(* Title: HOL/Orderings.thy
Author: Tobias Nipkow, Markus Wenzel, and Larry Paulson
*)
section \<open>Abstract orderings\<close>
theory Orderings
imports HOL
keywords "print_orders" :: diag
begin
subsection \<open>Abstract ordering\<close>
locale partial_preordering =
fixes less_eq :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold>\<le>\<close> 50)
assumes refl: \<open>a \<^bold>\<le> a\<close> \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
and trans: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>\<le> c\<close>
locale preordering = partial_preordering +
fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50)
assumes strict_iff_not: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close>
begin
lemma strict_implies_order:
\<open>a \<^bold>< b \<Longrightarrow> a \<^bold>\<le> b\<close>
by (simp add: strict_iff_not)
lemma irrefl: \<comment> \<open>not \<open>iff\<close>: makes problems due to multiple (dual) interpretations\<close>
\<open>\<not> a \<^bold>< a\<close>
by (simp add: strict_iff_not)
lemma asym:
\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< a \<Longrightarrow> False\<close>
by (auto simp add: strict_iff_not)
lemma strict_trans1:
\<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
by (auto simp add: strict_iff_not intro: trans)
lemma strict_trans2:
\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>\<le> c \<Longrightarrow> a \<^bold>< c\<close>
by (auto simp add: strict_iff_not intro: trans)
lemma strict_trans:
\<open>a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
by (auto intro: strict_trans1 strict_implies_order)
end
lemma preordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close>
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
and less (infix \<open>\<^bold><\<close> 50)
assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close>
assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close>
assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
proof
fix a b
show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close>
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show \<open>a \<^bold>\<le> a\<close>
by (auto simp add: less_eq_less)
next
fix a b c
assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close>
by (auto simp add: less_eq_less intro: trans)
qed
lemma preordering_dualI:
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
and less (infix \<open>\<^bold><\<close> 50)
assumes \<open>preordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close>
shows \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
proof -
from assms interpret preordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> .
show ?thesis
by standard (auto simp: strict_iff_not refl intro: trans)
qed
locale ordering = partial_preordering +
fixes less :: \<open>'a \<Rightarrow> 'a \<Rightarrow> bool\<close> (infix \<open>\<^bold><\<close> 50)
assumes strict_iff_order: \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close>
assumes antisym: \<open>a \<^bold>\<le> b \<Longrightarrow> b \<^bold>\<le> a \<Longrightarrow> a = b\<close>
begin
sublocale preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close>
proof
show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> \<not> b \<^bold>\<le> a\<close> for a b
by (auto simp add: strict_iff_order intro: antisym)
qed
lemma strict_implies_not_eq:
\<open>a \<^bold>< b \<Longrightarrow> a \<noteq> b\<close>
by (simp add: strict_iff_order)
lemma not_eq_order_implies_strict:
\<open>a \<noteq> b \<Longrightarrow> a \<^bold>\<le> b \<Longrightarrow> a \<^bold>< b\<close>
by (simp add: strict_iff_order)
lemma order_iff_strict:
\<open>a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
by (auto simp add: strict_iff_order refl)
lemma eq_iff: \<open>a = b \<longleftrightarrow> a \<^bold>\<le> b \<and> b \<^bold>\<le> a\<close>
by (auto simp add: refl intro: antisym)
end
lemma ordering_strictI: \<comment> \<open>Alternative introduction rule with bias towards strict order\<close>
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
and less (infix \<open>\<^bold><\<close> 50)
assumes less_eq_less: \<open>\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b\<close>
assumes asym: \<open>\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a\<close>
assumes irrefl: \<open>\<And>a. \<not> a \<^bold>< a\<close>
assumes trans: \<open>\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c\<close>
shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close>
proof
fix a b
show \<open>a \<^bold>< b \<longleftrightarrow> a \<^bold>\<le> b \<and> a \<noteq> b\<close>
by (auto simp add: less_eq_less asym irrefl)
next
fix a
show \<open>a \<^bold>\<le> a\<close>
by (auto simp add: less_eq_less)
next
fix a b c
assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> c\<close> then show \<open>a \<^bold>\<le> c\<close>
by (auto simp add: less_eq_less intro: trans)
next
fix a b
assume \<open>a \<^bold>\<le> b\<close> and \<open>b \<^bold>\<le> a\<close> then show \<open>a = b\<close>
by (auto simp add: less_eq_less asym)
qed
lemma ordering_dualI:
fixes less_eq (infix \<open>\<^bold>\<le>\<close> 50)
and less (infix \<open>\<^bold><\<close> 50)
assumes \<open>ordering (\<lambda>a b. b \<^bold>\<le> a) (\<lambda>a b. b \<^bold>< a)\<close>
shows \<open>ordering (\<^bold>\<le>) (\<^bold><)\<close>
proof -
from assms interpret ordering \<open>\<lambda>a b. b \<^bold>\<le> a\<close> \<open>\<lambda>a b. b \<^bold>< a\<close> .
show ?thesis
by standard (auto simp: strict_iff_order refl intro: antisym trans)
qed
locale ordering_top = ordering +
fixes top :: \<open>'a\<close> (\<open>\<^bold>\<top>\<close>)
assumes extremum [simp]: \<open>a \<^bold>\<le> \<^bold>\<top>\<close>
begin
lemma extremum_uniqueI:
\<open>\<^bold>\<top> \<^bold>\<le> a \<Longrightarrow> a = \<^bold>\<top>\<close>
by (rule antisym) auto
lemma extremum_unique:
\<open>\<^bold>\<top> \<^bold>\<le> a \<longleftrightarrow> a = \<^bold>\<top>\<close>
by (auto intro: antisym)
lemma extremum_strict [simp]:
\<open>\<not> (\<^bold>\<top> \<^bold>< a)\<close>
using extremum [of a] by (auto simp add: order_iff_strict intro: asym irrefl)
lemma not_eq_extremum:
\<open>a \<noteq> \<^bold>\<top> \<longleftrightarrow> a \<^bold>< \<^bold>\<top>\<close>
by (auto simp add: order_iff_strict intro: not_eq_order_implies_strict extremum)
end
subsection \<open>Syntactic orders\<close>
class ord =
fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"
begin
notation
less_eq ("'(\<le>')") and
less_eq ("(_/ \<le> _)" [51, 51] 50) and
less ("'(<')") and
less ("(_/ < _)" [51, 51] 50)
abbreviation (input)
greater_eq (infix "\<ge>" 50)
where "x \<ge> y \<equiv> y \<le> x"
abbreviation (input)
greater (infix ">" 50)
where "x > y \<equiv> y < x"
notation (ASCII)
less_eq ("'(<=')") and
less_eq ("(_/ <= _)" [51, 51] 50)
notation (input)
greater_eq (infix ">=" 50)
end
subsection \<open>Quasi orders\<close>
class preorder = ord +
assumes less_le_not_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> (y \<le> x)"
and order_refl [iff]: "x \<le> x"
and order_trans: "x \<le> y \<Longrightarrow> y \<le> z \<Longrightarrow> x \<le> z"
begin
sublocale order: preordering less_eq less + dual_order: preordering greater_eq greater
proof -
interpret preordering less_eq less
by standard (auto intro: order_trans simp add: less_le_not_le)
show \<open>preordering less_eq less\<close>
by (fact preordering_axioms)
then show \<open>preordering greater_eq greater\<close>
by (rule preordering_dualI)
qed
text \<open>Reflexivity.\<close>
lemma eq_refl: "x = y \<Longrightarrow> x \<le> y"
\<comment> \<open>This form is useful with the classical reasoner.\<close>
by (erule ssubst) (rule order_refl)
lemma less_irrefl [iff]: "\<not> x < x"
by (simp add: less_le_not_le)
lemma less_imp_le: "x < y \<Longrightarrow> x \<le> y"
by (simp add: less_le_not_le)
text \<open>Asymmetry.\<close>
lemma less_not_sym: "x < y \<Longrightarrow> \<not> (y < x)"
by (simp add: less_le_not_le)
lemma less_asym: "x < y \<Longrightarrow> (\<not> P \<Longrightarrow> y < x) \<Longrightarrow> P"
by (drule less_not_sym, erule contrapos_np) simp
text \<open>Transitivity.\<close>
lemma less_trans: "x < y \<Longrightarrow> y < z \<Longrightarrow> x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma le_less_trans: "x \<le> y \<Longrightarrow> y < z \<Longrightarrow> x < z"
by (auto simp add: less_le_not_le intro: order_trans)
lemma less_le_trans: "x < y \<Longrightarrow> y \<le> z \<Longrightarrow> x < z"
by (auto simp add: less_le_not_le intro: order_trans)
text \<open>Useful for simplification, but too risky to include by default.\<close>
lemma less_imp_not_less: "x < y \<Longrightarrow> (\<not> y < x) \<longleftrightarrow> True"
by (blast elim: less_asym)
lemma less_imp_triv: "x < y \<Longrightarrow> (y < x \<longrightarrow> P) \<longleftrightarrow> True"
by (blast elim: less_asym)
text \<open>Transitivity rules for calculational reasoning\<close>
lemma less_asym': "a < b \<Longrightarrow> b < a \<Longrightarrow> P"
by (rule less_asym)
text \<open>Dual order\<close>
lemma dual_preorder:
\<open>class.preorder (\<ge>) (>)\<close>
by standard (auto simp add: less_le_not_le intro: order_trans)
end
lemma preordering_preorderI:
\<open>class.preorder (\<^bold>\<le>) (\<^bold><)\<close> if \<open>preordering (\<^bold>\<le>) (\<^bold><)\<close>
for less_eq (infix \<open>\<^bold>\<le>\<close> 50) and less (infix \<open>\<^bold><\<close> 50)
proof -
from that interpret preordering \<open>(\<^bold>\<le>)\<close> \<open>(\<^bold><)\<close> .
show ?thesis
by standard (auto simp add: strict_iff_not refl intro: trans)
qed
subsection \<open>Partial orders\<close>
class order = preorder +
assumes order_antisym: "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x = y"
begin
lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> x \<noteq> y"
by (auto simp add: less_le_not_le intro: order_antisym)
sublocale order: ordering less_eq less + dual_order: ordering greater_eq greater
proof -
interpret ordering less_eq less
by standard (auto intro: order_antisym order_trans simp add: less_le)
show "ordering less_eq less"
by (fact ordering_axioms)
then show "ordering greater_eq greater"
by (rule ordering_dualI)
qed
text \<open>Reflexivity.\<close>
lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x = y"
\<comment> \<open>NOT suitable for iff, since it can cause PROOF FAILED.\<close>
by (fact order.order_iff_strict)
lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x = y"
by (simp add: less_le)
text \<open>Useful for simplification, but too risky to include by default.\<close>
lemma less_imp_not_eq: "x < y \<Longrightarrow> (x = y) \<longleftrightarrow> False"
by auto
lemma less_imp_not_eq2: "x < y \<Longrightarrow> (y = x) \<longleftrightarrow> False"
by auto
text \<open>Transitivity rules for calculational reasoning\<close>
lemma neq_le_trans: "a \<noteq> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
by (fact order.not_eq_order_implies_strict)
lemma le_neq_trans: "a \<le> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a < b"
by (rule order.not_eq_order_implies_strict)
text \<open>Asymmetry.\<close>
lemma order_eq_iff: "x = y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
by (fact order.eq_iff)
lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
by (simp add: order.eq_iff)
lemma less_imp_neq: "x < y \<Longrightarrow> x \<noteq> y"
by (fact order.strict_implies_not_eq)
lemma antisym_conv1: "\<not> x < y \<Longrightarrow> x \<le> y \<longleftrightarrow> x = y"
by (simp add: local.le_less)
lemma antisym_conv2: "x \<le> y \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
by (simp add: local.less_le)
lemma leD: "y \<le> x \<Longrightarrow> \<not> x < y"
by (auto simp: less_le order.antisym)
text \<open>Least value operator\<close>
definition (in ord)
Least :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "LEAST " 10) where
"Least P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y))"
lemma Least_equality:
assumes "P x"
and "\<And>y. P y \<Longrightarrow> x \<le> y"
shows "Least P = x"
unfolding Least_def by (rule the_equality)
(blast intro: assms order.antisym)+
lemma LeastI2_order:
assumes "P x"
and "\<And>y. P y \<Longrightarrow> x \<le> y"
and "\<And>x. P x \<Longrightarrow> \<forall>y. P y \<longrightarrow> x \<le> y \<Longrightarrow> Q x"
shows "Q (Least P)"
unfolding Least_def by (rule theI2)
(blast intro: assms order.antisym)+
lemma Least_ex1:
assumes "\<exists>!x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<le> y)"
shows Least1I: "P (Least P)" and Least1_le: "P z \<Longrightarrow> Least P \<le> z"
using theI'[OF assms]
unfolding Least_def
by auto
text \<open>Greatest value operator\<close>
definition Greatest :: "('a \<Rightarrow> bool) \<Rightarrow> 'a" (binder "GREATEST " 10) where
"Greatest P = (THE x. P x \<and> (\<forall>y. P y \<longrightarrow> x \<ge> y))"
lemma GreatestI2_order:
"\<lbrakk> P x;
\<And>y. P y \<Longrightarrow> x \<ge> y;
\<And>x. \<lbrakk> P x; \<forall>y. P y \<longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Q x \<rbrakk>
\<Longrightarrow> Q (Greatest P)"
unfolding Greatest_def
by (rule theI2) (blast intro: order.antisym)+
lemma Greatest_equality:
"\<lbrakk> P x; \<And>y. P y \<Longrightarrow> x \<ge> y \<rbrakk> \<Longrightarrow> Greatest P = x"
unfolding Greatest_def
by (rule the_equality) (blast intro: order.antisym)+
end
lemma ordering_orderI:
fixes less_eq (infix "\<^bold>\<le>" 50)
and less (infix "\<^bold><" 50)
assumes "ordering less_eq less"
shows "class.order less_eq less"
proof -
from assms interpret ordering less_eq less .
show ?thesis
by standard (auto intro: antisym trans simp add: refl strict_iff_order)
qed
lemma order_strictI:
fixes less (infix "\<^bold><" 50)
and less_eq (infix "\<^bold>\<le>" 50)
assumes "\<And>a b. a \<^bold>\<le> b \<longleftrightarrow> a \<^bold>< b \<or> a = b"
assumes "\<And>a b. a \<^bold>< b \<Longrightarrow> \<not> b \<^bold>< a"
assumes "\<And>a. \<not> a \<^bold>< a"
assumes "\<And>a b c. a \<^bold>< b \<Longrightarrow> b \<^bold>< c \<Longrightarrow> a \<^bold>< c"
shows "class.order less_eq less"
by (rule ordering_orderI) (rule ordering_strictI, (fact assms)+)
context order
begin
text \<open>Dual order\<close>
lemma dual_order:
"class.order (\<ge>) (>)"
using dual_order.ordering_axioms by (rule ordering_orderI)
end
subsection \<open>Linear (total) orders\<close>
class linorder = order +
assumes linear: "x \<le> y \<or> y \<le> x"
begin
lemma less_linear: "x < y \<or> x = y \<or> y < x"
unfolding less_le using less_le linear by blast
lemma le_less_linear: "x \<le> y \<or> y < x"
by (simp add: le_less less_linear)
lemma le_cases [case_names le ge]:
"(x \<le> y \<Longrightarrow> P) \<Longrightarrow> (y \<le> x \<Longrightarrow> P) \<Longrightarrow> P"
using linear by blast
lemma (in linorder) le_cases3:
"\<lbrakk>\<lbrakk>x \<le> y; y \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> x; x \<le> z\<rbrakk> \<Longrightarrow> P; \<lbrakk>x \<le> z; z \<le> y\<rbrakk> \<Longrightarrow> P;
\<lbrakk>z \<le> y; y \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> P; \<lbrakk>z \<le> x; x \<le> y\<rbrakk> \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
by (blast intro: le_cases)
lemma linorder_cases [case_names less equal greater]:
"(x < y \<Longrightarrow> P) \<Longrightarrow> (x = y \<Longrightarrow> P) \<Longrightarrow> (y < x \<Longrightarrow> P) \<Longrightarrow> P"
using less_linear by blast
lemma linorder_wlog[case_names le sym]:
"(\<And>a b. a \<le> b \<Longrightarrow> P a b) \<Longrightarrow> (\<And>a b. P b a \<Longrightarrow> P a b) \<Longrightarrow> P a b"
by (cases rule: le_cases[of a b]) blast+
lemma not_less: "\<not> x < y \<longleftrightarrow> y \<le> x"
unfolding less_le
using linear by (blast intro: order.antisym)
lemma not_less_iff_gr_or_eq: "\<not>(x < y) \<longleftrightarrow> (x > y \<or> x = y)"
by (auto simp add:not_less le_less)
lemma not_le: "\<not> x \<le> y \<longleftrightarrow> y < x"
unfolding less_le
using linear by (blast intro: order.antisym)
lemma neq_iff: "x \<noteq> y \<longleftrightarrow> x < y \<or> y < x"
by (cut_tac x = x and y = y in less_linear, auto)
lemma neqE: "x \<noteq> y \<Longrightarrow> (x < y \<Longrightarrow> R) \<Longrightarrow> (y < x \<Longrightarrow> R) \<Longrightarrow> R"
by (simp add: neq_iff) blast
lemma antisym_conv3: "\<not> y < x \<Longrightarrow> \<not> x < y \<longleftrightarrow> x = y"
by (blast intro: order.antisym dest: not_less [THEN iffD1])
lemma leI: "\<not> x < y \<Longrightarrow> y \<le> x"
unfolding not_less .
lemma not_le_imp_less: "\<not> y \<le> x \<Longrightarrow> x < y"
unfolding not_le .
lemma linorder_less_wlog[case_names less refl sym]:
"\<lbrakk>\<And>a b. a < b \<Longrightarrow> P a b; \<And>a. P a a; \<And>a b. P b a \<Longrightarrow> P a b\<rbrakk> \<Longrightarrow> P a b"
using antisym_conv3 by blast
text \<open>Dual order\<close>
lemma dual_linorder:
"class.linorder (\<ge>) (>)"
by (rule class.linorder.intro, rule dual_order) (unfold_locales, rule linear)
end
text \<open>Alternative introduction rule with bias towards strict order\<close>
lemma linorder_strictI:
fixes less_eq (infix "\<^bold>\<le>" 50)
and less (infix "\<^bold><" 50)
assumes "class.order less_eq less"
assumes trichotomy: "\<And>a b. a \<^bold>< b \<or> a = b \<or> b \<^bold>< a"
shows "class.linorder less_eq less"
proof -
interpret order less_eq less
by (fact \<open>class.order less_eq less\<close>)
show ?thesis
proof
fix a b
show "a \<^bold>\<le> b \<or> b \<^bold>\<le> a"
using trichotomy by (auto simp add: le_less)
qed
qed
subsection \<open>Reasoning tools setup\<close>
ML_file \<open>~~/src/Provers/order_procedure.ML\<close>
ML_file \<open>~~/src/Provers/order_tac.ML\<close>
ML \<open>
structure Logic_Signature : LOGIC_SIGNATURE = struct
val mk_Trueprop = HOLogic.mk_Trueprop
val dest_Trueprop = HOLogic.dest_Trueprop
val Trueprop_conv = HOLogic.Trueprop_conv
val Not = HOLogic.Not
val conj = HOLogic.conj
val disj = HOLogic.disj
val notI = @{thm notI}
val ccontr = @{thm ccontr}
val conjI = @{thm conjI}
val conjE = @{thm conjE}
val disjE = @{thm disjE}
val not_not_conv = Conv.rewr_conv @{thm eq_reflection[OF not_not]}
val de_Morgan_conj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_conj]}
val de_Morgan_disj_conv = Conv.rewr_conv @{thm eq_reflection[OF de_Morgan_disj]}
val conj_disj_distribL_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribL]}
val conj_disj_distribR_conv = Conv.rewr_conv @{thm eq_reflection[OF conj_disj_distribR]}
end
structure HOL_Base_Order_Tac = Base_Order_Tac(
structure Logic_Sig = Logic_Signature;
(* Exclude types with specialised solvers. *)
val excluded_types = [HOLogic.natT, HOLogic.intT, HOLogic.realT]
)
structure HOL_Order_Tac = Order_Tac(structure Base_Tac = HOL_Base_Order_Tac)
fun print_orders ctxt0 =
let
val ctxt = Config.put show_sorts true ctxt0
val orders = HOL_Order_Tac.Data.get (Context.Proof ctxt)
fun pretty_term t = Pretty.block
[Pretty.quote (Syntax.pretty_term ctxt t), Pretty.brk 1,
Pretty.str "::", Pretty.brk 1,
Pretty.quote (Syntax.pretty_typ ctxt (type_of t)), Pretty.brk 1]
fun pretty_order ({kind = kind, ops = ops, ...}, _) =
Pretty.block ([Pretty.str (@{make_string} kind), Pretty.str ":", Pretty.brk 1]
@ map pretty_term ops)
in
Pretty.writeln (Pretty.big_list "order structures:" (map pretty_order orders))
end
val _ =
Outer_Syntax.command \<^command_keyword>\<open>print_orders\<close>
"print order structures available to order reasoner"
(Scan.succeed (Toplevel.keep (print_orders o Toplevel.context_of)))
\<close>
method_setup order = \<open>
Scan.succeed (fn ctxt => SIMPLE_METHOD' (HOL_Order_Tac.tac [] ctxt))
\<close> "partial and linear order reasoner"
text \<open>Declarations to set up transitivity reasoner of partial and linear orders.\<close>
context order
begin
lemma nless_le: "(\<not> a < b) \<longleftrightarrow> (\<not> a \<le> b) \<or> a = b"
using local.dual_order.order_iff_strict by blast
local_setup \<open>
HOL_Order_Tac.declare_order {
ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF nless_le]}}
}
\<close>
end
context linorder
begin
lemma nle_le: "(\<not> a \<le> b) \<longleftrightarrow> b \<le> a \<and> b \<noteq> a"
using not_le less_le by simp
local_setup \<open>
HOL_Order_Tac.declare_linorder {
ops = {eq = @{term \<open>(=) :: 'a \<Rightarrow> 'a \<Rightarrow> bool\<close>}, le = @{term \<open>(\<le>)\<close>}, lt = @{term \<open>(<)\<close>}},
thms = {trans = @{thm order_trans}, refl = @{thm order_refl}, eqD1 = @{thm eq_refl},
eqD2 = @{thm eq_refl[OF sym]}, antisym = @{thm order_antisym}, contr = @{thm notE}},
conv_thms = {less_le = @{thm eq_reflection[OF less_le]},
nless_le = @{thm eq_reflection[OF not_less]},
nle_le = @{thm eq_reflection[OF nle_le]}}
}
\<close>
end
setup \<open>
map_theory_simpset (fn ctxt0 => ctxt0 addSolver
mk_solver "partial and linear orders" (fn ctxt => HOL_Order_Tac.tac (Simplifier.prems_of ctxt) ctxt))
\<close>
ML \<open>
local
fun prp t thm = Thm.prop_of thm = t; (* FIXME proper aconv!? *)
in
fun antisym_le_simproc ctxt ct =
(case Thm.term_of ct of
(le as Const (_, T)) $ r $ s =>
(let
val prems = Simplifier.prems_of ctxt;
val less = Const (\<^const_name>\<open>less\<close>, T);
val t = HOLogic.mk_Trueprop(le $ s $ r);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop(HOLogic.Not $ (less $ r $ s)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME(mk_meta_eq(thm RS @{thm antisym_conv1})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm order_class.antisym_conv})))
end handle THM _ => NONE)
| _ => NONE);
fun antisym_less_simproc ctxt ct =
(case Thm.term_of ct of
NotC $ ((less as Const(_,T)) $ r $ s) =>
(let
val prems = Simplifier.prems_of ctxt;
val le = Const (\<^const_name>\<open>less_eq\<close>, T);
val t = HOLogic.mk_Trueprop(le $ r $ s);
in
(case find_first (prp t) prems of
NONE =>
let val t = HOLogic.mk_Trueprop (NotC $ (less $ s $ r)) in
(case find_first (prp t) prems of
NONE => NONE
| SOME thm => SOME (mk_meta_eq(thm RS @{thm linorder_class.antisym_conv3})))
end
| SOME thm => SOME (mk_meta_eq (thm RS @{thm antisym_conv2})))
end handle THM _ => NONE)
| _ => NONE);
end;
\<close>
simproc_setup antisym_le ("(x::'a::order) \<le> y") = "K antisym_le_simproc"
simproc_setup antisym_less ("\<not> (x::'a::linorder) < y") = "K antisym_less_simproc"
subsection \<open>Bounded quantifiers\<close>
syntax (ASCII)
"_All_less" :: "[idt, 'a, bool] => bool" ("(3ALL _<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3EX _<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3EX _<=_./ _)" [0, 0, 10] 10)
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3ALL _>_./ _)" [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3EX _>_./ _)" [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3ALL _>=_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3EX _>=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3ALL _~=_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3EX _~=_./ _)" [0, 0, 10] 10)
syntax
"_All_less" :: "[idt, 'a, bool] => bool" ("(3\<forall>_<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3\<exists>_<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<le>_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<le>_./ _)" [0, 0, 10] 10)
"_All_greater" :: "[idt, 'a, bool] => bool" ("(3\<forall>_>_./ _)" [0, 0, 10] 10)
"_Ex_greater" :: "[idt, 'a, bool] => bool" ("(3\<exists>_>_./ _)" [0, 0, 10] 10)
"_All_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<ge>_./ _)" [0, 0, 10] 10)
"_Ex_greater_eq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<ge>_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3\<forall>_\<noteq>_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3\<exists>_\<noteq>_./ _)" [0, 0, 10] 10)
syntax (input)
"_All_less" :: "[idt, 'a, bool] => bool" ("(3! _<_./ _)" [0, 0, 10] 10)
"_Ex_less" :: "[idt, 'a, bool] => bool" ("(3? _<_./ _)" [0, 0, 10] 10)
"_All_less_eq" :: "[idt, 'a, bool] => bool" ("(3! _<=_./ _)" [0, 0, 10] 10)
"_Ex_less_eq" :: "[idt, 'a, bool] => bool" ("(3? _<=_./ _)" [0, 0, 10] 10)
"_All_neq" :: "[idt, 'a, bool] => bool" ("(3! _~=_./ _)" [0, 0, 10] 10)
"_Ex_neq" :: "[idt, 'a, bool] => bool" ("(3? _~=_./ _)" [0, 0, 10] 10)
translations
"\<forall>x<y. P" \<rightharpoonup> "\<forall>x. x < y \<longrightarrow> P"
"\<exists>x<y. P" \<rightharpoonup> "\<exists>x. x < y \<and> P"
"\<forall>x\<le>y. P" \<rightharpoonup> "\<forall>x. x \<le> y \<longrightarrow> P"
"\<exists>x\<le>y. P" \<rightharpoonup> "\<exists>x. x \<le> y \<and> P"
"\<forall>x>y. P" \<rightharpoonup> "\<forall>x. x > y \<longrightarrow> P"
"\<exists>x>y. P" \<rightharpoonup> "\<exists>x. x > y \<and> P"
"\<forall>x\<ge>y. P" \<rightharpoonup> "\<forall>x. x \<ge> y \<longrightarrow> P"
"\<exists>x\<ge>y. P" \<rightharpoonup> "\<exists>x. x \<ge> y \<and> P"
"\<forall>x\<noteq>y. P" \<rightharpoonup> "\<forall>x. x \<noteq> y \<longrightarrow> P"
"\<exists>x\<noteq>y. P" \<rightharpoonup> "\<exists>x. x \<noteq> y \<and> P"
print_translation \<open>
let
val All_binder = Mixfix.binder_name \<^const_syntax>\<open>All\<close>;
val Ex_binder = Mixfix.binder_name \<^const_syntax>\<open>Ex\<close>;
val impl = \<^const_syntax>\<open>HOL.implies\<close>;
val conj = \<^const_syntax>\<open>HOL.conj\<close>;
val less = \<^const_syntax>\<open>less\<close>;
val less_eq = \<^const_syntax>\<open>less_eq\<close>;
val trans =
[((All_binder, impl, less),
(\<^syntax_const>\<open>_All_less\<close>, \<^syntax_const>\<open>_All_greater\<close>)),
((All_binder, impl, less_eq),
(\<^syntax_const>\<open>_All_less_eq\<close>, \<^syntax_const>\<open>_All_greater_eq\<close>)),
((Ex_binder, conj, less),
(\<^syntax_const>\<open>_Ex_less\<close>, \<^syntax_const>\<open>_Ex_greater\<close>)),
((Ex_binder, conj, less_eq),
(\<^syntax_const>\<open>_Ex_less_eq\<close>, \<^syntax_const>\<open>_Ex_greater_eq\<close>))];
fun matches_bound v t =
(case t of
Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v', _) => v = v'
| _ => false);
fun contains_var v = Term.exists_subterm (fn Free (x, _) => x = v | _ => false);
fun mk x c n P = Syntax.const c $ Syntax_Trans.mark_bound_body x $ n $ P;
fun tr' q = (q, fn _ =>
(fn [Const (\<^syntax_const>\<open>_bound\<close>, _) $ Free (v, T),
Const (c, _) $ (Const (d, _) $ t $ u) $ P] =>
(case AList.lookup (=) trans (q, c, d) of
NONE => raise Match
| SOME (l, g) =>
if matches_bound v t andalso not (contains_var v u) then mk (v, T) l u P
else if matches_bound v u andalso not (contains_var v t) then mk (v, T) g t P
else raise Match)
| _ => raise Match));
in [tr' All_binder, tr' Ex_binder] end
\<close>
subsection \<open>Transitivity reasoning\<close>
context ord
begin
lemma ord_le_eq_trans: "a \<le> b \<Longrightarrow> b = c \<Longrightarrow> a \<le> c"
by (rule subst)
lemma ord_eq_le_trans: "a = b \<Longrightarrow> b \<le> c \<Longrightarrow> a \<le> c"
by (rule ssubst)
lemma ord_less_eq_trans: "a < b \<Longrightarrow> b = c \<Longrightarrow> a < c"
by (rule subst)
lemma ord_eq_less_trans: "a = b \<Longrightarrow> b < c \<Longrightarrow> a < c"
by (rule ssubst)
end
lemma order_less_subst2: "(a::'a::order) < b \<Longrightarrow> f b < (c::'c::order) \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b < c"
finally (less_trans) show ?thesis .
qed
lemma order_less_subst1: "(a::'a::order) < f b \<Longrightarrow> (b::'b::order) < c \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a < f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (less_trans) show ?thesis .
qed
lemma order_le_less_subst2: "(a::'a::order) <= b \<Longrightarrow> f b < (c::'c::order) \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a < c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b < c"
finally (le_less_trans) show ?thesis .
qed
lemma order_le_less_subst1: "(a::'a::order) <= f b \<Longrightarrow> (b::'b::order) < c \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a <= f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (le_less_trans) show ?thesis .
qed
lemma order_less_le_subst2: "(a::'a::order) < b \<Longrightarrow> f b <= (c::'c::order) \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b <= c"
finally (less_le_trans) show ?thesis .
qed
lemma order_less_le_subst1: "(a::'a::order) < f b \<Longrightarrow> (b::'b::order) <= c \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a < f c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a < f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (less_le_trans) show ?thesis .
qed
lemma order_subst1: "(a::'a::order) <= f b \<Longrightarrow> (b::'b::order) <= c \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a <= f c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a <= f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (order_trans) show ?thesis .
qed
lemma order_subst2: "(a::'a::order) <= b \<Longrightarrow> f b <= (c::'c::order) \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b <= c"
finally (order_trans) show ?thesis .
qed
lemma ord_le_eq_subst: "a <= b \<Longrightarrow> f b = c \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> f a <= c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a <= b" hence "f a <= f b" by (rule r)
also assume "f b = c"
finally (ord_le_eq_trans) show ?thesis .
qed
lemma ord_eq_le_subst: "a = f b \<Longrightarrow> b <= c \<Longrightarrow>
(!!x y. x <= y \<Longrightarrow> f x <= f y) \<Longrightarrow> a <= f c"
proof -
assume r: "!!x y. x <= y \<Longrightarrow> f x <= f y"
assume "a = f b"
also assume "b <= c" hence "f b <= f c" by (rule r)
finally (ord_eq_le_trans) show ?thesis .
qed
lemma ord_less_eq_subst: "a < b \<Longrightarrow> f b = c \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> f a < c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a < b" hence "f a < f b" by (rule r)
also assume "f b = c"
finally (ord_less_eq_trans) show ?thesis .
qed
lemma ord_eq_less_subst: "a = f b \<Longrightarrow> b < c \<Longrightarrow>
(!!x y. x < y \<Longrightarrow> f x < f y) \<Longrightarrow> a < f c"
proof -
assume r: "!!x y. x < y \<Longrightarrow> f x < f y"
assume "a = f b"
also assume "b < c" hence "f b < f c" by (rule r)
finally (ord_eq_less_trans) show ?thesis .
qed
text \<open>
Note that this list of rules is in reverse order of priorities.
\<close>
lemmas [trans] =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
lemmas (in order) [trans] =
neq_le_trans
le_neq_trans
lemmas (in preorder) [trans] =
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
lemmas (in order) [trans] =
order.antisym
lemmas (in ord) [trans] =
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
lemmas [trans] =
trans
lemmas order_trans_rules =
order_less_subst2
order_less_subst1
order_le_less_subst2
order_le_less_subst1
order_less_le_subst2
order_less_le_subst1
order_subst2
order_subst1
ord_le_eq_subst
ord_eq_le_subst
ord_less_eq_subst
ord_eq_less_subst
forw_subst
back_subst
rev_mp
mp
neq_le_trans
le_neq_trans
less_trans
less_asym'
le_less_trans
less_le_trans
order_trans
order.antisym
ord_le_eq_trans
ord_eq_le_trans
ord_less_eq_trans
ord_eq_less_trans
trans
text \<open>These support proving chains of decreasing inequalities
a \<open>\<ge>\<close> b \<open>\<ge>\<close> c ... in Isar proofs.\<close>
lemma xt1 [no_atp]:
"a = b \<Longrightarrow> b > c \<Longrightarrow> a > c"
"a > b \<Longrightarrow> b = c \<Longrightarrow> a > c"
"a = b \<Longrightarrow> b \<ge> c \<Longrightarrow> a \<ge> c"
"a \<ge> b \<Longrightarrow> b = c \<Longrightarrow> a \<ge> c"
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> x \<Longrightarrow> x = y"
"(x::'a::order) \<ge> y \<Longrightarrow> y \<ge> z \<Longrightarrow> x \<ge> z"
"(x::'a::order) > y \<Longrightarrow> y \<ge> z \<Longrightarrow> x > z"
"(x::'a::order) \<ge> y \<Longrightarrow> y > z \<Longrightarrow> x > z"
"(a::'a::order) > b \<Longrightarrow> b > a \<Longrightarrow> P"
"(x::'a::order) > y \<Longrightarrow> y > z \<Longrightarrow> x > z"
"(a::'a::order) \<ge> b \<Longrightarrow> a \<noteq> b \<Longrightarrow> a > b"
"(a::'a::order) \<noteq> b \<Longrightarrow> a \<ge> b \<Longrightarrow> a > b"
"a = f b \<Longrightarrow> b > c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> a > f c"
"a > b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x > y \<Longrightarrow> f x > f y) \<Longrightarrow> f a > c"
"a = f b \<Longrightarrow> b \<ge> c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> a \<ge> f c"
"a \<ge> b \<Longrightarrow> f b = c \<Longrightarrow> (\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y) \<Longrightarrow> f a \<ge> c"
by auto
lemma xt2 [no_atp]:
assumes "(a::'a::order) \<ge> f b"
and "b \<ge> c"
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y"
shows "a \<ge> f c"
using assms by force
lemma xt3 [no_atp]:
assumes "(a::'a::order) \<ge> b"
and "(f b::'b::order) \<ge> c"
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y"
shows "f a \<ge> c"
using assms by force
lemma xt4 [no_atp]:
assumes "(a::'a::order) > f b"
and "(b::'b::order) \<ge> c"
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y"
shows "a > f c"
using assms by force
lemma xt5 [no_atp]:
assumes "(a::'a::order) > b"
and "(f b::'b::order) \<ge> c"
and "\<And>x y. x > y \<Longrightarrow> f x > f y"
shows "f a > c"
using assms by force
lemma xt6 [no_atp]:
assumes "(a::'a::order) \<ge> f b"
and "b > c"
and "\<And>x y. x > y \<Longrightarrow> f x > f y"
shows "a > f c"
using assms by force
lemma xt7 [no_atp]:
assumes "(a::'a::order) \<ge> b"
and "(f b::'b::order) > c"
and "\<And>x y. x \<ge> y \<Longrightarrow> f x \<ge> f y"
shows "f a > c"
using assms by force
lemma xt8 [no_atp]:
assumes "(a::'a::order) > f b"
and "(b::'b::order) > c"
and "\<And>x y. x > y \<Longrightarrow> f x > f y"
shows "a > f c"
using assms by force
lemma xt9 [no_atp]:
assumes "(a::'a::order) > b"
and "(f b::'b::order) > c"
and "\<And>x y. x > y \<Longrightarrow> f x > f y"
shows "f a > c"
using assms by force
lemmas xtrans = xt1 xt2 xt3 xt4 xt5 xt6 xt7 xt8 xt9
(*
Since "a \<ge> b" abbreviates "b \<le> a", the abbreviation "..." stands
for the wrong thing in an Isar proof.
The extra transitivity rules can be used as follows:
lemma "(a::'a::order) > z"
proof -
have "a \<ge> b" (is "_ \<ge> ?rhs")
sorry
also have "?rhs \<ge> c" (is "_ \<ge> ?rhs")
sorry
also (xtrans) have "?rhs = d" (is "_ = ?rhs")
sorry
also (xtrans) have "?rhs \<ge> e" (is "_ \<ge> ?rhs")
sorry
also (xtrans) have "?rhs > f" (is "_ > ?rhs")
sorry
also (xtrans) have "?rhs > z"
sorry
finally (xtrans) show ?thesis .
qed
Alternatively, one can use "declare xtrans [trans]" and then
leave out the "(xtrans)" above.
*)
subsection \<open>min and max -- fundamental\<close>
definition (in ord) min :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"min a b = (if a \<le> b then a else b)"
definition (in ord) max :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"max a b = (if a \<le> b then b else a)"
lemma min_absorb1: "x \<le> y \<Longrightarrow> min x y = x"
by (simp add: min_def)
lemma max_absorb2: "x \<le> y \<Longrightarrow> max x y = y"
by (simp add: max_def)
lemma min_absorb2: "(y::'a::order) \<le> x \<Longrightarrow> min x y = y"
by (simp add:min_def)
lemma max_absorb1: "(y::'a::order) \<le> x \<Longrightarrow> max x y = x"
by (simp add: max_def)
lemma max_min_same [simp]:
fixes x y :: "'a :: linorder"
shows "max x (min x y) = x" "max (min x y) x = x" "max (min x y) y = y" "max y (min x y) = y"
by(auto simp add: max_def min_def)
subsection \<open>(Unique) top and bottom elements\<close>
class bot =
fixes bot :: 'a ("\<bottom>")
class order_bot = order + bot +
assumes bot_least: "\<bottom> \<le> a"
begin
sublocale bot: ordering_top greater_eq greater bot
by standard (fact bot_least)
lemma le_bot:
"a \<le> \<bottom> \<Longrightarrow> a = \<bottom>"
by (fact bot.extremum_uniqueI)
lemma bot_unique:
"a \<le> \<bottom> \<longleftrightarrow> a = \<bottom>"
by (fact bot.extremum_unique)
lemma not_less_bot:
"\<not> a < \<bottom>"
by (fact bot.extremum_strict)
lemma bot_less:
"a \<noteq> \<bottom> \<longleftrightarrow> \<bottom> < a"
by (fact bot.not_eq_extremum)
lemma max_bot[simp]: "max bot x = x"
by(simp add: max_def bot_unique)
lemma max_bot2[simp]: "max x bot = x"
by(simp add: max_def bot_unique)
lemma min_bot[simp]: "min bot x = bot"
by(simp add: min_def bot_unique)
lemma min_bot2[simp]: "min x bot = bot"
by(simp add: min_def bot_unique)
end
class top =
fixes top :: 'a ("\<top>")
class order_top = order + top +
assumes top_greatest: "a \<le> \<top>"
begin
sublocale top: ordering_top less_eq less top
by standard (fact top_greatest)
lemma top_le:
"\<top> \<le> a \<Longrightarrow> a = \<top>"
by (fact top.extremum_uniqueI)
lemma top_unique:
"\<top> \<le> a \<longleftrightarrow> a = \<top>"
by (fact top.extremum_unique)
lemma not_top_less:
"\<not> \<top> < a"
by (fact top.extremum_strict)
lemma less_top:
"a \<noteq> \<top> \<longleftrightarrow> a < \<top>"
by (fact top.not_eq_extremum)
lemma max_top[simp]: "max top x = top"
by(simp add: max_def top_unique)
lemma max_top2[simp]: "max x top = top"
by(simp add: max_def top_unique)
lemma min_top[simp]: "min top x = x"
by(simp add: min_def top_unique)
lemma min_top2[simp]: "min x top = x"
by(simp add: min_def top_unique)
end
subsection \<open>Dense orders\<close>
class dense_order = order +
assumes dense: "x < y \<Longrightarrow> (\<exists>z. x < z \<and> z < y)"
class dense_linorder = linorder + dense_order
begin
lemma dense_le:
fixes y z :: 'a
assumes "\<And>x. x < y \<Longrightarrow> x \<le> z"
shows "y \<le> z"
proof (rule ccontr)
assume "\<not> ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "x \<le> z" using assms[OF \<open>x < y\<close>] .
ultimately show False by auto
qed
lemma dense_le_bounded:
fixes x y z :: 'a
assumes "x < y"
assumes *: "\<And>w. \<lbrakk> x < w ; w < y \<rbrakk> \<Longrightarrow> w \<le> z"
shows "y \<le> z"
proof (rule dense_le)
fix w assume "w < y"
from dense[OF \<open>x < y\<close>] obtain u where "x < u" "u < y" by safe
from linear[of u w]
show "w \<le> z"
proof (rule disjE)
assume "u \<le> w"
from less_le_trans[OF \<open>x < u\<close> \<open>u \<le> w\<close>] \<open>w < y\<close>
show "w \<le> z" by (rule *)
next
assume "w \<le> u"
from \<open>w \<le> u\<close> *[OF \<open>x < u\<close> \<open>u < y\<close>]
show "w \<le> z" by (rule order_trans)
qed
qed
lemma dense_ge:
fixes y z :: 'a
assumes "\<And>x. z < x \<Longrightarrow> y \<le> x"
shows "y \<le> z"
proof (rule ccontr)
assume "\<not> ?thesis"
hence "z < y" by simp
from dense[OF this]
obtain x where "x < y" and "z < x" by safe
moreover have "y \<le> x" using assms[OF \<open>z < x\<close>] .
ultimately show False by auto
qed
lemma dense_ge_bounded:
fixes x y z :: 'a
assumes "z < x"
assumes *: "\<And>w. \<lbrakk> z < w ; w < x \<rbrakk> \<Longrightarrow> y \<le> w"
shows "y \<le> z"
proof (rule dense_ge)
fix w assume "z < w"
from dense[OF \<open>z < x\<close>] obtain u where "z < u" "u < x" by safe
from linear[of u w]
show "y \<le> w"
proof (rule disjE)
assume "w \<le> u"
from \<open>z < w\<close> le_less_trans[OF \<open>w \<le> u\<close> \<open>u < x\<close>]
show "y \<le> w" by (rule *)
next
assume "u \<le> w"
from *[OF \<open>z < u\<close> \<open>u < x\<close>] \<open>u \<le> w\<close>
show "y \<le> w" by (rule order_trans)
qed
qed
end
class no_top = order +
assumes gt_ex: "\<exists>y. x < y"
class no_bot = order +
assumes lt_ex: "\<exists>y. y < x"
class unbounded_dense_linorder = dense_linorder + no_top + no_bot
subsection \<open>Wellorders\<close>
class wellorder = linorder +
assumes less_induct [case_names less]: "(\<And>x. (\<And>y. y < x \<Longrightarrow> P y) \<Longrightarrow> P x) \<Longrightarrow> P a"
begin
lemma wellorder_Least_lemma:
fixes k :: 'a
assumes "P k"
shows LeastI: "P (LEAST x. P x)" and Least_le: "(LEAST x. P x) \<le> k"
proof -
have "P (LEAST x. P x) \<and> (LEAST x. P x) \<le> k"
using assms proof (induct k rule: less_induct)
case (less x) then have "P x" by simp
show ?case proof (rule classical)
assume assm: "\<not> (P (LEAST a. P a) \<and> (LEAST a. P a) \<le> x)"
have "\<And>y. P y \<Longrightarrow> x \<le> y"
proof (rule classical)
fix y
assume "P y" and "\<not> x \<le> y"
with less have "P (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
by (auto simp add: not_le)
with assm have "x < (LEAST a. P a)" and "(LEAST a. P a) \<le> y"
by auto
then show "x \<le> y" by auto
qed
with \<open>P x\<close> have Least: "(LEAST a. P a) = x"
by (rule Least_equality)
with \<open>P x\<close> show ?thesis by simp
qed
qed
then show "P (LEAST x. P x)" and "(LEAST x. P x) \<le> k" by auto
qed
\<comment> \<open>The following 3 lemmas are due to Brian Huffman\<close>
lemma LeastI_ex: "\<exists>x. P x \<Longrightarrow> P (Least P)"
by (erule exE) (erule LeastI)
lemma LeastI2:
"P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
by (blast intro: LeastI)
lemma LeastI2_ex:
"\<exists>a. P a \<Longrightarrow> (\<And>x. P x \<Longrightarrow> Q x) \<Longrightarrow> Q (Least P)"
by (blast intro: LeastI_ex)
lemma LeastI2_wellorder:
assumes "P a"
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
shows "Q (Least P)"
proof (rule LeastI2_order)
show "P (Least P)" using \<open>P a\<close> by (rule LeastI)
next
fix y assume "P y" thus "Least P \<le> y" by (rule Least_le)
next
fix x assume "P x" "\<forall>y. P y \<longrightarrow> x \<le> y" thus "Q x" by (rule assms(2))
qed
lemma LeastI2_wellorder_ex:
assumes "\<exists>x. P x"
and "\<And>a. \<lbrakk> P a; \<forall>b. P b \<longrightarrow> a \<le> b \<rbrakk> \<Longrightarrow> Q a"
shows "Q (Least P)"
using assms by clarify (blast intro!: LeastI2_wellorder)
lemma not_less_Least: "k < (LEAST x. P x) \<Longrightarrow> \<not> P k"
apply (simp add: not_le [symmetric])
apply (erule contrapos_nn)
apply (erule Least_le)
done
lemma exists_least_iff: "(\<exists>n. P n) \<longleftrightarrow> (\<exists>n. P n \<and> (\<forall>m < n. \<not> P m))" (is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?rhs thus ?lhs by blast
next
assume H: ?lhs then obtain n where n: "P n" by blast
let ?x = "Least P"
{ fix m assume m: "m < ?x"
from not_less_Least[OF m] have "\<not> P m" . }
with LeastI_ex[OF H] show ?rhs by blast
qed
end
subsection \<open>Order on \<^typ>\<open>bool\<close>\<close>
instantiation bool :: "{order_bot, order_top, linorder}"
begin
definition
le_bool_def [simp]: "P \<le> Q \<longleftrightarrow> P \<longrightarrow> Q"
definition
[simp]: "(P::bool) < Q \<longleftrightarrow> \<not> P \<and> Q"
definition
[simp]: "\<bottom> \<longleftrightarrow> False"
definition
[simp]: "\<top> \<longleftrightarrow> True"
instance proof
qed auto
end
lemma le_boolI: "(P \<Longrightarrow> Q) \<Longrightarrow> P \<le> Q"
by simp
lemma le_boolI': "P \<longrightarrow> Q \<Longrightarrow> P \<le> Q"
by simp
lemma le_boolE: "P \<le> Q \<Longrightarrow> P \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"
by simp
lemma le_boolD: "P \<le> Q \<Longrightarrow> P \<longrightarrow> Q"
by simp
lemma bot_boolE: "\<bottom> \<Longrightarrow> P"
by simp
lemma top_boolI: \<top>
by simp
lemma [code]:
"False \<le> b \<longleftrightarrow> True"
"True \<le> b \<longleftrightarrow> b"
"False < b \<longleftrightarrow> b"
"True < b \<longleftrightarrow> False"
by simp_all
subsection \<open>Order on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>
instantiation "fun" :: (type, ord) ord
begin
definition
le_fun_def: "f \<le> g \<longleftrightarrow> (\<forall>x. f x \<le> g x)"
definition
"(f::'a \<Rightarrow> 'b) < g \<longleftrightarrow> f \<le> g \<and> \<not> (g \<le> f)"
instance ..
end
instance "fun" :: (type, preorder) preorder proof
qed (auto simp add: le_fun_def less_fun_def
intro: order_trans order.antisym)
instance "fun" :: (type, order) order proof
qed (auto simp add: le_fun_def intro: order.antisym)
instantiation "fun" :: (type, bot) bot
begin
definition
"\<bottom> = (\<lambda>x. \<bottom>)"
instance ..
end
instantiation "fun" :: (type, order_bot) order_bot
begin
lemma bot_apply [simp, code]:
"\<bottom> x = \<bottom>"
by (simp add: bot_fun_def)
instance proof
qed (simp add: le_fun_def)
end
instantiation "fun" :: (type, top) top
begin
definition
[no_atp]: "\<top> = (\<lambda>x. \<top>)"
instance ..
end
instantiation "fun" :: (type, order_top) order_top
begin
lemma top_apply [simp, code]:
"\<top> x = \<top>"
by (simp add: top_fun_def)
instance proof
qed (simp add: le_fun_def)
end
lemma le_funI: "(\<And>x. f x \<le> g x) \<Longrightarrow> f \<le> g"
unfolding le_fun_def by simp
lemma le_funE: "f \<le> g \<Longrightarrow> (f x \<le> g x \<Longrightarrow> P) \<Longrightarrow> P"
unfolding le_fun_def by simp
lemma le_funD: "f \<le> g \<Longrightarrow> f x \<le> g x"
by (rule le_funE)
subsection \<open>Order on unary and binary predicates\<close>
lemma predicate1I:
assumes PQ: "\<And>x. P x \<Longrightarrow> Q x"
shows "P \<le> Q"
apply (rule le_funI)
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate1D:
"P \<le> Q \<Longrightarrow> P x \<Longrightarrow> Q x"
apply (erule le_funE)
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate1D:
"P x \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x"
by (rule predicate1D)
lemma predicate2I:
assumes PQ: "\<And>x y. P x y \<Longrightarrow> Q x y"
shows "P \<le> Q"
apply (rule le_funI)+
apply (rule le_boolI)
apply (rule PQ)
apply assumption
done
lemma predicate2D:
"P \<le> Q \<Longrightarrow> P x y \<Longrightarrow> Q x y"
apply (erule le_funE)+
apply (erule le_boolE)
apply assumption+
done
lemma rev_predicate2D:
"P x y \<Longrightarrow> P \<le> Q \<Longrightarrow> Q x y"
by (rule predicate2D)
lemma bot1E [no_atp]: "\<bottom> x \<Longrightarrow> P"
by (simp add: bot_fun_def)
lemma bot2E: "\<bottom> x y \<Longrightarrow> P"
by (simp add: bot_fun_def)
lemma top1I: "\<top> x"
by (simp add: top_fun_def)
lemma top2I: "\<top> x y"
by (simp add: top_fun_def)
subsection \<open>Name duplicates\<close>
lemmas antisym = order.antisym
lemmas eq_iff = order.eq_iff
lemmas order_eq_refl = preorder_class.eq_refl
lemmas order_less_irrefl = preorder_class.less_irrefl
lemmas order_less_imp_le = preorder_class.less_imp_le
lemmas order_less_not_sym = preorder_class.less_not_sym
lemmas order_less_asym = preorder_class.less_asym
lemmas order_less_trans = preorder_class.less_trans
lemmas order_le_less_trans = preorder_class.le_less_trans
lemmas order_less_le_trans = preorder_class.less_le_trans
lemmas order_less_imp_not_less = preorder_class.less_imp_not_less
lemmas order_less_imp_triv = preorder_class.less_imp_triv
lemmas order_less_asym' = preorder_class.less_asym'
lemmas order_less_le = order_class.less_le
lemmas order_le_less = order_class.le_less
lemmas order_le_imp_less_or_eq = order_class.le_imp_less_or_eq
lemmas order_less_imp_not_eq = order_class.less_imp_not_eq
lemmas order_less_imp_not_eq2 = order_class.less_imp_not_eq2
lemmas order_neq_le_trans = order_class.neq_le_trans
lemmas order_le_neq_trans = order_class.le_neq_trans
lemmas order_eq_iff = order_class.order.eq_iff
lemmas order_antisym_conv = order_class.antisym_conv
lemmas linorder_linear = linorder_class.linear
lemmas linorder_less_linear = linorder_class.less_linear
lemmas linorder_le_less_linear = linorder_class.le_less_linear
lemmas linorder_le_cases = linorder_class.le_cases
lemmas linorder_not_less = linorder_class.not_less
lemmas linorder_not_le = linorder_class.not_le
lemmas linorder_neq_iff = linorder_class.neq_iff
lemmas linorder_neqE = linorder_class.neqE
end