(* 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" ("\")
context
fixes f :: "nat \ 'a::{canonically_ordered_monoid_add, linorder_topology, complete_linorder}"
begin
lemma sums_SUP[simp, intro]: "f sums (SUP n. \i
unfolding sums_def by (intro LIMSEQ_SUP monoI sum_mono2 zero_le) auto
lemma suminf_eq_SUP: "suminf f = (SUP n. \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>\<open>of_nat\<close>\<close>
definition enat :: "nat \ enat" where
"enat n = Abs_enat (Some n)"
instantiation enat :: infinity
begin
definition "\ = Abs_enat None"
instance ..
end
instance enat :: countable
proof
show "\to_nat::enat \ nat. inj to_nat"
by (rule exI[of _ "to_nat \ Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
qed
old_rep_datatype enat "\ :: enat"
proof -
fix P i assume "\j. P (enat j)" "P \"
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\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 \ \) = (\i. x = enat i)"
by (cases x) auto
lemma not_enat_eq [iff]: "(\y. x \ enat y) = (x = \)"
by (cases x) auto
lemma enat_ex_split: "(\c::enat. P c) \ P \ \ (\c::nat. P c)"
by (metis enat.exhaust)
primrec the_enat :: "enat \ 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 \ enat" where
"eSuc i = (case i of enat n \ enat (Suc n) | \ \ \)"
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 \ x = 0" "0 = enat x \ x = 0"
by (auto simp add: zero_enat_def)
lemma enat_1_iff: "enat x = 1 \ x = 1" "1 = enat x \ 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]: "(\::enat) \ 0"
by (simp add: zero_enat_def)
lemma i0_ne_infinity [simp]: "0 \ (\::enat)"
by (simp add: zero_enat_def)
lemma zero_one_enat_neq:
"\ 0 = (1::enat)"
"\ 1 = (0::enat)"
unfolding zero_enat_def one_enat_def by simp_all
lemma infinity_ne_i1 [simp]: "(\::enat) \ 1"
by (simp add: one_enat_def)
lemma i1_ne_infinity [simp]: "1 \ (\::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 \ = \"
by (simp add: eSuc_def)
lemma eSuc_ne_0 [simp]: "eSuc n \ 0"
by (simp add: eSuc_def zero_enat_def split: enat.splits)
lemma zero_ne_eSuc [simp]: "0 \ eSuc n"
by (rule eSuc_ne_0 [symmetric])
lemma eSuc_inject [simp]: "eSuc m = eSuc n \ m = n"
by (simp add: eSuc_def split: enat.splits)
lemma eSuc_enat_iff: "eSuc x = enat y \ (\n. y = Suc n \ x = enat n)"
by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
lemma enat_eSuc_iff: "enat y = eSuc x \ (\n. y = Suc n \ 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 \ \ \ | enat m \ (case n of \ \ \ | enat n \ enat (m + n)))"
lemma plus_enat_simps [simp, code]:
fixes q :: enat
shows "enat m + enat n = enat (m + n)"
and "\ + q = \"
and "q + \ = \"
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 \ \ if n = 0 then 0 else \ | enat m \
(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)"
"\ * \ = (\::enat)"
"\ * enat n = (if n = 0 then 0 else \)"
"enat m * \ = (if m = 0 then 0 else \)"
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 \ 0 \ b \ 0 \ a * b \ 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 (\n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
qed
lemma imult_is_infinity: "((a::enat) * b = \) = (a = \ \ b \ 0 \ b = \ \ a \ 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]: "(\::enat) \ numeral k"
by (simp add: numeral_eq_enat)
lemma numeral_ne_infinity [simp]: "numeral k \ (\::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) \ (case b of (enat y) \ enat (x - y) | \ \ 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]: "\ - n = (\::enat)"
by (simp add: diff_enat_def)
lemma idiff_infinity_right [simp, code]: "enat a - \ = 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 \ \ \ (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 \ n = (case n of enat n1 \ (case m of enat m1 \ m1 \ n1 | \ \ False)
| \<infinity> \<Rightarrow> True)"
definition [nitpick_simp]:
"m < n = (case m of enat m1 \ (case n of enat n1 \ m1 < n1 | \ \ True)
| \<infinity> \<Rightarrow> False)"
lemma enat_ord_simps [simp]:
"enat m \ enat n \ m \ n"
"enat m < enat n \ m < n"
"q \ (\::enat)"
"q < (\::enat) \ q \ \"
"(\::enat) \ q \ q = \"
"(\::enat) < q \ False"
by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
lemma numeral_le_enat_iff[simp]:
shows "numeral m \ enat n \ numeral m \ n"
by (auto simp: numeral_eq_enat)
lemma numeral_less_enat_iff[simp]:
shows "numeral m < enat n \ numeral m < n"
by (auto simp: numeral_eq_enat)
lemma enat_ord_code [code]:
"enat m \ enat n \ m \ n"
"enat m < enat n \ m < n"
"q \ (\::enat) \ True"
"enat m < \ \ True"
"\ \ enat n \ False"
"(\::enat) < q \ 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 \ b) = (\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 \ b \ 0 \ c \c * a \ c * b"
unfolding times_enat_def less_eq_enat_def zero_enat_def
by (simp split: enat.splits)
show "a < b \ c < d \ 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 \ 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 add_diff_assoc_enat: "z \ y \ x + (y - z) = x + y - (z::enat)"
by(cases x)(auto simp add: diff_enat_def split: enat.split)
lemma enat_ord_number [simp]:
"(numeral m :: enat) \ numeral n \ (numeral m :: nat) \ numeral n"
"(numeral m :: enat) < numeral n \ (numeral m :: nat) < numeral n"
by (simp_all add: numeral_eq_enat)
lemma infinity_ileE [elim!]: "\ \ enat m \ R"
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
lemma infinity_ilessE [elim!]: "\ < enat m \ R"
by simp
lemma eSuc_ile_mono [simp]: "eSuc n \ eSuc m \ n \ m"
by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
lemma eSuc_mono [simp]: "eSuc n < eSuc m \ n < m"
by (simp add: eSuc_def less_enat_def split: enat.splits)
lemma ile_eSuc [simp]: "n \ eSuc n"
by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
lemma not_eSuc_ilei0 [simp]: "\ eSuc n \ 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 \ eSuc m \ n"
by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
lemma Suc_ile_eq: "enat (Suc m) \ n \ enat m < n"
by (cases n) auto
lemma iless_Suc_eq [simp]: "enat m < eSuc n \ enat m \ n"
by (auto simp add: eSuc_def less_enat_def split: enat.splits)
lemma imult_infinity: "(0::enat) < n \ \ * n = \"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma imult_infinity_right: "(0::enat) < n \ n * \ = \"
by (simp add: zero_enat_def less_enat_def split: enat.splits)
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \ 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 (\::enat) = q"
"min (\::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 \ = (\::enat)"
"max \ q = (\::enat)"
by (simp_all add: max_def)
lemma enat_ile: "n \ enat m \ \k. n = enat k"
by (cases n) simp_all
lemma enat_iless: "n < enat m \ \k. n = enat k"
by (cases n) simp_all
lemma iadd_le_enat_iff:
"x + y \ enat n \ (\y' x'. x = enat x' \ y = enat y' \ x' + y' \ n)"
by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all
lemma chain_incr: "\i. \j. Y i < Y j \ \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 \ {}"
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 = \"
instance
by standard (simp_all add: bot_enat_def top_enat_def)
end
lemma finite_enat_bounded:
assumes le_fin: "\y. y \ A \ y \ enat n"
shows "finite A"
proof (rule finite_subset)
show "finite (enat ` {..n})" by blast
have "A \ {..enat n}" using le_fin by fastforce
also have "\ \ enat ` {..n}"
apply (rule subsetI)
subgoal for x by (cases x) auto
done
finally show "A \ enat ` {..n}" .
qed
subsection \<open>Cancellation simprocs\<close>
lemma add_diff_cancel_enat[simp]: "x \ \ \ x + y - x = (y::enat)"
by (metis add.commute add.right_neutral add_diff_assoc_enat idiff_self order_refl)
lemma enat_add_left_cancel: "a + b = a + c \ a = (\::enat) \ b = c"
unfolding plus_enat_def by (simp split: enat.split)
lemma enat_add_left_cancel_le: "a + b \ a + c \ a = (\::enat) \ b \ c"
unfolding plus_enat_def by (simp split: enat.split)
lemma enat_add_left_cancel_less: "a + b < a + c \ a \ (\::enat) \ b < c"
unfolding plus_enat_def by (simp split: enat.split)
lemma plus_eq_infty_iff_enat: "(m::enat) + n = \ \ m=\ \ n=\"
using enat_add_left_cancel by fastforce
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>\<open>Groups.plus\<close>, _) $ 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>\<open>HOL.eq\<close> \<^typ>\<open>enat\<close>
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>\<open>Orderings.less_eq\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less_eq\<close> \<^typ>\<open>enat\<close>
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>\<open>Orderings.less\<close>
val dest_bal = HOLogic.dest_bin \<^const_name>\<open>Orderings.less\<close> \<^typ>\<open>enat\<close>
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 \ n" | "(l::enat) \ 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 < \; !!k. n = enat k ==> P |] ==> P"
by (induct n) auto
lemma enat_less_induct:
assumes prem: "\n. \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 \"
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: "(\n::enat. (\m::enat. m < n \ P m) \ 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 \ enat \ enat" where
"inf_enat = min"
definition sup_enat :: "enat \ enat \ enat" where
"sup_enat = max"
definition Inf_enat :: "enat set \ enat" where
"Inf_enat A = (if A = {} then \ else (LEAST x. x \ A))"
definition Sup_enat :: "enat set \ enat" where
"Sup_enat A = (if A = {} then 0 else if finite A then Max A else \)"
instance
proof
fix x :: "enat" and A :: "enat set"
{ assume "x \ A" then show "Inf A \ x"
unfolding Inf_enat_def by (auto intro: Least_le) }
{ assume "\y. y \ A \ x \ y" then show "x \ Inf A"
unfolding Inf_enat_def
by (cases "A = {}") (auto intro: LeastI2_ex) }
{ assume "x \ A" then show "x \ Sup A"
unfolding Sup_enat_def by (cases "finite A") auto }
{ assume "\y. y \ A \ y \ x" then show "Sup A \ 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 \ {} \ 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 \ sup_continuous (\x. eSuc (f x))"
using eSuc_Sup [of "_ ` UNIV"] by (auto simp: sup_continuous_def image_comp)
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 \ n = 0)"
by (rule add_eq_0_iff_both_eq_0)
lemma i0_lb : "(0::enat) \ n"
by (rule zero_le)
lemma ile0_eq: "n \ (0::enat) \ n = 0"
by (rule le_zero_eq)
lemma not_iless0: "\ n < (0::enat)"
by (rule not_less_zero)
lemma i0_less[simp]: "(0::enat) < n \ n \ 0"
by (rule zero_less_iff_neq_zero)
lemma imult_is_0: "((m::enat) * n = 0) = (m = 0 \ n = 0)"
by (rule mult_eq_0_iff)
end
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