(* 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 Plain Groebner_Basis

 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: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)

 lemma gather_simps: 

   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: "(\<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:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto

 lemma minf_gt: "\<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: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)

 lemma minf_ge: "\<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: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto

 lemma minf_neq: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto

 lemma minf_P: "\<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:  "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto

 lemma pinf_lt: "\<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: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)

 lemma pinf_le: "\<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: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto

 lemma pinf_neq: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto

 lemma pinf_P: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast

 lemma nmi_lt: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto

 lemma nmi_gt: "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: "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: "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: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto

 lemma  nmi_neq: "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: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto

 lemma  nmi_conj: "\<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: "\<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: "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: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto

 lemma  npi_le: "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: "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: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto

 lemma  npi_neq: "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: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto

 lemma  npi_conj: "\<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: "\<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: "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: "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: "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: "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: "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: "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: "\<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:

   "\<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:

   "\<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: "\<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:

   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:

   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_linear_order

 begin

 lemma interval_empty_iff:

   "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"

   by (auto dest: dense)

 lemma dlo_qe_bnds: 

   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: 

   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: 

   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: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 

   using gt_ex[of t] by auto

 lemmas dlo_simps = order_refl less_irrefl not_less not_le exists_neq 

   le_less neq_iff linear less_not_permute

 lemma axiom: "dense_linear_order (op \<le>) (op <)" by (rule dense_linear_order_axioms)

 lemma atoms:

   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:

   "(P & (Q | R)) = ((P&Q) | (P&R))" 

   "((Q | R) & P) = ((Q&P) | (R&P))"

   by blast+

 lemmas weak_dnf_simps = simp_thms dnf

 lemma nnf_simps:

     "(\<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: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast

 lemmas dnf_simps = weak_dnf_simps nnf_simps ex_distrib

 use "~~/src/HOL/Tools/Qelim/langford.ML"

 method_setup dlo = {*

   Method.ctxt_args (Method.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 -- see @{text "Arith_Tools.thy"} *}

 text {* Linear order without upper bounds *}

 class_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

 class_locale linorder_no_ub = linorder_stupid_syntax +

   assumes gt_ex: "\<exists>y. less x y"

 begin

 lemma ge_ex: "\<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:

   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:

   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: 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 *}

 class_locale linorder_no_lb = linorder_stupid_syntax +

   assumes lt_ex: "\<exists>y. less y x"

 begin

 lemma le_ex: "\<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:

   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:

   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: 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

 class_locale constr_dense_linear_order = 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"

 class_interpretation  constr_dense_linear_order < dense_linear_order 

   apply unfold_locales

   using gt_ex lt_ex between_less

     by (auto, rule_tac x="between x y" in exI, simp)

 context  constr_dense_linear_order

 begin

 lemma rinf_U:

   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:

   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 = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P

 lemmas pinf_thms = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P

 lemmas nmi_thms = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P

 lemmas npi_thms = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P

 lemmas lin_dense_thms = 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: "constr_dense_linear_order less_eq less between"

   by (rule constr_dense_linear_order_axioms)

 lemma atoms:

   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 = {*

   Method.ctxt_args (Method.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>ordered_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] ring_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>ordered_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] ring_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>ordered_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] ring_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>ordered_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] ring_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::pordered_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>ordered_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] ring_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>ordered_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] ring_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>ordered_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] ring_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>ordered_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] ring_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::pordered_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>ordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp

 lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>ordered_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] ring_simps)

   finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp

 qed

 lemma sum_eq:"((x::'a::pordered_ab_group_add) + t = 0) == (x = - t)"

   using eq_diff_eq[where a= x and b=t and c=0] by simp

 class_interpretation class_ordered_field_dense_linear_order: constr_dense_linear_order

  ["op <=" "op <"

    "\<lambda> x y. 1/2 * ((x::'a::{ordered_field,recpower,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 "HOL.divide"},_) $ a $ b=>

     Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))

  | 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 "HOL.plus"}, _)$(Const(@{const_name "HOL.times"},_)$_$y)$_ =>

      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 "HOL.plus"}, _)$y$_ =>

      if y aconv term_of x then ("x+t",[Thm.dest_arg ct])

      else ("Nox",[])

 | Const(@{const_name "HOL.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 HOL.less},_)$_$Const(@{const_name "HOL.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 (local_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 (local_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 HOL.less_eq},_)$_$Const(@{const_name "HOL.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 (local_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 (local_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 "HOL.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 (local_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 (local_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 HOL.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 HOL.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 HOL.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 HOL.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_ordered_field_dense_linear_order.ferrack_axiom"}

   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}

 end

 *}

 end