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 ("∞")
notation (HTML output)
Infty ("∞")
subsection {* Constructors and numbers *}
instantiation inat :: "{zero, one, number}"
begin
definition
"0 = Fin 0"
definition
[code inline]: "1 = Fin 1"
definition
[code inline, code del]: "number_of k = Fin (number_of k)"
instance ..
end
definition iSuc :: "inat => inat" where
"iSuc i = (case i of Fin n => Fin (Suc n) | ∞ => ∞)"
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]: "∞ ≠ 0"
by (simp add: zero_inat_def)
lemma i0_ne_Infty [simp]: "0 ≠ ∞"
by (simp add: zero_inat_def)
lemma zero_inat_eq [simp]:
"number_of k = (0::inat) <-> number_of k = (0::nat)"
"(0::inat) = number_of k <-> number_of k = (0::nat)"
unfolding zero_inat_def number_of_inat_def by simp_all
lemma one_inat_eq [simp]:
"number_of k = (1::inat) <-> number_of k = (1::nat)"
"(1::inat) = number_of k <-> number_of k = (1::nat)"
unfolding one_inat_def number_of_inat_def by simp_all
lemma zero_one_inat_neq [simp]:
"¬ 0 = (1::inat)"
"¬ 1 = (0::inat)"
unfolding zero_inat_def one_inat_def by simp_all
lemma Infty_ne_i1 [simp]: "∞ ≠ 1"
by (simp add: one_inat_def)
lemma i1_ne_Infty [simp]: "1 ≠ ∞"
by (simp add: one_inat_def)
lemma Infty_ne_number [simp]: "∞ ≠ number_of k"
by (simp add: number_of_inat_def)
lemma number_ne_Infty [simp]: "number_of k ≠ ∞"
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 ∞ = ∞"
by (simp add: iSuc_def)
lemma iSuc_ne_0 [simp]: "iSuc n ≠ 0"
by (simp add: iSuc_def zero_inat_def split: inat.splits)
lemma zero_ne_iSuc [simp]: "0 ≠ iSuc n"
by (rule iSuc_ne_0 [symmetric])
lemma iSuc_inject [simp]: "iSuc m = iSuc n <-> m = n"
by (simp add: iSuc_def split: inat.splits)
lemma number_of_inat_inject [simp]:
"(number_of k :: inat) = number_of l <-> (number_of k :: nat) = number_of l"
by (simp add: number_of_inat_def)
subsection {* Addition *}
instantiation inat :: comm_monoid_add
begin
definition
[code del]: "m + n = (case m of ∞ => ∞ | Fin m => (case n of ∞ => ∞ | Fin n => Fin (m + n)))"
lemma plus_inat_simps [simp, code]:
"Fin m + Fin n = Fin (m + n)"
"∞ + q = ∞"
"q + ∞ = ∞"
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::inat) = q"
"(q::inat) + 0 = q"
by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
lemma plus_inat_number [simp]:
"(number_of k :: 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 :: 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 [code del]:
"m * n = (case m of ∞ => if n = 0 then 0 else ∞ | Fin m =>
(case n of ∞ => if m = 0 then 0 else ∞ | Fin n => Fin (m * n)))"
lemma times_inat_simps [simp, code]:
"Fin m * Fin n = Fin (m * n)"
"∞ * ∞ = ∞"
"∞ * Fin n = (if n = 0 then 0 else ∞)"
"Fin m * ∞ = (if m = 0 then 0 else ∞)"
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) ≠ 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
by default (simp add: of_nat_eq_Fin)
subsection {* Ordering *}
instantiation inat :: ordered_ab_semigroup_add
begin
definition
[code del]: "m ≤ n = (case n of Fin n1 => (case m of Fin m1 => m1 ≤ n1 | ∞ => False)
| ∞ => True)"
definition
[code del]: "m < n = (case m of Fin m1 => (case n of Fin n1 => m1 < n1 | ∞ => True)
| ∞ => False)"
lemma inat_ord_simps [simp]:
"Fin m ≤ Fin n <-> m ≤ n"
"Fin m < Fin n <-> m < n"
"q ≤ ∞"
"q < ∞ <-> q ≠ ∞"
"∞ ≤ q <-> q = ∞"
"∞ < q <-> False"
by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
lemma inat_ord_code [code]:
"Fin m ≤ Fin n <-> m ≤ n"
"Fin m < Fin n <-> m < n"
"q ≤ ∞ <-> True"
"Fin m < ∞ <-> True"
"∞ ≤ Fin n <-> False"
"∞ < q <-> 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 :: pordered_comm_semiring
proof
fix a b c :: inat
assume "a ≤ b" and "0 ≤ c"
thus "c * a ≤ 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 :: inat) ≤ number_of n <-> (number_of m :: nat) ≤ number_of n"
"(number_of m :: inat) < number_of n <-> (number_of m :: nat) < number_of n"
by (simp_all add: number_of_inat_def)
lemma i0_lb [simp]: "(0::inat) ≤ n"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma i0_neq [simp]: "n ≤ (0::inat) <-> n = 0"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma Infty_ileE [elim!]: "∞ ≤ Fin m ==> R"
by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
lemma Infty_ilessE [elim!]: "∞ < Fin m ==> R"
by simp
lemma not_ilessi0 [simp]: "¬ n < (0::inat)"
by (simp add: zero_inat_def less_inat_def split: inat.splits)
lemma i0_eq [simp]: "(0::inat) < n <-> n ≠ 0"
by (simp add: zero_inat_def less_inat_def split: inat.splits)
lemma iSuc_ile_mono [simp]: "iSuc n ≤ iSuc m <-> n ≤ m"
by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
lemma iSuc_mono [simp]: "iSuc n < iSuc m <-> n < m"
by (simp add: iSuc_def less_inat_def split: inat.splits)
lemma ile_iSuc [simp]: "n ≤ iSuc n"
by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
lemma not_iSuc_ilei0 [simp]: "¬ iSuc n ≤ 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 ==> iSuc m ≤ n"
by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
lemma Suc_ile_eq: "Fin (Suc m) ≤ n <-> Fin m < n"
by (cases n) auto
lemma iless_Suc_eq [simp]: "Fin m < iSuc n <-> Fin m ≤ 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 ∞ = q"
"min ∞ 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 ∞ = ∞"
"max ∞ q = ∞"
by (simp_all add: max_def)
lemma Fin_ile: "n ≤ Fin m ==> ∃k. n = Fin k"
by (cases n) simp_all
lemma Fin_iless: "n < Fin m ==> ∃k. n = Fin k"
by (cases n) simp_all
lemma chain_incr: "∀i. ∃j. Y i < Y j ==> ∃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 = ∞"
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. ∀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: "(!!n::inat. (!!m::inat. m < n ==> P m) ==> 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