# Theory Dense_Linear_Order

Up to index of Isabelle/HOL/HOL-Multivariate_Analysis/HOL-Probability/Girth_Chromatic

theory Dense_Linear_Order
imports Main
`(*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy    Author      : Amine Chaieb, TU Muenchen*)header {* Dense linear order without endpoints  and a quantifier elimination procedure in Ferrante and Rackoff style *}theory Dense_Linear_Orderimports MainbeginML_file "langford_data.ML"ML_file "ferrante_rackoff_data.ML"setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *}context linorderbeginlemma less_not_permute[no_atp]: "¬ (x < y ∧ y < x)"  by (simp add: not_less linear)lemma gather_simps[no_atp]:   "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ x < u ∧ P x) <->    (∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ (insert u U). x < y) ∧ P x)"  "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ l < x ∧ P x) <->    (∃x. (∀y ∈ (insert l L). y < x) ∧ (∀y ∈ U. x < y) ∧ P x)"  "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ x < u) <->    (∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ (insert u U). x < y))"  "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y) ∧ l < x) <->    (∃x. (∀y ∈ (insert l L). y < x) ∧ (∀y ∈ U. x < y))"  by autolemma gather_start [no_atp]: "(∃x. P x) ≡ (∃x. (∀y ∈ {}. y < x) ∧ (∀y∈ {}. x < y) ∧ P x)"   by simptext{* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P⇘-∞⇙)"}*}lemma minf_lt[no_atp]:  "∃z . ∀x. x < z --> (x < t <-> True)" by autolemma minf_gt[no_atp]: "∃z . ∀x. x < z -->  (t < x <->  False)"  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)lemma minf_le[no_atp]: "∃z. ∀x. x < z --> (x ≤ t <-> True)" by (auto simp add: less_le)lemma minf_ge[no_atp]: "∃z. ∀x. x < z --> (t ≤ x <-> False)"  by (auto simp add: less_le not_less not_le)lemma minf_eq[no_atp]: "∃z. ∀x. x < z --> (x = t <-> False)" by autolemma minf_neq[no_atp]: "∃z. ∀x. x < z --> (x ≠ t <-> True)" by autolemma minf_P[no_atp]: "∃z. ∀x. x < z --> (P <-> P)" by blasttext{* Theorems for @{text "∃z. ∀x. x < z --> (P x <-> P⇘+∞⇙)"}*}lemma pinf_gt[no_atp]:  "∃z. ∀x. z < x --> (t < x <-> True)" by autolemma pinf_lt[no_atp]: "∃z. ∀x. z < x -->  (x < t <->  False)"  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)lemma pinf_ge[no_atp]: "∃z. ∀x. z < x --> (t ≤ x <-> True)" by (auto simp add: less_le)lemma pinf_le[no_atp]: "∃z. ∀x. z < x --> (x ≤ t <-> False)"  by (auto simp add: less_le not_less not_le)lemma pinf_eq[no_atp]: "∃z. ∀x. z < x --> (x = t <-> False)" by autolemma pinf_neq[no_atp]: "∃z. ∀x. z < x --> (x ≠ t <-> True)" by autolemma pinf_P[no_atp]: "∃z. ∀x. z < x --> (P <-> P)" by blastlemma nmi_lt[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ x < t -->  (∃ u∈ U. u ≤ x)" by autolemma nmi_gt[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ t < x -->  (∃ u∈ U. u ≤ x)"  by (auto simp add: le_less)lemma  nmi_le[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ x≤ t -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_ge[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ t≤ x -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_eq[no_atp]: "t ∈ U ==> ∀x. ¬False ∧  x = t -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_neq[no_atp]: "t ∈ U ==>∀x. ¬True ∧ x ≠ t -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_P[no_atp]: "∀ x. ~P ∧ P -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_conj[no_atp]: "[|∀x. ¬P1' ∧ P1 x -->  (∃ u∈ U. u ≤ x) ;  ∀x. ¬P2' ∧ P2 x -->  (∃ u∈ U. u ≤ x)|] ==>  ∀x. ¬(P1' ∧ P2') ∧ (P1 x ∧ P2 x) -->  (∃ u∈ U. u ≤ x)" by autolemma  nmi_disj[no_atp]: "[|∀x. ¬P1' ∧ P1 x -->  (∃ u∈ U. u ≤ x) ;  ∀x. ¬P2' ∧ P2 x -->  (∃ u∈ U. u ≤ x)|] ==>  ∀x. ¬(P1' ∨ P2') ∧ (P1 x ∨ P2 x) -->  (∃ u∈ U. u ≤ x)" by autolemma  npi_lt[no_atp]: "t ∈ U ==> ∀x. ¬False ∧  x < t -->  (∃ u∈ U. x ≤ u)" by (auto simp add: le_less)lemma  npi_gt[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ t < x -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_le[no_atp]: "t ∈ U ==> ∀x. ¬False ∧  x ≤ t -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_ge[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ t ≤ x -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_eq[no_atp]: "t ∈ U ==> ∀x. ¬False ∧  x = t -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_neq[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ x ≠ t -->  (∃ u∈ U. x ≤ u )" by autolemma  npi_P[no_atp]: "∀ x. ~P ∧ P -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_conj[no_atp]: "[|∀x. ¬P1' ∧ P1 x -->  (∃ u∈ U. x ≤ u) ;  ∀x. ¬P2' ∧ P2 x -->  (∃ u∈ U. x ≤ u)|]  ==>  ∀x. ¬(P1' ∧ P2') ∧ (P1 x ∧ P2 x) -->  (∃ u∈ U. x ≤ u)" by autolemma  npi_disj[no_atp]: "[|∀x. ¬P1' ∧ P1 x -->  (∃ u∈ U. x ≤ u) ; ∀x. ¬P2' ∧ P2 x -->  (∃ u∈ U. x ≤ u)|]  ==> ∀x. ¬(P1' ∨ P2') ∧ (P1 x ∨ P2 x) -->  (∃ u∈ U. x ≤ u)" by autolemma lin_dense_lt[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t < u --> t ∉ U) ∧ l< x ∧ x < u ∧ x < t --> (∀ y. l < y ∧ y < u --> y < t)"proof(clarsimp)  fix x l u y  assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x"    and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"  from tU noU ly yu have tny: "t≠y" by auto  { assume H: "t < y"    from less_trans[OF lx px] less_trans[OF H yu]    have "l < t ∧ t < u" by simp    with tU noU have "False" by auto }  then have "¬ t < y" by auto  then have "y ≤ t" by (simp add: not_less)  then show "y < t" using tny by (simp add: less_le)qedlemma lin_dense_gt[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l < x ∧ x < u ∧ t < x --> (∀ y. l < y ∧ y < u --> t < y)"proof(clarsimp)  fix x l u y  assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u"    and px: "t < x" and ly: "l<y" and yu:"y < u"  from tU noU ly yu have tny: "t≠y" by auto  { assume H: "y< t"    from less_trans[OF ly H] less_trans[OF px xu] have "l < t ∧ t < u" by simp    with tU noU have "False" by auto }  then have "¬ y<t" by auto  then have "t ≤ y" by (auto simp add: not_less)  then show "t < y" using tny by (simp add: less_le)qedlemma lin_dense_le[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x ≤ t --> (∀ y. l < y ∧ y < u --> y≤ t)"proof(clarsimp)  fix x l u y  assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u"    and px: "x ≤ t" and ly: "l<y" and yu:"y < u"  from tU noU ly yu have tny: "t≠y" by auto  { assume H: "t < y"    from less_le_trans[OF lx px] less_trans[OF H yu]    have "l < t ∧ t < u" by simp    with tU noU have "False" by auto }  then have "¬ t < y" by auto  then show "y ≤ t" by (simp add: not_less)qedlemma lin_dense_ge[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ t ≤ x --> (∀ y. l < y ∧ y < u --> t ≤ y)"proof(clarsimp)  fix x l u y  assume tU: "t ∈ U" and noU: "∀t. l < t ∧ t < u --> t ∉ U" and lx: "l < x" and xu: "x<u"    and px: "t ≤ x" and ly: "l<y" and yu:"y < u"  from tU noU ly yu have tny: "t≠y" by auto  { assume H: "y< t"    from less_trans[OF ly H] le_less_trans[OF px xu]    have "l < t ∧ t < u" by simp    with tU noU have "False" by auto }  then have "¬ y<t" by auto  then show "t ≤ y" by (simp add: not_less)qedlemma lin_dense_eq[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x = t   --> (∀ y. l < y ∧ y < u --> y= t)"  by autolemma lin_dense_neq[no_atp]:  "t ∈ U ==>    ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ x ≠ t   --> (∀ y. l < y ∧ y < u --> y≠ t)"  by autolemma lin_dense_P[no_atp]:  "∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P   --> (∀ y. l < y ∧ y < u --> P)"  by autolemma lin_dense_conj[no_atp]:  "[|∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P1 x  --> (∀ y. l < y ∧ y < u --> P1 y) ;  ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P2 x  --> (∀ y. l < y ∧ y < u --> P2 y)|] ==>  ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ (P1 x ∧ P2 x)  --> (∀ y. l < y ∧ y < u --> (P1 y ∧ P2 y))"  by blastlemma lin_dense_disj[no_atp]:  "[|∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P1 x  --> (∀ y. l < y ∧ y < u --> P1 y) ;  ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ P2 x  --> (∀ y. l < y ∧ y < u --> P2 y)|] ==>  ∀x l u. (∀ t. l < t ∧ t< u --> t ∉ U) ∧ l< x ∧ x < u ∧ (P1 x ∨ P2 x)  --> (∀ y. l < y ∧ y < u --> (P1 y ∨ P2 y))"  by blastlemma npmibnd[no_atp]: "[|∀x. ¬ MP ∧ P x --> (∃ u∈ U. u ≤ x); ∀x. ¬PP ∧ P x --> (∃ u∈ U. x ≤ u)|]  ==> ∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u ≤ x ∧ x ≤ u')"  by autolemma finite_set_intervals[no_atp]:  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  hence fMx: "finite ?Mx" using fS finite_subset by auto  from lx linS have linMx: "l ∈ ?Mx" by blast  hence Mxne: "?Mx ≠ {}" by blast  have xMS: "?xM ⊆ S" by blast  hence fxM: "finite ?xM" using fS finite_subset by auto  from xu uinS have linxM: "u ∈ ?xM" by blast  hence 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 hence ainS: "?a ∈ S" using MxS by blast  have "?b ∈ ?xM" using Min_in[OF fxM xMne] by simp hence 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 have "y∈ ?Mx ∨ y∈ ?xM" by (auto simp add: linear)    moreover {      assume "y ∈ ?Mx"      hence "y ≤ ?a" using Mxne fMx by auto      with ay have "False" by (simp add: not_le[symmetric])    }    moreover {      assume "y ∈ ?xM"      hence "?b ≤ y" using xMne fxM by auto      with yb have "False" by (simp add: not_le[symmetric])    }    ultimately show False by blast  qed  from ainS binS noy ax xb px show ?thesis by blastqedlemma finite_set_intervals2[no_atp]:  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 "(∃ s∈ S. P s) ∨ (∃ a ∈ S. ∃ b ∈ S. (∀ y. a < y ∧ y < b --> y ∉ S) ∧ a < x ∧ x < b ∧ P x)"proof-  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]  obtain a and b where as: "a∈ S" and bs: "b∈ S" and noS:"∀y. a < y ∧ y < b --> y ∉ S"    and axb: "a ≤ x ∧ x ≤ b ∧ P x" by auto  from axb have "x= a ∨ x= b ∨ (a < x ∧ x < b)" by (auto simp add: le_less)  thus ?thesis using px as bs noS by blastqedendsection {* The classical QE after Langford for dense linear orders *}context dense_linorderbeginlemma interval_empty_iff: "{y. x < y ∧ y < z} = {} <-> ¬ x < z"  by (auto dest: dense)lemma dlo_qe_bnds[no_atp]:   assumes ne: "L ≠ {}" and neU: "U ≠ {}" and fL: "finite L" and fU: "finite U"  shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ U. x < y)) ≡ (∀ l ∈ L. ∀u ∈ U. l < u)"proof (simp only: atomize_eq, rule iffI)  assume H: "∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y)"  then obtain x where xL: "∀y∈L. y < x" and xU: "∀y∈U. x < y" by blast  { fix l u assume l: "l ∈ L" and u: "u ∈ U"    have "l < x" using xL l by blast    also have "x < u" using xU u by blast    finally (less_trans) have "l < u" . }  then show "∀l∈L. ∀u∈U. l < u" by blastnext  assume H: "∀l∈L. ∀u∈U. l < u"  let ?ML = "Max L"  let ?MU = "Min U"    from fL ne have th1: "?ML ∈ L" and th1': "∀l∈L. l ≤ ?ML" by auto  from fU neU have th2: "?MU ∈ U" and th2': "∀u∈U. ?MU ≤ u" by auto  from th1 th2 H have "?ML < ?MU" by auto  with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast  from th3 th1' have "∀l ∈ L. l < w" by auto  moreover from th4 th2' have "∀u ∈ U. w < u" by auto  ultimately show "∃x. (∀y∈L. y < x) ∧ (∀y∈U. x < y)" by autoqedlemma dlo_qe_noub[no_atp]:   assumes ne: "L ≠ {}" and fL: "finite L"  shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ {}. x < y)) ≡ True"proof(simp add: atomize_eq)  from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast  from ne fL have "∀x ∈ L. x ≤ Max L" by simp  with M have "∀x∈L. x < M" by (auto intro: le_less_trans)  thus "∃x. ∀y∈L. y < x" by blastqedlemma dlo_qe_nolb[no_atp]:   assumes ne: "U ≠ {}" and fU: "finite U"  shows "(∃x. (∀y ∈ {}. y < x) ∧ (∀y ∈ U. x < y)) ≡ True"proof(simp add: atomize_eq)  from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast  from ne fU have "∀x ∈ U. Min U ≤ x" by simp  with M have "∀x∈U. M < x" by (auto intro: less_le_trans)  thus "∃x. ∀y∈U. x < y" by blastqedlemma exists_neq[no_atp]: "∃(x::'a). x ≠ t" "∃(x::'a). t ≠ x"   using gt_ex[of t] by autolemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq   le_less neq_iff linear less_not_permutelemma axiom[no_atp]: "class.dense_linorder (op ≤) (op <)" by (rule dense_linorder_axioms)lemma atoms[no_atp]:  shows "TERM (less :: 'a => _)"    and "TERM (less_eq :: 'a => _)"    and "TERM (op = :: 'a => _)" .declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]declare dlo_simps[langfordsimp]end(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribRlemmas weak_dnf_simps[no_atp] = simp_thms dnflemma nnf_simps[no_atp]:    "(¬(P ∧ Q)) = (¬P ∨ ¬Q)" "(¬(P ∨ Q)) = (¬P ∧ ¬Q)" "(P --> Q) = (¬P ∨ Q)"    "(P = Q) = ((P ∧ Q) ∨ (¬P ∧ ¬ Q))" "(¬ ¬(P)) = P"  by blast+lemma ex_distrib[no_atp]: "(∃x. P x ∨ Q x) <-> ((∃x. P x) ∨ (∃x. Q x))" by blastlemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distribML_file "langford.ML"method_setup dlo = {*  Scan.succeed (SIMPLE_METHOD' o LangfordQE.dlo_tac)*} "Langford's algorithm for quantifier elimination in dense linear orders"section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}text {* Linear order without upper bounds *}locale linorder_stupid_syntax = linorderbeginnotation  less_eq  ("op \<sqsubseteq>") and  less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and  less  ("op \<sqsubset>") and  less  ("(_/ \<sqsubset> _)"  [51, 51] 50)endlocale linorder_no_ub = linorder_stupid_syntax +  assumes gt_ex: "∃y. less x y"beginlemma ge_ex[no_atp]: "∃y. x \<sqsubseteq> y" using gt_ex by autotext {* Theorems for @{text "∃z. ∀x. z \<sqsubset> x --> (P x <-> P⇘+∞⇙)"} *}lemma pinf_conj[no_atp]:  assumes ex1: "∃z1. ∀x. z1 \<sqsubset> x --> (P1 x <-> P1')"  and ex2: "∃z2. ∀x. z2 \<sqsubset> x --> (P2 x <-> P2')"  shows "∃z. ∀x. z \<sqsubset>  x --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))"proof-  from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 \<sqsubset> x --> (P1 x <-> P1')"     and z2: "∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" by blast  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all  { fix x assume H: "z \<sqsubset> x"    from less_trans[OF zz1 H] less_trans[OF zz2 H]    have "(P1 x ∧ P2 x) <-> (P1' ∧ P2')"  using z1 zz1 z2 zz2 by auto  }  thus ?thesis by blastqedlemma pinf_disj[no_atp]:  assumes ex1: "∃z1. ∀x. z1 \<sqsubset> x --> (P1 x <-> P1')"    and ex2: "∃z2. ∀x. z2 \<sqsubset> x --> (P2 x <-> P2')"  shows "∃z. ∀x. z \<sqsubset>  x --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))"proof-  from ex1 ex2 obtain z1 and z2 where z1: "∀x. z1 \<sqsubset> x --> (P1 x <-> P1')"     and z2: "∀x. z2 \<sqsubset> x --> (P2 x <-> P2')" by blast  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all  { fix x assume H: "z \<sqsubset> x"    from less_trans[OF zz1 H] less_trans[OF zz2 H]    have "(P1 x ∨ P2 x) <-> (P1' ∨ P2')"  using z1 zz1 z2 zz2 by auto  }  thus ?thesis by blastqedlemma pinf_ex[no_atp]: assumes ex:"∃z. ∀x. z \<sqsubset> x --> (P x <-> P1)" and p1: P1 shows "∃ x. P x"proof -  from ex obtain z where z: "∀x. z \<sqsubset> x --> (P x <-> P1)" by blast  from gt_ex obtain x where x: "z \<sqsubset> x" by blast  from z x p1 show ?thesis by blastqedendtext {* Linear order without upper bounds *}locale linorder_no_lb = linorder_stupid_syntax +  assumes lt_ex: "∃y. less y x"beginlemma le_ex[no_atp]: "∃y. y \<sqsubseteq> x" using lt_ex by autotext {* Theorems for @{text "∃z. ∀x. x \<sqsubset> z --> (P x <-> P⇘-∞⇙)"} *}lemma minf_conj[no_atp]:  assumes ex1: "∃z1. ∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"    and ex2: "∃z2. ∀x. x \<sqsubset> z2 --> (P2 x <-> P2')"  shows "∃z. ∀x. x \<sqsubset>  z --> ((P1 x ∧ P2 x) <-> (P1' ∧ P2'))"proof-  from ex1 ex2 obtain z1 and z2 where z1: "∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"and z2: "∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" by blast  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all  { fix x assume H: "x \<sqsubset> z"    from less_trans[OF H zz1] less_trans[OF H zz2]    have "(P1 x ∧ P2 x) <-> (P1' ∧ P2')"  using z1 zz1 z2 zz2 by auto  }  thus ?thesis by blastqedlemma minf_disj[no_atp]:  assumes ex1: "∃z1. ∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"    and ex2: "∃z2. ∀x. x \<sqsubset> z2 --> (P2 x <-> P2')"  shows "∃z. ∀x. x \<sqsubset>  z --> ((P1 x ∨ P2 x) <-> (P1' ∨ P2'))"proof -  from ex1 ex2 obtain z1 and z2 where z1: "∀x. x \<sqsubset> z1 --> (P1 x <-> P1')"    and z2: "∀x. x \<sqsubset> z2 --> (P2 x <-> P2')" by blast  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all  { fix x assume H: "x \<sqsubset> z"    from less_trans[OF H zz1] less_trans[OF H zz2]    have "(P1 x ∨ P2 x) <-> (P1' ∨ P2')"  using z1 zz1 z2 zz2 by auto  }  thus ?thesis by blastqedlemma minf_ex[no_atp]:  assumes ex: "∃z. ∀x. x \<sqsubset> z --> (P x <-> P1)"    and p1: P1  shows "∃ x. P x"proof -  from ex obtain z where z: "∀x. x \<sqsubset> z --> (P x <-> P1)" by blast  from lt_ex obtain x where x: "x \<sqsubset> z" by blast  from z x p1 show ?thesis by blastqedendlocale constr_dense_linorder = linorder_no_lb + linorder_no_ub +  fixes between  assumes between_less: "less x y ==> less x (between x y) ∧ less (between x y) y"    and between_same: "between x x = x"sublocale  constr_dense_linorder < dlo: dense_linorder   apply unfold_locales  using gt_ex lt_ex between_less  apply auto  apply (rule_tac x="between x y" in exI)  apply simp  donecontext constr_dense_linorderbeginlemma rinf_U[no_atp]:  assumes fU: "finite U"    and lin_dense: "∀x l u. (∀ t. l \<sqsubset> t ∧ t\<sqsubset> u --> t ∉ U) ∧ l\<sqsubset> x ∧ x \<sqsubset> u ∧ P x      --> (∀ y. l \<sqsubset> y ∧ y \<sqsubset> u --> P y )"    and nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u \<sqsubseteq> x ∧ x \<sqsubseteq> u')"    and nmi: "¬ MP"  and npi: "¬ PP"  and ex: "∃ x.  P x"  shows "∃ u∈ U. ∃ u' ∈ U. P (between u u')"proof -  from ex obtain x where px: "P x" by blast  from px nmi npi nmpiU have "∃ u∈ U. ∃ u' ∈ U. u \<sqsubseteq> x ∧ x \<sqsubseteq> u'" by auto  then obtain u and u' where uU:"u∈ U" and uU': "u' ∈ U" and ux:"u \<sqsubseteq> x" and xu':"x \<sqsubseteq> u'" by auto  from uU have Une: "U ≠ {}" by auto  let ?l = "linorder.Min less_eq U"  let ?u = "linorder.Max less_eq U"  have linM: "?l ∈ U" using fU Une by simp  have uinM: "?u ∈ U" using fU Une by simp  have lM: "∀ t∈ U. ?l \<sqsubseteq> t" using Une fU by auto  have Mu: "∀ t∈ U. t \<sqsubseteq> ?u" using Une fU by auto  have th:"?l \<sqsubseteq> u" using uU Une lM by auto  from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .  have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp  from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .  from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]  have "(∃ s∈ U. P s) ∨      (∃ t1∈ U. ∃ t2 ∈ U. (∀ y. t1 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U) ∧ t1 \<sqsubset> x ∧ x \<sqsubset> t2 ∧ P x)" .  moreover {    fix u assume um: "u∈U" and pu: "P u"    have "between u u = u" by (simp add: between_same)    with um pu have "P (between u u)" by simp    with um have ?thesis by blast }  moreover {    assume "∃ t1∈ U. ∃ t2 ∈ U. (∀ y. t1 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U) ∧ t1 \<sqsubset> x ∧ x \<sqsubset> t2 ∧ P x"    then obtain t1 and t2 where t1M: "t1 ∈ U" and t2M: "t2∈ U"      and noM: "∀ y. t1 \<sqsubset> y ∧ y \<sqsubset> t2 --> y ∉ U"      and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" by blast    from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .    let ?u = "between t1 t2"    from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto    from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast    with t1M t2M have ?thesis by blast  }  ultimately show ?thesis by blastqedtheorem fr_eq[no_atp]:  assumes fU: "finite U"    and lin_dense: "∀x l u. (∀ t. l \<sqsubset> t ∧ t\<sqsubset> u --> t ∉ U) ∧ l\<sqsubset> x ∧ x \<sqsubset> u ∧ P x     --> (∀ y. l \<sqsubset> y ∧ y \<sqsubset> u --> P y )"    and nmibnd: "∀x. ¬ MP ∧ P x --> (∃ u∈ U. u \<sqsubseteq> x)"    and npibnd: "∀x. ¬PP ∧ P x --> (∃ u∈ U. x \<sqsubseteq> u)"    and mi: "∃z. ∀x. x \<sqsubset> z --> (P x = MP)"  and pi: "∃z. ∀x. z \<sqsubset> x --> (P x = PP)"  shows "(∃ x. P x) ≡ (MP ∨ PP ∨ (∃ u ∈ U. ∃ u'∈ U. P (between u u')))"  (is "_ ≡ (_ ∨ _ ∨ ?F)" is "?E ≡ ?D")proof -  { assume px: "∃ x. P x"    have "MP ∨ PP ∨ (¬ MP ∧ ¬ PP)" by blast    moreover { assume "MP ∨ PP" hence "?D" by blast }    moreover {      assume nmi: "¬ MP" and npi: "¬ PP"      from npmibnd[OF nmibnd npibnd]      have nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x --> (∃ u∈ U. ∃ u' ∈ U. u \<sqsubseteq> x ∧ x \<sqsubseteq> u')" .      from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast }    ultimately have "?D" by blast }  moreover  { assume "?D"    moreover { assume m:"MP" from minf_ex[OF mi m] have "?E" . }    moreover { assume p: "PP" from pinf_ex[OF pi p] have "?E" . }    moreover { assume f:"?F" hence "?E" by blast }    ultimately have "?E" by blast }  ultimately have "?E = ?D" by blast thus "?E ≡ ?D" by simpqedlemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_Plemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_Plemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_Plemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_Plemmas lin_dense_thms[no_atp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_Plemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between"  by (rule constr_dense_linorder_axioms)lemma atoms[no_atp]:  shows "TERM (less :: 'a => _)"    and "TERM (less_eq :: 'a => _)"    and "TERM (op = :: 'a => _)" .declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms    nmi: nmi_thms npi: npi_thms lindense:    lin_dense_thms qe: fr_eq atoms: atoms]declaration {*let  fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]  fun generic_whatis phi =    let      val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]      fun h x t =        case term_of t of          Const(@{const_name HOL.eq}, _)\$y\$z =>            if term_of x aconv y then Ferrante_Rackoff_Data.Eq            else Ferrante_Rackoff_Data.Nox       | @{term "Not"}\$(Const(@{const_name HOL.eq}, _)\$y\$z) =>            if term_of x aconv y then Ferrante_Rackoff_Data.NEq            else Ferrante_Rackoff_Data.Nox       | b\$y\$z => if Term.could_unify (b, lt) then                     if term_of x aconv y then Ferrante_Rackoff_Data.Lt                     else if term_of x aconv z then Ferrante_Rackoff_Data.Gt                     else Ferrante_Rackoff_Data.Nox                 else if Term.could_unify (b, le) then                     if term_of x aconv y then Ferrante_Rackoff_Data.Le                     else if term_of x aconv z then Ferrante_Rackoff_Data.Ge                     else Ferrante_Rackoff_Data.Nox                 else Ferrante_Rackoff_Data.Nox       | _ => Ferrante_Rackoff_Data.Nox  in h end  fun ss phi = HOL_ss addsimps (simps phi)in  Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}    {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}end*}endML_file "ferrante_rackoff.ML"method_setup ferrack = {*  Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"subsection {* Ferrante and Rackoff algorithm over ordered fields *}lemma neg_prod_lt:"(c::'a::linordered_field) < 0 ==> ((c*x < 0) == (x > 0))"proof -  assume H: "c < 0"  have "c*x < 0 = (0/c < x)"    by (simp only: neg_divide_less_eq[OF H] algebra_simps)  also have "… = (0 < x)" by simp  finally show  "(c*x < 0) == (x > 0)" by simpqedlemma pos_prod_lt:"(c::'a::linordered_field) > 0 ==> ((c*x < 0) == (x < 0))"proof -  assume H: "c > 0"  then have "c*x < 0 = (0/c > x)"    by (simp only: pos_less_divide_eq[OF H] algebra_simps)  also have "… = (0 > x)" by simp  finally show  "(c*x < 0) == (x < 0)" by simpqedlemma neg_prod_sum_lt: "(c::'a::linordered_field) < 0 ==> ((c*x + t< 0) == (x > (- 1/c)*t))"proof -  assume H: "c < 0"  have "c*x + t< 0 = (c*x < -t)"    by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)  also have "… = (-t/c < x)"    by (simp only: neg_divide_less_eq[OF H] algebra_simps)  also have "… = ((- 1/c)*t < x)" by simp  finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simpqedlemma pos_prod_sum_lt:"(c::'a::linordered_field) > 0 ==> ((c*x + t < 0) == (x < (- 1/c)*t))"proof -  assume H: "c > 0"  have "c*x + t< 0 = (c*x < -t)"    by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)  also have "… = (-t/c > x)"    by (simp only: pos_less_divide_eq[OF H] algebra_simps)  also have "… = ((- 1/c)*t > x)" by simp  finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simpqedlemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"  using less_diff_eq[where a= x and b=t and c=0] by simplemma neg_prod_le:"(c::'a::linordered_field) < 0 ==> ((c*x <= 0) == (x >= 0))"proof -  assume H: "c < 0"  have "c*x <= 0 = (0/c <= x)"    by (simp only: neg_divide_le_eq[OF H] algebra_simps)  also have "… = (0 <= x)" by simp  finally show  "(c*x <= 0) == (x >= 0)" by simpqedlemma pos_prod_le:"(c::'a::linordered_field) > 0 ==> ((c*x <= 0) == (x <= 0))"proof -  assume H: "c > 0"  hence "c*x <= 0 = (0/c >= x)"    by (simp only: pos_le_divide_eq[OF H] algebra_simps)  also have "… = (0 >= x)" by simp  finally show  "(c*x <= 0) == (x <= 0)" by simpqedlemma neg_prod_sum_le: "(c::'a::linordered_field) < 0 ==> ((c*x + t <= 0) == (x >= (- 1/c)*t))"proof -  assume H: "c < 0"  have "c*x + t <= 0 = (c*x <= -t)"    by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)  also have "… = (-t/c <= x)"    by (simp only: neg_divide_le_eq[OF H] algebra_simps)  also have "… = ((- 1/c)*t <= x)" by simp  finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simpqedlemma pos_prod_sum_le:"(c::'a::linordered_field) > 0 ==> ((c*x + t <= 0) == (x <= (- 1/c)*t))"proof -  assume H: "c > 0"  have "c*x + t <= 0 = (c*x <= -t)"    by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)  also have "… = (-t/c >= x)"    by (simp only: pos_le_divide_eq[OF H] algebra_simps)  also have "… = ((- 1/c)*t >= x)" by simp  finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simpqedlemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"  using le_diff_eq[where a= x and b=t and c=0] by simplemma nz_prod_eq:"(c::'a::linordered_field) ≠ 0 ==> ((c*x = 0) == (x = 0))" by simplemma nz_prod_sum_eq: "(c::'a::linordered_field) ≠ 0 ==> ((c*x + t = 0) == (x = (- 1/c)*t))"proof -  assume H: "c ≠ 0"  have "c*x + t = 0 = (c*x = -t)"    by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)  also have "… = (x = -t/c)"    by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)  finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simpqedlemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"  using eq_diff_eq[where a= x and b=t and c=0] by simpinterpretation class_dense_linordered_field: constr_dense_linorder "op <=" "op <"   "λ x y. 1/2 * ((x::'a::{linordered_field}) + y)"  by unfold_locales (dlo, dlo, auto)declaration{*let  fun earlier [] x y = false    | earlier (h::t) x y =        if h aconvc y then false else if h aconvc x then true else earlier t x y;fun dest_frac ct =  case term_of ct of    Const (@{const_name Fields.divide},_) \$ a \$ b=>      Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))  | Const(@{const_name inverse}, _)\$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)  | t => Rat.rat_of_int (snd (HOLogic.dest_number t))fun mk_frac phi cT x =  let val (a, b) = Rat.quotient_of_rat x  in if b = 1 then Numeral.mk_cnumber cT a    else Thm.apply         (Thm.apply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})                     (Numeral.mk_cnumber cT a))         (Numeral.mk_cnumber cT b) endfun whatis x ct = case term_of ct of  Const(@{const_name Groups.plus}, _)\$(Const(@{const_name Groups.times},_)\$_\$y)\$_ =>     if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])     else ("Nox",[])| Const(@{const_name Groups.plus}, _)\$y\$_ =>     if y aconv term_of x then ("x+t",[Thm.dest_arg ct])     else ("Nox",[])| Const(@{const_name Groups.times}, _)\$_\$y =>     if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])     else ("Nox",[])| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);fun xnormalize_conv ctxt [] ct = Thm.reflexive ct| xnormalize_conv ctxt (vs as (x::_)) ct =   case term_of ct of   Const(@{const_name Orderings.less},_)\$_\$Const(@{const_name Groups.zero},_) =>    (case whatis x (Thm.dest_arg1 ct) of    ("c*x+t",[c,t]) =>       let        val cr = dest_frac c        val clt = Thm.dest_fun2 ct        val cz = Thm.dest_arg ct        val neg = cr </ Rat.zero        val cthp = Simplifier.rewrite (simpset_of ctxt)               (Thm.apply @{cterm "Trueprop"}                  (if neg then Thm.apply (Thm.apply clt c) cz                    else Thm.apply (Thm.apply clt cz) c))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])             (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv                   (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th      in rth end    | ("x+t",[t]) =>       let        val T = ctyp_of_term x        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv              (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th       in  rth end    | ("c*x",[c]) =>       let        val cr = dest_frac c        val clt = Thm.dest_fun2 ct        val cz = Thm.dest_arg ct        val neg = cr </ Rat.zero        val cthp = Simplifier.rewrite (simpset_of ctxt)               (Thm.apply @{cterm "Trueprop"}                  (if neg then Thm.apply (Thm.apply clt c) cz                    else Thm.apply (Thm.apply clt cz) c))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])             (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth        val rth = th      in rth end    | _ => Thm.reflexive ct)|  Const(@{const_name Orderings.less_eq},_)\$_\$Const(@{const_name Groups.zero},_) =>   (case whatis x (Thm.dest_arg1 ct) of    ("c*x+t",[c,t]) =>       let        val T = ctyp_of_term x        val cr = dest_frac c        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}        val cz = Thm.dest_arg ct        val neg = cr </ Rat.zero        val cthp = Simplifier.rewrite (simpset_of ctxt)               (Thm.apply @{cterm "Trueprop"}                  (if neg then Thm.apply (Thm.apply clt c) cz                    else Thm.apply (Thm.apply clt cz) c))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val th = Thm.implies_elim (instantiate' [SOME T] (map SOME [c,x,t])             (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv                   (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th      in rth end    | ("x+t",[t]) =>       let        val T = ctyp_of_term x        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv              (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th       in  rth end    | ("c*x",[c]) =>       let        val T = ctyp_of_term x        val cr = dest_frac c        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}        val cz = Thm.dest_arg ct        val neg = cr </ Rat.zero        val cthp = Simplifier.rewrite (simpset_of ctxt)               (Thm.apply @{cterm "Trueprop"}                  (if neg then Thm.apply (Thm.apply clt c) cz                    else Thm.apply (Thm.apply clt cz) c))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])             (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth        val rth = th      in rth end    | _ => Thm.reflexive ct)|  Const(@{const_name HOL.eq},_)\$_\$Const(@{const_name Groups.zero},_) =>   (case whatis x (Thm.dest_arg1 ct) of    ("c*x+t",[c,t]) =>       let        val T = ctyp_of_term x        val cr = dest_frac c        val ceq = Thm.dest_fun2 ct        val cz = Thm.dest_arg ct        val cthp = Simplifier.rewrite (simpset_of ctxt)            (Thm.apply @{cterm "Trueprop"}             (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val th = Thm.implies_elim                 (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv                   (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th      in rth end    | ("x+t",[t]) =>       let        val T = ctyp_of_term x        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv              (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th       in  rth end    | ("c*x",[c]) =>       let        val T = ctyp_of_term x        val cr = dest_frac c        val ceq = Thm.dest_fun2 ct        val cz = Thm.dest_arg ct        val cthp = Simplifier.rewrite (simpset_of ctxt)            (Thm.apply @{cterm "Trueprop"}             (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))        val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI        val rth = Thm.implies_elim                 (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth      in rth end    | _ => Thm.reflexive ct);local  val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}  val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}  val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}infun field_isolate_conv phi ctxt vs ct = case term_of ct of  Const(@{const_name Orderings.less},_)\$a\$b =>   let val (ca,cb) = Thm.dest_binop ct       val T = ctyp_of_term ca       val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0       val nth = Conv.fconv_rule         (Conv.arg_conv (Conv.arg1_conv              (Semiring_Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th       val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))   in rth end| Const(@{const_name Orderings.less_eq},_)\$a\$b =>   let val (ca,cb) = Thm.dest_binop ct       val T = ctyp_of_term ca       val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0       val nth = Conv.fconv_rule         (Conv.arg_conv (Conv.arg1_conv              (Semiring_Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th       val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))   in rth end| Const(@{const_name HOL.eq},_)\$a\$b =>   let val (ca,cb) = Thm.dest_binop ct       val T = ctyp_of_term ca       val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0       val nth = Conv.fconv_rule         (Conv.arg_conv (Conv.arg1_conv              (Semiring_Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th       val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))   in rth end| @{term "Not"} \$(Const(@{const_name HOL.eq},_)\$a\$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct| _ => Thm.reflexive ctend;fun classfield_whatis phi = let  fun h x t =   case term_of t of     Const(@{const_name HOL.eq}, _)\$y\$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq                            else Ferrante_Rackoff_Data.Nox   | @{term "Not"}\$(Const(@{const_name HOL.eq}, _)\$y\$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq                            else Ferrante_Rackoff_Data.Nox   | Const(@{const_name Orderings.less},_)\$y\$z =>       if term_of x aconv y then Ferrante_Rackoff_Data.Lt        else if term_of x aconv z then Ferrante_Rackoff_Data.Gt        else Ferrante_Rackoff_Data.Nox   | Const (@{const_name Orderings.less_eq},_)\$y\$z =>         if term_of x aconv y then Ferrante_Rackoff_Data.Le         else if term_of x aconv z then Ferrante_Rackoff_Data.Ge         else Ferrante_Rackoff_Data.Nox   | _ => Ferrante_Rackoff_Data.Nox in h end;fun class_field_ss phi =   HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])   |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}]inFerrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}  {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}end*}(*lemma upper_bound_finite_set:  assumes fS: "finite S"  shows "∃(a::'a::linorder). ∀x ∈ S. f x ≤ a"proof(induct rule: finite_induct[OF fS])  case 1 thus ?case by simpnext  case (2 x F)  from "2.hyps" obtain a where a:"∀x ∈F. f x ≤ a" by blast  let ?a = "max a (f x)"  have m: "a ≤ ?a" "f x ≤ ?a" by simp_all  {fix y assume y: "y ∈ insert x F"    {assume "y = x" hence "f y ≤ ?a" using m by simp}    moreover    {assume yF: "y∈ F" from a[rule_format, OF yF] m have "f y ≤ ?a" by (simp add: max_def)}    ultimately have "f y ≤ ?a" using y by blast}  then show ?case by blastqedlemma lower_bound_finite_set:  assumes fS: "finite S"  shows "∃(a::'a::linorder). ∀x ∈ S. f x ≥ a"proof(induct rule: finite_induct[OF fS])  case 1 thus ?case by simpnext  case (2 x F)  from "2.hyps" obtain a where a:"∀x ∈F. f x ≥ a" by blast  let ?a = "min a (f x)"  have m: "a ≥ ?a" "f x ≥ ?a" by simp_all  {fix y assume y: "y ∈ insert x F"    {assume "y = x" hence "f y ≥ ?a" using m by simp}    moreover    {assume yF: "y∈ F" from a[rule_format, OF yF] m have "f y ≥ ?a" by (simp add: min_def)}    ultimately have "f y ≥ ?a" using y by blast}  then show ?case by blastqedlemma bound_finite_set: assumes f: "finite S"  shows "∃a. ∀x ∈S. (f x:: 'a::linorder) ≤ a"proof-  let ?F = "f ` S"  from f have fF: "finite ?F" by simp  let ?a = "Max ?F"  {assume "S = {}" hence ?thesis by blast}  moreover  {assume Se: "S ≠ {}" hence Fe: "?F ≠ {}" by simp  {fix x assume x: "x ∈ S"    hence th0: "f x ∈ ?F" by simp    hence "f x ≤ ?a" using Max_ge[OF fF th0] ..}  hence ?thesis by blast}ultimately show ?thesis by blastqed*)end `