hoelzl@43919: (*  Title:      HOL/Library/Extended_Nat.thy
haftmann@27110:     Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
nipkow@41853:     Contributions: David Trachtenherz, TU Muenchen
oheimb@11351: *)
oheimb@11351: 
hoelzl@43919: header {* Extended natural numbers (i.e. with infinity) *}
oheimb@11351: 
hoelzl@43919: theory Extended_Nat
haftmann@30663: imports Main
nipkow@15131: begin
oheimb@11351: 
hoelzl@43921: class infinity =
hoelzl@43921:   fixes infinity :: "'a"
hoelzl@43921: 
hoelzl@43921: notation (xsymbols)
hoelzl@43921:   infinity  ("\<infinity>")
hoelzl@43921: 
hoelzl@43921: notation (HTML output)
hoelzl@43921:   infinity  ("\<infinity>")
hoelzl@43921: 
haftmann@27110: subsection {* Type definition *}
oheimb@11351: 
oheimb@11351: text {*
wenzelm@11355:   We extend the standard natural numbers by a special value indicating
haftmann@27110:   infinity.
oheimb@11351: *}
oheimb@11351: 
hoelzl@43921: typedef (open) enat = "UNIV :: nat option set" ..
hoelzl@43921:  
hoelzl@43924: definition enat :: "nat \<Rightarrow> enat" where
hoelzl@43924:   "enat n = Abs_enat (Some n)"
hoelzl@43921:  
hoelzl@43921: instantiation enat :: infinity
hoelzl@43921: begin
hoelzl@43921:   definition "\<infinity> = Abs_enat None"
hoelzl@43921:   instance proof qed
hoelzl@43921: end
hoelzl@43921:  
hoelzl@43924: rep_datatype enat "\<infinity> :: enat"
hoelzl@43921: proof -
hoelzl@43924:   fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
hoelzl@43921:   then show "P i"
hoelzl@43921:   proof induct
hoelzl@43921:     case (Abs_enat y) then show ?case
hoelzl@43921:       by (cases y rule: option.exhaust)
hoelzl@43924:          (auto simp: enat_def infinity_enat_def)
hoelzl@43921:   qed
hoelzl@43924: qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
wenzelm@19736: 
hoelzl@43924: declare [[coercion "enat::nat\<Rightarrow>enat"]]
wenzelm@19736: 
noschinl@45934: lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
noschinl@45934: lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
noschinl@45934: 
huffman@44019: lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (EX i. x = enat i)"
huffman@44019:   by (cases x) auto
nipkow@31084: 
hoelzl@43924: lemma not_enat_eq [iff]: "(ALL y. x ~= enat y) = (x = \<infinity>)"
huffman@44019:   by (cases x) auto
nipkow@31077: 
hoelzl@43924: primrec the_enat :: "enat \<Rightarrow> nat"
huffman@44019:   where "the_enat (enat n) = n"
nipkow@41855: 
haftmann@27110: subsection {* Constructors and numbers *}
haftmann@27110: 
hoelzl@43919: instantiation enat :: "{zero, one, number}"
haftmann@25594: begin
haftmann@25594: 
haftmann@25594: definition
hoelzl@43924:   "0 = enat 0"
haftmann@25594: 
haftmann@25594: definition
hoelzl@43924:   [code_unfold]: "1 = enat 1"
haftmann@25594: 
haftmann@25594: definition
hoelzl@43924:   [code_unfold, code del]: "number_of k = enat (number_of k)"
oheimb@11351: 
haftmann@25594: instance ..
haftmann@25594: 
haftmann@25594: end
haftmann@25594: 
huffman@44019: definition eSuc :: "enat \<Rightarrow> enat" where
huffman@44019:   "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
oheimb@11351: 
hoelzl@43924: lemma enat_0: "enat 0 = 0"
hoelzl@43919:   by (simp add: zero_enat_def)
haftmann@27110: 
hoelzl@43924: lemma enat_1: "enat 1 = 1"
hoelzl@43919:   by (simp add: one_enat_def)
haftmann@27110: 
hoelzl@43924: lemma enat_number: "enat (number_of k) = number_of k"
hoelzl@43919:   by (simp add: number_of_enat_def)
haftmann@27110: 
huffman@44019: lemma one_eSuc: "1 = eSuc 0"
huffman@44019:   by (simp add: zero_enat_def one_enat_def eSuc_def)
oheimb@11351: 
huffman@44019: lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
hoelzl@43919:   by (simp add: zero_enat_def)
oheimb@11351: 
huffman@44019: lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
hoelzl@43919:   by (simp add: zero_enat_def)
haftmann@27110: 
hoelzl@43919: lemma zero_enat_eq [simp]:
hoelzl@43919:   "number_of k = (0\<Colon>enat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
hoelzl@43919:   "(0\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
hoelzl@43919:   unfolding zero_enat_def number_of_enat_def by simp_all
haftmann@27110: 
hoelzl@43919: lemma one_enat_eq [simp]:
hoelzl@43919:   "number_of k = (1\<Colon>enat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
hoelzl@43919:   "(1\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
hoelzl@43919:   unfolding one_enat_def number_of_enat_def by simp_all
haftmann@27110: 
hoelzl@43919: lemma zero_one_enat_neq [simp]:
hoelzl@43919:   "\<not> 0 = (1\<Colon>enat)"
hoelzl@43919:   "\<not> 1 = (0\<Colon>enat)"
hoelzl@43919:   unfolding zero_enat_def one_enat_def by simp_all
oheimb@11351: 
huffman@44019: lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
hoelzl@43919:   by (simp add: one_enat_def)
haftmann@27110: 
huffman@44019: lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
hoelzl@43919:   by (simp add: one_enat_def)
haftmann@27110: 
huffman@44019: lemma infinity_ne_number [simp]: "(\<infinity>::enat) \<noteq> number_of k"
hoelzl@43919:   by (simp add: number_of_enat_def)
haftmann@27110: 
huffman@44019: lemma number_ne_infinity [simp]: "number_of k \<noteq> (\<infinity>::enat)"
hoelzl@43919:   by (simp add: number_of_enat_def)
haftmann@27110: 
huffman@44019: lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
huffman@44019:   by (simp add: eSuc_def)
haftmann@27110: 
huffman@44019: lemma eSuc_number_of: "eSuc (number_of k) = enat (Suc (number_of k))"
huffman@44019:   by (simp add: eSuc_enat number_of_enat_def)
oheimb@11351: 
huffman@44019: lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
huffman@44019:   by (simp add: eSuc_def)
oheimb@11351: 
huffman@44019: lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
huffman@44019:   by (simp add: eSuc_def zero_enat_def split: enat.splits)
haftmann@27110: 
huffman@44019: lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
huffman@44019:   by (rule eSuc_ne_0 [symmetric])
oheimb@11351: 
huffman@44019: lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
huffman@44019:   by (simp add: eSuc_def split: enat.splits)
haftmann@27110: 
hoelzl@43919: lemma number_of_enat_inject [simp]:
hoelzl@43919:   "(number_of k \<Colon> enat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
hoelzl@43919:   by (simp add: number_of_enat_def)
oheimb@11351: 
oheimb@11351: 
haftmann@27110: subsection {* Addition *}
haftmann@27110: 
hoelzl@43919: instantiation enat :: comm_monoid_add
haftmann@27110: begin
haftmann@27110: 
blanchet@38167: definition [nitpick_simp]:
hoelzl@43924:   "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
oheimb@11351: 
hoelzl@43919: lemma plus_enat_simps [simp, code]:
hoelzl@43921:   fixes q :: enat
hoelzl@43924:   shows "enat m + enat n = enat (m + n)"
hoelzl@43921:     and "\<infinity> + q = \<infinity>"
hoelzl@43921:     and "q + \<infinity> = \<infinity>"
hoelzl@43919:   by (simp_all add: plus_enat_def split: enat.splits)
haftmann@27110: 
haftmann@27110: instance proof
hoelzl@43919:   fix n m q :: enat
haftmann@27110:   show "n + m + q = n + (m + q)"
noschinl@45934:     by (cases n m q rule: enat3_cases) auto
haftmann@27110:   show "n + m = m + n"
noschinl@45934:     by (cases n m rule: enat2_cases) auto
haftmann@27110:   show "0 + n = n"
hoelzl@43919:     by (cases n) (simp_all add: zero_enat_def)
huffman@26089: qed
huffman@26089: 
haftmann@27110: end
oheimb@11351: 
hoelzl@43919: lemma plus_enat_number [simp]:
hoelzl@43919:   "(number_of k \<Colon> enat) + number_of l = (if k < Int.Pls then number_of l
huffman@29012:     else if l < Int.Pls then number_of k else number_of (k + l))"
hoelzl@43924:   unfolding number_of_enat_def plus_enat_simps nat_arith(1) if_distrib [symmetric, of _ enat] ..
oheimb@11351: 
huffman@44019: lemma eSuc_number [simp]:
huffman@44019:   "eSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
huffman@44019:   unfolding eSuc_number_of
hoelzl@43919:   unfolding one_enat_def number_of_enat_def Suc_nat_number_of if_distrib [symmetric] ..
oheimb@11351: 
huffman@44019: lemma eSuc_plus_1:
huffman@44019:   "eSuc n = n + 1"
huffman@44019:   by (cases n) (simp_all add: eSuc_enat one_enat_def)
haftmann@27110:   
huffman@44019: lemma plus_1_eSuc:
huffman@44019:   "1 + q = eSuc q"
huffman@44019:   "q + 1 = eSuc q"
huffman@44019:   by (simp_all add: eSuc_plus_1 add_ac)
nipkow@41853: 
huffman@44019: lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
huffman@44019:   by (simp_all add: eSuc_plus_1 add_ac)
oheimb@11351: 
huffman@44019: lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
huffman@44019:   by (simp only: add_commute[of m] iadd_Suc)
nipkow@41853: 
hoelzl@43919: lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
huffman@44019:   by (cases m, cases n, simp_all add: zero_enat_def)
oheimb@11351: 
huffman@29014: subsection {* Multiplication *}
huffman@29014: 
hoelzl@43919: instantiation enat :: comm_semiring_1
huffman@29014: begin
huffman@29014: 
hoelzl@43919: definition times_enat_def [nitpick_simp]:
hoelzl@43924:   "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
hoelzl@43924:     (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
huffman@29014: 
hoelzl@43919: lemma times_enat_simps [simp, code]:
hoelzl@43924:   "enat m * enat n = enat (m * n)"
hoelzl@43921:   "\<infinity> * \<infinity> = (\<infinity>::enat)"
hoelzl@43924:   "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
hoelzl@43924:   "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
hoelzl@43919:   unfolding times_enat_def zero_enat_def
hoelzl@43919:   by (simp_all split: enat.split)
huffman@29014: 
huffman@29014: instance proof
hoelzl@43919:   fix a b c :: enat
huffman@29014:   show "(a * b) * c = a * (b * c)"
hoelzl@43919:     unfolding times_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.split)
huffman@29014:   show "a * b = b * a"
hoelzl@43919:     unfolding times_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.split)
huffman@29014:   show "1 * a = a"
hoelzl@43919:     unfolding times_enat_def zero_enat_def one_enat_def
hoelzl@43919:     by (simp split: enat.split)
huffman@29014:   show "(a + b) * c = a * c + b * c"
hoelzl@43919:     unfolding times_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.split add: left_distrib)
huffman@29014:   show "0 * a = 0"
hoelzl@43919:     unfolding times_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.split)
huffman@29014:   show "a * 0 = 0"
hoelzl@43919:     unfolding times_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.split)
hoelzl@43919:   show "(0::enat) \<noteq> 1"
hoelzl@43919:     unfolding zero_enat_def one_enat_def
huffman@29014:     by simp
huffman@29014: qed
huffman@29014: 
huffman@29014: end
huffman@29014: 
huffman@44019: lemma mult_eSuc: "eSuc m * n = n + m * n"
huffman@44019:   unfolding eSuc_plus_1 by (simp add: algebra_simps)
huffman@29014: 
huffman@44019: lemma mult_eSuc_right: "m * eSuc n = m + m * n"
huffman@44019:   unfolding eSuc_plus_1 by (simp add: algebra_simps)
huffman@29014: 
hoelzl@43924: lemma of_nat_eq_enat: "of_nat n = enat n"
huffman@29023:   apply (induct n)
hoelzl@43924:   apply (simp add: enat_0)
huffman@44019:   apply (simp add: plus_1_eSuc eSuc_enat)
huffman@29023:   done
huffman@29023: 
hoelzl@43919: instance enat :: number_semiring
huffman@43532: proof
hoelzl@43919:   fix n show "number_of (int n) = (of_nat n :: enat)"
hoelzl@43924:     unfolding number_of_enat_def number_of_int of_nat_id of_nat_eq_enat ..
huffman@43532: qed
huffman@43532: 
hoelzl@43919: instance enat :: semiring_char_0 proof
hoelzl@43924:   have "inj enat" by (rule injI) simp
hoelzl@43924:   then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
haftmann@38621: qed
huffman@29023: 
huffman@44019: lemma imult_is_0 [simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
huffman@44019:   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
nipkow@41853: 
huffman@44019: lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
huffman@44019:   by (auto simp add: times_enat_def zero_enat_def split: enat.split)
nipkow@41853: 
nipkow@41853: 
nipkow@41853: subsection {* Subtraction *}
nipkow@41853: 
hoelzl@43919: instantiation enat :: minus
nipkow@41853: begin
nipkow@41853: 
hoelzl@43919: definition diff_enat_def:
hoelzl@43924: "a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
nipkow@41853:           | \<infinity> \<Rightarrow> \<infinity>)"
nipkow@41853: 
nipkow@41853: instance ..
nipkow@41853: 
nipkow@41853: end
nipkow@41853: 
huffman@44019: lemma idiff_enat_enat [simp,code]: "enat a - enat b = enat (a - b)"
huffman@44019:   by (simp add: diff_enat_def)
nipkow@41853: 
huffman@44019: lemma idiff_infinity [simp,code]: "\<infinity> - n = (\<infinity>::enat)"
huffman@44019:   by (simp add: diff_enat_def)
nipkow@41853: 
huffman@44019: lemma idiff_infinity_right [simp,code]: "enat a - \<infinity> = 0"
huffman@44019:   by (simp add: diff_enat_def)
nipkow@41853: 
huffman@44019: lemma idiff_0 [simp]: "(0::enat) - n = 0"
huffman@44019:   by (cases n, simp_all add: zero_enat_def)
nipkow@41853: 
huffman@44019: lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
nipkow@41853: 
huffman@44019: lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
huffman@44019:   by (cases n) (simp_all add: zero_enat_def)
nipkow@41853: 
huffman@44019: lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
nipkow@41853: 
huffman@44019: lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
huffman@44019:   by (auto simp: zero_enat_def)
nipkow@41853: 
huffman@44019: lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
huffman@44019:   by (simp add: eSuc_def split: enat.split)
nipkow@41855: 
huffman@44019: lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
huffman@44019:   by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
nipkow@41855: 
hoelzl@43924: (*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
nipkow@41853: 
haftmann@27110: subsection {* Ordering *}
haftmann@27110: 
hoelzl@43919: instantiation enat :: linordered_ab_semigroup_add
haftmann@27110: begin
oheimb@11351: 
blanchet@38167: definition [nitpick_simp]:
hoelzl@43924:   "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
haftmann@27110:     | \<infinity> \<Rightarrow> True)"
oheimb@11351: 
blanchet@38167: definition [nitpick_simp]:
hoelzl@43924:   "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
haftmann@27110:     | \<infinity> \<Rightarrow> False)"
oheimb@11351: 
hoelzl@43919: lemma enat_ord_simps [simp]:
hoelzl@43924:   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924:   "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921:   "q \<le> (\<infinity>::enat)"
hoelzl@43921:   "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
hoelzl@43921:   "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
hoelzl@43921:   "(\<infinity>::enat) < q \<longleftrightarrow> False"
hoelzl@43919:   by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
oheimb@11351: 
noschinl@45934: lemma number_of_le_enat_iff[simp]:
noschinl@45934:   shows "number_of m \<le> enat n \<longleftrightarrow> number_of m \<le> n"
noschinl@45934: by (auto simp: number_of_enat_def)
noschinl@45934: 
noschinl@45934: lemma number_of_less_enat_iff[simp]:
noschinl@45934:   shows "number_of m < enat n \<longleftrightarrow> number_of m < n"
noschinl@45934: by (auto simp: number_of_enat_def)
noschinl@45934: 
hoelzl@43919: lemma enat_ord_code [code]:
hoelzl@43924:   "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924:   "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921:   "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
hoelzl@43924:   "enat m < \<infinity> \<longleftrightarrow> True"
hoelzl@43924:   "\<infinity> \<le> enat n \<longleftrightarrow> False"
hoelzl@43921:   "(\<infinity>::enat) < q \<longleftrightarrow> False"
haftmann@27110:   by simp_all
oheimb@11351: 
haftmann@27110: instance by default
hoelzl@43919:   (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
oheimb@11351: 
haftmann@27110: end
haftmann@27110: 
hoelzl@43919: instance enat :: ordered_comm_semiring
huffman@29014: proof
hoelzl@43919:   fix a b c :: enat
huffman@29014:   assume "a \<le> b" and "0 \<le> c"
huffman@29014:   thus "c * a \<le> c * b"
hoelzl@43919:     unfolding times_enat_def less_eq_enat_def zero_enat_def
hoelzl@43919:     by (simp split: enat.splits)
huffman@29014: qed
huffman@29014: 
hoelzl@43919: lemma enat_ord_number [simp]:
hoelzl@43919:   "(number_of m \<Colon> enat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
hoelzl@43919:   "(number_of m \<Colon> enat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
hoelzl@43919:   by (simp_all add: number_of_enat_def)
oheimb@11351: 
hoelzl@43919: lemma i0_lb [simp]: "(0\<Colon>enat) \<le> n"
hoelzl@43919:   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
oheimb@11351: 
hoelzl@43919: lemma ile0_eq [simp]: "n \<le> (0\<Colon>enat) \<longleftrightarrow> n = 0"
hoelzl@43919:   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
oheimb@11351: 
huffman@44019: lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
huffman@44019:   by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
huffman@44019: 
huffman@44019: lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
haftmann@27110:   by simp
oheimb@11351: 
hoelzl@43919: lemma not_iless0 [simp]: "\<not> n < (0\<Colon>enat)"
hoelzl@43919:   by (simp add: zero_enat_def less_enat_def split: enat.splits)
haftmann@27110: 
hoelzl@43919: lemma i0_less [simp]: "(0\<Colon>enat) < n \<longleftrightarrow> n \<noteq> 0"
huffman@44019:   by (simp add: zero_enat_def less_enat_def split: enat.splits)
oheimb@11351: 
huffman@44019: lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
huffman@44019:   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
haftmann@27110:  
huffman@44019: lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
huffman@44019:   by (simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351: 
huffman@44019: lemma ile_eSuc [simp]: "n \<le> eSuc n"
huffman@44019:   by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
oheimb@11351: 
huffman@44019: lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
huffman@44019:   by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
haftmann@27110: 
huffman@44019: lemma i0_iless_eSuc [simp]: "0 < eSuc n"
huffman@44019:   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
haftmann@27110: 
huffman@44019: lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
huffman@44019:   by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
nipkow@41853: 
huffman@44019: lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
huffman@44019:   by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
haftmann@27110: 
hoelzl@43924: lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
haftmann@27110:   by (cases n) auto
haftmann@27110: 
huffman@44019: lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
huffman@44019:   by (auto simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351: 
huffman@44019: lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
huffman@44019:   by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853: 
huffman@44019: lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
huffman@44019:   by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853: 
hoelzl@43919: lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
huffman@44019:   by (simp only: i0_less imult_is_0, simp)
nipkow@41853: 
huffman@44019: lemma mono_eSuc: "mono eSuc"
huffman@44019:   by (simp add: mono_def)
nipkow@41853: 
nipkow@41853: 
hoelzl@43919: lemma min_enat_simps [simp]:
hoelzl@43924:   "min (enat m) (enat n) = enat (min m n)"
haftmann@27110:   "min q 0 = 0"
haftmann@27110:   "min 0 q = 0"
hoelzl@43921:   "min q (\<infinity>::enat) = q"
hoelzl@43921:   "min (\<infinity>::enat) q = q"
haftmann@27110:   by (auto simp add: min_def)
oheimb@11351: 
hoelzl@43919: lemma max_enat_simps [simp]:
hoelzl@43924:   "max (enat m) (enat n) = enat (max m n)"
haftmann@27110:   "max q 0 = q"
haftmann@27110:   "max 0 q = q"
hoelzl@43921:   "max q \<infinity> = (\<infinity>::enat)"
hoelzl@43921:   "max \<infinity> q = (\<infinity>::enat)"
haftmann@27110:   by (simp_all add: max_def)
haftmann@27110: 
hoelzl@43924: lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110:   by (cases n) simp_all
haftmann@27110: 
hoelzl@43924: lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110:   by (cases n) simp_all
oheimb@11351: 
hoelzl@43924: lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
nipkow@25134: apply (induct_tac k)
hoelzl@43924:  apply (simp (no_asm) only: enat_0)
haftmann@27110:  apply (fast intro: le_less_trans [OF i0_lb])
nipkow@25134: apply (erule exE)
nipkow@25134: apply (drule spec)
nipkow@25134: apply (erule exE)
nipkow@25134: apply (drule ileI1)
huffman@44019: apply (rule eSuc_enat [THEN subst])
nipkow@25134: apply (rule exI)
haftmann@27110: apply (erule (1) le_less_trans)
nipkow@25134: done
oheimb@11351: 
hoelzl@43919: instantiation enat :: "{bot, top}"
haftmann@29337: begin
haftmann@29337: 
hoelzl@43919: definition bot_enat :: enat where
hoelzl@43919:   "bot_enat = 0"
haftmann@29337: 
hoelzl@43919: definition top_enat :: enat where
hoelzl@43919:   "top_enat = \<infinity>"
haftmann@29337: 
haftmann@29337: instance proof
hoelzl@43919: qed (simp_all add: bot_enat_def top_enat_def)
haftmann@29337: 
haftmann@29337: end
haftmann@29337: 
hoelzl@43924: lemma finite_enat_bounded:
hoelzl@43924:   assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
noschinl@42993:   shows "finite A"
noschinl@42993: proof (rule finite_subset)
hoelzl@43924:   show "finite (enat ` {..n})" by blast
noschinl@42993: 
nipkow@44890:   have "A \<subseteq> {..enat n}" using le_fin by fastforce
hoelzl@43924:   also have "\<dots> \<subseteq> enat ` {..n}"
noschinl@42993:     by (rule subsetI) (case_tac x, auto)
hoelzl@43924:   finally show "A \<subseteq> enat ` {..n}" .
noschinl@42993: qed
noschinl@42993: 
huffman@26089: 
huffman@45775: subsection {* Cancellation simprocs *}
huffman@45775: 
huffman@45775: lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
huffman@45775:   unfolding plus_enat_def by (simp split: enat.split)
huffman@45775: 
huffman@45775: lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
huffman@45775:   unfolding plus_enat_def by (simp split: enat.split)
huffman@45775: 
huffman@45775: lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
huffman@45775:   unfolding plus_enat_def by (simp split: enat.split)
huffman@45775: 
huffman@45775: ML {*
huffman@45775: structure Cancel_Enat_Common =
huffman@45775: struct
huffman@45775:   (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
huffman@45775:   fun find_first_t _    _ []         = raise TERM("find_first_t", [])
huffman@45775:     | find_first_t past u (t::terms) =
huffman@45775:           if u aconv t then (rev past @ terms)
huffman@45775:           else find_first_t (t::past) u terms
huffman@45775: 
huffman@45775:   val mk_sum = Arith_Data.long_mk_sum
huffman@45775:   val dest_sum = Arith_Data.dest_sum
huffman@45775:   val find_first = find_first_t []
huffman@45775:   val trans_tac = Numeral_Simprocs.trans_tac
huffman@45775:   val norm_ss = HOL_basic_ss addsimps
huffman@45775:     @{thms add_ac semiring_numeral_0_eq_0 add_0_left add_0_right}
huffman@45775:   fun norm_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss))
huffman@45775:   fun simplify_meta_eq ss cancel_th th =
huffman@45775:     Arith_Data.simplify_meta_eq @{thms semiring_numeral_0_eq_0} ss
huffman@45775:       ([th, cancel_th] MRS trans)
huffman@45775:   fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
huffman@45775: end
huffman@45775: 
huffman@45775: structure Eq_Enat_Cancel = ExtractCommonTermFun
huffman@45775: (open Cancel_Enat_Common
huffman@45775:   val mk_bal = HOLogic.mk_eq
huffman@45775:   val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
huffman@45775:   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
huffman@45775: )
huffman@45775: 
huffman@45775: structure Le_Enat_Cancel = ExtractCommonTermFun
huffman@45775: (open Cancel_Enat_Common
huffman@45775:   val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
huffman@45775:   val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
huffman@45775:   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
huffman@45775: )
huffman@45775: 
huffman@45775: structure Less_Enat_Cancel = ExtractCommonTermFun
huffman@45775: (open Cancel_Enat_Common
huffman@45775:   val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
huffman@45775:   val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
huffman@45775:   fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
huffman@45775: )
huffman@45775: *}
huffman@45775: 
huffman@45775: simproc_setup enat_eq_cancel
huffman@45775:   ("(l::enat) + m = n" | "(l::enat) = m + n") =
huffman@45775:   {* fn phi => fn ss => fn ct => Eq_Enat_Cancel.proc ss (term_of ct) *}
huffman@45775: 
huffman@45775: simproc_setup enat_le_cancel
huffman@45775:   ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
huffman@45775:   {* fn phi => fn ss => fn ct => Le_Enat_Cancel.proc ss (term_of ct) *}
huffman@45775: 
huffman@45775: simproc_setup enat_less_cancel
huffman@45775:   ("(l::enat) + m < n" | "(l::enat) < m + n") =
huffman@45775:   {* fn phi => fn ss => fn ct => Less_Enat_Cancel.proc ss (term_of ct) *}
huffman@45775: 
huffman@45775: text {* TODO: add regression tests for these simprocs *}
huffman@45775: 
huffman@45775: text {* TODO: add simprocs for combining and cancelling numerals *}
huffman@45775: 
huffman@45775: 
haftmann@27110: subsection {* Well-ordering *}
huffman@26089: 
hoelzl@43924: lemma less_enatE:
hoelzl@43924:   "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
huffman@26089: by (induct n) auto
huffman@26089: 
huffman@44019: lemma less_infinityE:
hoelzl@43924:   "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
huffman@26089: by (induct n) auto
huffman@26089: 
hoelzl@43919: lemma enat_less_induct:
hoelzl@43919:   assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
huffman@26089: proof -
hoelzl@43924:   have P_enat: "!!k. P (enat k)"
huffman@26089:     apply (rule nat_less_induct)
huffman@26089:     apply (rule prem, clarify)
hoelzl@43924:     apply (erule less_enatE, simp)
huffman@26089:     done
huffman@26089:   show ?thesis
huffman@26089:   proof (induct n)
huffman@26089:     fix nat
hoelzl@43924:     show "P (enat nat)" by (rule P_enat)
huffman@26089:   next
hoelzl@43921:     show "P \<infinity>"
huffman@26089:       apply (rule prem, clarify)
huffman@44019:       apply (erule less_infinityE)
hoelzl@43924:       apply (simp add: P_enat)
huffman@26089:       done
huffman@26089:   qed
huffman@26089: qed
huffman@26089: 
hoelzl@43919: instance enat :: wellorder
huffman@26089: proof
haftmann@27823:   fix P and n
hoelzl@43919:   assume hyp: "(\<And>n\<Colon>enat. (\<And>m\<Colon>enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
hoelzl@43919:   show "P n" by (blast intro: enat_less_induct hyp)
huffman@26089: qed
huffman@26089: 
noschinl@42993: subsection {* Complete Lattice *}
noschinl@42993: 
hoelzl@43919: instantiation enat :: complete_lattice
noschinl@42993: begin
noschinl@42993: 
hoelzl@43919: definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
hoelzl@43919:   "inf_enat \<equiv> min"
noschinl@42993: 
hoelzl@43919: definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
hoelzl@43919:   "sup_enat \<equiv> max"
noschinl@42993: 
hoelzl@43919: definition Inf_enat :: "enat set \<Rightarrow> enat" where
hoelzl@43919:   "Inf_enat A \<equiv> if A = {} then \<infinity> else (LEAST x. x \<in> A)"
noschinl@42993: 
hoelzl@43919: definition Sup_enat :: "enat set \<Rightarrow> enat" where
hoelzl@43919:   "Sup_enat A \<equiv> if A = {} then 0
noschinl@42993:     else if finite A then Max A
noschinl@42993:                      else \<infinity>"
noschinl@42993: instance proof
hoelzl@43919:   fix x :: "enat" and A :: "enat set"
noschinl@42993:   { assume "x \<in> A" then show "Inf A \<le> x"
hoelzl@43919:       unfolding Inf_enat_def by (auto intro: Least_le) }
noschinl@42993:   { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
hoelzl@43919:       unfolding Inf_enat_def
noschinl@42993:       by (cases "A = {}") (auto intro: LeastI2_ex) }
noschinl@42993:   { assume "x \<in> A" then show "x \<le> Sup A"
hoelzl@43919:       unfolding Sup_enat_def by (cases "finite A") auto }
noschinl@42993:   { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
hoelzl@43924:       unfolding Sup_enat_def using finite_enat_bounded by auto }
hoelzl@43919: qed (simp_all add: inf_enat_def sup_enat_def)
noschinl@42993: end
noschinl@42993: 
hoelzl@43978: instance enat :: complete_linorder ..
haftmann@27110: 
haftmann@27110: subsection {* Traditional theorem names *}
haftmann@27110: 
huffman@44019: lemmas enat_defs = zero_enat_def one_enat_def number_of_enat_def eSuc_def
hoelzl@43919:   plus_enat_def less_eq_enat_def less_enat_def
haftmann@27110: 
oheimb@11351: end