(*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy
    Author      : Amine Chaieb, TU Muenchen
*)

section ‹Dense linear order without endpoints
 and a quantifier elimination procedure in Ferrante and Rackoff style›

theory Dense_Linear_Order
imports Main
begin

ML_file ‹langford_data.ML›
ML_file ‹ferrante_rackoff_data.ML›

context linorder
begin

lemma 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 auto

lemma gather_start [no_atp]: "(∃x. P x) ⟷ (∃x. (∀y ∈ {}. y < x) ∧ (∀y∈ {}. x < y) ∧ P x)"
  by simp

text‹Theorems for ‹∃z. ∀x. x < z ⟶ (P x ⟷ P_-\∞)››
lemma minf_lt[no_atp]: "∃z. ∀x. x < z ⟶ (x < t ⟷ True)" 
  by auto
lemma 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 auto
lemma minf_neq[no_atp]: "∃z. ∀x. x < z ⟶ (x ≠ t ⟷ True)" 
  by auto
lemma minf_P[no_atp]: "∃z. ∀x. x < z ⟶ (P ⟷ P)" 
  by blast

text‹Theorems for ‹∃z. ∀x. x < z ⟶ (P x ⟷ P_+\∞)››
lemma pinf_gt[no_atp]:  "∃z. ∀x. z < x ⟶ (t < x ⟷ True)" 
  by auto
lemma 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 auto
lemma pinf_neq[no_atp]: "∃z. ∀x. z < x ⟶ (x ≠ t ⟷ True)" by auto
lemma pinf_P[no_atp]: "∃z. ∀x. z < x ⟶ (P ⟷ P)" by blast

lemma nmi_lt[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ x < t ⟶ (∃u∈ U. u ≤ x)" by auto
lemma 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 auto
lemma  nmi_ge[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ t≤ x ⟶ (∃u∈ U. u ≤ x)" by auto
lemma  nmi_eq[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ x = t ⟶ (∃u∈ U. u ≤ x)" by auto
lemma  nmi_neq[no_atp]: "t ∈ U ==>∀x. ¬True ∧ x ≠ t ⟶ (∃u∈ U. u ≤ x)" by auto
lemma  nmi_P[no_atp]: "∀x. ~P ∧ P ⟶ (∃u∈ U. u ≤ x)" by auto
lemma  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 auto
lemma  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 auto

lemma  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 auto
lemma  npi_le[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ x ≤ t ⟶ (∃u∈ U. x ≤ u)" by auto
lemma  npi_ge[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ t ≤ x ⟶ (∃u∈ U. x ≤ u)" by auto
lemma  npi_eq[no_atp]: "t ∈ U ==> ∀x. ¬False ∧ x = t ⟶ (∃u∈ U. x ≤ u)" by auto
lemma  npi_neq[no_atp]: "t ∈ U ==> ∀x. ¬True ∧ x ≠ t ⟶ (∃u∈ U. x ≤ u )" by auto
lemma  npi_P[no_atp]: "∀x. ~P ∧ P ⟶ (∃u∈ U. x ≤ u)" by auto
lemma  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 auto
lemma  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 auto

lemma 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)"
  by (metis antisym_conv3 order.strict_trans)

lemma 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)"
  by (metis antisym_conv3 order.strict_trans)

lemma 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)"
  by (metis local.less_le_trans local.less_trans local.not_less)

lemma 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)"
  by (metis local.le_less_trans local.nle_le not_le)

lemma 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 auto

lemma 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 auto

lemma 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 auto

lemma 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 blast

lemma 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 blast

lemma 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 auto

lemma 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
  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"
    using Mxne fMx fxM local.linear xMne by auto
  from ainS binS noy ax xb px show ?thesis
    by blast
qed

lemma 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)"
  using finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
  by (metis local.neq_le_trans)

end


section ‹The classical QE after Langford for dense linear orders›

context unbounded_dense_linorder
begin

lemma 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 
  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
  have "l < u" if l: "l ∈ L" and u: "u ∈ U" for l u
    using local.dual_order.strict_trans that(1) u xL xU by blast
  then show "∀l∈L. ∀u∈U. l < u" by blast
next
  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 auto
qed

lemma dlo_qe_noub[no_atp]:
  assumes ne: "L ≠ {}"
    and fL: "finite L"
  shows "(∃x. (∀y ∈ L. y < x) ∧ (∀y ∈ {}. x < y)) = True"
  using fL local.Max_less_iff local.gt_ex by fastforce

lemma dlo_qe_nolb[no_atp]:
  assumes ne: "U ≠ {}"
    and fU: "finite U"
  shows "(∃x. (∀y ∈ {}. y < x) ∧ (∀y ∈ U. x < y)) = True"
proof -
  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)
  then show ?thesis
    by blast
qed

lemma exists_neq[no_atp]: "∃(x::'a). x ≠ t" "∃(x::'a). t ≠ x"
  using gt_ex[of t] by auto

lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq
  le_less neq_iff linear less_not_permute

lemma axiom[no_atp]: "class.unbounded_dense_linorder (≤) (<)"
  by (rule unbounded_dense_linorder_axioms)

lemma atoms[no_atp]:
  shows "TERM (less :: 'a → _)"
    and "TERM (less_eq :: 'a → _)"
    and "TERM ((=) :: '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_distribR

lemmas weak_dnf_simps[no_atp] = simp_thms dnf

lemma 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 blast

lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib

ML_file ‹langford.ML›
method_setup dlo = ‹
 Scan.succeed (SIMPLE_METHOD' o Langford.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 = linorder
begin

notation
  less_eq  (‹'(⊑')›) and
  less_eq  (‹(_/ ⊑ _)› [51, 51] 50) and
  less  (‹'(⊏')›) and
  less  (‹(_/ ⊏ _)›  [51, 51] 50)

end

locale linorder_no_ub = linorder_stupid_syntax +
  assumes gt_ex: "∃y. less x y"
begin

lemma ge_ex[no_atp]: "∃y. x ⊑ y"
  using gt_ex by auto

text ‹Theorems for ‹∃z. ∀x. z ⊏ x ⟶ (P x ⟷ P_+\∞)››
lemma pinf_conj[no_atp]:
  assumes ex1: "∃z1. ∀x. z1 ⊏ x ⟶ (P1 x ⟷ P1')"
    and ex2: "∃z2. ∀x. z2 ⊏ x ⟶ (P2 x ⟷ P2')"
  shows "∃z. ∀x. z ⊏ x ⟶ ((P1 x ∧ P2 x) ⟷ (P1' ∧ P2'))"
  by (metis ex1 ex2 local.max_less_iff_conj)

lemma pinf_disj[no_atp]:
  assumes ex1: "∃z1. ∀x. z1 ⊏ x ⟶ (P1 x ⟷ P1')"
    and ex2: "∃z2. ∀x. z2 ⊏ x ⟶ (P2 x ⟷ P2')"
  shows "∃z. ∀x. z ⊏ x ⟶ ((P1 x ∨ P2 x) ⟷ (P1' ∨ P2'))"
  by (metis ex1 ex2 local.max.strict_boundedE)

lemma pinf_ex[no_atp]:
  assumes ex: "∃z. ∀x. z ⊏ x ⟶ (P x ⟷ P1)"
    and p1: P1
  shows "∃x. P x"
  using ex local.gt_ex p1 by auto

end

text ‹Linear order without upper bounds›

locale linorder_no_lb = linorder_stupid_syntax +
  assumes lt_ex: "∃y. less y x"
begin

lemma le_ex[no_atp]: "∃y. y ⊑ x"
  using lt_ex by auto


text ‹Theorems for ‹∃z. ∀x. x ⊏ z ⟶ (P x ⟷ P_-\∞)››
lemma minf_conj[no_atp]:
  assumes ex1: "∃z1. ∀x. x ⊏ z1 ⟶ (P1 x ⟷ P1')"
    and ex2: "∃z2. ∀x. x ⊏ z2 ⟶ (P2 x ⟷ P2')"
  shows "∃z. ∀x. x ⊏ z ⟶ ((P1 x ∧ P2 x) ⟷ (P1' ∧ P2'))"
  by (metis ex1 ex2 local.min_less_iff_conj)

lemma minf_disj[no_atp]:
  assumes ex1: "∃z1. ∀x. x ⊏ z1 ⟶ (P1 x ⟷ P1')"
    and ex2: "∃z2. ∀x. x ⊏ z2 ⟶ (P2 x ⟷ P2')"
  shows "∃z. ∀x. x ⊏ z ⟶ ((P1 x ∨ P2 x) ⟷ (P1' ∨ P2'))"
  by (metis ex1 ex2 local.min_less_iff_conj)

lemma minf_ex[no_atp]:
  assumes ex: "∃z. ∀x. x ⊏ z ⟶ (P x ⟷ P1)"
    and p1: P1
  shows "∃x. P x"
  using ex local.lt_ex p1 by auto

end


locale 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"
begin

sublocale dlo: unbounded_dense_linorder
proof (unfold_locales, goal_cases)
  case (1 x y)
  then show ?case
    using between_less [of x y] by auto
next
  case 2
  then show ?case by (rule lt_ex)
next
  case 3
  then show ?case by (rule gt_ex)
qed

lemma rinf_U[no_atp]:
  assumes fU: "finite U"
    and lin_dense: "∀x l u. (∀t. l ⊏ t ∧ t⊏ u ⟶ t ∉ U) ∧ l⊏ x ∧ x ⊏ u ∧ P x
      ⟶ (∀y. l ⊏ y ∧ y ⊏ u ⟶ P y )"
    and nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x ⟶ (∃u∈ U. ∃u' ∈ U. u ⊑ x ∧ x ⊑ 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 
  obtain u u' where uU: "u∈ U" and uU': "u' ∈ U" and ux: "u ⊑ x" and xu': "x ⊑ 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 ⊑ t"
    using Une fU by auto
  have Mu: "∀t∈ U. t ⊑ ?u"
    using Une fU by auto
  have th: "?l ⊑ u"
    using uU Une lM by auto
  from order_trans[OF th ux] have lx: "?l ⊑ x" .
  have th: "u' ⊑ ?u"
    using uU' Une Mu by simp
  from order_trans[OF xu' th] have xu: "x ⊑ ?u" .
  from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
  consider u where "u ∈ U" "P u" |
    t1 t2 where "t1 ∈ U" "t2 ∈ U" "∀y. t1 ⊏ y ∧ y ⊏ t2 ⟶ y ∉ U" "t1 ⊏ x" "x ⊏ t2" "P x"
    by blast
  then show ?thesis
  proof cases
    case 1 then show ?thesis
      by (metis between_same)
  next
    case 2
    then have t1t2: "t1 ⊏ t2"
      by order 
    let ?u = "between t1 t2"
    from between_less t1t2 have t1lu: "t1 ⊏ ?u" and ut2: "?u ⊏ t2" by auto
    then show ?thesis
      using "2" lin_dense px by blast
  qed
qed

theorem fr_eq[no_atp]:
  assumes fU: "finite U"
    and lin_dense: "∀x l u. (∀t. l ⊏ t ∧ t⊏ u ⟶ t ∉ U) ∧ l⊏ x ∧ x ⊏ u ∧ P x
     ⟶ (∀y. l ⊏ y ∧ y ⊏ u ⟶ P y )"
    and nmibnd: "∀x. ¬ MP ∧ P x ⟶ (∃u∈ U. u ⊑ x)"
    and npibnd: "∀x. ¬PP ∧ P x ⟶ (∃u∈ U. x ⊑ u)"
    and mi: "∃z. ∀x. x ⊏ z ⟶ (P x = MP)"  and pi: "∃z. ∀x. z ⊏ x ⟶ (P x = PP)"
  shows "(∃x. P x) = (MP ∨ PP ∨ (∃u ∈ U. ∃u'∈ U. P (between u u')))"
         (is"?E = ?D")
proof 
  show ?D if px: ?E
  proof -
    consider "MP ∨ PP" | "¬ MP" "¬ PP" by blast
    then show ?thesis
    proof cases
      case 1
      then show ?thesis by blast
    next
      case 2
      from npmibnd[OF nmibnd npibnd]
      have nmpiU: "∀x. ¬ MP ∧ ¬PP ∧ P x ⟶ (∃u∈ U. ∃u' ∈ U. u ⊑ x ∧ x ⊑ u')" .
      from rinf_U[OF fU lin_dense nmpiU ‹¬ MP› ‹¬ PP› px] show ?thesis
        by blast
    qed
  qed
  show ?E if ?D
    using local.gt_ex local.lt_ex mi pi that by blast
qed

lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P

lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
lemmas 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_P

lemma 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 ((=) :: '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 ‹
 
 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‹(⊏)›, term‹(⊑)›]
 fun h x t =
 case Thm.term_of t of
 🚫‹HOL.eq _ for y z› =>
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
 else Ferrante_Rackoff_Data.Nox
 | 🚫‹Not for Const‹HOL.eq _ for y z›› =>
 if Thm.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 Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt
 else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt
 else Ferrante_Rackoff_Data.Nox
 else if Term.could_unify (b, le) then
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le
 else if Thm.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 ctxt =
 simpset_of (put_simpset HOL_ss ctxt |> Simplifier.add_simps (simps phi))
 
 Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"}
 {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
 
 ›

end

ML_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:
  fixes c :: "'a::linordered_field"
  assumes "c < 0"
  shows "c * x < 0 ⟷ x > 0"
  by (metis assms mult_less_0_iff mult_neg_neg zero_less_mult_pos)

lemma pos_prod_lt:
  fixes c :: "'a::linordered_field"
  assumes "c > 0"
  shows "c * x < 0 ⟷ x < 0"
  by (meson assms mult_less_0_iff order_less_imp_not_less)

lemma neg_prod_sum_lt:
  fixes c :: "'a::linordered_field"
  assumes "c < 0"
  shows "c * x + t < 0 ⟷ x > (- 1 / c) * t"
  using assms by (auto simp add: mult.commute divide_simps)

lemma pos_prod_sum_lt:
  fixes c :: "'a::linordered_field"
  assumes "c > 0"
  shows "c * x + t < 0 ⟷ x < (- 1 / c) * t"
  using assms by (auto simp add: mult.commute divide_simps)

lemma sum_lt:
  fixes x :: "'a::ordered_ab_group_add"
  shows "x + t < 0 ⟷ x < - t"
  using less_diff_eq[where a= x and b=t and c=0] by simp

lemma neg_prod_le:
  fixes c :: "'a::linordered_field"
  assumes "c < 0"
  shows "c * x ≤ 0 ⟷ x ≥ 0"
  using assms linorder_not_less mult_le_0_iff by auto

lemma pos_prod_le:
  fixes c :: "'a::linordered_field"
  assumes "c > 0"
  shows "c * x ≤ 0 ⟷ x ≤ 0"
  using assms linorder_not_less mult_le_0_iff by auto

lemma neg_prod_sum_le:
  fixes c :: "'a::linordered_field"
  assumes "c < 0"
  shows "c * x + t ≤ 0 ⟷ x ≥ (- 1 / c) * t"
  using assms by (auto simp add: mult.commute divide_simps)

lemma pos_prod_sum_le:
  fixes c :: "'a::linordered_field"
  assumes "c > 0"
  shows "c * x + t ≤ 0 ⟷ x ≤ (- 1 / c) * t"
  using assms by (auto simp add: mult.commute divide_simps)

lemma sum_le:
  fixes x :: "'a::ordered_ab_group_add"
  shows "x + t ≤ 0 ⟷ x ≤ - t"
  using le_diff_eq[where a= x and b=t and c=0] by simp

lemma nz_prod_eq:
  fixes c :: "'a::linordered_field"
  assumes "c ≠ 0"
  shows "c * x = 0 ⟷ x = 0"
  using assms by simp

lemma nz_prod_sum_eq:
  fixes c :: "'a::linordered_field"
  assumes "c ≠ 0"
  shows "c * x + t = 0 ⟷ x = (- 1/c) * t"
  using assms by (auto simp add: mult.commute divide_simps)

lemma sum_eq:
  fixes x :: "'a::ordered_ab_group_add"
  shows "x + t = 0 ⟷ x = - t"
  using eq_diff_eq[where a= x and b=t and c=0] by simp

interpretation class_dense_linordered_field: constr_dense_linorder
  "(≤)" "(<)" "λx y. 1/2 * ((x::'a::linordered_field) + y)"
  by unfold_locales (auto simp add: gt_ex lt_ex)

declaration ‹
 
 fun earlier [] _ = 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 earlier_ord vs (x, y) =
 if x aconvc y then EQUAL
 else if earlier vs (x, y) then LESS
 else GREATER;

  dest_frac ct =
 case Thm.term_of ct of
 🚫‹Rings.divide _ for a b› =>
 Rat.make (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
 | 🚫‹inverse _ for a› => Rat.make(1, HOLogic.dest_number a |> snd)
 | t => Rat.of_int (snd (HOLogic.dest_number t))

  whatis x ct = case Thm.term_of ct of
 🚫‹plus _ for 🚫‹times _ for _ y› _› =>
 if y aconv Thm.term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
 else ("Nox",[])
  🚫‹plus _ for y _› =>
 if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct])
 else ("Nox",[])
  🚫‹times _ for _ y› =>
 if y aconv Thm.term_of x then ("c*x",[Thm.dest_arg1 ct])
 else ("Nox",[])
  t => if t aconv Thm.term_of x then ("x",[]) else ("Nox",[]);

 
  sum_lt = mk_meta_eq @{thm sum_lt}
  sum_le = mk_meta_eq @{thm sum_le}
  sum_eq = mk_meta_eq @{thm sum_eq}
  neg_prod_sum_lt = mk_meta_eq @{thm neg_prod_sum_lt}
  pos_prod_sum_lt = mk_meta_eq @{thm pos_prod_sum_lt}
  neg_prod_sum_le = mk_meta_eq @{thm neg_prod_sum_le}
  pos_prod_sum_le = mk_meta_eq @{thm pos_prod_sum_le}
  neg_prod_lt = mk_meta_eq @{thm neg_prod_lt}
  pos_prod_lt = mk_meta_eq @{thm pos_prod_lt}
  neg_prod_le = mk_meta_eq @{thm neg_prod_le}
  pos_prod_le = mk_meta_eq @{thm pos_prod_le}
  nz_prod_sum_eq = mk_meta_eq @{thm nz_prod_sum_eq}
  nz_prod_eq = mk_meta_eq @{thm nz_prod_eq}
 
  xnormalize_conv ctxt [] ct = Thm.reflexive ct
 | xnormalize_conv ctxt (vs as (x::_)) ct =
 case Thm.term_of ct of
 🚫‹less _ for _ 🚫‹zero_class.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 < @0
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x,t])
 (if neg then neg_prod_sum_lt else pos_prod_sum_lt)) cth
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
 in rth end
 | ("x+t",[t]) =>
 let
 val T = Thm.ctyp_of_cterm x
 val th = Thm.instantiate' [SOME T] [SOME x, SOME t] sum_lt
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord 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 < @0
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
 (if neg then neg_prod_lt else pos_prod_lt)) cth
 val rth = th
 in rth end
 | _ => Thm.reflexive ct)


  🚫‹less_eq _ for _ 🚫‹zero_class.zero _›› =>
 (case whatis x (Thm.dest_arg1 ct) of
 ("c*x+t",[c,t]) =>
 let
 val T = Thm.typ_of_cterm x
 val cT = Thm.ctyp_of_cterm x
 val cr = dest_frac c
 val clt = Thm.cterm_of ctxt Const‹less T›
 val cz = Thm.dest_arg ct
 val neg = cr < @0
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (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 (Thm.instantiate' [SOME cT] (map SOME [c,x,t])
 (if neg then neg_prod_sum_le else pos_prod_sum_le)) cth
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
 in rth end
 | ("x+t",[t]) =>
 let
 val T = Thm.ctyp_of_cterm x
 val th = Thm.instantiate' [SOME T] [SOME x, SOME t] sum_le
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
 in rth end
 | ("c*x",[c]) =>
 let
 val T = Thm.typ_of_cterm x
 val cT = Thm.ctyp_of_cterm x
 val cr = dest_frac c
 val clt = Thm.cterm_of ctxt Const‹less T›
 val cz = Thm.dest_arg ct
 val neg = cr < @0
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
 (if neg then neg_prod_le else pos_prod_le)) cth
 val rth = th
 in rth end
 | _ => Thm.reflexive ct)

  🚫‹HOL.eq _ for _ 🚫‹zero_class.zero _›› =>
 (case whatis x (Thm.dest_arg1 ct) of
 ("c*x+t",[c,t]) =>
 let
 val T = Thm.ctyp_of_cterm x
 val cr = dest_frac c
 val ceq = Thm.dest_fun2 ct
 val cz = Thm.dest_arg ct
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (Thm.apply 🚫‹Not› (Thm.apply (Thm.apply ceq c) cz)))
 val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
 val th = Thm.implies_elim
 (Thm.instantiate' [SOME T] (map SOME [c,x,t]) nz_prod_sum_eq) cth
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
 in rth end
 | ("x+t",[t]) =>
 let
 val T = Thm.ctyp_of_cterm x
 val th = Thm.instantiate' [SOME T] [SOME x, SOME t] sum_eq
 val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
 (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
 in rth end
 | ("c*x",[c]) =>
 let
 val T = Thm.ctyp_of_cterm x
 val cr = dest_frac c
 val ceq = Thm.dest_fun2 ct
 val cz = Thm.dest_arg ct
 val cthp = Simplifier.rewrite ctxt
 (HOLogic.mk_judgment
 (Thm.apply 🚫‹Not› (Thm.apply (Thm.apply ceq c) cz)))
 val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
 val rth = Thm.implies_elim
 (Thm.instantiate' [SOME T] (map SOME [c,x]) nz_prod_eq) cth
 in rth end
 | _ => Thm.reflexive ct);
 

 
 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"}
 val ss = simpset_of context
 
  field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of
 🚫‹less _ for a b› =>
 let val (ca,cb) = Thm.dest_binop ct
 val T = Thm.ctyp_of_cterm ca
 val th = Thm.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 (put_simpset ss ctxt) (earlier_ord vs)))) th
 val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
 in rth end
  🚫‹less_eq _ for a b› =>
 let val (ca,cb) = Thm.dest_binop ct
 val T = Thm.ctyp_of_cterm ca
 val th = Thm.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 (put_simpset ss ctxt) (earlier_ord vs)))) th
 val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
 in rth end

  🚫‹HOL.eq _ for a b› =>
 let val (ca,cb) = Thm.dest_binop ct
 val T = Thm.ctyp_of_cterm ca
 val th = Thm.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 (put_simpset ss ctxt) (earlier_ord vs)))) th
 val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
 in rth end
  🚫‹Not for 🚫‹HOL.eq _ for a b›› => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
  _ => Thm.reflexive ct
 ;

  classfield_whatis phi =
 let
 fun h x t =
 case Thm.term_of t of
 🚫‹HOL.eq _ for y z› =>
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
 else Ferrante_Rackoff_Data.Nox
 | 🚫‹Not for 🚫‹HOL.eq _ for y z›› =>
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
 else Ferrante_Rackoff_Data.Nox
 | 🚫‹less _ for y z› =>
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Lt
 else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Gt
 else Ferrante_Rackoff_Data.Nox
 | 🚫‹less_eq _ for y z› =>
 if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Le
 else if Thm.term_of x aconv z then Ferrante_Rackoff_Data.Ge
 else Ferrante_Rackoff_Data.Nox
 | _ => Ferrante_Rackoff_Data.Nox
 in h end;
  class_field_ss phi ctxt =
 simpset_of (put_simpset HOL_basic_ss ctxt
 |> Simplifier.add_simps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
 |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])

 
 .funs @{thm "class_dense_linordered_field.ferrack_axiom"}
 {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
 
 ›

end