# Theory HOL-Decision_Procs.Dense_Linear_Order

```(*  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)"

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)"

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)"

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)"
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)"
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
have False if H: "t < y"
proof -
from less_trans[OF lx px] less_trans[OF H yu] have "l < t ∧ t < u"
by simp
with tU noU show ?thesis
by auto
qed
then have "¬ t < y"
by auto
then have "y ≤ t"
then show "y < t"
using tny by (simp add: less_le)
qed

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)"
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
have False if H: "y < t"
proof -
from less_trans[OF ly H] less_trans[OF px xu] have "l < t ∧ t < u"
by simp
with tU noU show ?thesis
by auto
qed
then have "¬ y < t"
by auto
then have "t ≤ y"
then show "t < y"
using tny by (simp add: less_le)
qed

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)"
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
have False if H: "t < y"
proof -
from less_le_trans[OF lx px] less_trans[OF H yu]
have "l < t ∧ t < u" by simp
with tU noU show ?thesis by auto
qed
then have "¬ t < y" by auto
then show "y ≤ t" by (simp add: not_less)
qed

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)"
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
have False if H: "y < t"
proof -
from less_trans[OF ly H] le_less_trans[OF px xu]
have "l < t ∧ t < u" by simp
with tU noU show ?thesis by auto
qed
then have "¬ y < t" by auto
then show "t ≤ y" by (simp add: not_less)
qed

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"
proof clarsimp
fix y
assume ay: "?a < y" and yb: "y < ?b" and yS: "y ∈ S"
from yS have "y ∈ ?Mx ∨ y ∈ ?xM"
then show False
proof
assume "y ∈ ?Mx"
then have "y ≤ ?a"
using Mxne fMx by auto
with ay show ?thesis
next
assume "y ∈ ?xM"
then have "?b ≤ y"
using xMne fxM by auto
with yb show ?thesis
qed
qed
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)"
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)"
then show ?thesis
using px as bs noS by blast
qed

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 (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
have "l < u" if l: "l ∈ L" and u: "u ∈ U" for l u
proof -
have "l < x" using xL l by blast
also have "x < u" using xU u by blast
finally show ?thesis .
qed
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"
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)
then show "∃x. ∀y∈L. y < x"
by blast
qed

lemma dlo_qe_nolb[no_atp]:
assumes ne: "U ≠ {}"
and fU: "finite U"
shows "(∃x. (∀y ∈ {}. y < x) ∧ (∀y ∈ U. x < y)) ≡ True"
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 "∃x. ∀y∈U. x < y"
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'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "∀x. z1 ⊏ x ⟶ (P1 x ⟷ P1')"
and z2: "∀x. z2 ⊏ x ⟶ (P2 x ⟷ P2')"
by blast
from gt_ex obtain z where z:"ord.max less_eq z1 z2 ⊏ z"
by blast
from z have zz1: "z1 ⊏ z" and zz2: "z2 ⊏ z"
by simp_all
have "(P1 x ∧ P2 x) ⟷ (P1' ∧ P2')" if H: "z ⊏ x" for x
using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto
then show ?thesis
by blast
qed

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'))"
proof-
from ex1 ex2 obtain z1 and z2
where z1: "∀x. z1 ⊏ x ⟶ (P1 x ⟷ P1')"
and z2: "∀x. z2 ⊏ x ⟶ (P2 x ⟷ P2')"
by blast
from gt_ex obtain z where z: "ord.max less_eq z1 z2 ⊏ z"
by blast
from z have zz1: "z1 ⊏ z" and zz2: "z2 ⊏ z"
by simp_all
have "(P1 x ∨ P2 x) ⟷ (P1' ∨ P2')" if H: "z ⊏ x" for x
using less_trans[OF zz1 H] less_trans[OF zz2 H] z1 zz1 z2 zz2 by auto
then show ?thesis
by blast
qed

lemma pinf_ex[no_atp]:
assumes ex: "∃z. ∀x. z ⊏ x ⟶ (P x ⟷ P1)"
and p1: P1
shows "∃x. P x"
proof -
from ex obtain z where z: "∀x. z ⊏ x ⟶ (P x ⟷ P1)"
by blast
from gt_ex obtain x where x: "z ⊏ x"
by blast
from z x p1 show ?thesis
by blast
qed

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'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "∀x. x ⊏ z1 ⟶ (P1 x ⟷ P1')"
and z2: "∀x. x ⊏ z2 ⟶ (P2 x ⟷ P2')"
by blast
from lt_ex obtain z where z: "z ⊏ ord.min less_eq z1 z2"
by blast
from z have zz1: "z ⊏ z1" and zz2: "z ⊏ z2"
by simp_all
have "(P1 x ∧ P2 x) ⟷ (P1' ∧ P2')" if H: "x ⊏ z" for x
using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto
then show ?thesis
by blast
qed

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'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "∀x. x ⊏ z1 ⟶ (P1 x ⟷ P1')"
and z2: "∀x. x ⊏ z2 ⟶ (P2 x ⟷ P2')"
by blast
from lt_ex obtain z where z: "z ⊏ ord.min less_eq z1 z2"
by blast
from z have zz1: "z ⊏ z1" and zz2: "z ⊏ z2"
by simp_all
have "(P1 x ∨ P2 x) ⟷ (P1' ∨ P2')" if H: "x ⊏ z" for x
using less_trans[OF H zz1] less_trans[OF H zz2] z1 zz1 z2 zz2 by auto
then show ?thesis
by blast
qed

lemma minf_ex[no_atp]:
assumes ex: "∃z. ∀x. x ⊏ z ⟶ (P x ⟷ P1)"
and p1: P1
shows "∃x. P x"
proof -
from ex obtain z where z: "∀x. x ⊏ z ⟶ (P x ⟷ P1)"
by blast
from lt_ex obtain x where x: "x ⊏ z"
by blast
from z x p1 show ?thesis
by blast
qed

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 have "∃u∈ U. ∃u' ∈ U. u ⊑ x ∧ x ⊑ u'"
by auto
then obtain u and 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 u: 1
have "between u u = u" by (simp add: between_same)
with u have "P (between u u)" by simp
with u show ?thesis by blast
next
case 2
note t1M = ‹t1 ∈ U› and t2M = ‹t2∈ U›
and noM = ‹∀y. t1 ⊏ y ∧ y ⊏ t2 ⟶ y ∉ U›
and t1x = ‹t1 ⊏ x› and xt2 = ‹x ⊏ t2›
and px = ‹P x›
from less_trans[OF t1x xt2] have t1t2: "t1 ⊏ t2" .
let ?u = "between t1 t2"
from between_less t1t2 have t1lu: "t1 ⊏ ?u" and ut2: "?u ⊏ t2" by auto
from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
with t1M t2M show ?thesis 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 "_ ≡ (_ ∨ _ ∨ ?F)" is "?E ≡ ?D")
proof -
have "?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
proof -
from that consider MP | PP | ?F by blast
then show ?thesis
proof cases
case 1
from minf_ex[OF mi this] show ?thesis .
next
case 2
from pinf_ex[OF pi this] show ?thesis .
next
case 3
then show ?thesis by blast
qed
qed
qed
then show "?E ≡ ?D" by simp
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 ‹
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>‹(⊏)›, \<^term>‹(⊑)›]
fun h x t =
case Thm.term_of t of
\<^Const_>‹HOL.eq _ for y z› =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| \<^Const_>‹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 addsimps (simps phi))
in
Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
{isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
end
›

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"
proof -
have "c * x < 0 ⟷ 0 / c < x"
by (simp only: neg_divide_less_eq[OF ‹c < 0›] algebra_simps)
also have "… ⟷ 0 < x" by simp
finally show "PROP ?thesis" by simp
qed

lemma pos_prod_lt:
fixes c :: "'a::linordered_field"
assumes "c > 0"
shows "c * x < 0 ≡ x < 0"
proof -
have "c * x < 0 ⟷ 0 /c > x"
by (simp only: pos_less_divide_eq[OF ‹c > 0›] algebra_simps)
also have "… ⟷ 0 > x" by simp
finally show "PROP ?thesis" by simp
qed

lemma neg_prod_sum_lt:
fixes c :: "'a::linordered_field"
assumes "c < 0"
shows "c * x + t < 0 ≡ x > (- 1 / c) * t"
proof -
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 ‹c < 0›] algebra_simps)
also have "… ⟷ (- 1 / c) * t < x" by simp
finally show "PROP ?thesis" by simp
qed

lemma pos_prod_sum_lt:
fixes c :: "'a::linordered_field"
assumes "c > 0"
shows "c * x + t < 0 ≡ x < (- 1 / c) * t"
proof -
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 ‹c > 0›] algebra_simps)
also have "… ⟷ (- 1 / c) * t > x" by simp
finally show "PROP ?thesis" by simp
qed

lemma sum_lt:
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"
proof -
have "c * x ≤ 0 ⟷ 0 / c ≤ x"
by (simp only: neg_divide_le_eq[OF ‹c < 0›] algebra_simps)
also have "… ⟷ 0 ≤ x" by simp
finally show "PROP ?thesis" by simp
qed

lemma pos_prod_le:
fixes c :: "'a::linordered_field"
assumes "c > 0"
shows "c * x ≤ 0 ≡ x ≤ 0"
proof -
have "c * x ≤ 0 ⟷ 0 / c ≥ x"
by (simp only: pos_le_divide_eq[OF ‹c > 0›] algebra_simps)
also have "… ⟷ 0 ≥ x" by simp
finally show "PROP ?thesis" by simp
qed

lemma neg_prod_sum_le:
fixes c :: "'a::linordered_field"
assumes "c < 0"
shows "c * x + t ≤ 0 ≡ x ≥ (- 1 / c) * t"
proof -
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 ‹c < 0›] algebra_simps)
also have "… ⟷ (- 1 / c) * t ≤ x" by simp
finally show "PROP ?thesis" by simp
qed

lemma pos_prod_sum_le:
fixes c :: "'a::linordered_field"
assumes "c > 0"
shows "c * x + t ≤ 0 ≡ x ≤ (- 1 / c) * t"
proof -
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 ‹c > 0›] algebra_simps)
also have "… ⟷ (- 1 / c) * t ≥ x" by simp
finally show "PROP ?thesis" by simp
qed

lemma sum_le:
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"
proof -
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 ‹c ≠ 0›] algebra_simps)
finally show "PROP ?thesis" by simp
qed

lemma sum_eq:
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 (dlo, dlo, auto)

declaration ‹
let
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;

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

fun whatis x ct = case Thm.term_of ct of
\<^Const_>‹plus _ for \<^Const_>‹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",[])
| \<^Const_>‹plus _ for y _› =>
if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct])
else ("Nox",[])
| \<^Const_>‹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",[]);

fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
| xnormalize_conv ctxt (vs as (x::_)) ct =
case Thm.term_of ct of
\<^Const_>‹less _ for _ \<^Const_>‹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
(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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm 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_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] @{thm "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
(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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm 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_>‹less_eq _ for _ \<^Const_>‹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
(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 (Thm.instantiate' [SOME cT] (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_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] @{thm "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
(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 (Thm.instantiate' [SOME (Thm.ctyp_of_cterm 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_>‹HOL.eq _ for _ \<^Const_>‹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
(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
(Thm.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_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] @{thm "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
(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
(Thm.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"}
val ss = simpset_of \<^context>
in
fun field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of
\<^Const_>‹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
| \<^Const_>‹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

| \<^Const_>‹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
| \<^Const_>‹Not for \<^Const_>‹HOL.eq _ for a b›› => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => Thm.reflexive ct
end;

fun classfield_whatis phi =
let
fun h x t =
case Thm.term_of t of
\<^Const_>‹HOL.eq _ for y z› =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| \<^Const_>‹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
| \<^Const_>‹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
| \<^Const_>‹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;
fun class_field_ss phi ctxt =
simpset_of (put_simpset HOL_basic_ss ctxt