src/HOL/Library/Nat_Infinity.thy
changeset 27110 194aa674c2a1
parent 26089 373221497340
child 27368 9f90ac19e32b
     1.1 --- a/src/HOL/Library/Nat_Infinity.thy	Tue Jun 10 15:31:01 2008 +0200
     1.2 +++ b/src/HOL/Library/Nat_Infinity.thy	Tue Jun 10 15:31:02 2008 +0200
     1.3 @@ -1,6 +1,6 @@
     1.4  (*  Title:      HOL/Library/Nat_Infinity.thy
     1.5      ID:         $Id$
     1.6 -    Author:     David von Oheimb, TU Muenchen
     1.7 +    Author:     David von Oheimb, TU Muenchen;  Florian Haftmann, TU Muenchen
     1.8  *)
     1.9  
    1.10  header {* Natural numbers with infinity *}
    1.11 @@ -9,12 +9,11 @@
    1.12  imports ATP_Linkup
    1.13  begin
    1.14  
    1.15 -subsection "Definitions"
    1.16 +subsection {* Type definition *}
    1.17  
    1.18  text {*
    1.19    We extend the standard natural numbers by a special value indicating
    1.20 -  infinity.  This includes extending the ordering relations @{term "op
    1.21 -  <"} and @{term "op \<le>"}.
    1.22 +  infinity.
    1.23  *}
    1.24  
    1.25  datatype inat = Fin nat | Infty
    1.26 @@ -25,196 +24,267 @@
    1.27  notation (HTML output)
    1.28    Infty  ("\<infinity>")
    1.29  
    1.30 -definition
    1.31 -  iSuc :: "inat => inat" where
    1.32 -  "iSuc i = (case i of Fin n => Fin (Suc n) | \<infinity> => \<infinity>)"
    1.33  
    1.34 -instantiation inat :: "{ord, zero}"
    1.35 +subsection {* Constructors and numbers *}
    1.36 +
    1.37 +instantiation inat :: "{zero, one, number}"
    1.38  begin
    1.39  
    1.40  definition
    1.41 -  Zero_inat_def: "0 == Fin 0"
    1.42 +  "0 = Fin 0"
    1.43  
    1.44  definition
    1.45 -  iless_def: "m < n ==
    1.46 -    case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
    1.47 -    | \<infinity>  => False"
    1.48 +  [code inline]: "1 = Fin 1"
    1.49  
    1.50  definition
    1.51 -  ile_def: "m \<le> n ==
    1.52 -    case n of Fin n1 => (case m of Fin m1 => m1 \<le> n1 | \<infinity> => False)
    1.53 -    | \<infinity>  => True"
    1.54 +  [code inline, code func del]: "number_of k = Fin (number_of k)"
    1.55  
    1.56  instance ..
    1.57  
    1.58  end
    1.59  
    1.60 -lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def
    1.61 -lemmas inat_splits = inat.split inat.split_asm
    1.62 -
    1.63 -text {*
    1.64 -  Below is a not quite complete set of theorems.  Use the method
    1.65 -  @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
    1.66 -  new theorems or solve arithmetic subgoals involving @{typ inat} on
    1.67 -  the fly.
    1.68 -*}
    1.69 -
    1.70 -subsection "Constructors"
    1.71 +definition iSuc :: "inat \<Rightarrow> inat" where
    1.72 +  "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
    1.73  
    1.74  lemma Fin_0: "Fin 0 = 0"
    1.75 -by (simp add: inat_defs split:inat_splits)
    1.76 +  by (simp add: zero_inat_def)
    1.77 +
    1.78 +lemma Fin_1: "Fin 1 = 1"
    1.79 +  by (simp add: one_inat_def)
    1.80 +
    1.81 +lemma Fin_number: "Fin (number_of k) = number_of k"
    1.82 +  by (simp add: number_of_inat_def)
    1.83 +
    1.84 +lemma one_iSuc: "1 = iSuc 0"
    1.85 +  by (simp add: zero_inat_def one_inat_def iSuc_def)
    1.86  
    1.87  lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
    1.88 -by (simp add: inat_defs split:inat_splits)
    1.89 +  by (simp add: zero_inat_def)
    1.90  
    1.91  lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
    1.92 -by (simp add: inat_defs split:inat_splits)
    1.93 +  by (simp add: zero_inat_def)
    1.94 +
    1.95 +lemma zero_inat_eq [simp]:
    1.96 +  "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
    1.97 +  "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
    1.98 +  unfolding zero_inat_def number_of_inat_def by simp_all
    1.99 +
   1.100 +lemma one_inat_eq [simp]:
   1.101 +  "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
   1.102 +  "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
   1.103 +  unfolding one_inat_def number_of_inat_def by simp_all
   1.104 +
   1.105 +lemma zero_one_inat_neq [simp]:
   1.106 +  "\<not> 0 = (1\<Colon>inat)"
   1.107 +  "\<not> 1 = (0\<Colon>inat)"
   1.108 +  unfolding zero_inat_def one_inat_def by simp_all
   1.109  
   1.110 -lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
   1.111 -by (simp add: inat_defs split:inat_splits)
   1.112 +lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
   1.113 +  by (simp add: one_inat_def)
   1.114 +
   1.115 +lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
   1.116 +  by (simp add: one_inat_def)
   1.117 +
   1.118 +lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
   1.119 +  by (simp add: number_of_inat_def)
   1.120 +
   1.121 +lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
   1.122 +  by (simp add: number_of_inat_def)
   1.123 +
   1.124 +lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
   1.125 +  by (simp add: iSuc_def)
   1.126 +
   1.127 +lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
   1.128 +  by (simp add: iSuc_Fin number_of_inat_def)
   1.129  
   1.130  lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
   1.131 -by (simp add: inat_defs split:inat_splits)
   1.132 +  by (simp add: iSuc_def)
   1.133  
   1.134  lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
   1.135 -by (simp add: inat_defs split:inat_splits)
   1.136 +  by (simp add: iSuc_def zero_inat_def split: inat.splits)
   1.137 +
   1.138 +lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
   1.139 +  by (rule iSuc_ne_0 [symmetric])
   1.140  
   1.141 -lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
   1.142 -by (simp add: inat_defs split:inat_splits)
   1.143 +lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
   1.144 +  by (simp add: iSuc_def split: inat.splits)
   1.145 +
   1.146 +lemma number_of_inat_inject [simp]:
   1.147 +  "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
   1.148 +  by (simp add: number_of_inat_def)
   1.149  
   1.150  
   1.151 -subsection "Ordering relations"
   1.152 +subsection {* Addition *}
   1.153 +
   1.154 +instantiation inat :: comm_monoid_add
   1.155 +begin
   1.156 +
   1.157 +definition
   1.158 +  [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)))"
   1.159  
   1.160 -instance inat :: linorder
   1.161 -proof
   1.162 -  fix x :: inat
   1.163 -  show "x \<le> x"
   1.164 -    by (simp add: inat_defs split: inat_splits)
   1.165 -next
   1.166 -  fix x y :: inat
   1.167 -  assume "x \<le> y" and "y \<le> x" thus "x = y"
   1.168 -    by (simp add: inat_defs split: inat_splits)
   1.169 -next
   1.170 -  fix x y z :: inat
   1.171 -  assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
   1.172 -    by (simp add: inat_defs split: inat_splits)
   1.173 -next
   1.174 -  fix x y :: inat
   1.175 -  show "(x < y) = (x \<le> y \<and> x \<noteq> y)"
   1.176 -    by (simp add: inat_defs order_less_le split: inat_splits)
   1.177 -next
   1.178 -  fix x y :: inat
   1.179 -  show "x \<le> y \<or> y \<le> x"
   1.180 -    by (simp add: inat_defs linorder_linear split: inat_splits)
   1.181 +lemma plus_inat_simps [simp, code]:
   1.182 +  "Fin m + Fin n = Fin (m + n)"
   1.183 +  "\<infinity> + q = \<infinity>"
   1.184 +  "q + \<infinity> = \<infinity>"
   1.185 +  by (simp_all add: plus_inat_def split: inat.splits)
   1.186 +
   1.187 +instance proof
   1.188 +  fix n m q :: inat
   1.189 +  show "n + m + q = n + (m + q)"
   1.190 +    by (cases n, auto, cases m, auto, cases q, auto)
   1.191 +  show "n + m = m + n"
   1.192 +    by (cases n, auto, cases m, auto)
   1.193 +  show "0 + n = n"
   1.194 +    by (cases n) (simp_all add: zero_inat_def)
   1.195  qed
   1.196  
   1.197 -lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
   1.198 -by (simp add: inat_defs split:inat_splits)
   1.199 -
   1.200 -lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
   1.201 -by (rule linorder_less_linear)
   1.202 -
   1.203 -lemma iless_not_refl: "\<not> n < (n::inat)"
   1.204 -by (rule order_less_irrefl)
   1.205 +end
   1.206  
   1.207 -lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
   1.208 -by (rule order_less_trans)
   1.209 +lemma plus_inat_0 [simp]:
   1.210 +  "0 + (q\<Colon>inat) = q"
   1.211 +  "(q\<Colon>inat) + 0 = q"
   1.212 +  by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
   1.213  
   1.214 -lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
   1.215 -by (rule order_less_not_sym)
   1.216 -
   1.217 -lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
   1.218 -by (simp add: inat_defs split:inat_splits)
   1.219 +lemma plus_inat_number [simp]:
   1.220 +  "(number_of k \<Colon> inat) + number_of l = (if neg (number_of k \<Colon> int) then number_of l
   1.221 +    else if neg (number_of l \<Colon> int) then number_of k else number_of (k + l))"
   1.222 +  unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
   1.223  
   1.224 -lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
   1.225 -by (simp add: inat_defs split:inat_splits)
   1.226 -
   1.227 -lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
   1.228 -by (simp add: inat_defs split:inat_splits)
   1.229 -
   1.230 -lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
   1.231 -by (fastsimp simp: inat_defs split:inat_splits)
   1.232 +lemma iSuc_number [simp]:
   1.233 +  "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
   1.234 +  unfolding iSuc_number_of
   1.235 +  unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
   1.236  
   1.237 -lemma i0_iless_iSuc [simp]: "0 < iSuc n"
   1.238 -by (simp add: inat_defs split:inat_splits)
   1.239 -
   1.240 -lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
   1.241 -by (simp add: inat_defs split:inat_splits)
   1.242 -
   1.243 -lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
   1.244 -by (simp add: inat_defs split:inat_splits)
   1.245 -
   1.246 -lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
   1.247 -by (simp add: inat_defs split:inat_splits)
   1.248 -
   1.249 +lemma iSuc_plus_1:
   1.250 +  "iSuc n = n + 1"
   1.251 +  by (cases n) (simp_all add: iSuc_Fin one_inat_def)
   1.252 +  
   1.253 +lemma plus_1_iSuc:
   1.254 +  "1 + q = iSuc q"
   1.255 +  "q + 1 = iSuc q"
   1.256 +  unfolding iSuc_plus_1 by (simp_all add: add_ac)
   1.257  
   1.258  
   1.259 -lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
   1.260 -by (rule order_le_less)
   1.261 +subsection {* Ordering *}
   1.262 +
   1.263 +instantiation inat :: ordered_ab_semigroup_add
   1.264 +begin
   1.265  
   1.266 -lemma ile_refl [simp]: "n \<le> (n::inat)"
   1.267 -by (rule order_refl)
   1.268 +definition
   1.269 +  [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
   1.270 +    | \<infinity> \<Rightarrow> True)"
   1.271  
   1.272 -lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
   1.273 -by (rule order_trans)
   1.274 +definition
   1.275 +  [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
   1.276 +    | \<infinity> \<Rightarrow> False)"
   1.277  
   1.278 -lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
   1.279 -by (rule order_le_less_trans)
   1.280 -
   1.281 -lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
   1.282 -by (rule order_less_le_trans)
   1.283 +lemma inat_ord_simps [simp]:
   1.284 +  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
   1.285 +  "Fin m < Fin n \<longleftrightarrow> m < n"
   1.286 +  "q \<le> \<infinity>"
   1.287 +  "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
   1.288 +  "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
   1.289 +  "\<infinity> < q \<longleftrightarrow> False"
   1.290 +  by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
   1.291  
   1.292 -lemma Infty_ub [simp]: "n \<le> \<infinity>"
   1.293 -by (simp add: inat_defs split:inat_splits)
   1.294 +lemma inat_ord_code [code]:
   1.295 +  "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
   1.296 +  "Fin m < Fin n \<longleftrightarrow> m < n"
   1.297 +  "q \<le> \<infinity> \<longleftrightarrow> True"
   1.298 +  "Fin m < \<infinity> \<longleftrightarrow> True"
   1.299 +  "\<infinity> \<le> Fin n \<longleftrightarrow> False"
   1.300 +  "\<infinity> < q \<longleftrightarrow> False"
   1.301 +  by simp_all
   1.302  
   1.303 -lemma i0_lb [simp]: "(0::inat) \<le> n"
   1.304 -by (simp add: inat_defs split:inat_splits)
   1.305 +instance by default
   1.306 +  (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
   1.307  
   1.308 -lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
   1.309 -by (simp add: inat_defs split:inat_splits)
   1.310 +end
   1.311 +
   1.312 +lemma inat_ord_number [simp]:
   1.313 +  "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
   1.314 +  "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
   1.315 +  by (simp_all add: number_of_inat_def)
   1.316  
   1.317 -lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
   1.318 -by (simp add: inat_defs split:inat_splits)
   1.319 +lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
   1.320 +  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
   1.321  
   1.322 -lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
   1.323 -by (rule order_le_neq_trans)
   1.324 +lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
   1.325 +  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
   1.326 +
   1.327 +lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
   1.328 +  by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
   1.329  
   1.330 -lemma ileI1: "m < n ==> iSuc m \<le> n"
   1.331 -by (simp add: inat_defs split:inat_splits)
   1.332 +lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
   1.333 +  by simp
   1.334  
   1.335 -lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
   1.336 -by (simp add: inat_defs split:inat_splits, arith)
   1.337 +lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
   1.338 +  by (simp add: zero_inat_def less_inat_def split: inat.splits)
   1.339 +
   1.340 +lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
   1.341 +  by (simp add: zero_inat_def less_inat_def split: inat.splits)
   1.342  
   1.343 -lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
   1.344 -by (simp add: inat_defs split:inat_splits)
   1.345 +lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
   1.346 +  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
   1.347 + 
   1.348 +lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
   1.349 +  by (simp add: iSuc_def less_inat_def split: inat.splits)
   1.350  
   1.351 -lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
   1.352 -by (simp add: inat_defs split:inat_splits, arith)
   1.353 +lemma ile_iSuc [simp]: "n \<le> iSuc n"
   1.354 +  by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
   1.355  
   1.356  lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
   1.357 -by (simp add: inat_defs split:inat_splits)
   1.358 +  by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
   1.359 +
   1.360 +lemma i0_iless_iSuc [simp]: "0 < iSuc n"
   1.361 +  by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
   1.362 +
   1.363 +lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
   1.364 +  by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
   1.365 +
   1.366 +lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
   1.367 +  by (cases n) auto
   1.368 +
   1.369 +lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
   1.370 +  by (auto simp add: iSuc_def less_inat_def split: inat.splits)
   1.371  
   1.372 -lemma ile_iSuc [simp]: "n \<le> iSuc n"
   1.373 -by (simp add: inat_defs split:inat_splits)
   1.374 +lemma min_inat_simps [simp]:
   1.375 +  "min (Fin m) (Fin n) = Fin (min m n)"
   1.376 +  "min q 0 = 0"
   1.377 +  "min 0 q = 0"
   1.378 +  "min q \<infinity> = q"
   1.379 +  "min \<infinity> q = q"
   1.380 +  by (auto simp add: min_def)
   1.381  
   1.382 -lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
   1.383 -by (simp add: inat_defs split:inat_splits)
   1.384 +lemma max_inat_simps [simp]:
   1.385 +  "max (Fin m) (Fin n) = Fin (max m n)"
   1.386 +  "max q 0 = q"
   1.387 +  "max 0 q = q"
   1.388 +  "max q \<infinity> = \<infinity>"
   1.389 +  "max \<infinity> q = \<infinity>"
   1.390 +  by (simp_all add: max_def)
   1.391 +
   1.392 +lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
   1.393 +  by (cases n) simp_all
   1.394 +
   1.395 +lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
   1.396 +  by (cases n) simp_all
   1.397  
   1.398  lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
   1.399  apply (induct_tac k)
   1.400   apply (simp (no_asm) only: Fin_0)
   1.401 - apply (fast intro: ile_iless_trans [OF i0_lb])
   1.402 + apply (fast intro: le_less_trans [OF i0_lb])
   1.403  apply (erule exE)
   1.404  apply (drule spec)
   1.405  apply (erule exE)
   1.406  apply (drule ileI1)
   1.407  apply (rule iSuc_Fin [THEN subst])
   1.408  apply (rule exI)
   1.409 -apply (erule (1) ile_iless_trans)
   1.410 +apply (erule (1) le_less_trans)
   1.411  done
   1.412  
   1.413  
   1.414 -subsection "Well-ordering"
   1.415 +subsection {* Well-ordering *}
   1.416  
   1.417  lemma less_FinE:
   1.418    "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
   1.419 @@ -256,4 +326,12 @@
   1.420    qed
   1.421  qed
   1.422  
   1.423 +
   1.424 +subsection {* Traditional theorem names *}
   1.425 +
   1.426 +lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
   1.427 +  plus_inat_def less_eq_inat_def less_inat_def
   1.428 +
   1.429 +lemmas inat_splits = inat.splits
   1.430 +
   1.431  end