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" 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" proof (induct t rule: maxcoeff.induct) case (2 n c t) then have cnz: "c ≠ 0" and mx: "max ¦c¦ (maxcoeff t) = 0" by simp_all have "max ¦c¦ (maxcoeff t) ≥ ¦c¦" by simp with cnz have "max ¦c¦ (maxcoeff t) > 0" by arith with 2 show ?case by simp qed auto lemma numgcd_nz: assumes nz: "nozerocoeff t" and g0: "numgcd t = 0" shows "t = C 0" proof - from g0 have th:"numgcdh t (maxcoeff t) = 0" by (simp add: numgcd_def) from numgcdh0[OF th] have th:"maxcoeff t = 0" . from maxcoeff_nz[OF nz th] show ?thesis . qed definition simp_num_pair :: "(num × int) ⇒ num × int" where "simp_num_pair = (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 @{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