fM by auto
    have Mu: "∀t∈ ?M. t ≤ ?u"
      using Mne fM by auto
    have "?l ≤ ?N a t / real_of_int n"
      using tnM Mne by simp
    then have lx: "?l ≤ a"
      using tx by simp
    have "?N a s / real_of_int m ≤ ?u"
      using smM Mne by simp
    then have xu: "a ≤ ?u"
      using xs by simp
    from finite_set_intervals2[where P="λx. ?I x p",OF pa lx xu linM uinM fM lM Mu]
    consider u where "u ∈ ?M" "?I u p"
      | t1 t2 where "t1 ∈ ?M" "t2 ∈ ?M" "∀y. t1 < y ∧ y < t2 ⟶ y ∉ ?M" "t1 < a" "a < t2" "?I a p"
      by blast
    then show ?thesis
    proof cases
      case 1
      note um = ‹u ∈ ?M› and pu = ‹?I u p›
      then have "∃(tu,nu) ∈ ?U. u = ?N a tu / real_of_int nu"
        by auto
      then obtain tu nu where tuU: "(tu, nu) ∈ ?U" and tuu: "u= ?N a tu / real_of_int nu"
        by blast
      have "(u + u) / 2 = u"
        by auto
      with pu tuu have "?I (((?N a tu / real_of_int nu) + (?N a tu / real_of_int nu)) / 2) p"
        by simp
      with tuU show ?thesis by blast
    next
      case 2
      note t1M = ‹t1 ∈ ?M› and t2M = ‹t2∈ ?M›
        and noM = ‹∀y. t1 < y ∧ y < t2 ⟶ y ∉ ?M›
        and t1x = ‹t1 < a› and xt2 = ‹a < t2› and px = ‹?I a p›
      from t1M have "∃(t1u,t1n) ∈ ?U. t1 = ?N a t1u / real_of_int t1n"
        by auto
      then obtain t1u t1n where t1uU: "(t1u, t1n) ∈ ?U" and t1u: "t1 = ?N a t1u / real_of_int t1n"
        by blast
      from t2M have "∃(t2u,t2n) ∈ ?U. t2 = ?N a t2u / real_of_int t2n"
        by auto
      then obtain t2u t2n where t2uU: "(t2u, t2n) ∈ ?U" and t2u: "t2 = ?N a t2u / real_of_int 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 show ?thesis
        by blast
    qed
  qed
  then obtain l n s m where lnU: "(l, n) ∈ ?U" and smU:"(s, m) ∈ ?U"
    and pu: "?I ((?N a l / real_of_int n + ?N a s / real_of_int 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_of_int n + ?N x s / real_of_int 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 "(∃x. Ifm (x#bs) p) ⟷
    Ifm (x # bs) (minusinf p) ∨ Ifm (x # bs) (plusinf p) ∨
      (∃(t,n) ∈ set (uset p). ∃(s,m) ∈ set (uset p).
        Ifm ((((Inum (x # bs) t) / real_of_int n + (Inum (x # bs) s) / real_of_int m) / 2) # bs) p)"
  (is "(∃x. ?I x p) ⟷ (?M ∨ ?P ∨ ?F)" is "?E = ?D")
proof
  assume px: "∃x. ?I x p"
  consider "?M ∨ ?P" | "¬ ?M" "¬ ?P" by blast
  then show ?D
  proof cases
    case 1
    then show ?thesis by blast
  next
    case 2
    from rinf_uset[OF lp this] have ?F
      using px by blast
    then show ?thesis by blast
  qed
next
  assume ?D
  then consider ?M | ?P | ?F by blast
  then show ?E
  proof cases
    case 1
    from rminusinf_ex[OF lp this] show ?thesis .
  next
    case 2
    from rplusinf_ex[OF lp this] show ?thesis .
  next
    case 3
    then show ?thesis by blast
  qed
qed


lemma fr_equsubst:
  assumes lp: "isrlfm p"
  shows "(∃x. Ifm (x # bs) p) ⟷
    (Ifm (x # bs) (minusinf p) ∨ Ifm (x # bs) (plusinf p) ∨
      (∃(t,k) ∈ set (uset p). ∃(s,l) ∈ set (uset p).
        Ifm (x#bs) (usubst p (Add (Mul l t) (Mul k s), 2 * k * l))))"
  (is "(∃x. ?I x p) ⟷ ?M ∨ ?P ∨ ?F" is "?E = ?D")
proof
  assume px: "∃x. ?I x p"
  consider "?M ∨ ?P" | "¬ ?M" "¬ ?P" by blast
  then show ?D
  proof cases
    case 1
    then show ?thesis by blast
  next
    case 2
    let ?f = "λ(t,n). Inum (x # bs) t / real_of_int n"
    let ?N = "λt. Inum (x # bs) t"
    {
      fix t n s m
      assume "(t, n) ∈ set (uset p)" and "(s, m) ∈ set (uset p)"
      with uset_l[OF lp] have tnb: "numbound0 t"
        and np: "real_of_int n > 0" and snb: "numbound0 s" and mp: "real_of_int m > 0"
        by auto
      let ?st = "Add (Mul m t) (Mul n s)"
      from np mp have mnp: "real_of_int (2 * n * m) > 0"
        by (simp add: mult.commute)
      from tnb snb have st_nb: "numbound0 ?st"
        by simp
      have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
        using mnp mp np by (simp add: algebra_simps add_divide_distrib)
      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_of_int n + ?N s / real_of_int m) / 2) p"
        by (simp only: st[symmetric])
    }
    with rinf_uset[OF lp 2 px] have ?F
      by blast
    then show ?thesis
      by blast
  qed
next
  assume ?D
  then consider ?M | ?P | t k s l where "(t, k) ∈ set (uset p)" "(s, l) ∈ set (uset p)"
    "?I x (usubst p (Add (Mul l t) (Mul k s), 2 * k * l))"
    by blast
  then show ?E
  proof cases
    case 1
    from rminusinf_ex[OF lp this] show ?thesis .
  next
    case 2
    from rplusinf_ex[OF lp this] show ?thesis .
  next
    case 3
    with uset_l[OF lp] have tnb: "numbound0 t" and np: "real_of_int k > 0"
      and snb: "numbound0 s" and mp: "real_of_int l > 0"
      by auto
    let ?st = "Add (Mul l t) (Mul k s)"
    from np mp have mnp: "real_of_int (2 * k * l) > 0"
      by (simp add: mult.commute)
    from tnb snb have st_nb: "numbound0 ?st"
      by simp
    from usubst_I[OF lp mnp st_nb, where bs="bs"]
      ‹?I x (usubst p (Add (Mul l t) (Mul k s), 2 * k * l))› show ?thesis
      by auto
  qed
qed


    (* Implement the right hand side of Ferrante and Rackoff's Theorem. *)
definition ferrack :: "fm ⇒ fm"
where
  "ferrack p =
   (let
      p' = rlfm (simpfm p);
      mp = minusinf p';
      pp = plusinf p'
    in
      if mp = T ∨ pp = T then T
      else
       (let U = remdups (map simp_num_pair
         (map (λ((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 ∘ usubst p') U)))))"

lemma uset_cong_aux:
  assumes Ul: "∀(t,n) ∈ set U. numbound0 t ∧ n > 0"
  shows "((λ(t,n). Inum (x # bs) t / real_of_int n) `
    (set (map (λ((t,n),(s,m)). (Add (Mul m t) (Mul n s), 2 * n * m)) (alluopairs U)))) =
    ((λ((t,n),(s,m)). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (set U × set U))"
  (is "?lhs = ?rhs")
proof auto
  fix t n s m
  assume "((t, n), (s, m)) ∈ set (alluopairs U)"
  then have th: "((t, n), (s, m)) ∈ set U × set U"
    using alluopairs_set1[where xs="U"] by blast
  let ?N = "λt. Inum (x # bs) t"
  let ?st = "Add (Mul m t) (Mul n s)"
  from Ul th have mnz: "m ≠ 0"
    by auto
  from Ul th have nnz: "n ≠ 0"
    by auto
  have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
  then show "(real_of_int m *  Inum (x # bs) t + real_of_int n * Inum (x # bs) s) / (2 * real_of_int n * real_of_int m)
      ∈ (λ((t, n), s, m). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) `
         (set U × set U)"
    using mnz nnz th
    apply (auto simp add: th add_divide_distrib algebra_simps split_def image_def)
    apply (rule_tac x="(s,m)" in bexI)
    apply simp_all
    apply (rule_tac x="(t,n)" in bexI)
    apply (simp_all add: mult.commute)
    done
next
  fix t n s m
  assume tnU: "(t, n) ∈ set U" and smU: "(s, m) ∈ set U"
  let ?N = "λt. Inum (x # bs) t"
  let ?st = "Add (Mul m t) (Mul n s)"
  from Ul smU have mnz: "m ≠ 0"
    by auto
  from Ul tnU have nnz: "n ≠ 0"
    by auto
  have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
    using mnz nnz by (simp add: algebra_simps add_divide_distrib)
  let ?P = "λ(t',n') (s',m'). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m)/2 =
    (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m') / 2"
  have Pc:"∀a b. ?P a b = ?P b a"
    by auto
  from Ul alluopairs_set1 have Up:"∀((t,n),(s,m)) ∈ set (alluopairs U). n ≠ 0 ∧ m ≠ 0"
    by blast
  from alluopairs_ex[OF Pc, where xs="U"] tnU smU
  have th':"∃((t',n'),(s',m')) ∈ set (alluopairs U). ?P (t',n') (s',m')"
    by blast
  then obtain t' n' s' m' where ts'_U: "((t',n'),(s',m')) ∈ set (alluopairs U)"
    and Pts': "?P (t', n') (s', m')"
    by blast
  from ts'_U Up have mnz': "m' ≠ 0" and nnz': "n'≠ 0"
    by auto
  let ?st' = "Add (Mul m' t') (Mul n' s')"
  have st': "(?N t' / real_of_int n' + ?N s' / real_of_int m') / 2 = ?N ?st' / real_of_int (2 * n' * m')"
    using mnz' nnz' by (simp add: algebra_simps add_divide_distrib)
  from Pts' have "(Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2 =
    (Inum (x # bs) t' / real_of_int n' + Inum (x # bs) s' / real_of_int m') / 2"
    by simp
  also have "… = (λ(t, n). Inum (x # bs) t / real_of_int n)
      ((λ((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_of_int n + Inum (x # bs) s / real_of_int m) / 2
    ∈ (λ(t, n). Inum (x # bs) t / real_of_int n) `
      (λ((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': "((λ(t,n). Inum (x # bs) t / real_of_int n) ` U') =
      ((λ((t,n),(s,m)). (Inum (x # bs) t / real_of_int n + Inum (x # bs) s / real_of_int m) / 2) ` (U × U))"
      (is "?f ` U' = ?g ` (U × U)")
    and U: "∀(t,n) ∈ U. numbound0 t ∧ n > 0"
    and U': "∀(t,n) ∈ U'. numbound0 t ∧ n > 0"
  shows "(∃(t,n) ∈ U. ∃(s,m) ∈ U. Ifm (x # bs) (usubst p (Add (Mul m t) (Mul n s), 2 * n * m))) =
    (∃(t,n) ∈ U'. Ifm (x # bs) (usubst p (t, n)))"
    (is "?lhs ⟷ ?rhs")
proof
  show ?rhs if ?lhs
  proof -
    from that obtain t n s m where tnU: "(t, n) ∈ U" and smU: "(s, m) ∈ U"
      and Pst: "Ifm (x # bs) (usubst p (Add (Mul m t) (Mul n s), 2 * n * m))"
      by blast
    let ?N = "λt. 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 np mp have mnp: "real_of_int (2 * n * m) > 0"
      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
    from tnb snb have stnb: "numbound0 ?st"
      by simp
    have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
      using mp np by (simp add: algebra_simps add_divide_distrib)
    from tnU smU UU' have "?g ((t, n), (s, m)) ∈ ?f ` U'"
      by blast
    then have "∃(t',n') ∈ U'. ?g ((t, n), (s, m)) = ?f (t', n')"
      apply auto
      apply (rule_tac x="(a, b)" in bexI)
      apply auto
      done
    then obtain t' n' where tnU': "(t',n') ∈ U'" and th: "?g ((t, n), (s, m)) = ?f (t', n')"
      by blast
    from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int 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_of_int (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 ?thesis
      using tnU' by auto
  qed
  show ?lhs if ?rhs
  proof -
    from that obtain t' n' where tnU': "(t', n') ∈ U'" and Pt': "Ifm (x # bs) (usubst p (t', n'))"
      by blast
    from tnU' UU' have "?f (t', n') ∈ ?g ` (U × U)"
      by blast
    then have "∃((t,n),(s,m)) ∈ U × U. ?f (t', n') = ?g ((t, n), (s, m))"
      apply auto
      apply (rule_tac x="(a,b)" in bexI)
      apply auto
      done
    then obtain t n s m where tnU: "(t, n) ∈ U" and smU: "(s, m) ∈ U" and
      th: "?f (t', n') = ?g ((t, n), (s, m))"
      by blast
    let ?N = "λt. 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 np mp have mnp: "real_of_int (2 * n * m) > 0"
      by (simp add: mult.commute of_int_mult[symmetric] del: of_int_mult)
    from tnb snb have stnb: "numbound0 ?st"
      by simp
    have st: "(?N t / real_of_int n + ?N s / real_of_int m) / 2 = ?N ?st / real_of_int (2 * n * m)"
      using mp np by (simp add: algebra_simps add_divide_distrib)
    from U' tnU' have tnb': "numbound0 t'" and np': "real_of_int 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_of_int (2 * n * m) # bs) p"
      by simp
    with usubst_I[OF lp mnp stnb, where x="x" and bs="bs"] tnU smU
    show ?thesis by blast
  qed
qed

lemma ferrack:
  assumes qf: "qfree p"
  shows "qfree (ferrack p) ∧ (Ifm bs (ferrack p) ⟷ (∃x. Ifm (x # bs) p))"
  (is "_ ∧ (?rhs ⟷ ?lhs)")
proof -
  let ?I = "λx p. Ifm (x # bs) p"
  fix x
  let ?N = "λt. Inum (x # bs) t"
  let ?q = "rlfm (simpfm p)"
  let ?U = "uset ?q"
  let ?Up = "alluopairs ?U"
  let ?g = "λ((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 = "remdups ?SS"
  let ?f = "λ(t,n). ?N t / real_of_int n"
  let ?h = "λ((t,n),(s,m)). (?N t / real_of_int n + ?N s / real_of_int m) / 2"
  let ?F = "λp. ∃a ∈ set (uset p). ∃b ∈ set (uset p). ?I x (usubst p (?g (a, b)))"
  let ?ep = "evaldjf (simpfm ∘ (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 ⊆ set ?U × set ?U"
    by simp
  from uset_l[OF lq] have U_l: "∀(t,n) ∈ set ?U. numbound0 t ∧ n > 0" .
  from U_l UpU
  have "∀((t,n),(s,m)) ∈ set ?Up. numbound0 t ∧ n> 0 ∧ numbound0 s ∧ m > 0"
    by auto
  then have Snb: "∀(t,n) ∈ set ?S. numbound0 t ∧ n > 0 "
    by auto
  have Y_l: "∀(t,n) ∈ set ?Y. numbound0 t ∧ n > 0"
  proof -
    have "numbound0 t ∧ n > 0" if tnY: "(t, n) ∈ set ?Y" for t n
    proof -
      from that have "(t,n) ∈ set ?SS"
        by simp
      then have "∃(t',n') ∈ set ?S. simp_num_pair (t', n') = (t, n)"
        apply (auto simp add: split_def simp del: map_map)
        apply (rule_tac x="((aa,ba),(ab,bb))" in bexI)
        apply simp_all
        done
      then obtain t' n' where tn'S: "(t', n') ∈ 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] show ?thesis .
    qed
    then show ?thesis by blast
  qed

  have YU: "(?f ` set ?Y) = (?h ` (set ?U × set ?U))"
  proof -
    from simp_num_pair_ci[where bs="x#bs"] have "∀x. (?f ∘ simp_num_pair) x = ?f x"
      by auto
    then have th: "?f ∘ simp_num_pair = ?f"
      by auto
    have "(?f ` set ?Y) = ((?f ∘ simp_num_pair) ` set ?S)"
      by (simp add: comp_assoc image_comp)
    also have "… = ?f ` set ?S"
      by (simp add: th)
    also have "… = (?f ∘ ?g) ` set ?Up"
      by (simp only: set_map o_def image_comp)
    also have "… = ?h ` (set ?U × set ?U)"
      using uset_cong_aux[OF U_l, where x="x" and bs="bs", simplified set_map image_comp]
      by blast
    finally show ?thesis .
  qed
  have "∀(t,n) ∈ set ?Y. bound0 (simpfm (usubst ?q (t, n)))"
  proof -
    have "bound0 (simpfm (usubst ?q (t, n)))" if tnY: "(t,n) ∈ set ?Y" for t n
    proof -
      from Y_l that have tnb: "numbound0 t" and np: "real_of_int n > 0"
        by auto
      from usubst_I[OF lq np tnb] have "bound0 (usubst ?q (t, n))"
        by simp
      then show ?thesis
        using simpfm_bound0 by simp
    qed
    then show ?thesis by blast
  qed
  then have ep_nb: "bound0 ?ep"
    using evaldjf_bound0[where xs="?Y" and f="simpfm ∘ (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 = (∃x. ?I x ?q)"
    by auto
  from th fr_equsubst[OF lq, where bs="bs" and x="x"] have lhfr: "?lhs = (?M ∨ ?P ∨ ?F ?q)"
    by (simp only: split_def fst_conv snd_conv)
  also have "… = (?M ∨ ?P ∨ (∃(t,n) ∈ 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 "… = (Ifm (x#bs) ?res)"
    using evaldjf_ex[where ps="?Y" and bs = "x#bs" and f="simpfm ∘ (usubst ?q)",symmetric]
    by (simp add: split_def prod.collapse)
  finally have lheq: "?lhs = Ifm bs (decr ?res)"
    using decr[OF nbth] by blast
  then have lr: "?lhs = ?rhs"
    unfolding ferrack_def Let_def
    by (cases "?mp = T ∨ ?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 ⇒ fm"
  where "linrqe p = qelim (prep p) ferrack"

theorem linrqe: "Ifm bs (linrqe p) = Ifm bs p ∧ qfree (linrqe p)"
  using ferrack qelim_ci prep
  unfolding linrqe_def by auto

definition ferrack_test :: "unit ⇒ 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_val ‹@{code ferrack_test} ()›

oracle linr_oracle = ‹
let

val mk_C = @{code C} o @{code int_of_integer};
val mk_Bound = @{code Bound} o @{code nat_of_integer};

fun num_of_term vs (Free vT) = mk_Bound (find_index (fn vT' => vT = vT') vs)
  | num_of_term vs \<^term>‹real_of_int (0::int)› = mk_C 0
  | num_of_term vs \<^term>‹real_of_int (1::int)› = mk_C 1
  | num_of_term vs \<^Const_>‹zero_class.zero \<^Type>‹real›› = mk_C 0
  | num_of_term vs \<^Const_>‹one_class.one \<^Type>‹real›› = mk_C 1
  | num_of_term vs (Bound i) = mk_Bound i
  | num_of_term vs \<^Const_>‹uminus \<^Type>‹real› for t'› = @{code Neg} (num_of_term vs t')
  | num_of_term vs \<^Const_>‹plus \<^Type>‹real› for t1 t2› =
     @{code Add} (num_of_term vs t1, num_of_term vs t2)
  | num_of_term vs \<^Const_>‹minus \<^Type>‹real› for t1 t2› =
     @{code Sub} (num_of_term vs t1, num_of_term vs t2)
  | num_of_term vs \<^Const_>‹times \<^Type>‹real› for 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_of_int :: int ⇒ real› $ t') =
     (mk_C (snd (HOLogic.dest_number t'))
       handle TERM _ => error ("num_of_term: unknown term"))
  | num_of_term vs t' =
     (mk_C (snd (HOLogic.dest_number t'))
       handle TERM _ => error ("num_of_term: unknown term"));

fun fm_of_term vs \<^Const_>‹True› = @{code T}
  | fm_of_term vs \<^Const_>‹False› = @{code F}
  | fm_of_term vs \<^Const_>‹less \<^Type>‹real› for t1 t2› =
      @{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  | fm_of_term vs \<^Const_>‹less_eq \<^Type>‹real› for t1 t2› =
      @{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  | fm_of_term vs \<^Const_>‹HOL.eq \<^Type>‹real› for t1 t2› =
      @{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
  | fm_of_term vs \<^Const_>‹HOL.eq \<^Type>‹bool› for t1 t2› =
      @{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs \<^Const_>‹HOL.conj for t1 t2› = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs \<^Const_>‹HOL.disj for t1 t2› = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs \<^Const_>‹HOL.implies for t1 t2› = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
  | fm_of_term vs \<^Const_>‹HOL.Not for t'› = @{code Not} (fm_of_term vs t')
  | fm_of_term vs \<^Const_>‹Ex _ for ‹Abs (xn, xT, p)›› = @{code E} (fm_of_term (("", dummyT) :: vs) p)
  | fm_of_term vs \<^Const_>‹All _ for ‹Abs (xn, xT, p)›› = @{code A} (fm_of_term (("", dummyT) ::  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_of_int :: int ⇒ real› $
      HOLogic.mk_number HOLogic.intT (@{code integer_of_int} i)
  | term_of_num vs (@{code Bound} n) = Free (nth vs (@{code integer_of_nat} n))
  | term_of_num vs (@{code Neg} t') = \<^Const>‹uminus \<^Type>‹real› for ‹term_of_num vs t'››
  | term_of_num vs (@{code Add} (t1, t2)) =
      \<^Const>‹plus \<^Type>‹real› for ‹term_of_num vs t1› ‹term_of_num vs t2››
  | term_of_num vs (@{code Sub} (t1, t2)) =
      \<^Const>‹minus \<^Type>‹real› for ‹term_of_num vs t1› ‹term_of_num vs t2››
  | term_of_num vs (@{code Mul} (i, t2)) =
      \<^Const>‹times \<^Type>‹real› for ‹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} = \<^Const>‹True›
  | term_of_fm vs @{code F} = \<^Const>‹False›
  | term_of_fm vs (@{code Lt} t) = \<^Const>‹less \<^Type>‹real› for ‹term_of_num vs t› \<^term>‹0::real››
  | term_of_fm vs (@{code Le} t) = \<^Const>‹less_eq \<^Type>‹real› for ‹term_of_num vs t› \<^term>‹0::real››
  | term_of_fm vs (@{code Gt} t) = \<^Const>‹less \<^Type>‹real› for \<^term>‹0::real› ‹term_of_num vs t››
  | term_of_fm vs (@{code Ge} t) = \<^Const>‹less_eq \<^Type>‹real› for \<^term>‹0::real› ‹term_of_num vs t››
  | term_of_fm vs (@{code Eq} t) = \<^Const>‹HOL.eq \<^Type>‹real› for ‹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') = \<^Const>‹HOL.Not for ‹term_of_fm vs t'››
  | term_of_fm vs (@{code And} (t1, t2)) = \<^Const>‹HOL.conj for ‹term_of_fm vs t1› ‹term_of_fm vs t2››
  | term_of_fm vs (@{code Or} (t1, t2)) = \<^Const>‹HOL.disj for ‹term_of_fm vs t1› ‹term_of_fm vs t2››
  | term_of_fm vs (@{code Imp} (t1, t2)) = \<^Const>‹HOL.implies for ‹term_of_fm vs t1› ‹term_of_fm vs t2››
  | term_of_fm vs (@{code Iff} (t1, t2)) = \<^Const>‹HOL.eq \<^Type>‹bool› for ‹term_of_fm vs t1› ‹term_of_fm vs t2››

in fn (ctxt, t) =>
  let
    val vs = Term.add_frees t [];
    val t' = (term_of_fm vs o @{code linrqe} o fm_of_term vs) t;
  in (Thm.cterm_of ctxt o HOLogic.mk_Trueprop o HOLogic.mk_eq) (t, t') end
end
›

ML_file ‹ferrack_tac.ML›

method_setup rferrack = ‹
  Scan.lift (Args.mode "no_quantify") >>
    (fn q => fn ctxt => SIMPLE_METHOD' (Ferrack_Tac.linr_tac ctxt (not q)))
› "decision procedure for linear real arithmetic"


lemma
  fixes x :: real
  shows "2 * x ≤ 2 * x ∧ 2 * x ≤ 2 * x + 1"
  by rferrack

lemma
  fixes x :: real
  shows "∃y ≤ x. x = y + 1"
  by rferrack

lemma
  fixes x :: real
  shows "¬ (∃z. x + z = x + z + 1)"
  by rferrack

end