(* 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 "langford_data.ML"
ML_file "ferrante_rackoff_data.ML"
context linorder
begin
lemma less_not_permute[no_atp]: "\<not> (x < y \<and> y < x)"
by (simp add: not_less linear)
lemma gather_simps[no_atp]:
"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow>
(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow>
(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow>
(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow>
(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))"
by auto
lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> 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]: "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow> (t < x \<longleftrightarrow> False)"
by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
lemma minf_le[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
lemma minf_ge[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
by (auto simp add: less_le not_less not_le)
lemma minf_eq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> 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]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
lemma pinf_lt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x < t \<longleftrightarrow> False)"
by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
lemma pinf_ge[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
lemma pinf_le[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
by (auto simp add: less_le not_less not_le)
lemma pinf_eq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
lemma pinf_neq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
lemma pinf_P[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
lemma nmi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)"
by (auto simp add: le_less)
lemma nmi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_neq[no_atp]: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_P[no_atp]: "\<forall>x. ~P \<and> P \<longrightarrow> (\<exists>u\<in> U. u \<le> x)" by auto
lemma nmi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. u \<le> 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]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. u \<le> 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 \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x < t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by (auto simp add: le_less)
lemma npi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x \<le> t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_neq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow> (\<exists>u\<in> U. x \<le> u )" by auto
lemma npi_P[no_atp]: "\<forall>x. ~P \<and> P \<longrightarrow> (\<exists>u\<in> U. x \<le> u)" by auto
lemma npi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk>
\<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]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk>
\<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 \<in> U \<Longrightarrow>
\<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 \<in> U"
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> 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 \<noteq> y" by auto
have False if H: "t < y"
proof -
from less_trans[OF lx px] less_trans[OF H yu] have "l < t \<and> t < u"
by simp
with tU noU show ?thesis
by auto
qed
then have "\<not> t < y"
by auto
then have "y \<le> 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 \<in> U \<Longrightarrow>
\<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 \<in> U"
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> 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 \<noteq> y" by auto
have False if H: "y < t"
proof -
from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u"
by simp
with tU noU show ?thesis
by auto
qed
then have "\<not> y < t"
by auto
then have "t \<le> 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 \<in> U \<Longrightarrow>
\<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 \<in> U"
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
and lx: "l < x"
and xu: "x < u"
and px: "x \<le> t"
and ly: "l < y"
and yu: "y < u"
from tU noU ly yu have tny: "t \<noteq> 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 \<and> t < u" by simp
with tU noU show ?thesis by auto
qed
then have "\<not> t < y" by auto
then show "y \<le> t" by (simp add: not_less)
qed
lemma lin_dense_ge[no_atp]:
"t \<in> U \<Longrightarrow>
\<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 \<in> U"
and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U"
and lx: "l < x"
and xu: "x < u"
and px: "t \<le> x"
and ly: "l < y"
and yu: "y < u"
from tU noU ly yu have tny: "t \<noteq> 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 \<and> t < u" by simp
with tU noU show ?thesis by auto
qed
then have "\<not> y < t" by auto
then show "t \<le> y" by (simp add: not_less)
qed
lemma lin_dense_eq[no_atp]:
"t \<in> U \<Longrightarrow>
\<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 \<in> U \<Longrightarrow>
\<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]:
"\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> P \<longrightarrow> (\<forall>y. l < y \<and> y < u \<longrightarrow> P)"
by auto
lemma lin_dense_conj[no_atp]:
"\<lbrakk>\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> 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]:
"\<lbrakk>\<forall>x l u. (\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> 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]: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. x \<le> u)\<rbrakk>
\<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 \<le> x"
and xu: "x \<le> u"
and linS: "l\<in> S"
and uinS: "u \<in> S"
and fS:"finite S"
and lS: "\<forall>x\<in> S. l \<le> x"
and Su: "\<forall>x\<in> S. x \<le> u"
shows "\<exists>a \<in> S. \<exists>b \<in> S. (\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a \<le> x \<and> x \<le> b \<and> P x"
proof -
let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
let ?xM = "{y. y\<in> S \<and> x \<le> y}"
let ?a = "Max ?Mx"
let ?b = "Min ?xM"
have MxS: "?Mx \<subseteq> S"
by blast
then have fMx: "finite ?Mx"
using fS finite_subset by auto
from lx linS have linMx: "l \<in> ?Mx"
by blast
then have Mxne: "?Mx \<noteq> {}"
by blast
have xMS: "?xM \<subseteq> S"
by blast
then have fxM: "finite ?xM"
using fS finite_subset by auto
from xu uinS have linxM: "u \<in> ?xM"
by blast
then have xMne: "?xM \<noteq> {}"
by blast
have ax: "?a \<le> x"
using Mxne fMx by auto
have xb: "x \<le> ?b"
using xMne fxM by auto
have "?a \<in> ?Mx"
using Max_in[OF fMx Mxne] by simp
then have ainS: "?a \<in> S"
using MxS by blast
have "?b \<in> ?xM"
using Min_in[OF fxM xMne] by simp
then have binS: "?b \<in> S"
using xMS by blast
have noy: "\<forall>y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
proof clarsimp
fix y
assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
from yS have "y \<in> ?Mx \<or> y \<in> ?xM"
by (auto simp add: linear)
then show False
proof
assume "y \<in> ?Mx"
then have "y \<le> ?a"
using Mxne fMx by auto
with ay show ?thesis
by (simp add: not_le[symmetric])
next
assume "y \<in> ?xM"
then have "?b \<le> 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 \<le> x"
and xu: "x \<le> u"
and linS: "l\<in> S"
and uinS: "u \<in> S"
and fS: "finite S"
and lS: "\<forall>x\<in> S. l \<le> x"
and Su: "\<forall>x\<in> S. x \<le> u"
shows "(\<exists>s\<in> S. P s) \<or> (\<exists>a \<in> S. \<exists>b \<in> S. (\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S) \<and> a < x \<and> x < b \<and> 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 \<in> S" and bs: "b \<in> S"
and noS: "\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
and axb: "a \<le> x \<and> x \<le> b \<and> P x"
by auto
from axb have "x = a \<or> x = b \<or> (a < x \<and> 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 \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
by (auto dest: dense)
lemma dlo_qe_bnds[no_atp]:
assumes ne: "L \<noteq> {}"
and neU: "U \<noteq> {}"
and fL: "finite L"
and fU: "finite U"
shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> (\<forall>l \<in> L. \<forall>u \<in> U. l < u)"
proof (simp only: atomize_eq, rule iffI)
assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y"
by blast
have "l < u" if l: "l \<in> L" and u: "u \<in> 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 "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
next
assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
let ?ML = "Max L"
let ?MU = "Min U"
from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML"
by auto
from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> 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 "\<forall>l \<in> L. l < w"
by auto
moreover from th4 th2' have "\<forall>u \<in> U. w < u"
by auto
ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
by auto
qed
lemma dlo_qe_noub[no_atp]:
assumes ne: "L \<noteq> {}"
and fL: "finite L"
shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> 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 "\<forall>x \<in> L. x \<le> Max L"
by simp
with M have "\<forall>x\<in>L. x < M"
by (auto intro: le_less_trans)
then show "\<exists>x. \<forall>y\<in>L. y < x"
by blast
qed
lemma dlo_qe_nolb[no_atp]:
assumes ne: "U \<noteq> {}"
and fU: "finite U"
shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> 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 "\<forall>x \<in> U. Min U \<le> x"
by simp
with M have "\<forall>x\<in>U. M < x"
by (auto intro: less_le_trans)
then show "\<exists>x. \<forall>y\<in>U. x < y"
by blast
qed
lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> 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 (op \<le>) (op <)"
by (rule unbounded_dense_linorder_axioms)
lemma atoms[no_atp]:
shows "TERM (less :: 'a \<Rightarrow> _)"
and "TERM (less_eq :: 'a \<Rightarrow> _)"
and "TERM (op = :: 'a \<Rightarrow> _)" .
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]:
"(\<not> (P \<and> Q)) \<longleftrightarrow> (\<not> P \<or> \<not> Q)"
"(\<not> (P \<or> Q)) \<longleftrightarrow> (\<not> P \<and> \<not> Q)"
"(P \<longrightarrow> Q) \<longleftrightarrow> (\<not> P \<or> Q)"
"(P \<longleftrightarrow> Q) \<longleftrightarrow> ((P \<and> Q) \<or> (\<not> P \<and> \<not> Q))"
"(\<not> \<not> P) \<longleftrightarrow> P"
by blast+
lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))"
by blast
lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib
ML_file "langford.ML"
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 ("op \<sqsubseteq>") and
less_eq ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
less ("op \<sqsubset>") and
less ("(_/ \<sqsubset> _)" [51, 51] 50)
end
locale linorder_no_ub = linorder_stupid_syntax +
assumes gt_ex: "\<exists>y. less x y"
begin
lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> 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: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
by blast
from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z"
by blast
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z"
by simp_all
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "z \<sqsubset> 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: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
shows "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
proof-
from ex1 ex2 obtain z1 and z2
where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
by blast
from gt_ex obtain z where z: "ord.max less_eq z1 z2 \<sqsubset> z"
by blast
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z"
by simp_all
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "z \<sqsubset> 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: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)"
and p1: P1
shows "\<exists>x. P x"
proof -
from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)"
by blast
from gt_ex obtain x where x: "z \<sqsubset> 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: "\<exists>y. less y x"
begin
lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> 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: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
by blast
from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2"
by blast
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2"
by simp_all
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" if H: "x \<sqsubset> 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: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
proof -
from ex1 ex2 obtain z1 and z2
where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
by blast
from lt_ex obtain z where z: "z \<sqsubset> ord.min less_eq z1 z2"
by blast
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2"
by simp_all
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" if H: "x \<sqsubset> 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: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
and p1: P1
shows "\<exists>x. P x"
proof -
from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
by blast
from lt_ex obtain x where x: "x \<sqsubset> 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 \<Longrightarrow> less x (between x y) \<and> 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: "\<forall>x l u. (\<forall>t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
and nmpiU: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u')"
and nmi: "\<not> MP" and npi: "\<not> PP" and ex: "\<exists>x. P x"
shows "\<exists>u\<in> U. \<exists>u' \<in> U. P (between u u')"
proof -
from ex obtain x where px: "P x"
by blast
from px nmi npi nmpiU have "\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'"
by auto
then obtain u and u' where uU: "u\<in> U" and uU': "u' \<in> U" and ux: "u \<sqsubseteq> x" and xu': "x \<sqsubseteq> u'"
by auto
from uU have Une: "U \<noteq> {}"
by auto
let ?l = "linorder.Min less_eq U"
let ?u = "linorder.Max less_eq U"
have linM: "?l \<in> U"
using fU Une by simp
have uinM: "?u \<in> U"
using fU Une by simp
have lM: "\<forall>t\<in> U. ?l \<sqsubseteq> t"
using Une fU by auto
have Mu: "\<forall>t\<in> U. t \<sqsubseteq> ?u"
using Une fU by auto
have th: "?l \<sqsubseteq> u"
using uU Une lM by auto
from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
have th: "u' \<sqsubseteq> ?u"
using uU' Une Mu by simp
from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
consider u where "u \<in> U" "P u" |
t1 t2 where "t1 \<in> U" "t2 \<in> U" "\<forall>y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" "t1 \<sqsubset> x" "x \<sqsubset> 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 \<sqsubset> t2" .
let ?u = "between t1 t2"
from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
with t1M t2M show ?thesis by blast
qed
qed
theorem fr_eq[no_atp]:
assumes fU: "finite U"
and lin_dense: "\<forall>x l u. (\<forall>t. l \<sqsubset> t \<and> t\<sqsubset> u \<longrightarrow> t \<notin> U) \<and> l\<sqsubset> x \<and> x \<sqsubset> u \<and> P x
\<longrightarrow> (\<forall>y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists>u\<in> U. u \<sqsubseteq> x)"
and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. x \<sqsubseteq> u)"
and mi: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x = MP)" and pi: "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x = PP)"
shows "(\<exists>x. P x) \<equiv> (MP \<or> PP \<or> (\<exists>u \<in> U. \<exists>u'\<in> U. P (between u u')))"
(is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
proof -
have "?E \<longleftrightarrow> ?D"
proof
show ?D if px: ?E
proof -
consider "MP \<or> PP" | "\<not> MP" "\<not> 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: "\<forall>x. \<not> MP \<and> \<not>PP \<and> P x \<longrightarrow> (\<exists>u\<in> U. \<exists>u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> 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 \<equiv> ?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 \<Rightarrow> _)"
and "TERM (less_eq :: 'a \<Rightarrow> _)"
and "TERM (op = :: 'a \<Rightarrow> _)" .
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 "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
fun h x t =
case Thm.term_of t of
Const(@{const_name HOL.eq}, _)$y$z =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
else Ferrante_Rackoff_Data.Nox
| b$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 "ferrante_rackoff.ML"
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 \<equiv> x > 0"
proof -
have "c * x < 0 \<longleftrightarrow> 0 / c < x"
by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> 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 \<equiv> x < 0"
proof -
have "c * x < 0 \<longleftrightarrow> 0 /c > x"
by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> 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 \<equiv> x > (- 1 / c) * t"
proof -
have "c * x + t < 0 \<longleftrightarrow> c * x < - t"
by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp
also have "\<dots> \<longleftrightarrow> - t / c < x"
by (simp only: neg_divide_less_eq[OF \<open>c < 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> (- 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 \<equiv> x < (- 1 / c) * t"
proof -
have "c * x + t < 0 \<longleftrightarrow> c * x < - t"
by (subst less_iff_diff_less_0 [of "c * x" "- t"]) simp
also have "\<dots> \<longleftrightarrow> - t / c > x"
by (simp only: pos_less_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> (- 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 \<equiv> 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 \<le> 0 \<equiv> x \<ge> 0"
proof -
have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<le> x"
by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> 0 \<le> 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 \<le> 0 \<equiv> x \<le> 0"
proof -
have "c * x \<le> 0 \<longleftrightarrow> 0 / c \<ge> x"
by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> 0 \<ge> 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 \<le> 0 \<equiv> x \<ge> (- 1 / c) * t"
proof -
have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> -t"
by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp
also have "\<dots> \<longleftrightarrow> - t / c \<le> x"
by (simp only: neg_divide_le_eq[OF \<open>c < 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<le> 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 \<le> 0 \<equiv> x \<le> (- 1 / c) * t"
proof -
have "c * x + t \<le> 0 \<longleftrightarrow> c * x \<le> - t"
by (subst le_iff_diff_le_0 [of "c*x" "-t"]) simp
also have "\<dots> \<longleftrightarrow> - t / c \<ge> x"
by (simp only: pos_le_divide_eq[OF \<open>c > 0\<close>] algebra_simps)
also have "\<dots> \<longleftrightarrow> (- 1 / c) * t \<ge> x" by simp
finally show "PROP ?thesis" by simp
qed
lemma sum_le:
fixes x :: "'a::ordered_ab_group_add"
shows "x + t \<le> 0 \<equiv> x \<le> - 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 \<noteq> 0"
shows "c * x = 0 \<equiv> x = 0"
using assms by simp
lemma nz_prod_sum_eq:
fixes c :: "'a::linordered_field"
assumes "c \<noteq> 0"
shows "c * x + t = 0 \<equiv> x = (- 1/c) * t"
proof -
have "c * x + t = 0 \<longleftrightarrow> c * x = - t"
by (subst eq_iff_diff_eq_0 [of "c*x" "-t"]) simp
also have "\<dots> \<longleftrightarrow> 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 \<equiv> 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
"op \<le>" "op <" "\<lambda>x y. 1/2 * ((x::'a::linordered_field) + y)"
by unfold_locales (dlo, dlo, auto)
declaration \<open>
let
fun earlier [] x y = false
| earlier (h::t) x y =
if h aconvc y then false else if h aconvc x then true else earlier t x y;
fun dest_frac ct =
case Thm.term_of ct of
Const (@{const_name Rings.divide},_) $ a $ b=>
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
| Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
| t => Rat.rat_of_int (snd (HOLogic.dest_number t))
fun whatis x ct = case Thm.term_of ct of
Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ =>
if y aconv Thm.term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
else ("Nox",[])
| Const(@{const_name Groups.plus}, _)$y$_ =>
if y aconv Thm.term_of x then ("x+t",[Thm.dest_arg ct])
else ("Nox",[])
| Const(@{const_name Groups.times}, _)$_$y =>
if y aconv Thm.term_of x then ("c*x",[Thm.dest_arg1 ct])
else ("Nox",[])
| t => if t aconv Thm.term_of x then ("x",[]) else ("Nox",[]);
fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
| xnormalize_conv ctxt (vs as (x::_)) ct =
case Thm.term_of ct of
Const(@{const_name Orderings.less},_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr < Rat.zero
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr < Rat.zero
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
(if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
val rth = th
in rth end
| _ => Thm.reflexive ct)
| Const(@{const_name Orderings.less_eq},_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = Thm.typ_of_cterm x
val cT = Thm.ctyp_of_cterm x
val cr = dest_frac c
val clt = Thm.cterm_of ctxt (Const (@{const_name ord_class.less}, T --> T --> @{typ bool}))
val cz = Thm.dest_arg ct
val neg = cr < Rat.zero
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME cT] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = Thm.typ_of_cterm x
val cT = Thm.ctyp_of_cterm x
val cr = dest_frac c
val clt = Thm.cterm_of ctxt (Const (@{const_name ord_class.less}, T --> T --> @{typ bool}))
val cz = Thm.dest_arg ct
val neg = cr < Rat.zero
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(if neg then Thm.apply (Thm.apply clt c) cz
else Thm.apply (Thm.apply clt cz) c))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim (Thm.instantiate' [SOME (Thm.ctyp_of_cterm x)] (map SOME [c,x])
(if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
val rth = th
in rth end
| _ => Thm.reflexive ct)
| Const(@{const_name HOL.eq},_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = Thm.ctyp_of_cterm x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val th = Thm.implies_elim
(Thm.instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = Thm.ctyp_of_cterm x
val th = Thm.instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = Thm.ctyp_of_cterm x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite ctxt
(Thm.apply @{cterm "Trueprop"}
(Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
val rth = Thm.implies_elim
(Thm.instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
in rth end
| _ => Thm.reflexive ct);
local
val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
val ss = simpset_of @{context}
in
fun field_isolate_conv phi ctxt vs ct = case Thm.term_of ct of
Const(@{const_name Orderings.less},_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = Thm.ctyp_of_cterm ca
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const(@{const_name Orderings.less_eq},_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = Thm.ctyp_of_cterm ca
val th = Thm.instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const(@{const_name HOL.eq},_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = 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 vs)))) th
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| @{term "Not"} $(Const(@{const_name HOL.eq},_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => Thm.reflexive ct
end;
fun classfield_whatis phi =
let
fun h x t =
case Thm.term_of t of
Const(@{const_name HOL.eq}, _)$y$z =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) =>
if Thm.term_of x aconv y then Ferrante_Rackoff_Data.NEq
else Ferrante_Rackoff_Data.Nox
| Const(@{const_name Orderings.less},_)$y$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 Orderings.less_eq},_)$y$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