author  wenzelm 
Tue, 17 Apr 2007 00:30:44 +0200  
changeset 22718  936f7580937d 
parent 22483  86064f2f2188 
child 22744  5cbe966d67a2 
permissions  rwrr 
923  1 
(* Title: HOL/Nat.thy 
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ID: $Id$ 

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Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel 
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Type "nat" is a linear order, and a datatype; arithmetic operators +  
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and * (for div, mod and dvd, see theory Divides). 
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*) 
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13449  9 
header {* Natural numbers *} 
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15131  11 
theory Nat 
15140  12 
imports Wellfounded_Recursion Ring_and_Field 
21243  13 
uses ("arith_data.ML") 
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begin 
13449  15 

16 
subsection {* Type @{text ind} *} 

17 

18 
typedecl ind 

19 

19573  20 
axiomatization 
21 
Zero_Rep :: ind and 

22 
Suc_Rep :: "ind => ind" 

23 
where 

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 {* the axiom of infinity in 2 parts *} 
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inj_Suc_Rep: "inj Suc_Rep" and 
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Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" 
19573  27 

13449  28 

29 
subsection {* Type nat *} 

30 

31 
text {* Type definition *} 

32 

22262  33 
inductive2 Nat :: "ind \<Rightarrow> bool" 
34 
where 

35 
Zero_RepI: "Nat Zero_Rep" 

36 
 Suc_RepI: "Nat i ==> Nat (Suc_Rep i)" 

13449  37 

38 
global 

39 

40 
typedef (open Nat) 

22262  41 
nat = "Collect Nat" 
21243  42 
proof 
22262  43 
from Nat.Zero_RepI 
44 
show "Zero_Rep : Collect Nat" .. 

21243  45 
qed 
13449  46 

47 
text {* Abstract constants and syntax *} 

48 

49 
consts 

50 
Suc :: "nat => nat" 

51 

52 
local 

53 

54 
defs 

18648  55 
Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 
22718  56 

57 
definition 

58 
pred_nat :: "(nat * nat) set" where 

59 
"pred_nat = {(m, n). n = Suc m}" 

13449  60 

21456  61 
instance nat :: "{ord, zero, one}" 
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Zero_nat_def: "0 == Abs_Nat Zero_Rep" 

63 
One_nat_def [simp]: "1 == Suc 0" 

22262  64 
less_def: "m < n == (m, n) : pred_nat^+" 
21456  65 
le_def: "m \<le> (n::nat) == ~ (n < m)" .. 
13449  66 

67 
text {* Induction *} 

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22718  69 
lemma Rep_Nat': "Nat (Rep_Nat x)" 
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by (rule Rep_Nat [simplified mem_Collect_eq]) 

71 

72 
lemma Abs_Nat_inverse': "Nat y \<Longrightarrow> Rep_Nat (Abs_Nat y) = y" 

73 
by (rule Abs_Nat_inverse [simplified mem_Collect_eq]) 

22262  74 

13449  75 
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" 
76 
apply (unfold Zero_nat_def Suc_def) 

77 
apply (rule Rep_Nat_inverse [THEN subst])  {* types force good instantiation *} 

22262  78 
apply (erule Rep_Nat' [THEN Nat.induct]) 
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apply (iprover elim: Abs_Nat_inverse' [THEN subst]) 

13449  80 
done 
81 

82 
text {* Distinctness of constructors *} 

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lemma Suc_not_Zero [iff]: "Suc m \<noteq> 0" 
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by (simp add: Zero_nat_def Suc_def Abs_Nat_inject Rep_Nat' Suc_RepI Zero_RepI 
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Suc_Rep_not_Zero_Rep) 
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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m" 
13449  89 
by (rule not_sym, rule Suc_not_Zero not_sym) 
90 

91 
lemma Suc_neq_Zero: "Suc m = 0 ==> R" 

92 
by (rule notE, rule Suc_not_Zero) 

93 

94 
lemma Zero_neq_Suc: "0 = Suc m ==> R" 

95 
by (rule Suc_neq_Zero, erule sym) 

96 

97 
text {* Injectiveness of @{term Suc} *} 

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lemma inj_Suc[simp]: "inj_on Suc N" 
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by (simp add: Suc_def inj_on_def Abs_Nat_inject Rep_Nat' Suc_RepI 
101 
inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 

13449  102 

103 
lemma Suc_inject: "Suc x = Suc y ==> x = y" 

104 
by (rule inj_Suc [THEN injD]) 

105 

106 
lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)" 

15413  107 
by (rule inj_Suc [THEN inj_eq]) 
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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False" 
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by auto 
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text {* size of a datatype value *} 
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22473  114 
class size = type + 
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fixes size :: "'a \<Rightarrow> nat" 
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13449  117 
text {* @{typ nat} is a datatype *} 
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rep_datatype nat 
13449  120 
distinct Suc_not_Zero Zero_not_Suc 
121 
inject Suc_Suc_eq 

21411  122 
induction nat_induct 
123 

124 
declare nat.induct [case_names 0 Suc, induct type: nat] 

125 
declare nat.exhaust [case_names 0 Suc, cases type: nat] 

13449  126 

21672  127 
lemmas nat_rec_0 = nat.recs(1) 
128 
and nat_rec_Suc = nat.recs(2) 

129 

130 
lemmas nat_case_0 = nat.cases(1) 

131 
and nat_case_Suc = nat.cases(2) 

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lemma n_not_Suc_n: "n \<noteq> Suc n" 
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by (induct n) simp_all 
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lemma Suc_n_not_n: "Suc t \<noteq> t" 
13449  138 
by (rule not_sym, rule n_not_Suc_n) 
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140 
text {* A special form of induction for reasoning 

141 
about @{term "m < n"} and @{term "m  n"} *} 

142 

143 
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 

144 
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 

14208  145 
apply (rule_tac x = m in spec) 
15251  146 
apply (induct n) 
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prefer 2 
148 
apply (rule allI) 

17589  149 
apply (induct_tac x, iprover+) 
13449  150 
done 
151 

152 
subsection {* Basic properties of "less than" *} 

153 

154 
lemma wf_pred_nat: "wf pred_nat" 

14208  155 
apply (unfold wf_def pred_nat_def, clarify) 
156 
apply (induct_tac x, blast+) 

13449  157 
done 
158 

159 
lemma wf_less: "wf {(x, y::nat). x < y}" 

160 
apply (unfold less_def) 

14208  161 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast) 
13449  162 
done 
163 

164 
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" 

165 
apply (unfold less_def) 

166 
apply (rule refl) 

167 
done 

168 

169 
subsubsection {* Introduction properties *} 

170 

171 
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

172 
apply (unfold less_def) 

14208  173 
apply (rule trans_trancl [THEN transD], assumption+) 
13449  174 
done 
175 

176 
lemma lessI [iff]: "n < Suc n" 

177 
apply (unfold less_def pred_nat_def) 

178 
apply (simp add: r_into_trancl) 

179 
done 

180 

181 
lemma less_SucI: "i < j ==> i < Suc j" 

14208  182 
apply (rule less_trans, assumption) 
13449  183 
apply (rule lessI) 
184 
done 

185 

186 
lemma zero_less_Suc [iff]: "0 < Suc n" 

187 
apply (induct n) 

188 
apply (rule lessI) 

189 
apply (erule less_trans) 

190 
apply (rule lessI) 

191 
done 

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193 
subsubsection {* Elimination properties *} 

194 

195 
lemma less_not_sym: "n < m ==> ~ m < (n::nat)" 

196 
apply (unfold less_def) 

197 
apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) 

198 
done 

199 

200 
lemma less_asym: 

201 
assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P 

202 
apply (rule contrapos_np) 

203 
apply (rule less_not_sym) 

204 
apply (rule h1) 

205 
apply (erule h2) 

206 
done 

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208 
lemma less_not_refl: "~ n < (n::nat)" 

209 
apply (unfold less_def) 

210 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl]) 

211 
done 

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213 
lemma less_irrefl [elim!]: "(n::nat) < n ==> R" 

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by (rule notE, rule less_not_refl) 

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lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" by blast 
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lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t" 
13449  219 
by (rule not_sym, rule less_not_refl2) 
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lemma lessE: 

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assumes major: "i < k" 

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and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" 

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shows P 

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apply (rule major [unfolded less_def pred_nat_def, THEN tranclE], simp_all) 
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apply (erule p1) 
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apply (rule p2) 

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apply (simp add: less_def pred_nat_def, assumption) 
13449  229 
done 
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231 
lemma not_less0 [iff]: "~ n < (0::nat)" 

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by (blast elim: lessE) 

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lemma less_zeroE: "(n::nat) < 0 ==> R" 

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by (rule notE, rule not_less0) 

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lemma less_SucE: assumes major: "m < Suc n" 

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and less: "m < n ==> P" and eq: "m = n ==> P" shows P 

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apply (rule major [THEN lessE]) 

14208  240 
apply (rule eq, blast) 
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apply (rule less, blast) 

13449  242 
done 
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lemma less_Suc_eq: "(m < Suc n) = (m < n  m = n)" 

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by (blast elim!: less_SucE intro: less_trans) 

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lemma less_one [iff]: "(n < (1::nat)) = (n = 0)" 

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by (simp add: less_Suc_eq) 

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lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" 

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by (simp add: less_Suc_eq) 

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lemma Suc_mono: "m < n ==> Suc m < Suc n" 

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by (induct n) (fast elim: less_trans lessE)+ 

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text {* "Less than" is a linear ordering *} 

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lemma less_linear: "m < n  m = n  n < (m::nat)" 

15251  258 
apply (induct m) 
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apply (induct n) 

13449  260 
apply (rule refl [THEN disjI1, THEN disjI2]) 
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apply (rule zero_less_Suc [THEN disjI1]) 

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apply (blast intro: Suc_mono less_SucI elim: lessE) 

263 
done 

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text {* "Less than" is antisymmetric, sort of *} 
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lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n" 

22718  267 
apply(simp only:less_Suc_eq) 
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apply blast 

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done 

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lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n  n < m)" 
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using less_linear by blast 
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274 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

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and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

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shows "P n m" 

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apply (rule less_linear [THEN disjE]) 

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apply (erule_tac [2] disjE) 

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apply (erule lessCase) 

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apply (erule sym [THEN eqCase]) 

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apply (erule major) 

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done 

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284 

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subsubsection {* Inductive (?) properties *} 

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lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n" 
13449  288 
apply (simp add: nat_neq_iff) 
289 
apply (blast elim!: less_irrefl less_SucE elim: less_asym) 

290 
done 

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292 
lemma Suc_lessD: "Suc m < n ==> m < n" 

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apply (induct n) 

294 
apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+ 

295 
done 

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297 
lemma Suc_lessE: assumes major: "Suc i < k" 

298 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

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apply (rule major [THEN lessE]) 

300 
apply (erule lessI [THEN minor]) 

14208  301 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  302 
done 
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304 
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

305 
by (blast elim: lessE dest: Suc_lessD) 

4104  306 

16635  307 
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)" 
13449  308 
apply (rule iffI) 
309 
apply (erule Suc_less_SucD) 

310 
apply (erule Suc_mono) 

311 
done 

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313 
lemma less_trans_Suc: 

314 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  315 
apply (induct k, simp_all) 
13449  316 
apply (insert le) 
317 
apply (simp add: less_Suc_eq) 

318 
apply (blast dest: Suc_lessD) 

319 
done 

320 

16635  321 
lemma [code]: "((n::nat) < 0) = False" by simp 
322 
lemma [code]: "(0 < Suc n) = True" by simp 

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13449  324 
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m  n < m"} *} 
325 
lemma not_less_eq: "(~ m < n) = (n < Suc m)" 

22718  326 
by (induct m n rule: diff_induct) simp_all 
13449  327 

328 
text {* Complete induction, aka courseofvalues induction *} 

329 
lemma nat_less_induct: 

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assumes prem: "!!n. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
22718  331 
apply (induct n rule: wf_induct [OF wf_pred_nat [THEN wf_trancl]]) 
13449  332 
apply (rule prem) 
14208  333 
apply (unfold less_def, assumption) 
13449  334 
done 
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14131  336 
lemmas less_induct = nat_less_induct [rule_format, case_names less] 
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21243  338 

14131  339 
subsection {* Properties of "less than or equal" *} 
13449  340 

341 
text {* Was @{text le_eq_less_Suc}, but this orientation is more useful *} 

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lemma less_Suc_eq_le: "(m < Suc n) = (m \<le> n)" 
22718  343 
unfolding le_def by (rule not_less_eq [symmetric]) 
13449  344 

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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
13449  346 
by (rule less_Suc_eq_le [THEN iffD2]) 
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lemma le0 [iff]: "(0::nat) \<le> n" 
22718  349 
unfolding le_def by (rule not_less0) 
13449  350 

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lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
13449  352 
by (simp add: le_def) 
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)" 
13449  355 
by (induct i) (simp_all add: le_def) 
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lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n  m = Suc n)" 
13449  358 
by (simp del: less_Suc_eq_le add: less_Suc_eq_le [symmetric] less_Suc_eq) 
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lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R" 
17589  361 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  362 

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lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
13449  364 
apply (simp add: le_def less_Suc_eq) 
365 
apply (blast elim!: less_irrefl less_asym) 

366 
done  {* formerly called lessD *} 

367 

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lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n" 
13449  369 
by (simp add: le_def less_Suc_eq) 
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text {* Stronger version of @{text Suc_leD} *} 

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lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" 
13449  373 
apply (simp add: le_def less_Suc_eq) 
374 
using less_linear 

375 
apply blast 

376 
done 

377 

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lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)" 
13449  379 
by (blast intro: Suc_leI Suc_le_lessD) 
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lemma le_SucI: "m \<le> n ==> m \<le> Suc n" 
13449  382 
by (unfold le_def) (blast dest: Suc_lessD) 
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lemma less_imp_le: "m < n ==> m \<le> (n::nat)" 
13449  385 
by (unfold le_def) (blast elim: less_asym) 
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text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *} 
13449  388 
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 
389 

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text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  392 

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lemma le_imp_less_or_eq: "m \<le> n ==> m < n  m = (n::nat)" 
22718  394 
unfolding le_def 
13449  395 
using less_linear 
22718  396 
by (blast elim: less_irrefl less_asym) 
13449  397 

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lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
22718  399 
unfolding le_def 
13449  400 
using less_linear 
22718  401 
by (blast elim!: less_irrefl elim: less_asym) 
13449  402 

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403 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
17589  404 
by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq) 
13449  405 

22718  406 
text {* Useful with @{text blast}. *} 
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407 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
22718  408 
by (rule less_or_eq_imp_le) (rule disjI2) 
13449  409 

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410 
lemma le_refl: "n \<le> (n::nat)" 
13449  411 
by (simp add: le_eq_less_or_eq) 
412 

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413 
lemma le_less_trans: "[ i \<le> j; j < k ] ==> i < (k::nat)" 
13449  414 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
415 

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416 
lemma less_le_trans: "[ i < j; j \<le> k ] ==> i < (k::nat)" 
13449  417 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
418 

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419 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::nat)" 
13449  420 
by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) 
421 

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422 
lemma le_anti_sym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
13449  423 
by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) 
424 

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425 
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)" 
13449  426 
by (simp add: le_simps) 
427 

428 
text {* Axiom @{text order_less_le} of class @{text order}: *} 

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429 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
13449  430 
by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) 
431 

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432 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
13449  433 
by (rule iffD2, rule nat_less_le, rule conjI) 
434 

435 
text {* Axiom @{text linorder_linear} of class @{text linorder}: *} 

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436 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
13449  437 
apply (simp add: le_eq_less_or_eq) 
22718  438 
using less_linear by blast 
13449  439 

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440 
text {* Type {@typ nat} is a wellfounded linear order *} 
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441 

22318  442 
instance nat :: wellorder 
14691  443 
by intro_classes 
444 
(assumption  

445 
rule le_refl le_trans le_anti_sym nat_less_le nat_le_linear wf_less)+ 

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446 

22718  447 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  448 

13449  449 
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 
450 
by (blast elim!: less_SucE) 

451 

452 
text {* 

453 
Rewrite @{term "n < Suc m"} to @{term "n = m"} 

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454 
if @{term "~ n < m"} or @{term "m \<le> n"} hold. 
13449  455 
Not suitable as default simprules because they often lead to looping 
456 
*} 

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457 
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" 
13449  458 
by (rule not_less_less_Suc_eq, rule leD) 
459 

460 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

461 

462 

463 
text {* 

22718  464 
Reorientation of the equations @{text "0 = x"} and @{text "1 = x"}. 
465 
No longer added as simprules (they loop) 

13449  466 
but via @{text reorient_simproc} in Bin 
467 
*} 

468 

469 
text {* Polymorphic, not just for @{typ nat} *} 

470 
lemma zero_reorient: "(0 = x) = (x = 0)" 

471 
by auto 

472 

473 
lemma one_reorient: "(1 = x) = (x = 1)" 

474 
by auto 

475 

21243  476 

13449  477 
subsection {* Arithmetic operators *} 
1660  478 

22473  479 
class power = type + 
21411  480 
fixes power :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "\<^loc>^" 80) 
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481 

13449  482 
text {* arithmetic operators @{text "+ "} and @{text "*"} *} 
483 

21456  484 
instance nat :: "{plus, minus, times}" .. 
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485 

13449  486 
primrec 
487 
add_0: "0 + n = n" 

488 
add_Suc: "Suc m + n = Suc (m + n)" 

489 

490 
primrec 

491 
diff_0: "m  0 = m" 

492 
diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 

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493 

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494 
primrec 
13449  495 
mult_0: "0 * n = 0" 
496 
mult_Suc: "Suc m * n = n + (m * n)" 

497 

22718  498 
text {* These two rules ease the use of primitive recursion. 
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499 
NOTE USE OF @{text "=="} *} 
13449  500 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
501 
by simp 

502 

503 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

504 
by simp 

505 

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506 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
22718  507 
by (cases n) simp_all 
13449  508 

22718  509 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
510 
by (cases n) simp_all 

13449  511 

22718  512 
lemma neq0_conv [iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
513 
by (cases n) simp_all 

13449  514 

515 
text {* This theorem is useful with @{text blast} *} 

516 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

17589  517 
by (rule iffD1, rule neq0_conv, iprover) 
13449  518 

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519 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
13449  520 
by (fast intro: not0_implies_Suc) 
521 

522 
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)" 

523 
apply (rule iffI) 

22718  524 
apply (rule ccontr) 
525 
apply simp_all 

13449  526 
done 
527 

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528 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
13449  529 
by (induct m') simp_all 
530 

531 
text {* Useful in certain inductive arguments *} 

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532 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
22718  533 
by (cases m) simp_all 
13449  534 

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535 
lemma nat_induct2: "[P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))] ==> P n" 
13449  536 
apply (rule nat_less_induct) 
537 
apply (case_tac n) 

538 
apply (case_tac [2] nat) 

539 
apply (blast intro: less_trans)+ 

540 
done 

541 

21243  542 

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543 
subsection {* @{text LEAST} theorems for type @{typ nat}*} 
13449  544 

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545 
lemma Least_Suc: 
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546 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 
14208  547 
apply (case_tac "n", auto) 
13449  548 
apply (frule LeastI) 
549 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

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550 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 
13449  551 
apply (erule_tac [2] Least_le) 
14208  552 
apply (case_tac "LEAST x. P x", auto) 
13449  553 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
554 
apply (blast intro: order_antisym) 

555 
done 

556 

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557 
lemma Least_Suc2: 
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558 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 
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559 
by (erule (1) Least_Suc [THEN ssubst], simp) 
13449  560 

561 

562 
subsection {* @{term min} and @{term max} *} 

563 

564 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

565 
by (rule min_leastL) simp 

566 

567 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

568 
by (rule min_leastR) simp 

569 

570 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

571 
by (simp add: min_of_mono) 

572 

22191  573 
lemma min_Suc1: 
574 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

22718  575 
by (simp split: nat.split) 
22191  576 

577 
lemma min_Suc2: 

578 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

579 
by (simp split: nat.split) 

580 

13449  581 
lemma max_0L [simp]: "max 0 n = (n::nat)" 
582 
by (rule max_leastL) simp 

583 

584 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

585 
by (rule max_leastR) simp 

586 

587 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

588 
by (simp add: max_of_mono) 

589 

22191  590 
lemma max_Suc1: 
591 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

22718  592 
by (simp split: nat.split) 
22191  593 

594 
lemma max_Suc2: 

595 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

596 
by (simp split: nat.split) 

597 

13449  598 

599 
subsection {* Basic rewrite rules for the arithmetic operators *} 

600 

601 
text {* Difference *} 

602 

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603 
lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
15251  604 
by (induct n) simp_all 
13449  605 

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606 
lemma diff_Suc_Suc [simp, code]: "Suc(m)  Suc(n) = m  n" 
15251  607 
by (induct n) simp_all 
13449  608 

609 

610 
text {* 

611 
Could be (and is, below) generalized in various ways 

612 
However, none of the generalizations are currently in the simpset, 

613 
and I dread to think what happens if I put them in 

614 
*} 

615 
lemma Suc_pred [simp]: "0 < n ==> Suc (n  Suc 0) = n" 

616 
by (simp split add: nat.split) 

617 

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618 
declare diff_Suc [simp del, code del] 
13449  619 

620 

621 
subsection {* Addition *} 

622 

623 
lemma add_0_right [simp]: "m + 0 = (m::nat)" 

624 
by (induct m) simp_all 

625 

626 
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 

627 
by (induct m) simp_all 

628 

19890  629 
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" 
630 
by simp 

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631 

13449  632 

633 
text {* Associative law for addition *} 

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634 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
13449  635 
by (induct m) simp_all 
636 

637 
text {* Commutative law for addition *} 

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638 
lemma nat_add_commute: "m + n = n + (m::nat)" 
13449  639 
by (induct m) simp_all 
640 

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641 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
13449  642 
apply (rule mk_left_commute [of "op +"]) 
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643 
apply (rule nat_add_assoc) 
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644 
apply (rule nat_add_commute) 
13449  645 
done 
646 

14331  647 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
13449  648 
by (induct k) simp_all 
649 

14331  650 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
13449  651 
by (induct k) simp_all 
652 

14331  653 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
13449  654 
by (induct k) simp_all 
655 

14331  656 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
13449  657 
by (induct k) simp_all 
658 

659 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 

660 

22718  661 
lemma add_is_0 [iff]: fixes m :: nat shows "(m + n = 0) = (m = 0 & n = 0)" 
662 
by (cases m) simp_all 

13449  663 

664 
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 

22718  665 
by (cases m) simp_all 
13449  666 

667 
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 

668 
by (rule trans, rule eq_commute, rule add_is_1) 

669 

670 
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m  0 < n)" 

671 
by (simp del: neq0_conv add: neq0_conv [symmetric]) 

672 

673 
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" 

674 
apply (drule add_0_right [THEN ssubst]) 

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675 
apply (simp add: nat_add_assoc del: add_0_right) 
13449  676 
done 
677 

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678 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
22718  679 
apply (induct k) 
680 
apply simp 

681 
apply(drule comp_inj_on[OF _ inj_Suc]) 

682 
apply (simp add:o_def) 

683 
done 

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684 

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685 

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686 
subsection {* Multiplication *} 
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687 

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688 
text {* right annihilation in product *} 
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689 
lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 
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690 
by (induct m) simp_all 
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691 

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692 
text {* right successor law for multiplication *} 
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693 
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
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694 
by (induct m) (simp_all add: nat_add_left_commute) 
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695 

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696 
text {* Commutative law for multiplication *} 
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14266
diff
changeset

697 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

698 
by (induct m) simp_all 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

699 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

700 
text {* addition distributes over multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

701 
lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

702 
by (induct m) (simp_all add: nat_add_assoc nat_add_left_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

703 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

704 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

705 
by (induct m) (simp_all add: nat_add_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

706 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

707 
text {* Associative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

708 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

709 
by (induct m) (simp_all add: add_mult_distrib) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

710 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

711 

14740  712 
text{*The naturals form a @{text comm_semiring_1_cancel}*} 
14738  713 
instance nat :: comm_semiring_1_cancel 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

714 
proof 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

715 
fix i j k :: nat 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

716 
show "(i + j) + k = i + (j + k)" by (rule nat_add_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

717 
show "i + j = j + i" by (rule nat_add_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

718 
show "0 + i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

719 
show "(i * j) * k = i * (j * k)" by (rule nat_mult_assoc) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

720 
show "i * j = j * i" by (rule nat_mult_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

721 
show "1 * i = i" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

722 
show "(i + j) * k = i * k + j * k" by (simp add: add_mult_distrib) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

723 
show "0 \<noteq> (1::nat)" by simp 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

724 
assume "k+i = k+j" thus "i=j" by simp 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

725 
qed 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

726 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

727 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
15251  728 
apply (induct m) 
22718  729 
apply (induct_tac [2] n) 
730 
apply simp_all 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

731 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

732 

21243  733 

14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

734 
subsection {* Monotonicity of Addition *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

735 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

736 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

737 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

738 
by (induct k) simp_all 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

739 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

740 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

741 
lemma add_less_mono: "[i < j; k < l] ==> i + k < j + (l::nat)" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

742 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  743 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

744 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

745 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

746 
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

747 
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

748 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

749 
apply (simp_all add: order_le_less) 
22718  750 
apply (blast elim!: less_SucE 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

751 
intro!: add_0_right [symmetric] add_Suc_right [symmetric]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

752 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

753 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

754 
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

755 
lemma mult_less_mono2: "(i::nat) < j ==> 0 < k ==> k * i < k * j" 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

756 
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp) 
22718  757 
apply (induct_tac x) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

758 
apply (simp_all add: add_less_mono) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

759 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

760 

a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

761 

14740  762 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
14738  763 
instance nat :: ordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

764 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

765 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

766 
show "0 < (1::nat)" by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

767 
show "i \<le> j ==> k + i \<le> k + j" by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

768 
show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

769 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

770 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

771 
lemma nat_mult_1: "(1::nat) * n = n" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

772 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

773 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

774 
lemma nat_mult_1_right: "n * (1::nat) = n" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

775 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

776 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

777 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

778 
subsection {* Additional theorems about "less than" *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

779 

19870  780 
text{*An induction rule for estabilishing binary relations*} 
22718  781 
lemma less_Suc_induct: 
19870  782 
assumes less: "i < j" 
783 
and step: "!!i. P i (Suc i)" 

784 
and trans: "!!i j k. P i j ==> P j k ==> P i k" 

785 
shows "P i j" 

786 
proof  

22718  787 
from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 
788 
have "P i (Suc (i + k))" 

19870  789 
proof (induct k) 
22718  790 
case 0 
791 
show ?case by (simp add: step) 

19870  792 
next 
793 
case (Suc k) 

22718  794 
thus ?case by (auto intro: assms) 
19870  795 
qed 
22718  796 
thus "P i j" by (simp add: j) 
19870  797 
qed 
798 

799 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

800 
text {* A [clumsy] way of lifting @{text "<"} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

801 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

802 
lemma less_mono_imp_le_mono: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

803 
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" 
22718  804 
and le: "i \<le> j" 
805 
shows "f i \<le> ((f j)::nat)" 

806 
using le 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

807 
apply (simp add: order_le_less) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

808 
apply (blast intro!: lt_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

809 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

810 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

811 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

812 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

813 
by (rule add_right_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

814 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

815 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

816 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

817 
by (rule add_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

818 

b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

819 
lemma le_add2: "n \<le> ((m + n)::nat)" 
22718  820 
by (insert add_right_mono [of 0 m n], simp) 
13449  821 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

822 
lemma le_add1: "n \<le> ((n + m)::nat)" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

823 
by (simp add: add_commute, rule le_add2) 
13449  824 

825 
lemma less_add_Suc1: "i < Suc (i + m)" 

826 
by (rule le_less_trans, rule le_add1, rule lessI) 

827 

828 
lemma less_add_Suc2: "i < Suc (m + i)" 

829 
by (rule le_less_trans, rule le_add2, rule lessI) 

830 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

831 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
17589  832 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  833 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

834 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
13449  835 
by (rule le_trans, assumption, rule le_add1) 
836 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

837 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
13449  838 
by (rule le_trans, assumption, rule le_add2) 
839 

840 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

841 
by (rule less_le_trans, assumption, rule le_add1) 

842 

843 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

844 
by (rule less_le_trans, assumption, rule le_add2) 

845 

846 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

22718  847 
apply (rule le_less_trans [of _ "i+j"]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

848 
apply (simp_all add: le_add1) 
13449  849 
done 
850 

851 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

852 
apply (rule notI) 

853 
apply (erule add_lessD1 [THEN less_irrefl]) 

854 
done 

855 

856 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

857 
by (simp add: add_commute not_add_less1) 

858 

14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

859 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
22718  860 
apply (rule order_trans [of _ "m+k"]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

861 
apply (simp_all add: le_add1) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

862 
done 
13449  863 

14267
b963e9cee2a0
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864 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
13449  865 
apply (simp add: add_commute) 
866 
apply (erule add_leD1) 

867 
done 

868 

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869 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
13449  870 
by (blast dest: add_leD1 add_leD2) 
871 

872 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

873 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

874 
by (force simp del: add_Suc_right 

875 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 

876 

877 

878 
subsection {* Difference *} 

879 

880 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

881 
by (induct m) simp_all 

882 

883 
text {* Addition is the inverse of subtraction: 

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884 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  885 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
886 
by (induct m n rule: diff_induct) simp_all 

887 

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888 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
16796  889 
by (simp add: add_diff_inverse linorder_not_less) 
13449  890 

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891 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
13449  892 
by (simp add: le_add_diff_inverse add_commute) 
893 

894 

895 
subsection {* More results about difference *} 

896 

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897 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
13449  898 
by (induct m n rule: diff_induct) simp_all 
899 

900 
lemma diff_less_Suc: "m  n < Suc m" 

901 
apply (induct m n rule: diff_induct) 

902 
apply (erule_tac [3] less_SucE) 

903 
apply (simp_all add: less_Suc_eq) 

904 
done 

905 

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906 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
13449  907 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
908 

909 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 

910 
by (rule le_less_trans, rule diff_le_self) 

911 

912 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 

913 
by (induct i j rule: diff_induct) simp_all 

914 

915 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 

916 
by (simp add: diff_diff_left) 

917 

918 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

22718  919 
by (cases n) (auto simp add: le_simps) 
13449  920 

921 
text {* This and the next few suggested by Florian Kammueller *} 

922 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 

923 
by (simp add: diff_diff_left add_commute) 

924 

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925 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
13449  926 
by (induct j k rule: diff_induct) simp_all 
927 

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928 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
13449  929 
by (simp add: add_commute diff_add_assoc) 
930 

931 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 

932 
by (induct n) simp_all 

933 

934 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 

935 
by (simp add: diff_add_assoc) 

936 

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937 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
22718  938 
by (auto simp add: diff_add_inverse2) 
13449  939 

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940 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
13449  941 
by (induct m n rule: diff_induct) simp_all 
942 

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943 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
13449  944 
by (rule iffD2, rule diff_is_0_eq) 
945 

946 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

947 
by (induct m n rule: diff_induct) simp_all 

948 

22718  949 
lemma less_imp_add_positive: 
950 
assumes "i < j" 

951 
shows "\<exists>k::nat. 0 < k & i + k = j" 

952 
proof 

953 
from assms show "0 < j  i & i + (j  i) = j" 

954 
by (simp add: add_diff_inverse less_not_sym) 

955 
qed 

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956 

13449  957 
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k  i)" 
958 
apply (induct k i rule: diff_induct) 

959 
apply (simp_all (no_asm)) 

17589  960 
apply iprover 
13449  961 
done 
962 

963 
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" 

964 
apply (rule diff_self_eq_0 [THEN subst]) 

17589  965 
apply (rule zero_induct_lemma, iprover+) 
13449  966 
done 
967 

968 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 

969 
by (induct k) simp_all 

970 

971 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 

972 
by (simp add: diff_cancel add_commute) 

973 

974 
lemma diff_add_0: "n  (n + m) = (0::nat)" 

975 
by (induct n) simp_all 

976 

977 

978 
text {* Difference distributes over multiplication *} 

979 

980 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 

981 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 

982 

983 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 

984 
by (simp add: diff_mult_distrib mult_commute [of k]) 

985 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 

986 

987 
lemmas nat_distrib = 

988 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 

989 

990 

991 
subsection {* Monotonicity of Multiplication *} 

992 

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993 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
22718  994 
by (simp add: mult_right_mono) 
13449  995 

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996 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
22718  997 
by (simp add: mult_left_mono) 
13449  998 

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999 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
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1000 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
22718  1001 
by (simp add: mult_mono) 
13449  1002 

1003 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

22718  1004 
by (simp add: mult_strict_right_mono) 
13449  1005 

14266  1006 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1007 
there are no negative numbers.*} 

1008 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1009 
apply (induct m) 
22718  1010 
apply simp 
1011 
apply (case_tac n) 

1012 
apply simp_all 

13449  1013 
done 
1014 

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1015 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" 
13449  1016 
apply (induct m) 
22718  1017 
apply simp 
1018 
apply (case_tac n) 

1019 
apply simp_all 

13449  1020 
done 
1021 

1022 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 

22718  1023 
apply (induct m) 
1024 
apply simp 

1025 
apply (induct n) 

1026 
apply auto 

13449  1027 
done 
1028 

1029 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 

1030 
apply (rule trans) 

14208  1031 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
13449  1032 
done 
1033 

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1034 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1035 
apply (safe intro!: mult_less_mono1) 
14208  1036 
apply (case_tac k, auto) 
13449  1037 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1038 
apply (blast intro: mult_le_mono1) 

1039 
done 

1040 

1041 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

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diff
changeset

1042 
by (simp add: mult_commute [of k]) 
13449  1043 

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1044 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
22718  1045 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1046 

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1047 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
22718  1048 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1049 

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1050 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
14208  1051 
apply (cut_tac less_linear, safe, auto) 
13449  1052 
apply (drule mult_less_mono1, assumption, simp)+ 
1053 
done 

1054 

1055 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 

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Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
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diff
changeset

1056 
by (simp add: mult_commute [of k]) 
13449  1057 

1058 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

1059 
by (subst mult_less_cancel1) simp 

1060 

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1061 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
13449  1062 
by (subst mult_le_cancel1) simp 
1063 

1064 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 

1065 
by (subst mult_cancel1) simp 

1066 

1067 
text {* Lemma for @{text gcd} *} 

1068 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1069 
apply (drule sym) 

1070 
apply (rule disjCI) 

1071 
apply (rule nat_less_cases, erule_tac [2] _) 

1072 
apply (fastsimp elim!: less_SucE) 

1073 
apply (fastsimp dest: mult_less_mono2) 

1074 
done 

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1075 

20588  1076 

18702  1077 
subsection {* Code generator setup *} 
1078 

22718  1079 
lemma one_is_Suc_zero [code inline]: "1 = Suc 0" 
20355  1080 
by simp 
1081 

20588  1082 
instance nat :: eq .. 
1083 

1084 
lemma [code func]: 

22718  1085 
"(0\<Colon>nat) = 0 \<longleftrightarrow> True" 
1086 
"Suc n = Suc m \<longleftrightarrow> n = m" 

1087 
"Suc n = 0 \<longleftrightarrow> False" 

1088 
"0 = Suc m \<longleftrightarrow> False" 

22348  1089 
by auto 
20588  1090 

1091 
lemma [code func]: 

22718  1092 
"(0\<Colon>nat) \<le> m \<longleftrightarrow> True" 
1093 
"Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m" 

1094 
"(n\<Colon>nat) < 0 \<longleftrightarrow> False" 

1095 
"(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m" 

22348  1096 
using Suc_le_eq less_Suc_eq_le by simp_all 
20588  1097 

21243  1098 

1099 
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *} 

1100 

1101 
use "arith_data.ML" 

1102 
setup arith_setup 

1103 

1104 
text{*The following proofs may rely on the arithmetic proof procedures.*} 

1105 

1106 
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)" 

1107 
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add) 

1108 

1109 
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)" 

22718  1110 
by (simp add: less_eq reflcl_trancl [symmetric] del: reflcl_trancl, arith) 
21243  1111 

1112 
lemma nat_diff_split: 

22718  1113 
"P(a  b::nat) = ((a<b > P 0) & (ALL d. a = b + d > P d))" 
21243  1114 
 {* elimination of @{text } on @{text nat} *} 
22718  1115 
by (cases "a<b" rule: case_split) (auto simp add: diff_is_0_eq [THEN iffD2]) 
21243  1116 

1117 
lemma nat_diff_split_asm: 

1118 
"P(a  b::nat) = (~ (a < b & ~ P 0  (EX d. a = b + d & ~ P d)))" 

1119 
 {* elimination of @{text } on @{text nat} in assumptions *} 

1120 
by (simp split: nat_diff_split) 

1121 

1122 
lemmas [arith_split] = nat_diff_split split_min split_max 

1123 

1124 

1125 
lemma le_square: "m \<le> m * (m::nat)" 

1126 
by (induct m) auto 

1127 

1128 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 

1129 
by (induct m) auto 

1130 

1131 

1132 
text{*Subtraction laws, mostly by Clemens Ballarin*} 

1133 

1134 
lemma diff_less_mono: "[ a < (b::nat); c \<le> a ] ==> ac < bc" 

22718  1135 
by arith 
21243  1136 

1137 
lemma less_diff_conv: "(i < jk) = (i+k < (j::nat))" 

22718  1138 
by arith 
21243  1139 

1140 
lemma le_diff_conv: "(jk \<le> (i::nat)) = (j \<le> i+k)" 

22718  1141 
by arith 
21243  1142 

1143 
lemma le_diff_conv2: "k \<le> j ==> (i \<le> jk) = (i+k \<le> (j::nat))" 

22718  1144 
by arith 
21243  1145 

1146 
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n  (n  i) = i" 

22718  1147 
by arith 
21243  1148 

1149 
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m  k" 

22718  1150 
by arith 
21243  1151 

1152 
(*Replaces the previous diff_less and le_diff_less, which had the stronger 

1153 
second premise n\<le>m*) 

1154 
lemma diff_less[simp]: "!!m::nat. [ 0<n; 0<m ] ==> m  n < m" 

22718  1155 
by arith 
21243  1156 

1157 

1158 
(** Simplification of relational expressions involving subtraction **) 

1159 

1160 
lemma diff_diff_eq: "[ k \<le> m; k \<le> (n::nat) ] ==> ((mk)  (nk)) = (mn)" 

22718  1161 
by (simp split add: nat_diff_split) 
21243  1162 

1163 
lemma eq_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk = nk) = (m=n)" 

22718  1164 
by (auto split add: nat_diff_split) 
21243  1165 

1166 
lemma less_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk < nk) = (m<n)" 

22718  1167 
by (auto split add: nat_diff_split) 
21243  1168 

1169 
lemma le_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk \<le> nk) = (m\<le>n)" 

22718  1170 
by (auto split add: nat_diff_split) 
21243  1171 

1172 

1173 
text{*(Anti)Monotonicity of subtraction  by Stephan Merz*} 

1174 

1175 
(* Monotonicity of subtraction in first argument *) 

1176 
lemma diff_le_mono: "m \<le> (n::nat) ==> (ml) \<le> (nl)" 

22718  1177 
by (simp split add: nat_diff_split) 
21243  1178 

1179 
lemma diff_le_mono2: "m \<le> (n::nat) ==> (ln) \<le> (lm)" 

22718  1180 
by (simp split add: nat_diff_split) 
21243  1181 

1182 
lemma diff_less_mono2: "[ m < (n::nat); m<l ] ==> (ln) < (lm)" 

22718  1183 
by (simp split add: nat_diff_split) 
21243  1184 

1185 
lemma diffs0_imp_equal: "!!m::nat. [ mn = 0; nm = 0 ] ==> m=n" 

22718  1186 
by (simp split add: nat_diff_split) 
21243  1187 

1188 
text{*Lemmas for ex/Factorization*} 

1189 

1190 
lemma one_less_mult: "[ Suc 0 < n; Suc 0 < m ] ==> Suc 0 < m*n" 

22718  1191 
by (cases m) auto 
21243  1192 

1193 
lemma n_less_m_mult_n: "[ Suc 0 < n; Suc 0 < m ] ==> n<m*n" 

22718  1194 
by (cases m) auto 
21243  1195 

1196 
lemma n_less_n_mult_m: "[ Suc 0 < n; Suc 0 < m ] ==> n<n*m" 

22718  1197 
by (cases m) auto 
21243  1198 

1199 

1200 
text{*Rewriting to pull differences out*} 

1201 

1202 
lemma diff_diff_right [simp]: "k\<le>j > i  (j  k) = i + (k::nat)  j" 

22718  1203 
by arith 
21243  1204 

1205 
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m  Suc (j  k) = m + k  Suc j" 

22718  1206 
by arith 
21243  1207 

1208 
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j  k)  m = Suc j  (k + m)" 

22718  1209 
by arith 
21243  1210 

1211 
(*The others are 

1212 
i  j  k = i  (j + k), 

1213 
k \<le> j ==> j  k + i = j + i  k, 

1214 
k \<le> j ==> i + (j  k) = i + j  k *) 

1215 
lemmas add_diff_assoc = diff_add_assoc [symmetric] 

1216 
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric] 

1217 
declare diff_diff_left [simp] add_diff_assoc [simp] add_diff_assoc2[simp] 

1218 

1219 
text{*At present we prove no analogue of @{text not_less_Least} or @{text 

1220 
Least_Suc}, since there appears to be no need.*} 

1221 

1222 
ML 

1223 
{* 

1224 
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le"; 

1225 
val nat_diff_split = thm "nat_diff_split"; 

1226 
val nat_diff_split_asm = thm "nat_diff_split_asm"; 

1227 
val le_square = thm "le_square"; 

1228 
val le_cube = thm "le_cube"; 

1229 
val diff_less_mono = thm "diff_less_mono"; 

1230 
val less_diff_conv = thm "less_diff_conv"; 

1231 
val le_diff_conv = thm "le_diff_conv"; 

1232 
val le_diff_conv2 = thm "le_diff_conv2"; 

1233 
val diff_diff_cancel = thm "diff_diff_cancel"; 

1234 
val le_add_diff = thm "le_add_diff"; 

1235 
val diff_less = thm "diff_less"; 

1236 
val diff_diff_eq = thm "diff_diff_eq"; 

1237 
val eq_diff_iff = thm "eq_diff_iff"; 

1238 
val less_diff_iff = thm "less_diff_iff"; 

1239 
val le_diff_iff = thm "le_diff_iff"; 

1240 
val diff_le_mono = thm "diff_le_mono"; 

1241 
val diff_le_mono2 = thm "diff_le_mono2"; 

1242 
val diff_less_mono2 = thm "diff_less_mono2"; 

1243 
val diffs0_imp_equal = thm "diffs0_imp_equal"; 

1244 
val one_less_mult = thm "one_less_mult"; 

1245 
val n_less_m_mult_n = thm "n_less_m_mult_n"; 

1246 
val n_less_n_mult_m = thm "n_less_n_mult_m"; 

1247 
val diff_diff_right = thm "diff_diff_right"; 

1248 
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1"; 

1249 
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2"; 

1250 
*} 

1251 

22718  1252 

1253 
subsection{*Embedding of the Naturals into any 

1254 
@{text semiring_1_cancel}: @{term of_nat}*} 

21243  1255 

1256 
consts of_nat :: "nat => 'a::semiring_1_cancel" 

1257 

1258 
primrec 

1259 
of_nat_0: "of_nat 0 = 0" 

1260 
of_nat_Suc: "of_nat (Suc m) = of_nat m + 1" 

1261 

1262 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 

22718  1263 
by simp 
21243  1264 

1265 
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n" 

22718  1266 
by (induct m) (simp_all add: add_ac) 
21243  1267 

1268 
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n" 

22718  1269 
by (induct m) (simp_all add: add_ac left_distrib) 
21243  1270 

1271 
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)" 

22718  1272 
apply (induct m, simp_all) 
1273 
apply (erule order_trans) 

1274 
apply (rule less_add_one [THEN order_less_imp_le]) 

1275 
done 

21243  1276 

1277 
lemma less_imp_of_nat_less: 

22718  1278 
"m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)" 
1279 
apply (induct m n rule: diff_induct, simp_all) 

1280 
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force) 

1281 
done 

21243  1282 

1283 
lemma of_nat_less_imp_less: 

22718  1284 
"of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n" 
1285 
apply (induct m n rule: diff_induct, simp_all) 

1286 
apply (insert zero_le_imp_of_nat) 

1287 
apply (force simp add: linorder_not_less [symmetric]) 

1288 
done 

21243  1289 

1290 
lemma of_nat_less_iff [simp]: 

22718  1291 
"(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)" 
1292 
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) 

21243  1293 

1294 
text{*Special cases where either operand is zero*} 

22718  1295 

1296 
lemma of_nat_0_less_iff [simp]: "((0::'a::ordered_semidom) < of_nat n) = (0 < n)" 

1297 
by (rule of_nat_less_iff [of 0, simplified]) 

1298 

1299 
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < (0::'a::ordered_semidom)" 

1300 
by (rule of_nat_less_iff [of _ 0, simplified]) 

21243  1301 

1302 
lemma of_nat_le_iff [simp]: 

22718  1303 
"(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)" 
1304 
by (simp add: linorder_not_less [symmetric]) 

21243  1305 

1306 
text{*Special cases where either operand is zero*} 

22718  1307 
lemma of_nat_0_le_iff [simp]: "(0::'a::ordered_semidom) \<le> of_nat n" 
1308 
by (rule of_nat_le_iff [of 0, simplified]) 

1309 
lemma of_nat_le_0_iff [simp]: "(of_nat m \<le> (0::'a::ordered_semidom)) = (m = 0)" 

1310 
by (rule of_nat_le_iff [of _ 0, simplified]) 

21243  1311 

1312 
text{*The ordering on the @{text semiring_1_cancel} is necessary 

1313 
to exclude the possibility of a finite field, which indeed wraps back to 

1314 
zero.*} 

1315 
lemma of_nat_eq_iff [simp]: 

22718  1316 
"(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)" 
1317 
by (simp add: order_eq_iff) 

21243  1318 

1319 
text{*Special cases where either operand is zero*} 

22718  1320 
lemma of_nat_0_eq_iff [simp]: "((0::'a::ordered_semidom) = of_nat n) = (0 = n)" 
1321 
by (rule of_nat_eq_iff [of 0, simplified]) 

1322 
lemma of_nat_eq_0_iff [simp]: "(of_nat m = (0::'a::ordered_semidom)) = (m = 0)" 

1323 
by (rule of_nat_eq_iff [of _ 0, simplified]) 

21243  1324 

1325 
lemma of_nat_diff [simp]: 

22718  1326 
"n \<le> m ==> of_nat (m  n) = of_nat m  (of_nat n :: 'a::ring_1)" 
1327 
by (simp del: of_nat_add 

1328 
add: compare_rls of_nat_add [symmetric] split add: nat_diff_split) 

21243  1329 

22483  1330 
instance nat :: distrib_lattice 
1331 
"inf \<equiv> min" 

1332 
"sup \<equiv> max" 

1333 
by intro_classes (auto simp add: inf_nat_def sup_nat_def) 

1334 

22157  1335 

1336 
subsection {* Size function *} 

1337 

22718  1338 
lemma nat_size [simp]: "size (n::nat) = n" 
22157  1339 
by (induct n) simp_all 
1340 

923  1341 
end 