Implemented trick (due to Tobias Nipkow) for fine-tuning simplification
of premises of congruence rules.
(* Title: HOL/Library/Nat_Infinity.thy
ID: $Id$
Author: David von Oheimb, TU Muenchen
*)
header {* Natural numbers with infinity *}
theory Nat_Infinity
imports Main
begin
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, 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