(* Author: Tobias Nipkow, TU München
A theory of types extended with a greatest and a least element.
Oriented towards numeric types, hence "\<infinity>" and "-\<infinity>".
*)
theory Extended
imports Simps_Case_Conv
begin
datatype 'a extended = Fin 'a | Pinf ("\") | Minf ("-\")
instantiation extended :: (order)order
begin
fun less_eq_extended :: "'a extended \ 'a extended \ bool" where
"Fin x \ Fin y = (x \ y)" |
"_ \ Pinf = True" |
"Minf \ _ = True" |
"(_::'a extended) \ _ = False"
case_of_simps less_eq_extended_case: less_eq_extended.simps
definition less_extended :: "'a extended \ 'a extended \ bool" where
"((x::'a extended) < y) = (x \ y \ \ y \ x)"
instance
by intro_classes (auto simp: less_extended_def less_eq_extended_case split: extended.splits)
end
instance extended :: (linorder)linorder
by intro_classes (auto simp: less_eq_extended_case split:extended.splits)
lemma Minf_le[simp]: "Minf \ y"
by(cases y) auto
lemma le_Pinf[simp]: "x \ Pinf"
by(cases x) auto
lemma le_Minf[simp]: "x \ Minf \ x = Minf"
by(cases x) auto
lemma Pinf_le[simp]: "Pinf \ x \ x = Pinf"
by(cases x) auto
lemma less_extended_simps[simp]:
"Fin x < Fin y = (x < y)"
"Fin x < Pinf = True"
"Fin x < Minf = False"
"Pinf < h = False"
"Minf < Fin x = True"
"Minf < Pinf = True"
"l < Minf = False"
by (auto simp add: less_extended_def)
lemma min_extended_simps[simp]:
"min (Fin x) (Fin y) = Fin(min x y)"
"min xx Pinf = xx"
"min xx Minf = Minf"
"min Pinf yy = yy"
"min Minf yy = Minf"
by (auto simp add: min_def)
lemma max_extended_simps[simp]:
"max (Fin x) (Fin y) = Fin(max x y)"
"max xx Pinf = Pinf"
"max xx Minf = xx"
"max Pinf yy = Pinf"
"max Minf yy = yy"
by (auto simp add: max_def)
instantiation extended :: (zero)zero
begin
definition "0 = Fin(0::'a)"
instance ..
end
declare zero_extended_def[symmetric, code_post]
instantiation extended :: (one)one
begin
definition "1 = Fin(1::'a)"
instance ..
end
declare one_extended_def[symmetric, code_post]
instantiation extended :: (plus)plus
begin
text \<open>The following definition of of addition is totalized
to make it asociative and commutative. Normally the sum of plus and minus infinity is undefined.\<close>
fun plus_extended where
"Fin x + Fin y = Fin(x+y)" |
"Fin x + Pinf = Pinf" |
"Pinf + Fin x = Pinf" |
"Pinf + Pinf = Pinf" |
"Minf + Fin y = Minf" |
"Fin x + Minf = Minf" |
"Minf + Minf = Minf" |
"Minf + Pinf = Pinf" |
"Pinf + Minf = Pinf"
case_of_simps plus_case: plus_extended.simps
instance ..
end
instance extended :: (ab_semigroup_add)ab_semigroup_add
by intro_classes (simp_all add: ac_simps plus_case split: extended.splits)
instance extended :: (ordered_ab_semigroup_add)ordered_ab_semigroup_add
by intro_classes (auto simp: add_left_mono plus_case split: extended.splits)
instance extended :: (comm_monoid_add)comm_monoid_add
proof
fix x :: "'a extended" show "0 + x = x" unfolding zero_extended_def by(cases x)auto
qed
instantiation extended :: (uminus)uminus
begin
fun uminus_extended where
"- (Fin x) = Fin (- x)" |
"- Pinf = Minf" |
"- Minf = Pinf"
instance ..
end
instantiation extended :: (ab_group_add)minus
begin
definition "x - y = x + -(y::'a extended)"
instance ..
end
lemma minus_extended_simps[simp]:
"Fin x - Fin y = Fin(x - y)"
"Fin x - Pinf = Minf"
"Fin x - Minf = Pinf"
"Pinf - Fin y = Pinf"
"Pinf - Minf = Pinf"
"Minf - Fin y = Minf"
"Minf - Pinf = Minf"
"Minf - Minf = Pinf"
"Pinf - Pinf = Pinf"
by (simp_all add: minus_extended_def)
text\<open>Numerals:\<close>
instance extended :: ("{ab_semigroup_add,one}")numeral ..
lemma Fin_numeral[code_post]: "Fin(numeral w) = numeral w"
apply (induct w rule: num_induct)
apply (simp only: numeral_One one_extended_def)
apply (simp only: numeral_inc one_extended_def plus_extended.simps(1)[symmetric])
done
lemma Fin_neg_numeral[code_post]: "Fin (- numeral w) = - numeral w"
by (simp only: Fin_numeral uminus_extended.simps[symmetric])
instantiation extended :: (lattice)bounded_lattice
begin
definition "bot = Minf"
definition "top = Pinf"
fun inf_extended :: "'a extended \ 'a extended \ 'a extended" where
"inf_extended (Fin i) (Fin j) = Fin (inf i j)" |
"inf_extended a Minf = Minf" |
"inf_extended Minf a = Minf" |
"inf_extended Pinf a = a" |
"inf_extended a Pinf = a"
fun sup_extended :: "'a extended \ 'a extended \ 'a extended" where
"sup_extended (Fin i) (Fin j) = Fin (sup i j)" |
"sup_extended a Pinf = Pinf" |
"sup_extended Pinf a = Pinf" |
"sup_extended Minf a = a" |
"sup_extended a Minf = a"
case_of_simps inf_extended_case: inf_extended.simps
case_of_simps sup_extended_case: sup_extended.simps
instance
by (intro_classes) (auto simp: inf_extended_case sup_extended_case less_eq_extended_case
bot_extended_def top_extended_def split: extended.splits)
end
end
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