wenzelm@11355: (*  Title:      HOL/Library/Nat_Infinity.thy
wenzelm@11355:     ID:         $Id$
wenzelm@11355:     Author:     David von Oheimb, TU Muenchen
oheimb@11351: *)
oheimb@11351: 
wenzelm@14706: header {* Natural numbers with infinity *}
oheimb@11351: 
nipkow@15131: theory Nat_Infinity
haftmann@25594: imports PreList
nipkow@15131: begin
oheimb@11351: 
oheimb@11351: subsection "Definitions"
oheimb@11351: 
oheimb@11351: text {*
wenzelm@11355:   We extend the standard natural numbers by a special value indicating
wenzelm@11355:   infinity.  This includes extending the ordering relations @{term "op
wenzelm@11355:   <"} and @{term "op \<le>"}.
oheimb@11351: *}
oheimb@11351: 
oheimb@11351: datatype inat = Fin nat | Infty
oheimb@11351: 
wenzelm@21210: notation (xsymbols)
wenzelm@19736:   Infty  ("\<infinity>")
wenzelm@19736: 
wenzelm@21210: notation (HTML output)
wenzelm@19736:   Infty  ("\<infinity>")
wenzelm@19736: 
wenzelm@19736: definition
wenzelm@21404:   iSuc :: "inat => inat" where
wenzelm@19736:   "iSuc i = (case i of Fin n => Fin (Suc n) | \<infinity> => \<infinity>)"
oheimb@11351: 
haftmann@25594: instantiation inat :: "{ord, zero}"
haftmann@25594: begin
haftmann@25594: 
haftmann@25594: definition
wenzelm@11701:   Zero_inat_def: "0 == Fin 0"
haftmann@25594: 
haftmann@25594: definition
wenzelm@11355:   iless_def: "m < n ==
wenzelm@11355:     case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
wenzelm@11355:     | \<infinity>  => False"
haftmann@25594: 
haftmann@25594: definition
wenzelm@11355:   ile_def: "(m::inat) \<le> n == \<not> (n < m)"
oheimb@11351: 
haftmann@25594: instance ..
haftmann@25594: 
haftmann@25594: end
haftmann@25594: 
wenzelm@11701: lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def
oheimb@11351: lemmas inat_splits = inat.split inat.split_asm
oheimb@11351: 
wenzelm@11355: text {*
wenzelm@11357:   Below is a not quite complete set of theorems.  Use the method
wenzelm@11357:   @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
wenzelm@11357:   new theorems or solve arithmetic subgoals involving @{typ inat} on
wenzelm@11357:   the fly.
oheimb@11351: *}
oheimb@11351: 
oheimb@11351: subsection "Constructors"
oheimb@11351: 
oheimb@11351: lemma Fin_0: "Fin 0 = 0"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: 
oheimb@11351: subsection "Ordering relations"
oheimb@11351: 
oheimb@11351: lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits, arith)
oheimb@11351: 
oheimb@11351: lemma iless_not_refl [simp]: "\<not> n < (n::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11655: lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
nipkow@25134: by (fastsimp simp: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma i0_iless_iSuc [simp]: "0 < iSuc n"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11655: lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: 
oheimb@11351: 
wenzelm@11655: lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
nipkow@25134: by (simp add: inat_defs split:inat_splits, arith)
oheimb@11351: 
wenzelm@11355: lemma ile_refl [simp]: "n \<le> (n::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma Infty_ub [simp]: "n \<le> \<infinity>"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma i0_lb [simp]: "(0::inat) \<le> n"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
nipkow@25134: by (simp add: inat_defs split:inat_splits, arith)
oheimb@11351: 
wenzelm@11355: lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma ileI1: "m < n ==> iSuc m \<le> n"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11655: lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
nipkow@25134: by (simp add: inat_defs split:inat_splits, arith)
oheimb@11351: 
wenzelm@11655: lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11655: lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
nipkow@25134: by (simp add: inat_defs split:inat_splits, arith)
oheimb@11351: 
wenzelm@11355: lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma ile_iSuc [simp]: "n \<le> iSuc n"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
wenzelm@11355: lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
nipkow@25134: by (simp add: inat_defs split:inat_splits)
oheimb@11351: 
oheimb@11351: lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
nipkow@25134: apply (induct_tac k)
nipkow@25134:  apply (simp (no_asm) only: Fin_0)
nipkow@25134:  apply (fast intro: ile_iless_trans i0_lb)
nipkow@25134: apply (erule exE)
nipkow@25134: apply (drule spec)
nipkow@25134: apply (erule exE)
nipkow@25134: apply (drule ileI1)
nipkow@25134: apply (rule iSuc_Fin [THEN subst])
nipkow@25134: apply (rule exI)
nipkow@25134: apply (erule (1) ile_iless_trans)
nipkow@25134: done
oheimb@11351: 
oheimb@11351: end