src/HOL/Library/Extended_Nat.thy
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(*  Title:      HOL/Library/Extended_Nat.thy
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    Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
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    Contributions: David Trachtenherz, TU Muenchen
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*)
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section {* Extended natural numbers (i.e. with infinity) *}
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theory Extended_Nat
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imports Main Countable
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begin
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class infinity =
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  fixes infinity :: "'a"
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notation (xsymbols)
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  infinity  ("\<infinity>")
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notation (HTML output)
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  infinity  ("\<infinity>")
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subsection {* Type definition *}
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text {*
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  We extend the standard natural numbers by a special value indicating
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  infinity.
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*}
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typedef enat = "UNIV :: nat option set" ..
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text {* TODO: introduce enat as coinductive datatype, enat is just @{const of_nat} *}
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definition enat :: "nat \<Rightarrow> enat" where
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  "enat n = Abs_enat (Some n)"
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instantiation enat :: infinity
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begin
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  definition "\<infinity> = Abs_enat None"
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  instance proof qed
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end
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instance enat :: countable
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proof
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  show "\<exists>to_nat::enat \<Rightarrow> nat. inj to_nat"
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    by (rule exI[of _ "to_nat \<circ> Rep_enat"]) (simp add: inj_on_def Rep_enat_inject)
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qed
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old_rep_datatype enat "\<infinity> :: enat"
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proof -
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  fix P i assume "\<And>j. P (enat j)" "P \<infinity>"
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  then show "P i"
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  proof induct
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    case (Abs_enat y) then show ?case
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      by (cases y rule: option.exhaust)
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         (auto simp: enat_def infinity_enat_def)
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  qed
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qed (auto simp add: enat_def infinity_enat_def Abs_enat_inject)
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declare [[coercion "enat::nat\<Rightarrow>enat"]]
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lemmas enat2_cases = enat.exhaust[case_product enat.exhaust]
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lemmas enat3_cases = enat.exhaust[case_product enat.exhaust enat.exhaust]
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lemma not_infinity_eq [iff]: "(x \<noteq> \<infinity>) = (\<exists>i. x = enat i)"
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  by (cases x) auto
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lemma not_enat_eq [iff]: "(\<forall>y. x \<noteq> enat y) = (x = \<infinity>)"
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  by (cases x) auto
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primrec the_enat :: "enat \<Rightarrow> nat"
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  where "the_enat (enat n) = n"
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subsection {* Constructors and numbers *}
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instantiation enat :: "{zero, one}"
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begin
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definition
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  "0 = enat 0"
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definition
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  "1 = enat 1"
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instance ..
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end
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definition eSuc :: "enat \<Rightarrow> enat" where
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  "eSuc i = (case i of enat n \<Rightarrow> enat (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
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lemma enat_0 [code_post]: "enat 0 = 0"
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  by (simp add: zero_enat_def)
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lemma enat_1 [code_post]: "enat 1 = 1"
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  by (simp add: one_enat_def)
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lemma enat_0_iff: "enat x = 0 \<longleftrightarrow> x = 0" "0 = enat x \<longleftrightarrow> x = 0"
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  by (auto simp add: zero_enat_def)
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lemma enat_1_iff: "enat x = 1 \<longleftrightarrow> x = 1" "1 = enat x \<longleftrightarrow> x = 1"
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  by (auto simp add: one_enat_def)
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lemma one_eSuc: "1 = eSuc 0"
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  by (simp add: zero_enat_def one_enat_def eSuc_def)
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lemma infinity_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0"
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  by (simp add: zero_enat_def)
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lemma i0_ne_infinity [simp]: "0 \<noteq> (\<infinity>::enat)"
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  by (simp add: zero_enat_def)
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lemma zero_one_enat_neq [simp]:
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  "\<not> 0 = (1\<Colon>enat)"
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  "\<not> 1 = (0\<Colon>enat)"
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  unfolding zero_enat_def one_enat_def by simp_all
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lemma infinity_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1"
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  by (simp add: one_enat_def)
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lemma i1_ne_infinity [simp]: "1 \<noteq> (\<infinity>::enat)"
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  by (simp add: one_enat_def)
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lemma eSuc_enat: "eSuc (enat n) = enat (Suc n)"
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  by (simp add: eSuc_def)
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lemma eSuc_infinity [simp]: "eSuc \<infinity> = \<infinity>"
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  by (simp add: eSuc_def)
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lemma eSuc_ne_0 [simp]: "eSuc n \<noteq> 0"
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  by (simp add: eSuc_def zero_enat_def split: enat.splits)
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lemma zero_ne_eSuc [simp]: "0 \<noteq> eSuc n"
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  by (rule eSuc_ne_0 [symmetric])
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lemma eSuc_inject [simp]: "eSuc m = eSuc n \<longleftrightarrow> m = n"
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  by (simp add: eSuc_def split: enat.splits)
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subsection {* Addition *}
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instantiation enat :: comm_monoid_add
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begin
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definition [nitpick_simp]:
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  "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | enat m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | enat n \<Rightarrow> enat (m + n)))"
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lemma plus_enat_simps [simp, code]:
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  fixes q :: enat
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  shows "enat m + enat n = enat (m + n)"
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    and "\<infinity> + q = \<infinity>"
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    and "q + \<infinity> = \<infinity>"
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  by (simp_all add: plus_enat_def split: enat.splits)
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instance proof
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  fix n m q :: enat
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  show "n + m + q = n + (m + q)"
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    by (cases n m q rule: enat3_cases) auto
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  show "n + m = m + n"
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    by (cases n m rule: enat2_cases) auto
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  show "0 + n = n"
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    by (cases n) (simp_all add: zero_enat_def)
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qed
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end
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lemma eSuc_plus_1:
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  "eSuc n = n + 1"
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  by (cases n) (simp_all add: eSuc_enat one_enat_def)
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lemma plus_1_eSuc:
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  "1 + q = eSuc q"
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  "q + 1 = eSuc q"
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  by (simp_all add: eSuc_plus_1 ac_simps)
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lemma iadd_Suc: "eSuc m + n = eSuc (m + n)"
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  by (simp_all add: eSuc_plus_1 ac_simps)
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lemma iadd_Suc_right: "m + eSuc n = eSuc (m + n)"
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  by (simp only: add.commute[of m] iadd_Suc)
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lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)"
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  by (cases m, cases n, simp_all add: zero_enat_def)
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subsection {* Multiplication *}
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instantiation enat :: comm_semiring_1
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begin
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definition times_enat_def [nitpick_simp]:
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  "m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | enat m \<Rightarrow>
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    (case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | enat n \<Rightarrow> enat (m * n)))"
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lemma times_enat_simps [simp, code]:
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  "enat m * enat n = enat (m * n)"
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  "\<infinity> * \<infinity> = (\<infinity>::enat)"
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  "\<infinity> * enat n = (if n = 0 then 0 else \<infinity>)"
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  "enat m * \<infinity> = (if m = 0 then 0 else \<infinity>)"
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  unfolding times_enat_def zero_enat_def
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  by (simp_all split: enat.split)
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instance proof
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  fix a b c :: enat
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  show "(a * b) * c = a * (b * c)"
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    unfolding times_enat_def zero_enat_def
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    by (simp split: enat.split)
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  show "a * b = b * a"
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    unfolding times_enat_def zero_enat_def
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    by (simp split: enat.split)
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  show "1 * a = a"
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    unfolding times_enat_def zero_enat_def one_enat_def
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    by (simp split: enat.split)
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  show "(a + b) * c = a * c + b * c"
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    unfolding times_enat_def zero_enat_def
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    by (simp split: enat.split add: distrib_right)
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  show "0 * a = 0"
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    unfolding times_enat_def zero_enat_def
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    by (simp split: enat.split)
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  show "a * 0 = 0"
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    unfolding times_enat_def zero_enat_def
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    by (simp split: enat.split)
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  show "(0::enat) \<noteq> 1"
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    unfolding zero_enat_def one_enat_def
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    by simp
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qed
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end
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lemma mult_eSuc: "eSuc m * n = n + m * n"
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  unfolding eSuc_plus_1 by (simp add: algebra_simps)
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lemma mult_eSuc_right: "m * eSuc n = m + m * n"
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  unfolding eSuc_plus_1 by (simp add: algebra_simps)
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lemma of_nat_eq_enat: "of_nat n = enat n"
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  apply (induct n)
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  apply (simp add: enat_0)
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  apply (simp add: plus_1_eSuc eSuc_enat)
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  done
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instance enat :: semiring_char_0 proof
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  have "inj enat" by (rule injI) simp
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  then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_enat)
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qed
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lemma imult_is_0 [simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)"
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  by (auto simp add: times_enat_def zero_enat_def split: enat.split)
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lemma imult_is_infinity: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)"
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  by (auto simp add: times_enat_def zero_enat_def split: enat.split)
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subsection {* Numerals *}
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lemma numeral_eq_enat:
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  "numeral k = enat (numeral k)"
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  using of_nat_eq_enat [of "numeral k"] by simp
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lemma enat_numeral [code_abbrev]:
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  "enat (numeral k) = numeral k"
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  using numeral_eq_enat ..
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lemma infinity_ne_numeral [simp]: "(\<infinity>::enat) \<noteq> numeral k"
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  by (simp add: numeral_eq_enat)
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lemma numeral_ne_infinity [simp]: "numeral k \<noteq> (\<infinity>::enat)"
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  by (simp add: numeral_eq_enat)
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lemma eSuc_numeral [simp]: "eSuc (numeral k) = numeral (k + Num.One)"
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  by (simp only: eSuc_plus_1 numeral_plus_one)
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subsection {* Subtraction *}
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instantiation enat :: minus
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begin
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definition diff_enat_def:
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"a - b = (case a of (enat x) \<Rightarrow> (case b of (enat y) \<Rightarrow> enat (x - y) | \<infinity> \<Rightarrow> 0)
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          | \<infinity> \<Rightarrow> \<infinity>)"
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instance ..
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end
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lemma idiff_enat_enat [simp, code]: "enat a - enat b = enat (a - b)"
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  by (simp add: diff_enat_def)
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lemma idiff_infinity [simp, code]: "\<infinity> - n = (\<infinity>::enat)"
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  by (simp add: diff_enat_def)
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lemma idiff_infinity_right [simp, code]: "enat a - \<infinity> = 0"
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  by (simp add: diff_enat_def)
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lemma idiff_0 [simp]: "(0::enat) - n = 0"
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  by (cases n, simp_all add: zero_enat_def)
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lemmas idiff_enat_0 [simp] = idiff_0 [unfolded zero_enat_def]
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lemma idiff_0_right [simp]: "(n::enat) - 0 = n"
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  by (cases n) (simp_all add: zero_enat_def)
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lemmas idiff_enat_0_right [simp] = idiff_0_right [unfolded zero_enat_def]
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lemma idiff_self [simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0"
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  by (auto simp: zero_enat_def)
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   304
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lemma eSuc_minus_eSuc [simp]: "eSuc n - eSuc m = n - m"
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  by (simp add: eSuc_def split: enat.split)
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   307
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lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
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   309
  by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
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   310
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(*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
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subsection {* Ordering *}
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instantiation enat :: linordered_ab_semigroup_add
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begin
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38167
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definition [nitpick_simp]:
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  "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
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    | \<infinity> \<Rightarrow> True)"
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definition [nitpick_simp]:
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  "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
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    | \<infinity> \<Rightarrow> False)"
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lemma enat_ord_simps [simp]:
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  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
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   328
  "enat m < enat n \<longleftrightarrow> m < n"
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   329
  "q \<le> (\<infinity>::enat)"
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   330
  "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
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   331
  "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
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   332
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
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   333
  by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
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parents:
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   334
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lemma numeral_le_enat_iff[simp]:
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  shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
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   337
by (auto simp: numeral_eq_enat)
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diff changeset
   338
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lemma numeral_less_enat_iff[simp]:
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  shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
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   341
by (auto simp: numeral_eq_enat)
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parents: 45775
diff changeset
   342
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   343
lemma enat_ord_code [code]:
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   344
  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
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   345
  "enat m < enat n \<longleftrightarrow> m < n"
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   346
  "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
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   347
  "enat m < \<infinity> \<longleftrightarrow> True"
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   348
  "\<infinity> \<le> enat n \<longleftrightarrow> False"
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   349
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
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  by simp_all
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parents:
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instance by default
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   353
  (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
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parents:
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   354
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end
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   356
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   357
instance enat :: ordered_comm_semiring
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   358
proof
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parents: 43532
diff changeset
   359
  fix a b c :: enat
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parents: 29012
diff changeset
   360
  assume "a \<le> b" and "0 \<le> c"
e515f42d1db7 multiplication for type inat
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parents: 29012
diff changeset
   361
  thus "c * a \<le> c * b"
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parents: 43532
diff changeset
   362
    unfolding times_enat_def less_eq_enat_def zero_enat_def
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
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parents: 43532
diff changeset
   363
    by (simp split: enat.splits)
29014
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parents: 29012
diff changeset
   364
qed
e515f42d1db7 multiplication for type inat
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parents: 29012
diff changeset
   365
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   366
(* BH: These equations are already proven generally for any type in
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diff changeset
   367
class linordered_semidom. However, enat is not in that class because
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45934
diff changeset
   368
it does not have the cancellation property. Would it be worthwhile to
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45934
diff changeset
   369
a generalize linordered_semidom to a new class that includes enat? *)
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diff changeset
   370
43919
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diff changeset
   371
lemma enat_ord_number [simp]:
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diff changeset
   372
  "(numeral m \<Colon> enat) \<le> numeral n \<longleftrightarrow> (numeral m \<Colon> nat) \<le> numeral n"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45934
diff changeset
   373
  "(numeral m \<Colon> enat) < numeral n \<longleftrightarrow> (numeral m \<Colon> nat) < numeral n"
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45934
diff changeset
   374
  by (simp_all add: numeral_eq_enat)
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oheimb
parents:
diff changeset
   375
43919
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parents: 43532
diff changeset
   376
lemma i0_lb [simp]: "(0\<Colon>enat) \<le> n"
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   377
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
11351
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oheimb
parents:
diff changeset
   378
43919
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hoelzl
parents: 43532
diff changeset
   379
lemma ile0_eq [simp]: "n \<le> (0\<Colon>enat) \<longleftrightarrow> n = 0"
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   380
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   381
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   382
lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   383
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   384
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   385
lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   386
  by simp
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   387
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   388
lemma not_iless0 [simp]: "\<not> n < (0\<Colon>enat)"
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   389
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   390
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   391
lemma i0_less [simp]: "(0\<Colon>enat) < n \<longleftrightarrow> n \<noteq> 0"
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   392
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   393
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   394
lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   395
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   396
 
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   397
lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   398
  by (simp add: eSuc_def less_enat_def split: enat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   399
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   400
lemma ile_eSuc [simp]: "n \<le> eSuc n"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   401
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   402
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   403
lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   404
  by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   405
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   406
lemma i0_iless_eSuc [simp]: "0 < eSuc n"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   407
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   408
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   409
lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   410
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
41853
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   411
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   412
lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   413
  by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   414
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   415
lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   416
  by (cases n) auto
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   417
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   418
lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   419
  by (auto simp add: eSuc_def less_enat_def split: enat.splits)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   420
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   421
lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   422
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
41853
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   423
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   424
lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   425
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
41853
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   426
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   427
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   428
  by (simp only: i0_less imult_is_0, simp)
41853
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   429
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   430
lemma mono_eSuc: "mono eSuc"
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   431
  by (simp add: mono_def)
41853
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   432
258a489c24b2 Added material by David Trachtenherz
nipkow
parents: 38621
diff changeset
   433
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   434
lemma min_enat_simps [simp]:
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   435
  "min (enat m) (enat n) = enat (min m n)"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   436
  "min q 0 = 0"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   437
  "min 0 q = 0"
43921
e8511be08ddd Introduce infinity type class
hoelzl
parents: 43919
diff changeset
   438
  "min q (\<infinity>::enat) = q"
e8511be08ddd Introduce infinity type class
hoelzl
parents: 43919
diff changeset
   439
  "min (\<infinity>::enat) q = q"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   440
  by (auto simp add: min_def)
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   441
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   442
lemma max_enat_simps [simp]:
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   443
  "max (enat m) (enat n) = enat (max m n)"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   444
  "max q 0 = q"
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   445
  "max 0 q = q"
43921
e8511be08ddd Introduce infinity type class
hoelzl
parents: 43919
diff changeset
   446
  "max q \<infinity> = (\<infinity>::enat)"
e8511be08ddd Introduce infinity type class
hoelzl
parents: 43919
diff changeset
   447
  "max \<infinity> q = (\<infinity>::enat)"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   448
  by (simp_all add: max_def)
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   449
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   450
lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   451
  by (cases n) simp_all
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   452
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   453
lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   454
  by (cases n) simp_all
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   455
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   456
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   457
apply (induct_tac k)
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   458
 apply (simp (no_asm) only: enat_0)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   459
 apply (fast intro: le_less_trans [OF i0_lb])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   460
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   461
apply (drule spec)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   462
apply (erule exE)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   463
apply (drule ileI1)
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   464
apply (rule eSuc_enat [THEN subst])
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   465
apply (rule exI)
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   466
apply (erule (1) le_less_trans)
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25112
diff changeset
   467
done
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   468
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 51717
diff changeset
   469
instantiation enat :: "{order_bot, order_top}"
29337
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   470
begin
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   471
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   472
definition bot_enat :: enat where
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   473
  "bot_enat = 0"
29337
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   474
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   475
definition top_enat :: enat where
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   476
  "top_enat = \<infinity>"
29337
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   477
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   478
instance proof
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   479
qed (simp_all add: bot_enat_def top_enat_def)
29337
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   480
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   481
end
450805a4a91f added instance for bot, top
haftmann
parents: 29023
diff changeset
   482
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   483
lemma finite_enat_bounded:
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   484
  assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   485
  shows "finite A"
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   486
proof (rule finite_subset)
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   487
  show "finite (enat ` {..n})" by blast
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   488
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44019
diff changeset
   489
  have "A \<subseteq> {..enat n}" using le_fin by fastforce
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   490
  also have "\<dots> \<subseteq> enat ` {..n}"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   491
    by (rule subsetI) (case_tac x, auto)
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   492
  finally show "A \<subseteq> enat ` {..n}" .
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   493
qed
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   494
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   495
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   496
subsection {* Cancellation simprocs *}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   497
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   498
lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   499
  unfolding plus_enat_def by (simp split: enat.split)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   500
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   501
lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   502
  unfolding plus_enat_def by (simp split: enat.split)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   503
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   504
lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   505
  unfolding plus_enat_def by (simp split: enat.split)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   506
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   507
ML {*
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   508
structure Cancel_Enat_Common =
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   509
struct
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   510
  (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   511
  fun find_first_t _    _ []         = raise TERM("find_first_t", [])
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   512
    | find_first_t past u (t::terms) =
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   513
          if u aconv t then (rev past @ terms)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   514
          else find_first_t (t::past) u terms
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   515
51366
abdcf1a7cabf avoid using Arith_Data.dest_sum in extended-nat simprocs (it treats 'x - y' as 'x + - y', which is not valid for enat)
huffman
parents: 51301
diff changeset
   516
  fun dest_summing (Const (@{const_name Groups.plus}, _) $ t $ u, ts) =
abdcf1a7cabf avoid using Arith_Data.dest_sum in extended-nat simprocs (it treats 'x - y' as 'x + - y', which is not valid for enat)
huffman
parents: 51301
diff changeset
   517
        dest_summing (t, dest_summing (u, ts))
abdcf1a7cabf avoid using Arith_Data.dest_sum in extended-nat simprocs (it treats 'x - y' as 'x + - y', which is not valid for enat)
huffman
parents: 51301
diff changeset
   518
    | dest_summing (t, ts) = t :: ts
abdcf1a7cabf avoid using Arith_Data.dest_sum in extended-nat simprocs (it treats 'x - y' as 'x + - y', which is not valid for enat)
huffman
parents: 51301
diff changeset
   519
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   520
  val mk_sum = Arith_Data.long_mk_sum
51366
abdcf1a7cabf avoid using Arith_Data.dest_sum in extended-nat simprocs (it treats 'x - y' as 'x + - y', which is not valid for enat)
huffman
parents: 51301
diff changeset
   521
  fun dest_sum t = dest_summing (t, [])
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   522
  val find_first = find_first_t []
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   523
  val trans_tac = Numeral_Simprocs.trans_tac
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   524
  val norm_ss =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   525
    simpset_of (put_simpset HOL_basic_ss @{context}
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
   526
      addsimps @{thms ac_simps add_0_left add_0_right})
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   527
  fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   528
  fun simplify_meta_eq ctxt cancel_th th =
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   529
    Arith_Data.simplify_meta_eq [] ctxt
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   530
      ([th, cancel_th] MRS trans)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   531
  fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   532
end
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   533
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   534
structure Eq_Enat_Cancel = ExtractCommonTermFun
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   535
(open Cancel_Enat_Common
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   536
  val mk_bal = HOLogic.mk_eq
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   537
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   538
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   539
)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   540
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   541
structure Le_Enat_Cancel = ExtractCommonTermFun
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   542
(open Cancel_Enat_Common
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   543
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   544
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   545
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   546
)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   547
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   548
structure Less_Enat_Cancel = ExtractCommonTermFun
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   549
(open Cancel_Enat_Common
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   550
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   551
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   552
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   553
)
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   554
*}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   555
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   556
simproc_setup enat_eq_cancel
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   557
  ("(l::enat) + m = n" | "(l::enat) = m + n") =
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   558
  {* fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (term_of ct) *}
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   559
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   560
simproc_setup enat_le_cancel
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   561
  ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   562
  {* fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (term_of ct) *}
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   563
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   564
simproc_setup enat_less_cancel
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   565
  ("(l::enat) + m < n" | "(l::enat) < m + n") =
51717
9e7d1c139569 simplifier uses proper Proof.context instead of historic type simpset;
wenzelm
parents: 51366
diff changeset
   566
  {* fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (term_of ct) *}
45775
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   567
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   568
text {* TODO: add regression tests for these simprocs *}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   569
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   570
text {* TODO: add simprocs for combining and cancelling numerals *}
6c340de26a0d add cancellation simprocs for type enat
huffman
parents: 45539
diff changeset
   571
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   572
subsection {* Well-ordering *}
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   573
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   574
lemma less_enatE:
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   575
  "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   576
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   577
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   578
lemma less_infinityE:
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   579
  "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   580
by (induct n) auto
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   581
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   582
lemma enat_less_induct:
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   583
  assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   584
proof -
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   585
  have P_enat: "!!k. P (enat k)"
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   586
    apply (rule nat_less_induct)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   587
    apply (rule prem, clarify)
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   588
    apply (erule less_enatE, simp)
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   589
    done
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   590
  show ?thesis
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   591
  proof (induct n)
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   592
    fix nat
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   593
    show "P (enat nat)" by (rule P_enat)
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   594
  next
43921
e8511be08ddd Introduce infinity type class
hoelzl
parents: 43919
diff changeset
   595
    show "P \<infinity>"
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   596
      apply (rule prem, clarify)
44019
ee784502aed5 Extended_Nat.thy: renamed iSuc to eSuc, standardized theorem names
huffman
parents: 43978
diff changeset
   597
      apply (erule less_infinityE)
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   598
      apply (simp add: P_enat)
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   599
      done
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   600
  qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   601
qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   602
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   603
instance enat :: wellorder
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   604
proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27487
diff changeset
   605
  fix P and n
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   606
  assume hyp: "(\<And>n\<Colon>enat. (\<And>m\<Colon>enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   607
  show "P n" by (blast intro: enat_less_induct hyp)
26089
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   608
qed
373221497340 proved linorder and wellorder class instances
huffman
parents: 25691
diff changeset
   609
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   610
subsection {* Complete Lattice *}
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   611
54415
eaf25431d4c4 enat is countable
hoelzl
parents: 52729
diff changeset
   612
text {* TODO: enat as order topology? *}
eaf25431d4c4 enat is countable
hoelzl
parents: 52729
diff changeset
   613
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   614
instantiation enat :: complete_lattice
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   615
begin
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   616
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   617
definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
56777
wenzelm
parents: 54419
diff changeset
   618
  "inf_enat = min"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   619
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   620
definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
56777
wenzelm
parents: 54419
diff changeset
   621
  "sup_enat = max"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   622
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   623
definition Inf_enat :: "enat set \<Rightarrow> enat" where
56777
wenzelm
parents: 54419
diff changeset
   624
  "Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   625
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   626
definition Sup_enat :: "enat set \<Rightarrow> enat" where
56777
wenzelm
parents: 54419
diff changeset
   627
  "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
wenzelm
parents: 54419
diff changeset
   628
instance
wenzelm
parents: 54419
diff changeset
   629
proof
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   630
  fix x :: "enat" and A :: "enat set"
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   631
  { assume "x \<in> A" then show "Inf A \<le> x"
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   632
      unfolding Inf_enat_def by (auto intro: Least_le) }
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   633
  { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   634
      unfolding Inf_enat_def
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   635
      by (cases "A = {}") (auto intro: LeastI2_ex) }
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   636
  { assume "x \<in> A" then show "x \<le> Sup A"
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   637
      unfolding Sup_enat_def by (cases "finite A") auto }
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   638
  { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
43924
1165fe965da8 rename Fin to enat
hoelzl
parents: 43923
diff changeset
   639
      unfolding Sup_enat_def using finite_enat_bounded by auto }
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 51717
diff changeset
   640
qed (simp_all add:
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 51717
diff changeset
   641
 inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
42993
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   642
end
da014b00d7a4 instance inat for complete_lattice
noschinl
parents: 41855
diff changeset
   643
43978
da7d04d4023c enat is a complete_linorder instance
hoelzl
parents: 43924
diff changeset
   644
instance enat :: complete_linorder ..
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   645
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   646
subsection {* Traditional theorem names *}
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   647
47108
2a1953f0d20d merged fork with new numeral representation (see NEWS)
huffman
parents: 45934
diff changeset
   648
lemmas enat_defs = zero_enat_def one_enat_def eSuc_def
43919
a7e4fb1a0502 rename Nat_Infinity (inat) to Extended_Nat (enat)
hoelzl
parents: 43532
diff changeset
   649
  plus_enat_def less_eq_enat_def less_enat_def
27110
194aa674c2a1 refactoring; addition, numerals
haftmann
parents: 26089
diff changeset
   650
11351
c5c403d30c77 added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
parents:
diff changeset
   651
end