author | haftmann |
Wed, 28 Jan 2009 11:02:12 +0100 | |
changeset 29652 | f4c6e546b7fe |
parent 29337 | 450805a4a91f |
child 29668 | 33ba3faeaa0e |
permissions | -rw-r--r-- |
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(* Title: HOL/Library/Nat_Infinity.thy |
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Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen |
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*) |
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header {* Natural numbers with infinity *} |
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theory Nat_Infinity |
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imports Plain "~~/src/HOL/Presburger" |
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begin |
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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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datatype inat = Fin nat | Infty |
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notation (xsymbols) |
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Infty ("\<infinity>") |
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notation (HTML output) |
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Infty ("\<infinity>") |
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subsection {* Constructors and numbers *} |
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instantiation inat :: "{zero, one, number}" |
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begin |
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definition |
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"0 = Fin 0" |
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definition |
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[code inline]: "1 = Fin 1" |
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definition |
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[code inline, code del]: "number_of k = Fin (number_of k)" |
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instance .. |
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end |
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definition iSuc :: "inat \<Rightarrow> inat" where |
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"iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)" |
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lemma Fin_0: "Fin 0 = 0" |
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by (simp add: zero_inat_def) |
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lemma Fin_1: "Fin 1 = 1" |
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by (simp add: one_inat_def) |
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lemma Fin_number: "Fin (number_of k) = number_of k" |
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by (simp add: number_of_inat_def) |
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lemma one_iSuc: "1 = iSuc 0" |
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by (simp add: zero_inat_def one_inat_def iSuc_def) |
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lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0" |
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by (simp add: zero_inat_def) |
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lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>" |
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by (simp add: zero_inat_def) |
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lemma zero_inat_eq [simp]: |
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"number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
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"(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
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unfolding zero_inat_def number_of_inat_def by simp_all |
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lemma one_inat_eq [simp]: |
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"number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
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"(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
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unfolding one_inat_def number_of_inat_def by simp_all |
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lemma zero_one_inat_neq [simp]: |
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"\<not> 0 = (1\<Colon>inat)" |
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"\<not> 1 = (0\<Colon>inat)" |
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unfolding zero_inat_def one_inat_def by simp_all |
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lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1" |
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by (simp add: one_inat_def) |
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lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>" |
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by (simp add: one_inat_def) |
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lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k" |
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by (simp add: number_of_inat_def) |
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lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>" |
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by (simp add: number_of_inat_def) |
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lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)" |
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by (simp add: iSuc_def) |
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lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))" |
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by (simp add: iSuc_Fin number_of_inat_def) |
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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>" |
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by (simp add: iSuc_def) |
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0" |
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by (simp add: iSuc_def zero_inat_def split: inat.splits) |
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lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n" |
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by (rule iSuc_ne_0 [symmetric]) |
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lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n" |
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by (simp add: iSuc_def split: inat.splits) |
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lemma number_of_inat_inject [simp]: |
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"(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l" |
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by (simp add: number_of_inat_def) |
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subsection {* Addition *} |
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instantiation inat :: comm_monoid_add |
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begin |
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definition |
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[code del]: "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))" |
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lemma plus_inat_simps [simp, code]: |
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"Fin m + Fin n = Fin (m + n)" |
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"\<infinity> + q = \<infinity>" |
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"q + \<infinity> = \<infinity>" |
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by (simp_all add: plus_inat_def split: inat.splits) |
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instance proof |
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fix n m q :: inat |
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show "n + m + q = n + (m + q)" |
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by (cases n, auto, cases m, auto, cases q, auto) |
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show "n + m = m + n" |
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by (cases n, auto, cases m, auto) |
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show "0 + n = n" |
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by (cases n) (simp_all add: zero_inat_def) |
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qed |
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end |
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lemma plus_inat_0 [simp]: |
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"0 + (q\<Colon>inat) = q" |
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"(q\<Colon>inat) + 0 = q" |
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by (simp_all add: plus_inat_def zero_inat_def split: inat.splits) |
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lemma plus_inat_number [simp]: |
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"(number_of k \<Colon> inat) + number_of l = (if k < Int.Pls then number_of l |
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else if l < Int.Pls then number_of k else number_of (k + l))" |
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unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] .. |
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lemma iSuc_number [simp]: |
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"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))" |
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unfolding iSuc_number_of |
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unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] .. |
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lemma iSuc_plus_1: |
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"iSuc n = n + 1" |
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by (cases n) (simp_all add: iSuc_Fin one_inat_def) |
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lemma plus_1_iSuc: |
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"1 + q = iSuc q" |
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"q + 1 = iSuc q" |
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unfolding iSuc_plus_1 by (simp_all add: add_ac) |
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subsection {* Multiplication *} |
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instantiation inat :: comm_semiring_1 |
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begin |
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definition |
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times_inat_def [code del]: |
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"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow> |
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(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))" |
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lemma times_inat_simps [simp, code]: |
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"Fin m * Fin n = Fin (m * n)" |
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"\<infinity> * \<infinity> = \<infinity>" |
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"\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)" |
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"Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)" |
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unfolding times_inat_def zero_inat_def |
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by (simp_all split: inat.split) |
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instance proof |
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fix a b c :: inat |
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show "(a * b) * c = a * (b * c)" |
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unfolding times_inat_def zero_inat_def |
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by (simp split: inat.split) |
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show "a * b = b * a" |
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unfolding times_inat_def zero_inat_def |
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by (simp split: inat.split) |
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show "1 * a = a" |
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unfolding times_inat_def zero_inat_def one_inat_def |
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by (simp split: inat.split) |
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show "(a + b) * c = a * c + b * c" |
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unfolding times_inat_def zero_inat_def |
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by (simp split: inat.split add: left_distrib) |
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show "0 * a = 0" |
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unfolding times_inat_def zero_inat_def |
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by (simp split: inat.split) |
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show "a * 0 = 0" |
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unfolding times_inat_def zero_inat_def |
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by (simp split: inat.split) |
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show "(0::inat) \<noteq> 1" |
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unfolding zero_inat_def one_inat_def |
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by simp |
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qed |
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end |
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lemma mult_iSuc: "iSuc m * n = n + m * n" |
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unfolding iSuc_plus_1 by (simp add: ring_simps) |
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lemma mult_iSuc_right: "m * iSuc n = m + m * n" |
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unfolding iSuc_plus_1 by (simp add: ring_simps) |
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lemma of_nat_eq_Fin: "of_nat n = Fin n" |
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apply (induct n) |
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apply (simp add: Fin_0) |
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apply (simp add: plus_1_iSuc iSuc_Fin) |
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done |
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instance inat :: semiring_char_0 |
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by default (simp add: of_nat_eq_Fin) |
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subsection {* Ordering *} |
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instantiation inat :: ordered_ab_semigroup_add |
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begin |
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definition |
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[code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False) |
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| \<infinity> \<Rightarrow> True)" |
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definition |
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[code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True) |
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| \<infinity> \<Rightarrow> False)" |
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lemma inat_ord_simps [simp]: |
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"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
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"Fin m < Fin n \<longleftrightarrow> m < n" |
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"q \<le> \<infinity>" |
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"q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>" |
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"\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>" |
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"\<infinity> < q \<longleftrightarrow> False" |
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by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits) |
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lemma inat_ord_code [code]: |
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"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
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"Fin m < Fin n \<longleftrightarrow> m < n" |
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"q \<le> \<infinity> \<longleftrightarrow> True" |
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"Fin m < \<infinity> \<longleftrightarrow> True" |
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"\<infinity> \<le> Fin n \<longleftrightarrow> False" |
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"\<infinity> < q \<longleftrightarrow> False" |
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by simp_all |
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instance by default |
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(auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits) |
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end |
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instance inat :: pordered_comm_semiring |
265 |
proof |
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fix a b c :: inat |
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assume "a \<le> b" and "0 \<le> c" |
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thus "c * a \<le> c * b" |
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unfolding times_inat_def less_eq_inat_def zero_inat_def |
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by (simp split: inat.splits) |
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qed |
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lemma inat_ord_number [simp]: |
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"(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n" |
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"(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n" |
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by (simp_all add: number_of_inat_def) |
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lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n" |
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by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
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lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0" |
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by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
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lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R" |
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by (simp add: zero_inat_def less_eq_inat_def split: inat.splits) |
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lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R" |
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by simp |
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lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)" |
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by (simp add: zero_inat_def less_inat_def split: inat.splits) |
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lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0" |
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by (simp add: zero_inat_def less_inat_def split: inat.splits) |
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|
27110 | 296 |
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m" |
297 |
by (simp add: iSuc_def less_eq_inat_def split: inat.splits) |
|
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||
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lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m" |
|
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by (simp add: iSuc_def less_inat_def split: inat.splits) |
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|
27110 | 302 |
lemma ile_iSuc [simp]: "n \<le> iSuc n" |
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by (simp add: iSuc_def less_eq_inat_def split: inat.splits) |
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11355 | 305 |
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0" |
27110 | 306 |
by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits) |
307 |
||
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lemma i0_iless_iSuc [simp]: "0 < iSuc n" |
|
309 |
by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits) |
|
310 |
||
311 |
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n" |
|
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by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits) |
|
313 |
||
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lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n" |
|
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by (cases n) auto |
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316 |
||
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lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n" |
|
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by (auto simp add: iSuc_def less_inat_def split: inat.splits) |
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27110 | 320 |
lemma min_inat_simps [simp]: |
321 |
"min (Fin m) (Fin n) = Fin (min m n)" |
|
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"min q 0 = 0" |
|
323 |
"min 0 q = 0" |
|
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"min q \<infinity> = q" |
|
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"min \<infinity> q = q" |
|
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by (auto simp add: min_def) |
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|
27110 | 328 |
lemma max_inat_simps [simp]: |
329 |
"max (Fin m) (Fin n) = Fin (max m n)" |
|
330 |
"max q 0 = q" |
|
331 |
"max 0 q = q" |
|
332 |
"max q \<infinity> = \<infinity>" |
|
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"max \<infinity> q = \<infinity>" |
|
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by (simp_all add: max_def) |
|
335 |
||
336 |
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
337 |
by (cases n) simp_all |
|
338 |
||
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lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
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by (cases n) simp_all |
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lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j" |
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apply (induct_tac k) |
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apply (simp (no_asm) only: Fin_0) |
27110 | 345 |
apply (fast intro: le_less_trans [OF i0_lb]) |
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346 |
apply (erule exE) |
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347 |
apply (drule spec) |
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348 |
apply (erule exE) |
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changeset
|
349 |
apply (drule ileI1) |
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350 |
apply (rule iSuc_Fin [THEN subst]) |
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|
351 |
apply (rule exI) |
27110 | 352 |
apply (erule (1) le_less_trans) |
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|
353 |
done |
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|
29337 | 355 |
instantiation inat :: "{bot, top}" |
356 |
begin |
|
357 |
||
358 |
definition bot_inat :: inat where |
|
359 |
"bot_inat = 0" |
|
360 |
||
361 |
definition top_inat :: inat where |
|
362 |
"top_inat = \<infinity>" |
|
363 |
||
364 |
instance proof |
|
365 |
qed (simp_all add: bot_inat_def top_inat_def) |
|
366 |
||
367 |
end |
|
368 |
||
26089 | 369 |
|
27110 | 370 |
subsection {* Well-ordering *} |
26089 | 371 |
|
372 |
lemma less_FinE: |
|
373 |
"[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P" |
|
374 |
by (induct n) auto |
|
375 |
||
376 |
lemma less_InftyE: |
|
377 |
"[| n < Infty; !!k. n = Fin k ==> P |] ==> P" |
|
378 |
by (induct n) auto |
|
379 |
||
380 |
lemma inat_less_induct: |
|
381 |
assumes prem: "!!n. \<forall>m::inat. m < n --> P m ==> P n" shows "P n" |
|
382 |
proof - |
|
383 |
have P_Fin: "!!k. P (Fin k)" |
|
384 |
apply (rule nat_less_induct) |
|
385 |
apply (rule prem, clarify) |
|
386 |
apply (erule less_FinE, simp) |
|
387 |
done |
|
388 |
show ?thesis |
|
389 |
proof (induct n) |
|
390 |
fix nat |
|
391 |
show "P (Fin nat)" by (rule P_Fin) |
|
392 |
next |
|
393 |
show "P Infty" |
|
394 |
apply (rule prem, clarify) |
|
395 |
apply (erule less_InftyE) |
|
396 |
apply (simp add: P_Fin) |
|
397 |
done |
|
398 |
qed |
|
399 |
qed |
|
400 |
||
401 |
instance inat :: wellorder |
|
402 |
proof |
|
27823 | 403 |
fix P and n |
404 |
assume hyp: "(\<And>n\<Colon>inat. (\<And>m\<Colon>inat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)" |
|
405 |
show "P n" by (blast intro: inat_less_induct hyp) |
|
26089 | 406 |
qed |
407 |
||
27110 | 408 |
|
409 |
subsection {* Traditional theorem names *} |
|
410 |
||
411 |
lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def |
|
412 |
plus_inat_def less_eq_inat_def less_inat_def |
|
413 |
||
414 |
lemmas inat_splits = inat.splits |
|
415 |
||
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416 |
end |