(* Title: HOL/Library/Extended_Nat.thy
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen
Contributions: David Trachtenherz, TU Muenchen
*)
header {* Extended natural numbers (i.e. with infinity) *}
theory Extended_Nat
imports Main
begin
class infinity =
fixes infinity :: "'a"
notation (xsymbols)
infinity ("\<infinity>")
notation (HTML output)
infinity ("\<infinity>")
subsection {* Type definition *}
text {*
We extend the standard natural numbers by a special value indicating
infinity.
*}
typedef (open) enat = "UNIV :: nat option set" ..
definition enat :: "nat \<Rightarrow> enat" where
"enat n = Abs_enat (Some n)"
instantiation enat :: infinity
begin
definition "\<infinity> = Abs_enat None"
instance proof qed
end
rep_datatype enat "\<infinity> :: enat"
proof -
fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
then show "P i"
proof induct
case (Abs_enat y) then show ?case
by (cases y rule: option.exhaust)
(auto simp: enat_def infinity_enat_def)
qed
qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
declare [[coercion "enat::nat\<Rightarrow>enat"]]
lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (EX i. x = enat i)"
by (cases x) auto
lemma not_enat_eq [iff]: "(ALL y. x ~= enat y) = (x = \<infinity>)"
by (cases x) auto
primrec the_enat :: "enat \<Rightarrow> nat"
where "the_enat (enat n) = n"
subsection {* Constructors and numbers *}
instantiation enat :: "{zero, one, number}"
begin
definition
"0 = enat 0"
definition
[code_unfold]: "1 = enat 1"
definition
[code_unfold, code del]: "number_of k = enat (number_of k)"
instance ..
end
definition eSuc :: "enat \<Rightarrow> enat" where
"eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
lemma enat_0: "enat 0 = 0"
by (simp add: zero_enat_def)
lemma enat_1: "enat 1 = 1"
by (simp add: one_enat_def)
lemma enat_number: "enat (number_of k) = number_of k"
by (simp add: number_of_enat_def)
lemma one_eSuc: "1 = eSuc 0"
by (simp add: zero_enat_def one_enat_def eSuc_def)
lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
by (simp add: zero_enat_def)
lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
by (simp add: zero_enat_def)
lemma zero_enat_eq [simp]:
"number_of k = (0\<Colon>enat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
"(0\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
unfolding zero_enat_def number_of_enat_def by simp_all
lemma one_enat_eq [simp]:
"number_of k = (1\<Colon>enat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
"(1\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
unfolding one_enat_def number_of_enat_def by simp_all
lemma zero_one_enat_neq [simp]:
"\<not> 0 = (1\<Colon>enat)"
"\<not> 1 = (0\<Colon>enat)"
unfolding zero_enat_def one_enat_def by simp_all
lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
by (simp add: one_enat_def)
lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
by (simp add: one_enat_def)
lemma infinity_ne_number [simp]: "(\<infinity>::enat) \<noteq> number_of k"
by (simp add: number_of_enat_def)
lemma number_ne_infinity [simp]: "number_of k \<noteq> (\<infinity>::enat)"
by (simp add: number_of_enat_def)
lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
by (simp add: eSuc_def)
lemma eSuc_number_of: "eSuc (number_of k) = enat (Suc (number_of k))"
by (simp add: eSuc_enat number_of_enat_def)
lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
by (simp add: eSuc_def)
lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
by (simp add: eSuc_def zero_enat_def split: enat.splits)
lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
by (rule eSuc_ne_0 [symmetric])
lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
by (simp add: eSuc_def split: enat.splits)
lemma number_of_enat_inject [simp]:
"(number_of k \<Colon> enat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
by (simp add: number_of_enat_def)
subsection {* Addition *}
instantiation enat :: comm_monoid_add
begin
definition [nitpick_simp]:
"m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
lemma plus_enat_simps [simp, code]:
fixes q :: enat
shows "enat m + enat n = enat (m + n)"
and "\<infinity> + q = \<infinity>"
and "q + \<infinity> = \<infinity>"
by (simp_all add: plus_enat_def split: enat.splits)
instance proof
fix n m q :: enat
show "n + m + q = n + (m + q)"
by (cases n, auto, cases m, auto, cases q, auto)
show "n + m = m + n"
by (cases n, auto, cases m, auto)
show "0 + n = n"
by (cases n) (simp_all add: zero_enat_def)
qed
end
lemma plus_enat_0 [simp]:
"0 + (q\<Colon>enat) = q"
"(q\<Colon>enat) + 0 = q"
by (simp_all add: plus_enat_def zero_enat_def split: enat.splits)
lemma plus_enat_number [simp]:
"(number_of k \<Colon> enat) + number_of l = (if k < Int.Pls then number_of l
else if l < Int.Pls then number_of k else number_of (k + l))"
unfolding number_of_enat_def plus_enat_simps nat_arith(1) if_distrib [symmetric, of _ enat] ..
lemma eSuc_number [simp]:
"eSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
unfolding eSuc_number_of
unfolding one_enat_def number_of_enat_def Suc_nat_number_of if_distrib [symmetric] ..
lemma eSuc_plus_1:
"eSuc n = n + 1"
by (cases n) (simp_all add: eSuc_enat one_enat_def)
lemma plus_1_eSuc:
"1 + q = eSuc q"
"q + 1 = eSuc q"
by (simp_all add: eSuc_plus_1 add_ac)
lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
by (simp_all add: eSuc_plus_1 add_ac)
lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
by (simp only: add_commute[of m] iadd_Suc)
lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
by (cases m, cases n, simp_all add: zero_enat_def)
subsection {* Multiplication *}
instantiation enat :: comm_semiring_1
begin
definition times_enat_def [nitpick_simp]:
"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
lemma times_enat_simps [simp, code]:
"enat m * enat n = enat (m * n)"
"\<infinity> * \<infinity> = (\<infinity>::enat)"
"\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
"enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
unfolding times_enat_def zero_enat_def
by (simp_all split: enat.split)
instance proof
fix a b c :: enat
show "(a * b) * c = a * (b * c)"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split)
show "a * b = b * a"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split)
show "1 * a = a"
unfolding times_enat_def zero_enat_def one_enat_def
by (simp split: enat.split)
show "(a + b) * c = a * c + b * c"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split add: left_distrib)
show "0 * a = 0"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split)
show "a * 0 = 0"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split)
show "(0::enat) \<noteq> 1"
unfolding zero_enat_def one_enat_def
by simp
qed
end
lemma mult_eSuc: "eSuc m * n = n + m * n"
unfolding eSuc_plus_1 by (simp add: algebra_simps)
lemma mult_eSuc_right: "m * eSuc n = m + m * n"
unfolding eSuc_plus_1 by (simp add: algebra_simps)
lemma of_nat_eq_enat: "of_nat n = enat n"
apply (induct n)
apply (simp add: enat_0)
apply (simp add: plus_1_eSuc eSuc_enat)
done
instance enat :: number_semiring
proof
fix n show "number_of (int n) = (of_nat n :: enat)"
unfolding number_of_enat_def number_of_int of_nat_id of_nat_eq_enat ..
qed
instance enat :: semiring_char_0 proof
have "inj enat" by (rule injI) simp
then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
qed
lemma imult_is_0 [simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
by (auto simp add: times_enat_def zero_enat_def split: enat.split)
lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
by (auto simp add: times_enat_def zero_enat_def split: enat.split)
subsection {* Subtraction *}
instantiation enat :: minus
begin
definition diff_enat_def:
"a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
| \<infinity> \<Rightarrow> \<infinity>)"
instance ..
end
lemma idiff_enat_enat [simp,code]: "enat a - enat b = enat (a - b)"
by (simp add: diff_enat_def)
lemma idiff_infinity [simp,code]: "\<infinity> - n = (\<infinity>::enat)"
by (simp add: diff_enat_def)
lemma idiff_infinity_right [simp,code]: "enat a - \<infinity> = 0"
by (simp add: diff_enat_def)
lemma idiff_0 [simp]: "(0::enat) - n = 0"
by (cases n, simp_all add: zero_enat_def)
lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
by (cases n) (simp_all add: zero_enat_def)
lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
by (auto simp: zero_enat_def)
lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
by (simp add: eSuc_def split: enat.split)
lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
(*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
subsection {* Ordering *}
instantiation enat :: linordered_ab_semigroup_add
begin
definition [nitpick_simp]:
"m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
| \<infinity> \<Rightarrow> True)"
definition [nitpick_simp]:
"m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
| \<infinity> \<Rightarrow> False)"
lemma enat_ord_simps [simp]:
"enat m \<le> enat n \<longleftrightarrow> m \<le> n"
"enat m < enat n \<longleftrightarrow> m < n"
"q \<le> (\<infinity>::enat)"
"q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
"(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
"(\<infinity>::enat) < q \<longleftrightarrow> False"
by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
lemma enat_ord_code [code]:
"enat m \<le> enat n \<longleftrightarrow> m \<le> n"
"enat m < enat n \<longleftrightarrow> m < n"
"q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
"enat m < \<infinity> \<longleftrightarrow> True"
"\<infinity> \<le> enat n \<longleftrightarrow> False"
"(\<infinity>::enat) < q \<longleftrightarrow> False"
by simp_all
instance by default
(auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
end
instance enat :: ordered_comm_semiring
proof
fix a b c :: enat
assume "a \<le> b" and "0 \<le> c"
thus "c * a \<le> c * b"
unfolding times_enat_def less_eq_enat_def zero_enat_def
by (simp split: enat.splits)
qed
lemma enat_ord_number [simp]:
"(number_of m \<Colon> enat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
"(number_of m \<Colon> enat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
by (simp_all add: number_of_enat_def)
lemma i0_lb [simp]: "(0\<Colon>enat) \<le> n"
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
lemma ile0_eq [simp]: "n \<le> (0\<Colon>enat) \<longleftrightarrow> n = 0"
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
by simp
lemma not_iless0 [simp]: "\<not> n < (0\<Colon>enat)"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma i0_less [simp]: "(0\<Colon>enat) < n \<longleftrightarrow> n \<noteq> 0"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
by (simp add: eSuc_def less_enat_def split: enat.splits)
lemma ile_eSuc [simp]: "n \<le> eSuc n"
by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
lemma i0_iless_eSuc [simp]: "0 < eSuc n"
by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
by (cases n) auto
lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
by (auto simp add: eSuc_def less_enat_def split: enat.splits)
lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
by (simp only: i0_less imult_is_0, simp)
lemma mono_eSuc: "mono eSuc"
by (simp add: mono_def)
lemma min_enat_simps [simp]:
"min (enat m) (enat n) = enat (min m n)"
"min q 0 = 0"
"min 0 q = 0"
"min q (\<infinity>::enat) = q"
"min (\<infinity>::enat) q = q"
by (auto simp add: min_def)
lemma max_enat_simps [simp]:
"max (enat m) (enat n) = enat (max m n)"
"max q 0 = q"
"max 0 q = q"
"max q \<infinity> = (\<infinity>::enat)"
"max \<infinity> q = (\<infinity>::enat)"
by (simp_all add: max_def)
lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
by (cases n) simp_all
lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
by (cases n) simp_all
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
apply (induct_tac k)
apply (simp (no_asm) only: enat_0)
apply (fast intro: le_less_trans [OF i0_lb])
apply (erule exE)
apply (drule spec)
apply (erule exE)
apply (drule ileI1)
apply (rule eSuc_enat [THEN subst])
apply (rule exI)
apply (erule (1) le_less_trans)
done
instantiation enat :: "{bot, top}"
begin
definition bot_enat :: enat where
"bot_enat = 0"
definition top_enat :: enat where
"top_enat = \<infinity>"
instance proof
qed (simp_all add: bot_enat_def top_enat_def)
end
lemma finite_enat_bounded:
assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
shows "finite A"
proof (rule finite_subset)
show "finite (enat ` {..n})" by blast
have "A \<subseteq> {..enat n}" using le_fin by fastforce
also have "\<dots> \<subseteq> enat ` {..n}"
by (rule subsetI) (case_tac x, auto)
finally show "A \<subseteq> enat ` {..n}" .
qed
subsection {* Well-ordering *}
lemma less_enatE:
"[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
by (induct n) auto
lemma less_infinityE:
"[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
by (induct n) auto
lemma enat_less_induct:
assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
proof -
have P_enat: "!!k. P (enat k)"
apply (rule nat_less_induct)
apply (rule prem, clarify)
apply (erule less_enatE, simp)
done
show ?thesis
proof (induct n)
fix nat
show "P (enat nat)" by (rule P_enat)
next
show "P \<infinity>"
apply (rule prem, clarify)
apply (erule less_infinityE)
apply (simp add: P_enat)
done
qed
qed
instance enat :: wellorder
proof
fix P and n
assume hyp: "(\<And>n\<Colon>enat. (\<And>m\<Colon>enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
show "P n" by (blast intro: enat_less_induct hyp)
qed
subsection {* Complete Lattice *}
instantiation enat :: complete_lattice
begin
definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
"inf_enat \<equiv> min"
definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
"sup_enat \<equiv> max"
definition Inf_enat :: "enat set \<Rightarrow> enat" where
"Inf_enat A \<equiv> if A = {} then \<infinity> else (LEAST x. x \<in> A)"
definition Sup_enat :: "enat set \<Rightarrow> enat" where
"Sup_enat A \<equiv> if A = {} then 0
else if finite A then Max A
else \<infinity>"
instance proof
fix x :: "enat" and A :: "enat set"
{ assume "x \<in> A" then show "Inf A \<le> x"
unfolding Inf_enat_def by (auto intro: Least_le) }
{ assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
unfolding Inf_enat_def
by (cases "A = {}") (auto intro: LeastI2_ex) }
{ assume "x \<in> A" then show "x \<le> Sup A"
unfolding Sup_enat_def by (cases "finite A") auto }
{ assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
unfolding Sup_enat_def using finite_enat_bounded by auto }
qed (simp_all add: inf_enat_def sup_enat_def)
end
instance enat :: complete_linorder ..
subsection {* Traditional theorem names *}
lemmas enat_defs = zero_enat_def one_enat_def number_of_enat_def eSuc_def
plus_enat_def less_eq_enat_def less_enat_def
end