--- a/src/HOL/Library/Nat_Infinity.thy Tue Jun 10 15:31:01 2008 +0200
+++ b/src/HOL/Library/Nat_Infinity.thy Tue Jun 10 15:31:02 2008 +0200
@@ -1,6 +1,6 @@
(* Title: HOL/Library/Nat_Infinity.thy
ID: $Id$
- Author: David von Oheimb, TU Muenchen
+ Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen
*)
header {* Natural numbers with infinity *}
@@ -9,12 +9,11 @@
imports ATP_Linkup
begin
-subsection "Definitions"
+subsection {* Type definition *}
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>"}.
+ infinity.
*}
datatype inat = Fin nat | Infty
@@ -25,196 +24,267 @@
notation (HTML output)
Infty ("\<infinity>")
-definition
- iSuc :: "inat => inat" where
- "iSuc i = (case i of Fin n => Fin (Suc n) | \<infinity> => \<infinity>)"
-instantiation inat :: "{ord, zero}"
+subsection {* Constructors and numbers *}
+
+instantiation inat :: "{zero, one, number}"
begin
definition
- Zero_inat_def: "0 == Fin 0"
+ "0 = Fin 0"
definition
- iless_def: "m < n ==
- case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
- | \<infinity> => False"
+ [code inline]: "1 = Fin 1"
definition
- ile_def: "m \<le> n ==
- case n of Fin n1 => (case m of Fin m1 => m1 \<le> n1 | \<infinity> => False)
- | \<infinity> => True"
+ [code inline, code func del]: "number_of k = Fin (number_of k)"
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"
+definition iSuc :: "inat \<Rightarrow> inat" where
+ "iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)"
lemma Fin_0: "Fin 0 = 0"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: zero_inat_def)
+
+lemma Fin_1: "Fin 1 = 1"
+ by (simp add: one_inat_def)
+
+lemma Fin_number: "Fin (number_of k) = number_of k"
+ by (simp add: number_of_inat_def)
+
+lemma one_iSuc: "1 = iSuc 0"
+ by (simp add: zero_inat_def one_inat_def iSuc_def)
lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: zero_inat_def)
lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: zero_inat_def)
+
+lemma zero_inat_eq [simp]:
+ "number_of k = (0\<Colon>inat) \<longleftrightarrow> number_of k = (0\<Colon>nat)"
+ "(0\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)"
+ unfolding zero_inat_def number_of_inat_def by simp_all
+
+lemma one_inat_eq [simp]:
+ "number_of k = (1\<Colon>inat) \<longleftrightarrow> number_of k = (1\<Colon>nat)"
+ "(1\<Colon>inat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)"
+ unfolding one_inat_def number_of_inat_def by simp_all
+
+lemma zero_one_inat_neq [simp]:
+ "\<not> 0 = (1\<Colon>inat)"
+ "\<not> 1 = (0\<Colon>inat)"
+ unfolding zero_inat_def one_inat_def by simp_all
-lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
-by (simp add: inat_defs split:inat_splits)
+lemma Infty_ne_i1 [simp]: "\<infinity> \<noteq> 1"
+ by (simp add: one_inat_def)
+
+lemma i1_ne_Infty [simp]: "1 \<noteq> \<infinity>"
+ by (simp add: one_inat_def)
+
+lemma Infty_ne_number [simp]: "\<infinity> \<noteq> number_of k"
+ by (simp add: number_of_inat_def)
+
+lemma number_ne_Infty [simp]: "number_of k \<noteq> \<infinity>"
+ by (simp add: number_of_inat_def)
+
+lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)"
+ by (simp add: iSuc_def)
+
+lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))"
+ by (simp add: iSuc_Fin number_of_inat_def)
lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: iSuc_def)
lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: iSuc_def zero_inat_def split: inat.splits)
+
+lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n"
+ by (rule iSuc_ne_0 [symmetric])
-lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
-by (simp add: inat_defs split:inat_splits)
+lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n"
+ by (simp add: iSuc_def split: inat.splits)
+
+lemma number_of_inat_inject [simp]:
+ "(number_of k \<Colon> inat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l"
+ by (simp add: number_of_inat_def)
-subsection "Ordering relations"
+subsection {* Addition *}
+
+instantiation inat :: comm_monoid_add
+begin
+
+definition
+ [code del]: "m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))"
-instance inat :: linorder
-proof
- fix x :: inat
- show "x \<le> x"
- by (simp add: inat_defs split: inat_splits)
-next
- fix x y :: inat
- assume "x \<le> y" and "y \<le> x" thus "x = y"
- by (simp add: inat_defs split: inat_splits)
-next
- fix x y z :: inat
- assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
- by (simp add: inat_defs split: inat_splits)
-next
- fix x y :: inat
- show "(x < y) = (x \<le> y \<and> x \<noteq> y)"
- by (simp add: inat_defs order_less_le split: inat_splits)
-next
- fix x y :: inat
- show "x \<le> y \<or> y \<le> x"
- by (simp add: inat_defs linorder_linear split: inat_splits)
+lemma plus_inat_simps [simp, code]:
+ "Fin m + Fin n = Fin (m + n)"
+ "\<infinity> + q = \<infinity>"
+ "q + \<infinity> = \<infinity>"
+ by (simp_all add: plus_inat_def split: inat.splits)
+
+instance proof
+ fix n m q :: inat
+ show "n + m + q = n + (m + q)"
+ by (cases n, auto, cases m, auto, cases q, auto)
+ show "n + m = m + n"
+ by (cases n, auto, cases m, auto)
+ show "0 + n = n"
+ by (cases n) (simp_all add: zero_inat_def)
qed
-lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
-by (simp add: inat_defs split:inat_splits)
-
-lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
-by (rule linorder_less_linear)
-
-lemma iless_not_refl: "\<not> n < (n::inat)"
-by (rule order_less_irrefl)
+end
-lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
-by (rule order_less_trans)
+lemma plus_inat_0 [simp]:
+ "0 + (q\<Colon>inat) = q"
+ "(q\<Colon>inat) + 0 = q"
+ by (simp_all add: plus_inat_def zero_inat_def split: inat.splits)
-lemma iless_not_sym: "n < m ==> \<not> 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 plus_inat_number [simp]:
+ "(number_of k \<Colon> inat) + number_of l = (if neg (number_of k \<Colon> int) then number_of l
+ else if neg (number_of l \<Colon> int) then number_of k else number_of (k + l))"
+ unfolding number_of_inat_def plus_inat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] ..
-lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
-by (simp add: inat_defs split:inat_splits)
-
-lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
-by (simp add: inat_defs split:inat_splits)
-
-lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
-by (fastsimp simp: inat_defs split:inat_splits)
+lemma iSuc_number [simp]:
+ "iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))"
+ unfolding iSuc_number_of
+ unfolding one_inat_def number_of_inat_def Suc_nat_number_of if_distrib [symmetric] ..
-lemma i0_iless_iSuc [simp]: "0 < iSuc n"
-by (simp add: inat_defs split:inat_splits)
-
-lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
-by (simp add: inat_defs split:inat_splits)
-
-lemma Fin_iless: "n < Fin m ==> \<exists>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 iSuc_plus_1:
+ "iSuc n = n + 1"
+ by (cases n) (simp_all add: iSuc_Fin one_inat_def)
+
+lemma plus_1_iSuc:
+ "1 + q = iSuc q"
+ "q + 1 = iSuc q"
+ unfolding iSuc_plus_1 by (simp_all add: add_ac)
-lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
-by (rule order_le_less)
+subsection {* Ordering *}
+
+instantiation inat :: ordered_ab_semigroup_add
+begin
-lemma ile_refl [simp]: "n \<le> (n::inat)"
-by (rule order_refl)
+definition
+ [code del]: "m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False)
+ | \<infinity> \<Rightarrow> True)"
-lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
-by (rule order_trans)
+definition
+ [code del]: "m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True)
+ | \<infinity> \<Rightarrow> False)"
-lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
-by (rule order_le_less_trans)
-
-lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
-by (rule order_less_le_trans)
+lemma inat_ord_simps [simp]:
+ "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
+ "Fin m < Fin n \<longleftrightarrow> m < n"
+ "q \<le> \<infinity>"
+ "q < \<infinity> \<longleftrightarrow> q \<noteq> \<infinity>"
+ "\<infinity> \<le> q \<longleftrightarrow> q = \<infinity>"
+ "\<infinity> < q \<longleftrightarrow> False"
+ by (simp_all add: less_eq_inat_def less_inat_def split: inat.splits)
-lemma Infty_ub [simp]: "n \<le> \<infinity>"
-by (simp add: inat_defs split:inat_splits)
+lemma inat_ord_code [code]:
+ "Fin m \<le> Fin n \<longleftrightarrow> m \<le> n"
+ "Fin m < Fin n \<longleftrightarrow> m < n"
+ "q \<le> \<infinity> \<longleftrightarrow> True"
+ "Fin m < \<infinity> \<longleftrightarrow> True"
+ "\<infinity> \<le> Fin n \<longleftrightarrow> False"
+ "\<infinity> < q \<longleftrightarrow> False"
+ by simp_all
-lemma i0_lb [simp]: "(0::inat) \<le> n"
-by (simp add: inat_defs split:inat_splits)
+instance by default
+ (auto simp add: less_eq_inat_def less_inat_def plus_inat_def split: inat.splits)
-lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
-by (simp add: inat_defs split:inat_splits)
+end
+
+lemma inat_ord_number [simp]:
+ "(number_of m \<Colon> inat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n"
+ "(number_of m \<Colon> inat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n"
+ by (simp_all add: number_of_inat_def)
-lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
-by (simp add: inat_defs split:inat_splits)
+lemma i0_lb [simp]: "(0\<Colon>inat) \<le> n"
+ by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
-lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
-by (rule order_le_neq_trans)
+lemma i0_neq [simp]: "n \<le> (0\<Colon>inat) \<longleftrightarrow> n = 0"
+ by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
+
+lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R"
+ by (simp add: zero_inat_def less_eq_inat_def split: inat.splits)
-lemma ileI1: "m < n ==> iSuc m \<le> n"
-by (simp add: inat_defs split:inat_splits)
+lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R"
+ by simp
-lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
-by (simp add: inat_defs split:inat_splits, arith)
+lemma not_ilessi0 [simp]: "\<not> n < (0\<Colon>inat)"
+ by (simp add: zero_inat_def less_inat_def split: inat.splits)
+
+lemma i0_eq [simp]: "(0\<Colon>inat) < n \<longleftrightarrow> n \<noteq> 0"
+ by (simp add: zero_inat_def less_inat_def split: inat.splits)
-lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
-by (simp add: inat_defs split:inat_splits)
+lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m"
+ by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
+
+lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m"
+ by (simp add: iSuc_def less_inat_def split: inat.splits)
-lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
-by (simp add: inat_defs split:inat_splits, arith)
+lemma ile_iSuc [simp]: "n \<le> iSuc n"
+ by (simp add: iSuc_def less_eq_inat_def split: inat.splits)
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
-by (simp add: inat_defs split:inat_splits)
+ by (simp add: zero_inat_def iSuc_def less_eq_inat_def split: inat.splits)
+
+lemma i0_iless_iSuc [simp]: "0 < iSuc n"
+ by (simp add: zero_inat_def iSuc_def less_inat_def split: inat.splits)
+
+lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n"
+ by (simp add: iSuc_def less_eq_inat_def less_inat_def split: inat.splits)
+
+lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n"
+ by (cases n) auto
+
+lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n"
+ by (auto simp add: iSuc_def less_inat_def split: inat.splits)
-lemma ile_iSuc [simp]: "n \<le> iSuc n"
-by (simp add: inat_defs split:inat_splits)
+lemma min_inat_simps [simp]:
+ "min (Fin m) (Fin n) = Fin (min m n)"
+ "min q 0 = 0"
+ "min 0 q = 0"
+ "min q \<infinity> = q"
+ "min \<infinity> q = q"
+ by (auto simp add: min_def)
-lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
-by (simp add: inat_defs split:inat_splits)
+lemma max_inat_simps [simp]:
+ "max (Fin m) (Fin n) = Fin (max m n)"
+ "max q 0 = q"
+ "max 0 q = q"
+ "max q \<infinity> = \<infinity>"
+ "max \<infinity> q = \<infinity>"
+ by (simp_all add: max_def)
+
+lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k"
+ by (cases n) simp_all
+
+lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k"
+ by (cases n) simp_all
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 [OF i0_lb])
+ apply (fast intro: le_less_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)
+apply (erule (1) le_less_trans)
done
-subsection "Well-ordering"
+subsection {* Well-ordering *}
lemma less_FinE:
"[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P"
@@ -256,4 +326,12 @@
qed
qed
+
+subsection {* Traditional theorem names *}
+
+lemmas inat_defs = zero_inat_def one_inat_def number_of_inat_def iSuc_def
+ plus_inat_def less_eq_inat_def less_inat_def
+
+lemmas inat_splits = inat.splits
+
end