
hoelzl@30439: (*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy
haftmann@29655:     Author      : Amine Chaieb, TU Muenchen
haftmann@29655: *)
chaieb@26161: 
chaieb@26161: header {* Dense linear order without endpoints
chaieb@26161:   and a quantifier elimination procedure in Ferrante and Rackoff style *}
chaieb@26161: 
chaieb@26161: theory Dense_Linear_Order
haftmann@30738: imports Main
chaieb@26161: begin
chaieb@26161: 
wenzelm@48891: ML_file "langford_data.ML"
wenzelm@48891: ML_file "ferrante_rackoff_data.ML"
wenzelm@48891: 
chaieb@26161: context linorder
chaieb@26161: begin
chaieb@26161: 
wenzelm@49519: lemma less_not_permute[no_atp]: "\<not> (x < y \<and> y < x)"
wenzelm@49519:   by (simp add: not_less linear)
chaieb@26161: 
blanchet@35828: lemma gather_simps[no_atp]: 
wenzelm@49519:   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u \<and> P x) \<longleftrightarrow>
wenzelm@49519:     (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y) \<and> P x)"
wenzelm@49519:   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x \<and> P x) \<longleftrightarrow>
wenzelm@49519:     (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y) \<and> P x)"
wenzelm@49519:   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> x < u) \<longleftrightarrow>
wenzelm@49519:     (\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))"
wenzelm@49519:   "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow>
wenzelm@49519:     (\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))"
wenzelm@49519:   by auto
chaieb@26161: 
wenzelm@49519: lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" 
chaieb@26161:   by simp
chaieb@26161: 
wenzelm@53072: text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"}*}
blanchet@35828: lemma minf_lt[no_atp]:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
blanchet@35828: lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
chaieb@26161:   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
chaieb@26161: 
blanchet@35828: lemma minf_le[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
blanchet@35828: lemma minf_ge[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
chaieb@26161:   by (auto simp add: less_le not_less not_le)
blanchet@35828: lemma minf_eq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
blanchet@35828: lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
blanchet@35828: lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
chaieb@26161: 
wenzelm@53072: text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"}*}
wenzelm@49519: lemma pinf_gt[no_atp]:  "\<exists>z. \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
wenzelm@49519: lemma pinf_lt[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
chaieb@26161:   by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
chaieb@26161: 
blanchet@35828: lemma pinf_ge[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
blanchet@35828: lemma pinf_le[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
chaieb@26161:   by (auto simp add: less_le not_less not_le)
blanchet@35828: lemma pinf_eq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
blanchet@35828: lemma pinf_neq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
blanchet@35828: lemma pinf_P[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
chaieb@26161: 
blanchet@35828: lemma nmi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x < t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma nmi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
chaieb@26161:   by (auto simp add: le_less)
blanchet@35828: lemma  nmi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x\<le> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t\<le> x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_neq[no_atp]: "t \<in> U \<Longrightarrow>\<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
chaieb@26161:   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
chaieb@26161:   \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
blanchet@35828: lemma  nmi_disj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
chaieb@26161:   \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
chaieb@26161:   \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
chaieb@26161: 
blanchet@35828: lemma  npi_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x < t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by (auto simp add: le_less)
blanchet@35828: lemma  npi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x \<le> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> t \<le> x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and>  x = t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_neq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>True \<and> x \<noteq> t \<longrightarrow>  (\<exists> u\<in> U. x \<le> u )" by auto
blanchet@35828: lemma  npi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ;  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
chaieb@26161:   \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
blanchet@35828: lemma  npi_disj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u) ; \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)\<rbrakk>
chaieb@26161:   \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
chaieb@26161: 
wenzelm@49519: lemma lin_dense_lt[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
chaieb@26161: proof(clarsimp)
wenzelm@49519:   fix x l u y
wenzelm@49519:   assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x"
chaieb@26161:     and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"
chaieb@26161:   from tU noU ly yu have tny: "t\<noteq>y" by auto
wenzelm@49519:   { assume H: "t < y"
chaieb@26161:     from less_trans[OF lx px] less_trans[OF H yu]
wenzelm@49519:     have "l < t \<and> t < u" by simp
wenzelm@49519:     with tU noU have "False" by auto }
wenzelm@49519:   then have "\<not> t < y" by auto
wenzelm@49519:   then have "y \<le> t" by (simp add: not_less)
wenzelm@49519:   then show "y < t" using tny by (simp add: less_le)
chaieb@26161: qed
chaieb@26161: 
wenzelm@49519: lemma lin_dense_gt[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
wenzelm@49519: proof(clarsimp)
wenzelm@49519:   fix x l u y
wenzelm@49519:   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"
wenzelm@49519:     and px: "t < x" and ly: "l<y" and yu:"y < u"
wenzelm@49519:   from tU noU ly yu have tny: "t\<noteq>y" by auto
wenzelm@49519:   { assume H: "y< t"
wenzelm@49519:     from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
wenzelm@49519:     with tU noU have "False" by auto }
wenzelm@49519:   then have "\<not> y<t" by auto
wenzelm@49519:   then have "t \<le> y" by (auto simp add: not_less)
wenzelm@49519:   then show "t < y" using tny by (simp add: less_le)
wenzelm@49519: qed
wenzelm@49519: 
wenzelm@49519: lemma lin_dense_le[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
chaieb@26161: proof(clarsimp)
chaieb@26161:   fix x l u y
chaieb@26161:   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"
wenzelm@49519:     and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
chaieb@26161:   from tU noU ly yu have tny: "t\<noteq>y" by auto
wenzelm@49519:   { assume H: "t < y"
wenzelm@49519:     from less_le_trans[OF lx px] less_trans[OF H yu]
wenzelm@49519:     have "l < t \<and> t < u" by simp
wenzelm@49519:     with tU noU have "False" by auto }
wenzelm@49519:   then have "\<not> t < y" by auto
wenzelm@49519:   then show "y \<le> t" by (simp add: not_less)
chaieb@26161: qed
chaieb@26161: 
wenzelm@49519: lemma lin_dense_ge[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
chaieb@26161: proof(clarsimp)
chaieb@26161:   fix x l u y
chaieb@26161:   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"
wenzelm@49519:     and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
chaieb@26161:   from tU noU ly yu have tny: "t\<noteq>y" by auto
wenzelm@49519:   { assume H: "y< t"
wenzelm@49519:     from less_trans[OF ly H] le_less_trans[OF px xu]
chaieb@26161:     have "l < t \<and> t < u" by simp
wenzelm@49519:     with tU noU have "False" by auto }
wenzelm@49519:   then have "\<not> y<t" by auto
wenzelm@49519:   then show "t \<le> y" by (simp add: not_less)
chaieb@26161: qed
chaieb@26161: 
wenzelm@49519: lemma lin_dense_eq[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
wenzelm@49519:   by auto
wenzelm@49519: 
wenzelm@49519: lemma lin_dense_neq[no_atp]:
wenzelm@49519:   "t \<in> U \<Longrightarrow>
wenzelm@49519:     \<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)"
wenzelm@49519:   by auto
wenzelm@49519: 
wenzelm@49519: lemma lin_dense_P[no_atp]:
wenzelm@49519:   "\<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)"
wenzelm@49519:   by auto
chaieb@26161: 
blanchet@35828: lemma lin_dense_conj[no_atp]:
chaieb@26161:   "\<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
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
chaieb@26161:   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
chaieb@26161:   \<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)
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
chaieb@26161:   by blast
blanchet@35828: lemma lin_dense_disj[no_atp]:
chaieb@26161:   "\<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
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
chaieb@26161:   \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
chaieb@26161:   \<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)
chaieb@26161:   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
chaieb@26161:   by blast
chaieb@26161: 
blanchet@35828: lemma npmibnd[no_atp]: "\<lbrakk>\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<le> x); \<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<le> u)\<rbrakk>
chaieb@26161:   \<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')"
wenzelm@49519:   by auto
chaieb@26161: 
blanchet@35828: lemma finite_set_intervals[no_atp]:
chaieb@26161:   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
wenzelm@49519:     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"
chaieb@26161:   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"
wenzelm@49519: proof -
chaieb@26161:   let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
chaieb@26161:   let ?xM = "{y. y\<in> S \<and> x \<le> y}"
chaieb@26161:   let ?a = "Max ?Mx"
chaieb@26161:   let ?b = "Min ?xM"
chaieb@26161:   have MxS: "?Mx \<subseteq> S" by blast
chaieb@26161:   hence fMx: "finite ?Mx" using fS finite_subset by auto
chaieb@26161:   from lx linS have linMx: "l \<in> ?Mx" by blast
chaieb@26161:   hence Mxne: "?Mx \<noteq> {}" by blast
chaieb@26161:   have xMS: "?xM \<subseteq> S" by blast
chaieb@26161:   hence fxM: "finite ?xM" using fS finite_subset by auto
chaieb@26161:   from xu uinS have linxM: "u \<in> ?xM" by blast
chaieb@26161:   hence xMne: "?xM \<noteq> {}" by blast
chaieb@26161:   have ax:"?a \<le> x" using Mxne fMx by auto
chaieb@26161:   have xb:"x \<le> ?b" using xMne fxM by auto
chaieb@26161:   have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
chaieb@26161:   have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
chaieb@26161:   have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
chaieb@26161:   proof(clarsimp)
wenzelm@49519:     fix y
wenzelm@49519:     assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
chaieb@26161:     from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
wenzelm@49519:     moreover {
wenzelm@49519:       assume "y \<in> ?Mx"
wenzelm@49519:       hence "y \<le> ?a" using Mxne fMx by auto
wenzelm@49519:       with ay have "False" by (simp add: not_le[symmetric])
wenzelm@49519:     }
wenzelm@49519:     moreover {
wenzelm@49519:       assume "y \<in> ?xM"
wenzelm@49519:       hence "?b \<le> y" using xMne fxM by auto
wenzelm@49519:       with yb have "False" by (simp add: not_le[symmetric])
wenzelm@49519:     }
wenzelm@49519:     ultimately show False by blast
chaieb@26161:   qed
chaieb@26161:   from ainS binS noy ax xb px show ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma finite_set_intervals2[no_atp]:
chaieb@26161:   assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
wenzelm@49519:     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"
chaieb@26161:   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)"
chaieb@26161: proof-
chaieb@26161:   from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
wenzelm@49519:   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"
wenzelm@49519:     and axb: "a \<le> x \<and> x \<le> b \<and> P x" by auto
chaieb@26161:   from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less)
chaieb@26161:   thus ?thesis using px as bs noS by blast
chaieb@26161: qed
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
wenzelm@49519: 
chaieb@26161: section {* The classical QE after Langford for dense linear orders *}
chaieb@26161: 
hoelzl@53215: context unbounded_dense_linorder
chaieb@26161: begin
chaieb@26161: 
wenzelm@49519: lemma interval_empty_iff: "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
haftmann@27825:   by (auto dest: dense)
haftmann@27825: 
blanchet@35828: lemma dlo_qe_bnds[no_atp]: 
chaieb@26161:   assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
chaieb@26161:   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)"
chaieb@26161: proof (simp only: atomize_eq, rule iffI)
chaieb@26161:   assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
chaieb@26161:   then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
wenzelm@49519:   { fix l u assume l: "l \<in> L" and u: "u \<in> U"
chaieb@26161:     have "l < x" using xL l by blast
chaieb@26161:     also have "x < u" using xU u by blast
wenzelm@49519:     finally (less_trans) have "l < u" . }
wenzelm@49519:   then show "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
chaieb@26161: next
chaieb@26161:   assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
chaieb@26161:   let ?ML = "Max L"
chaieb@26161:   let ?MU = "Min U"  
chaieb@26161:   from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
chaieb@26161:   from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
chaieb@26161:   from th1 th2 H have "?ML < ?MU" by auto
chaieb@26161:   with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
chaieb@26161:   from th3 th1' have "\<forall>l \<in> L. l < w" by auto
chaieb@26161:   moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
chaieb@26161:   ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma dlo_qe_noub[no_atp]: 
chaieb@26161:   assumes ne: "L \<noteq> {}" and fL: "finite L"
chaieb@26161:   shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
chaieb@26161: proof(simp add: atomize_eq)
chaieb@26161:   from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
chaieb@26161:   from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
chaieb@26161:   with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
chaieb@26161:   thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma dlo_qe_nolb[no_atp]: 
chaieb@26161:   assumes ne: "U \<noteq> {}" and fU: "finite U"
chaieb@26161:   shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
chaieb@26161: proof(simp add: atomize_eq)
chaieb@26161:   from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
chaieb@26161:   from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
chaieb@26161:   with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
chaieb@26161:   thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
chaieb@26161:   using gt_ex[of t] by auto
chaieb@26161: 
blanchet@35828: lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq 
chaieb@26161:   le_less neq_iff linear less_not_permute
chaieb@26161: 
hoelzl@53215: lemma axiom[no_atp]: "class.unbounded_dense_linorder (op \<le>) (op <)"
hoelzl@53215:   by (rule unbounded_dense_linorder_axioms)
blanchet@35828: lemma atoms[no_atp]:
chaieb@26161:   shows "TERM (less :: 'a \<Rightarrow> _)"
chaieb@26161:     and "TERM (less_eq :: 'a \<Rightarrow> _)"
chaieb@26161:     and "TERM (op = :: 'a \<Rightarrow> _)" .
chaieb@26161: 
chaieb@26161: declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
chaieb@26161: declare dlo_simps[langfordsimp]
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
chaieb@26161: (* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
nipkow@41849: lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribR
chaieb@26161: 
blanchet@35828: lemmas weak_dnf_simps[no_atp] = simp_thms dnf
chaieb@26161: 
blanchet@35828: lemma nnf_simps[no_atp]:
chaieb@26161:     "(\<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)"
chaieb@26161:     "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
chaieb@26161:   by blast+
chaieb@26161: 
blanchet@35828: lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
chaieb@26161: 
blanchet@35828: lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib
chaieb@26161: 
wenzelm@48891: ML_file "langford.ML"
chaieb@26161: method_setup dlo = {*
wenzelm@55848:   Scan.succeed (SIMPLE_METHOD' o Langford.dlo_tac)
chaieb@26161: *} "Langford's algorithm for quantifier elimination in dense linear orders"
chaieb@26161: 
chaieb@26161: 
haftmann@30652: section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}
chaieb@26161: 
chaieb@26161: text {* Linear order without upper bounds *}
chaieb@26161: 
haftmann@29509: locale linorder_stupid_syntax = linorder
chaieb@26161: begin
wenzelm@49519: 
chaieb@26161: notation
chaieb@26161:   less_eq  ("op \<sqsubseteq>") and
chaieb@26161:   less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
chaieb@26161:   less  ("op \<sqsubset>") and
chaieb@26161:   less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
haftmann@29509: locale linorder_no_ub = linorder_stupid_syntax +
chaieb@26161:   assumes gt_ex: "\<exists>y. less x y"
chaieb@26161: begin
wenzelm@49519: 
blanchet@35828: lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
chaieb@26161: 
wenzelm@53072: text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"} *}
blanchet@35828: lemma pinf_conj[no_atp]:
chaieb@26161:   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
chaieb@26161:   and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
chaieb@26161:   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
chaieb@26161: proof-
chaieb@26161:   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
chaieb@26161:      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
chaieb@26161:   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
chaieb@26161:   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
wenzelm@49519:   { fix x assume H: "z \<sqsubset> x"
chaieb@26161:     from less_trans[OF zz1 H] less_trans[OF zz2 H]
chaieb@26161:     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
chaieb@26161:   }
chaieb@26161:   thus ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma pinf_disj[no_atp]:
chaieb@26161:   assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
wenzelm@49519:     and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
chaieb@26161:   shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
chaieb@26161: proof-
chaieb@26161:   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
chaieb@26161:      and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
chaieb@26161:   from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
chaieb@26161:   from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
wenzelm@49519:   { fix x assume H: "z \<sqsubset> x"
chaieb@26161:     from less_trans[OF zz1 H] less_trans[OF zz2 H]
chaieb@26161:     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
chaieb@26161:   }
chaieb@26161:   thus ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma pinf_ex[no_atp]: assumes ex:"\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
wenzelm@49519: proof -
chaieb@26161:   from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
chaieb@26161:   from gt_ex obtain x where x: "z \<sqsubset> x" by blast
chaieb@26161:   from z x p1 show ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
chaieb@26161: text {* Linear order without upper bounds *}
chaieb@26161: 
haftmann@29509: locale linorder_no_lb = linorder_stupid_syntax +
chaieb@26161:   assumes lt_ex: "\<exists>y. less y x"
chaieb@26161: begin
wenzelm@49519: 
blanchet@35828: lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
chaieb@26161: 
chaieb@26161: 
wenzelm@53072: text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"} *}
blanchet@35828: lemma minf_conj[no_atp]:
chaieb@26161:   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
wenzelm@49519:     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
chaieb@26161:   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
chaieb@26161: proof-
chaieb@26161:   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
chaieb@26161:   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
chaieb@26161:   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
wenzelm@49519:   { fix x assume H: "x \<sqsubset> z"
chaieb@26161:     from less_trans[OF H zz1] less_trans[OF H zz2]
chaieb@26161:     have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
chaieb@26161:   }
chaieb@26161:   thus ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemma minf_disj[no_atp]:
chaieb@26161:   assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
wenzelm@49519:     and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
chaieb@26161:   shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
wenzelm@49519: proof -
wenzelm@49519:   from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
wenzelm@49519:     and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
chaieb@26161:   from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
chaieb@26161:   from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
wenzelm@49519:   { fix x assume H: "x \<sqsubset> z"
chaieb@26161:     from less_trans[OF H zz1] less_trans[OF H zz2]
chaieb@26161:     have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
chaieb@26161:   }
chaieb@26161:   thus ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
wenzelm@49519: lemma minf_ex[no_atp]:
wenzelm@49519:   assumes ex: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)"
wenzelm@49519:     and p1: P1
wenzelm@49519:   shows "\<exists> x. P x"
wenzelm@49519: proof -
chaieb@26161:   from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
chaieb@26161:   from lt_ex obtain x where x: "x \<sqsubset> z" by blast
chaieb@26161:   from z x p1 show ?thesis by blast
chaieb@26161: qed
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
chaieb@26161: 
haftmann@35028: locale constr_dense_linorder = linorder_no_lb + linorder_no_ub +
chaieb@26161:   fixes between
chaieb@26161:   assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
wenzelm@49519:     and between_same: "between x x = x"
haftmann@54868: begin
chaieb@26161: 
haftmann@54868: sublocale dlo: unbounded_dense_linorder 
chaieb@26161:   apply unfold_locales
chaieb@26161:   using gt_ex lt_ex between_less
wenzelm@49519:   apply auto
wenzelm@49519:   apply (rule_tac x="between x y" in exI)
wenzelm@49519:   apply simp
wenzelm@49519:   done
chaieb@26161: 
blanchet@35828: lemma rinf_U[no_atp]:
chaieb@26161:   assumes fU: "finite U"
wenzelm@49519:     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
wenzelm@49519:       \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
wenzelm@49519:     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')"
wenzelm@49519:     and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
chaieb@26161:   shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
wenzelm@49519: proof -
chaieb@26161:   from ex obtain x where px: "P x" by blast
chaieb@26161:   from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
chaieb@26161:   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
chaieb@26161:   from uU have Une: "U \<noteq> {}" by auto
chaieb@26161:   let ?l = "linorder.Min less_eq U"
chaieb@26161:   let ?u = "linorder.Max less_eq U"
chaieb@26161:   have linM: "?l \<in> U" using fU Une by simp
chaieb@26161:   have uinM: "?u \<in> U" using fU Une by simp
chaieb@26161:   have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
chaieb@26161:   have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
chaieb@26161:   have th:"?l \<sqsubseteq> u" using uU Une lM by auto
chaieb@26161:   from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
chaieb@26161:   have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
chaieb@26161:   from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
chaieb@26161:   from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
chaieb@26161:   have "(\<exists> s\<in> U. P s) \<or>
chaieb@26161:       (\<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)" .
wenzelm@49519:   moreover {
wenzelm@49519:     fix u assume um: "u\<in>U" and pu: "P u"
chaieb@26161:     have "between u u = u" by (simp add: between_same)
chaieb@26161:     with um pu have "P (between u u)" by simp
wenzelm@49519:     with um have ?thesis by blast }
wenzelm@49519:   moreover {
chaieb@26161:     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"
wenzelm@49519:     then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
wenzelm@49519:       and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U"
wenzelm@49519:       and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" by blast
wenzelm@49519:     from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
wenzelm@49519:     let ?u = "between t1 t2"
wenzelm@49519:     from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
wenzelm@49519:     from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
wenzelm@49519:     with t1M t2M have ?thesis by blast
wenzelm@49519:   }
wenzelm@49519:   ultimately show ?thesis by blast
wenzelm@49519: qed
chaieb@26161: 
blanchet@35828: theorem fr_eq[no_atp]:
chaieb@26161:   assumes fU: "finite U"
wenzelm@49519:     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
wenzelm@49519:      \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
wenzelm@49519:     and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
wenzelm@49519:     and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
wenzelm@49519:     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)"
chaieb@26161:   shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
chaieb@26161:   (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
wenzelm@49519: proof -
wenzelm@49519:   { assume px: "\<exists> x. P x"
wenzelm@49519:     have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
wenzelm@49519:     moreover { assume "MP \<or> PP" hence "?D" by blast }
wenzelm@49519:     moreover {
wenzelm@49519:       assume nmi: "\<not> MP" and npi: "\<not> PP"
wenzelm@49519:       from npmibnd[OF nmibnd npibnd]
wenzelm@49519:       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')" .
wenzelm@49519:       from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast }
wenzelm@49519:     ultimately have "?D" by blast }
wenzelm@49519:   moreover
wenzelm@49519:   { assume "?D"
wenzelm@49519:     moreover { assume m:"MP" from minf_ex[OF mi m] have "?E" . }
wenzelm@49519:     moreover { assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
wenzelm@49519:     moreover { assume f:"?F" hence "?E" by blast }
wenzelm@49519:     ultimately have "?E" by blast }
wenzelm@49519:   ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
chaieb@26161: qed
chaieb@26161: 
blanchet@35828: lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
blanchet@35828: lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
chaieb@26161: 
blanchet@35828: lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
blanchet@35828: lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
blanchet@35828: lemmas lin_dense_thms[no_atp] = lin_dense_conj lin_dense_disj lin_dense_eq lin_dense_neq lin_dense_lt lin_dense_le lin_dense_gt lin_dense_ge lin_dense_P
chaieb@26161: 
blanchet@35828: lemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between"
haftmann@35028:   by (rule constr_dense_linorder_axioms)
wenzelm@49519: 
blanchet@35828: lemma atoms[no_atp]:
chaieb@26161:   shows "TERM (less :: 'a \<Rightarrow> _)"
chaieb@26161:     and "TERM (less_eq :: 'a \<Rightarrow> _)"
chaieb@26161:     and "TERM (op = :: 'a \<Rightarrow> _)" .
chaieb@26161: 
chaieb@26161: declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
chaieb@26161:     nmi: nmi_thms npi: npi_thms lindense:
chaieb@26161:     lin_dense_thms qe: fr_eq atoms: atoms]
chaieb@26161: 
chaieb@26161: declaration {*
chaieb@26161: let
wenzelm@49519:   fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
wenzelm@49519:   fun generic_whatis phi =
wenzelm@49519:     let
wenzelm@49519:       val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
wenzelm@49519:       fun h x t =
wenzelm@49519:         case term_of t of
wenzelm@49519:           Const(@{const_name HOL.eq}, _)$y$z =>
wenzelm@49519:             if term_of x aconv y then Ferrante_Rackoff_Data.Eq
wenzelm@49519:             else Ferrante_Rackoff_Data.Nox
wenzelm@49519:        | @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) =>
wenzelm@49519:             if term_of x aconv y then Ferrante_Rackoff_Data.NEq
wenzelm@49519:             else Ferrante_Rackoff_Data.Nox
wenzelm@49519:        | b$y$z => if Term.could_unify (b, lt) then
wenzelm@49519:                      if term_of x aconv y then Ferrante_Rackoff_Data.Lt
wenzelm@49519:                      else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
wenzelm@49519:                      else Ferrante_Rackoff_Data.Nox
wenzelm@49519:                  else if Term.could_unify (b, le) then
wenzelm@49519:                      if term_of x aconv y then Ferrante_Rackoff_Data.Le
wenzelm@49519:                      else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
wenzelm@49519:                      else Ferrante_Rackoff_Data.Nox
chaieb@26161:                  else Ferrante_Rackoff_Data.Nox
wenzelm@49519:        | _ => Ferrante_Rackoff_Data.Nox
wenzelm@49519:   in h end
wenzelm@51717:   fun ss phi =
wenzelm@51717:     simpset_of (put_simpset HOL_ss @{context} addsimps (simps phi))
chaieb@26161: in
wenzelm@49519:   Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
wenzelm@49519:     {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
chaieb@26161: end
chaieb@26161: *}
chaieb@26161: 
chaieb@26161: end
chaieb@26161: 
wenzelm@48891: ML_file "ferrante_rackoff.ML"
chaieb@26161: 
chaieb@26161: method_setup ferrack = {*
wenzelm@30549:   Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
chaieb@26161: *} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
chaieb@26161: 
wenzelm@49519: 
chaieb@26161: subsection {* Ferrante and Rackoff algorithm over ordered fields *}
chaieb@26161: 
haftmann@35028: lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c < 0"
wenzelm@49519:   have "c*x < 0 = (0/c < x)"
wenzelm@49519:     by (simp only: neg_divide_less_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = (0 < x)" by simp
chaieb@26161:   finally show  "(c*x < 0) == (x > 0)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c > 0"
wenzelm@49519:   then have "c*x < 0 = (0/c > x)"
wenzelm@49519:     by (simp only: pos_less_divide_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = (0 > x)" by simp
chaieb@26161:   finally show  "(c*x < 0) == (x < 0)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c < 0"
wenzelm@49519:   have "c*x + t< 0 = (c*x < -t)"
wenzelm@49519:     by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
wenzelm@49519:   also have "\<dots> = (-t/c < x)"
wenzelm@49519:     by (simp only: neg_divide_less_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = ((- 1/c)*t < x)" by simp
wenzelm@49519:   finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c > 0"
wenzelm@49519:   have "c*x + t< 0 = (c*x < -t)"
wenzelm@49519:     by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
wenzelm@49519:   also have "\<dots> = (-t/c > x)"
wenzelm@49519:     by (simp only: pos_less_divide_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = ((- 1/c)*t > x)" by simp
chaieb@26161:   finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"
chaieb@26161:   using less_diff_eq[where a= x and b=t and c=0] by simp
chaieb@26161: 
haftmann@35028: lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c < 0"
wenzelm@49519:   have "c*x <= 0 = (0/c <= x)"
wenzelm@49519:     by (simp only: neg_divide_le_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = (0 <= x)" by simp
chaieb@26161:   finally show  "(c*x <= 0) == (x >= 0)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c > 0"
wenzelm@49519:   hence "c*x <= 0 = (0/c >= x)"
wenzelm@49519:     by (simp only: pos_le_divide_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = (0 >= x)" by simp
chaieb@26161:   finally show  "(c*x <= 0) == (x <= 0)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c < 0"
wenzelm@49519:   have "c*x + t <= 0 = (c*x <= -t)"
wenzelm@49519:     by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
wenzelm@49519:   also have "\<dots> = (-t/c <= x)"
wenzelm@49519:     by (simp only: neg_divide_le_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = ((- 1/c)*t <= x)" by simp
wenzelm@49519:   finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c > 0"
wenzelm@49519:   have "c*x + t <= 0 = (c*x <= -t)"
wenzelm@49519:     by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
wenzelm@49519:   also have "\<dots> = (-t/c >= x)"
wenzelm@49519:     by (simp only: pos_le_divide_eq[OF H] algebra_simps)
chaieb@26161:   also have "\<dots> = ((- 1/c)*t >= x)" by simp
wenzelm@49519:   finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
chaieb@26161: qed
chaieb@26161: 
haftmann@35028: lemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"
chaieb@26161:   using le_diff_eq[where a= x and b=t and c=0] by simp
chaieb@26161: 
haftmann@35028: lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
wenzelm@49519: 
haftmann@35028: lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
wenzelm@49519: proof -
chaieb@26161:   assume H: "c \<noteq> 0"
wenzelm@49519:   have "c*x + t = 0 = (c*x = -t)"
wenzelm@49519:     by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
wenzelm@49519:   also have "\<dots> = (x = -t/c)"
wenzelm@49519:     by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
chaieb@26161:   finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
chaieb@26161: qed
wenzelm@49519: 
haftmann@35028: lemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"
chaieb@26161:   using eq_diff_eq[where a= x and b=t and c=0] by simp
chaieb@26161: 
chaieb@26161: 
haftmann@35043: interpretation class_dense_linordered_field: constr_dense_linorder
haftmann@29509:  "op <=" "op <"
huffman@47108:    "\<lambda> x y. 1/2 * ((x::'a::{linordered_field}) + y)"
wenzelm@49519:   by unfold_locales (dlo, dlo, auto)
chaieb@26161: 
chaieb@26161: declaration{*
chaieb@26161: let
wenzelm@49519:   fun earlier [] x y = false
wenzelm@49519:     | earlier (h::t) x y =
wenzelm@49519:         if h aconvc y then false else if h aconvc x then true else earlier t x y;
chaieb@26161: 
wenzelm@49519: fun dest_frac ct =
wenzelm@49519:   case term_of ct of
wenzelm@49519:     Const (@{const_name Fields.divide},_) $ a $ b=>
wenzelm@49519:       Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
wenzelm@49519:   | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
wenzelm@49519:   | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
chaieb@26161: 
chaieb@26161: fun mk_frac phi cT x =
wenzelm@49519:   let val (a, b) = Rat.quotient_of_rat x
wenzelm@49519:   in if b = 1 then Numeral.mk_cnumber cT a
wenzelm@46497:     else Thm.apply
wenzelm@46497:          (Thm.apply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
chaieb@26161:                      (Numeral.mk_cnumber cT a))
chaieb@26161:          (Numeral.mk_cnumber cT b)
chaieb@26161:  end
chaieb@26161: 
chaieb@26161: fun whatis x ct = case term_of ct of
haftmann@35267:   Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ =>
chaieb@26161:      if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
chaieb@26161:      else ("Nox",[])
haftmann@35267: | Const(@{const_name Groups.plus}, _)$y$_ =>
chaieb@26161:      if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
chaieb@26161:      else ("Nox",[])
haftmann@35267: | Const(@{const_name Groups.times}, _)$_$y =>
chaieb@26161:      if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
chaieb@26161:      else ("Nox",[])
chaieb@26161: | t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
chaieb@26161: 
wenzelm@36945: fun xnormalize_conv ctxt [] ct = Thm.reflexive ct
chaieb@26161: | xnormalize_conv ctxt (vs as (x::_)) ct =
chaieb@26161:    case term_of ct of
haftmann@35267:    Const(@{const_name Orderings.less},_)$_$Const(@{const_name Groups.zero},_) =>
chaieb@26161:     (case whatis x (Thm.dest_arg1 ct) of
chaieb@26161:     ("c*x+t",[c,t]) =>
chaieb@26161:        let
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val clt = Thm.dest_fun2 ct
chaieb@26161:         val cz = Thm.dest_arg ct
chaieb@26161:         val neg = cr </ Rat.zero
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:                (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:                   (if neg then Thm.apply (Thm.apply clt c) cz
wenzelm@46497:                     else Thm.apply (Thm.apply clt cz) c))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
chaieb@26161:              (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:       in rth end
chaieb@26161:     | ("x+t",[t]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:        in  rth end
chaieb@26161:     | ("c*x",[c]) =>
chaieb@26161:        let
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val clt = Thm.dest_fun2 ct
chaieb@26161:         val cz = Thm.dest_arg ct
chaieb@26161:         val neg = cr </ Rat.zero
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:                (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:                   (if neg then Thm.apply (Thm.apply clt c) cz
wenzelm@46497:                     else Thm.apply (Thm.apply clt cz) c))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
chaieb@26161:              (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
chaieb@26161:         val rth = th
chaieb@26161:       in rth end
wenzelm@36945:     | _ => Thm.reflexive ct)
chaieb@26161: 
chaieb@26161: 
haftmann@35267: |  Const(@{const_name Orderings.less_eq},_)$_$Const(@{const_name Groups.zero},_) =>
chaieb@26161:    (case whatis x (Thm.dest_arg1 ct) of
chaieb@26161:     ("c*x+t",[c,t]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
chaieb@26161:         val cz = Thm.dest_arg ct
chaieb@26161:         val neg = cr </ Rat.zero
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:                (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:                   (if neg then Thm.apply (Thm.apply clt c) cz
wenzelm@46497:                     else Thm.apply (Thm.apply clt cz) c))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val th = Thm.implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
chaieb@26161:              (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:       in rth end
chaieb@26161:     | ("x+t",[t]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:        in  rth end
chaieb@26161:     | ("c*x",[c]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
chaieb@26161:         val cz = Thm.dest_arg ct
chaieb@26161:         val neg = cr </ Rat.zero
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:                (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:                   (if neg then Thm.apply (Thm.apply clt c) cz
wenzelm@46497:                     else Thm.apply (Thm.apply clt cz) c))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
chaieb@26161:              (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
chaieb@26161:         val rth = th
chaieb@26161:       in rth end
wenzelm@36945:     | _ => Thm.reflexive ct)
chaieb@26161: 
haftmann@38864: |  Const(@{const_name HOL.eq},_)$_$Const(@{const_name Groups.zero},_) =>
chaieb@26161:    (case whatis x (Thm.dest_arg1 ct) of
chaieb@26161:     ("c*x+t",[c,t]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val ceq = Thm.dest_fun2 ct
chaieb@26161:         val cz = Thm.dest_arg ct
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:             (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val th = Thm.implies_elim
chaieb@26161:                  (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:                    (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:       in rth end
chaieb@26161:     | ("x+t",[t]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
chaieb@26161:         val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
haftmann@36753:               (Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
chaieb@26161:        in  rth end
chaieb@26161:     | ("c*x",[c]) =>
chaieb@26161:        let
chaieb@26161:         val T = ctyp_of_term x
chaieb@26161:         val cr = dest_frac c
chaieb@26161:         val ceq = Thm.dest_fun2 ct
chaieb@26161:         val cz = Thm.dest_arg ct
wenzelm@51717:         val cthp = Simplifier.rewrite ctxt
wenzelm@46497:             (Thm.apply @{cterm "Trueprop"}
wenzelm@46497:              (Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz)))
wenzelm@36945:         val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI
wenzelm@36945:         val rth = Thm.implies_elim
chaieb@26161:                  (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
chaieb@26161:       in rth end
wenzelm@36945:     | _ => Thm.reflexive ct);
chaieb@26161: 
chaieb@26161: local
chaieb@26161:   val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
chaieb@26161:   val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
chaieb@26161:   val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
wenzelm@52090:   val ss = simpset_of @{context}
chaieb@26161: in
chaieb@26161: fun field_isolate_conv phi ctxt vs ct = case term_of ct of
haftmann@35092:   Const(@{const_name Orderings.less},_)$a$b =>
chaieb@26161:    let val (ca,cb) = Thm.dest_binop ct
chaieb@26161:        val T = ctyp_of_term ca
chaieb@26161:        val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
chaieb@26161:        val nth = Conv.fconv_rule
chaieb@26161:          (Conv.arg_conv (Conv.arg1_conv
wenzelm@52090:               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
wenzelm@36945:        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
chaieb@26161:    in rth end
haftmann@35092: | Const(@{const_name Orderings.less_eq},_)$a$b =>
chaieb@26161:    let val (ca,cb) = Thm.dest_binop ct
chaieb@26161:        val T = ctyp_of_term ca
chaieb@26161:        val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
chaieb@26161:        val nth = Conv.fconv_rule
chaieb@26161:          (Conv.arg_conv (Conv.arg1_conv
wenzelm@52090:               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
wenzelm@36945:        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
chaieb@26161:    in rth end
chaieb@26161: 
haftmann@38864: | Const(@{const_name HOL.eq},_)$a$b =>
chaieb@26161:    let val (ca,cb) = Thm.dest_binop ct
chaieb@26161:        val T = ctyp_of_term ca
chaieb@26161:        val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
chaieb@26161:        val nth = Conv.fconv_rule
chaieb@26161:          (Conv.arg_conv (Conv.arg1_conv
wenzelm@52090:               (Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th
wenzelm@36945:        val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
chaieb@26161:    in rth end
haftmann@38864: | @{term "Not"} $(Const(@{const_name HOL.eq},_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
wenzelm@36945: | _ => Thm.reflexive ct
chaieb@26161: end;
chaieb@26161: 
chaieb@26161: fun classfield_whatis phi =
chaieb@26161:  let
chaieb@26161:   fun h x t =
chaieb@26161:    case term_of t of
haftmann@38864:      Const(@{const_name HOL.eq}, _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
chaieb@26161:                             else Ferrante_Rackoff_Data.Nox
haftmann@38864:    | @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
chaieb@26161:                             else Ferrante_Rackoff_Data.Nox
haftmann@35092:    | Const(@{const_name Orderings.less},_)$y$z =>
chaieb@26161:        if term_of x aconv y then Ferrante_Rackoff_Data.Lt
chaieb@26161:         else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
chaieb@26161:         else Ferrante_Rackoff_Data.Nox
haftmann@35092:    | Const (@{const_name Orderings.less_eq},_)$y$z =>
chaieb@26161:          if term_of x aconv y then Ferrante_Rackoff_Data.Le
chaieb@26161:          else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
chaieb@26161:          else Ferrante_Rackoff_Data.Nox
chaieb@26161:    | _ => Ferrante_Rackoff_Data.Nox
chaieb@26161:  in h end;
chaieb@26161: fun class_field_ss phi =
wenzelm@51717:   simpset_of (put_simpset HOL_basic_ss @{context}
wenzelm@51717:     addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
wenzelm@51717:     |> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}])
chaieb@26161: 
chaieb@26161: in
haftmann@35043: Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}
chaieb@26161:   {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
chaieb@26161: end
chaieb@26161: *}
nipkow@41849: (*
chaieb@29841: lemma upper_bound_finite_set:
chaieb@29841:   assumes fS: "finite S"
chaieb@29841:   shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<le> a"
chaieb@29841: proof(induct rule: finite_induct[OF fS])
chaieb@29841:   case 1 thus ?case by simp
chaieb@29841: next
chaieb@29841:   case (2 x F)
chaieb@29841:   from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<le> a" by blast
chaieb@29841:   let ?a = "max a (f x)"
chaieb@29841:   have m: "a \<le> ?a" "f x \<le> ?a" by simp_all
chaieb@29841:   {fix y assume y: "y \<in> insert x F"
chaieb@29841:     {assume "y = x" hence "f y \<le> ?a" using m by simp}
chaieb@29841:     moreover
chaieb@29841:     {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<le> ?a" by (simp add: max_def)}
chaieb@29841:     ultimately have "f y \<le> ?a" using y by blast}
chaieb@29841:   then show ?case by blast
chaieb@29841: qed
chaieb@29841: 
chaieb@29841: lemma lower_bound_finite_set:
chaieb@29841:   assumes fS: "finite S"
chaieb@29841:   shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<ge> a"
chaieb@29841: proof(induct rule: finite_induct[OF fS])
chaieb@29841:   case 1 thus ?case by simp
chaieb@29841: next
chaieb@29841:   case (2 x F)
chaieb@29841:   from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<ge> a" by blast
chaieb@29841:   let ?a = "min a (f x)"
chaieb@29841:   have m: "a \<ge> ?a" "f x \<ge> ?a" by simp_all
chaieb@29841:   {fix y assume y: "y \<in> insert x F"
chaieb@29841:     {assume "y = x" hence "f y \<ge> ?a" using m by simp}
chaieb@29841:     moreover
chaieb@29841:     {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<ge> ?a" by (simp add: min_def)}
chaieb@29841:     ultimately have "f y \<ge> ?a" using y by blast}
chaieb@29841:   then show ?case by blast
chaieb@29841: qed
chaieb@29841: 
chaieb@29841: lemma bound_finite_set: assumes f: "finite S"
chaieb@29841:   shows "\<exists>a. \<forall>x \<in>S. (f x:: 'a::linorder) \<le> a"
chaieb@29841: proof-
chaieb@29841:   let ?F = "f ` S"
chaieb@29841:   from f have fF: "finite ?F" by simp
chaieb@29841:   let ?a = "Max ?F"
chaieb@29841:   {assume "S = {}" hence ?thesis by blast}
chaieb@29841:   moreover
chaieb@29841:   {assume Se: "S \<noteq> {}" hence Fe: "?F \<noteq> {}" by simp
chaieb@29841:   {fix x assume x: "x \<in> S"
chaieb@29841:     hence th0: "f x \<in> ?F" by simp
chaieb@29841:     hence "f x \<le> ?a" using Max_ge[OF fF th0] ..}
chaieb@29841:   hence ?thesis by blast}
chaieb@29841: ultimately show ?thesis by blast
chaieb@29841: qed
nipkow@41849: *)
chaieb@26161: 
chaieb@26161: end 




