(* Title : HOL/Decision_Procs/Dense_Linear_Order.thy
Author : Amine Chaieb, TU Muenchen
*)
header {* Dense linear order without endpoints
and a quantifier elimination procedure in Ferrante and Rackoff style *}
theory Dense_Linear_Order
imports Main
uses
"~~/src/HOL/Tools/Qelim/langford_data.ML"
"~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML"
("~~/src/HOL/Tools/Qelim/langford.ML")
("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML")
begin
setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *}
context linorder
begin
lemma less_not_permute[noatp]: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)
lemma gather_simps[noatp]:
shows
"(\<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)"
and "(\<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))"
and "(\<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[noatp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)"
by simp
text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
lemma minf_lt[noatp]: "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
lemma minf_gt[noatp]: "\<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[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
lemma minf_ge[noatp]: "\<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[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
lemma minf_neq[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
lemma minf_P[noatp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
lemma pinf_gt[noatp]: "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
lemma pinf_lt[noatp]: "\<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[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
lemma pinf_le[noatp]: "\<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[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
lemma pinf_neq[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
lemma pinf_P[noatp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
lemma nmi_lt[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
lemma nmi_gt[noatp]: "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[noatp]: "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[noatp]: "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[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
lemma nmi_neq[noatp]: "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[noatp]: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. u \<le> x)" by auto
lemma nmi_conj[noatp]: "\<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[noatp]: "\<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[noatp]: "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[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
lemma npi_le[noatp]: "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[noatp]: "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[noatp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> x = t \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
lemma npi_neq[noatp]: "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[noatp]: "\<forall> x. ~P \<and> P \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto
lemma npi_conj[noatp]: "\<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[noatp]: "\<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[noatp]: "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
{assume H: "t < y"
from less_trans[OF lx px] less_trans[OF H yu]
have "l < t \<and> t < u" by simp
with tU noU have "False" by auto}
hence "\<not> t < y" by auto hence "y \<le> t" by (simp add: not_less)
thus "y < t" using tny by (simp add: less_le)
qed
lemma lin_dense_gt[noatp]: "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
{assume H: "y< t"
from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
with tU noU have "False" by auto}
hence "\<not> y<t" by auto hence "t \<le> y" by (auto simp add: not_less)
thus "t < y" using tny by (simp add:less_le)
qed
lemma lin_dense_le[noatp]: "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
{assume H: "t < y"
from less_le_trans[OF lx px] less_trans[OF H yu]
have "l < t \<and> t < u" by simp
with tU noU have "False" by auto}
hence "\<not> t < y" by auto thus "y \<le> t" by (simp add: not_less)
qed
lemma lin_dense_ge[noatp]: "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
{assume H: "y< t"
from less_trans[OF ly H] le_less_trans[OF px xu]
have "l < t \<and> t < u" by simp
with tU noU have "False" by auto}
hence "\<not> y<t" by auto thus "t \<le> y" by (simp add: not_less)
qed
lemma lin_dense_eq[noatp]: "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[noatp]: "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[noatp]: "\<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[noatp]:
"\<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[noatp]:
"\<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[noatp]: "\<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[noatp]:
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
hence fMx: "finite ?Mx" using fS finite_subset by auto
from lx linS have linMx: "l \<in> ?Mx" by blast
hence Mxne: "?Mx \<noteq> {}" by blast
have xMS: "?xM \<subseteq> S" by blast
hence fxM: "finite ?xM" using fS finite_subset by auto
from xu uinS have linxM: "u \<in> ?xM" by blast
hence 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 hence ainS: "?a \<in> S" using MxS by blast
have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence 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)
moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
ultimately show "False" by blast
qed
from ainS binS noy ax xb px show ?thesis by blast
qed
lemma finite_set_intervals2[noatp]:
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)
thus ?thesis using px as bs noS by blast
qed
end
section {* The classical QE after Langford for dense linear orders *}
context 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[noatp]:
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
{fix l u assume l: "l \<in> L" and u: "u \<in> U"
have "l < x" using xL l by blast
also have "x < u" using xU u by blast
finally (less_trans) have "l < u" .}
thus "\<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[noatp]:
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)
thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
qed
lemma dlo_qe_nolb[noatp]:
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)
thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
qed
lemma exists_neq[noatp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x"
using gt_ex[of t] by auto
lemmas dlo_simps[noatp] = order_refl less_irrefl not_less not_le exists_neq
le_less neq_iff linear less_not_permute
lemma axiom[noatp]: "dense_linorder (op \<le>) (op <)" by (rule dense_linorder_axioms)
lemma atoms[noatp]:
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 *)
lemma dnf[noatp]:
"(P & (Q | R)) = ((P&Q) | (P&R))"
"((Q | R) & P) = ((Q&P) | (R&P))"
by blast+
lemmas weak_dnf_simps[noatp] = simp_thms dnf
lemma nnf_simps[noatp]:
"(\<not>(P \<and> Q)) = (\<not>P \<or> \<not>Q)" "(\<not>(P \<or> Q)) = (\<not>P \<and> \<not>Q)" "(P \<longrightarrow> Q) = (\<not>P \<or> Q)"
"(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
by blast+
lemma ex_distrib[noatp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
lemmas dnf_simps[noatp] = weak_dnf_simps nnf_simps ex_distrib
use "~~/src/HOL/Tools/Qelim/langford.ML"
method_setup dlo = {*
Scan.succeed (SIMPLE_METHOD' o LangfordQE.dlo_tac)
*} "Langford's algorithm for quantifier elimination in dense linear orders"
section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}
text {* Linear order without upper bounds *}
locale linorder_stupid_syntax = 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[noatp]: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
lemma pinf_conj[noatp]:
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
{fix x assume H: "z \<sqsubset> x"
from less_trans[OF zz1 H] less_trans[OF zz2 H]
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto
}
thus ?thesis by blast
qed
lemma pinf_disj[noatp]:
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
{fix x assume H: "z \<sqsubset> x"
from less_trans[OF zz1 H] less_trans[OF zz2 H]
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto
}
thus ?thesis by blast
qed
lemma pinf_ex[noatp]: 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 {* Linear order without upper bounds *}
locale linorder_no_lb = linorder_stupid_syntax +
assumes lt_ex: "\<exists>y. less y x"
begin
lemma le_ex[noatp]: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
lemma minf_conj[noatp]:
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
{fix x assume H: "x \<sqsubset> z"
from less_trans[OF H zz1] less_trans[OF H zz2]
have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')" using z1 zz1 z2 zz2 by auto
}
thus ?thesis by blast
qed
lemma minf_disj[noatp]:
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
{fix x assume H: "x \<sqsubset> z"
from less_trans[OF H zz1] less_trans[OF H zz2]
have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')" using z1 zz1 z2 zz2 by auto
}
thus ?thesis by blast
qed
lemma minf_ex[noatp]: 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"
sublocale constr_dense_linorder < dense_linorder
apply unfold_locales
using gt_ex lt_ex between_less
by (auto, rule_tac x="between x y" in exI, simp)
context constr_dense_linorder
begin
lemma rinf_U[noatp]:
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
term "linorder.Min less_eq"
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]
have "(\<exists> s\<in> U. P s) \<or>
(\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x)" .
moreover { fix u assume um: "u\<in>U" and pu: "P u"
have "between u u = u" by (simp add: between_same)
with um pu have "P (between u u)" by simp
with um have ?thesis by blast}
moreover{
assume "\<exists> t1\<in> U. \<exists> t2 \<in> U. (\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U) \<and> t1 \<sqsubset> x \<and> x \<sqsubset> t2 \<and> P x"
then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x"
by blast
from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
let ?u = "between t1 t2"
from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
with t1M t2M have ?thesis by blast}
ultimately show ?thesis by blast
qed
theorem fr_eq[noatp]:
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-
{
assume px: "\<exists> x. P x"
have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
moreover {assume "MP \<or> PP" hence "?D" by blast}
moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
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 nmi npi px] have "?D" by blast}
ultimately have "?D" by blast}
moreover
{ assume "?D"
moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
moreover {assume f:"?F" hence "?E" by blast}
ultimately have "?E" by blast}
ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
qed
lemmas minf_thms[noatp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
lemmas pinf_thms[noatp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
lemmas nmi_thms[noatp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
lemmas npi_thms[noatp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
lemmas lin_dense_thms[noatp] = 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[noatp]: "constr_dense_linorder less_eq less between"
by (rule constr_dense_linorder_axioms)
lemma atoms[noatp]:
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 {*
let
fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
fun generic_whatis phi =
let
val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
fun h x t =
case term_of t of
Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
else Ferrante_Rackoff_Data.Nox
| b$y$z => if Term.could_unify (b, lt) then
if term_of x aconv y then Ferrante_Rackoff_Data.Lt
else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
else Ferrante_Rackoff_Data.Nox
else if Term.could_unify (b, le) then
if term_of x aconv y then Ferrante_Rackoff_Data.Le
else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
else Ferrante_Rackoff_Data.Nox
else Ferrante_Rackoff_Data.Nox
| _ => Ferrante_Rackoff_Data.Nox
in h end
fun ss phi = HOL_ss addsimps (simps phi)
in
Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"}
{isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
end
*}
end
use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML"
method_setup ferrack = {*
Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
subsection {* Ferrante and Rackoff algorithm over ordered fields *}
lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
proof-
assume H: "c < 0"
have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
also have "\<dots> = (0 < x)" by simp
finally show "(c*x < 0) == (x > 0)" by simp
qed
lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
proof-
assume H: "c > 0"
hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
also have "\<dots> = (0 > x)" by simp
finally show "(c*x < 0) == (x < 0)" by simp
qed
lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
proof-
assume H: "c < 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t < x)" by simp
finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
qed
lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
proof-
assume H: "c > 0"
have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t > x)" by simp
finally show "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
qed
lemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"
using less_diff_eq[where a= x and b=t and c=0] by simp
lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
proof-
assume H: "c < 0"
have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
also have "\<dots> = (0 <= x)" by simp
finally show "(c*x <= 0) == (x >= 0)" by simp
qed
lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
proof-
assume H: "c > 0"
hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
also have "\<dots> = (0 >= x)" by simp
finally show "(c*x <= 0) == (x <= 0)" by simp
qed
lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
proof-
assume H: "c < 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t <= x)" by simp
finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
qed
lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
proof-
assume H: "c > 0"
have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
also have "\<dots> = ((- 1/c)*t >= x)" by simp
finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
qed
lemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"
using le_diff_eq[where a= x and b=t and c=0] by simp
lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
proof-
assume H: "c \<noteq> 0"
have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
finally show "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
qed
lemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"
using eq_diff_eq[where a= x and b=t and c=0] by simp
interpretation class_dense_linordered_field: constr_dense_linorder
"op <=" "op <"
"\<lambda> x y. 1/2 * ((x::'a::{linordered_field,number_ring}) + y)"
proof (unfold_locales, dlo, dlo, auto)
fix x y::'a assume lt: "x < y"
from less_half_sum[OF lt] show "x < (x + y) /2" by simp
next
fix x y::'a assume lt: "x < y"
from gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
qed
declaration{*
let
fun earlier [] x y = false
| earlier (h::t) x y =
if h aconvc y then false else if h aconvc x then true else earlier t x y;
fun dest_frac ct = case term_of ct of
Const (@{const_name Rings.divide},_) $ a $ b=>
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
| Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
| t => Rat.rat_of_int (snd (HOLogic.dest_number t))
fun mk_frac phi cT x =
let val (a, b) = Rat.quotient_of_rat x
in if b = 1 then Numeral.mk_cnumber cT a
else Thm.capply
(Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
(Numeral.mk_cnumber cT a))
(Numeral.mk_cnumber cT b)
end
fun whatis x ct = case term_of ct of
Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ =>
if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
else ("Nox",[])
| Const(@{const_name Groups.plus}, _)$y$_ =>
if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
else ("Nox",[])
| Const(@{const_name Groups.times}, _)$_$y =>
if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
else ("Nox",[])
| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
fun xnormalize_conv ctxt [] ct = reflexive ct
| xnormalize_conv ctxt (vs as (x::_)) ct =
case term_of ct of
Const(@{const_name Orderings.less},_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr </ Rat.zero
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(if neg then Thm.capply (Thm.capply clt c) cz
else Thm.capply (Thm.capply clt cz) c))
val cth = equal_elim (symmetric cthp) TrueI
val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = ctyp_of_term x
val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val cr = dest_frac c
val clt = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val neg = cr </ Rat.zero
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(if neg then Thm.capply (Thm.capply clt c) cz
else Thm.capply (Thm.capply clt cz) c))
val cth = equal_elim (symmetric cthp) TrueI
val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
(if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
val rth = th
in rth end
| _ => reflexive ct)
| Const(@{const_name Orderings.less_eq},_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = ctyp_of_term x
val cr = dest_frac c
val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
val cz = Thm.dest_arg ct
val neg = cr </ Rat.zero
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(if neg then Thm.capply (Thm.capply clt c) cz
else Thm.capply (Thm.capply clt cz) c))
val cth = equal_elim (symmetric cthp) TrueI
val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
(if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = ctyp_of_term x
val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = ctyp_of_term x
val cr = dest_frac c
val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
val cz = Thm.dest_arg ct
val neg = cr </ Rat.zero
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(if neg then Thm.capply (Thm.capply clt c) cz
else Thm.capply (Thm.capply clt cz) c))
val cth = equal_elim (symmetric cthp) TrueI
val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
(if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
val rth = th
in rth end
| _ => reflexive ct)
| Const("op =",_)$_$Const(@{const_name Groups.zero},_) =>
(case whatis x (Thm.dest_arg1 ct) of
("c*x+t",[c,t]) =>
let
val T = ctyp_of_term x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
val cth = equal_elim (symmetric cthp) TrueI
val th = implies_elim
(instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("x+t",[t]) =>
let
val T = ctyp_of_term x
val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
(Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
in rth end
| ("c*x",[c]) =>
let
val T = ctyp_of_term x
val cr = dest_frac c
val ceq = Thm.dest_fun2 ct
val cz = Thm.dest_arg ct
val cthp = Simplifier.rewrite (simpset_of ctxt)
(Thm.capply @{cterm "Trueprop"}
(Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
val cth = equal_elim (symmetric cthp) TrueI
val rth = implies_elim
(instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
in rth end
| _ => reflexive ct);
local
val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
in
fun field_isolate_conv phi ctxt vs ct = case term_of ct of
Const(@{const_name Orderings.less},_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = ctyp_of_term ca
val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const(@{const_name Orderings.less_eq},_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = ctyp_of_term ca
val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| Const("op =",_)$a$b =>
let val (ca,cb) = Thm.dest_binop ct
val T = ctyp_of_term ca
val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
val nth = Conv.fconv_rule
(Conv.arg_conv (Conv.arg1_conv
(Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
in rth end
| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
| _ => reflexive ct
end;
fun classfield_whatis phi =
let
fun h x t =
case term_of t of
Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
else Ferrante_Rackoff_Data.Nox
| @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
else Ferrante_Rackoff_Data.Nox
| Const(@{const_name Orderings.less},_)$y$z =>
if term_of x aconv y then Ferrante_Rackoff_Data.Lt
else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
else Ferrante_Rackoff_Data.Nox
| Const (@{const_name Orderings.less_eq},_)$y$z =>
if term_of x aconv y then Ferrante_Rackoff_Data.Le
else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
else Ferrante_Rackoff_Data.Nox
| _ => Ferrante_Rackoff_Data.Nox
in h end;
fun class_field_ss phi =
HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
addsplits [@{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
*}
lemma upper_bound_finite_set:
assumes fS: "finite S"
shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<le> a"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 x F)
from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<le> a" by blast
let ?a = "max a (f x)"
have m: "a \<le> ?a" "f x \<le> ?a" by simp_all
{fix y assume y: "y \<in> insert x F"
{assume "y = x" hence "f y \<le> ?a" using m by simp}
moreover
{assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<le> ?a" by (simp add: max_def)}
ultimately have "f y \<le> ?a" using y by blast}
then show ?case by blast
qed
lemma lower_bound_finite_set:
assumes fS: "finite S"
shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<ge> a"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by simp
next
case (2 x F)
from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<ge> a" by blast
let ?a = "min a (f x)"
have m: "a \<ge> ?a" "f x \<ge> ?a" by simp_all
{fix y assume y: "y \<in> insert x F"
{assume "y = x" hence "f y \<ge> ?a" using m by simp}
moreover
{assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<ge> ?a" by (simp add: min_def)}
ultimately have "f y \<ge> ?a" using y by blast}
then show ?case by blast
qed
lemma bound_finite_set: assumes f: "finite S"
shows "\<exists>a. \<forall>x \<in>S. (f x:: 'a::linorder) \<le> a"
proof-
let ?F = "f ` S"
from f have fF: "finite ?F" by simp
let ?a = "Max ?F"
{assume "S = {}" hence ?thesis by blast}
moreover
{assume Se: "S \<noteq> {}" hence Fe: "?F \<noteq> {}" by simp
{fix x assume x: "x \<in> S"
hence th0: "f x \<in> ?F" by simp
hence "f x \<le> ?a" using Max_ge[OF fF th0] ..}
hence ?thesis by blast}
ultimately show ?thesis by blast
qed
end