(* Title: HOL/ex/LocaleTest2.thy
Author: Clemens Ballarin
Copyright (c) 2007 by Clemens Ballarin
More regression tests for locales.
Definitions are less natural in FOL, since there is no selection operator.
Hence we do them here in HOL, not in the main test suite for locales,
which is FOL/ex/LocaleTest.thy
*)
section \<open>Test of Locale Interpretation\<close>
theory LocaleTest2
imports Main
begin
section \<open>Interpretation of Defined Concepts\<close>
text \<open>Naming convention for global objects: prefixes D and d\<close>
subsection \<open>Lattices\<close>
text \<open>Much of the lattice proofs are from HOL/Lattice.\<close>
subsubsection \<open>Definitions\<close>
locale dpo =
fixes le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>" 50)
assumes refl [intro, simp]: "x \<sqsubseteq> x"
and antisym [intro]: "[| x \<sqsubseteq> y; y \<sqsubseteq> x |] ==> x = y"
and trans [trans]: "[| x \<sqsubseteq> y; y \<sqsubseteq> z |] ==> x \<sqsubseteq> z"
begin
theorem circular:
"[| x \<sqsubseteq> y; y \<sqsubseteq> z; z \<sqsubseteq> x |] ==> x = y & y = z"
by (blast intro: trans)
definition
less :: "['a, 'a] => bool" (infixl "\<sqsubset>" 50)
where "(x \<sqsubset> y) = (x \<sqsubseteq> y & x ~= y)"
theorem abs_test:
"(\<sqsubset>) = (\<lambda>x y. x \<sqsubset> y)"
by simp
definition
is_inf :: "['a, 'a, 'a] => bool"
where "is_inf x y i = (i \<sqsubseteq> x \<and> i \<sqsubseteq> y \<and> (\<forall>z. z \<sqsubseteq> x \<and> z \<sqsubseteq> y \<longrightarrow> z \<sqsubseteq> i))"
definition
is_sup :: "['a, 'a, 'a] => bool"
where "is_sup x y s = (x \<sqsubseteq> s \<and> y \<sqsubseteq> s \<and> (\<forall>z. x \<sqsubseteq> z \<and> y \<sqsubseteq> z \<longrightarrow> s \<sqsubseteq> z))"
end
locale dlat = dpo +
assumes ex_inf: "\<exists>inf. dpo.is_inf le x y inf"
and ex_sup: "\<exists>sup. dpo.is_sup le x y sup"
begin
definition
meet :: "['a, 'a] => 'a" (infixl "\<sqinter>" 70)
where "x \<sqinter> y = (THE inf. is_inf x y inf)"
definition
join :: "['a, 'a] => 'a" (infixl "\<squnion>" 65)
where "x \<squnion> y = (THE sup. is_sup x y sup)"
lemma is_infI [intro?]: "i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow>
(\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i) \<Longrightarrow> is_inf x y i"
by (unfold is_inf_def) blast
lemma is_inf_lower [elim?]:
"is_inf x y i \<Longrightarrow> (i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold is_inf_def) blast
lemma is_inf_greatest [elim?]:
"is_inf x y i \<Longrightarrow> z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i"
by (unfold is_inf_def) blast
theorem is_inf_uniq: "is_inf x y i \<Longrightarrow> is_inf x y i' \<Longrightarrow> i = i'"
proof -
assume inf: "is_inf x y i"
assume inf': "is_inf x y i'"
show ?thesis
proof (rule antisym)
from inf' show "i \<sqsubseteq> i'"
proof (rule is_inf_greatest)
from inf show "i \<sqsubseteq> x" ..
from inf show "i \<sqsubseteq> y" ..
qed
from inf show "i' \<sqsubseteq> i"
proof (rule is_inf_greatest)
from inf' show "i' \<sqsubseteq> x" ..
from inf' show "i' \<sqsubseteq> y" ..
qed
qed
qed
theorem is_inf_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_inf x y x"
proof -
assume "x \<sqsubseteq> y"
show ?thesis
proof
show "x \<sqsubseteq> x" ..
show "x \<sqsubseteq> y" by fact
fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> y" show "z \<sqsubseteq> x" by fact
qed
qed
lemma meet_equality [elim?]: "is_inf x y i \<Longrightarrow> x \<sqinter> y = i"
proof (unfold meet_def)
assume "is_inf x y i"
then show "(THE i. is_inf x y i) = i"
by (rule the_equality) (rule is_inf_uniq [OF _ \<open>is_inf x y i\<close>])
qed
lemma meetI [intro?]:
"i \<sqsubseteq> x \<Longrightarrow> i \<sqsubseteq> y \<Longrightarrow> (\<And>z. z \<sqsubseteq> x \<Longrightarrow> z \<sqsubseteq> y \<Longrightarrow> z \<sqsubseteq> i) \<Longrightarrow> x \<sqinter> y = i"
by (rule meet_equality, rule is_infI) blast+
lemma is_inf_meet [intro?]: "is_inf x y (x \<sqinter> y)"
proof (unfold meet_def)
from ex_inf obtain i where "is_inf x y i" ..
then show "is_inf x y (THE i. is_inf x y i)"
by (rule theI) (rule is_inf_uniq [OF _ \<open>is_inf x y i\<close>])
qed
lemma meet_left [intro?]:
"x \<sqinter> y \<sqsubseteq> x"
by (rule is_inf_lower) (rule is_inf_meet)
lemma meet_right [intro?]:
"x \<sqinter> y \<sqsubseteq> y"
by (rule is_inf_lower) (rule is_inf_meet)
lemma meet_le [intro?]:
"[| z \<sqsubseteq> x; z \<sqsubseteq> y |] ==> z \<sqsubseteq> x \<sqinter> y"
by (rule is_inf_greatest) (rule is_inf_meet)
lemma is_supI [intro?]: "x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow>
(\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z) \<Longrightarrow> is_sup x y s"
by (unfold is_sup_def) blast
lemma is_sup_least [elim?]:
"is_sup x y s \<Longrightarrow> x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z"
by (unfold is_sup_def) blast
lemma is_sup_upper [elim?]:
"is_sup x y s \<Longrightarrow> (x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow> C) \<Longrightarrow> C"
by (unfold is_sup_def) blast
theorem is_sup_uniq: "is_sup x y s \<Longrightarrow> is_sup x y s' \<Longrightarrow> s = s'"
proof -
assume sup: "is_sup x y s"
assume sup': "is_sup x y s'"
show ?thesis
proof (rule antisym)
from sup show "s \<sqsubseteq> s'"
proof (rule is_sup_least)
from sup' show "x \<sqsubseteq> s'" ..
from sup' show "y \<sqsubseteq> s'" ..
qed
from sup' show "s' \<sqsubseteq> s"
proof (rule is_sup_least)
from sup show "x \<sqsubseteq> s" ..
from sup show "y \<sqsubseteq> s" ..
qed
qed
qed
theorem is_sup_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> is_sup x y y"
proof -
assume "x \<sqsubseteq> y"
show ?thesis
proof
show "x \<sqsubseteq> y" by fact
show "y \<sqsubseteq> y" ..
fix z assume "x \<sqsubseteq> z" and "y \<sqsubseteq> z"
show "y \<sqsubseteq> z" by fact
qed
qed
lemma join_equality [elim?]: "is_sup x y s \<Longrightarrow> x \<squnion> y = s"
proof (unfold join_def)
assume "is_sup x y s"
then show "(THE s. is_sup x y s) = s"
by (rule the_equality) (rule is_sup_uniq [OF _ \<open>is_sup x y s\<close>])
qed
lemma joinI [intro?]: "x \<sqsubseteq> s \<Longrightarrow> y \<sqsubseteq> s \<Longrightarrow>
(\<And>z. x \<sqsubseteq> z \<Longrightarrow> y \<sqsubseteq> z \<Longrightarrow> s \<sqsubseteq> z) \<Longrightarrow> x \<squnion> y = s"
by (rule join_equality, rule is_supI) blast+
lemma is_sup_join [intro?]: "is_sup x y (x \<squnion> y)"
proof (unfold join_def)
from ex_sup obtain s where "is_sup x y s" ..
then show "is_sup x y (THE s. is_sup x y s)"
by (rule theI) (rule is_sup_uniq [OF _ \<open>is_sup x y s\<close>])
qed
lemma join_left [intro?]:
"x \<sqsubseteq> x \<squnion> y"
by (rule is_sup_upper) (rule is_sup_join)
lemma join_right [intro?]:
"y \<sqsubseteq> x \<squnion> y"
by (rule is_sup_upper) (rule is_sup_join)
lemma join_le [intro?]:
"[| x \<sqsubseteq> z; y \<sqsubseteq> z |] ==> x \<squnion> y \<sqsubseteq> z"
by (rule is_sup_least) (rule is_sup_join)
theorem meet_assoc: "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
proof (rule meetI)
show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> x \<sqinter> y"
proof
show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> x" ..
show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y"
proof -
have "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y \<sqinter> z" ..
also have "\<dots> \<sqsubseteq> y" ..
finally show ?thesis .
qed
qed
show "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> z"
proof -
have "x \<sqinter> (y \<sqinter> z) \<sqsubseteq> y \<sqinter> z" ..
also have "\<dots> \<sqsubseteq> z" ..
finally show ?thesis .
qed
fix w assume "w \<sqsubseteq> x \<sqinter> y" and "w \<sqsubseteq> z"
show "w \<sqsubseteq> x \<sqinter> (y \<sqinter> z)"
proof
show "w \<sqsubseteq> x"
proof -
have "w \<sqsubseteq> x \<sqinter> y" by fact
also have "\<dots> \<sqsubseteq> x" ..
finally show ?thesis .
qed
show "w \<sqsubseteq> y \<sqinter> z"
proof
show "w \<sqsubseteq> y"
proof -
have "w \<sqsubseteq> x \<sqinter> y" by fact
also have "\<dots> \<sqsubseteq> y" ..
finally show ?thesis .
qed
show "w \<sqsubseteq> z" by fact
qed
qed
qed
theorem meet_commute: "x \<sqinter> y = y \<sqinter> x"
proof (rule meetI)
show "y \<sqinter> x \<sqsubseteq> x" ..
show "y \<sqinter> x \<sqsubseteq> y" ..
fix z assume "z \<sqsubseteq> y" and "z \<sqsubseteq> x"
then show "z \<sqsubseteq> y \<sqinter> x" ..
qed
theorem meet_join_absorb: "x \<sqinter> (x \<squnion> y) = x"
proof (rule meetI)
show "x \<sqsubseteq> x" ..
show "x \<sqsubseteq> x \<squnion> y" ..
fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> x \<squnion> y"
show "z \<sqsubseteq> x" by fact
qed
theorem join_assoc: "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
proof (rule joinI)
show "x \<squnion> y \<sqsubseteq> x \<squnion> (y \<squnion> z)"
proof
show "x \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
show "y \<sqsubseteq> x \<squnion> (y \<squnion> z)"
proof -
have "y \<sqsubseteq> y \<squnion> z" ..
also have "... \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
finally show ?thesis .
qed
qed
show "z \<sqsubseteq> x \<squnion> (y \<squnion> z)"
proof -
have "z \<sqsubseteq> y \<squnion> z" ..
also have "... \<sqsubseteq> x \<squnion> (y \<squnion> z)" ..
finally show ?thesis .
qed
fix w assume "x \<squnion> y \<sqsubseteq> w" and "z \<sqsubseteq> w"
show "x \<squnion> (y \<squnion> z) \<sqsubseteq> w"
proof
show "x \<sqsubseteq> w"
proof -
have "x \<sqsubseteq> x \<squnion> y" ..
also have "\<dots> \<sqsubseteq> w" by fact
finally show ?thesis .
qed
show "y \<squnion> z \<sqsubseteq> w"
proof
show "y \<sqsubseteq> w"
proof -
have "y \<sqsubseteq> x \<squnion> y" ..
also have "... \<sqsubseteq> w" by fact
finally show ?thesis .
qed
show "z \<sqsubseteq> w" by fact
qed
qed
qed
theorem join_commute: "x \<squnion> y = y \<squnion> x"
proof (rule joinI)
show "x \<sqsubseteq> y \<squnion> x" ..
show "y \<sqsubseteq> y \<squnion> x" ..
fix z assume "y \<sqsubseteq> z" and "x \<sqsubseteq> z"
then show "y \<squnion> x \<sqsubseteq> z" ..
qed
theorem join_meet_absorb: "x \<squnion> (x \<sqinter> y) = x"
proof (rule joinI)
show "x \<sqsubseteq> x" ..
show "x \<sqinter> y \<sqsubseteq> x" ..
fix z assume "x \<sqsubseteq> z" and "x \<sqinter> y \<sqsubseteq> z"
show "x \<sqsubseteq> z" by fact
qed
theorem meet_idem: "x \<sqinter> x = x"
proof -
have "x \<sqinter> (x \<squnion> (x \<sqinter> x)) = x" by (rule meet_join_absorb)
also have "x \<squnion> (x \<sqinter> x) = x" by (rule join_meet_absorb)
finally show ?thesis .
qed
theorem meet_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
proof (rule meetI)
assume "x \<sqsubseteq> y"
show "x \<sqsubseteq> x" ..
show "x \<sqsubseteq> y" by fact
fix z assume "z \<sqsubseteq> x" and "z \<sqsubseteq> y"
show "z \<sqsubseteq> x" by fact
qed
theorem meet_related2 [elim?]: "y \<sqsubseteq> x \<Longrightarrow> x \<sqinter> y = y"
by (drule meet_related) (simp add: meet_commute)
theorem join_related [elim?]: "x \<sqsubseteq> y \<Longrightarrow> x \<squnion> y = y"
proof (rule joinI)
assume "x \<sqsubseteq> y"
show "y \<sqsubseteq> y" ..
show "x \<sqsubseteq> y" by fact
fix z assume "x \<sqsubseteq> z" and "y \<sqsubseteq> z"
show "y \<sqsubseteq> z" by fact
qed
theorem join_related2 [elim?]: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
by (drule join_related) (simp add: join_commute)
text \<open>Additional theorems\<close>
theorem meet_connection: "(x \<sqsubseteq> y) = (x \<sqinter> y = x)"
proof
assume "x \<sqsubseteq> y"
then have "is_inf x y x" ..
then show "x \<sqinter> y = x" ..
next
have "x \<sqinter> y \<sqsubseteq> y" ..
also assume "x \<sqinter> y = x"
finally show "x \<sqsubseteq> y" .
qed
theorem meet_connection2: "(x \<sqsubseteq> y) = (y \<sqinter> x = x)"
using meet_commute meet_connection by simp
theorem join_connection: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
proof
assume "x \<sqsubseteq> y"
then have "is_sup x y y" ..
then show "x \<squnion> y = y" ..
next
have "x \<sqsubseteq> x \<squnion> y" ..
also assume "x \<squnion> y = y"
finally show "x \<sqsubseteq> y" .
qed
theorem join_connection2: "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
using join_commute join_connection by simp
text \<open>Naming according to Jacobson I, p.\ 459.\<close>
lemmas L1 = join_commute meet_commute
lemmas L2 = join_assoc meet_assoc
(*lemmas L3 = join_idem meet_idem*)
lemmas L4 = join_meet_absorb meet_join_absorb
end
locale ddlat = dlat +
assumes meet_distr:
"dlat.meet le x (dlat.join le y z) =
dlat.join le (dlat.meet le x y) (dlat.meet le x z)"
begin
lemma join_distr:
"x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
txt \<open>Jacobson I, p.\ 462\<close>
proof -
have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by (simp add: L4)
also have "... = x \<squnion> ((x \<sqinter> z) \<squnion> (y \<sqinter> z))" by (simp add: L2)
also have "... = x \<squnion> ((x \<squnion> y) \<sqinter> z)" by (simp add: L1 meet_distr)
also have "... = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)" by (simp add: L1 L4)
also have "... = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by (simp add: meet_distr)
finally show ?thesis .
qed
end
locale dlo = dpo +
assumes total: "x \<sqsubseteq> y | y \<sqsubseteq> x"
begin
lemma less_total: "x \<sqsubset> y | x = y | y \<sqsubset> x"
using total
by (unfold less_def) blast
end
sublocale dlo < dlat
proof
fix x y
from total have "is_inf x y (if x \<sqsubseteq> y then x else y)" by (auto simp: is_inf_def)
then show "\<exists>inf. is_inf x y inf" by blast
next
fix x y
from total have "is_sup x y (if x \<sqsubseteq> y then y else x)" by (auto simp: is_sup_def)
then show "\<exists>sup. is_sup x y sup" by blast
qed
sublocale dlo < ddlat
proof
fix x y z
show "x \<sqinter> (y \<squnion> z) = x \<sqinter> y \<squnion> x \<sqinter> z" (is "?l = ?r")
txt \<open>Jacobson I, p.\ 462\<close>
proof -
{ assume c: "y \<sqsubseteq> x" "z \<sqsubseteq> x"
from c have "?l = y \<squnion> z"
by (metis c (*join_commute*) join_connection2 join_related2 (*meet_commute*) meet_connection meet_related2 total)
also from c have "... = ?r" by (metis (*c*) (*join_commute*) meet_related2)
finally have "?l = ?r" . }
moreover
{ assume c: "x \<sqsubseteq> y | x \<sqsubseteq> z"
from c have "?l = x"
by (metis (*antisym*) (*c*) (*circular*) (*join_assoc*)(* join_commute *) join_connection2 (*join_left*) join_related2 meet_connection(* meet_related2*) total trans)
also from c have "... = ?r"
by (metis join_commute join_related2 meet_connection meet_related2 total)
finally have "?l = ?r" . }
moreover note total
ultimately show ?thesis by blast
qed
qed
subsubsection \<open>Total order \<open><=\<close> on \<^typ>\<open>int\<close>\<close>
interpretation int: dpo "(<=) :: [int, int] => bool"
rewrites "(dpo.less (<=) (x::int) y) = (x < y)"
txt \<open>We give interpretation for less, but not \<open>is_inf\<close> and \<open>is_sub\<close>.\<close>
proof -
show "dpo ((<=) :: [int, int] => bool)"
proof qed auto
then interpret int: dpo "(<=) :: [int, int] => bool" .
txt \<open>Gives interpreted version of \<open>less_def\<close> (without condition).\<close>
show "(dpo.less (<=) (x::int) y) = (x < y)"
by (unfold int.less_def) auto
qed
thm int.circular
lemma "\<lbrakk> (x::int) \<le> y; y \<le> z; z \<le> x\<rbrakk> \<Longrightarrow> x = y \<and> y = z"
apply (rule int.circular) apply assumption apply assumption apply assumption done
thm int.abs_test
lemma "((<) :: [int, int] => bool) = (<)"
apply (rule int.abs_test) done
interpretation int: dlat "(<=) :: [int, int] => bool"
rewrites meet_eq: "dlat.meet (<=) (x::int) y = min x y"
and join_eq: "dlat.join (<=) (x::int) y = max x y"
proof -
show "dlat ((<=) :: [int, int] => bool)"
apply unfold_locales
apply (unfold int.is_inf_def int.is_sup_def)
apply arith+
done
then interpret int: dlat "(<=) :: [int, int] => bool" .
txt \<open>Interpretation to ease use of definitions, which are
conditional in general but unconditional after interpretation.\<close>
show "dlat.meet (<=) (x::int) y = min x y"
apply (unfold int.meet_def)
apply (rule the_equality)
apply (unfold int.is_inf_def)
by auto
show "dlat.join (<=) (x::int) y = max x y"
apply (unfold int.join_def)
apply (rule the_equality)
apply (unfold int.is_sup_def)
by auto
qed
interpretation int: dlo "(<=) :: [int, int] => bool"
proof qed arith
text \<open>Interpreted theorems from the locales, involving defined terms.\<close>
thm int.less_def text \<open>from dpo\<close>
thm int.meet_left text \<open>from dlat\<close>
thm int.meet_distr text \<open>from ddlat\<close>
thm int.less_total text \<open>from dlo\<close>
subsubsection \<open>Total order \<open><=\<close> on \<^typ>\<open>nat\<close>\<close>
interpretation nat: dpo "(<=) :: [nat, nat] => bool"
rewrites "dpo.less (<=) (x::nat) y = (x < y)"
txt \<open>We give interpretation for less, but not \<open>is_inf\<close> and \<open>is_sub\<close>.\<close>
proof -
show "dpo ((<=) :: [nat, nat] => bool)"
proof qed auto
then interpret nat: dpo "(<=) :: [nat, nat] => bool" .
txt \<open>Gives interpreted version of \<open>less_def\<close> (without condition).\<close>
show "dpo.less (<=) (x::nat) y = (x < y)"
apply (unfold nat.less_def)
apply auto
done
qed
interpretation nat: dlat "(<=) :: [nat, nat] => bool"
rewrites "dlat.meet (<=) (x::nat) y = min x y"
and "dlat.join (<=) (x::nat) y = max x y"
proof -
show "dlat ((<=) :: [nat, nat] => bool)"
apply unfold_locales
apply (unfold nat.is_inf_def nat.is_sup_def)
apply arith+
done
then interpret nat: dlat "(<=) :: [nat, nat] => bool" .
txt \<open>Interpretation to ease use of definitions, which are
conditional in general but unconditional after interpretation.\<close>
show "dlat.meet (<=) (x::nat) y = min x y"
apply (unfold nat.meet_def)
apply (rule the_equality)
apply (unfold nat.is_inf_def)
by auto
show "dlat.join (<=) (x::nat) y = max x y"
apply (unfold nat.join_def)
apply (rule the_equality)
apply (unfold nat.is_sup_def)
by auto
qed
interpretation nat: dlo "(<=) :: [nat, nat] => bool"
proof qed arith
text \<open>Interpreted theorems from the locales, involving defined terms.\<close>
thm nat.less_def text \<open>from dpo\<close>
thm nat.meet_left text \<open>from dlat\<close>
thm nat.meet_distr text \<open>from ddlat\<close>
thm nat.less_total text \<open>from ldo\<close>
subsubsection \<open>Lattice \<open>dvd\<close> on \<^typ>\<open>nat\<close>\<close>
interpretation nat_dvd: dpo "(dvd) :: [nat, nat] => bool"
rewrites "dpo.less (dvd) (x::nat) y = (x dvd y & x ~= y)"
txt \<open>We give interpretation for less, but not \<open>is_inf\<close> and \<open>is_sub\<close>.\<close>
proof -
show "dpo ((dvd) :: [nat, nat] => bool)"
proof qed (auto simp: dvd_def)
then interpret nat_dvd: dpo "(dvd) :: [nat, nat] => bool" .
txt \<open>Gives interpreted version of \<open>less_def\<close> (without condition).\<close>
show "dpo.less (dvd) (x::nat) y = (x dvd y & x ~= y)"
apply (unfold nat_dvd.less_def)
apply auto
done
qed
interpretation nat_dvd: dlat "(dvd) :: [nat, nat] => bool"
rewrites "dlat.meet (dvd) (x::nat) y = gcd x y"
and "dlat.join (dvd) (x::nat) y = lcm x y"
proof -
show "dlat ((dvd) :: [nat, nat] => bool)"
apply unfold_locales
apply (unfold nat_dvd.is_inf_def nat_dvd.is_sup_def)
apply (rule_tac x = "gcd x y" in exI)
apply auto [1]
apply (rule_tac x = "lcm x y" in exI)
apply (auto intro: dvd_lcm1 dvd_lcm2 lcm_least)
done
then interpret nat_dvd: dlat "(dvd) :: [nat, nat] => bool" .
txt \<open>Interpretation to ease use of definitions, which are
conditional in general but unconditional after interpretation.\<close>
show "dlat.meet (dvd) (x::nat) y = gcd x y"
apply (unfold nat_dvd.meet_def)
apply (rule the_equality)
apply (unfold nat_dvd.is_inf_def)
by auto
show "dlat.join (dvd) (x::nat) y = lcm x y"
apply (unfold nat_dvd.join_def)
apply (rule the_equality)
apply (unfold nat_dvd.is_sup_def)
by (auto intro: dvd_lcm1 dvd_lcm2 lcm_least)
qed
text \<open>Interpreted theorems from the locales, involving defined terms.\<close>
thm nat_dvd.less_def text \<open>from dpo\<close>
lemma "((x::nat) dvd y & x ~= y) = (x dvd y & x ~= y)"
apply (rule nat_dvd.less_def) done
thm nat_dvd.meet_left text \<open>from dlat\<close>
lemma "gcd x y dvd (x::nat)"
apply (rule nat_dvd.meet_left) done
subsection \<open>Group example with defined operations \<open>inv\<close> and \<open>unit\<close>\<close>
subsubsection \<open>Locale declarations and lemmas\<close>
locale Dsemi =
fixes prod (infixl "**" 65)
assumes assoc: "(x ** y) ** z = x ** (y ** z)"
locale Dmonoid = Dsemi +
fixes one
assumes l_one [simp]: "one ** x = x"
and r_one [simp]: "x ** one = x"
begin
definition
inv where "inv x = (THE y. x ** y = one & y ** x = one)"
definition
unit where "unit x = (\<exists>y. x ** y = one & y ** x = one)"
lemma inv_unique:
assumes eq: "y ** x = one" "x ** y' = one"
shows "y = y'"
proof -
from eq have "y = y ** (x ** y')" by (simp add: r_one)
also have "... = (y ** x) ** y'" by (simp add: assoc)
also from eq have "... = y'" by (simp add: l_one)
finally show ?thesis .
qed
lemma unit_one [intro, simp]:
"unit one"
by (unfold unit_def) auto
lemma unit_l_inv_ex:
"unit x ==> \<exists>y. y ** x = one"
by (unfold unit_def) auto
lemma unit_r_inv_ex:
"unit x ==> \<exists>y. x ** y = one"
by (unfold unit_def) auto
lemma unit_l_inv:
"unit x ==> inv x ** x = one"
apply (simp add: unit_def inv_def) apply (erule exE)
apply (rule theI2, fast)
apply (rule inv_unique)
apply fast+
done
lemma unit_r_inv:
"unit x ==> x ** inv x = one"
apply (simp add: unit_def inv_def) apply (erule exE)
apply (rule theI2, fast)
apply (rule inv_unique)
apply fast+
done
lemma unit_inv_unit [intro, simp]:
"unit x ==> unit (inv x)"
proof -
assume x: "unit x"
show "unit (inv x)"
by (auto simp add: unit_def
intro: unit_l_inv unit_r_inv x)
qed
lemma unit_l_cancel [simp]:
"unit x ==> (x ** y = x ** z) = (y = z)"
proof
assume eq: "x ** y = x ** z"
and G: "unit x"
then have "(inv x ** x) ** y = (inv x ** x) ** z"
by (simp add: assoc)
with G show "y = z" by (simp add: unit_l_inv)
next
assume eq: "y = z"
and G: "unit x"
then show "x ** y = x ** z" by simp
qed
lemma unit_inv_inv [simp]:
"unit x ==> inv (inv x) = x"
proof -
assume x: "unit x"
then have "inv x ** inv (inv x) = inv x ** x"
by (simp add: unit_l_inv unit_r_inv)
with x show ?thesis by simp
qed
lemma inv_inj_on_unit:
"inj_on inv {x. unit x}"
proof (rule inj_onI, simp)
fix x y
assume G: "unit x" "unit y" and eq: "inv x = inv y"
then have "inv (inv x) = inv (inv y)" by simp
with G show "x = y" by simp
qed
lemma unit_inv_comm:
assumes inv: "x ** y = one"
and G: "unit x" "unit y"
shows "y ** x = one"
proof -
from G have "x ** y ** x = x ** one" by (auto simp add: inv)
with G show ?thesis by (simp del: r_one add: assoc)
qed
end
locale Dgrp = Dmonoid +
assumes unit [intro, simp]: "Dmonoid.unit ((**)) one x"
begin
lemma l_inv_ex [simp]:
"\<exists>y. y ** x = one"
using unit_l_inv_ex by simp
lemma r_inv_ex [simp]:
"\<exists>y. x ** y = one"
using unit_r_inv_ex by simp
lemma l_inv [simp]:
"inv x ** x = one"
using unit_l_inv by simp
lemma l_cancel [simp]:
"(x ** y = x ** z) = (y = z)"
using unit_l_inv by simp
lemma r_inv [simp]:
"x ** inv x = one"
proof -
have "inv x ** (x ** inv x) = inv x ** one"
by (simp add: assoc [symmetric] l_inv)
then show ?thesis by (simp del: r_one)
qed
lemma r_cancel [simp]:
"(y ** x = z ** x) = (y = z)"
proof
assume eq: "y ** x = z ** x"
then have "y ** (x ** inv x) = z ** (x ** inv x)"
by (simp add: assoc [symmetric] del: r_inv)
then show "y = z" by simp
qed simp
lemma inv_one [simp]:
"inv one = one"
proof -
have "inv one = one ** (inv one)" by (simp del: r_inv)
moreover have "... = one" by simp
finally show ?thesis .
qed
lemma inv_inv [simp]:
"inv (inv x) = x"
using unit_inv_inv by simp
lemma inv_inj:
"inj_on inv UNIV"
using inv_inj_on_unit by simp
lemma inv_mult_group:
"inv (x ** y) = inv y ** inv x"
proof -
have "inv (x ** y) ** (x ** y) = (inv y ** inv x) ** (x ** y)"
by (simp add: assoc l_inv) (simp add: assoc [symmetric])
then show ?thesis by (simp del: l_inv)
qed
lemma inv_comm:
"x ** y = one ==> y ** x = one"
by (rule unit_inv_comm) auto
lemma inv_equality:
"y ** x = one ==> inv x = y"
apply (simp add: inv_def)
apply (rule the_equality)
apply (simp add: inv_comm [of y x])
apply (rule r_cancel [THEN iffD1], auto)
done
end
locale Dhom = prod: Dgrp prod one + sum: Dgrp sum zero
for prod (infixl "**" 65) and one and sum (infixl "+++" 60) and zero +
fixes hom
assumes hom_mult [simp]: "hom (x ** y) = hom x +++ hom y"
begin
lemma hom_one [simp]:
"hom one = zero"
proof -
have "hom one +++ zero = hom one +++ hom one"
by (simp add: hom_mult [symmetric] del: hom_mult)
then show ?thesis by (simp del: sum.r_one)
qed
end
subsubsection \<open>Interpretation of Functions\<close>
interpretation Dfun: Dmonoid "(o)" "id :: 'a => 'a"
rewrites "Dmonoid.unit (o) id f = bij (f::'a => 'a)"
(* and "Dmonoid.inv (o) id" = "inv :: ('a => 'a) => ('a => 'a)" *)
proof -
show "Dmonoid (o) (id :: 'a => 'a)" proof qed (simp_all add: o_assoc)
note Dmonoid = this
(*
from this interpret Dmonoid "(o)" "id :: 'a => 'a" .
*)
show "Dmonoid.unit (o) (id :: 'a => 'a) f = bij f"
apply (unfold Dmonoid.unit_def [OF Dmonoid])
apply rule apply clarify
proof -
fix f g
assume id1: "f o g = id" and id2: "g o f = id"
show "bij f"
proof (rule bijI)
show "inj f"
proof (rule inj_onI)
fix x y
assume "f x = f y"
then have "(g o f) x = (g o f) y" by simp
with id2 show "x = y" by simp
qed
next
show "surj f"
proof (rule surjI)
fix x
from id1 have "(f o g) x = x" by simp
then show "f (g x) = x" by simp
qed
qed
next
fix f
assume bij: "bij f"
then
have inv: "f o Hilbert_Choice.inv f = id & Hilbert_Choice.inv f o f = id"
by (simp add: bij_def surj_iff inj_iff)
show "\<exists>g. f \<circ> g = id \<and> g \<circ> f = id" by rule (rule inv)
qed
qed
thm Dmonoid.unit_def Dfun.unit_def
thm Dmonoid.inv_inj_on_unit Dfun.inv_inj_on_unit
lemma unit_id:
"(f :: unit => unit) = id"
by rule simp
interpretation Dfun: Dgrp "(o)" "id :: unit => unit"
rewrites "Dmonoid.inv (o) id f = inv (f :: unit => unit)"
proof -
have "Dmonoid (o) (id :: 'a => 'a)" ..
note Dmonoid = this
show "Dgrp (o) (id :: unit => unit)"
apply unfold_locales
apply (unfold Dmonoid.unit_def [OF Dmonoid])
apply (insert unit_id)
apply simp
done
show "Dmonoid.inv (o) id f = inv (f :: unit \<Rightarrow> unit)"
apply (unfold Dmonoid.inv_def [OF Dmonoid])
apply (insert unit_id)
apply simp
apply (rule the_equality)
apply rule
apply rule
apply simp
done
qed
thm Dfun.unit_l_inv Dfun.l_inv
thm Dfun.inv_equality
thm Dfun.inv_equality
end