Theory Nat_Infinity

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theory Nat_Infinity
imports Main
begin

(*  Title:      HOL/Library/Nat_Infinity.thy
    ID:         $Id: Nat_Infinity.thy,v 1.17 2007/10/21 12:53:46 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

notation (xsymbols)
  Infty  ("∞")

notation (HTML output)
  Infty  ("∞")

instance inat :: "{ord, zero}" ..

definition
  iSuc :: "inat => inat" where
  "iSuc i = (case i of Fin n => Fin (Suc n) | ∞ => ∞)"

defs (overloaded)
  Zero_inat_def: "0 == Fin 0"
  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)

lemma Infty_ne_i0 [simp]: "∞ ≠ 0"
by (simp add: inat_defs split:inat_splits)

lemma i0_ne_Infty [simp]: "0 ≠ ∞"
by (simp add: inat_defs split:inat_splits)

lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
by (simp add: inat_defs split:inat_splits)

lemma iSuc_Infty [simp]: "iSuc ∞ = ∞"
by (simp add: inat_defs split:inat_splits)

lemma iSuc_ne_0 [simp]: "iSuc n ≠ 0"
by (simp add: inat_defs split:inat_splits)

lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
by (simp add: inat_defs split:inat_splits)


subsection "Ordering relations"

lemma Infty_ilessE [elim!]: "∞ < Fin m ==> R"
by (simp add: inat_defs split:inat_splits)

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)

lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
by (simp add: inat_defs split:inat_splits)

lemma iless_not_sym: "n < m ==> ¬ m < (n::inat)"
by (simp add: inat_defs split:inat_splits)

lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
by (simp add: inat_defs split:inat_splits)

lemma Fin_iless_Infty [simp]: "Fin n < ∞"
by (simp add: inat_defs split:inat_splits)

lemma Infty_eq [simp]: "(n < ∞) = (n ≠ ∞)"
by (simp add: inat_defs split:inat_splits)

lemma i0_eq [simp]: "((0::inat) < n) = (n ≠ 0)"
by (fastsimp simp: inat_defs split:inat_splits)

lemma i0_iless_iSuc [simp]: "0 < iSuc n"
by (simp add: inat_defs split:inat_splits)

lemma not_ilessi0 [simp]: "¬ n < (0::inat)"
by (simp add: inat_defs split:inat_splits)

lemma Fin_iless: "n < Fin m ==> ∃k. n = Fin k"
by (simp add: inat_defs split:inat_splits)

lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
by (simp add: inat_defs split:inat_splits)



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)

lemma ile_trans: "i ≤ j ==> j ≤ k ==> i ≤ (k::inat)"
by (simp add: inat_defs split:inat_splits)

lemma ile_iless_trans: "i ≤ j ==> j < k ==> i < (k::inat)"
by (simp add: inat_defs split:inat_splits)

lemma iless_ile_trans: "i < j ==> j ≤ k ==> i < (k::inat)"
by (simp add: inat_defs split:inat_splits)

lemma Infty_ub [simp]: "n ≤ ∞"
by (simp add: inat_defs split:inat_splits)

lemma i0_lb [simp]: "(0::inat) ≤ n"
by (simp add: inat_defs split:inat_splits)

lemma Infty_ileE [elim!]: "∞ ≤ Fin m ==> R"
by (simp add: inat_defs split:inat_splits)

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)

lemma ileI1: "m < n ==> iSuc m ≤ n"
by (simp add: inat_defs split:inat_splits)

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)

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)

lemma ile_iSuc [simp]: "n ≤ iSuc n"
by (simp add: inat_defs split:inat_splits)

lemma Fin_ile: "n ≤ Fin m ==> ∃k. n = Fin k"
by (simp add: inat_defs split:inat_splits)

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

Definitions

lemma 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

lemma 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))

Constructors

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)

Ordering relations

lemma Infty_ilessE:

  < Fin m ==> R

lemma iless_linear:

  m < nm = nn < 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 < nm = 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