(*  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>")

 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

 primrec the_enat :: "enat \<Rightarrow> nat"

   where "the_enat (enat n) = n"

 subsection \<open>Constructors and numbers\<close>

 instantiation enat :: "{zero, one}"

 begin

 definition

   "0 = enat 0"

 definition

   "1 = enat 1"

 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 [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 [simp]:

   "\<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)

 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 \<open>Multiplication\<close>

 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: 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 "(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 :: 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 \<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 eSuc_enat[symmetric] zero_enat_def[symmetric])

 (*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 :: 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

 (* 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 i0_lb [simp]: "(0::enat) \<le> n"

   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)

 lemma ile0_eq [simp]: "n \<le> (0::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::enat)"

   by (simp add: zero_enat_def less_enat_def split: enat.splits)

 lemma i0_less [simp]: "(0::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 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 ==> \<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

 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: "!!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::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

 end