(* Title : HOL/Decision_Procs/Dense_Linear_Order.thy
Author : Amine Chaieb, TU Muenchen
*)
section \<open>Dense linear order without endpoints
and a quantifier elimination procedure in Ferrante and Rackoff style\<close>
theory Dense_Linear_Order
imports Main
begin
ML_file \<open>langford_data.ML\<close>
ML_file \<open>ferrante_rackoff_data.ML\<close>
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) \
(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
"(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x \ P x) \
(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
"(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ x < u) \
(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
"(\x. (\y \ L. y < x) \ (\y \ U. x < y) \ l < x) \
(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> 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\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
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\<open>Theorems for \<open>\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
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) ;
\<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
\<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_disj[no_atp]: "\\x. \P1' \ P1 x \ (\u\ U. u \ x) ;
\<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
\<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists>u\<in> U. u \<le> 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)\
\<Longrightarrow> \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_disj[no_atp]: "\\x. \P1' \ P1 x \ (\u\ U. x \ u) ; \x. \P2' \ P2 x \ (\u\ U. x \ u)\
\<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma lin_dense_lt[no_atp]:
"t \ U \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x < t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> 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"
by (simp add: not_less)
then show "y < t"
using tny by (simp add: less_le)
qed
lemma lin_dense_gt[no_atp]:
"t \ U \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> 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"
by (auto simp add: not_less)
then show "t < y"
using tny by (simp add: less_le)
qed
lemma lin_dense_le[no_atp]:
"t \ U \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y \<le> 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 \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> t \<le> 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 \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x = t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y = t)"
by auto
lemma lin_dense_neq[no_atp]:
"t \ U \
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> x \<noteq> t \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> y \<noteq> 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
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ;
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> (P1 x \<and> P2 x)
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> 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
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P1 y) ;
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P2 x
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> (P1 x \<or> P2 x)
\<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
by blast
lemma npmibnd[no_atp]: "\\x. \ MP \ P x \ (\u\ U. u \ x); \x. \PP \ P x \ (\u\ U. x \ u)\
\<Longrightarrow> \<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<le> x \<and> x \<le> 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"
by (auto simp add: linear)
then show False
proof
assume "y \ ?Mx"
then have "y \ ?a"
using Mxne fMx by auto
with ay show ?thesis
by (simp add: not_le[symmetric])
next
assume "y \ ?xM"
then have "?b \ y"
using xMne fxM by auto
with yb show ?thesis
by (simp add: not_le[symmetric])
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)"
by (auto simp add: le_less)
then show ?thesis
using px as bs noS by blast
qed
end
section \<open>The classical QE after Langford for dense linear orders\<close>
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"
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)
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"
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)
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 \<open>langford.ML\<close>
method_setup dlo = \<open>
Scan.succeed (SIMPLE_METHOD' o Langford.dlo_tac)
\<close> "Langford's algorithm for quantifier elimination in dense linear orders"
section \<open>Contructive dense linear orders yield QE for linear arithmetic over ordered Fields\<close>
text \<open>Linear order without upper bounds\<close>
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 \<open>Theorems for \<open>\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)\<close>\<close>
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 \<open>Linear order without upper bounds\<close>
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 \<open>Theorems for \<open>\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)\<close>\<close>
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
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> 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 = \<open>t1 \<in> U\<close> and t2M = \<open>t2\<in> U\<close>
and noM = \<open>\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U\<close>
and t1x = \<open>t1 \<sqsubset> x\<close> and xt2 = \<open>x \<sqsubset> t2\<close>
and px = \<open>P x\<close>
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
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> 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 \<open>\<not> MP\<close> \<open>\<not> PP\<close> 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 \<open>
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>\<open>(\<sqsubset>)\<close>, \<^term>\<open>(\<sqsubseteq>)\<close>]
fun h x t =
case Thm.term_of t of
Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| \<^term>\<open>Not\<close>$(Const(\<^const_name>\<open>HOL.eq\<close>, _)$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
\<close>
end
ML_file \<open>ferrante_rackoff.ML\<close>
method_setup ferrack = \<open>
Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
\<close> "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
subsection \<open>Ferrante and Rackoff algorithm over ordered fields\<close>
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 \<open>c < 0\<close>] 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 \<open>c > 0\<close>] 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 \<open>c < 0\<close>] 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 \<open>c > 0\<close>] algebra_simps)
also have "\ \ (- 1 / c) * t > x" by simp
finally show "PROP ?thesis" by simp
qed
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"
proof -
have "c * x \ 0 \ 0 / c \ x"
by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] 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 \<open>c > 0\<close>] 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 \<open>c < 0\<close>] 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 \<open>c > 0\<close>] algebra_simps)
also have "\ \ (- 1 / c) * t \ x" by simp
finally show "PROP ?thesis" by simp
qed
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"
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 \<open>c \<noteq> 0\<close>] algebra_simps)
finally show "PROP ?thesis" by simp
qed
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
"(\)" "(<)" "\x y. 1/2 * ((x::'a::linordered_field) + y)"
by unfold_locales (dlo, dlo, auto)
declaration \<open>
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 (\<^const_name>\<open>Rings.divide\<close>,_) $ a $ b=>
Rat.make (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
| Const(\<^const_name>\<open>inverse\<close>, _)$a => Rat.make(1, HOLogic.dest_number a |> snd)
| t => Rat.of_int (snd (HOLogic.dest_number t))
fun whatis x ct = case Thm.term_of ct of
Const(\<^const_name>\<open>Groups.plus\<close>, _)$(Const(\<^const_name>\<open>Groups.times\<close>,_)$_$y)$_ =>
if y aconv Thm.term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
else ("Nox",[])
| Const(\<^const_name>\<open>Groups.plus\<close>, _)$y$_ =>
if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct])
else ("Nox",[])
| Const(\<^const_name>\<open>Groups.times\<close>, _)$_$y =>
if y aconv Thm.term_of x then ("c*x",[Thm.dest_arg1 ct])
else ("Nox",[])
| t => if t aconv Thm.term_of x then ("x",[]) else ("Nox",[]);
fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
| xnormalize_conv ctxt (vs as (x::_)) ct =
case Thm.term_of ct of
Const(\<^const_name>\<open>Orderings.less\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr < @0
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("c*x",[c]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr < @0
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
(if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
val rth = th
in rth end
| _ => Thm.reflexive ct)
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = Thm.typ_of_cterm x
val cT = Thm.ctyp_of_cterm x
val cr = dest_frac c
val clt = Thm.cterm_of ctxt (Const (\<^const_name>\<open>ord_class.less\<close>, T --> T --> \<^typ>\<open>bool\<close>))
val cz = Thm.dest_arg ct
val neg = cr < @0
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME cT] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = Thm.typ_of_cterm x
val cT = Thm.ctyp_of_cterm x
val cr = dest_frac c
val clt = Thm.cterm_of ctxt (Const (\<^const_name>\<open>ord_class.less\<close>, T --> T --> \<^typ>\<open>bool\<close>))
val cz = Thm.dest_arg ct
val neg = cr < @0
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
(if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
val rth = th
in rth end
| _ => Thm.reflexive ct)
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$_$Const(\<^const_name>\<open>Groups.zero\<close>,_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = Thm.ctyp_of_cterm x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(Thm.apply \<^cterm>\<open>Not\<close> (Thm.apply (Thm.apply ceq c) cz)))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim
(Thm.instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier_ord vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = Thm.ctyp_of_cterm x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite ctxt
(Thm.apply \<^cterm>\<open>Trueprop\<close>
(Thm.apply \<^cterm>\<open>Not\<close> (Thm.apply (Thm.apply ceq c) cz)))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val rth = Thm.implies_elim
(Thm.instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
in rth end
| _ => Thm.reflexive ct);
local
val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
val ss = simpset_of \<^context>
in
fun field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of
Const(\<^const_name>\<open>Orderings.less\<close>,_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = Thm.ctyp_of_cterm ca
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const(\<^const_name>\<open>Orderings.less_eq\<close>,_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = Thm.ctyp_of_cterm ca
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier_ord vs)))) th
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const(\<^const_name>\<open>HOL.eq\<close>,_)$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
| \<^term>\<open>Not\<close> $(Const(\<^const_name>\<open>HOL.eq\<close>,_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => Thm.reflexive ct
end;
fun classfield_whatis phi =
let
fun h x t =
case Thm.term_of t of
Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| \<^term>\<open>Not\<close>$(Const(\<^const_name>\<open>HOL.eq\<close>, _)$y$z) =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
else Ferrante_Rackoff_Data.Nox
| Const(\<^const_name>\<open>Orderings.less\<close>,_)$y$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 (\<^const_name>\<open>Orderings.less_eq\<close>,_)$y$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
addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
|> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])
in
Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}
{isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
end
\<close>
end
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