author  haftmann 
Tue, 20 Mar 2007 08:27:15 +0100  
changeset 22473  753123c89d72 
parent 22348  ab505d281015 
child 22483  86064f2f2188 
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") 
15131  14 
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 *} 
19573  25 
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 
pred_nat :: "(nat * nat) set" 

52 

53 
local 

54 

55 
defs 

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Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 
13449  57 
pred_nat_def: "pred_nat == {(m, n). n = Suc m}" 
58 

21456  59 
instance nat :: "{ord, zero, one}" 
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Zero_nat_def: "0 == Abs_Nat Zero_Rep" 

61 
One_nat_def [simp]: "1 == Suc 0" 

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less_def: "m < n == (m, n) : pred_nat^+" 
21456  63 
le_def: "m \<le> (n::nat) == ~ (n < m)" .. 
13449  64 

65 
text {* Induction *} 

923  66 

22262  67 
lemmas Rep_Nat' = Rep_Nat [simplified mem_Collect_eq] 
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lemmas Abs_Nat_inverse' = Abs_Nat_inverse [simplified mem_Collect_eq] 

69 

13449  70 
theorem nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" 
71 
apply (unfold Zero_nat_def Suc_def) 

72 
apply (rule Rep_Nat_inverse [THEN subst])  {* types force good instantiation *} 

22262  73 
apply (erule Rep_Nat' [THEN Nat.induct]) 
74 
apply (iprover elim: Abs_Nat_inverse' [THEN subst]) 

13449  75 
done 
76 

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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  84 
by (rule not_sym, rule Suc_not_Zero not_sym) 
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86 
lemma Suc_neq_Zero: "Suc m = 0 ==> R" 

87 
by (rule notE, rule Suc_not_Zero) 

88 

89 
lemma Zero_neq_Suc: "0 = Suc m ==> R" 

90 
by (rule Suc_neq_Zero, erule sym) 

91 

92 
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 
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inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 
13449  97 

98 
lemma Suc_inject: "Suc x = Suc y ==> x = y" 

99 
by (rule inj_Suc [THEN injD]) 

100 

101 
lemma Suc_Suc_eq [iff]: "(Suc m = Suc n) = (m = n)" 

15413  102 
by (rule inj_Suc [THEN inj_eq]) 
13449  103 

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lemma nat_not_singleton: "(\<forall>x. x = (0::nat)) = False" 
13449  105 
by auto 
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text {* size of a datatype value *} 
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22473  109 
class size = type + 
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fixes size :: "'a \<Rightarrow> nat" 
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13449  112 
text {* @{typ nat} is a datatype *} 
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rep_datatype nat 
13449  115 
distinct Suc_not_Zero Zero_not_Suc 
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inject Suc_Suc_eq 

21411  117 
induction nat_induct 
118 

119 
declare nat.induct [case_names 0 Suc, induct type: nat] 

120 
declare nat.exhaust [case_names 0 Suc, cases type: nat] 

13449  121 

21672  122 
lemmas nat_rec_0 = nat.recs(1) 
123 
and nat_rec_Suc = nat.recs(2) 

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125 
lemmas nat_case_0 = nat.cases(1) 

126 
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  133 
by (rule not_sym, rule n_not_Suc_n) 
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135 
text {* A special form of induction for reasoning 

136 
about @{term "m < n"} and @{term "m  n"} *} 

137 

138 
theorem diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 

139 
(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 

14208  140 
apply (rule_tac x = m in spec) 
15251  141 
apply (induct n) 
13449  142 
prefer 2 
143 
apply (rule allI) 

17589  144 
apply (induct_tac x, iprover+) 
13449  145 
done 
146 

147 
subsection {* Basic properties of "less than" *} 

148 

149 
lemma wf_pred_nat: "wf pred_nat" 

14208  150 
apply (unfold wf_def pred_nat_def, clarify) 
151 
apply (induct_tac x, blast+) 

13449  152 
done 
153 

154 
lemma wf_less: "wf {(x, y::nat). x < y}" 

155 
apply (unfold less_def) 

14208  156 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_subset], blast) 
13449  157 
done 
158 

159 
lemma less_eq: "((m, n) : pred_nat^+) = (m < n)" 

160 
apply (unfold less_def) 

161 
apply (rule refl) 

162 
done 

163 

164 
subsubsection {* Introduction properties *} 

165 

166 
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

167 
apply (unfold less_def) 

14208  168 
apply (rule trans_trancl [THEN transD], assumption+) 
13449  169 
done 
170 

171 
lemma lessI [iff]: "n < Suc n" 

172 
apply (unfold less_def pred_nat_def) 

173 
apply (simp add: r_into_trancl) 

174 
done 

175 

176 
lemma less_SucI: "i < j ==> i < Suc j" 

14208  177 
apply (rule less_trans, assumption) 
13449  178 
apply (rule lessI) 
179 
done 

180 

181 
lemma zero_less_Suc [iff]: "0 < Suc n" 

182 
apply (induct n) 

183 
apply (rule lessI) 

184 
apply (erule less_trans) 

185 
apply (rule lessI) 

186 
done 

187 

188 
subsubsection {* Elimination properties *} 

189 

190 
lemma less_not_sym: "n < m ==> ~ m < (n::nat)" 

191 
apply (unfold less_def) 

192 
apply (blast intro: wf_pred_nat wf_trancl [THEN wf_asym]) 

193 
done 

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195 
lemma less_asym: 

196 
assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P 

197 
apply (rule contrapos_np) 

198 
apply (rule less_not_sym) 

199 
apply (rule h1) 

200 
apply (erule h2) 

201 
done 

202 

203 
lemma less_not_refl: "~ n < (n::nat)" 

204 
apply (unfold less_def) 

205 
apply (rule wf_pred_nat [THEN wf_trancl, THEN wf_not_refl]) 

206 
done 

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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  214 
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) 

14208  223 
apply (simp add: less_def pred_nat_def, assumption) 
13449  224 
done 
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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  235 
apply (rule eq, blast) 
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apply (rule less, blast) 

13449  237 
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  253 
apply (induct m) 
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apply (induct n) 

13449  255 
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) 

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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" 

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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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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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subsubsection {* Inductive (?) properties *} 

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

285 
done 

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287 
lemma Suc_lessD: "Suc m < n ==> m < n" 

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apply (induct n) 

289 
apply (fast intro!: lessI [THEN less_SucI] elim: less_trans lessE)+ 

290 
done 

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

293 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

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apply (rule major [THEN lessE]) 

295 
apply (erule lessI [THEN minor]) 

14208  296 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  297 
done 
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299 
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

300 
by (blast elim: lessE dest: Suc_lessD) 

4104  301 

16635  302 
lemma Suc_less_eq [iff, code]: "(Suc m < Suc n) = (m < n)" 
13449  303 
apply (rule iffI) 
304 
apply (erule Suc_less_SucD) 

305 
apply (erule Suc_mono) 

306 
done 

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308 
lemma less_trans_Suc: 

309 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  310 
apply (induct k, simp_all) 
13449  311 
apply (insert le) 
312 
apply (simp add: less_Suc_eq) 

313 
apply (blast dest: Suc_lessD) 

314 
done 

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16635  316 
lemma [code]: "((n::nat) < 0) = False" by simp 
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lemma [code]: "(0 < Suc n) = True" by simp 

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13449  319 
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m  n < m"} *} 
320 
lemma not_less_eq: "(~ m < n) = (n < Suc m)" 

14208  321 
by (rule_tac m = m and n = n in diff_induct, simp_all) 
13449  322 

323 
text {* Complete induction, aka courseofvalues induction *} 

324 
lemma nat_less_induct: 

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assumes prem: "!!n. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
13449  326 
apply (rule_tac a=n in wf_induct) 
327 
apply (rule wf_pred_nat [THEN wf_trancl]) 

328 
apply (rule prem) 

14208  329 
apply (unfold less_def, assumption) 
13449  330 
done 
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14131  332 
lemmas less_induct = nat_less_induct [rule_format, case_names less] 
333 

21243  334 

14131  335 
subsection {* Properties of "less than or equal" *} 
13449  336 

337 
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)" 
13449  339 
by (unfold le_def, rule not_less_eq [symmetric]) 
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lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
13449  342 
by (rule less_Suc_eq_le [THEN iffD2]) 
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lemma le0 [iff]: "(0::nat) \<le> n" 
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by (unfold le_def, rule not_less0) 
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lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
13449  348 
by (simp add: le_def) 
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lemma le_0_eq [iff]: "((i::nat) \<le> 0) = (i = 0)" 
13449  351 
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  354 
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  357 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  358 

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lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
13449  360 
apply (simp add: le_def less_Suc_eq) 
361 
apply (blast elim!: less_irrefl less_asym) 

362 
done  {* formerly called lessD *} 

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lemma Suc_leD: "Suc(m) \<le> n ==> m \<le> n" 
13449  365 
by (simp add: le_def less_Suc_eq) 
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367 
text {* Stronger version of @{text Suc_leD} *} 

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lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" 
13449  369 
apply (simp add: le_def less_Suc_eq) 
370 
using less_linear 

371 
apply blast 

372 
done 

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lemma Suc_le_eq: "(Suc m \<le> n) = (m < n)" 
13449  375 
by (blast intro: Suc_leI Suc_le_lessD) 
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lemma le_SucI: "m \<le> n ==> m \<le> Suc n" 
13449  378 
by (unfold le_def) (blast dest: Suc_lessD) 
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lemma less_imp_le: "m < n ==> m \<le> (n::nat)" 
13449  381 
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  384 
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 
385 

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text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  388 

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lemma le_imp_less_or_eq: "m \<le> n ==> m < n  m = (n::nat)" 
13449  390 
apply (unfold le_def) 
391 
using less_linear 

392 
apply (blast elim: less_irrefl less_asym) 

393 
done 

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lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
13449  396 
apply (unfold le_def) 
397 
using less_linear 

398 
apply (blast elim!: less_irrefl elim: less_asym) 

399 
done 

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401 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
17589  402 
by (iprover intro: less_or_eq_imp_le le_imp_less_or_eq) 
13449  403 

404 
text {* Useful with @{text Blast}. *} 

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405 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
13449  406 
by (rule less_or_eq_imp_le, rule disjI2) 
407 

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408 
lemma le_refl: "n \<le> (n::nat)" 
13449  409 
by (simp add: le_eq_less_or_eq) 
410 

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411 
lemma le_less_trans: "[ i \<le> j; j < k ] ==> i < (k::nat)" 
13449  412 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
413 

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414 
lemma less_le_trans: "[ i < j; j \<le> k ] ==> i < (k::nat)" 
13449  415 
by (blast dest!: le_imp_less_or_eq intro: less_trans) 
416 

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417 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::nat)" 
13449  418 
by (blast dest!: le_imp_less_or_eq intro: less_or_eq_imp_le less_trans) 
419 

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420 
lemma le_anti_sym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
13449  421 
by (blast dest!: le_imp_less_or_eq elim!: less_irrefl elim: less_asym) 
422 

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423 
lemma Suc_le_mono [iff]: "(Suc n \<le> Suc m) = (n \<le> m)" 
13449  424 
by (simp add: le_simps) 
425 

426 
text {* Axiom @{text order_less_le} of class @{text order}: *} 

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427 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
13449  428 
by (simp add: le_def nat_neq_iff) (blast elim!: less_asym) 
429 

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430 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
13449  431 
by (rule iffD2, rule nat_less_le, rule conjI) 
432 

433 
text {* Axiom @{text linorder_linear} of class @{text linorder}: *} 

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434 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
13449  435 
apply (simp add: le_eq_less_or_eq) 
436 
using less_linear 

437 
apply blast 

438 
done 

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 

15921  447 
lemmas linorder_neqE_nat = linorder_neqE[where 'a = nat] 
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 {* 

464 
Reorientation of the equations @{text "0 = x"} and @{text "1 = x"}. 

465 
No longer added as simprules (they loop) 

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 

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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" 
13449  507 
by (case_tac n) simp_all 
508 

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509 
lemma gr_implies_not0: "!!n::nat. m<n ==> n \<noteq> 0" 
13449  510 
by (case_tac n) simp_all 
511 

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512 
lemma neq0_conv [iff]: "!!n::nat. (n \<noteq> 0) = (0 < n)" 
13449  513 
by (case_tac n) simp_all 
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) 

14208  524 
apply (rule ccontr, simp_all) 
13449  525 
done 
526 

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527 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
13449  528 
by (induct m') simp_all 
529 

530 
text {* Useful in certain inductive arguments *} 

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531 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
13449  532 
by (case_tac m) simp_all 
533 

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534 
lemma nat_induct2: "[P 0; P (Suc 0); !!k. P k ==> P (Suc (Suc k))] ==> P n" 
13449  535 
apply (rule nat_less_induct) 
536 
apply (case_tac n) 

537 
apply (case_tac [2] nat) 

538 
apply (blast intro: less_trans)+ 

539 
done 

540 

21243  541 

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542 
subsection {* @{text LEAST} theorems for type @{typ nat}*} 
13449  543 

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544 
lemma Least_Suc: 
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545 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 
14208  546 
apply (case_tac "n", auto) 
13449  547 
apply (frule LeastI) 
548 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

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549 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 
13449  550 
apply (erule_tac [2] Least_le) 
14208  551 
apply (case_tac "LEAST x. P x", auto) 
13449  552 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
553 
apply (blast intro: order_antisym) 

554 
done 

555 

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556 
lemma Least_Suc2: 
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557 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 
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558 
by (erule (1) Least_Suc [THEN ssubst], simp) 
13449  559 

560 

561 
subsection {* @{term min} and @{term max} *} 

562 

563 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

564 
by (rule min_leastL) simp 

565 

566 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

567 
by (rule min_leastR) simp 

568 

569 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

570 
by (simp add: min_of_mono) 

571 

22191  572 
lemma min_Suc1: 
573 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

574 
by (simp split: nat.split) 

575 

576 
lemma min_Suc2: 

577 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

578 
by (simp split: nat.split) 

579 

13449  580 
lemma max_0L [simp]: "max 0 n = (n::nat)" 
581 
by (rule max_leastL) simp 

582 

583 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

584 
by (rule max_leastR) simp 

585 

586 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

587 
by (simp add: max_of_mono) 

588 

22191  589 
lemma max_Suc1: 
590 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

591 
by (simp split: nat.split) 

592 

593 
lemma max_Suc2: 

594 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

595 
by (simp split: nat.split) 

596 

13449  597 

598 
subsection {* Basic rewrite rules for the arithmetic operators *} 

599 

600 
text {* Difference *} 

601 

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602 
lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
15251  603 
by (induct n) simp_all 
13449  604 

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605 
lemma diff_Suc_Suc [simp, code]: "Suc(m)  Suc(n) = m  n" 
15251  606 
by (induct n) simp_all 
13449  607 

608 

609 
text {* 

610 
Could be (and is, below) generalized in various ways 

611 
However, none of the generalizations are currently in the simpset, 

612 
and I dread to think what happens if I put them in 

613 
*} 

614 
lemma Suc_pred [simp]: "0 < n ==> Suc (n  Suc 0) = n" 

615 
by (simp split add: nat.split) 

616 

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617 
declare diff_Suc [simp del, code del] 
13449  618 

619 

620 
subsection {* Addition *} 

621 

622 
lemma add_0_right [simp]: "m + 0 = (m::nat)" 

623 
by (induct m) simp_all 

624 

625 
lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 

626 
by (induct m) simp_all 

627 

19890  628 
lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" 
629 
by simp 

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630 

13449  631 

632 
text {* Associative law for addition *} 

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633 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
13449  634 
by (induct m) simp_all 
635 

636 
text {* Commutative law for addition *} 

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637 
lemma nat_add_commute: "m + n = n + (m::nat)" 
13449  638 
by (induct m) simp_all 
639 

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640 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
13449  641 
apply (rule mk_left_commute [of "op +"]) 
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642 
apply (rule nat_add_assoc) 
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643 
apply (rule nat_add_commute) 
13449  644 
done 
645 

14331  646 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
13449  647 
by (induct k) simp_all 
648 

14331  649 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
13449  650 
by (induct k) simp_all 
651 

14331  652 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
13449  653 
by (induct k) simp_all 
654 

14331  655 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
13449  656 
by (induct k) simp_all 
657 

658 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 

659 

660 
lemma add_is_0 [iff]: "!!m::nat. (m + n = 0) = (m = 0 & n = 0)" 

661 
by (case_tac m) simp_all 

662 

663 
lemma add_is_1: "(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 

664 
by (case_tac m) simp_all 

665 

666 
lemma one_is_add: "(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 

667 
by (rule trans, rule eq_commute, rule add_is_1) 

668 

669 
lemma add_gr_0 [iff]: "!!m::nat. (0 < m + n) = (0 < m  0 < n)" 

670 
by (simp del: neq0_conv add: neq0_conv [symmetric]) 

671 

672 
lemma add_eq_self_zero: "!!m::nat. m + n = m ==> n = 0" 

673 
apply (drule add_0_right [THEN ssubst]) 

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674 
apply (simp add: nat_add_assoc del: add_0_right) 
13449  675 
done 
676 

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677 

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678 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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679 
apply(induct k) 
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680 
apply simp 
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681 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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682 
apply (simp add:o_def) 
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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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changeset

692 
text {* right successor law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

693 
lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

694 
by (induct m) (simp_all add: nat_add_left_commute) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

695 

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

696 
text {* Commutative law for multiplication *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
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) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

729 
apply (induct_tac [2] n, simp_all) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

730 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

731 

21243  732 

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

733 
subsection {* Monotonicity of Addition *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

734 

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

735 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

736 
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

737 
by (induct k) simp_all 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

738 

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

739 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

740 
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

741 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  742 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

743 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

744 

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

745 
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

746 
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

747 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

748 
apply (simp_all add: order_le_less) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

749 
apply (blast elim!: less_SucE 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

750 
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

751 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

752 

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

753 
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

754 
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

755 
apply (erule_tac m1 = 0 in less_imp_Suc_add [THEN exE], simp) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

756 
apply (induct_tac x) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

757 
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

758 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

759 

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

760 

14740  761 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
14738  762 
instance nat :: ordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

763 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

764 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

765 
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

766 
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

767 
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

768 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

769 

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

770 
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

771 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

772 

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

773 
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

774 
by simp 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

775 

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

778 

19870  779 
text{*An induction rule for estabilishing binary relations*} 
780 
lemma less_Suc_induct: 

781 
assumes less: "i < j" 

782 
and step: "!!i. P i (Suc i)" 

783 
and trans: "!!i j k. P i j ==> P j k ==> P i k" 

784 
shows "P i j" 

785 
proof  

786 
from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 

787 
have "P i (Suc(i+k))" 

788 
proof (induct k) 

789 
case 0 

790 
show ?case by (simp add: step) 

791 
next 

792 
case (Suc k) 

793 
thus ?case by (auto intro: prems) 

794 
qed 

795 
thus "P i j" by (simp add: j) 

796 
qed 

797 

798 

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

799 
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

800 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

801 
lemma less_mono_imp_le_mono: 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

802 
assumes lt_mono: "!!i j::nat. i < j ==> f i < f j" 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

803 
and le: "i \<le> j" shows "f i \<le> ((f j)::nat)" using le 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

804 
apply (simp add: order_le_less) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

805 
apply (blast intro!: lt_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

806 
done 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

807 

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

808 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

809 
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

810 
by (rule add_right_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

811 

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

812 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

813 
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

814 
by (rule add_mono) 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

815 

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

816 
lemma le_add2: "n \<le> ((m + n)::nat)" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

817 
by (insert add_right_mono [of 0 m n], simp) 
13449  818 

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

819 
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

820 
by (simp add: add_commute, rule le_add2) 
13449  821 

822 
lemma less_add_Suc1: "i < Suc (i + m)" 

823 
by (rule le_less_trans, rule le_add1, rule lessI) 

824 

825 
lemma less_add_Suc2: "i < Suc (m + i)" 

826 
by (rule le_less_trans, rule le_add2, rule lessI) 

827 

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

828 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
17589  829 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  830 

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

831 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
13449  832 
by (rule le_trans, assumption, rule le_add1) 
833 

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

834 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
13449  835 
by (rule le_trans, assumption, rule le_add2) 
836 

837 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

838 
by (rule less_le_trans, assumption, rule le_add1) 

839 

840 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

841 
by (rule less_le_trans, assumption, rule le_add2) 

842 

843 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

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

844 
apply (rule le_less_trans [of _ "i+j"]) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

845 
apply (simp_all add: le_add1) 
13449  846 
done 
847 

848 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

849 
apply (rule notI) 

850 
apply (erule add_lessD1 [THEN less_irrefl]) 

851 
done 

852 

853 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

854 
by (simp add: add_commute not_add_less1) 

855 

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856 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

857 
apply (rule order_trans [of _ "m+k"]) 
a09441bd4f1e
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paulson
parents:
14331
diff
changeset

858 
apply (simp_all add: le_add1) 
a09441bd4f1e
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paulson
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diff
changeset

859 
done 
13449  860 

14267
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861 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
13449  862 
apply (simp add: add_commute) 
863 
apply (erule add_leD1) 

864 
done 

865 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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866 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
13449  867 
by (blast dest: add_leD1 add_leD2) 
868 

869 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

870 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

871 
by (force simp del: add_Suc_right 

872 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 

873 

874 

875 
subsection {* Difference *} 

876 

877 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

878 
by (induct m) simp_all 

879 

880 
text {* Addition is the inverse of subtraction: 

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881 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  882 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
883 
by (induct m n rule: diff_induct) simp_all 

884 

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885 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
16796  886 
by (simp add: add_diff_inverse linorder_not_less) 
13449  887 

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888 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
13449  889 
by (simp add: le_add_diff_inverse add_commute) 
890 

891 

892 
subsection {* More results about difference *} 

893 

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894 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
13449  895 
by (induct m n rule: diff_induct) simp_all 
896 

897 
lemma diff_less_Suc: "m  n < Suc m" 

898 
apply (induct m n rule: diff_induct) 

899 
apply (erule_tac [3] less_SucE) 

900 
apply (simp_all add: less_Suc_eq) 

901 
done 

902 

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903 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
13449  904 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
905 

906 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 

907 
by (rule le_less_trans, rule diff_le_self) 

908 

909 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 

910 
by (induct i j rule: diff_induct) simp_all 

911 

912 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 

913 
by (simp add: diff_diff_left) 

914 

915 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

14208  916 
apply (case_tac "n", safe) 
13449  917 
apply (simp add: le_simps) 
918 
done 

919 

920 
text {* This and the next few suggested by Florian Kammueller *} 

921 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 

922 
by (simp add: diff_diff_left add_commute) 

923 

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924 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
13449  925 
by (induct j k rule: diff_induct) simp_all 
926 

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927 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
13449  928 
by (simp add: add_commute diff_add_assoc) 
929 

930 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 

931 
by (induct n) simp_all 

932 

933 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 

934 
by (simp add: diff_add_assoc) 

935 

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936 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
13449  937 
apply safe 
938 
apply (simp_all add: diff_add_inverse2) 

939 
done 

940 

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941 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
13449  942 
by (induct m n rule: diff_induct) simp_all 
943 

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944 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
13449  945 
by (rule iffD2, rule diff_is_0_eq) 
946 

947 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

948 
by (induct m n rule: diff_induct) simp_all 

949 

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950 
lemma less_imp_add_positive: "i < j ==> \<exists>k::nat. 0 < k & i + k = j" 
13449  951 
apply (rule_tac x = "j  i" in exI) 
952 
apply (simp (no_asm_simp) add: add_diff_inverse less_not_sym) 

953 
done 

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rearranged setup of arithmetic procedures, avoiding global reference values;
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954 

13449  955 
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k  i)" 
956 
apply (induct k i rule: diff_induct) 

957 
apply (simp_all (no_asm)) 

17589  958 
apply iprover 
13449  959 
done 
960 

961 
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0" 

962 
apply (rule diff_self_eq_0 [THEN subst]) 

17589  963 
apply (rule zero_induct_lemma, iprover+) 
13449  964 
done 
965 

966 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 

967 
by (induct k) simp_all 

968 

969 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 

970 
by (simp add: diff_cancel add_commute) 

971 

972 
lemma diff_add_0: "n  (n + m) = (0::nat)" 

973 
by (induct n) simp_all 

974 

975 

976 
text {* Difference distributes over multiplication *} 

977 

978 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 

979 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 

980 

981 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 

982 
by (simp add: diff_mult_distrib mult_commute [of k]) 

983 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 

984 

985 
lemmas nat_distrib = 

986 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 

987 

988 

989 
subsection {* Monotonicity of Multiplication *} 

990 

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991 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
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parents:
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diff
changeset

992 
by (simp add: mult_right_mono) 
13449  993 

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994 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
14341
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Ring_and_Field now requires axiom add_left_imp_eq for semirings.
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parents:
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diff
changeset

995 
by (simp add: mult_left_mono) 
13449  996 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset

997 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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998 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

999 
by (simp add: mult_mono) 
13449  1000 

1001 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

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

1002 
by (simp add: mult_strict_right_mono) 
13449  1003 

14266  1004 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1005 
there are no negative numbers.*} 

1006 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1007 
apply (induct m) 
14208  1008 
apply (case_tac [2] n, simp_all) 
13449  1009 
done 
1010 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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1011 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" 
13449  1012 
apply (induct m) 
14208  1013 
apply (case_tac [2] n, simp_all) 
13449  1014 
done 
1015 

1016 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 

15251  1017 
apply (induct m, simp) 
1018 
apply (induct n, simp, fastsimp) 

13449  1019 
done 
1020 

1021 
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 

1022 
apply (rule trans) 

14208  1023 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
13449  1024 
done 
1025 

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

1026 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1027 
apply (safe intro!: mult_less_mono1) 
14208  1028 
apply (case_tac k, auto) 
13449  1029 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1030 
apply (blast intro: mult_le_mono1) 

1031 
done 

1032 

1033 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

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

1034 
by (simp add: mult_commute [of k]) 
13449  1035 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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changeset

1036 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
14208  1037 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1038 

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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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1039 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
14208  1040 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1041 

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

1042 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
14208  1043 
apply (cut_tac less_linear, safe, auto) 
13449  1044 
apply (drule mult_less_mono1, assumption, simp)+ 
1045 
done 

1046 

1047 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 

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

1048 
by (simp add: mult_commute [of k]) 
13449  1049 

1050 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

1051 
by (subst mult_less_cancel1) simp 

1052 

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1053 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
13449  1054 
by (subst mult_le_cancel1) simp 
1055 

1056 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 

1057 
by (subst mult_cancel1) simp 

1058 

1059 
text {* Lemma for @{text gcd} *} 

1060 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1061 
apply (drule sym) 

1062 
apply (rule disjCI) 

1063 
apply (rule nat_less_cases, erule_tac [2] _) 

1064 
apply (fastsimp elim!: less_SucE) 

1065 
apply (fastsimp dest: mult_less_mono2) 

1066 
done 

9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1067 

20588  1068 

18702  1069 
subsection {* Code generator setup *} 
1070 

22348  1071 
lemma one_is_Suc_zero [code inline]: 
20355  1072 
"1 = Suc 0" 
1073 
by simp 

1074 

20588  1075 
instance nat :: eq .. 
1076 

1077 
lemma [code func]: 

22348  1078 
"(0\<Colon>nat) = 0 \<longleftrightarrow> True" 
1079 
"Suc n = Suc m \<longleftrightarrow> n = m" 

1080 
"Suc n = 0 \<longleftrightarrow> False" 

1081 
"0 = Suc m \<longleftrightarrow> False" 

1082 
by auto 

20588  1083 

1084 
lemma [code func]: 

22348  1085 
"(0\<Colon>nat) \<le> m \<longleftrightarrow> True" 
1086 
"Suc (n\<Colon>nat) \<le> m \<longleftrightarrow> n < m" 

1087 
"(n\<Colon>nat) < 0 \<longleftrightarrow> False" 

1088 
"(n\<Colon>nat) < Suc m \<longleftrightarrow> n \<le> m" 

1089 
using Suc_le_eq less_Suc_eq_le by simp_all 

20588  1090 

21243  1091 

1092 
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *} 

1093 

1094 
use "arith_data.ML" 

1095 
setup arith_setup 

1096 

1097 
text{*The following proofs may rely on the arithmetic proof procedures.*} 

1098 

1099 
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)" 

1100 
by (auto simp: le_eq_less_or_eq dest: less_imp_Suc_add) 

1101 

1102 
lemma pred_nat_trancl_eq_le: "((m, n) : pred_nat^*) = (m \<le> n)" 

1103 
by (simp add: less_eq reflcl_trancl [symmetric] 

1104 
del: reflcl_trancl, arith) 

1105 

1106 
lemma nat_diff_split: 

1107 
"P(a  b::nat) = ((a<b > P 0) & (ALL d. a = b + d > P d))" 

1108 
 {* elimination of @{text } on @{text nat} *} 

1109 
by (cases "a<b" rule: case_split) 

1110 
(auto simp add: diff_is_0_eq [THEN iffD2]) 

1111 

1112 
lemma nat_diff_split_asm: 

1113 
"P(a  b::nat) = (~ (a < b & ~ P 0  (EX d. a = b + d & ~ P d)))" 

1114 
 {* elimination of @{text } on @{text nat} in assumptions *} 

1115 
by (simp split: nat_diff_split) 

1116 

1117 
lemmas [arith_split] = nat_diff_split split_min split_max 

1118 

1119 

1120 

1121 
lemma le_square: "m \<le> m * (m::nat)" 

1122 
by (induct m) auto 

1123 

1124 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 

1125 
by (induct m) auto 

1126 

1127 

1128 
text{*Subtraction laws, mostly by Clemens Ballarin*} 

1129 

1130 
lemma diff_less_mono: "[ a < (b::nat); c \<le> a ] ==> ac < bc" 

1131 
by arith 

1132 

1133 
lemma less_diff_conv: "(i < jk) = (i+k < (j::nat))" 

1134 
by arith 

1135 

1136 
lemma le_diff_conv: "(jk \<le> (i::nat)) = (j \<le> i+k)" 

1137 
by arith 

1138 

1139 
lemma le_diff_conv2: "k \<le> j ==> (i \<le> jk) = (i+k \<le> (j::nat))" 

1140 
by arith 

1141 

1142 
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n  (n  i) = i" 

1143 
by arith 

1144 

1145 
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m  k" 

1146 
by arith 

1147 

1148 
(*Replaces the previous diff_less and le_diff_less, which had the stronger 

1149 
second premise n\<le>m*) 

1150 
lemma diff_less[simp]: "!!m::nat. [ 0<n; 0<m ] ==> m  n < m" 

1151 
by arith 

1152 

1153 

1154 
(** Simplification of relational expressions involving subtraction **) 

1155 

1156 
lemma diff_diff_eq: "[ k \<le> m; k \<le> (n::nat) ] ==> ((mk)  (nk)) = (mn)" 

1157 
by (simp split add: nat_diff_split) 

1158 

1159 
lemma eq_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk = nk) = (m=n)" 

1160 
by (auto split add: nat_diff_split) 

1161 

1162 
lemma less_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk < nk) = (m<n)" 

1163 
by (auto split add: nat_diff_split) 

1164 

1165 
lemma le_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk \<le> nk) = (m\<le>n)" 

1166 
by (auto split add: nat_diff_split) 

1167 

1168 

1169 
text{*(Anti)Monotonicity of subtraction  by Stephan Merz*} 

1170 

1171 
(* Monotonicity of subtraction in first argument *) 

1172 
lemma diff_le_mono: "m \<le> (n::nat) ==> (ml) \<le> (nl)" 

1173 
by (simp split add: nat_diff_split) 

1174 

1175 
lemma diff_le_mono2: "m \<le> (n::nat) ==> (ln) \<le> (lm)" 

1176 
by (simp split add: nat_diff_split) 

1177 

1178 
lemma diff_less_mono2: "[ m < (n::nat); m<l ] ==> (ln) < (lm)" 

1179 
by (simp split add: nat_diff_split) 

1180 

1181 
lemma diffs0_imp_equal: "!!m::nat. [ mn = 0; nm = 0 ] ==> m=n" 

1182 
by (simp split add: nat_diff_split) 

1183 

1184 
text{*Lemmas for ex/Factorization*} 

1185 

1186 
lemma one_less_mult: "[ Suc 0 < n; Suc 0 < m ] ==> Suc 0 < m*n" 

1187 
by (case_tac "m", auto) 

1188 

1189 
lemma n_less_m_mult_n: "[ Suc 0 < n; Suc 0 < m ] ==> n<m*n" 

1190 
by (case_tac "m", auto) 

1191 

1192 
lemma n_less_n_mult_m: "[ Suc 0 < n; Suc 0 < m ] ==> n<n*m" 

1193 
by (case_tac "m", auto) 

1194 

1195 

1196 
text{*Rewriting to pull differences out*} 

1197 

1198 
lemma diff_diff_right [simp]: "k\<le>j > i  (j  k) = i + (k::nat)  j" 

1199 
by arith 

1200 

1201 
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m  Suc (j  k) = m + k  Suc j" 

1202 
by arith 

1203 

1204 
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j  k)  m = Suc j  (k + m)" 

1205 
by arith 

1206 

1207 
(*The others are 

1208 
i  j  k = i  (j + k), 

1209 
k \<le> j ==> j  k + i = j + i  k, 

1210 
k \<le> j ==> i + (j  k) = i + j  k *) 

1211 
lemmas add_diff_assoc = diff_add_assoc [symmetric] 

1212 
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric] 

1213 
declare diff_diff_left [simp] add_diff_assoc [simp] add_diff_assoc2[simp] 

1214 

1215 
text{*At present we prove no analogue of @{text not_less_Least} or @{text 

1216 
Least_Suc}, since there appears to be no need.*} 

1217 

1218 
ML 

1219 
{* 

1220 
val pred_nat_trancl_eq_le = thm "pred_nat_trancl_eq_le"; 

1221 
val nat_diff_split = thm "nat_diff_split"; 

1222 
val nat_diff_split_asm = thm "nat_diff_split_asm"; 

1223 
val le_square = thm "le_square"; 

1224 
val le_cube = thm "le_cube"; 

1225 
val diff_less_mono = thm "diff_less_mono"; 

1226 
val less_diff_conv = thm "less_diff_conv"; 

1227 
val le_diff_conv = thm "le_diff_conv"; 

1228 
val le_diff_conv2 = thm "le_diff_conv2"; 

1229 
val diff_diff_cancel = thm "diff_diff_cancel"; 

1230 
val le_add_diff = thm "le_add_diff"; 

1231 
val diff_less = thm "diff_less"; 

1232 
val diff_diff_eq = thm "diff_diff_eq"; 

1233 
val eq_diff_iff = thm "eq_diff_iff"; 

1234 
val less_diff_iff = thm "less_diff_iff"; 

1235 
val le_diff_iff = thm "le_diff_iff"; 

1236 
val diff_le_mono = thm "diff_le_mono"; 

1237 
val diff_le_mono2 = thm "diff_le_mono2"; 

1238 
val diff_less_mono2 = thm "diff_less_mono2"; 

1239 
val diffs0_imp_equal = thm "diffs0_imp_equal"; 

1240 
val one_less_mult = thm "one_less_mult"; 

1241 
val n_less_m_mult_n = thm "n_less_m_mult_n"; 

1242 
val n_less_n_mult_m = thm "n_less_n_mult_m"; 

1243 
val diff_diff_right = thm "diff_diff_right"; 

1244 
val diff_Suc_diff_eq1 = thm "diff_Suc_diff_eq1"; 

1245 
val diff_Suc_diff_eq2 = thm "diff_Suc_diff_eq2"; 

1246 
*} 

1247 

1248 
subsection{*Embedding of the Naturals into any @{text 

1249 
semiring_1_cancel}: @{term of_nat}*} 

1250 

1251 
consts of_nat :: "nat => 'a::semiring_1_cancel" 

1252 

1253 
primrec 

1254 
of_nat_0: "of_nat 0 = 0" 

1255 
of_nat_Suc: "of_nat (Suc m) = of_nat m + 1" 

1256 

1257 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 

1258 
by simp 

1259 

1260 
lemma of_nat_add [simp]: "of_nat (m+n) = of_nat m + of_nat n" 

1261 
apply (induct m) 

1262 
apply (simp_all add: add_ac) 

1263 
done 

1264 

1265 
lemma of_nat_mult [simp]: "of_nat (m*n) = of_nat m * of_nat n" 

1266 
apply (induct m) 

1267 
apply (simp_all add: add_ac left_distrib) 

1268 
done 

1269 

1270 
lemma zero_le_imp_of_nat: "0 \<le> (of_nat m::'a::ordered_semidom)" 

1271 
apply (induct m, simp_all) 

1272 
apply (erule order_trans) 

1273 
apply (rule less_add_one [THEN order_less_imp_le]) 

1274 
done 

1275 

1276 
lemma less_imp_of_nat_less: 

1277 
"m < n ==> of_nat m < (of_nat n::'a::ordered_semidom)" 

1278 
apply (induct m n rule: diff_induct, simp_all) 

1279 
apply (insert add_le_less_mono [OF zero_le_imp_of_nat zero_less_one], force) 

1280 
done 

1281 

1282 
lemma of_nat_less_imp_less: 

1283 
"of_nat m < (of_nat n::'a::ordered_semidom) ==> m < n" 

1284 
apply (induct m n rule: diff_induct, simp_all) 

1285 
apply (insert zero_le_imp_of_nat) 

1286 
apply (force simp add: linorder_not_less [symmetric]) 

1287 
done 

1288 

1289 
lemma of_nat_less_iff [simp]: 

1290 
"(of_nat m < (of_nat n::'a::ordered_semidom)) = (m<n)" 

1291 
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) 

1292 

1293 
text{*Special cases where either operand is zero*} 

1294 
lemmas of_nat_0_less_iff = of_nat_less_iff [of 0, simplified] 

1295 
lemmas of_nat_less_0_iff = of_nat_less_iff [of _ 0, simplified] 

1296 
declare of_nat_0_less_iff [simp] 

1297 
declare of_nat_less_0_iff [simp] 

1298 

1299 
lemma of_nat_le_iff [simp]: 

1300 
"(of_nat m \<le> (of_nat n::'a::ordered_semidom)) = (m \<le> n)" 

1301 
by (simp add: linorder_not_less [symmetric]) 

1302 

1303 
text{*Special cases where either operand is zero*} 

1304 
lemmas of_nat_0_le_iff = of_nat_le_iff [of 0, simplified] 

1305 
lemmas of_nat_le_0_iff = of_nat_le_iff [of _ 0, simplified] 

1306 
declare of_nat_0_le_iff [simp] 

1307 
declare of_nat_le_0_iff [simp] 

1308 

1309 
text{*The ordering on the @{text semiring_1_cancel} is necessary 

1310 
to exclude the possibility of a finite field, which indeed wraps back to 

1311 
zero.*} 

1312 
lemma of_nat_eq_iff [simp]: 

1313 
"(of_nat m = (of_nat n::'a::ordered_semidom)) = (m = n)" 

1314 
by (simp add: order_eq_iff) 

1315 

1316 
text{*Special cases where either operand is zero*} 

1317 
lemmas of_nat_0_eq_iff = of_nat_eq_iff [of 0, simplified] 

1318 
lemmas of_nat_eq_0_iff = of_nat_eq_iff [of _ 0, simplified] 

1319 
declare of_nat_0_eq_iff [simp] 

1320 
declare of_nat_eq_0_iff [simp] 

1321 

1322 
lemma of_nat_diff [simp]: 

1323 
"n \<le> m ==> of_nat (m  n) = of_nat m  (of_nat n :: 'a::ring_1)" 

1324 
by (simp del: of_nat_add 

1325 
add: compare_rls of_nat_add [symmetric] split add: nat_diff_split) 

1326 

22157  1327 

1328 
subsection {* Size function *} 

1329 

1330 
lemma nat_size[simp]: "size (n::nat) = n" 

1331 
by (induct n) simp_all 

1332 

923  1333 
end 