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theory Nat_Infinity(* Title: HOL/Library/Nat_Infinity.thy ID: $Id: Nat_Infinity.thy,v 1.20 2008/02/18 01:10:55 huffman Exp $ Author: David von Oheimb, TU Muenchen *) header {* Natural numbers with infinity *} theory Nat_Infinity imports ATP_Linkup 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 ("∞") definition iSuc :: "inat => inat" where "iSuc i = (case i of Fin n => Fin (Suc n) | ∞ => ∞)" instantiation inat :: "{ord, zero}" begin definition Zero_inat_def: "0 == Fin 0" definition iless_def: "m < n == case m of Fin m1 => (case n of Fin n1 => m1 < n1 | ∞ => True) | ∞ => False" definition ile_def: "m ≤ n == case n of Fin n1 => (case m of Fin m1 => m1 ≤ n1 | ∞ => False) | ∞ => True" instance .. end 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" instance inat :: linorder proof fix x :: inat show "x ≤ x" by (simp add: inat_defs split: inat_splits) next fix x y :: inat assume "x ≤ y" and "y ≤ x" thus "x = y" by (simp add: inat_defs split: inat_splits) next fix x y z :: inat assume "x ≤ y" and "y ≤ z" thus "x ≤ z" by (simp add: inat_defs split: inat_splits) next fix x y :: inat show "(x < y) = (x ≤ y ∧ x ≠ y)" by (simp add: inat_defs order_less_le split: inat_splits) next fix x y :: inat show "x ≤ y ∨ y ≤ x" by (simp add: inat_defs linorder_linear split: inat_splits) qed 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 (rule linorder_less_linear) lemma iless_not_refl: "¬ n < (n::inat)" by (rule order_less_irrefl) lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)" by (rule order_less_trans) lemma iless_not_sym: "n < m ==> ¬ m < (n::inat)" by (rule order_less_not_sym) 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 (rule order_le_less) lemma ile_refl [simp]: "n ≤ (n::inat)" by (rule order_refl) lemma ile_trans: "i ≤ j ==> j ≤ k ==> i ≤ (k::inat)" by (rule order_trans) lemma ile_iless_trans: "i ≤ j ==> j < k ==> i < (k::inat)" by (rule order_le_less_trans) lemma iless_ile_trans: "i < j ==> j ≤ k ==> i < (k::inat)" by (rule order_less_le_trans) 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) lemma ilessI1: "n ≤ m ==> n ≠ m ==> n < (m::inat)" by (rule order_le_neq_trans) 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 [OF 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 subsection "Well-ordering" lemma less_FinE: "[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P" by (induct n) auto lemma less_InftyE: "[| n < Infty; !!k. n = Fin k ==> P |] ==> P" by (induct n) auto lemma inat_less_induct: assumes prem: "!!n. ∀m::inat. m < n --> P m ==> P n" shows "P n" proof - have P_Fin: "!!k. P (Fin k)" apply (rule nat_less_induct) apply (rule prem, clarify) apply (erule less_FinE, simp) done show ?thesis proof (induct n) fix nat show "P (Fin nat)" by (rule P_Fin) next show "P Infty" apply (rule prem, clarify) apply (erule less_InftyE) apply (simp add: P_Fin) done qed qed instance inat :: wellorder proof show "wf {(x::inat, y::inat). x < y}" proof (rule wfUNIVI) fix P and x :: inat assume "∀x::inat. (∀y. (y, x) ∈ {(x, y). x < y} --> P y) --> P x" hence 1: "!!x::inat. ALL y. y < x --> P y ==> P x" by fast thus "P x" by (rule inat_less_induct) qed qed end
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 ==
case n of Fin n1 => case m of Fin m1 => m1 ≤ n1 | ∞ => False | ∞ => True
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))
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
lemma less_FinE:
[| n < Fin m; !!k. [| n = Fin k; k < m |] ==> P |] ==> P
lemma less_InftyE:
[| n < ∞; !!k. n = Fin k ==> P |] ==> P
lemma inat_less_induct:
(!!n. ∀m<n. P m ==> P n) ==> P n