# Theory Ferrack

theory Ferrack
imports Complex_Main Dense_Linear_Order DP_Library Code_Target_Numeral
```(*  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"

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"

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"

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 (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)"

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))"
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)"
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))"
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])

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
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
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
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"
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
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)"

lemma numadd_nb [simp]: "numbound0 t ⟹ numbound0 s ⟹ numbound0 (numadd t s)"

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 (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)"

lemma nummul_nz : "⋀i. i≠0 ⟹ nozerocoeff a ⟹ nozerocoeff (nummul a i)"

lemma numneg_nz : "nozerocoeff a ⟹ nozerocoeff (numneg a)"

lemma numsub_nz: "nozerocoeff a ⟹ nozerocoeff b ⟹ nozerocoeff (numsub a b)"

lemma simpnum_nz: "nozerocoeff (simpnum t)"

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"
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
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
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')"
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
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 (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

(* 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)"

lemma disj_lin: "isrlfm p ⟹ isrlfm q ⟹ isrlfm (disj p q)"

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"
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"
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"
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"
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"
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"
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"
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"
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"
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"
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"
then show ?case using numbound0_I[OF nb, where bs="bs" and b="x" and b'="y"]
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"
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)"
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"
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)"
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 (rule_tac x="(s,m)" in bexI)
apply simp_all
apply (rule_tac x="(t,n)" in bexI)
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)"
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')"
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')))"
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)"
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)"
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)"
also have "… = ?f ` set ?S"
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]
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 @{term "0::real"} = mk_C 0
| num_of_term vs @{term "1::real"} = mk_C 1
| num_of_term vs (Bound i) = mk_Bound i
| num_of_term vs (@{term "uminus :: real ⇒ real"} \$ t') = @{code Neg} (num_of_term vs t')
| num_of_term vs (@{term "(+) :: real ⇒ real ⇒ real"} \$ t1 \$ t2) =
@{code Add} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "(-) :: real ⇒ real ⇒ real"} \$ t1 \$ t2) =
@{code Sub} (num_of_term vs t1, num_of_term vs t2)
| num_of_term vs (@{term "( * ) :: real ⇒ real ⇒ real"} \$ t1 \$ t2) = (case num_of_term vs t1
of @{code C} i => @{code Mul} (i, num_of_term vs t2)
| _ => error "num_of_term: unsupported multiplication")
| num_of_term vs (@{term "real_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 @{term True} = @{code T}
| fm_of_term vs @{term False} = @{code F}
| fm_of_term vs (@{term "(<) :: real ⇒ real ⇒ bool"} \$ t1 \$ t2) =
@{code Lt} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "(≤) :: real ⇒ real ⇒ bool"} \$ t1 \$ t2) =
@{code Le} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "(=) :: real ⇒ real ⇒ bool"} \$ t1 \$ t2) =
@{code Eq} (@{code Sub} (num_of_term vs t1, num_of_term vs t2))
| fm_of_term vs (@{term "(⟷) :: bool ⇒ bool ⇒ bool"} \$ t1 \$ t2) =
@{code Iff} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term HOL.conj} \$ t1 \$ t2) = @{code And} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term HOL.disj} \$ t1 \$ t2) = @{code Or} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term HOL.implies} \$ t1 \$ t2) = @{code Imp} (fm_of_term vs t1, fm_of_term vs t2)
| fm_of_term vs (@{term "Not"} \$ t') = @{code NOT} (fm_of_term vs t')
| fm_of_term vs (Const (@{const_name Ex}, _) \$ Abs (xn, xT, p)) =
@{code E} (fm_of_term (("", dummyT) :: vs) p)
| fm_of_term vs (Const (@{const_name All}, _) \$ 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') = @{term "uminus :: real ⇒ real"} \$ term_of_num vs t'
| term_of_num vs (@{code Add} (t1, t2)) = @{term "(+) :: real ⇒ real ⇒ real"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Sub} (t1, t2)) = @{term "(-) :: real ⇒ real ⇒ real"} \$
term_of_num vs t1 \$ term_of_num vs t2
| term_of_num vs (@{code Mul} (i, t2)) = @{term "( * ) :: real ⇒ real ⇒ real"} \$
term_of_num vs (@{code C} i) \$ term_of_num vs t2
| term_of_num vs (@{code CN} (n, i, t)) = term_of_num vs (@{code Add} (@{code Mul} (i, @{code Bound} n), t));

fun term_of_fm vs @{code T} = @{term True}
| term_of_fm vs @{code F} = @{term False}
| term_of_fm vs (@{code Lt} t) = @{term "(<) :: real ⇒ real ⇒ bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code Le} t) = @{term "(≤) :: real ⇒ real ⇒ bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code Gt} t) = @{term "(<) :: real ⇒ real ⇒ bool"} \$
@{term "0::real"} \$ term_of_num vs t
| term_of_fm vs (@{code Ge} t) = @{term "(≤) :: real ⇒ real ⇒ bool"} \$
@{term "0::real"} \$ term_of_num vs t
| term_of_fm vs (@{code Eq} t) = @{term "(=) :: real ⇒ real ⇒ bool"} \$
term_of_num vs t \$ @{term "0::real"}
| term_of_fm vs (@{code NEq} t) = term_of_fm vs (@{code NOT} (@{code Eq} t))
| term_of_fm vs (@{code NOT} t') = HOLogic.Not \$ term_of_fm vs t'
| term_of_fm vs (@{code And} (t1, t2)) = HOLogic.conj \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Or} (t1, t2)) = HOLogic.disj \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Imp}  (t1, t2)) = HOLogic.imp \$ term_of_fm vs t1 \$ term_of_fm vs t2
| term_of_fm vs (@{code Iff} (t1, t2)) = @{term "(⟷) :: bool ⇒ bool ⇒ bool"} \$
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
```