(* 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 \"}. *} datatype inat = Fin nat | Infty instance inat :: ord .. instance inat :: zero .. consts iSuc :: "inat => inat" syntax (xsymbols) Infty :: inat ("\") defs iZero_def: "0 == Fin 0" iSuc_def: "iSuc i == case i of Fin n => Fin (Suc n) | \ => \" iless_def: "m < n == case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \ => True) | \ => False" ile_def: "(m::inat) \ n == \ (n < m)" lemmas inat_defs = iZero_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]: "\ \ 0" by (simp add: inat_defs split:inat_splits, arith?) lemma i0_ne_Infty [simp]: "0 \ \" 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 \ = \" by (simp add: inat_defs split:inat_splits, arith?) lemma iSuc_ne_0 [simp]: "iSuc n \ 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!]: "\ < Fin m ==> R" by (simp add: inat_defs split:inat_splits, arith?) lemma iless_linear: "m < n \ m = n \ n < (m::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma iless_not_refl [simp]: "\ 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 ==> \ 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 < \" by (simp add: inat_defs split:inat_splits, arith?) lemma Infty_eq [simp]: "n < \ = (n \ \)" by (simp add: inat_defs split:inat_splits, arith?) lemma i0_eq [simp]: "((0::inat) < n) = (n \ 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]: "\ n < (0::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma Fin_iless: "n < Fin m ==> \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 \ n = (m < n \ m = (n::inat))" by (simp add: inat_defs split:inat_splits, arith?) lemma ile_refl [simp]: "n \ (n::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma ile_trans: "i \ j ==> j \ k ==> i \ (k::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma ile_iless_trans: "i \ j ==> j < k ==> i < (k::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma iless_ile_trans: "i < j ==> j \ k ==> i < (k::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma Infty_ub [simp]: "n \ \" by (simp add: inat_defs split:inat_splits, arith?) lemma i0_lb [simp]: "(0::inat) \ n" by (simp add: inat_defs split:inat_splits, arith?) lemma Infty_ileE [elim!]: "\ \ Fin m ==> R" by (simp add: inat_defs split:inat_splits, arith?) lemma Fin_ile_mono [simp]: "(Fin n \ Fin m) = (n \ m)" by (simp add: inat_defs split:inat_splits, arith?) lemma ilessI1: "n \ m ==> n \ m ==> n < (m::inat)" by (simp add: inat_defs split:inat_splits, arith?) lemma ileI1: "m < n ==> iSuc m \ n" by (simp add: inat_defs split:inat_splits, arith?) lemma Suc_ile_eq: "Fin (Suc m) \ n = (Fin m < n)" by (simp add: inat_defs split:inat_splits, arith?) lemma iSuc_ile_mono [simp]: "iSuc n \ iSuc m = (n \ m)" by (simp add: inat_defs split:inat_splits, arith?) lemma iless_Suc_eq [simp]: "Fin m < iSuc n = (Fin m \ n)" by (simp add: inat_defs split:inat_splits, arith?) lemma not_iSuc_ilei0 [simp]: "\ iSuc n \ 0" by (simp add: inat_defs split:inat_splits, arith?) lemma ile_iSuc [simp]: "n \ iSuc n" by (simp add: inat_defs split:inat_splits, arith?) lemma Fin_ile: "n \ Fin m ==> \k. n = Fin k" by (simp add: inat_defs split:inat_splits, arith?) lemma chain_incr: "\i. \j. Y i < Y j ==> \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