bot comes before top, inf before sup etc.
(* Title: HOL/Library/Nat_Infinity.thy
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen
*)
header {* Natural numbers with infinity *}
theory Nat_Infinity
imports Main
begin
subsection {* Type definition *}
text {*
We extend the standard natural numbers by a special value indicating
infinity.
*}
datatype inat = Fin nat | Infty
notation (xsymbols)
Infty ("\<infinity>")
notation (HTML output)
Infty ("\<infinity>")
lemma not_Infty_eq[iff]: "(x ~= Infty) = (EX i. x = Fin i)"
by (cases x) auto
lemma not_Fin_eq [iff]: "(ALL y. x ~= Fin y) = (x = Infty)"
by (cases x) auto
subsection {* Constructors and numbers *}
instantiation inat :: "{zero, one, number}"
begin
definition
"0 = Fin 0"
definition
[code_unfold]: "1 = Fin 1"
definition
[code_unfold, code del]: "number_of k = Fin (number_of k)"
instance ..
end
definition iSuc :: "inat \<Rightarrow> inat" where
"iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
lemma Fin_0: "Fin 0 = 0"
by (simp add: zero_inat_def)
lemma Fin_1: "Fin 1 = 1"
by (simp add: one_inat_def)
lemma Fin_number: "Fin (number_of k) = number_of k"
by (simp add: number_of_inat_def)
lemma one_iSuc: "1 = iSuc 0"
by (simp add: zero_inat_def one_inat_def iSuc_def)
lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
by (simp add: zero_inat_def)
lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
by (simp add: zero_inat_def)
lemma zero_inat_eq [simp]:
"number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
"(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
unfolding zero_inat_def number_of_inat_def by simp_all
lemma one_inat_eq [simp]:
"number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
"(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
unfolding one_inat_def number_of_inat_def by simp_all
lemma zero_one_inat_neq [simp]:
"\<not> 0 = (1\<Colon>inat)"
"\<not> 1 = (0\<Colon>inat)"
unfolding zero_inat_def one_inat_def by simp_all
lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
by (simp add: one_inat_def)
lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
by (simp add: one_inat_def)
lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
by (simp add: number_of_inat_def)
lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
by (simp add: number_of_inat_def)
lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
by (simp add: iSuc_def)
lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
by (simp add: iSuc_Fin number_of_inat_def)
lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
by (simp add: iSuc_def)
lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
by (simp add: iSuc_def zero_inat_def split: inat.splits)
lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
by (rule iSuc_ne_0 [symmetric])
lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
by (simp add: iSuc_def split: inat.splits)
lemma number_of_inat_inject [simp]:
"(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
by (simp add: number_of_inat_def)
subsection {* Addition *}
instantiation inat :: comm_monoid_add
begin
definition [nitpick_simp]:
"m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))"
lemma plus_inat_simps [simp, code]:
"Fin m + Fin n = Fin (m + n)"
"\<infinity> + q = \<infinity>"
"q + \<infinity> = \<infinity>"
by (simp_all add: plus_inat_def split: inat.splits)
instance proof
fix n m q :: inat
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_inat_def)
qed
end
lemma plus_inat_0 [simp]:
"0 + (q\<Colon>inat) = q"
"(q\<Colon>inat) + 0 = q"
by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
lemma plus_inat_number [simp]:
"(number_of k \<Colon> inat) + 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_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
lemma iSuc_number [simp]:
"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
unfolding iSuc_number_of
unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
lemma iSuc_plus_1:
"iSuc n = n + 1"
by (cases n) (simp_all add: iSuc_Fin one_inat_def)
lemma plus_1_iSuc:
"1 + q = iSuc q"
"q + 1 = iSuc q"
unfolding iSuc_plus_1 by (simp_all add: add_ac)
subsection {* Multiplication *}
instantiation inat :: comm_semiring_1
begin
definition times_inat_def [nitpick_simp]:
"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow>
(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))"
lemma times_inat_simps [simp, code]:
"Fin m * Fin n = Fin (m * n)"
"\<infinity> * \<infinity> = \<infinity>"
"\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)"
"Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
unfolding times_inat_def zero_inat_def
by (simp_all split: inat.split)
instance proof
fix a b c :: inat
show "(a * b) * c = a * (b * c)"
unfolding times_inat_def zero_inat_def
by (simp split: inat.split)
show "a * b = b * a"
unfolding times_inat_def zero_inat_def
by (simp split: inat.split)
show "1 * a = a"
unfolding times_inat_def zero_inat_def one_inat_def
by (simp split: inat.split)
show "(a + b) * c = a * c + b * c"
unfolding times_inat_def zero_inat_def
by (simp split: inat.split add: left_distrib)
show "0 * a = 0"
unfolding times_inat_def zero_inat_def
by (simp split: inat.split)
show "a * 0 = 0"
unfolding times_inat_def zero_inat_def
by (simp split: inat.split)
show "(0::inat) \<noteq> 1"
unfolding zero_inat_def one_inat_def
by simp
qed
end
lemma mult_iSuc: "iSuc m * n = n + m * n"
unfolding iSuc_plus_1 by (simp add: algebra_simps)
lemma mult_iSuc_right: "m * iSuc n = m + m * n"
unfolding iSuc_plus_1 by (simp add: algebra_simps)
lemma of_nat_eq_Fin: "of_nat n = Fin n"
apply (induct n)
apply (simp add: Fin_0)
apply (simp add: plus_1_iSuc iSuc_Fin)
done
instance inat :: semiring_char_0 proof
have "inj Fin" by (rule injI) simp
then show "inj (\<lambda>n. of_nat n :: inat)" by (simp add: of_nat_eq_Fin)
qed
subsection {* Ordering *}
instantiation inat :: linordered_ab_semigroup_add
begin
definition [nitpick_simp]:
"m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
| \<infinity> \<Rightarrow> True)"
definition [nitpick_simp]:
"m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
| \<infinity> \<Rightarrow> False)"
lemma inat_ord_simps [simp]:
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
"Fin m < Fin n \<longleftrightarrow> m < n"
"q \<le> \<infinity>"
"q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
"\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
"\<infinity> < q \<longleftrightarrow> False"
by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
lemma inat_ord_code [code]:
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
"Fin m < Fin n \<longleftrightarrow> m < n"
"q \<le> \<infinity> \<longleftrightarrow> True"
"Fin m < \<infinity> \<longleftrightarrow> True"
"\<infinity> \<le> Fin n \<longleftrightarrow> False"
"\<infinity> < q \<longleftrightarrow> False"
by simp_all
instance by default
(auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
end
instance inat :: ordered_comm_semiring
proof
fix a b c :: inat
assume "a \<le> b" and "0 \<le> c"
thus "c * a \<le> c * b"
unfolding times_inat_def less_eq_inat_def zero_inat_def
by (simp split: inat.splits)
qed
lemma inat_ord_number [simp]:
"(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
"(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
by (simp_all add: number_of_inat_def)
lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
by simp
lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
by (simp add: zero_inat_def less_inat_def split: inat.splits)
lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
by (simp add: zero_inat_def less_inat_def split: inat.splits)
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
by (simp add: iSuc_def less_inat_def split: inat.splits)
lemma ile_iSuc [simp]: "n \<le> iSuc n"
by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
lemma i0_iless_iSuc [simp]: "0 < iSuc n"
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
by (cases n) auto
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
by (auto simp add: iSuc_def less_inat_def split: inat.splits)
lemma min_inat_simps [simp]:
"min (Fin m) (Fin n) = Fin (min m n)"
"min q 0 = 0"
"min 0 q = 0"
"min q \<infinity> = q"
"min \<infinity> q = q"
by (auto simp add: min_def)
lemma max_inat_simps [simp]:
"max (Fin m) (Fin n) = Fin (max m n)"
"max q 0 = q"
"max 0 q = q"
"max q \<infinity> = \<infinity>"
"max \<infinity> q = \<infinity>"
by (simp_all add: max_def)
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
by (cases n) simp_all
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
by (cases n) simp_all
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
apply (induct_tac k)
apply (simp (no_asm) only: Fin_0)
apply (fast intro: le_less_trans [OF i0_lb])
apply (erule exE)
apply (drule spec)
apply (erule exE)
apply (drule ileI1)
apply (rule iSuc_Fin [THEN subst])
apply (rule exI)
apply (erule (1) le_less_trans)
done
instantiation inat :: "{bot, top}"
begin
definition bot_inat :: inat where
"bot_inat = 0"
definition top_inat :: inat where
"top_inat = \<infinity>"
instance proof
qed (simp_all add: bot_inat_def top_inat_def)
end
subsection {* Well-ordering *}
lemma less_FinE:
"[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
by (induct n) auto
lemma less_InftyE:
"[| n < Infty; !!k. n = Fin k ==> P |] ==> P"
by (induct n) auto
lemma inat_less_induct:
assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n"
proof -
have P_Fin: "!!k. P (Fin k)"
apply (rule nat_less_induct)
apply (rule prem, clarify)
apply (erule less_FinE, simp)
done
show ?thesis
proof (induct n)
fix nat
show "P (Fin nat)" by (rule P_Fin)
next
show "P Infty"
apply (rule prem, clarify)
apply (erule less_InftyE)
apply (simp add: P_Fin)
done
qed
qed
instance inat :: wellorder
proof
fix P and n
assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
show "P n" by (blast intro: inat_less_induct hyp)
qed
subsection {* Traditional theorem names *}
lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
plus_inat_def less_eq_inat_def less_inat_def
lemmas inat_splits = inat.splits
end