src/HOL/Library/Extended_Nat.thy
author wenzelm
Tue Sep 01 22:32:58 2015 +0200 (2015-09-01)
changeset 61076 bdc1e2f0a86a
parent 60679 ade12ef2773c
child 61378 3e04c9ca001a
permissions -rw-r--r--
eliminated \<Colon>;
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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 \<open>Extended natural numbers (i.e. with infinity)\<close>
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theory Extended_Nat
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imports Main Countable Order_Continuity
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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 \<open>Type definition\<close>
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text \<open>
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  We extend the standard natural numbers by a special value indicating
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  infinity.
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\<close>
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typedef enat = "UNIV :: nat option set" ..
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text \<open>TODO: introduce enat as coinductive datatype, enat is just @{const of_nat}\<close>
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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 ..
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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 \<open>Constructors and numbers\<close>
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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::enat)"
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  "\<not> 1 = (0::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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lemma eSuc_enat_iff: "eSuc x = enat y \<longleftrightarrow> (\<exists>n. y = Suc n \<and> x = enat n)"
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  by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
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lemma enat_eSuc_iff: "enat y = eSuc x \<longleftrightarrow> (\<exists>n. y = Suc n \<and> enat n = x)"
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  by (cases y) (auto simp: enat_0 eSuc_enat[symmetric])
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subsection \<open>Addition\<close>
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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
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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 \<open>Multiplication\<close>
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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
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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
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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 \<open>Numerals\<close>
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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 \<open>Subtraction\<close>
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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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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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lemma eSuc_minus_1 [simp]: "eSuc n - 1 = n"
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  by (simp add: one_enat_def eSuc_enat[symmetric] zero_enat_def[symmetric])
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(*lemmas idiff_self_eq_0_enat = idiff_self_eq_0[unfolded zero_enat_def]*)
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subsection \<open>Ordering\<close>
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instantiation enat :: linordered_ab_semigroup_add
haftmann@27110
   327
begin
oheimb@11351
   328
blanchet@38167
   329
definition [nitpick_simp]:
hoelzl@43924
   330
  "m \<le> n = (case n of enat n1 \<Rightarrow> (case m of enat m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
haftmann@27110
   331
    | \<infinity> \<Rightarrow> True)"
oheimb@11351
   332
blanchet@38167
   333
definition [nitpick_simp]:
hoelzl@43924
   334
  "m < n = (case m of enat m1 \<Rightarrow> (case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
haftmann@27110
   335
    | \<infinity> \<Rightarrow> False)"
oheimb@11351
   336
hoelzl@43919
   337
lemma enat_ord_simps [simp]:
hoelzl@43924
   338
  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924
   339
  "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921
   340
  "q \<le> (\<infinity>::enat)"
hoelzl@43921
   341
  "q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>"
hoelzl@43921
   342
  "(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>"
hoelzl@43921
   343
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
hoelzl@43919
   344
  by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits)
oheimb@11351
   345
huffman@47108
   346
lemma numeral_le_enat_iff[simp]:
huffman@47108
   347
  shows "numeral m \<le> enat n \<longleftrightarrow> numeral m \<le> n"
huffman@47108
   348
by (auto simp: numeral_eq_enat)
noschinl@45934
   349
huffman@47108
   350
lemma numeral_less_enat_iff[simp]:
huffman@47108
   351
  shows "numeral m < enat n \<longleftrightarrow> numeral m < n"
huffman@47108
   352
by (auto simp: numeral_eq_enat)
noschinl@45934
   353
hoelzl@43919
   354
lemma enat_ord_code [code]:
hoelzl@43924
   355
  "enat m \<le> enat n \<longleftrightarrow> m \<le> n"
hoelzl@43924
   356
  "enat m < enat n \<longleftrightarrow> m < n"
hoelzl@43921
   357
  "q \<le> (\<infinity>::enat) \<longleftrightarrow> True"
hoelzl@43924
   358
  "enat m < \<infinity> \<longleftrightarrow> True"
hoelzl@43924
   359
  "\<infinity> \<le> enat n \<longleftrightarrow> False"
hoelzl@43921
   360
  "(\<infinity>::enat) < q \<longleftrightarrow> False"
haftmann@27110
   361
  by simp_all
oheimb@11351
   362
wenzelm@60679
   363
instance
wenzelm@60679
   364
  by standard (auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits)
oheimb@11351
   365
haftmann@27110
   366
end
haftmann@27110
   367
hoelzl@43919
   368
instance enat :: ordered_comm_semiring
huffman@29014
   369
proof
hoelzl@43919
   370
  fix a b c :: enat
huffman@29014
   371
  assume "a \<le> b" and "0 \<le> c"
huffman@29014
   372
  thus "c * a \<le> c * b"
hoelzl@43919
   373
    unfolding times_enat_def less_eq_enat_def zero_enat_def
hoelzl@43919
   374
    by (simp split: enat.splits)
huffman@29014
   375
qed
huffman@29014
   376
huffman@47108
   377
(* BH: These equations are already proven generally for any type in
huffman@47108
   378
class linordered_semidom. However, enat is not in that class because
huffman@47108
   379
it does not have the cancellation property. Would it be worthwhile to
huffman@47108
   380
a generalize linordered_semidom to a new class that includes enat? *)
huffman@47108
   381
hoelzl@43919
   382
lemma enat_ord_number [simp]:
wenzelm@61076
   383
  "(numeral m :: enat) \<le> numeral n \<longleftrightarrow> (numeral m :: nat) \<le> numeral n"
wenzelm@61076
   384
  "(numeral m :: enat) < numeral n \<longleftrightarrow> (numeral m :: nat) < numeral n"
huffman@47108
   385
  by (simp_all add: numeral_eq_enat)
oheimb@11351
   386
wenzelm@61076
   387
lemma i0_lb [simp]: "(0::enat) \<le> n"
hoelzl@43919
   388
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
oheimb@11351
   389
wenzelm@61076
   390
lemma ile0_eq [simp]: "n \<le> (0::enat) \<longleftrightarrow> n = 0"
hoelzl@43919
   391
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
oheimb@11351
   392
huffman@44019
   393
lemma infinity_ileE [elim!]: "\<infinity> \<le> enat m \<Longrightarrow> R"
huffman@44019
   394
  by (simp add: zero_enat_def less_eq_enat_def split: enat.splits)
huffman@44019
   395
huffman@44019
   396
lemma infinity_ilessE [elim!]: "\<infinity> < enat m \<Longrightarrow> R"
haftmann@27110
   397
  by simp
oheimb@11351
   398
wenzelm@61076
   399
lemma not_iless0 [simp]: "\<not> n < (0::enat)"
hoelzl@43919
   400
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
haftmann@27110
   401
wenzelm@61076
   402
lemma i0_less [simp]: "(0::enat) < n \<longleftrightarrow> n \<noteq> 0"
huffman@44019
   403
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
oheimb@11351
   404
huffman@44019
   405
lemma eSuc_ile_mono [simp]: "eSuc n \<le> eSuc m \<longleftrightarrow> n \<le> m"
huffman@44019
   406
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
haftmann@27110
   407
 
huffman@44019
   408
lemma eSuc_mono [simp]: "eSuc n < eSuc m \<longleftrightarrow> n < m"
huffman@44019
   409
  by (simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351
   410
huffman@44019
   411
lemma ile_eSuc [simp]: "n \<le> eSuc n"
huffman@44019
   412
  by (simp add: eSuc_def less_eq_enat_def split: enat.splits)
oheimb@11351
   413
huffman@44019
   414
lemma not_eSuc_ilei0 [simp]: "\<not> eSuc n \<le> 0"
huffman@44019
   415
  by (simp add: zero_enat_def eSuc_def less_eq_enat_def split: enat.splits)
haftmann@27110
   416
huffman@44019
   417
lemma i0_iless_eSuc [simp]: "0 < eSuc n"
huffman@44019
   418
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.splits)
haftmann@27110
   419
huffman@44019
   420
lemma iless_eSuc0[simp]: "(n < eSuc 0) = (n = 0)"
huffman@44019
   421
  by (simp add: zero_enat_def eSuc_def less_enat_def split: enat.split)
nipkow@41853
   422
huffman@44019
   423
lemma ileI1: "m < n \<Longrightarrow> eSuc m \<le> n"
huffman@44019
   424
  by (simp add: eSuc_def less_eq_enat_def less_enat_def split: enat.splits)
haftmann@27110
   425
hoelzl@43924
   426
lemma Suc_ile_eq: "enat (Suc m) \<le> n \<longleftrightarrow> enat m < n"
haftmann@27110
   427
  by (cases n) auto
haftmann@27110
   428
huffman@44019
   429
lemma iless_Suc_eq [simp]: "enat m < eSuc n \<longleftrightarrow> enat m \<le> n"
huffman@44019
   430
  by (auto simp add: eSuc_def less_enat_def split: enat.splits)
oheimb@11351
   431
huffman@44019
   432
lemma imult_infinity: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>"
huffman@44019
   433
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853
   434
huffman@44019
   435
lemma imult_infinity_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>"
huffman@44019
   436
  by (simp add: zero_enat_def less_enat_def split: enat.splits)
nipkow@41853
   437
hoelzl@43919
   438
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)"
huffman@44019
   439
  by (simp only: i0_less imult_is_0, simp)
nipkow@41853
   440
huffman@44019
   441
lemma mono_eSuc: "mono eSuc"
huffman@44019
   442
  by (simp add: mono_def)
nipkow@41853
   443
nipkow@41853
   444
hoelzl@43919
   445
lemma min_enat_simps [simp]:
hoelzl@43924
   446
  "min (enat m) (enat n) = enat (min m n)"
haftmann@27110
   447
  "min q 0 = 0"
haftmann@27110
   448
  "min 0 q = 0"
hoelzl@43921
   449
  "min q (\<infinity>::enat) = q"
hoelzl@43921
   450
  "min (\<infinity>::enat) q = q"
haftmann@27110
   451
  by (auto simp add: min_def)
oheimb@11351
   452
hoelzl@43919
   453
lemma max_enat_simps [simp]:
hoelzl@43924
   454
  "max (enat m) (enat n) = enat (max m n)"
haftmann@27110
   455
  "max q 0 = q"
haftmann@27110
   456
  "max 0 q = q"
hoelzl@43921
   457
  "max q \<infinity> = (\<infinity>::enat)"
hoelzl@43921
   458
  "max \<infinity> q = (\<infinity>::enat)"
haftmann@27110
   459
  by (simp_all add: max_def)
haftmann@27110
   460
hoelzl@43924
   461
lemma enat_ile: "n \<le> enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110
   462
  by (cases n) simp_all
haftmann@27110
   463
hoelzl@43924
   464
lemma enat_iless: "n < enat m \<Longrightarrow> \<exists>k. n = enat k"
haftmann@27110
   465
  by (cases n) simp_all
oheimb@11351
   466
hoelzl@43924
   467
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. enat k < Y j"
nipkow@25134
   468
apply (induct_tac k)
hoelzl@43924
   469
 apply (simp (no_asm) only: enat_0)
haftmann@27110
   470
 apply (fast intro: le_less_trans [OF i0_lb])
nipkow@25134
   471
apply (erule exE)
nipkow@25134
   472
apply (drule spec)
nipkow@25134
   473
apply (erule exE)
nipkow@25134
   474
apply (drule ileI1)
huffman@44019
   475
apply (rule eSuc_enat [THEN subst])
nipkow@25134
   476
apply (rule exI)
haftmann@27110
   477
apply (erule (1) le_less_trans)
nipkow@25134
   478
done
oheimb@11351
   479
hoelzl@60636
   480
lemma eSuc_max: "eSuc (max x y) = max (eSuc x) (eSuc y)"
hoelzl@60636
   481
  by (simp add: eSuc_def split: enat.split)
hoelzl@60636
   482
hoelzl@60636
   483
lemma eSuc_Max: 
hoelzl@60636
   484
  assumes "finite A" "A \<noteq> {}"
hoelzl@60636
   485
  shows "eSuc (Max A) = Max (eSuc ` A)"
hoelzl@60636
   486
using assms proof induction
hoelzl@60636
   487
  case (insert x A)
hoelzl@60636
   488
  thus ?case by(cases "A = {}")(simp_all add: eSuc_max)
hoelzl@60636
   489
qed simp
hoelzl@60636
   490
haftmann@52729
   491
instantiation enat :: "{order_bot, order_top}"
haftmann@29337
   492
begin
haftmann@29337
   493
wenzelm@60679
   494
definition bot_enat :: enat where "bot_enat = 0"
wenzelm@60679
   495
definition top_enat :: enat where "top_enat = \<infinity>"
haftmann@29337
   496
wenzelm@60679
   497
instance
wenzelm@60679
   498
  by standard (simp_all add: bot_enat_def top_enat_def)
haftmann@29337
   499
haftmann@29337
   500
end
haftmann@29337
   501
hoelzl@43924
   502
lemma finite_enat_bounded:
hoelzl@43924
   503
  assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> enat n"
noschinl@42993
   504
  shows "finite A"
noschinl@42993
   505
proof (rule finite_subset)
hoelzl@43924
   506
  show "finite (enat ` {..n})" by blast
nipkow@44890
   507
  have "A \<subseteq> {..enat n}" using le_fin by fastforce
hoelzl@43924
   508
  also have "\<dots> \<subseteq> enat ` {..n}"
wenzelm@60679
   509
    apply (rule subsetI)
wenzelm@60679
   510
    subgoal for x by (cases x) auto
wenzelm@60679
   511
    done
hoelzl@43924
   512
  finally show "A \<subseteq> enat ` {..n}" .
noschinl@42993
   513
qed
noschinl@42993
   514
huffman@26089
   515
wenzelm@60500
   516
subsection \<open>Cancellation simprocs\<close>
huffman@45775
   517
huffman@45775
   518
lemma enat_add_left_cancel: "a + b = a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b = c"
huffman@45775
   519
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   520
huffman@45775
   521
lemma enat_add_left_cancel_le: "a + b \<le> a + c \<longleftrightarrow> a = (\<infinity>::enat) \<or> b \<le> c"
huffman@45775
   522
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   523
huffman@45775
   524
lemma enat_add_left_cancel_less: "a + b < a + c \<longleftrightarrow> a \<noteq> (\<infinity>::enat) \<and> b < c"
huffman@45775
   525
  unfolding plus_enat_def by (simp split: enat.split)
huffman@45775
   526
wenzelm@60500
   527
ML \<open>
huffman@45775
   528
structure Cancel_Enat_Common =
huffman@45775
   529
struct
huffman@45775
   530
  (* copied from src/HOL/Tools/nat_numeral_simprocs.ML *)
huffman@45775
   531
  fun find_first_t _    _ []         = raise TERM("find_first_t", [])
huffman@45775
   532
    | find_first_t past u (t::terms) =
huffman@45775
   533
          if u aconv t then (rev past @ terms)
huffman@45775
   534
          else find_first_t (t::past) u terms
huffman@45775
   535
huffman@51366
   536
  fun dest_summing (Const (@{const_name Groups.plus}, _) $ t $ u, ts) =
huffman@51366
   537
        dest_summing (t, dest_summing (u, ts))
huffman@51366
   538
    | dest_summing (t, ts) = t :: ts
huffman@51366
   539
huffman@45775
   540
  val mk_sum = Arith_Data.long_mk_sum
huffman@51366
   541
  fun dest_sum t = dest_summing (t, [])
huffman@45775
   542
  val find_first = find_first_t []
huffman@45775
   543
  val trans_tac = Numeral_Simprocs.trans_tac
wenzelm@51717
   544
  val norm_ss =
wenzelm@51717
   545
    simpset_of (put_simpset HOL_basic_ss @{context}
haftmann@57514
   546
      addsimps @{thms ac_simps add_0_left add_0_right})
wenzelm@51717
   547
  fun norm_tac ctxt = ALLGOALS (simp_tac (put_simpset norm_ss ctxt))
wenzelm@51717
   548
  fun simplify_meta_eq ctxt cancel_th th =
wenzelm@51717
   549
    Arith_Data.simplify_meta_eq [] ctxt
huffman@45775
   550
      ([th, cancel_th] MRS trans)
huffman@45775
   551
  fun mk_eq (a, b) = HOLogic.mk_Trueprop (HOLogic.mk_eq (a, b))
huffman@45775
   552
end
huffman@45775
   553
huffman@45775
   554
structure Eq_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   555
(open Cancel_Enat_Common
huffman@45775
   556
  val mk_bal = HOLogic.mk_eq
huffman@45775
   557
  val dest_bal = HOLogic.dest_bin @{const_name HOL.eq} @{typ enat}
huffman@45775
   558
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel}
huffman@45775
   559
)
huffman@45775
   560
huffman@45775
   561
structure Le_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   562
(open Cancel_Enat_Common
huffman@45775
   563
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less_eq}
huffman@45775
   564
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} @{typ enat}
huffman@45775
   565
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_le}
huffman@45775
   566
)
huffman@45775
   567
huffman@45775
   568
structure Less_Enat_Cancel = ExtractCommonTermFun
huffman@45775
   569
(open Cancel_Enat_Common
huffman@45775
   570
  val mk_bal = HOLogic.mk_binrel @{const_name Orderings.less}
huffman@45775
   571
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} @{typ enat}
huffman@45775
   572
  fun simp_conv _ _ = SOME @{thm enat_add_left_cancel_less}
huffman@45775
   573
)
wenzelm@60500
   574
\<close>
huffman@45775
   575
huffman@45775
   576
simproc_setup enat_eq_cancel
huffman@45775
   577
  ("(l::enat) + m = n" | "(l::enat) = m + n") =
wenzelm@60500
   578
  \<open>fn phi => fn ctxt => fn ct => Eq_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   579
huffman@45775
   580
simproc_setup enat_le_cancel
huffman@45775
   581
  ("(l::enat) + m \<le> n" | "(l::enat) \<le> m + n") =
wenzelm@60500
   582
  \<open>fn phi => fn ctxt => fn ct => Le_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   583
huffman@45775
   584
simproc_setup enat_less_cancel
huffman@45775
   585
  ("(l::enat) + m < n" | "(l::enat) < m + n") =
wenzelm@60500
   586
  \<open>fn phi => fn ctxt => fn ct => Less_Enat_Cancel.proc ctxt (Thm.term_of ct)\<close>
huffman@45775
   587
wenzelm@60500
   588
text \<open>TODO: add regression tests for these simprocs\<close>
huffman@45775
   589
wenzelm@60500
   590
text \<open>TODO: add simprocs for combining and cancelling numerals\<close>
huffman@45775
   591
wenzelm@60500
   592
subsection \<open>Well-ordering\<close>
huffman@26089
   593
hoelzl@43924
   594
lemma less_enatE:
hoelzl@43924
   595
  "[| n < enat m; !!k. n = enat k ==> k < m ==> P |] ==> P"
huffman@26089
   596
by (induct n) auto
huffman@26089
   597
huffman@44019
   598
lemma less_infinityE:
hoelzl@43924
   599
  "[| n < \<infinity>; !!k. n = enat k ==> P |] ==> P"
huffman@26089
   600
by (induct n) auto
huffman@26089
   601
hoelzl@43919
   602
lemma enat_less_induct:
hoelzl@43919
   603
  assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n"
huffman@26089
   604
proof -
hoelzl@43924
   605
  have P_enat: "!!k. P (enat k)"
huffman@26089
   606
    apply (rule nat_less_induct)
huffman@26089
   607
    apply (rule prem, clarify)
hoelzl@43924
   608
    apply (erule less_enatE, simp)
huffman@26089
   609
    done
huffman@26089
   610
  show ?thesis
huffman@26089
   611
  proof (induct n)
huffman@26089
   612
    fix nat
hoelzl@43924
   613
    show "P (enat nat)" by (rule P_enat)
huffman@26089
   614
  next
hoelzl@43921
   615
    show "P \<infinity>"
huffman@26089
   616
      apply (rule prem, clarify)
huffman@44019
   617
      apply (erule less_infinityE)
hoelzl@43924
   618
      apply (simp add: P_enat)
huffman@26089
   619
      done
huffman@26089
   620
  qed
huffman@26089
   621
qed
huffman@26089
   622
hoelzl@43919
   623
instance enat :: wellorder
huffman@26089
   624
proof
haftmann@27823
   625
  fix P and n
wenzelm@61076
   626
  assume hyp: "(\<And>n::enat. (\<And>m::enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)"
hoelzl@43919
   627
  show "P n" by (blast intro: enat_less_induct hyp)
huffman@26089
   628
qed
huffman@26089
   629
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subsection \<open>Complete Lattice\<close>
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instantiation enat :: complete_lattice
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begin
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definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
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  "inf_enat = min"
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definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where
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  "sup_enat = max"
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definition Inf_enat :: "enat set \<Rightarrow> enat" where
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  "Inf_enat A = (if A = {} then \<infinity> else (LEAST x. x \<in> A))"
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definition Sup_enat :: "enat set \<Rightarrow> enat" where
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  "Sup_enat A = (if A = {} then 0 else if finite A then Max A else \<infinity>)"
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instance
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proof
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  fix x :: "enat" and A :: "enat set"
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  { assume "x \<in> A" then show "Inf A \<le> x"
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      unfolding Inf_enat_def by (auto intro: Least_le) }
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  { assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A"
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      unfolding Inf_enat_def
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      by (cases "A = {}") (auto intro: LeastI2_ex) }
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  { assume "x \<in> A" then show "x \<le> Sup A"
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      unfolding Sup_enat_def by (cases "finite A") auto }
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  { assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x"
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      unfolding Sup_enat_def using finite_enat_bounded by auto }
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qed (simp_all add:
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 inf_enat_def sup_enat_def bot_enat_def top_enat_def Inf_enat_def Sup_enat_def)
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end
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instance enat :: complete_linorder ..
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lemma eSuc_Sup: "A \<noteq> {} \<Longrightarrow> eSuc (Sup A) = Sup (eSuc ` A)"
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  by(auto simp add: Sup_enat_def eSuc_Max inj_on_def dest: finite_imageD)
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lemma sup_continuous_eSuc: "sup_continuous f \<Longrightarrow> sup_continuous (\<lambda>x. eSuc (f x))"
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  using  eSuc_Sup[of "_ ` UNIV"] by (auto simp: sup_continuous_def)
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subsection \<open>Traditional theorem names\<close>
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lemmas enat_defs = zero_enat_def one_enat_def eSuc_def
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  plus_enat_def less_eq_enat_def less_enat_def
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end