(* Title: HOL/Library/Extended_Nat.thy
Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen
Contributions: David Trachtenherz, TU Muenchen
*)
section \<open>Extended natural numbers (i.e. with infinity)\<close>
theory Extended_Nat
imports Main Countable Order_Continuity
begin
class infinity =
fixes infinity :: "'a" ("\<infinity>")
context
fixes f :: "nat \<Rightarrow> 'a::{canonically_ordered_monoid_add, linorder_topology, complete_linorder}"
begin
lemma sums_SUP[simp, intro]: "f sums (SUP n. \<Sum>i<n. f i)"
unfolding sums_def by (intro LIMSEQ_SUP monoI sum_mono2 zero_le) auto
lemma suminf_eq_SUP: "suminf f = (SUP n. \<Sum>i<n. f i)"
using sums_SUP by (rule sums_unique[symmetric])
end
subsection \<open>Type definition\<close>
text \<open>
We extend the standard natural numbers by a special value indicating
infinity.
\<close>
typedef enat = "UNIV :: nat option set" ..
text \<open>TODO: introduce enat as coinductive datatype, enat is just @{const of_nat}\<close>
definition enat :: "nat \<Rightarrow> enat" where
"enat n = Abs_enat (Some n)"
instantiation enat :: infinity
begin
definition "\<infinity> = Abs_enat None"
instance ..
end
instance enat :: countable
proof
show "\<exists>to_nat::enat \<Rightarrow> nat. inj to_nat"
by (rule exI[of _ "to_nat \<circ> Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
qed
old_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"]]
lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
by (cases x) auto
lemma not_enat_eq [iff]: "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
by (cases x) auto
lemma enat_ex_split: "(\<exists>c::enat. P c) \<longleftrightarrow> P \<infinity> \<or> (\<exists>c::nat. P c)"
by (metis enat.exhaust)
primrec the_enat :: "enat \<Rightarrow> nat"
where "the_enat (enat n) = n"
subsection \<open>Constructors and numbers\<close>
instantiation enat :: zero_neq_one
begin
definition
"0 = enat 0"
definition
"1 = enat 1"
instance
proof qed (simp add: zero_enat_def one_enat_def)
end
definition eSuc :: "enat \<Rightarrow> enat" where
"eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
lemma enat_0 [code_post]: "enat 0 = 0"
by (simp add: zero_enat_def)
lemma enat_1 [code_post]: "enat 1 = 1"
by (simp add: one_enat_def)
lemma enat_0_iff: "enat x = 0 \<longleftrightarrow> x = 0" "0 = enat x \<longleftrightarrow> x = 0"
by (auto simp add: zero_enat_def)
lemma enat_1_iff: "enat x = 1 \<longleftrightarrow> x = 1" "1 = enat x \<longleftrightarrow> x = 1"
by (auto simp add: one_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_one_enat_neq:
"\<not> 0 = (1::enat)"
"\<not> 1 = (0::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 eSuc_enat: "eSuc (enat n) = enat (Suc n)"
by (simp add: eSuc_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 eSuc_enat_iff: "eSuc x = enat y \<longleftrightarrow> (\<exists>n. y = Suc n \<and> x = enat n)"
by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
lemma enat_eSuc_iff: "enat y = eSuc x \<longleftrightarrow> (\<exists>n. y = Suc n \<and> enat n = x)"
by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
subsection \<open>Addition\<close>
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 m q rule: enat3_cases) auto
show "n + m = m + n"
by (cases n m rule: enat2_cases) auto
show "0 + n = n"
by (cases n) (simp_all add: zero_enat_def)
qed
end
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 ac_simps)
lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
by (simp_all add: eSuc_plus_1 ac_simps)
lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
by (simp only: add.commute[of m] iadd_Suc)
subsection \<open>Multiplication\<close>
instantiation enat :: "{comm_semiring_1, semiring_no_zero_divisors}"
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 comm: "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 distr: "(a + b) * c = a * c + b * c"
unfolding times_enat_def zero_enat_def
by (simp split: enat.split add: distrib_right)
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 "a * (b + c) = a * b + a * c"
by (cases a b c rule: enat3_cases) (auto simp: times_enat_def zero_enat_def distrib_left)
show "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> a * b \<noteq> 0"
by (cases a b rule: enat2_cases) (auto simp: times_enat_def zero_enat_def)
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 :: 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_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 \<open>Numerals\<close>
lemma numeral_eq_enat:
"numeral k = enat (numeral k)"
using of_nat_eq_enat [of "numeral k"] by simp
lemma enat_numeral [code_abbrev]:
"enat (numeral k) = numeral k"
using numeral_eq_enat ..
lemma infinity_ne_numeral [simp]: "(\<infinity>::enat) \<noteq> numeral k"
by (simp add: numeral_eq_enat)
lemma numeral_ne_infinity [simp]: "numeral k \<noteq> (\<infinity>::enat)"
by (simp add: numeral_eq_enat)
lemma eSuc_numeral [simp]: "eSuc (numeral k) = numeral (k + Num.One)"
by (simp only: eSuc_plus_1 numeral_plus_one)
subsection \<open>Subtraction\<close>
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 flip: eSuc_enat zero_enat_def)
(*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
subsection \<open>Ordering\<close>
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 numeral_le_enat_iff[simp]:
shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
by (auto simp: numeral_eq_enat)
lemma numeral_less_enat_iff[simp]:
shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
by (auto simp: numeral_eq_enat)
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 standard (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
end
instance enat :: dioid
proof
fix a b :: enat show "(a \<le> b) = (\<exists>c. b = a + c)"
by (cases a b rule: enat2_cases) (auto simp: le_iff_add enat_ex_split)
qed
instance enat :: "{linordered_nonzero_semiring, strict_ordered_comm_monoid_add}"
proof
fix a b c :: enat
show "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow>c * a \<le> c * b"
unfolding times_enat_def less_eq_enat_def zero_enat_def
by (simp split: enat.splits)
show "a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d" for a b c d :: enat
by (cases a b c d rule: enat2_cases[case_product enat2_cases]) auto
show "a < b \<Longrightarrow> a + 1 < b + 1"
by (metis add_right_mono eSuc_minus_1 eSuc_plus_1 less_le)
qed (simp add: zero_enat_def one_enat_def)
(* BH: These equations are already proven generally for any type in
class linordered_semidom. However, enat is not in that class because
it does not have the cancellation property. Would it be worthwhile to
a generalize linordered_semidom to a new class that includes enat? *)
lemma enat_ord_number [simp]:
"(numeral m :: enat) \<le> numeral n \<longleftrightarrow> (numeral m :: nat) \<le> numeral n"
"(numeral m :: enat) < numeral n \<longleftrightarrow> (numeral m :: nat) < numeral n"
by (simp_all add: numeral_eq_enat)
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 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: zero_less_iff_neq_zero mult_eq_0_iff, 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 iadd_le_enat_iff:
"x + y \<le> enat n \<longleftrightarrow> (\<exists>y' x'. x = enat x' \<and> y = enat y' \<and> x' + y' \<le> n)"
by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j \<Longrightarrow> \<exists>j. enat k < Y j"
apply (induct_tac k)
apply (simp (no_asm) only: enat_0)
apply (fast intro: le_less_trans [OF zero_le])
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
lemma eSuc_max: "eSuc (max x y) = max (eSuc x) (eSuc y)"
by (simp add: eSuc_def split: enat.split)
lemma eSuc_Max:
assumes "finite A" "A \<noteq> {}"
shows "eSuc (Max A) = Max (eSuc ` A)"
using assms proof induction
case (insert x A)
thus ?case by(cases "A = {}")(simp_all add: eSuc_max)
qed simp
instantiation enat :: "{order_bot, order_top}"
begin
definition bot_enat :: enat where "bot_enat = 0"
definition top_enat :: enat where "top_enat = \<infinity>"
instance
by standard (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}"
apply (rule subsetI)
subgoal for x by (cases x) auto
done
finally show "A \<subseteq> enat ` {..n}" .
qed
subsection \<open>Cancellation simprocs\<close>
lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
unfolding plus_enat_def by (simp split: enat.split)
lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
unfolding plus_enat_def by (simp split: enat.split)
lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
unfolding plus_enat_def by (simp split: enat.split)
ML \<open>
structure Cancel_Enat_Common =
struct
(* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
fun find_first_t _ _ [] = raise TERM("find_first_t", [])
| find_first_t past u (t::terms) =
if u aconv t then (rev past @ terms)
else find_first_t (t::past) u terms
fun dest_summing (Const (@{const_name Groups.plus}, _) $ t $ u, ts) =
dest_summing (t, dest_summing (u, ts))
| dest_summing (t, ts) = t :: ts
val mk_sum = Arith_Data.long_mk_sum
fun dest_sum t = dest_summing (t, [])
val find_first = find_first_t []
val trans_tac = Numeral_Simprocs.trans_tac
val norm_ss =
simpset_of (put_simpset HOL_basic_ss @{context}
addsimps @{thms ac_simps add_0_left add_0_right})
fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
fun simplify_meta_eq ctxt cancel_th th =
Arith_Data.simplify_meta_eq [] ctxt
([th, cancel_th] MRS trans)
fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
end
structure Eq_Enat_Cancel = ExtractCommonTermFun
(open Cancel_Enat_Common
val mk_bal = HOLogic.mk_eq
val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
)
structure Le_Enat_Cancel = ExtractCommonTermFun
(open Cancel_Enat_Common
val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
)
structure Less_Enat_Cancel = ExtractCommonTermFun
(open Cancel_Enat_Common
val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
)
\<close>
simproc_setup enat_eq_cancel
("(l::enat) + m = n" | "(l::enat) = m + n") =
\<open>fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
simproc_setup enat_le_cancel
("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
\<open>fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
simproc_setup enat_less_cancel
("(l::enat) + m < n" | "(l::enat) < m + n") =
\<open>fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
text \<open>TODO: add regression tests for these simprocs\<close>
text \<open>TODO: add simprocs for combining and cancelling numerals\<close>
subsection \<open>Well-ordering\<close>
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: "\<And>n. \<forall>m::enat. m < n \<longrightarrow> P m \<Longrightarrow> P n" shows "P n"
proof -
have P_enat: "\<And>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::enat. (\<And>m::enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
show "P n" by (blast intro: enat_less_induct hyp)
qed
subsection \<open>Complete Lattice\<close>
instantiation enat :: complete_lattice
begin
definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
"inf_enat = min"
definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
"sup_enat = max"
definition Inf_enat :: "enat set \<Rightarrow> enat" where
"Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
definition Sup_enat :: "enat set \<Rightarrow> enat" where
"Sup_enat A = (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 bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
end
instance enat :: complete_linorder ..
lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"
by(auto simp add: Sup_enat_def eSuc_Max inj_on_def dest: finite_imageD)
lemma sup_continuous_eSuc: "sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. eSuc (f x))"
using eSuc_Sup[of "_ ` UNIV"] by (auto simp: sup_continuous_def)
subsection \<open>Traditional theorem names\<close>
lemmas enat_defs = zero_enat_def one_enat_def eSuc_def
plus_enat_def less_eq_enat_def less_enat_def
lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
by (rule add_eq_0_iff_both_eq_0)
lemma i0_lb : "(0::enat) \<le> n"
by (rule zero_le)
lemma ile0_eq: "n \<le> (0::enat) \<longleftrightarrow> n = 0"
by (rule le_zero_eq)
lemma not_iless0: "\<not> n < (0::enat)"
by (rule not_less_zero)
lemma i0_less[simp]: "(0::enat) < n \<longleftrightarrow> n \<noteq> 0"
by (rule zero_less_iff_neq_zero)
lemma imult_is_0: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
by (rule mult_eq_0_iff)
end