src/HOL/Decision_Procs/Dense_Linear_Order.thy
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(*  Title       : HOL/Decision_Procs/Dense_Linear_Order.thy
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    Author      : Amine Chaieb, TU Muenchen
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*)
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header {* Dense linear order without endpoints
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  and a quantifier elimination procedure in Ferrante and Rackoff style *}
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theory Dense_Linear_Order
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imports Main
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uses
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  "~~/src/HOL/Tools/Qelim/langford_data.ML"
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  "~~/src/HOL/Tools/Qelim/ferrante_rackoff_data.ML"
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  ("~~/src/HOL/Tools/Qelim/langford.ML")
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  ("~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML")
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begin
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setup {* Langford_Data.setup #> Ferrante_Rackoff_Data.setup *}
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context linorder
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begin
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lemma less_not_permute[no_atp]: "\<not> (x < y \<and> y < x)" by (simp add: not_less linear)
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lemma gather_simps[no_atp]: 
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  shows 
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  "(\<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)"
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  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)"
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  "(\<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))"
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  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
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lemma 
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  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)" 
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  by simp
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text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"}*}
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lemma minf_lt[no_atp]:  "\<exists>z . \<forall>x. x < z \<longrightarrow> (x < t \<longleftrightarrow> True)" by auto
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lemma minf_gt[no_atp]: "\<exists>z . \<forall>x. x < z \<longrightarrow>  (t < x \<longleftrightarrow>  False)"
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  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
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lemma minf_le[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<le> t \<longleftrightarrow> True)" by (auto simp add: less_le)
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lemma minf_ge[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (t \<le> x \<longleftrightarrow> False)"
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  by (auto simp add: less_le not_less not_le)
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lemma minf_eq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
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lemma minf_neq[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
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lemma minf_P[no_atp]: "\<exists>z. \<forall>x. x < z \<longrightarrow> (P \<longleftrightarrow> P)" by blast
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text{* Theorems for @{text "\<exists>z. \<forall>x. x < z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"}*}
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lemma pinf_gt[no_atp]:  "\<exists>z . \<forall>x. z < x \<longrightarrow> (t < x \<longleftrightarrow> True)" by auto
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lemma pinf_lt[no_atp]: "\<exists>z . \<forall>x. z < x \<longrightarrow>  (x < t \<longleftrightarrow>  False)"
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  by (simp add: not_less) (rule exI[where x="t"], auto simp add: less_le)
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lemma pinf_ge[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (t \<le> x \<longleftrightarrow> True)" by (auto simp add: less_le)
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lemma pinf_le[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<le> t \<longleftrightarrow> False)"
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  by (auto simp add: less_le not_less not_le)
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lemma pinf_eq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x = t \<longleftrightarrow> False)" by auto
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lemma pinf_neq[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (x \<noteq> t \<longleftrightarrow> True)" by auto
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lemma pinf_P[no_atp]: "\<exists>z. \<forall>x. z < x \<longrightarrow> (P \<longleftrightarrow> P)" by blast
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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
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lemma nmi_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x. \<not>False \<and> t < x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)"
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  by (auto simp add: le_less)
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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
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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
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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
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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
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lemma  nmi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
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lemma  nmi_conj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
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  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
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  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
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lemma  nmi_disj[no_atp]: "\<lbrakk>\<forall>x. \<not>P1' \<and> P1 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x) ;
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  \<forall>x. \<not>P2' \<and> P2 x \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)\<rbrakk> \<Longrightarrow>
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  \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. u \<le> x)" by auto
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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)
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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
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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
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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
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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
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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
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lemma  npi_P[no_atp]: "\<forall> x. ~P \<and> P \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
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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>
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  \<Longrightarrow>  \<forall>x. \<not>(P1' \<and> P2') \<and> (P1 x \<and> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
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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>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    84
  \<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow>  (\<exists> u\<in> U. x \<le> u)" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    85
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
    86
lemma lin_dense_lt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t < u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x < t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y < t)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    87
proof(clarsimp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    88
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    89
    and xu: "x<u"  and px: "x < t" and ly: "l<y" and yu:"y < u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    90
  from tU noU ly yu have tny: "t\<noteq>y" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    91
  {assume H: "t < y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    92
    from less_trans[OF lx px] less_trans[OF H yu]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    93
    have "l < t \<and> t < u"  by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    94
    with tU noU have "False" by auto}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    95
  hence "\<not> t < y"  by auto hence "y \<le> t" by (simp add: not_less)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    96
  thus "y < t" using tny by (simp add: less_le)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    97
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
    98
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
    99
lemma lin_dense_gt[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l < x \<and> x < u \<and> t < x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t < y)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   100
proof(clarsimp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   101
  fix x l u y
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   102
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   103
  and px: "t < x" and ly: "l<y" and yu:"y < u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   104
  from tU noU ly yu have tny: "t\<noteq>y" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   105
  {assume H: "y< t"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   106
    from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   107
    with tU noU have "False" by auto}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   108
  hence "\<not> y<t"  by auto hence "t \<le> y" by (auto simp add: not_less)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   109
  thus "t < y" using tny by (simp add:less_le)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   110
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   111
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   112
lemma lin_dense_le[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<le> t \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<le> t)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   113
proof(clarsimp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   114
  fix x l u y
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   115
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   116
  and px: "x \<le> t" and ly: "l<y" and yu:"y < u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   117
  from tU noU ly yu have tny: "t\<noteq>y" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   118
  {assume H: "t < y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   119
    from less_le_trans[OF lx px] less_trans[OF H yu]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   120
    have "l < t \<and> t < u" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   121
    with tU noU have "False" by auto}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   122
  hence "\<not> t < y"  by auto thus "y \<le> t" by (simp add: not_less)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   123
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   124
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   125
lemma lin_dense_ge[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> t \<le> x \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> t \<le> y)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   126
proof(clarsimp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   127
  fix x l u y
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   128
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   129
  and px: "t \<le> x" and ly: "l<y" and yu:"y < u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   130
  from tU noU ly yu have tny: "t\<noteq>y" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   131
  {assume H: "y< t"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   132
    from less_trans[OF ly H] le_less_trans[OF px xu]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   133
    have "l < t \<and> t < u" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   134
    with tU noU have "False" by auto}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   135
  hence "\<not> y<t"  by auto thus "t \<le> y" by (simp add: not_less)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   136
qed
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   137
lemma lin_dense_eq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x = t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y= t)"  by auto
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   138
lemma lin_dense_neq[no_atp]: "t \<in> U \<Longrightarrow> \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> x \<noteq> t   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> y\<noteq> t)"  by auto
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   139
lemma lin_dense_P[no_atp]: "\<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P   \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P)"  by auto
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   140
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   141
lemma lin_dense_conj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   142
  "\<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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   143
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   144
  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   145
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   146
  \<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)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   147
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<and> P2 y))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   148
  by blast
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   149
lemma lin_dense_disj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   150
  "\<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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   151
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P1 y) ;
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   152
  \<forall>x l u. (\<forall> t. l < t \<and> t< u \<longrightarrow> t \<notin> U) \<and> l< x \<and> x < u \<and> P2 x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   153
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> P2 y)\<rbrakk> \<Longrightarrow>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   154
  \<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)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   155
  \<longrightarrow> (\<forall> y. l < y \<and> y < u \<longrightarrow> (P1 y \<or> P2 y))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   156
  by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   157
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   158
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>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   159
  \<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')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   160
by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   161
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   162
lemma finite_set_intervals[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   163
  assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   164
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   165
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   166
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   167
  let ?Mx = "{y. y\<in> S \<and> y \<le> x}"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   168
  let ?xM = "{y. y\<in> S \<and> x \<le> y}"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   169
  let ?a = "Max ?Mx"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   170
  let ?b = "Min ?xM"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   171
  have MxS: "?Mx \<subseteq> S" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   172
  hence fMx: "finite ?Mx" using fS finite_subset by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   173
  from lx linS have linMx: "l \<in> ?Mx" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   174
  hence Mxne: "?Mx \<noteq> {}" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   175
  have xMS: "?xM \<subseteq> S" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   176
  hence fxM: "finite ?xM" using fS finite_subset by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   177
  from xu uinS have linxM: "u \<in> ?xM" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   178
  hence xMne: "?xM \<noteq> {}" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   179
  have ax:"?a \<le> x" using Mxne fMx by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   180
  have xb:"x \<le> ?b" using xMne fxM by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   181
  have "?a \<in> ?Mx" using Max_in[OF fMx Mxne] by simp hence ainS: "?a \<in> S" using MxS by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   182
  have "?b \<in> ?xM" using Min_in[OF fxM xMne] by simp hence binS: "?b \<in> S" using xMS by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   183
  have noy:"\<forall> y. ?a < y \<and> y < ?b \<longrightarrow> y \<notin> S"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   184
  proof(clarsimp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   185
    fix y   assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   186
    from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   187
    moreover {assume "y \<in> ?Mx" hence "y \<le> ?a" using Mxne fMx by auto with ay have "False" by (simp add: not_le[symmetric])}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   188
    moreover {assume "y \<in> ?xM" hence "?b \<le> y" using xMne fxM by auto with yb have "False" by (simp add: not_le[symmetric])}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   189
    ultimately show "False" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   190
  qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   191
  from ainS binS noy ax xb px show ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   192
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   193
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   194
lemma finite_set_intervals2[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   195
  assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   196
  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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   197
  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)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   198
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   199
  from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   200
  obtain a and b where
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   201
    as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   202
    and axb: "a \<le> x \<and> x \<le> b \<and> P x"  by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   203
  from axb have "x= a \<or> x= b \<or> (a < x \<and> x < b)" by (auto simp add: le_less)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   204
  thus ?thesis using px as bs noS by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   205
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   206
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   207
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   208
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   209
section {* The classical QE after Langford for dense linear orders *}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   210
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   211
context dense_linorder
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   212
begin
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   213
27825
12254665fc41 re-arranged class dense_linear_order
haftmann
parents: 27487
diff changeset
   214
lemma interval_empty_iff:
12254665fc41 re-arranged class dense_linear_order
haftmann
parents: 27487
diff changeset
   215
  "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z"
12254665fc41 re-arranged class dense_linear_order
haftmann
parents: 27487
diff changeset
   216
  by (auto dest: dense)
12254665fc41 re-arranged class dense_linear_order
haftmann
parents: 27487
diff changeset
   217
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   218
lemma dlo_qe_bnds[no_atp]: 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   219
  assumes ne: "L \<noteq> {}" and neU: "U \<noteq> {}" and fL: "finite L" and fU: "finite U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   220
  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)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   221
proof (simp only: atomize_eq, rule iffI)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   222
  assume H: "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   223
  then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   224
  {fix l u assume l: "l \<in> L" and u: "u \<in> U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   225
    have "l < x" using xL l by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   226
    also have "x < u" using xU u by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   227
    finally (less_trans) have "l < u" .}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   228
  thus "\<forall>l\<in>L. \<forall>u\<in>U. l < u" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   229
next
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   230
  assume H: "\<forall>l\<in>L. \<forall>u\<in>U. l < u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   231
  let ?ML = "Max L"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   232
  let ?MU = "Min U"  
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   233
  from fL ne have th1: "?ML \<in> L" and th1': "\<forall>l\<in>L. l \<le> ?ML" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   234
  from fU neU have th2: "?MU \<in> U" and th2': "\<forall>u\<in>U. ?MU \<le> u" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   235
  from th1 th2 H have "?ML < ?MU" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   236
  with dense obtain w where th3: "?ML < w" and th4: "w < ?MU" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   237
  from th3 th1' have "\<forall>l \<in> L. l < w" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   238
  moreover from th4 th2' have "\<forall>u \<in> U. w < u" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   239
  ultimately show "\<exists>x. (\<forall>y\<in>L. y < x) \<and> (\<forall>y\<in>U. x < y)" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   240
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   241
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   242
lemma dlo_qe_noub[no_atp]: 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   243
  assumes ne: "L \<noteq> {}" and fL: "finite L"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   244
  shows "(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> {}. x < y)) \<equiv> True"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   245
proof(simp add: atomize_eq)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   246
  from gt_ex[of "Max L"] obtain M where M: "Max L < M" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   247
  from ne fL have "\<forall>x \<in> L. x \<le> Max L" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   248
  with M have "\<forall>x\<in>L. x < M" by (auto intro: le_less_trans)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   249
  thus "\<exists>x. \<forall>y\<in>L. y < x" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   250
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   251
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   252
lemma dlo_qe_nolb[no_atp]: 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   253
  assumes ne: "U \<noteq> {}" and fU: "finite U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   254
  shows "(\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y \<in> U. x < y)) \<equiv> True"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   255
proof(simp add: atomize_eq)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   256
  from lt_ex[of "Min U"] obtain M where M: "M < Min U" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   257
  from ne fU have "\<forall>x \<in> U. Min U \<le> x" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   258
  with M have "\<forall>x\<in>U. M < x" by (auto intro: less_le_trans)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   259
  thus "\<exists>x. \<forall>y\<in>U. x < y" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   260
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   261
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   262
lemma exists_neq[no_atp]: "\<exists>(x::'a). x \<noteq> t" "\<exists>(x::'a). t \<noteq> x" 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   263
  using gt_ex[of t] by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   264
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   265
lemmas dlo_simps[no_atp] = order_refl less_irrefl not_less not_le exists_neq 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   266
  le_less neq_iff linear less_not_permute
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   267
36635
080b755377c0 locale predicates of classes carry a mandatory "class" prefix
haftmann
parents: 35828
diff changeset
   268
lemma axiom[no_atp]: "class.dense_linorder (op \<le>) (op <)" by (rule dense_linorder_axioms)
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   269
lemma atoms[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   270
  shows "TERM (less :: 'a \<Rightarrow> _)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   271
    and "TERM (less_eq :: 'a \<Rightarrow> _)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   272
    and "TERM (op = :: 'a \<Rightarrow> _)" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   273
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   274
declare axiom[langford qe: dlo_qe_bnds dlo_qe_nolb dlo_qe_noub gather: gather_start gather_simps atoms: atoms]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   275
declare dlo_simps[langfordsimp]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   276
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   277
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   278
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   279
(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *)
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   280
lemma dnf[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   281
  "(P & (Q | R)) = ((P&Q) | (P&R))" 
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   282
  "((Q | R) & P) = ((Q&P) | (R&P))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   283
  by blast+
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   284
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   285
lemmas weak_dnf_simps[no_atp] = simp_thms dnf
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   286
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   287
lemma nnf_simps[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   288
    "(\<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)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   289
    "(P = Q) = ((P \<and> Q) \<or> (\<not>P \<and> \<not> Q))" "(\<not> \<not>(P)) = P"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   290
  by blast+
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   291
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   292
lemma ex_distrib[no_atp]: "(\<exists>x. P x \<or> Q x) \<longleftrightarrow> ((\<exists>x. P x) \<or> (\<exists>x. Q x))" by blast
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   293
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   294
lemmas dnf_simps[no_atp] = weak_dnf_simps nnf_simps ex_distrib
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   295
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   296
use "~~/src/HOL/Tools/Qelim/langford.ML"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   297
method_setup dlo = {*
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   298
  Scan.succeed (SIMPLE_METHOD' o LangfordQE.dlo_tac)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   299
*} "Langford's algorithm for quantifier elimination in dense linear orders"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   300
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   301
30652
752329615264 distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents: 30549
diff changeset
   302
section {* Contructive dense linear orders yield QE for linear arithmetic over ordered Fields *}
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   303
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   304
text {* Linear order without upper bounds *}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   305
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
   306
locale linorder_stupid_syntax = linorder
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   307
begin
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   308
notation
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   309
  less_eq  ("op \<sqsubseteq>") and
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   310
  less_eq  ("(_/ \<sqsubseteq> _)" [51, 51] 50) and
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   311
  less  ("op \<sqsubset>") and
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   312
  less  ("(_/ \<sqsubset> _)"  [51, 51] 50)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   313
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   314
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   315
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
   316
locale linorder_no_ub = linorder_stupid_syntax +
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   317
  assumes gt_ex: "\<exists>y. less x y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   318
begin
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   319
lemma ge_ex[no_atp]: "\<exists>y. x \<sqsubseteq> y" using gt_ex by auto
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   320
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   321
text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>+\<infinity>\<^esub>)"} *}
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   322
lemma pinf_conj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   323
  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   324
  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   325
  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   326
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   327
  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   328
     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   329
  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   330
  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   331
  {fix x assume H: "z \<sqsubset> x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   332
    from less_trans[OF zz1 H] less_trans[OF zz2 H]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   333
    have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   334
  }
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   335
  thus ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   336
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   337
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   338
lemma pinf_disj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   339
  assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   340
  and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   341
  shows "\<exists>z. \<forall>x. z \<sqsubset>  x \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   342
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   343
  from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   344
     and z2: "\<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   345
  from gt_ex obtain z where z:"ord.max less_eq z1 z2 \<sqsubset> z" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   346
  from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   347
  {fix x assume H: "z \<sqsubset> x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   348
    from less_trans[OF zz1 H] less_trans[OF zz2 H]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   349
    have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   350
  }
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   351
  thus ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   352
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   353
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   354
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"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   355
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   356
  from ex obtain z where z: "\<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   357
  from gt_ex obtain x where x: "z \<sqsubset> x" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   358
  from z x p1 show ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   359
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   360
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   361
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   362
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   363
text {* Linear order without upper bounds *}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   364
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
   365
locale linorder_no_lb = linorder_stupid_syntax +
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   366
  assumes lt_ex: "\<exists>y. less y x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   367
begin
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   368
lemma le_ex[no_atp]: "\<exists>y. y \<sqsubseteq> x" using lt_ex by auto
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   369
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   370
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   371
text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^bsub>-\<infinity>\<^esub>)"} *}
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   372
lemma minf_conj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   373
  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   374
  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   375
  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2'))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   376
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   377
  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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   378
  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   379
  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   380
  {fix x assume H: "x \<sqsubset> z"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   381
    from less_trans[OF H zz1] less_trans[OF H zz2]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   382
    have "(P1 x \<and> P2 x) \<longleftrightarrow> (P1' \<and> P2')"  using z1 zz1 z2 zz2 by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   383
  }
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   384
  thus ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   385
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   386
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   387
lemma minf_disj[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   388
  assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   389
  and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   390
  shows "\<exists>z. \<forall>x. x \<sqsubset>  z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   391
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   392
  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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   393
  from lt_ex obtain z where z:"z \<sqsubset> ord.min less_eq z1 z2" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   394
  from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   395
  {fix x assume H: "x \<sqsubset> z"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   396
    from less_trans[OF H zz1] less_trans[OF H zz2]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   397
    have "(P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2')"  using z1 zz1 z2 zz2 by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   398
  }
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   399
  thus ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   400
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   401
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   402
lemma minf_ex[no_atp]: assumes ex:"\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" and p1: P1 shows "\<exists> x. P x"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   403
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   404
  from ex obtain z where z: "\<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   405
  from lt_ex obtain x where x: "x \<sqsubset> z" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   406
  from z x p1 show ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   407
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   408
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   409
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   410
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   411
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   412
locale constr_dense_linorder = linorder_no_lb + linorder_no_ub +
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   413
  fixes between
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   414
  assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   415
     and  between_same: "between x x = x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   416
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   417
sublocale  constr_dense_linorder < dense_linorder 
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   418
  apply unfold_locales
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   419
  using gt_ex lt_ex between_less
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   420
    by (auto, rule_tac x="between x y" in exI, simp)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   421
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   422
context  constr_dense_linorder
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   423
begin
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   424
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   425
lemma rinf_U[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   426
  assumes fU: "finite U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   427
  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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   428
  \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   429
  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')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   430
  and nmi: "\<not> MP"  and npi: "\<not> PP"  and ex: "\<exists> x.  P x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   431
  shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   432
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   433
  from ex obtain x where px: "P x" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   434
  from px nmi npi nmpiU have "\<exists> u\<in> U. \<exists> u' \<in> U. u \<sqsubseteq> x \<and> x \<sqsubseteq> u'" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   435
  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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   436
  from uU have Une: "U \<noteq> {}" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   437
  term "linorder.Min less_eq"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   438
  let ?l = "linorder.Min less_eq U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   439
  let ?u = "linorder.Max less_eq U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   440
  have linM: "?l \<in> U" using fU Une by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   441
  have uinM: "?u \<in> U" using fU Une by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   442
  have lM: "\<forall> t\<in> U. ?l \<sqsubseteq> t" using Une fU by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   443
  have Mu: "\<forall> t\<in> U. t \<sqsubseteq> ?u" using Une fU by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   444
  have th:"?l \<sqsubseteq> u" using uU Une lM by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   445
  from order_trans[OF th ux] have lx: "?l \<sqsubseteq> x" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   446
  have th: "u' \<sqsubseteq> ?u" using uU' Une Mu by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   447
  from order_trans[OF xu' th] have xu: "x \<sqsubseteq> ?u" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   448
  from finite_set_intervals2[where P="P",OF px lx xu linM uinM fU lM Mu]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   449
  have "(\<exists> s\<in> U. P s) \<or>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   450
      (\<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)" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   451
  moreover { fix u assume um: "u\<in>U" and pu: "P u"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   452
    have "between u u = u" by (simp add: between_same)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   453
    with um pu have "P (between u u)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   454
    with um have ?thesis by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   455
  moreover{
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   456
    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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   457
      then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   458
        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"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   459
        by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   460
      from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   461
      let ?u = "between t1 t2"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   462
      from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   463
      from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   464
      with t1M t2M have ?thesis by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   465
    ultimately show ?thesis by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   466
  qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   467
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   468
theorem fr_eq[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   469
  assumes fU: "finite U"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   470
  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
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   471
   \<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   472
  and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   473
  and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   474
  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)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   475
  shows "(\<exists> x. P x) \<equiv> (MP \<or> PP \<or> (\<exists> u \<in> U. \<exists> u'\<in> U. P (between u u')))"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   476
  (is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D")
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   477
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   478
 {
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   479
   assume px: "\<exists> x. P x"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   480
   have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   481
   moreover {assume "MP \<or> PP" hence "?D" by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   482
   moreover {assume nmi: "\<not> MP" and npi: "\<not> PP"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   483
     from npmibnd[OF nmibnd npibnd]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   484
     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')" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   485
     from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   486
   ultimately have "?D" by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   487
 moreover
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   488
 { assume "?D"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   489
   moreover {assume m:"MP" from minf_ex[OF mi m] have "?E" .}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   490
   moreover {assume p: "PP" from pinf_ex[OF pi p] have "?E" . }
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   491
   moreover {assume f:"?F" hence "?E" by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   492
   ultimately have "?E" by blast}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   493
 ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   494
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   495
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   496
lemmas minf_thms[no_atp] = minf_conj minf_disj minf_eq minf_neq minf_lt minf_le minf_gt minf_ge minf_P
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   497
lemmas pinf_thms[no_atp] = pinf_conj pinf_disj pinf_eq pinf_neq pinf_lt pinf_le pinf_gt pinf_ge pinf_P
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   498
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   499
lemmas nmi_thms[no_atp] = nmi_conj nmi_disj nmi_eq nmi_neq nmi_lt nmi_le nmi_gt nmi_ge nmi_P
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   500
lemmas npi_thms[no_atp] = npi_conj npi_disj npi_eq npi_neq npi_lt npi_le npi_gt npi_ge npi_P
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   501
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
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   502
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   503
lemma ferrack_axiom[no_atp]: "constr_dense_linorder less_eq less between"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   504
  by (rule constr_dense_linorder_axioms)
35828
46cfc4b8112e now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents: 35267
diff changeset
   505
lemma atoms[no_atp]:
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   506
  shows "TERM (less :: 'a \<Rightarrow> _)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   507
    and "TERM (less_eq :: 'a \<Rightarrow> _)"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   508
    and "TERM (op = :: 'a \<Rightarrow> _)" .
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   509
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   510
declare ferrack_axiom [ferrack minf: minf_thms pinf: pinf_thms
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   511
    nmi: nmi_thms npi: npi_thms lindense:
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   512
    lin_dense_thms qe: fr_eq atoms: atoms]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   513
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   514
declaration {*
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   515
let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   516
fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   517
fun generic_whatis phi =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   518
 let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   519
  val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   520
  fun h x t =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   521
   case term_of t of
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   522
     Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   523
                            else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   524
   | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   525
                            else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   526
   | b$y$z => if Term.could_unify (b, lt) then
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   527
                 if term_of x aconv y then Ferrante_Rackoff_Data.Lt
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   528
                 else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   529
                 else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   530
             else if Term.could_unify (b, le) then
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   531
                 if term_of x aconv y then Ferrante_Rackoff_Data.Le
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   532
                 else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   533
                 else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   534
             else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   535
   | _ => Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   536
 in h end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   537
 fun ss phi = HOL_ss addsimps (simps phi)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   538
in
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   539
 Ferrante_Rackoff_Data.funs  @{thm "ferrack_axiom"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   540
  {isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   541
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   542
*}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   543
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   544
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   545
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   546
use "~~/src/HOL/Tools/Qelim/ferrante_rackoff.ML"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   547
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   548
method_setup ferrack = {*
30549
d2d7874648bd simplified method setup;
wenzelm
parents: 30510
diff changeset
   549
  Scan.succeed (SIMPLE_METHOD' o FerranteRackoff.dlo_tac)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   550
*} "Ferrante and Rackoff's algorithm for quantifier elimination in dense linear orders"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   551
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   552
subsection {* Ferrante and Rackoff algorithm over ordered fields *}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   553
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   554
lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   555
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   556
  assume H: "c < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   557
  have "c*x < 0 = (0/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   558
  also have "\<dots> = (0 < x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   559
  finally show  "(c*x < 0) == (x > 0)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   560
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   561
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   562
lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   563
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   564
  assume H: "c > 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   565
  hence "c*x < 0 = (0/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   566
  also have "\<dots> = (0 > x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   567
  finally show  "(c*x < 0) == (x < 0)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   568
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   569
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   570
lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   571
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   572
  assume H: "c < 0"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   573
  have "c*x + t< 0 = (c*x < -t)" by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   574
  also have "\<dots> = (-t/c < x)" by (simp only: neg_divide_less_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   575
  also have "\<dots> = ((- 1/c)*t < x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   576
  finally show  "(c*x + t < 0) == (x > (- 1/c)*t)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   577
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   578
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   579
lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   580
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   581
  assume H: "c > 0"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   582
  have "c*x + t< 0 = (c*x < -t)"  by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   583
  also have "\<dots> = (-t/c > x)" by (simp only: pos_less_divide_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   584
  also have "\<dots> = ((- 1/c)*t > x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   585
  finally show  "(c*x + t < 0) == (x < (- 1/c)*t)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   586
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   587
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   588
lemma sum_lt:"((x::'a::ordered_ab_group_add) + t < 0) == (x < - t)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   589
  using less_diff_eq[where a= x and b=t and c=0] by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   590
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   591
lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   592
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   593
  assume H: "c < 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   594
  have "c*x <= 0 = (0/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   595
  also have "\<dots> = (0 <= x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   596
  finally show  "(c*x <= 0) == (x >= 0)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   597
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   598
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   599
lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   600
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   601
  assume H: "c > 0"
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   602
  hence "c*x <= 0 = (0/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   603
  also have "\<dots> = (0 >= x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   604
  finally show  "(c*x <= 0) == (x <= 0)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   605
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   606
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   607
lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   608
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   609
  assume H: "c < 0"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   610
  have "c*x + t <= 0 = (c*x <= -t)"  by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   611
  also have "\<dots> = (-t/c <= x)" by (simp only: neg_divide_le_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   612
  also have "\<dots> = ((- 1/c)*t <= x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   613
  finally show  "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   614
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   615
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   616
lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   617
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   618
  assume H: "c > 0"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   619
  have "c*x + t <= 0 = (c*x <= -t)" by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   620
  also have "\<dots> = (-t/c >= x)" by (simp only: pos_le_divide_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   621
  also have "\<dots> = ((- 1/c)*t >= x)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   622
  finally show  "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   623
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   624
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   625
lemma sum_le:"((x::'a::ordered_ab_group_add) + t <= 0) == (x <= - t)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   626
  using le_diff_eq[where a= x and b=t and c=0] by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   627
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   628
lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   629
lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   630
proof-
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   631
  assume H: "c \<noteq> 0"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   632
  have "c*x + t = 0 = (c*x = -t)" by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp)
29667
53103fc8ffa3 Replaced group_ and ring_simps by algebra_simps;
nipkow
parents: 29509
diff changeset
   633
  also have "\<dots> = (x = -t/c)" by (simp only: nonzero_eq_divide_eq[OF H] algebra_simps)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   634
  finally show  "(c*x + t = 0) == (x = (- 1/c)*t)" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   635
qed
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   636
lemma sum_eq:"((x::'a::ordered_ab_group_add) + t = 0) == (x = - t)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   637
  using eq_diff_eq[where a= x and b=t and c=0] by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   638
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   639
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35028
diff changeset
   640
interpretation class_dense_linordered_field: constr_dense_linorder
29509
1ff0f3f08a7b migrated class package to new locale implementation
haftmann
parents: 29252
diff changeset
   641
 "op <=" "op <"
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34974
diff changeset
   642
   "\<lambda> x y. 1/2 * ((x::'a::{linordered_field,number_ring}) + y)"
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   643
proof (unfold_locales, dlo, dlo, auto)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   644
  fix x y::'a assume lt: "x < y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   645
  from  less_half_sum[OF lt] show "x < (x + y) /2" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   646
next
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   647
  fix x y::'a assume lt: "x < y"
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   648
  from  gt_half_sum[OF lt] show "(x + y) /2 < y" by simp
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   649
qed
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   650
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   651
declaration{*
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   652
let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   653
fun earlier [] x y = false
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   654
        | earlier (h::t) x y =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   655
    if h aconvc y then false else if h aconvc x then true else earlier t x y;
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   656
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   657
fun dest_frac ct = case term_of ct of
35084
e25eedfc15ce moved constants inverse and divide to Ring.thy
haftmann
parents: 35043
diff changeset
   658
   Const (@{const_name Rings.divide},_) $ a $ b=>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   659
    Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b))
30868
1040425c86a2 fixed usage of rational constants
chaieb
parents: 30741
diff changeset
   660
 | Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   661
 | t => Rat.rat_of_int (snd (HOLogic.dest_number t))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   662
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   663
fun mk_frac phi cT x =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   664
 let val (a, b) = Rat.quotient_of_rat x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   665
 in if b = 1 then Numeral.mk_cnumber cT a
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   666
    else Thm.capply
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   667
         (Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"})
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   668
                     (Numeral.mk_cnumber cT a))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   669
         (Numeral.mk_cnumber cT b)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   670
 end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   671
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   672
fun whatis x ct = case term_of ct of
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   673
  Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   674
     if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   675
     else ("Nox",[])
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   676
| Const(@{const_name Groups.plus}, _)$y$_ =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   677
     if y aconv term_of x then ("x+t",[Thm.dest_arg ct])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   678
     else ("Nox",[])
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   679
| Const(@{const_name Groups.times}, _)$_$y =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   680
     if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   681
     else ("Nox",[])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   682
| t => if t aconv term_of x then ("x",[]) else ("Nox",[]);
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   683
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   684
fun xnormalize_conv ctxt [] ct = reflexive ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   685
| xnormalize_conv ctxt (vs as (x::_)) ct =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   686
   case term_of ct of
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   687
   Const(@{const_name Orderings.less},_)$_$Const(@{const_name Groups.zero},_) =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   688
    (case whatis x (Thm.dest_arg1 ct) of
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   689
    ("c*x+t",[c,t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   690
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   691
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   692
        val clt = Thm.dest_fun2 ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   693
        val cz = Thm.dest_arg ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   694
        val neg = cr </ Rat.zero
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   695
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   696
               (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   697
                  (if neg then Thm.capply (Thm.capply clt c) cz
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   698
                    else Thm.capply (Thm.capply clt cz) c))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   699
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   700
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x,t])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   701
             (if neg then @{thm neg_prod_sum_lt} else @{thm pos_prod_sum_lt})) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   702
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   703
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   704
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   705
    | ("x+t",[t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   706
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   707
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   708
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_lt"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   709
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   710
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   711
       in  rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   712
    | ("c*x",[c]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   713
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   714
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   715
        val clt = Thm.dest_fun2 ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   716
        val cz = Thm.dest_arg ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   717
        val neg = cr </ Rat.zero
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   718
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   719
               (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   720
                  (if neg then Thm.capply (Thm.capply clt c) cz
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   721
                    else Thm.capply (Thm.capply clt cz) c))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   722
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   723
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   724
             (if neg then @{thm neg_prod_lt} else @{thm pos_prod_lt})) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   725
        val rth = th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   726
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   727
    | _ => reflexive ct)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   728
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   729
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   730
|  Const(@{const_name Orderings.less_eq},_)$_$Const(@{const_name Groups.zero},_) =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   731
   (case whatis x (Thm.dest_arg1 ct) of
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   732
    ("c*x+t",[c,t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   733
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   734
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   735
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   736
        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   737
        val cz = Thm.dest_arg ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   738
        val neg = cr </ Rat.zero
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   739
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   740
               (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   741
                  (if neg then Thm.capply (Thm.capply clt c) cz
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   742
                    else Thm.capply (Thm.capply clt cz) c))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   743
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   744
        val th = implies_elim (instantiate' [SOME T] (map SOME [c,x,t])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   745
             (if neg then @{thm neg_prod_sum_le} else @{thm pos_prod_sum_le})) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   746
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   747
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   748
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   749
    | ("x+t",[t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   750
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   751
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   752
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_le"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   753
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   754
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   755
       in  rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   756
    | ("c*x",[c]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   757
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   758
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   759
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   760
        val clt = Drule.cterm_rule (instantiate' [SOME T] []) @{cpat "op <"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   761
        val cz = Thm.dest_arg ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   762
        val neg = cr </ Rat.zero
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   763
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   764
               (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   765
                  (if neg then Thm.capply (Thm.capply clt c) cz
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   766
                    else Thm.capply (Thm.capply clt cz) c))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   767
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   768
        val th = implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,x])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   769
             (if neg then @{thm neg_prod_le} else @{thm pos_prod_le})) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   770
        val rth = th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   771
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   772
    | _ => reflexive ct)
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   773
35267
8dfd816713c6 moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents: 35092
diff changeset
   774
|  Const("op =",_)$_$Const(@{const_name Groups.zero},_) =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   775
   (case whatis x (Thm.dest_arg1 ct) of
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   776
    ("c*x+t",[c,t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   777
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   778
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   779
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   780
        val ceq = Thm.dest_fun2 ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   781
        val cz = Thm.dest_arg ct
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   782
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   783
            (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   784
             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   785
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   786
        val th = implies_elim
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   787
                 (instantiate' [SOME T] (map SOME [c,x,t]) @{thm nz_prod_sum_eq}) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   788
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   789
                   (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   790
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   791
    | ("x+t",[t]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   792
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   793
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   794
        val th = instantiate' [SOME T] [SOME x, SOME t] @{thm "sum_eq"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   795
        val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   796
              (Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   797
       in  rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   798
    | ("c*x",[c]) =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   799
       let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   800
        val T = ctyp_of_term x
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   801
        val cr = dest_frac c
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   802
        val ceq = Thm.dest_fun2 ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   803
        val cz = Thm.dest_arg ct
32149
ef59550a55d3 renamed simpset_of to global_simpset_of, and local_simpset_of to simpset_of -- same for claset and clasimpset;
wenzelm
parents: 31021
diff changeset
   804
        val cthp = Simplifier.rewrite (simpset_of ctxt)
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   805
            (Thm.capply @{cterm "Trueprop"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   806
             (Thm.capply @{cterm "Not"} (Thm.capply (Thm.capply ceq c) cz)))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   807
        val cth = equal_elim (symmetric cthp) TrueI
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   808
        val rth = implies_elim
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   809
                 (instantiate' [SOME T] (map SOME [c,x]) @{thm nz_prod_eq}) cth
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   810
      in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   811
    | _ => reflexive ct);
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   812
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   813
local
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   814
  val less_iff_diff_less_0 = mk_meta_eq @{thm "less_iff_diff_less_0"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   815
  val le_iff_diff_le_0 = mk_meta_eq @{thm "le_iff_diff_le_0"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   816
  val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   817
in
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   818
fun field_isolate_conv phi ctxt vs ct = case term_of ct of
35092
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35084
diff changeset
   819
  Const(@{const_name Orderings.less},_)$a$b =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   820
   let val (ca,cb) = Thm.dest_binop ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   821
       val T = ctyp_of_term ca
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   822
       val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   823
       val nth = Conv.fconv_rule
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   824
         (Conv.arg_conv (Conv.arg1_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   825
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   826
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   827
   in rth end
35092
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35084
diff changeset
   828
| Const(@{const_name Orderings.less_eq},_)$a$b =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   829
   let val (ca,cb) = Thm.dest_binop ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   830
       val T = ctyp_of_term ca
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   831
       val th = instantiate' [SOME T] [SOME ca, SOME cb] le_iff_diff_le_0
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   832
       val nth = Conv.fconv_rule
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   833
         (Conv.arg_conv (Conv.arg1_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   834
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   835
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   836
   in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   837
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   838
| Const("op =",_)$a$b =>
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   839
   let val (ca,cb) = Thm.dest_binop ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   840
       val T = ctyp_of_term ca
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   841
       val th = instantiate' [SOME T] [SOME ca, SOME cb] eq_iff_diff_eq_0
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   842
       val nth = Conv.fconv_rule
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   843
         (Conv.arg_conv (Conv.arg1_conv
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   844
              (Normalizer.semiring_normalize_ord_conv @{context} (earlier vs)))) th
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   845
       val rth = transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth))
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   846
   in rth end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   847
| @{term "Not"} $(Const("op =",_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   848
| _ => reflexive ct
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   849
end;
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   850
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   851
fun classfield_whatis phi =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   852
 let
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   853
  fun h x t =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   854
   case term_of t of
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   855
     Const("op =", _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   856
                            else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   857
   | @{term "Not"}$(Const("op =", _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.NEq
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   858
                            else Ferrante_Rackoff_Data.Nox
35092
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35084
diff changeset
   859
   | Const(@{const_name Orderings.less},_)$y$z =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   860
       if term_of x aconv y then Ferrante_Rackoff_Data.Lt
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   861
        else if term_of x aconv z then Ferrante_Rackoff_Data.Gt
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   862
        else Ferrante_Rackoff_Data.Nox
35092
cfe605c54e50 moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents: 35084
diff changeset
   863
   | Const (@{const_name Orderings.less_eq},_)$y$z =>
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   864
         if term_of x aconv y then Ferrante_Rackoff_Data.Le
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   865
         else if term_of x aconv z then Ferrante_Rackoff_Data.Ge
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   866
         else Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   867
   | _ => Ferrante_Rackoff_Data.Nox
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   868
 in h end;
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   869
fun class_field_ss phi =
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   870
   HOL_basic_ss addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}])
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   871
   addsplits [@{thm "abs_split"},@{thm "split_max"}, @{thm "split_min"}]
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   872
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   873
in
35043
07dbdf60d5ad dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents: 35028
diff changeset
   874
Ferrante_Rackoff_Data.funs @{thm "class_dense_linordered_field.ferrack_axiom"}
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   875
  {isolate_conv = field_isolate_conv, whatis = classfield_whatis, simpset = class_field_ss}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   876
end
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   877
*}
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   878
29841
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   879
lemma upper_bound_finite_set:
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   880
  assumes fS: "finite S"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   881
  shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<le> a"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   882
proof(induct rule: finite_induct[OF fS])
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   883
  case 1 thus ?case by simp
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   884
next
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   885
  case (2 x F)
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   886
  from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<le> a" by blast
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   887
  let ?a = "max a (f x)"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   888
  have m: "a \<le> ?a" "f x \<le> ?a" by simp_all
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   889
  {fix y assume y: "y \<in> insert x F"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   890
    {assume "y = x" hence "f y \<le> ?a" using m by simp}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   891
    moreover
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   892
    {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<le> ?a" by (simp add: max_def)}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   893
    ultimately have "f y \<le> ?a" using y by blast}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   894
  then show ?case by blast
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   895
qed
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   896
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   897
lemma lower_bound_finite_set:
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   898
  assumes fS: "finite S"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   899
  shows "\<exists>(a::'a::linorder). \<forall>x \<in> S. f x \<ge> a"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   900
proof(induct rule: finite_induct[OF fS])
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   901
  case 1 thus ?case by simp
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   902
next
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   903
  case (2 x F)
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   904
  from "2.hyps" obtain a where a:"\<forall>x \<in>F. f x \<ge> a" by blast
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   905
  let ?a = "min a (f x)"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   906
  have m: "a \<ge> ?a" "f x \<ge> ?a" by simp_all
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   907
  {fix y assume y: "y \<in> insert x F"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   908
    {assume "y = x" hence "f y \<ge> ?a" using m by simp}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   909
    moreover
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   910
    {assume yF: "y\<in> F" from a[rule_format, OF yF] m have "f y \<ge> ?a" by (simp add: min_def)}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   911
    ultimately have "f y \<ge> ?a" using y by blast}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   912
  then show ?case by blast
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   913
qed
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   914
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   915
lemma bound_finite_set: assumes f: "finite S"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   916
  shows "\<exists>a. \<forall>x \<in>S. (f x:: 'a::linorder) \<le> a"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   917
proof-
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   918
  let ?F = "f ` S"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   919
  from f have fF: "finite ?F" by simp
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   920
  let ?a = "Max ?F"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   921
  {assume "S = {}" hence ?thesis by blast}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   922
  moreover
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   923
  {assume Se: "S \<noteq> {}" hence Fe: "?F \<noteq> {}" by simp
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   924
  {fix x assume x: "x \<in> S"
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   925
    hence th0: "f x \<in> ?F" by simp
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   926
    hence "f x \<le> ?a" using Max_ge[OF fF th0] ..}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   927
  hence ?thesis by blast}
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   928
ultimately show ?thesis by blast
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   929
qed
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   930
86d94bb79226 A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents: 29823
diff changeset
   931
26161
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   932
34cb0b457dcc Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff changeset
   933
end