src/HOL/Library/Nat_Infinity.thy
 author paulson Tue Jun 28 15:27:45 2005 +0200 (2005-06-28) changeset 16587 b34c8aa657a5 parent 15140 322485b816ac child 19736 d8d0f8f51d69 permissions -rw-r--r--
Constant "If" is now local
1 (*  Title:      HOL/Library/Nat_Infinity.thy
2     ID:         \$Id\$
3     Author:     David von Oheimb, TU Muenchen
4 *)
6 header {* Natural numbers with infinity *}
8 theory Nat_Infinity
9 imports Main
10 begin
12 subsection "Definitions"
14 text {*
15   We extend the standard natural numbers by a special value indicating
16   infinity.  This includes extending the ordering relations @{term "op
17   <"} and @{term "op \<le>"}.
18 *}
20 datatype inat = Fin nat | Infty
22 instance inat :: "{ord, zero}" ..
24 consts
25   iSuc :: "inat => inat"
27 syntax (xsymbols)
28   Infty :: inat    ("\<infinity>")
30 syntax (HTML output)
31   Infty :: inat    ("\<infinity>")
33 defs
34   Zero_inat_def: "0 == Fin 0"
35   iSuc_def: "iSuc i == case i of Fin n  => Fin (Suc n) | \<infinity> => \<infinity>"
36   iless_def: "m < n ==
37     case m of Fin m1 => (case n of Fin n1 => m1 < n1 | \<infinity> => True)
38     | \<infinity>  => False"
39   ile_def: "(m::inat) \<le> n == \<not> (n < m)"
41 lemmas inat_defs = Zero_inat_def iSuc_def iless_def ile_def
42 lemmas inat_splits = inat.split inat.split_asm
44 text {*
45   Below is a not quite complete set of theorems.  Use the method
46   @{text "(simp add: inat_defs split:inat_splits, arith?)"} to prove
47   new theorems or solve arithmetic subgoals involving @{typ inat} on
48   the fly.
49 *}
51 subsection "Constructors"
53 lemma Fin_0: "Fin 0 = 0"
54   by (simp add: inat_defs split:inat_splits, arith?)
56 lemma Infty_ne_i0 [simp]: "\<infinity> \<noteq> 0"
57   by (simp add: inat_defs split:inat_splits, arith?)
59 lemma i0_ne_Infty [simp]: "0 \<noteq> \<infinity>"
60   by (simp add: inat_defs split:inat_splits, arith?)
62 lemma iSuc_Fin [simp]: "iSuc (Fin n) = Fin (Suc n)"
63   by (simp add: inat_defs split:inat_splits, arith?)
65 lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>"
66   by (simp add: inat_defs split:inat_splits, arith?)
68 lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0"
69   by (simp add: inat_defs split:inat_splits, arith?)
71 lemma iSuc_inject [simp]: "(iSuc x = iSuc y) = (x = y)"
72   by (simp add: inat_defs split:inat_splits, arith?)
75 subsection "Ordering relations"
77 lemma Infty_ilessE [elim!]: "\<infinity> < Fin m ==> R"
78   by (simp add: inat_defs split:inat_splits, arith?)
80 lemma iless_linear: "m < n \<or> m = n \<or> n < (m::inat)"
81   by (simp add: inat_defs split:inat_splits, arith?)
83 lemma iless_not_refl [simp]: "\<not> n < (n::inat)"
84   by (simp add: inat_defs split:inat_splits, arith?)
86 lemma iless_trans: "i < j ==> j < k ==> i < (k::inat)"
87   by (simp add: inat_defs split:inat_splits, arith?)
89 lemma iless_not_sym: "n < m ==> \<not> m < (n::inat)"
90   by (simp add: inat_defs split:inat_splits, arith?)
92 lemma Fin_iless_mono [simp]: "(Fin n < Fin m) = (n < m)"
93   by (simp add: inat_defs split:inat_splits, arith?)
95 lemma Fin_iless_Infty [simp]: "Fin n < \<infinity>"
96   by (simp add: inat_defs split:inat_splits, arith?)
98 lemma Infty_eq [simp]: "(n < \<infinity>) = (n \<noteq> \<infinity>)"
99   by (simp add: inat_defs split:inat_splits, arith?)
101 lemma i0_eq [simp]: "((0::inat) < n) = (n \<noteq> 0)"
102   by (simp add: inat_defs split:inat_splits, arith?)
104 lemma i0_iless_iSuc [simp]: "0 < iSuc n"
105   by (simp add: inat_defs split:inat_splits, arith?)
107 lemma not_ilessi0 [simp]: "\<not> n < (0::inat)"
108   by (simp add: inat_defs split:inat_splits, arith?)
110 lemma Fin_iless: "n < Fin m ==> \<exists>k. n = Fin k"
111   by (simp add: inat_defs split:inat_splits, arith?)
113 lemma iSuc_mono [simp]: "(iSuc n < iSuc m) = (n < m)"
114   by (simp add: inat_defs split:inat_splits, arith?)
117 (* ----------------------------------------------------------------------- *)
119 lemma ile_def2: "(m \<le> n) = (m < n \<or> m = (n::inat))"
120   by (simp add: inat_defs split:inat_splits, arith?)
122 lemma ile_refl [simp]: "n \<le> (n::inat)"
123   by (simp add: inat_defs split:inat_splits, arith?)
125 lemma ile_trans: "i \<le> j ==> j \<le> k ==> i \<le> (k::inat)"
126   by (simp add: inat_defs split:inat_splits, arith?)
128 lemma ile_iless_trans: "i \<le> j ==> j < k ==> i < (k::inat)"
129   by (simp add: inat_defs split:inat_splits, arith?)
131 lemma iless_ile_trans: "i < j ==> j \<le> k ==> i < (k::inat)"
132   by (simp add: inat_defs split:inat_splits, arith?)
134 lemma Infty_ub [simp]: "n \<le> \<infinity>"
135   by (simp add: inat_defs split:inat_splits, arith?)
137 lemma i0_lb [simp]: "(0::inat) \<le> n"
138   by (simp add: inat_defs split:inat_splits, arith?)
140 lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m ==> R"
141   by (simp add: inat_defs split:inat_splits, arith?)
143 lemma Fin_ile_mono [simp]: "(Fin n \<le> Fin m) = (n \<le> m)"
144   by (simp add: inat_defs split:inat_splits, arith?)
146 lemma ilessI1: "n \<le> m ==> n \<noteq> m ==> n < (m::inat)"
147   by (simp add: inat_defs split:inat_splits, arith?)
149 lemma ileI1: "m < n ==> iSuc m \<le> n"
150   by (simp add: inat_defs split:inat_splits, arith?)
152 lemma Suc_ile_eq: "(Fin (Suc m) \<le> n) = (Fin m < n)"
153   by (simp add: inat_defs split:inat_splits, arith?)
155 lemma iSuc_ile_mono [simp]: "(iSuc n \<le> iSuc m) = (n \<le> m)"
156   by (simp add: inat_defs split:inat_splits, arith?)
158 lemma iless_Suc_eq [simp]: "(Fin m < iSuc n) = (Fin m \<le> n)"
159   by (simp add: inat_defs split:inat_splits, arith?)
161 lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0"
162   by (simp add: inat_defs split:inat_splits, arith?)
164 lemma ile_iSuc [simp]: "n \<le> iSuc n"
165   by (simp add: inat_defs split:inat_splits, arith?)
167 lemma Fin_ile: "n \<le> Fin m ==> \<exists>k. n = Fin k"
168   by (simp add: inat_defs split:inat_splits, arith?)
170 lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j"
171   apply (induct_tac k)
172    apply (simp (no_asm) only: Fin_0)
173    apply (fast intro: ile_iless_trans i0_lb)
174   apply (erule exE)
175   apply (drule spec)
176   apply (erule exE)
177   apply (drule ileI1)
178   apply (rule iSuc_Fin [THEN subst])
179   apply (rule exI)
180   apply (erule (1) ile_iless_trans)
181   done
183 end