author | hoelzl |
Tue, 19 Jul 2011 14:38:29 +0200 | |
changeset 43923 | ab93d0190a5d |
parent 43922 | c6f35921056e |
child 43924 | 1165fe965da8 |
permissions | -rw-r--r-- |
43919 | 1 |
(* Title: HOL/Library/Extended_Nat.thy |
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Author: David von Oheimb, TU Muenchen; Florian Haftmann, TU Muenchen |
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Contributions: David Trachtenherz, TU Muenchen |
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*) |
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|
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header {* Extended natural numbers (i.e. with infinity) *} |
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theory Extended_Nat |
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Main is (Complex_Main) base entry point in library theories
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imports Main |
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begin |
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class infinity = |
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fixes infinity :: "'a" |
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||
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notation (xsymbols) |
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infinity ("\<infinity>") |
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||
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notation (HTML output) |
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infinity ("\<infinity>") |
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||
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subsection {* Type definition *} |
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text {* |
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We extend the standard natural numbers by a special value indicating |
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infinity. |
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*} |
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typedef (open) enat = "UNIV :: nat option set" .. |
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||
30 |
definition Fin :: "nat \<Rightarrow> enat" where |
|
31 |
"Fin n = Abs_enat (Some n)" |
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32 |
||
33 |
instantiation enat :: infinity |
|
34 |
begin |
|
35 |
definition "\<infinity> = Abs_enat None" |
|
36 |
instance proof qed |
|
37 |
end |
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38 |
||
39 |
rep_datatype Fin "\<infinity> :: enat" |
|
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proof - |
|
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fix P i assume "\<And>j. P (Fin j)" "P \<infinity>" |
|
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then show "P i" |
|
43 |
proof induct |
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case (Abs_enat y) then show ?case |
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45 |
by (cases y rule: option.exhaust) |
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(auto simp: Fin_def infinity_enat_def) |
|
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qed |
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qed (auto simp add: Fin_def infinity_enat_def Abs_enat_inject) |
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|
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declare [[coercion "Fin::nat\<Rightarrow>enat"]] |
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|
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lemma not_Infty_eq[iff]: "(x \<noteq> \<infinity>) = (EX i. x = Fin i)" |
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by (cases x) auto |
54 |
||
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lemma not_Fin_eq [iff]: "(ALL y. x ~= Fin y) = (x = \<infinity>)" |
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by (cases x) auto |
57 |
||
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primrec the_Fin :: "enat \<Rightarrow> nat" |
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where "the_Fin (Fin n) = n" |
60 |
||
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subsection {* Constructors and numbers *} |
62 |
||
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instantiation enat :: "{zero, one, number}" |
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begin |
65 |
||
66 |
definition |
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"0 = Fin 0" |
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definition |
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[code_unfold]: "1 = Fin 1" |
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definition |
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[code_unfold, code del]: "number_of k = Fin (number_of k)" |
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instance .. |
76 |
||
77 |
end |
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||
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definition iSuc :: "enat \<Rightarrow> enat" where |
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"iSuc i = (case i of Fin n \<Rightarrow> Fin (Suc n) | \<infinity> \<Rightarrow> \<infinity>)" |
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lemma Fin_0: "Fin 0 = 0" |
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by (simp add: zero_enat_def) |
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|
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lemma Fin_1: "Fin 1 = 1" |
|
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by (simp add: one_enat_def) |
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lemma Fin_number: "Fin (number_of k) = number_of k" |
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by (simp add: number_of_enat_def) |
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lemma one_iSuc: "1 = iSuc 0" |
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by (simp add: zero_enat_def one_enat_def iSuc_def) |
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lemma Infty_ne_i0 [simp]: "(\<infinity>::enat) \<noteq> 0" |
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by (simp add: zero_enat_def) |
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lemma i0_ne_Infty [simp]: "0 \<noteq> (\<infinity>::enat)" |
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by (simp add: zero_enat_def) |
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|
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lemma zero_enat_eq [simp]: |
101 |
"number_of k = (0\<Colon>enat) \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
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"(0\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (0\<Colon>nat)" |
|
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unfolding zero_enat_def number_of_enat_def by simp_all |
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|
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lemma one_enat_eq [simp]: |
106 |
"number_of k = (1\<Colon>enat) \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
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"(1\<Colon>enat) = number_of k \<longleftrightarrow> number_of k = (1\<Colon>nat)" |
|
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unfolding one_enat_def number_of_enat_def by simp_all |
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lemma zero_one_enat_neq [simp]: |
111 |
"\<not> 0 = (1\<Colon>enat)" |
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"\<not> 1 = (0\<Colon>enat)" |
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unfolding zero_enat_def one_enat_def by simp_all |
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lemma Infty_ne_i1 [simp]: "(\<infinity>::enat) \<noteq> 1" |
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by (simp add: one_enat_def) |
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|
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lemma i1_ne_Infty [simp]: "1 \<noteq> (\<infinity>::enat)" |
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by (simp add: one_enat_def) |
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|
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lemma Infty_ne_number [simp]: "(\<infinity>::enat) \<noteq> number_of k" |
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by (simp add: number_of_enat_def) |
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|
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lemma number_ne_Infty [simp]: "number_of k \<noteq> (\<infinity>::enat)" |
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by (simp add: number_of_enat_def) |
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|
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lemma iSuc_Fin: "iSuc (Fin n) = Fin (Suc n)" |
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128 |
by (simp add: iSuc_def) |
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lemma iSuc_number_of: "iSuc (number_of k) = Fin (Suc (number_of k))" |
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by (simp add: iSuc_Fin number_of_enat_def) |
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lemma iSuc_Infty [simp]: "iSuc \<infinity> = \<infinity>" |
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by (simp add: iSuc_def) |
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lemma iSuc_ne_0 [simp]: "iSuc n \<noteq> 0" |
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by (simp add: iSuc_def zero_enat_def split: enat.splits) |
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|
139 |
lemma zero_ne_iSuc [simp]: "0 \<noteq> iSuc n" |
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by (rule iSuc_ne_0 [symmetric]) |
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141 |
|
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lemma iSuc_inject [simp]: "iSuc m = iSuc n \<longleftrightarrow> m = n" |
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by (simp add: iSuc_def split: enat.splits) |
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|
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lemma number_of_enat_inject [simp]: |
146 |
"(number_of k \<Colon> enat) = number_of l \<longleftrightarrow> (number_of k \<Colon> nat) = number_of l" |
|
147 |
by (simp add: number_of_enat_def) |
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subsection {* Addition *} |
151 |
||
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instantiation enat :: comm_monoid_add |
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begin |
154 |
||
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definition [nitpick_simp]: |
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"m + n = (case m of \<infinity> \<Rightarrow> \<infinity> | Fin m \<Rightarrow> (case n of \<infinity> \<Rightarrow> \<infinity> | Fin n \<Rightarrow> Fin (m + n)))" |
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|
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lemma plus_enat_simps [simp, code]: |
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fixes q :: enat |
160 |
shows "Fin m + Fin n = Fin (m + n)" |
|
161 |
and "\<infinity> + q = \<infinity>" |
|
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and "q + \<infinity> = \<infinity>" |
|
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by (simp_all add: plus_enat_def split: enat.splits) |
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|
165 |
instance proof |
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fix n m q :: enat |
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show "n + m + q = n + (m + q)" |
168 |
by (cases n, auto, cases m, auto, cases q, auto) |
|
169 |
show "n + m = m + n" |
|
170 |
by (cases n, auto, cases m, auto) |
|
171 |
show "0 + n = n" |
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by (cases n) (simp_all add: zero_enat_def) |
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qed |
174 |
||
27110 | 175 |
end |
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lemma plus_enat_0 [simp]: |
178 |
"0 + (q\<Colon>enat) = q" |
|
179 |
"(q\<Colon>enat) + 0 = q" |
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180 |
by (simp_all add: plus_enat_def zero_enat_def split: enat.splits) |
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181 |
|
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lemma plus_enat_number [simp]: |
183 |
"(number_of k \<Colon> enat) + number_of l = (if k < Int.Pls then number_of l |
|
29012 | 184 |
else if l < Int.Pls then number_of k else number_of (k + l))" |
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unfolding number_of_enat_def plus_enat_simps nat_arith(1) if_distrib [symmetric, of _ Fin] .. |
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186 |
|
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lemma iSuc_number [simp]: |
188 |
"iSuc (number_of k) = (if neg (number_of k \<Colon> int) then 1 else number_of (Int.succ k))" |
|
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unfolding iSuc_number_of |
|
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unfolding one_enat_def number_of_enat_def Suc_nat_number_of if_distrib [symmetric] .. |
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191 |
|
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lemma iSuc_plus_1: |
193 |
"iSuc n = n + 1" |
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by (cases n) (simp_all add: iSuc_Fin one_enat_def) |
27110 | 195 |
|
196 |
lemma plus_1_iSuc: |
|
197 |
"1 + q = iSuc q" |
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"q + 1 = iSuc q" |
|
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by (simp_all add: iSuc_plus_1 add_ac) |
200 |
||
201 |
lemma iadd_Suc: "iSuc m + n = iSuc (m + n)" |
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by (simp_all add: iSuc_plus_1 add_ac) |
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203 |
|
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lemma iadd_Suc_right: "m + iSuc n = iSuc (m + n)" |
205 |
by (simp only: add_commute[of m] iadd_Suc) |
|
206 |
||
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lemma iadd_is_0: "(m + n = (0::enat)) = (m = 0 \<and> n = 0)" |
208 |
by (cases m, cases n, simp_all add: zero_enat_def) |
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209 |
|
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subsection {* Multiplication *} |
211 |
||
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instantiation enat :: comm_semiring_1 |
29014 | 213 |
begin |
214 |
||
43919 | 215 |
definition times_enat_def [nitpick_simp]: |
29014 | 216 |
"m * n = (case m of \<infinity> \<Rightarrow> if n = 0 then 0 else \<infinity> | Fin m \<Rightarrow> |
217 |
(case n of \<infinity> \<Rightarrow> if m = 0 then 0 else \<infinity> | Fin n \<Rightarrow> Fin (m * n)))" |
|
218 |
||
43919 | 219 |
lemma times_enat_simps [simp, code]: |
29014 | 220 |
"Fin m * Fin n = Fin (m * n)" |
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"\<infinity> * \<infinity> = (\<infinity>::enat)" |
29014 | 222 |
"\<infinity> * Fin n = (if n = 0 then 0 else \<infinity>)" |
223 |
"Fin m * \<infinity> = (if m = 0 then 0 else \<infinity>)" |
|
43919 | 224 |
unfolding times_enat_def zero_enat_def |
225 |
by (simp_all split: enat.split) |
|
29014 | 226 |
|
227 |
instance proof |
|
43919 | 228 |
fix a b c :: enat |
29014 | 229 |
show "(a * b) * c = a * (b * c)" |
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unfolding times_enat_def zero_enat_def |
231 |
by (simp split: enat.split) |
|
29014 | 232 |
show "a * b = b * a" |
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unfolding times_enat_def zero_enat_def |
234 |
by (simp split: enat.split) |
|
29014 | 235 |
show "1 * a = a" |
43919 | 236 |
unfolding times_enat_def zero_enat_def one_enat_def |
237 |
by (simp split: enat.split) |
|
29014 | 238 |
show "(a + b) * c = a * c + b * c" |
43919 | 239 |
unfolding times_enat_def zero_enat_def |
240 |
by (simp split: enat.split add: left_distrib) |
|
29014 | 241 |
show "0 * a = 0" |
43919 | 242 |
unfolding times_enat_def zero_enat_def |
243 |
by (simp split: enat.split) |
|
29014 | 244 |
show "a * 0 = 0" |
43919 | 245 |
unfolding times_enat_def zero_enat_def |
246 |
by (simp split: enat.split) |
|
247 |
show "(0::enat) \<noteq> 1" |
|
248 |
unfolding zero_enat_def one_enat_def |
|
29014 | 249 |
by simp |
250 |
qed |
|
251 |
||
252 |
end |
|
253 |
||
254 |
lemma mult_iSuc: "iSuc m * n = n + m * n" |
|
29667 | 255 |
unfolding iSuc_plus_1 by (simp add: algebra_simps) |
29014 | 256 |
|
257 |
lemma mult_iSuc_right: "m * iSuc n = m + m * n" |
|
29667 | 258 |
unfolding iSuc_plus_1 by (simp add: algebra_simps) |
29014 | 259 |
|
29023 | 260 |
lemma of_nat_eq_Fin: "of_nat n = Fin n" |
261 |
apply (induct n) |
|
262 |
apply (simp add: Fin_0) |
|
263 |
apply (simp add: plus_1_iSuc iSuc_Fin) |
|
264 |
done |
|
265 |
||
43919 | 266 |
instance enat :: number_semiring |
43532 | 267 |
proof |
43919 | 268 |
fix n show "number_of (int n) = (of_nat n :: enat)" |
269 |
unfolding number_of_enat_def number_of_int of_nat_id of_nat_eq_Fin .. |
|
43532 | 270 |
qed |
271 |
||
43919 | 272 |
instance enat :: semiring_char_0 proof |
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273 |
have "inj Fin" by (rule injI) simp |
43919 | 274 |
then show "inj (\<lambda>n. of_nat n :: enat)" by (simp add: of_nat_eq_Fin) |
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275 |
qed |
29023 | 276 |
|
43919 | 277 |
lemma imult_is_0[simp]: "((m::enat) * n = 0) = (m = 0 \<or> n = 0)" |
278 |
by(auto simp add: times_enat_def zero_enat_def split: enat.split) |
|
41853 | 279 |
|
43919 | 280 |
lemma imult_is_Infty: "((a::enat) * b = \<infinity>) = (a = \<infinity> \<and> b \<noteq> 0 \<or> b = \<infinity> \<and> a \<noteq> 0)" |
281 |
by(auto simp add: times_enat_def zero_enat_def split: enat.split) |
|
41853 | 282 |
|
283 |
||
284 |
subsection {* Subtraction *} |
|
285 |
||
43919 | 286 |
instantiation enat :: minus |
41853 | 287 |
begin |
288 |
||
43919 | 289 |
definition diff_enat_def: |
41853 | 290 |
"a - b = (case a of (Fin x) \<Rightarrow> (case b of (Fin y) \<Rightarrow> Fin (x - y) | \<infinity> \<Rightarrow> 0) |
291 |
| \<infinity> \<Rightarrow> \<infinity>)" |
|
292 |
||
293 |
instance .. |
|
294 |
||
295 |
end |
|
296 |
||
297 |
lemma idiff_Fin_Fin[simp,code]: "Fin a - Fin b = Fin (a - b)" |
|
43919 | 298 |
by(simp add: diff_enat_def) |
41853 | 299 |
|
43921 | 300 |
lemma idiff_Infty[simp,code]: "\<infinity> - n = (\<infinity>::enat)" |
43919 | 301 |
by(simp add: diff_enat_def) |
41853 | 302 |
|
303 |
lemma idiff_Infty_right[simp,code]: "Fin a - \<infinity> = 0" |
|
43919 | 304 |
by(simp add: diff_enat_def) |
41853 | 305 |
|
43919 | 306 |
lemma idiff_0[simp]: "(0::enat) - n = 0" |
307 |
by (cases n, simp_all add: zero_enat_def) |
|
41853 | 308 |
|
43919 | 309 |
lemmas idiff_Fin_0[simp] = idiff_0[unfolded zero_enat_def] |
41853 | 310 |
|
43919 | 311 |
lemma idiff_0_right[simp]: "(n::enat) - 0 = n" |
312 |
by (cases n) (simp_all add: zero_enat_def) |
|
41853 | 313 |
|
43919 | 314 |
lemmas idiff_Fin_0_right[simp] = idiff_0_right[unfolded zero_enat_def] |
41853 | 315 |
|
43919 | 316 |
lemma idiff_self[simp]: "n \<noteq> \<infinity> \<Longrightarrow> (n::enat) - n = 0" |
317 |
by(auto simp: zero_enat_def) |
|
41853 | 318 |
|
41855 | 319 |
lemma iSuc_minus_iSuc [simp]: "iSuc n - iSuc m = n - m" |
43919 | 320 |
by(simp add: iSuc_def split: enat.split) |
41855 | 321 |
|
322 |
lemma iSuc_minus_1 [simp]: "iSuc n - 1 = n" |
|
43919 | 323 |
by(simp add: one_enat_def iSuc_Fin[symmetric] zero_enat_def[symmetric]) |
41855 | 324 |
|
43919 | 325 |
(*lemmas idiff_self_eq_0_Fin = idiff_self_eq_0[unfolded zero_enat_def]*) |
41853 | 326 |
|
27110 | 327 |
subsection {* Ordering *} |
328 |
||
43919 | 329 |
instantiation enat :: linordered_ab_semigroup_add |
27110 | 330 |
begin |
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|
331 |
|
38167 | 332 |
definition [nitpick_simp]: |
37765 | 333 |
"m \<le> n = (case n of Fin n1 \<Rightarrow> (case m of Fin m1 \<Rightarrow> m1 \<le> n1 | \<infinity> \<Rightarrow> False) |
27110 | 334 |
| \<infinity> \<Rightarrow> True)" |
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|
335 |
|
38167 | 336 |
definition [nitpick_simp]: |
37765 | 337 |
"m < n = (case m of Fin m1 \<Rightarrow> (case n of Fin n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True) |
27110 | 338 |
| \<infinity> \<Rightarrow> False)" |
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|
339 |
|
43919 | 340 |
lemma enat_ord_simps [simp]: |
27110 | 341 |
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
342 |
"Fin m < Fin n \<longleftrightarrow> m < n" |
|
43921 | 343 |
"q \<le> (\<infinity>::enat)" |
344 |
"q < (\<infinity>::enat) \<longleftrightarrow> q \<noteq> \<infinity>" |
|
345 |
"(\<infinity>::enat) \<le> q \<longleftrightarrow> q = \<infinity>" |
|
346 |
"(\<infinity>::enat) < q \<longleftrightarrow> False" |
|
43919 | 347 |
by (simp_all add: less_eq_enat_def less_enat_def split: enat.splits) |
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|
348 |
|
43919 | 349 |
lemma enat_ord_code [code]: |
27110 | 350 |
"Fin m \<le> Fin n \<longleftrightarrow> m \<le> n" |
351 |
"Fin m < Fin n \<longleftrightarrow> m < n" |
|
43921 | 352 |
"q \<le> (\<infinity>::enat) \<longleftrightarrow> True" |
27110 | 353 |
"Fin m < \<infinity> \<longleftrightarrow> True" |
354 |
"\<infinity> \<le> Fin n \<longleftrightarrow> False" |
|
43921 | 355 |
"(\<infinity>::enat) < q \<longleftrightarrow> False" |
27110 | 356 |
by simp_all |
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|
357 |
|
27110 | 358 |
instance by default |
43919 | 359 |
(auto simp add: less_eq_enat_def less_enat_def plus_enat_def split: enat.splits) |
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|
360 |
|
27110 | 361 |
end |
362 |
||
43919 | 363 |
instance enat :: ordered_comm_semiring |
29014 | 364 |
proof |
43919 | 365 |
fix a b c :: enat |
29014 | 366 |
assume "a \<le> b" and "0 \<le> c" |
367 |
thus "c * a \<le> c * b" |
|
43919 | 368 |
unfolding times_enat_def less_eq_enat_def zero_enat_def |
369 |
by (simp split: enat.splits) |
|
29014 | 370 |
qed |
371 |
||
43919 | 372 |
lemma enat_ord_number [simp]: |
373 |
"(number_of m \<Colon> enat) \<le> number_of n \<longleftrightarrow> (number_of m \<Colon> nat) \<le> number_of n" |
|
374 |
"(number_of m \<Colon> enat) < number_of n \<longleftrightarrow> (number_of m \<Colon> nat) < number_of n" |
|
375 |
by (simp_all add: number_of_enat_def) |
|
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|
376 |
|
43919 | 377 |
lemma i0_lb [simp]: "(0\<Colon>enat) \<le> n" |
378 |
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits) |
|
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|
379 |
|
43919 | 380 |
lemma ile0_eq [simp]: "n \<le> (0\<Colon>enat) \<longleftrightarrow> n = 0" |
381 |
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits) |
|
27110 | 382 |
|
383 |
lemma Infty_ileE [elim!]: "\<infinity> \<le> Fin m \<Longrightarrow> R" |
|
43919 | 384 |
by (simp add: zero_enat_def less_eq_enat_def split: enat.splits) |
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|
385 |
|
27110 | 386 |
lemma Infty_ilessE [elim!]: "\<infinity> < Fin m \<Longrightarrow> R" |
387 |
by simp |
|
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|
388 |
|
43919 | 389 |
lemma not_iless0 [simp]: "\<not> n < (0\<Colon>enat)" |
390 |
by (simp add: zero_enat_def less_enat_def split: enat.splits) |
|
27110 | 391 |
|
43919 | 392 |
lemma i0_less [simp]: "(0\<Colon>enat) < n \<longleftrightarrow> n \<noteq> 0" |
393 |
by (simp add: zero_enat_def less_enat_def split: enat.splits) |
|
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|
394 |
|
27110 | 395 |
lemma iSuc_ile_mono [simp]: "iSuc n \<le> iSuc m \<longleftrightarrow> n \<le> m" |
43919 | 396 |
by (simp add: iSuc_def less_eq_enat_def split: enat.splits) |
27110 | 397 |
|
398 |
lemma iSuc_mono [simp]: "iSuc n < iSuc m \<longleftrightarrow> n < m" |
|
43919 | 399 |
by (simp add: iSuc_def less_enat_def split: enat.splits) |
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|
400 |
|
27110 | 401 |
lemma ile_iSuc [simp]: "n \<le> iSuc n" |
43919 | 402 |
by (simp add: iSuc_def less_eq_enat_def split: enat.splits) |
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|
403 |
|
11355 | 404 |
lemma not_iSuc_ilei0 [simp]: "\<not> iSuc n \<le> 0" |
43919 | 405 |
by (simp add: zero_enat_def iSuc_def less_eq_enat_def split: enat.splits) |
27110 | 406 |
|
407 |
lemma i0_iless_iSuc [simp]: "0 < iSuc n" |
|
43919 | 408 |
by (simp add: zero_enat_def iSuc_def less_enat_def split: enat.splits) |
27110 | 409 |
|
41853 | 410 |
lemma iless_iSuc0[simp]: "(n < iSuc 0) = (n = 0)" |
43919 | 411 |
by (simp add: zero_enat_def iSuc_def less_enat_def split: enat.split) |
41853 | 412 |
|
27110 | 413 |
lemma ileI1: "m < n \<Longrightarrow> iSuc m \<le> n" |
43919 | 414 |
by (simp add: iSuc_def less_eq_enat_def less_enat_def split: enat.splits) |
27110 | 415 |
|
416 |
lemma Suc_ile_eq: "Fin (Suc m) \<le> n \<longleftrightarrow> Fin m < n" |
|
417 |
by (cases n) auto |
|
418 |
||
419 |
lemma iless_Suc_eq [simp]: "Fin m < iSuc n \<longleftrightarrow> Fin m \<le> n" |
|
43919 | 420 |
by (auto simp add: iSuc_def less_enat_def split: enat.splits) |
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|
421 |
|
43919 | 422 |
lemma imult_Infty: "(0::enat) < n \<Longrightarrow> \<infinity> * n = \<infinity>" |
423 |
by (simp add: zero_enat_def less_enat_def split: enat.splits) |
|
41853 | 424 |
|
43919 | 425 |
lemma imult_Infty_right: "(0::enat) < n \<Longrightarrow> n * \<infinity> = \<infinity>" |
426 |
by (simp add: zero_enat_def less_enat_def split: enat.splits) |
|
41853 | 427 |
|
43919 | 428 |
lemma enat_0_less_mult_iff: "(0 < (m::enat) * n) = (0 < m \<and> 0 < n)" |
41853 | 429 |
by (simp only: i0_less imult_is_0, simp) |
430 |
||
431 |
lemma mono_iSuc: "mono iSuc" |
|
432 |
by(simp add: mono_def) |
|
433 |
||
434 |
||
43919 | 435 |
lemma min_enat_simps [simp]: |
27110 | 436 |
"min (Fin m) (Fin n) = Fin (min m n)" |
437 |
"min q 0 = 0" |
|
438 |
"min 0 q = 0" |
|
43921 | 439 |
"min q (\<infinity>::enat) = q" |
440 |
"min (\<infinity>::enat) q = q" |
|
27110 | 441 |
by (auto simp add: min_def) |
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|
442 |
|
43919 | 443 |
lemma max_enat_simps [simp]: |
27110 | 444 |
"max (Fin m) (Fin n) = Fin (max m n)" |
445 |
"max q 0 = q" |
|
446 |
"max 0 q = q" |
|
43921 | 447 |
"max q \<infinity> = (\<infinity>::enat)" |
448 |
"max \<infinity> q = (\<infinity>::enat)" |
|
27110 | 449 |
by (simp_all add: max_def) |
450 |
||
451 |
lemma Fin_ile: "n \<le> Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
452 |
by (cases n) simp_all |
|
453 |
||
454 |
lemma Fin_iless: "n < Fin m \<Longrightarrow> \<exists>k. n = Fin k" |
|
455 |
by (cases n) simp_all |
|
11351
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changeset
|
456 |
|
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changeset
|
457 |
lemma chain_incr: "\<forall>i. \<exists>j. Y i < Y j ==> \<exists>j. Fin k < Y j" |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
458 |
apply (induct_tac k) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
459 |
apply (simp (no_asm) only: Fin_0) |
27110 | 460 |
apply (fast intro: le_less_trans [OF i0_lb]) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
461 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
462 |
apply (drule spec) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
463 |
apply (erule exE) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
464 |
apply (drule ileI1) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
465 |
apply (rule iSuc_Fin [THEN subst]) |
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
466 |
apply (rule exI) |
27110 | 467 |
apply (erule (1) le_less_trans) |
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25112
diff
changeset
|
468 |
done |
11351
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added Library/Nat_Infinity.thy and Library/Continuity.thy
oheimb
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diff
changeset
|
469 |
|
43919 | 470 |
instantiation enat :: "{bot, top}" |
29337 | 471 |
begin |
472 |
||
43919 | 473 |
definition bot_enat :: enat where |
474 |
"bot_enat = 0" |
|
29337 | 475 |
|
43919 | 476 |
definition top_enat :: enat where |
477 |
"top_enat = \<infinity>" |
|
29337 | 478 |
|
479 |
instance proof |
|
43919 | 480 |
qed (simp_all add: bot_enat_def top_enat_def) |
29337 | 481 |
|
482 |
end |
|
483 |
||
42993 | 484 |
lemma finite_Fin_bounded: |
485 |
assumes le_fin: "\<And>y. y \<in> A \<Longrightarrow> y \<le> Fin n" |
|
486 |
shows "finite A" |
|
487 |
proof (rule finite_subset) |
|
488 |
show "finite (Fin ` {..n})" by blast |
|
489 |
||
490 |
have "A \<subseteq> {..Fin n}" using le_fin by fastsimp |
|
491 |
also have "\<dots> \<subseteq> Fin ` {..n}" |
|
492 |
by (rule subsetI) (case_tac x, auto) |
|
493 |
finally show "A \<subseteq> Fin ` {..n}" . |
|
494 |
qed |
|
495 |
||
26089 | 496 |
|
27110 | 497 |
subsection {* Well-ordering *} |
26089 | 498 |
|
499 |
lemma less_FinE: |
|
500 |
"[| n < Fin m; !!k. n = Fin k ==> k < m ==> P |] ==> P" |
|
501 |
by (induct n) auto |
|
502 |
||
503 |
lemma less_InftyE: |
|
43921 | 504 |
"[| n < \<infinity>; !!k. n = Fin k ==> P |] ==> P" |
26089 | 505 |
by (induct n) auto |
506 |
||
43919 | 507 |
lemma enat_less_induct: |
508 |
assumes prem: "!!n. \<forall>m::enat. m < n --> P m ==> P n" shows "P n" |
|
26089 | 509 |
proof - |
510 |
have P_Fin: "!!k. P (Fin k)" |
|
511 |
apply (rule nat_less_induct) |
|
512 |
apply (rule prem, clarify) |
|
513 |
apply (erule less_FinE, simp) |
|
514 |
done |
|
515 |
show ?thesis |
|
516 |
proof (induct n) |
|
517 |
fix nat |
|
518 |
show "P (Fin nat)" by (rule P_Fin) |
|
519 |
next |
|
43921 | 520 |
show "P \<infinity>" |
26089 | 521 |
apply (rule prem, clarify) |
522 |
apply (erule less_InftyE) |
|
523 |
apply (simp add: P_Fin) |
|
524 |
done |
|
525 |
qed |
|
526 |
qed |
|
527 |
||
43919 | 528 |
instance enat :: wellorder |
26089 | 529 |
proof |
27823 | 530 |
fix P and n |
43919 | 531 |
assume hyp: "(\<And>n\<Colon>enat. (\<And>m\<Colon>enat. m < n \<Longrightarrow> P m) \<Longrightarrow> P n)" |
532 |
show "P n" by (blast intro: enat_less_induct hyp) |
|
26089 | 533 |
qed |
534 |
||
42993 | 535 |
subsection {* Complete Lattice *} |
536 |
||
43919 | 537 |
instantiation enat :: complete_lattice |
42993 | 538 |
begin |
539 |
||
43919 | 540 |
definition inf_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where |
541 |
"inf_enat \<equiv> min" |
|
42993 | 542 |
|
43919 | 543 |
definition sup_enat :: "enat \<Rightarrow> enat \<Rightarrow> enat" where |
544 |
"sup_enat \<equiv> max" |
|
42993 | 545 |
|
43919 | 546 |
definition Inf_enat :: "enat set \<Rightarrow> enat" where |
547 |
"Inf_enat A \<equiv> if A = {} then \<infinity> else (LEAST x. x \<in> A)" |
|
42993 | 548 |
|
43919 | 549 |
definition Sup_enat :: "enat set \<Rightarrow> enat" where |
550 |
"Sup_enat A \<equiv> if A = {} then 0 |
|
42993 | 551 |
else if finite A then Max A |
552 |
else \<infinity>" |
|
553 |
instance proof |
|
43919 | 554 |
fix x :: "enat" and A :: "enat set" |
42993 | 555 |
{ assume "x \<in> A" then show "Inf A \<le> x" |
43919 | 556 |
unfolding Inf_enat_def by (auto intro: Least_le) } |
42993 | 557 |
{ assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" then show "x \<le> Inf A" |
43919 | 558 |
unfolding Inf_enat_def |
42993 | 559 |
by (cases "A = {}") (auto intro: LeastI2_ex) } |
560 |
{ assume "x \<in> A" then show "x \<le> Sup A" |
|
43919 | 561 |
unfolding Sup_enat_def by (cases "finite A") auto } |
42993 | 562 |
{ assume "\<And>y. y \<in> A \<Longrightarrow> y \<le> x" then show "Sup A \<le> x" |
43919 | 563 |
unfolding Sup_enat_def using finite_Fin_bounded by auto } |
564 |
qed (simp_all add: inf_enat_def sup_enat_def) |
|
42993 | 565 |
end |
566 |
||
27110 | 567 |
|
568 |
subsection {* Traditional theorem names *} |
|
569 |
||
43919 | 570 |
lemmas enat_defs = zero_enat_def one_enat_def number_of_enat_def iSuc_def |
571 |
plus_enat_def less_eq_enat_def less_enat_def |
|
27110 | 572 |
|
11351
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|
573 |
end |