(*  Title:      HOL/Library/Nat_Infinity.thy

     ID:         $Id$

     Author:     David von Oheimb, TU Muenchen

     License:    GPL (GNU GENERAL PUBLIC LICENSE)

 *)

 header {*

   \title{Natural numbers with infinity}

   \author{David von Oheimb}

 *}

 theory Nat_Infinity = Main:

 subsection "Definitions"

 text {*

   We extend the standard natural numbers by a special value indicating

   infinity.  This includes extending the ordering relations @{term "op

   <"} and @{term "op \<le>"}.

 *}

 datatype inat = Fin nat | Infty

 instance inat :: ord ..

 instance inat :: zero ..

 consts

   iSuc :: "inat => inat"

 syntax (xsymbols)

   Infty :: inat    ("\<infinity>")

 syntax (HTML output)

   Infty :: inat    ("\<infinity>")

 defs

   Zero_inat_def: "0 == Fin 0"

   iSuc_def: "iSuc i == case i of Fin n  => Fin (Suc n) | \<infinity> => \<infinity>"

   iless_def: "m < n ==

     case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)

     | \<infinity>  => False"

   ile_def: "(m::inat) \<le> n == \<not> (n < m)"

 lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def

 lemmas inat_splits = inat.split inat.split_asm

 text {*

   Below is a not quite complete set of theorems.  Use the method

   @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove

   new theorems or solve arithmetic subgoals involving @{typ inat} on

   the fly.

 *}

 subsection "Constructors"

 lemma Fin_0: "Fin 0 = 0"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"

   by (simp add: inat_defs split:inat_splits, arith?)

 subsection "Ordering relations"

 lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_not_refl [simp]: "\<not> n < (n::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma i0_iless_iSuc [simp]: "0 < iSuc n"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"

   by (simp add: inat_defs split:inat_splits, arith?)

 (* ----------------------------------------------------------------------- *)

 lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ile_refl [simp]: "n \<le> (n::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Infty_ub [simp]: "n \<le> \<infinity>"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma i0_lb [simp]: "(0::inat) \<le> n"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ileI1: "m < n ==> iSuc m \<le> n"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma ile_iSuc [simp]: "n \<le> iSuc n"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"

   by (simp add: inat_defs split:inat_splits, arith?)

 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"

   apply (induct_tac k)

    apply (simp (no_asm) only: Fin_0)

    apply (fast intro: ile_iless_trans i0_lb)

   apply (erule exE)

   apply (drule spec)

   apply (erule exE)

   apply (drule ileI1)

   apply (rule iSuc_Fin [THEN subst])

   apply (rule exI)

   apply (erule (1) ile_iless_trans)

   done

 end