author | wenzelm |
Fri, 22 Nov 2013 20:37:19 +0100 | |
changeset 54560 | 7f36da77130d |
parent 53215 | 5e47c31c6f7c |
child 54868 | bab6cade3cc5 |
permissions | -rw-r--r-- |
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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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begin |
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ML_file "langford_data.ML" |
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ML_file "ferrante_rackoff_data.ML" |
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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)" |
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by (simp add: not_less linear) |
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lemma gather_simps[no_atp]: |
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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> |
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(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u 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> l < x \<and> P x) \<longleftrightarrow> |
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(\<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> |
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(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> (insert u U). x < y))" |
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"(\<exists>x. (\<forall>y \<in> L. y < x) \<and> (\<forall>y \<in> U. x < y) \<and> l < x) \<longleftrightarrow> |
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(\<exists>x. (\<forall>y \<in> (insert l L). y < x) \<and> (\<forall>y \<in> U. x < y))" |
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by auto |
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lemma gather_start [no_atp]: "(\<exists>x. P x) \<equiv> (\<exists>x. (\<forall>y \<in> {}. y < x) \<and> (\<forall>y\<in> {}. x < y) \<and> P x)" |
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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\<^sub>-\<^sub>\<infinity>)"}*} |
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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\<^sub>+\<^sub>\<infinity>)"}*} |
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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> |
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\<Longrightarrow> \<forall>x. \<not>(P1' \<or> P2') \<and> (P1 x \<or> P2 x) \<longrightarrow> (\<exists> u\<in> U. x \<le> u)" by auto |
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lemma lin_dense_lt[no_atp]: |
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"t \<in> U \<Longrightarrow> |
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\<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)" |
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proof(clarsimp) |
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fix x l u y |
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assume tU: "t \<in> U" and noU: "\<forall>t. l < t \<and> t < u \<longrightarrow> t \<notin> U" and lx: "l < x" |
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parents:
diff
changeset
|
94 |
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
|
95 |
from tU noU ly yu have tny: "t\<noteq>y" by auto |
49519 | 96 |
{ assume H: "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
|
97 |
from less_trans[OF lx px] less_trans[OF H yu] |
49519 | 98 |
have "l < t \<and> t < u" by simp |
99 |
with tU noU have "False" by auto } |
|
100 |
then have "\<not> t < y" by auto |
|
101 |
then have "y \<le> t" by (simp add: not_less) |
|
102 |
then show "y < t" using tny by (simp add: less_le) |
|
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
|
103 |
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
|
104 |
|
49519 | 105 |
lemma lin_dense_gt[no_atp]: |
106 |
"t \<in> U \<Longrightarrow> |
|
107 |
\<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)" |
|
108 |
proof(clarsimp) |
|
109 |
fix x l u y |
|
110 |
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" |
|
111 |
and px: "t < x" and ly: "l<y" and yu:"y < u" |
|
112 |
from tU noU ly yu have tny: "t\<noteq>y" by auto |
|
113 |
{ assume H: "y< t" |
|
114 |
from less_trans[OF ly H] less_trans[OF px xu] have "l < t \<and> t < u" by simp |
|
115 |
with tU noU have "False" by auto } |
|
116 |
then have "\<not> y<t" by auto |
|
117 |
then have "t \<le> y" by (auto simp add: not_less) |
|
118 |
then show "t < y" using tny by (simp add: less_le) |
|
119 |
qed |
|
120 |
||
121 |
lemma lin_dense_le[no_atp]: |
|
122 |
"t \<in> U \<Longrightarrow> |
|
123 |
\<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
|
124 |
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
|
125 |
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
|
126 |
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" |
49519 | 127 |
and px: "x \<le> t" and ly: "l<y" and yu:"y < u" |
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
|
128 |
from tU noU ly yu have tny: "t\<noteq>y" by auto |
49519 | 129 |
{ assume H: "t < y" |
130 |
from less_le_trans[OF lx px] less_trans[OF H yu] |
|
131 |
have "l < t \<and> t < u" by simp |
|
132 |
with tU noU have "False" by auto } |
|
133 |
then have "\<not> t < y" by auto |
|
134 |
then show "y \<le> t" by (simp add: not_less) |
|
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
|
135 |
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
|
136 |
|
49519 | 137 |
lemma lin_dense_ge[no_atp]: |
138 |
"t \<in> U \<Longrightarrow> |
|
139 |
\<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
|
140 |
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
|
141 |
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
|
142 |
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" |
49519 | 143 |
and px: "t \<le> x" and ly: "l<y" and yu:"y < u" |
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
|
144 |
from tU noU ly yu have tny: "t\<noteq>y" by auto |
49519 | 145 |
{ assume H: "y< t" |
146 |
from less_trans[OF ly H] le_less_trans[OF px xu] |
|
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
|
147 |
have "l < t \<and> t < u" by simp |
49519 | 148 |
with tU noU have "False" by auto } |
149 |
then have "\<not> y<t" by auto |
|
150 |
then show "t \<le> y" by (simp add: not_less) |
|
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
|
151 |
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
|
152 |
|
49519 | 153 |
lemma lin_dense_eq[no_atp]: |
154 |
"t \<in> U \<Longrightarrow> |
|
155 |
\<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)" |
|
156 |
by auto |
|
157 |
||
158 |
lemma lin_dense_neq[no_atp]: |
|
159 |
"t \<in> U \<Longrightarrow> |
|
160 |
\<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)" |
|
161 |
by auto |
|
162 |
||
163 |
lemma lin_dense_P[no_atp]: |
|
164 |
"\<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)" |
|
165 |
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
|
166 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
167 |
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
|
168 |
"\<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
|
169 |
\<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
|
170 |
\<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
|
171 |
\<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
|
172 |
\<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
|
173 |
\<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
|
174 |
by blast |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
175 |
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
|
176 |
"\<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
|
177 |
\<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
|
178 |
\<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
|
179 |
\<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
|
180 |
\<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
|
181 |
\<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
|
182 |
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 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
184 |
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
|
185 |
\<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')" |
49519 | 186 |
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
|
187 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
188 |
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
|
189 |
assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" |
49519 | 190 |
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" |
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
|
191 |
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" |
49519 | 192 |
proof - |
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
|
193 |
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
|
194 |
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
|
195 |
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
|
196 |
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
|
197 |
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
|
198 |
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
|
199 |
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
|
200 |
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
|
201 |
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
|
202 |
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
|
203 |
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
|
204 |
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
|
205 |
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
|
206 |
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
|
207 |
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
|
208 |
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
|
209 |
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
|
210 |
proof(clarsimp) |
49519 | 211 |
fix y |
212 |
assume ay: "?a < y" and yb: "y < ?b" and yS: "y \<in> S" |
|
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
|
213 |
from yS have "y\<in> ?Mx \<or> y\<in> ?xM" by (auto simp add: linear) |
49519 | 214 |
moreover { |
215 |
assume "y \<in> ?Mx" |
|
216 |
hence "y \<le> ?a" using Mxne fMx by auto |
|
217 |
with ay have "False" by (simp add: not_le[symmetric]) |
|
218 |
} |
|
219 |
moreover { |
|
220 |
assume "y \<in> ?xM" |
|
221 |
hence "?b \<le> y" using xMne fxM by auto |
|
222 |
with yb have "False" by (simp add: not_le[symmetric]) |
|
223 |
} |
|
224 |
ultimately show False 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
|
225 |
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
|
226 |
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
|
227 |
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
|
228 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
229 |
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
|
230 |
assumes px: "P x" and lx: "l \<le> x" and xu: "x \<le> u" and linS: "l\<in> S" |
49519 | 231 |
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" |
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
|
232 |
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
|
233 |
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
|
234 |
from finite_set_intervals[where P="P", OF px lx xu linS uinS fS lS Su] |
49519 | 235 |
obtain a and b where as: "a\<in> S" and bs: "b\<in> S" and noS:"\<forall>y. a < y \<and> y < b \<longrightarrow> y \<notin> S" |
236 |
and axb: "a \<le> x \<and> x \<le> b \<and> P x" 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
|
237 |
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
|
238 |
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
|
239 |
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
|
240 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
242 |
|
49519 | 243 |
|
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
|
244 |
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
|
245 |
|
53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
246 |
context unbounded_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
|
247 |
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
|
248 |
|
49519 | 249 |
lemma interval_empty_iff: "{y. x < y \<and> y < z} = {} \<longleftrightarrow> \<not> x < z" |
27825 | 250 |
by (auto dest: dense) |
251 |
||
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
252 |
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
|
253 |
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
|
254 |
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
|
255 |
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
|
256 |
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
|
257 |
then obtain x where xL: "\<forall>y\<in>L. y < x" and xU: "\<forall>y\<in>U. x < y" by blast |
49519 | 258 |
{ fix l u assume l: "l \<in> L" and u: "u \<in> U" |
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
|
259 |
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
|
260 |
also have "x < u" using xU u by blast |
49519 | 261 |
finally (less_trans) have "l < u" . } |
262 |
then show "\<forall>l\<in>L. \<forall>u\<in>U. l < u" 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
|
263 |
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
|
264 |
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
|
265 |
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
|
266 |
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
|
267 |
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
|
268 |
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
|
269 |
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
|
270 |
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
|
271 |
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
|
272 |
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
|
273 |
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
|
274 |
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
|
275 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
276 |
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
|
277 |
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
|
278 |
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
|
279 |
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
|
280 |
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
|
281 |
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
|
282 |
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
|
283 |
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
|
284 |
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
|
285 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
286 |
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
|
287 |
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
|
288 |
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
|
289 |
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
|
290 |
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
|
291 |
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
|
292 |
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
|
293 |
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
|
294 |
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
|
295 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
296 |
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
|
297 |
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
|
298 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
299 |
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
|
300 |
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
|
301 |
|
53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
302 |
lemma axiom[no_atp]: "class.unbounded_dense_linorder (op \<le>) (op <)" |
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
303 |
by (rule unbounded_dense_linorder_axioms) |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
304 |
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
|
305 |
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
|
306 |
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
|
307 |
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
|
308 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
310 |
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
|
311 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
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 |
(* FIXME: Move to HOL -- together with the conj_aci_rule in langford.ML *) |
41849 | 315 |
lemmas dnf[no_atp] = conj_disj_distribL conj_disj_distribR |
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
|
316 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
317 |
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
|
318 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
319 |
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
|
320 |
"(\<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
|
321 |
"(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
|
322 |
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
|
323 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
324 |
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
|
325 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
326 |
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
|
327 |
|
48891 | 328 |
ML_file "langford.ML" |
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
|
329 |
method_setup dlo = {* |
30549 | 330 |
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
|
331 |
*} "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
|
332 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
30652
752329615264
distributed contents of theory Arith_Tools to theories Int, IntDiv and NatBin accordingly
haftmann
parents:
30549
diff
changeset
|
334 |
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
|
335 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
337 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
338 |
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
|
339 |
begin |
49519 | 340 |
|
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
|
341 |
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
|
342 |
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
|
343 |
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
|
344 |
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
|
345 |
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
|
346 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
348 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
349 |
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
|
350 |
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
|
351 |
begin |
49519 | 352 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
353 |
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
|
354 |
|
53072 | 355 |
text {* Theorems for @{text "\<exists>z. \<forall>x. z \<sqsubset> x \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>+\<^sub>\<infinity>)"} *} |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
356 |
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
|
357 |
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
|
358 |
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
|
359 |
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
|
360 |
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
|
361 |
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
|
362 |
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
|
363 |
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
|
364 |
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all |
49519 | 365 |
{ fix x assume H: "z \<sqsubset> 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
|
366 |
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
|
367 |
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
|
368 |
} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
370 |
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
|
371 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
372 |
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
|
373 |
assumes ex1: "\<exists>z1. \<forall>x. z1 \<sqsubset> x \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 374 |
and ex2: "\<exists>z2. \<forall>x. z2 \<sqsubset> x \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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
|
375 |
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
|
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. 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
|
378 |
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
|
379 |
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
|
380 |
from z have zz1: "z1 \<sqsubset> z" and zz2: "z2 \<sqsubset> z" by simp_all |
49519 | 381 |
{ fix x assume H: "z \<sqsubset> 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
|
382 |
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
|
383 |
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
|
384 |
} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
386 |
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
|
387 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
388 |
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" |
49519 | 389 |
proof - |
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
|
390 |
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
|
391 |
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
|
392 |
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
|
393 |
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
|
394 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
396 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
398 |
|
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
399 |
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
|
400 |
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
|
401 |
begin |
49519 | 402 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
403 |
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
|
404 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
53072 | 406 |
text {* Theorems for @{text "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P\<^sub>-\<^sub>\<infinity>)"} *} |
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
407 |
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
|
408 |
assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 409 |
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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
|
410 |
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
|
411 |
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
|
412 |
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
|
413 |
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
|
414 |
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all |
49519 | 415 |
{ fix x assume H: "x \<sqsubset> 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
|
416 |
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
|
417 |
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
|
418 |
} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
420 |
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
|
421 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
422 |
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
|
423 |
assumes ex1: "\<exists>z1. \<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
49519 | 424 |
and ex2: "\<exists>z2. \<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" |
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
|
425 |
shows "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> ((P1 x \<or> P2 x) \<longleftrightarrow> (P1' \<or> P2'))" |
49519 | 426 |
proof - |
427 |
from ex1 ex2 obtain z1 and z2 where z1: "\<forall>x. x \<sqsubset> z1 \<longrightarrow> (P1 x \<longleftrightarrow> P1')" |
|
428 |
and z2: "\<forall>x. x \<sqsubset> z2 \<longrightarrow> (P2 x \<longleftrightarrow> P2')" 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
|
429 |
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
|
430 |
from z have zz1: "z \<sqsubset> z1" and zz2: "z \<sqsubset> z2" by simp_all |
49519 | 431 |
{ fix x assume H: "x \<sqsubset> 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
|
432 |
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
|
433 |
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
|
434 |
} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
436 |
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
|
437 |
|
49519 | 438 |
lemma minf_ex[no_atp]: |
439 |
assumes ex: "\<exists>z. \<forall>x. x \<sqsubset> z \<longrightarrow> (P x \<longleftrightarrow> P1)" |
|
440 |
and p1: P1 |
|
441 |
shows "\<exists> x. P x" |
|
442 |
proof - |
|
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
|
443 |
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
|
444 |
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
|
445 |
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
|
446 |
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
|
447 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
449 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
451 |
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
|
452 |
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
|
453 |
assumes between_less: "less x y \<Longrightarrow> less x (between x y) \<and> less (between x y) y" |
49519 | 454 |
and between_same: "between x x = 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
|
455 |
|
53215
5e47c31c6f7c
renamed typeclass dense_linorder to unbounded_dense_linorder
hoelzl
parents:
53072
diff
changeset
|
456 |
sublocale constr_dense_linorder < dlo: unbounded_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
|
457 |
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
|
458 |
using gt_ex lt_ex between_less |
49519 | 459 |
apply auto |
460 |
apply (rule_tac x="between x y" in exI) |
|
461 |
apply simp |
|
462 |
done |
|
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
|
463 |
|
49519 | 464 |
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
|
465 |
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
|
466 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
467 |
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
|
468 |
assumes fU: "finite U" |
49519 | 469 |
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 |
470 |
\<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
|
471 |
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')" |
|
472 |
and nmi: "\<not> MP" and npi: "\<not> PP" and ex: "\<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
|
473 |
shows "\<exists> u\<in> U. \<exists> u' \<in> U. P (between u u')" |
49519 | 474 |
proof - |
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
|
475 |
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
|
476 |
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
|
477 |
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
|
478 |
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
|
479 |
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
|
480 |
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
|
481 |
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
|
482 |
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
|
483 |
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
|
484 |
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
|
485 |
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
|
486 |
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
|
487 |
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
|
488 |
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
|
489 |
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
|
490 |
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
|
491 |
(\<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)" . |
49519 | 492 |
moreover { |
493 |
fix u assume um: "u\<in>U" and pu: "P u" |
|
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
|
494 |
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
|
495 |
with um pu have "P (between u u)" by simp |
49519 | 496 |
with um have ?thesis by blast } |
497 |
moreover { |
|
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 |
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" |
49519 | 499 |
then obtain t1 and t2 where t1M: "t1 \<in> U" and t2M: "t2\<in> U" |
500 |
and noM: "\<forall> y. t1 \<sqsubset> y \<and> y \<sqsubset> t2 \<longrightarrow> y \<notin> U" |
|
501 |
and t1x: "t1 \<sqsubset> x" and xt2: "x \<sqsubset> t2" and px: "P x" by blast |
|
502 |
from less_trans[OF t1x xt2] have t1t2: "t1 \<sqsubset> t2" . |
|
503 |
let ?u = "between t1 t2" |
|
504 |
from between_less t1t2 have t1lu: "t1 \<sqsubset> ?u" and ut2: "?u \<sqsubset> t2" by auto |
|
505 |
from lin_dense noM t1x xt2 px t1lu ut2 have "P ?u" by blast |
|
506 |
with t1M t2M have ?thesis by blast |
|
507 |
} |
|
508 |
ultimately show ?thesis by blast |
|
509 |
qed |
|
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
|
510 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
511 |
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
|
512 |
assumes fU: "finite U" |
49519 | 513 |
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 |
514 |
\<longrightarrow> (\<forall> y. l \<sqsubset> y \<and> y \<sqsubset> u \<longrightarrow> P y )" |
|
515 |
and nmibnd: "\<forall>x. \<not> MP \<and> P x \<longrightarrow> (\<exists> u\<in> U. u \<sqsubseteq> x)" |
|
516 |
and npibnd: "\<forall>x. \<not>PP \<and> P x \<longrightarrow> (\<exists> u\<in> U. x \<sqsubseteq> u)" |
|
517 |
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)" |
|
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
|
518 |
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
|
519 |
(is "_ \<equiv> (_ \<or> _ \<or> ?F)" is "?E \<equiv> ?D") |
49519 | 520 |
proof - |
521 |
{ assume px: "\<exists> x. P x" |
|
522 |
have "MP \<or> PP \<or> (\<not> MP \<and> \<not> PP)" by blast |
|
523 |
moreover { assume "MP \<or> PP" hence "?D" by blast } |
|
524 |
moreover { |
|
525 |
assume nmi: "\<not> MP" and npi: "\<not> PP" |
|
526 |
from npmibnd[OF nmibnd npibnd] |
|
527 |
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')" . |
|
528 |
from rinf_U[OF fU lin_dense nmpiU nmi npi px] have "?D" by blast } |
|
529 |
ultimately have "?D" by blast } |
|
530 |
moreover |
|
531 |
{ assume "?D" |
|
532 |
moreover { assume m:"MP" from minf_ex[OF mi m] have "?E" . } |
|
533 |
moreover { assume p: "PP" from pinf_ex[OF pi p] have "?E" . } |
|
534 |
moreover { assume f:"?F" hence "?E" by blast } |
|
535 |
ultimately have "?E" by blast } |
|
536 |
ultimately have "?E = ?D" by blast thus "?E \<equiv> ?D" by simp |
|
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
|
537 |
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
|
538 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
539 |
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
|
540 |
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
|
541 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
542 |
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
|
543 |
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
|
544 |
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
|
545 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
546 |
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
|
547 |
by (rule constr_dense_linorder_axioms) |
49519 | 548 |
|
35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35267
diff
changeset
|
549 |
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
|
550 |
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
|
551 |
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
|
552 |
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
|
553 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
554 |
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
|
555 |
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
|
556 |
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
|
557 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
559 |
let |
49519 | 560 |
fun simps phi = map (Morphism.thm phi) [@{thm "not_less"}, @{thm "not_le"}] |
561 |
fun generic_whatis phi = |
|
562 |
let |
|
563 |
val [lt, le] = map (Morphism.term phi) [@{term "op \<sqsubset>"}, @{term "op \<sqsubseteq>"}] |
|
564 |
fun h x t = |
|
565 |
case term_of t of |
|
566 |
Const(@{const_name HOL.eq}, _)$y$z => |
|
567 |
if term_of x aconv y then Ferrante_Rackoff_Data.Eq |
|
568 |
else Ferrante_Rackoff_Data.Nox |
|
569 |
| @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) => |
|
570 |
if term_of x aconv y then Ferrante_Rackoff_Data.NEq |
|
571 |
else Ferrante_Rackoff_Data.Nox |
|
572 |
| b$y$z => if Term.could_unify (b, lt) then |
|
573 |
if term_of x aconv y then Ferrante_Rackoff_Data.Lt |
|
574 |
else if term_of x aconv z then Ferrante_Rackoff_Data.Gt |
|
575 |
else Ferrante_Rackoff_Data.Nox |
|
576 |
else if Term.could_unify (b, le) then |
|
577 |
if term_of x aconv y then Ferrante_Rackoff_Data.Le |
|
578 |
else if term_of x aconv z then Ferrante_Rackoff_Data.Ge |
|
579 |
else Ferrante_Rackoff_Data.Nox |
|
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 |
else Ferrante_Rackoff_Data.Nox |
49519 | 581 |
| _ => Ferrante_Rackoff_Data.Nox |
582 |
in h end |
|
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
583 |
fun ss phi = |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
584 |
simpset_of (put_simpset HOL_ss @{context} addsimps (simps phi)) |
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
|
585 |
in |
49519 | 586 |
Ferrante_Rackoff_Data.funs @{thm "ferrack_axiom"} |
587 |
{isolate_conv = K (K (K Thm.reflexive)), whatis = generic_whatis, simpset = ss} |
|
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
|
588 |
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
|
589 |
*} |
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
591 |
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
|
592 |
|
48891 | 593 |
ML_file "ferrante_rackoff.ML" |
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
|
594 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
method_setup ferrack = {* |
30549 | 596 |
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
|
597 |
*} "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
|
598 |
|
49519 | 599 |
|
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 |
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
|
601 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
602 |
lemma neg_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x < 0) == (x > 0))" |
49519 | 603 |
proof - |
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
|
604 |
assume H: "c < 0" |
49519 | 605 |
have "c*x < 0 = (0/c < x)" |
606 |
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
|
607 |
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
|
608 |
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
|
609 |
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
|
610 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
611 |
lemma pos_prod_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x < 0) == (x < 0))" |
49519 | 612 |
proof - |
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
|
613 |
assume H: "c > 0" |
49519 | 614 |
then have "c*x < 0 = (0/c > x)" |
615 |
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
|
616 |
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
|
617 |
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
|
618 |
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
|
619 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
620 |
lemma neg_prod_sum_lt: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t< 0) == (x > (- 1/c)*t))" |
49519 | 621 |
proof - |
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
|
622 |
assume H: "c < 0" |
49519 | 623 |
have "c*x + t< 0 = (c*x < -t)" |
624 |
by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) |
|
625 |
also have "\<dots> = (-t/c < x)" |
|
626 |
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
|
627 |
also have "\<dots> = ((- 1/c)*t < x)" by simp |
49519 | 628 |
finally show "(c*x + t < 0) == (x > (- 1/c)*t)" by simp |
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
|
629 |
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
|
630 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
631 |
lemma pos_prod_sum_lt:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t < 0) == (x < (- 1/c)*t))" |
49519 | 632 |
proof - |
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
|
633 |
assume H: "c > 0" |
49519 | 634 |
have "c*x + t< 0 = (c*x < -t)" |
635 |
by (subst less_iff_diff_less_0 [of "c*x" "-t"], simp) |
|
636 |
also have "\<dots> = (-t/c > x)" |
|
637 |
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
|
638 |
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
|
639 |
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
|
640 |
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
|
641 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
642 |
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
|
643 |
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
|
644 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
645 |
lemma neg_prod_le:"(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x <= 0) == (x >= 0))" |
49519 | 646 |
proof - |
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
|
647 |
assume H: "c < 0" |
49519 | 648 |
have "c*x <= 0 = (0/c <= x)" |
649 |
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
|
650 |
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
|
651 |
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
|
652 |
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
|
653 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
654 |
lemma pos_prod_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x <= 0) == (x <= 0))" |
49519 | 655 |
proof - |
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
|
656 |
assume H: "c > 0" |
49519 | 657 |
hence "c*x <= 0 = (0/c >= x)" |
658 |
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
|
659 |
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
|
660 |
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
|
661 |
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
|
662 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
663 |
lemma neg_prod_sum_le: "(c\<Colon>'a\<Colon>linordered_field) < 0 \<Longrightarrow> ((c*x + t <= 0) == (x >= (- 1/c)*t))" |
49519 | 664 |
proof - |
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
|
665 |
assume H: "c < 0" |
49519 | 666 |
have "c*x + t <= 0 = (c*x <= -t)" |
667 |
by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) |
|
668 |
also have "\<dots> = (-t/c <= x)" |
|
669 |
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
|
670 |
also have "\<dots> = ((- 1/c)*t <= x)" by simp |
49519 | 671 |
finally show "(c*x + t <= 0) == (x >= (- 1/c)*t)" by simp |
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
|
672 |
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
|
673 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
674 |
lemma pos_prod_sum_le:"(c\<Colon>'a\<Colon>linordered_field) > 0 \<Longrightarrow> ((c*x + t <= 0) == (x <= (- 1/c)*t))" |
49519 | 675 |
proof - |
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
|
676 |
assume H: "c > 0" |
49519 | 677 |
have "c*x + t <= 0 = (c*x <= -t)" |
678 |
by (subst le_iff_diff_le_0 [of "c*x" "-t"], simp) |
|
679 |
also have "\<dots> = (-t/c >= x)" |
|
680 |
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
|
681 |
also have "\<dots> = ((- 1/c)*t >= x)" by simp |
49519 | 682 |
finally show "(c*x + t <= 0) == (x <= (- 1/c)*t)" by simp |
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
|
683 |
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
|
684 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
685 |
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
|
686 |
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
|
687 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
688 |
lemma nz_prod_eq:"(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x = 0) == (x = 0))" by simp |
49519 | 689 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
690 |
lemma nz_prod_sum_eq: "(c\<Colon>'a\<Colon>linordered_field) \<noteq> 0 \<Longrightarrow> ((c*x + t = 0) == (x = (- 1/c)*t))" |
49519 | 691 |
proof - |
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
|
692 |
assume H: "c \<noteq> 0" |
49519 | 693 |
have "c*x + t = 0 = (c*x = -t)" |
694 |
by (subst eq_iff_diff_eq_0 [of "c*x" "-t"], simp) |
|
695 |
also have "\<dots> = (x = -t/c)" |
|
696 |
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
|
697 |
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
|
698 |
qed |
49519 | 699 |
|
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents:
34974
diff
changeset
|
700 |
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
|
701 |
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
|
702 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35028
diff
changeset
|
704 |
interpretation class_dense_linordered_field: constr_dense_linorder |
29509
1ff0f3f08a7b
migrated class package to new locale implementation
haftmann
parents:
29252
diff
changeset
|
705 |
"op <=" "op <" |
47108
2a1953f0d20d
merged fork with new numeral representation (see NEWS)
huffman
parents:
46497
diff
changeset
|
706 |
"\<lambda> x y. 1/2 * ((x::'a::{linordered_field}) + y)" |
49519 | 707 |
by unfold_locales (dlo, dlo, 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
|
708 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
710 |
let |
49519 | 711 |
fun earlier [] x y = false |
712 |
| earlier (h::t) x y = |
|
713 |
if h aconvc y then false else if h aconvc x then true else earlier t x 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
|
714 |
|
49519 | 715 |
fun dest_frac ct = |
716 |
case term_of ct of |
|
717 |
Const (@{const_name Fields.divide},_) $ a $ b=> |
|
718 |
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) |
|
719 |
| Const(@{const_name inverse}, _)$a => Rat.rat_of_quotient(1, HOLogic.dest_number a |> snd) |
|
720 |
| t => Rat.rat_of_int (snd (HOLogic.dest_number 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
|
721 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
fun mk_frac phi cT x = |
49519 | 723 |
let val (a, b) = Rat.quotient_of_rat x |
724 |
in if b = 1 then Numeral.mk_cnumber cT a |
|
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
725 |
else Thm.apply |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
726 |
(Thm.apply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) |
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
|
727 |
(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
|
728 |
(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
|
729 |
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
|
730 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
732 |
Const(@{const_name Groups.plus}, _)$(Const(@{const_name Groups.times},_)$_$y)$_ => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
733 |
if y aconv term_of x then ("c*x+t",[(funpow 2 Thm.dest_arg1) ct, Thm.dest_arg ct]) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
734 |
else ("Nox",[]) |
35267
8dfd816713c6
moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents:
35092
diff
changeset
|
735 |
| Const(@{const_name Groups.plus}, _)$y$_ => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
736 |
if y aconv term_of x then ("x+t",[Thm.dest_arg ct]) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
737 |
else ("Nox",[]) |
35267
8dfd816713c6
moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents:
35092
diff
changeset
|
738 |
| Const(@{const_name Groups.times}, _)$_$y => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
739 |
if y aconv term_of x then ("c*x",[Thm.dest_arg1 ct]) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
740 |
else ("Nox",[]) |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
741 |
| t => if t aconv term_of x then ("x",[]) else ("Nox",[]); |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
742 |
|
36945 | 743 |
fun xnormalize_conv ctxt [] ct = Thm.reflexive ct |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
744 |
| xnormalize_conv ctxt (vs as (x::_)) ct = |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
745 |
case term_of ct of |
35267
8dfd816713c6
moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents:
35092
diff
changeset
|
746 |
Const(@{const_name Orderings.less},_)$_$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
|
747 |
(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
|
748 |
("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
|
749 |
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
|
750 |
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
|
751 |
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
|
752 |
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
|
753 |
val neg = cr </ Rat.zero |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
754 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
755 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
756 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
757 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 758 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
759 |
val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,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
|
760 |
(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
|
761 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
762 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
763 |
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
|
764 |
| ("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
|
765 |
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
|
766 |
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
|
767 |
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
|
768 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
769 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
770 |
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
|
771 |
| ("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
|
772 |
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
|
773 |
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
|
774 |
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
|
775 |
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
|
776 |
val neg = cr </ Rat.zero |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
777 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
778 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
779 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
780 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 781 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
782 |
val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,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
|
783 |
(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
|
784 |
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
|
785 |
in rth end |
36945 | 786 |
| _ => Thm.reflexive ct) |
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
|
787 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
|
35267
8dfd816713c6
moved remaning class operations from Algebras.thy to Groups.thy
haftmann
parents:
35092
diff
changeset
|
789 |
| 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
|
790 |
(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
|
791 |
("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
|
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 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
|
795 |
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
|
796 |
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
|
797 |
val neg = cr </ Rat.zero |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
798 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
799 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
800 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
801 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 802 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
803 |
val th = Thm.implies_elim (instantiate' [SOME T] (map SOME [c,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
|
804 |
(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
|
805 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
806 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
807 |
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
|
808 |
| ("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
|
809 |
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
|
810 |
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
|
811 |
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
|
812 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
813 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
814 |
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
|
815 |
| ("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
|
816 |
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
|
817 |
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
|
818 |
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
|
819 |
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
|
820 |
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
|
821 |
val neg = cr </ Rat.zero |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
822 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
823 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
824 |
(if neg then Thm.apply (Thm.apply clt c) cz |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
825 |
else Thm.apply (Thm.apply clt cz) c)) |
36945 | 826 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
827 |
val th = Thm.implies_elim (instantiate' [SOME (ctyp_of_term x)] (map SOME [c,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
|
828 |
(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
|
829 |
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
|
830 |
in rth end |
36945 | 831 |
| _ => Thm.reflexive ct) |
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
|
832 |
|
38864
4abe644fcea5
formerly unnamed infix equality now named HOL.eq
haftmann
parents:
38549
diff
changeset
|
833 |
| Const(@{const_name HOL.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
|
834 |
(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
|
835 |
("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
|
836 |
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
|
837 |
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
|
838 |
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
|
839 |
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
|
840 |
val cz = Thm.dest_arg ct |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
841 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
842 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
843 |
(Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz))) |
36945 | 844 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
845 |
val th = Thm.implies_elim |
|
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
|
846 |
(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
|
847 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
848 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
849 |
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
|
850 |
| ("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
|
851 |
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
|
852 |
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
|
853 |
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
|
854 |
val rth = Conv.fconv_rule (Conv.arg_conv (Conv.binop_conv |
36753
5cf4e9128f22
renamed Normalizer to the more specific Semiring_Normalizer
haftmann
parents:
36635
diff
changeset
|
855 |
(Semiring_Normalizer.semiring_normalize_ord_conv ctxt (earlier vs)))) th |
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
|
856 |
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
|
857 |
| ("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
|
858 |
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
|
859 |
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
|
860 |
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
|
861 |
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
|
862 |
val cz = Thm.dest_arg ct |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
863 |
val cthp = Simplifier.rewrite ctxt |
46497
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
864 |
(Thm.apply @{cterm "Trueprop"} |
89ccf66aa73d
renamed Thm.capply to Thm.apply, and Thm.cabs to Thm.lambda in conformance with similar operations in structure Term and Logic;
wenzelm
parents:
45620
diff
changeset
|
865 |
(Thm.apply @{cterm "Not"} (Thm.apply (Thm.apply ceq c) cz))) |
36945 | 866 |
val cth = Thm.equal_elim (Thm.symmetric cthp) TrueI |
867 |
val rth = Thm.implies_elim |
|
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
|
868 |
(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
|
869 |
in rth end |
36945 | 870 |
| _ => Thm.reflexive ct); |
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
|
871 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.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 |
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
|
873 |
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
|
874 |
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
|
875 |
val eq_iff_diff_eq_0 = mk_meta_eq @{thm "eq_iff_diff_eq_0"} |
52090 | 876 |
val ss = simpset_of @{context} |
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
|
877 |
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
|
878 |
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
|
879 |
Const(@{const_name Orderings.less},_)$a$b => |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
880 |
let val (ca,cb) = Thm.dest_binop ct |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
881 |
val T = ctyp_of_term ca |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
882 |
val th = instantiate' [SOME T] [SOME ca, SOME cb] less_iff_diff_less_0 |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
883 |
val nth = Conv.fconv_rule |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
884 |
(Conv.arg_conv (Conv.arg1_conv |
52090 | 885 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th |
36945 | 886 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
887 |
in rth end |
35092
cfe605c54e50
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
haftmann
parents:
35084
diff
changeset
|
888 |
| Const(@{const_name Orderings.less_eq},_)$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
|
889 |
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
|
890 |
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
|
891 |
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
|
892 |
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
|
893 |
(Conv.arg_conv (Conv.arg1_conv |
52090 | 894 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th |
36945 | 895 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
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
|
896 |
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
|
897 |
|
38864
4abe644fcea5
formerly unnamed infix equality now named HOL.eq
haftmann
parents:
38549
diff
changeset
|
898 |
| Const(@{const_name HOL.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
|
899 |
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
|
900 |
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
|
901 |
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
|
902 |
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
|
903 |
(Conv.arg_conv (Conv.arg1_conv |
52090 | 904 |
(Semiring_Normalizer.semiring_normalize_ord_conv (put_simpset ss ctxt) (earlier vs)))) th |
36945 | 905 |
val rth = Thm.transitive nth (xnormalize_conv ctxt vs (Thm.rhs_of nth)) |
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
|
906 |
in rth end |
38864
4abe644fcea5
formerly unnamed infix equality now named HOL.eq
haftmann
parents:
38549
diff
changeset
|
907 |
| @{term "Not"} $(Const(@{const_name HOL.eq},_)$a$b) => Conv.arg_conv (field_isolate_conv phi ctxt vs) ct |
36945 | 908 |
| _ => Thm.reflexive ct |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
909 |
end; |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
910 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
911 |
fun classfield_whatis phi = |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
912 |
let |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
913 |
fun h x t = |
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
914 |
case term_of t of |
38864
4abe644fcea5
formerly unnamed infix equality now named HOL.eq
haftmann
parents:
38549
diff
changeset
|
915 |
Const(@{const_name HOL.eq}, _)$y$z => if term_of x aconv y then Ferrante_Rackoff_Data.Eq |
26161
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
916 |
else Ferrante_Rackoff_Data.Nox |
38864
4abe644fcea5
formerly unnamed infix equality now named HOL.eq
haftmann
parents:
38549
diff
changeset
|
917 |
| @{term "Not"}$(Const(@{const_name HOL.eq}, _)$y$z) => if term_of x aconv y then Ferrante_Rackoff_Data.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
|
918 |
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
|
919 |
| 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
|
920 |
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
|
921 |
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
|
922 |
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
|
923 |
| 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
|
924 |
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
|
925 |
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
|
926 |
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
|
927 |
| _ => 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
|
928 |
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
|
929 |
fun class_field_ss phi = |
51717
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
930 |
simpset_of (put_simpset HOL_basic_ss @{context} |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
931 |
addsimps ([@{thm "linorder_not_less"}, @{thm "linorder_not_le"}]) |
9e7d1c139569
simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents:
49756
diff
changeset
|
932 |
|> fold Splitter.add_split [@{thm "abs_split"}, @{thm "split_max"}, @{thm "split_min"}]) |
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
|
933 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
934 |
in |
35043
07dbdf60d5ad
dropped accidental duplication of "lin" prefix from cs. 108662d50512
haftmann
parents:
35028
diff
changeset
|
935 |
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
|
936 |
{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
|
937 |
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
|
938 |
*} |
41849 | 939 |
(* |
29841
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
940 |
lemma upper_bound_finite_set: |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
941 |
assumes fS: "finite S" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
942 |
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
|
943 |
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
|
944 |
case 1 thus ?case by simp |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
945 |
next |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
946 |
case (2 x F) |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
947 |
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
|
948 |
let ?a = "max a (f x)" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
949 |
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
|
950 |
{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
|
951 |
{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
|
952 |
moreover |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
953 |
{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
|
954 |
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
|
955 |
then show ?case by blast |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
956 |
qed |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
957 |
|
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
958 |
lemma lower_bound_finite_set: |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
959 |
assumes fS: "finite S" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
960 |
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
|
961 |
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
|
962 |
case 1 thus ?case by simp |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
963 |
next |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
964 |
case (2 x F) |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
965 |
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
|
966 |
let ?a = "min a (f x)" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
967 |
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
|
968 |
{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
|
969 |
{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
|
970 |
moreover |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
971 |
{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
|
972 |
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
|
973 |
then show ?case by blast |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
974 |
qed |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
975 |
|
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
976 |
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
|
977 |
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
|
978 |
proof- |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
979 |
let ?F = "f ` S" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
980 |
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
|
981 |
let ?a = "Max ?F" |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
982 |
{assume "S = {}" hence ?thesis by blast} |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
983 |
moreover |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
984 |
{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
|
985 |
{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
|
986 |
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
|
987 |
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
|
988 |
hence ?thesis by blast} |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
989 |
ultimately show ?thesis by blast |
86d94bb79226
A generic decision procedure for linear rea arithmetic and normed vector spaces
chaieb
parents:
29823
diff
changeset
|
990 |
qed |
41849 | 991 |
*) |
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
|
992 |
|
34cb0b457dcc
Old HOL/Dense_Linear_Order.thy and the setup in Arith_Tools for Ferrante and Rackoff's Quantifier elimination for linear arithmetic over ordered Fields.
chaieb
parents:
diff
changeset
|
993 |
end |