Quelle Lattices.thy Sprache: Isabelle

(* Title: HOL/Lattices.thy
 Author: Tobias Nipkow
*)

section \<open>Abstract lattices\<close>

theory Lattices
imports Groups
begin

subsection \<open>Abstract semilattice\<close>

text \<open>
 These locales provide a basic structure for interpretation into
 bigger structures; extensions require careful thinking, otherwise
 undesired effects may occur due to interpretation.
\<close>

locale semilattice = abel_semigroup +
 assumes idem [simp]: "a \<^bold>* a = a"
begin

lemma left_idem [simp]: "a \<^bold>* (a \<^bold>* b) = a \<^bold>* b"
 by (simp add: assoc [symmetric])

lemma right_idem [simp]: "(a \<^bold>* b) \<^bold>* b = a \<^bold>* b"
 by (simp add: assoc)

end

locale semilattice_neutr = semilattice + comm_monoid

locale semilattice_order = semilattice +
 fixes less_eq :: "'a \ 'a \ bool" (infix \\<^bold>\\ 50)
 and less :: "'a \ 'a \ bool" (infix \\<^bold><\ 50)
 assumes order_iff: "a \<^bold>\ b \ a = a \<^bold>* b"
 and strict_order_iff: "a \<^bold>* b \ a \ b"
begin

lemma orderI: "a = a \<^bold>* b \ a \<^bold>\ b"
 by (simp add: order_iff)

lemma orderE:
 assumes "a \<^bold>\ b"
 obtains "a = a \<^bold>* b"
 using assms by (unfold order_iff)

sublocale ordering less_eq less
proof
 show "a \<^bold>\ b \ a \ b" for a b
 by (simp add: order_iff strict_order_iff)
next
 show "a \<^bold>\ a" for a
 by (simp add: order_iff)
next
 fix a b
 assume "a \<^bold>\ b" "b \<^bold>\ a"
 then have "a = a \<^bold>* b" "a \<^bold>* b = b"
 by (simp_all add: order_iff commute)
 then show "a = b" by simp
next
 fix a b c
 assume "a \<^bold>\ b" "b \<^bold>\ c"
 then have "a = a \<^bold>* b" "b = b \<^bold>* c"
 by (simp_all add: order_iff commute)
 then have "a = a \<^bold>* (b \<^bold>* c)"
 by simp
 then have "a = (a \<^bold>* b) \<^bold>* c"
 by (simp add: assoc)
 with \<open>a = a \<^bold>* b\<close> [symmetric] have "a = a \<^bold>* c" by simp
 then show "a \<^bold>\ c" by (rule orderI)
qed

lemma cobounded1 [simp]: "a \<^bold>* b \<^bold>\ a"
 by (simp add: order_iff commute)

lemma cobounded2 [simp]: "a \<^bold>* b \<^bold>\ b"
 by (simp add: order_iff)

lemma boundedI:
 assumes "a \<^bold>\ b" and "a \<^bold>\ c"
 shows "a \<^bold>\ b \<^bold>* c"
proof (rule orderI)
 from assms obtain "a \<^bold>* b = a" and "a \<^bold>* c = a"
 by (auto elim!: orderE)
 then show "a = a \<^bold>* (b \<^bold>* c)"
 by (simp add: assoc [symmetric])
qed

lemma boundedE:
 assumes "a \<^bold>\ b \<^bold>* c"
 obtains "a \<^bold>\ b" and "a \<^bold>\ c"
 using assms by (blast intro: trans cobounded1 cobounded2)

lemma bounded_iff [simp]: "a \<^bold>\ b \<^bold>* c \ a \<^bold>\ b \ a \<^bold>\ c"
 by (blast intro: boundedI elim: boundedE)

lemma strict_boundedE:
 assumes "a \<^bold>* c"
 obtains "a \<^bold>< b" and "a \<^bold>< c"
 using assms by (auto simp add: commute strict_iff_order elim: orderE intro!: that)+

lemma coboundedI1: "a \<^bold>\ c \ a \<^bold>* b \<^bold>\ c"
 by (rule trans) auto

lemma coboundedI2: "b \<^bold>\ c \ a \<^bold>* b \<^bold>\ c"
 by (rule trans) auto

lemma strict_coboundedI1: "a \<^bold>< c \ a \<^bold>* b \<^bold>< c"
 using irrefl
 by (auto intro: not_eq_order_implies_strict coboundedI1 strict_implies_order
 elim: strict_boundedE)

lemma strict_coboundedI2: "b \<^bold>< c \ a \<^bold>* b \<^bold>< c"
 using strict_coboundedI1 [of b c a] by (simp add: commute)

lemma mono: "a \<^bold>\ c \ b \<^bold>\ d \ a \<^bold>* b \<^bold>\ c \<^bold>* d"
 by (blast intro: boundedI coboundedI1 coboundedI2)

lemma absorb1: "a \<^bold>\ b \ a \<^bold>* b = a"
 by (rule antisym) (auto simp: refl)

lemma absorb2: "b \<^bold>\ a \ a \<^bold>* b = b"
 by (rule antisym) (auto simp: refl)

lemma absorb3: "a \<^bold>* b = a"
 by (rule absorb1) (rule strict_implies_order)

lemma absorb4: "b \<^bold>< a \ a \<^bold>* b = b"
 by (rule absorb2) (rule strict_implies_order)

lemma absorb_iff1: "a \<^bold>\ b \ a \<^bold>* b = a"
 using order_iff by auto

lemma absorb_iff2: "b \<^bold>\ a \ a \<^bold>* b = b"
 using order_iff by (auto simp add: commute)

end

locale semilattice_neutr_order = semilattice_neutr + semilattice_order
begin

sublocale ordering_top less_eq less "\<^bold>1"
 by standard (simp add: order_iff)

lemma eq_neutr_iff [simp]: \<open>a \<^bold>* b = \<^bold>1 \<longleftrightarrow> a = \<^bold>1 \<and> b = \<^bold>1\<close>
 by (simp add: eq_iff)

lemma neutr_eq_iff [simp]: \<open>\<^bold>1 = a \<^bold>* b \<longleftrightarrow> a = \<^bold>1 \<and> b = \<^bold>1\<close>
 by (simp add: eq_iff)

end

text \<open>Interpretations for boolean operators\<close>

interpretation conj: semilattice_neutr \<open>(\<and>)\<close> True
 by standard auto

interpretation disj: semilattice_neutr \<open>(\<or>)\<close> False
 by standard auto

declare conj_assoc [ac_simps del] disj_assoc [ac_simps del] \<comment> \<open>already simp by default\<close>

subsection \<open>Syntactic infimum and supremum operations\<close>

class inf =
 fixes inf :: "'a \ 'a \ 'a" (infixl \\\ 70)

class sup =
 fixes sup :: "'a \ 'a \ 'a" (infixl \\\ 65)

subsection \<open>Concrete lattices\<close>

class semilattice_inf = order + inf +
 assumes inf_le1 [simp]: "x \ y \ x"
 and inf_le2 [simp]: "x \ y \ y"
 and inf_greatest: "x \ y \ x \ z \ x \ y \ z"

class semilattice_sup = order + sup +
 assumes sup_ge1 [simp]: "x \ x \ y"
 and sup_ge2 [simp]: "y \ x \ y"
 and sup_least: "y \ x \ z \ x \ y \ z \ x"
begin

text \<open>Dual lattice.\<close>
lemma dual_semilattice: "class.semilattice_inf sup greater_eq greater"
 by (rule class.semilattice_inf.intro, rule dual_order)
 (unfold_locales, simp_all add: sup_least)

end

class lattice = semilattice_inf + semilattice_sup

subsubsection \<open>Intro and elim rules\<close>

context semilattice_inf
begin

lemma le_infI1: "a \ x \ a \ b \ x"
 by (rule order_trans) auto

lemma le_infI2: "b \ x \ a \ b \ x"
 by (rule order_trans) auto

lemma le_infI: "x \ a \ x \ b \ x \ a \ b"
 by (fact inf_greatest) (* FIXME: duplicate lemma *)

lemma le_infE: "x \ a \ b \ (x \ a \ x \ b \ P) \ P"
 by (blast intro: order_trans inf_le1 inf_le2)

lemma le_inf_iff: "x \ y \ z \ x \ y \ x \ z"
 by (blast intro: le_infI elim: le_infE) (* [simp] via bounded_iff *)

lemma le_iff_inf: "x \ y \ x \ y = x"
 by (auto intro: le_infI1 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_inf_iff)

lemma inf_mono: "a \ c \ b \ d \ a \ b \ c \ d"
 by (fast intro: inf_greatest le_infI1 le_infI2)

end

context semilattice_sup
begin

lemma le_supI1: "x \ a \ x \ a \ b"
 by (rule order_trans) auto

lemma le_supI2: "x \ b \ x \ a \ b"
 by (rule order_trans) auto

lemma le_supI: "a \ x \ b \ x \ a \ b \ x"
 by (fact sup_least) (* FIXME: duplicate lemma *)

lemma le_supE: "a \ b \ x \ (a \ x \ b \ x \ P) \ P"
 by (blast intro: order_trans sup_ge1 sup_ge2)

lemma le_sup_iff: "x \ y \ z \ x \ z \ y \ z"
 by (blast intro: le_supI elim: le_supE) (* [simp] via bounded_iff *)

lemma le_iff_sup: "x \ y \ x \ y = y"
 by (auto intro: le_supI2 order.antisym dest: order.eq_iff [THEN iffD1] simp add: le_sup_iff)

lemma sup_mono: "a \ c \ b \ d \ a \ b \ c \ d"
 by (fast intro: sup_least le_supI1 le_supI2)

end

subsubsection \<open>Equational laws\<close>

context semilattice_inf
begin

sublocale inf: semilattice inf
proof
 fix a b c
 show "(a \ b) \ c = a \ (b \ c)"
 by (rule order.antisym) (auto intro: le_infI1 le_infI2 simp add: le_inf_iff)
 show "a \ b = b \ a"
 by (rule order.antisym) (auto simp add: le_inf_iff)
 show "a \ a = a"
 by (rule order.antisym) (auto simp add: le_inf_iff)
qed

sublocale inf: semilattice_order inf less_eq less
 by standard (auto simp add: le_iff_inf less_le)

lemma inf_assoc: "(x \ y) \ z = x \ (y \ z)"
 by (fact inf.assoc)

lemma inf_commute: "(x \ y) = (y \ x)"
 by (fact inf.commute)

lemma inf_left_commute: "x \ (y \ z) = y \ (x \ z)"
 by (fact inf.left_commute)

lemma inf_idem: "x \ x = x"
 by (fact inf.idem) (* already simp *)

lemma inf_left_idem: "x \ (x \ y) = x \ y"
 by (fact inf.left_idem) (* already simp *)

lemma inf_right_idem: "(x \ y) \ y = x \ y"
 by (fact inf.right_idem) (* already simp *)

lemma inf_absorb1: "x \ y \ x \ y = x"
 by (rule order.antisym) auto

lemma inf_absorb2: "y \ x \ x \ y = y"
 by (rule order.antisym) auto

lemmas inf_aci = inf_commute inf_assoc inf_left_commute inf_left_idem

end

context semilattice_sup
begin

sublocale sup: semilattice sup
proof
 fix a b c
 show "(a \ b) \ c = a \ (b \ c)"
 by (rule order.antisym) (auto intro: le_supI1 le_supI2 simp add: le_sup_iff)
 show "a \ b = b \ a"
 by (rule order.antisym) (auto simp add: le_sup_iff)
 show "a \ a = a"
 by (rule order.antisym) (auto simp add: le_sup_iff)
qed

sublocale sup: semilattice_order sup greater_eq greater
 by standard (auto simp add: le_iff_sup sup.commute less_le)

lemma sup_assoc: "(x \ y) \ z = x \ (y \ z)"
 by (fact sup.assoc)

lemma sup_commute: "(x \ y) = (y \ x)"
 by (fact sup.commute)

lemma sup_left_commute: "x \ (y \ z) = y \ (x \ z)"
 by (fact sup.left_commute)

lemma sup_idem: "x \ x = x"
 by (fact sup.idem) (* already simp *)

lemma sup_left_idem [simp]: "x \ (x \ y) = x \ y"
 by (fact sup.left_idem)

lemma sup_absorb1: "y \ x \ x \ y = x"
 by (rule order.antisym) auto

lemma sup_absorb2: "x \ y \ x \ y = y"
 by (rule order.antisym) auto

lemmas sup_aci = sup_commute sup_assoc sup_left_commute sup_left_idem

end

context lattice
begin

lemma dual_lattice: "class.lattice sup (\) (>) inf"
 by (rule class.lattice.intro,
 rule dual_semilattice,
 rule class.semilattice_sup.intro,
 rule dual_order)
 (unfold_locales, auto)

lemma inf_sup_absorb [simp]: "x \ (x \ y) = x"
 by (blast intro: order.antisym inf_le1 inf_greatest sup_ge1)

lemma sup_inf_absorb [simp]: "x \ (x \ y) = x"
 by (blast intro: order.antisym sup_ge1 sup_least inf_le1)

lemmas inf_sup_aci = inf_aci sup_aci

lemmas inf_sup_ord = inf_le1 inf_le2 sup_ge1 sup_ge2

text \<open>Towards distributivity.\<close>

lemma distrib_sup_le: "x \ (y \ z) \ (x \ y) \ (x \ z)"
 by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

lemma distrib_inf_le: "(x \ y) \ (x \ z) \ x \ (y \ z)"
 by (auto intro: le_infI1 le_infI2 le_supI1 le_supI2)

text \<open>If you have one of them, you have them all.\<close>

lemma distrib_imp1:
 assumes distrib: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)"
 shows "x \ (y \ z) = (x \ y) \ (x \ z)"
proof-
 have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)"
 by simp
 also have "\ = x \ (z \ (x \ y))"
 by (simp add: distrib inf_commute sup_assoc del: sup_inf_absorb)
 also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)"
 by (simp add: inf_commute)
 also have "\ = (x \ y) \ (x \ z)" by(simp add:distrib)
 finally show ?thesis .
qed

lemma distrib_imp2:
 assumes distrib: "\x y z. x \ (y \ z) = (x \ y) \ (x \ z)"
 shows "x \ (y \ z) = (x \ y) \ (x \ z)"
proof-
 have "x \ (y \ z) = (x \ (x \ z)) \ (y \ z)"
 by simp
 also have "\ = x \ (z \ (x \ y))"
 by (simp add: distrib sup_commute inf_assoc del: inf_sup_absorb)
 also have "\ = ((x \ y) \ x) \ ((x \ y) \ z)"
 by (simp add: sup_commute)
 also have "\ = (x \ y) \ (x \ z)" by (simp add:distrib)
 finally show ?thesis .
qed

end

subsubsection \<open>Strict order\<close>

context semilattice_inf
begin

lemma less_infI1: "a < x \ a \ b < x"
 by (auto simp add: less_le inf_absorb1 intro: le_infI1)

lemma less_infI2: "b < x \ a \ b < x"
 by (auto simp add: less_le inf_absorb2 intro: le_infI2)

end

context semilattice_sup
begin

lemma less_supI1: "x < a \ x < a \ b"
 using dual_semilattice
 by (rule semilattice_inf.less_infI1)

lemma less_supI2: "x Distributive lattices\<close>

class distrib_lattice = lattice +
 assumes sup_inf_distrib1: "x \ (y \ z) = (x \ y) \ (x \ z)"

context distrib_lattice
begin

lemma sup_inf_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)"
 by (simp add: sup_commute sup_inf_distrib1)

lemma inf_sup_distrib1: "x \ (y \ z) = (x \ y) \ (x \ z)"
 by (rule distrib_imp2 [OF sup_inf_distrib1])

lemma inf_sup_distrib2: "(y \ z) \ x = (y \ x) \ (z \ x)"
 by (simp add: inf_commute inf_sup_distrib1)

lemma dual_distrib_lattice: "class.distrib_lattice sup (\) (>) inf"
 by (rule class.distrib_lattice.intro, rule dual_lattice)
 (unfold_locales, fact inf_sup_distrib1)

lemmas sup_inf_distrib = sup_inf_distrib1 sup_inf_distrib2

lemmas inf_sup_distrib = inf_sup_distrib1 inf_sup_distrib2

lemmas distrib = sup_inf_distrib1 sup_inf_distrib2 inf_sup_distrib1 inf_sup_distrib2

end

subsection \<open>Bounded lattices\<close>

class bounded_semilattice_inf_top = semilattice_inf + order_top
begin

sublocale inf_top: semilattice_neutr inf top
 + inf_top: semilattice_neutr_order inf top less_eq less
proof
 show "x \ \ = x" for x
 by (rule inf_absorb1) simp
qed

lemma inf_top_left: "\ \ x = x"
 by (fact inf_top.left_neutral)

lemma inf_top_right: "x \ \ = x"
 by (fact inf_top.right_neutral)

lemma inf_eq_top_iff: "x \ y = \ \ x = \ \ y = \"
 by (fact inf_top.eq_neutr_iff)

lemma top_eq_inf_iff: "\ = x \ y \ x = \ \ y = \"
 by (fact inf_top.neutr_eq_iff)

end

class bounded_semilattice_sup_bot = semilattice_sup + order_bot
begin

sublocale sup_bot: semilattice_neutr sup bot
 + sup_bot: semilattice_neutr_order sup bot greater_eq greater
proof
 show "x \ \ = x" for x
 by (rule sup_absorb1) simp
qed

lemma sup_bot_left: "\ \ x = x"
 by (fact sup_bot.left_neutral)

lemma sup_bot_right: "x \ \ = x"
 by (fact sup_bot.right_neutral)

lemma sup_eq_bot_iff: "x \ y = \ \ x = \ \ y = \"
 by (fact sup_bot.eq_neutr_iff)

lemma bot_eq_sup_iff: "\ = x \ y \ x = \ \ y = \"
 by (fact sup_bot.neutr_eq_iff)

end

class bounded_lattice_bot = lattice + order_bot
begin

subclass bounded_semilattice_sup_bot ..

lemma inf_bot_left [simp]: "\ \ x = \"
 by (rule inf_absorb1) simp

lemma inf_bot_right [simp]: "x \ \ = \"
 by (rule inf_absorb2) simp

end

class bounded_lattice_top = lattice + order_top
begin

subclass bounded_semilattice_inf_top ..

lemma sup_top_left [simp]: "\ \ x = \"
 by (rule sup_absorb1) simp

lemma sup_top_right [simp]: "x \ \ = \"
 by (rule sup_absorb2) simp

end

class bounded_lattice = lattice + order_bot + order_top
begin

subclass bounded_lattice_bot ..
subclass bounded_lattice_top ..

lemma dual_bounded_lattice: "class.bounded_lattice sup greater_eq greater inf \ \"
 by unfold_locales (auto simp add: less_le_not_le)

end

subsection \<open>\<open>min/max\<close> as special case of lattice\<close>

context linorder
begin

sublocale min: semilattice_order min less_eq less
 + max: semilattice_order max greater_eq greater
 by standard (auto simp add: min_def max_def)

declare min.absorb1 [simp] min.absorb2 [simp]
 min.absorb3 [simp] min.absorb4 [simp]
 max.absorb1 [simp] max.absorb2 [simp]
 max.absorb3 [simp] max.absorb4 [simp]

lemma min_le_iff_disj: "min x y \ z \ x \ z \ y \ z"
 unfolding min_def using linear by (auto intro: order_trans)

lemma le_max_iff_disj: "z \ max x y \ z \ x \ z \ y"
 unfolding max_def using linear by (auto intro: order_trans)

lemma min_less_iff_disj: "min x y < z \ x < z \ y < z"
 unfolding min_def le_less using less_linear by (auto intro: less_trans)

lemma less_max_iff_disj: "z < max x y \ z < x \ z < y"
 unfolding max_def le_less using less_linear by (auto intro: less_trans)

lemma min_less_iff_conj [simp]: "z < min x y \ z < x \ z < y"
 unfolding min_def le_less using less_linear by (auto intro: less_trans)

lemma max_less_iff_conj [simp]: "max x y < z \ x < z \ y < z"
 unfolding max_def le_less using less_linear by (auto intro: less_trans)

lemma min_max_distrib1: "min (max b c) a = max (min b a) (min c a)"
 by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma min_max_distrib2: "min a (max b c) = max (min a b) (min a c)"
 by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma max_min_distrib1: "max (min b c) a = min (max b a) (max c a)"
 by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemma max_min_distrib2: "max a (min b c) = min (max a b) (max a c)"
 by (auto simp add: min_def max_def not_le dest: le_less_trans less_trans intro: antisym)

lemmas min_max_distribs = min_max_distrib1 min_max_distrib2 max_min_distrib1 max_min_distrib2

lemma split_min [no_atp]: "P (min i j) \ (i \ j \ P i) \ (\ i \ j \ P j)"
 by (simp add: min_def)

lemma split_max [no_atp]: "P (max i j) \ (i \ j \ P j) \ (\ i \ j \ P i)"
 by (simp add: max_def)

lemma split_min_lin [no_atp]:
 \<open>P (min a b) \<longleftrightarrow> (b = a \<longrightarrow> P a) \<and> (a P a) \<and> (b < a \<longrightarrow> P b)\<close>
 by (cases a b rule: linorder_cases) auto

lemma split_max_lin [no_atp]:
 \<open>P (max a b) \<longleftrightarrow> (b = a \<longrightarrow> P a) \<and> (a P b) \<and> (b < a \<longrightarrow> P a)\<close>
 by (cases a b rule: linorder_cases) auto

end

lemma inf_min: "inf = (min :: 'a::{semilattice_inf,linorder} \ 'a \ 'a)"
 by (auto intro: antisym simp add: min_def fun_eq_iff)

lemma sup_max: "sup = (max :: 'a::{semilattice_sup,linorder} \ 'a \ 'a)"
 by (auto intro: antisym simp add: max_def fun_eq_iff)

subsection \<open>Uniqueness of inf and sup\<close>

lemma (in semilattice_inf) inf_unique:
 fixes f (infixl \<open>\<triangle>\<close> 70)
 assumes le1: "\x y. x \ y \ x"
 and le2: "\x y. x \ y \ y"
 and greatest: "\x y z. x \ y \ x \ z \ x \ y \ z"
 shows "x \ y = x \ y"
proof (rule order.antisym)
 show "x \ y \ x \ y"
 by (rule le_infI) (rule le1, rule le2)
 have leI: "\x y z. x \ y \ x \ z \ x \ y \ z"
 by (blast intro: greatest)
 show "x \ y \ x \ y"
 by (rule leI) simp_all
qed

lemma (in semilattice_sup) sup_unique:
 fixes f (infixl \<open>\<nabla>\<close> 70)
 assumes ge1 [simp]: "\x y. x \ x \ y"
 and ge2: "\x y. y \ x \ y"
 and least: "\x y z. y \ x \ z \ x \ y \ z \ x"
 shows "x \ y = x \ y"
proof (rule order.antisym)
 show "x \ y \ x \ y"
 by (rule le_supI) (rule ge1, rule ge2)
 have leI: "\x y z. x \ z \ y \ z \ x \ y \ z"
 by (blast intro: least)
 show "x \ y \ x \ y"
 by (rule leI) simp_all
qed

subsection \<open>Lattice on \<^typ>\<open>_ \<Rightarrow> _\<close>\<close>

instantiation "fun" :: (type, semilattice_sup) semilattice_sup
begin

definition "f \ g = (\x. f x \ g x)"

lemma sup_apply [simp, code]: "(f \ g) x = f x \ g x"
 by (simp add: sup_fun_def)

instance
 by standard (simp_all add: le_fun_def)

end

instantiation "fun" :: (type, semilattice_inf) semilattice_inf
begin

definition "f \ g = (\x. f x \ g x)"

lemma inf_apply [simp, code]: "(f \ g) x = f x \ g x"
 by (simp add: inf_fun_def)

instance by standard (simp_all add: le_fun_def)

end

instance "fun" :: (type, lattice) lattice ..

instance "fun" :: (type, distrib_lattice) distrib_lattice
 by standard (rule ext, simp add: sup_inf_distrib1)

instance "fun" :: (type, bounded_lattice) bounded_lattice ..

instantiation "fun" :: (type, uminus) uminus
begin

definition fun_Compl_def: "- A = (\x. - A x)"

lemma uminus_apply [simp, code]: "(- A) x = - (A x)"
 by (simp add: fun_Compl_def)

instance ..

end

instantiation "fun" :: (type, minus) minus
begin

definition fun_diff_def: "A - B = (\x. A x - B x)"

lemma minus_apply [simp, code]: "(A - B) x = A x - B x"
 by (simp add: fun_diff_def)

instance ..

end

end

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Bemerkung:

Die farbliche Syntaxdarstellung ist noch experimentell.