(* Title: HOL/Library/Nat_Infinity.thy ID: $Id: Nat_Infinity.thy,v 1.11 2004/08/18 09:10:01 nipkow Exp $ 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 ≤"}. *} datatype inat = Fin nat | Infty instance inat :: "{ord, zero}" .. consts iSuc :: "inat => inat" syntax (xsymbols) Infty :: inat ("∞") syntax (HTML output) Infty :: inat ("∞") defs Zero_inat_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 = 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]: "∞ ≠ 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
lemmas inat_defs:
0 == Fin 0
iSuc i == case i of Fin n => Fin (Suc n) | ∞ => ∞
m < n == case m of Fin m1 => case n of Fin n1 => m1 < n1 | ∞ => True | ∞ => False
m ≤ n == ¬ n < m
lemmas inat_defs:
0 == Fin 0
iSuc i == case i of Fin n => Fin (Suc n) | ∞ => ∞
m < n == case m of Fin m1 => case n of Fin n1 => m1 < n1 | ∞ => True | ∞ => False
m ≤ n == ¬ n < m
lemmas inat_splits:
P (inat_case f1.0 f2.0 x) = ((∀nat. x = Fin nat --> P (f1.0 nat)) ∧ (x = ∞ --> P f2.0))
P (inat_case f1.0 f2.0 x) = (¬ ((∃nat. x = Fin nat ∧ ¬ P (f1.0 nat)) ∨ x = ∞ ∧ ¬ P f2.0))
lemmas inat_splits:
P (inat_case f1.0 f2.0 x) = ((∀nat. x = Fin nat --> P (f1.0 nat)) ∧ (x = ∞ --> P f2.0))
P (inat_case f1.0 f2.0 x) = (¬ ((∃nat. x = Fin nat ∧ ¬ P (f1.0 nat)) ∨ x = ∞ ∧ ¬ P f2.0))
lemma Fin_0:
Fin 0 = 0
lemma Infty_ne_i0:
∞ ≠ 0
lemma i0_ne_Infty:
0 ≠ ∞
lemma iSuc_Fin:
iSuc (Fin n) = Fin (Suc n)
lemma iSuc_Infty:
iSuc ∞ = ∞
lemma iSuc_ne_0:
iSuc n ≠ 0
lemma iSuc_inject:
(iSuc x = iSuc y) = (x = y)
lemma Infty_ilessE:
∞ < Fin m ==> R
lemma iless_linear:
m < n ∨ m = n ∨ n < m
lemma iless_not_refl:
¬ n < n
lemma iless_trans:
[| i < j; j < k |] ==> i < k
lemma iless_not_sym:
n < m ==> ¬ m < n
lemma Fin_iless_mono:
(Fin n < Fin m) = (n < m)
lemma Fin_iless_Infty:
Fin n < ∞
lemma Infty_eq:
(n < ∞) = (n ≠ ∞)
lemma i0_eq:
(0 < n) = (n ≠ 0)
lemma i0_iless_iSuc:
0 < iSuc n
lemma not_ilessi0:
¬ n < 0
lemma Fin_iless:
n < Fin m ==> ∃k. n = Fin k
lemma iSuc_mono:
(iSuc n < iSuc m) = (n < m)
lemma ile_def2:
(m ≤ n) = (m < n ∨ m = n)
lemma ile_refl:
n ≤ n
lemma ile_trans:
[| i ≤ j; j ≤ k |] ==> i ≤ k
lemma ile_iless_trans:
[| i ≤ j; j < k |] ==> i < k
lemma iless_ile_trans:
[| i < j; j ≤ k |] ==> i < k
lemma Infty_ub:
n ≤ ∞
lemma i0_lb:
0 ≤ n
lemma Infty_ileE:
∞ ≤ Fin m ==> R
lemma Fin_ile_mono:
(Fin n ≤ Fin m) = (n ≤ m)
lemma ilessI1:
[| n ≤ m; n ≠ m |] ==> n < m
lemma ileI1:
m < n ==> iSuc m ≤ n
lemma Suc_ile_eq:
(Fin (Suc m) ≤ n) = (Fin m < n)
lemma iSuc_ile_mono:
(iSuc n ≤ iSuc m) = (n ≤ m)
lemma iless_Suc_eq:
(Fin m < iSuc n) = (Fin m ≤ n)
lemma not_iSuc_ilei0:
¬ iSuc n ≤ 0
lemma ile_iSuc:
n ≤ iSuc n
lemma Fin_ile:
n ≤ Fin m ==> ∃k. n = Fin k
lemma chain_incr:
∀i. ∃j. Y i < Y j ==> ∃j. Fin k < Y j