author  bulwahn 
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permissions  rwrr 
923  1 
(* Title: HOL/Nat.thy 
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ID: $Id$ 

21243  3 
Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel 
923  4 

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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). 
923  7 
*) 
8 

13449  9 
header {* Natural numbers *} 
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15131  11 
theory Nat 
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imports Inductive Ring_and_Field 
23263  13 
uses 
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"~~/src/Tools/rat.ML" 

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"~~/src/Provers/Arith/cancel_sums.ML" 

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("arith_data.ML") 

24091  17 
"~~/src/Provers/Arith/fast_lin_arith.ML" 
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("Tools/lin_arith.ML") 

15131  19 
begin 
13449  20 

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subsection {* Type @{text ind} *} 

22 

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typedecl ind 

24 

19573  25 
axiomatization 
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Zero_Rep :: ind and 

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Suc_Rep :: "ind => ind" 

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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  32 

13449  33 

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subsection {* Type nat *} 

35 

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text {* Type definition *} 

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inductive Nat :: "ind \<Rightarrow> bool" 
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where 
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Zero_RepI: "Nat Zero_Rep" 
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 Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)" 
13449  42 

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global 

44 

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typedef (open Nat) 

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nat = "Collect Nat" 
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by (rule exI, rule CollectI, rule Nat.Zero_RepI) 
13449  48 

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constdefs 
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Suc :: "nat => nat" 
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Suc_def: "Suc == (%n. Abs_Nat (Suc_Rep (Rep_Nat n)))" 
13449  52 

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local 

54 

25510  55 
instantiation nat :: zero 
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begin 

57 

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definition Zero_nat_def [code func del]: 

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"0 = Abs_Nat Zero_Rep" 

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instance .. 

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end 

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lemma nat_induct: "P 0 ==> (!!n. P n ==> P (Suc n)) ==> P n" 
13449  66 
apply (unfold Zero_nat_def Suc_def) 
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apply (rule Rep_Nat_inverse [THEN subst])  {* types force good instantiation *} 

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apply (erule Rep_Nat [THEN CollectD, THEN Nat.induct]) 
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apply (iprover elim: Abs_Nat_inverse [OF CollectI, THEN subst]) 
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done 
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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 
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Abs_Nat_inject Rep_Nat [THEN CollectD] Suc_RepI Zero_RepI 
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Suc_Rep_not_Zero_Rep) 
13449  76 

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lemma Zero_not_Suc [iff]: "0 \<noteq> Suc m" 
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by (rule not_sym, rule Suc_not_Zero not_sym) 
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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 [THEN CollectD] Suc_RepI 
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inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 
13449  83 

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lemma Suc_Suc_eq [iff]: "Suc m = Suc n \<longleftrightarrow> m = n" 
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by (rule inj_Suc [THEN inj_eq]) 
13449  86 

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rep_datatype nat 
13449  88 
distinct Suc_not_Zero Zero_not_Suc 
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inject Suc_Suc_eq 

21411  90 
induction nat_induct 
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declare nat.induct [case_names 0 Suc, induct type: nat] 

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

13449  94 

21672  95 
lemmas nat_rec_0 = nat.recs(1) 
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and nat_rec_Suc = nat.recs(2) 

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lemmas nat_case_0 = nat.cases(1) 

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and nat_case_Suc = nat.cases(2) 

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24995  101 

102 
text {* Injectiveness and distinctness lemmas *} 

103 

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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R" 
25162  105 
by (rule notE, rule Suc_not_Zero) 
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R" 
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by (rule Suc_neq_Zero, erule sym) 
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y" 
25162  111 
by (rule inj_Suc [THEN injD]) 
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lemma n_not_Suc_n: "n \<noteq> Suc n" 
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by (induct n) simp_all 
13449  115 

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lemma Suc_n_not_n: "Suc n \<noteq> n" 
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by (rule not_sym, rule n_not_Suc_n) 
13449  118 

119 
text {* A special form of induction for reasoning 

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

121 

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lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==> 
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(!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n" 
14208  124 
apply (rule_tac x = m in spec) 
15251  125 
apply (induct n) 
13449  126 
prefer 2 
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apply (rule allI) 

17589  128 
apply (induct_tac x, iprover+) 
13449  129 
done 
130 

24995  131 

132 
subsection {* Arithmetic operators *} 

133 

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instantiation nat :: "{minus, comm_monoid_add}" 
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begin 
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primrec plus_nat 
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where 
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add_0: "0 + n = (n\<Colon>nat)" 
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 add_Suc: "Suc m + n = Suc (m + n)" 
24995  141 

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lemma add_0_right [simp]: "m + 0 = (m::nat)" 
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by (induct m) simp_all 
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)" 
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by (induct m) simp_all 
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n" 
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by simp 
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primrec minus_nat 
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where 
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diff_0: "m  0 = (m\<Colon>nat)" 
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 diff_Suc: "m  Suc n = (case m  n of 0 => 0  Suc k => k)" 
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declare diff_Suc [simp del, code del] 
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lemma diff_0_eq_0 [simp, code]: "0  n = (0::nat)" 
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by (induct n) (simp_all add: diff_Suc) 
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lemma diff_Suc_Suc [simp, code]: "Suc m  Suc n = m  n" 
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by (induct n) (simp_all add: diff_Suc) 
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instance proof 
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fix n m q :: nat 
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show "(n + m) + q = n + (m + q)" by (induct n) simp_all 
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show "n + m = m + n" by (induct n) simp_all 
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show "0 + n = n" by simp 
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qed 
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end 
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instantiation nat :: comm_semiring_1_cancel 
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begin 
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definition 
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One_nat_def [simp]: "1 = Suc 0" 
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primrec times_nat 
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where 
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mult_0: "0 * n = (0\<Colon>nat)" 
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 mult_Suc: "Suc m * n = n + (m * n)" 
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lemma mult_0_right [simp]: "(m::nat) * 0 = 0" 
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by (induct m) simp_all 
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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)" 
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by (induct m) (simp_all add: add_left_commute) 
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lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)" 
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by (induct m) (simp_all add: add_assoc) 
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instance proof 
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fix n m q :: nat 
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show "0 \<noteq> (1::nat)" by simp 
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show "1 * n = n" by simp 
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show "n * m = m * n" by (induct n) simp_all 
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show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib) 
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show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib) 
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assume "n + m = n + q" thus "m = q" by (induct n) simp_all 
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qed 
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end 
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subsubsection {* Addition *} 
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lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
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by (rule add_assoc) 
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lemma nat_add_commute: "m + n = n + (m::nat)" 
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by (rule add_commute) 
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lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
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by (rule add_left_commute) 
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lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
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217 
by (rule add_left_cancel) 
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218 

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219 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
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220 
by (rule add_right_cancel) 
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221 

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222 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 
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223 

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224 
lemma add_is_0 [iff]: 
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225 
fixes m n :: nat 
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226 
shows "(m + n = 0) = (m = 0 & n = 0)" 
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227 
by (cases m) simp_all 
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228 

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229 
lemma add_is_1: 
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230 
"(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 
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231 
by (cases m) simp_all 
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232 

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233 
lemma one_is_add: 
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234 
"(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 
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235 
by (rule trans, rule eq_commute, rule add_is_1) 
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236 

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237 
lemma add_eq_self_zero: 
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238 
fixes m n :: nat 
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239 
shows "m + n = m \<Longrightarrow> n = 0" 
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240 
by (induct m) simp_all 
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241 

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242 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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243 
apply (induct k) 
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244 
apply simp 
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245 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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246 
apply (simp add:o_def) 
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247 
done 
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248 

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249 

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250 
subsubsection {* Difference *} 
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251 

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252 
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat)  m = 0" 
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253 
by (induct m) simp_all 
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254 

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255 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 
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256 
by (induct i j rule: diff_induct) simp_all 
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257 

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258 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 
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259 
by (simp add: diff_diff_left) 
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260 

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261 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 
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262 
by (simp add: diff_diff_left add_commute) 
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263 

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264 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 
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265 
by (induct n) simp_all 
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266 

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267 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 
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268 
by (simp add: diff_add_inverse add_commute [of m n]) 
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269 

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270 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 
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271 
by (induct k) simp_all 
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272 

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273 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 
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274 
by (simp add: diff_cancel add_commute) 
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275 

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276 
lemma diff_add_0: "n  (n + m) = (0::nat)" 
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277 
by (induct n) simp_all 
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278 

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279 
text {* Difference distributes over multiplication *} 
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280 

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281 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 
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282 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 
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283 

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284 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 
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285 
by (simp add: diff_mult_distrib mult_commute [of k]) 
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286 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 
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287 

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288 

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289 
subsubsection {* Multiplication *} 
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290 

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291 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
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292 
by (rule mult_assoc) 
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293 

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294 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
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295 
by (rule mult_commute) 
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296 

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297 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
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298 
by (rule right_distrib) 
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299 

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300 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
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301 
by (induct m) auto 
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302 

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303 
lemmas nat_distrib = 
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304 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 
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305 

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306 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = 1 & n = 1)" 
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307 
apply (induct m) 
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308 
apply simp 
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309 
apply (induct n) 
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310 
apply auto 
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311 
done 
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312 

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313 
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = 1 & n = 1)" 
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314 
apply (rule trans) 
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315 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
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316 
done 
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317 

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318 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 
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319 
proof  
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320 
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n" 
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321 
proof (induct n arbitrary: m) 
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322 
case 0 then show "m = 0" by simp 
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323 
next 
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324 
case (Suc n) then show "m = Suc n" 
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325 
by (cases m) (simp_all add: eq_commute [of "0"]) 
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326 
qed 
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327 
then show ?thesis by auto 
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328 
qed 
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329 

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330 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
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331 
by (simp add: mult_commute) 
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changeset

332 

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parents:
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333 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 
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334 
by (subst mult_cancel1) simp 
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335 

24995  336 

337 
subsection {* Orders on @{typ nat} *} 

338 

26072
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339 
subsubsection {* Operation definition *} 
24995  340 

26072
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341 
instantiation nat :: linorder 
25510  342 
begin 
343 

26072
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344 
primrec less_eq_nat where 
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345 
"(0\<Colon>nat) \<le> n \<longleftrightarrow> True" 
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346 
 "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False  Suc n \<Rightarrow> m \<le> n)" 
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347 

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348 
declare less_eq_nat.simps [simp del, code del] 
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349 
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps) 
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350 
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps) 
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351 

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352 
definition less_nat where 
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353 
less_eq_Suc_le [code func del]: "n < m \<longleftrightarrow> Suc n \<le> m" 
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354 

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changeset

355 
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

356 
by (simp add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

357 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

358 
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

359 
unfolding less_eq_Suc_le .. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

360 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

361 
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

362 
by (induct n) (simp_all add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

363 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

364 
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

365 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

366 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

367 
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

368 
by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

369 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

370 
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

371 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

372 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

373 
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

374 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

375 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

376 
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

377 
by (induct m arbitrary: n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

378 
(simp_all add: less_eq_nat.simps(2) split: nat.splits) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

379 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

380 
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

381 
by (cases n) (auto intro: le_SucI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

382 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

383 
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

384 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
24995  385 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

386 
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

387 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
25510  388 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

389 
instance proof 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

390 
fix n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

391 
have less_imp_le: "n < m \<Longrightarrow> n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

392 
unfolding less_eq_Suc_le by (erule Suc_leD) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

393 
have irrefl: "\<not> m < m" by (induct m) auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

394 
have strict: "n \<le> m \<Longrightarrow> n \<noteq> m \<Longrightarrow> n < m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

395 
proof (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

396 
case 0 then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

397 
by (cases m) (simp_all add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

398 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

399 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

400 
by (cases m) (simp_all add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

401 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

402 
show "n < m \<longleftrightarrow> n \<le> m \<and> n \<noteq> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

403 
by (auto simp add: irrefl intro: less_imp_le strict) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

404 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

405 
fix n :: nat show "n \<le> n" by (induct n) simp_all 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

406 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

407 
fix n m :: nat assume "n \<le> m" and "m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

408 
then show "n = m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

409 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

410 
(simp_all add: less_eq_nat.simps(2) split: nat.splits) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

411 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

412 
fix n m q :: nat assume "n \<le> m" and "m \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

413 
then show "n \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

414 
proof (induct n arbitrary: m q) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

415 
case 0 show ?case by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

416 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

417 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

418 
by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

419 
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify, 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

420 
simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

421 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

422 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

423 
fix n m :: nat show "n \<le> m \<or> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

424 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

425 
(simp_all add: less_eq_nat.simps(2) split: nat.splits) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

426 
qed 
25510  427 

428 
end 

13449  429 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

430 
subsubsection {* Introduction properties *} 
13449  431 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

432 
lemma lessI [iff]: "n < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

433 
by (simp add: less_Suc_eq_le) 
13449  434 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

435 
lemma zero_less_Suc [iff]: "0 < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

436 
by (simp add: less_Suc_eq_le) 
13449  437 

438 
lemma less_trans: "i < j ==> j < k ==> i < (k::nat)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

439 
by (rule order_less_trans) 
13449  440 

441 
subsubsection {* Elimination properties *} 

442 

443 
lemma less_not_sym: "n < m ==> ~ m < (n::nat)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

444 
by (rule order_less_not_sym) 
13449  445 

446 
lemma less_asym: 

447 
assumes h1: "(n::nat) < m" and h2: "~ P ==> m < n" shows P 

448 
apply (rule contrapos_np) 

449 
apply (rule less_not_sym) 

450 
apply (rule h1) 

451 
apply (erule h2) 

452 
done 

453 

454 
lemma less_not_refl: "~ n < (n::nat)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

455 
by (rule order_less_irrefl) 
13449  456 

457 
lemma less_irrefl [elim!]: "(n::nat) < n ==> R" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

458 
by (rule notE, rule less_not_refl) 
13449  459 

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

460 
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

461 
by (rule less_imp_neq) 
13449  462 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

463 
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

464 
by (rule not_sym) (rule less_imp_neq) 
13449  465 

466 
lemma less_zeroE: "(n::nat) < 0 ==> R" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

467 
by (rule notE) (rule not_less0) 
13449  468 

469 
lemma less_Suc_eq: "(m < Suc n) = (m < n  m = n)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

470 
unfolding less_Suc_eq_le le_less .. 
13449  471 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

472 
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

473 
by (simp add: less_Suc_eq) 
13449  474 

475 
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

476 
by (simp add: less_Suc_eq) 
13449  477 

478 
lemma Suc_mono: "m < n ==> Suc m < Suc n" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

479 
by simp 
13449  480 

481 
lemma less_linear: "m < n  m = n  n < (m::nat)" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

482 
by (rule less_linear) 
13449  483 

14302  484 
text {* "Less than" is antisymmetric, sort of *} 
485 
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

486 
unfolding not_less less_Suc_eq_le by (rule antisym) 
14302  487 

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

488 
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n  n < m)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

489 
by (rule linorder_neq_iff) 
13449  490 

491 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

492 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

493 
shows "P n m" 

494 
apply (rule less_linear [THEN disjE]) 

495 
apply (erule_tac [2] disjE) 

496 
apply (erule lessCase) 

497 
apply (erule sym [THEN eqCase]) 

498 
apply (erule major) 

499 
done 

500 

501 

502 
subsubsection {* Inductive (?) properties *} 

503 

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

504 
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

505 
unfolding less_eq_Suc_le [of m] le_less by simp 
13449  506 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

507 
lemma lessE: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

508 
assumes major: "i < k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

509 
and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

510 
shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

511 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

512 
from major have "\<exists>j. i \<le> j \<and> k = Suc j" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

513 
unfolding less_eq_Suc_le by (induct k) simp_all 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

514 
then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

515 
by (clarsimp simp add: less_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

516 
with p1 p2 show P by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

517 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

518 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

519 
lemma less_SucE: assumes major: "m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

520 
and less: "m < n ==> P" and eq: "m = n ==> P" shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

521 
apply (rule major [THEN lessE]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

522 
apply (rule eq, blast) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

523 
apply (rule less, blast) 
13449  524 
done 
525 

526 
lemma Suc_lessE: assumes major: "Suc i < k" 

527 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

528 
apply (rule major [THEN lessE]) 

529 
apply (erule lessI [THEN minor]) 

14208  530 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  531 
done 
532 

533 
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n" 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

534 
by simp 
13449  535 

536 
lemma less_trans_Suc: 

537 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  538 
apply (induct k, simp_all) 
13449  539 
apply (insert le) 
540 
apply (simp add: less_Suc_eq) 

541 
apply (blast dest: Suc_lessD) 

542 
done 

543 

544 
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m  n < m"} *} 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

545 
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

546 
unfolding not_less less_Suc_eq_le .. 
13449  547 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

548 
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

549 
unfolding not_le Suc_le_eq .. 
21243  550 

24995  551 
text {* Properties of "less than or equal" *} 
13449  552 

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

553 
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

554 
unfolding less_Suc_eq_le . 
13449  555 

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

556 
lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

557 
unfolding not_le less_Suc_eq_le .. 
13449  558 

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

559 
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n  m = Suc n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

560 
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) 
13449  561 

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

562 
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

563 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  564 

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

565 
lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

566 
unfolding Suc_le_eq . 
13449  567 

568 
text {* Stronger version of @{text Suc_leD} *} 

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

569 
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

570 
unfolding Suc_le_eq . 
13449  571 

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

572 
lemma less_imp_le: "m < n ==> m \<le> (n::nat)" 
26072
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haftmann
parents:
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diff
changeset

573 
unfolding less_eq_Suc_le by (rule Suc_leD) 
13449  574 

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

575 
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *} 
13449  576 
lemmas le_simps = less_imp_le less_Suc_eq_le Suc_le_eq 
577 

578 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

579 
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  580 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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parents:
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changeset

581 
lemma le_imp_less_or_eq: "m \<le> n ==> m < n  m = (n::nat)" 
26072
f65a7fa2da6c
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haftmann
parents:
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diff
changeset

582 
unfolding le_less . 
13449  583 

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

584 
lemma less_or_eq_imp_le: "m < n  m = n ==> m \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

585 
unfolding le_less . 
13449  586 

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

587 
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n  m=n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

588 
by (rule le_less) 
13449  589 

22718  590 
text {* Useful with @{text blast}. *} 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

591 
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

592 
by auto 
13449  593 

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

594 
lemma le_refl: "n \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

595 
by simp 
13449  596 

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

597 
lemma le_less_trans: "[ i \<le> j; j < k ] ==> i < (k::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

598 
by (rule order_le_less_trans) 
13449  599 

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

600 
lemma less_le_trans: "[ i < j; j \<le> k ] ==> i < (k::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

601 
by (rule order_less_le_trans) 
13449  602 

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

603 
lemma le_trans: "[ i \<le> j; j \<le> k ] ==> i \<le> (k::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

604 
by (rule order_trans) 
13449  605 

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

606 
lemma le_anti_sym: "[ m \<le> n; n \<le> m ] ==> m = (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

607 
by (rule antisym) 
13449  608 

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

609 
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

610 
by (rule less_le) 
13449  611 

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

612 
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

613 
unfolding less_le .. 
13449  614 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

615 
lemma nat_le_linear: "(m::nat) \<le> n  n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

616 
by (rule linear) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

617 

22718  618 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  619 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

620 
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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changeset

621 
unfolding less_Suc_eq_le by auto 
13449  622 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

623 
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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changeset

624 
unfolding not_less by (rule le_less_Suc_eq) 
13449  625 

626 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

627 

22718  628 
text {* These two rules ease the use of primitive recursion. 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

629 
NOTE USE OF @{text "=="} *} 
13449  630 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
25162  631 
by simp 
13449  632 

633 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

25162  634 
by simp 
13449  635 

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

636 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
25162  637 
by (cases n) simp_all 
638 

639 
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m" 

640 
by (cases n) simp_all 

13449  641 

22718  642 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
25162  643 
by (cases n) simp_all 
13449  644 

25162  645 
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
646 
by (cases n) simp_all 

25140  647 

13449  648 
text {* This theorem is useful with @{text blast} *} 
649 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

25162  650 
by (rule neq0_conv[THEN iffD1], iprover) 
13449  651 

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

652 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
25162  653 
by (fast intro: not0_implies_Suc) 
13449  654 

24286
7619080e49f0
ATP blacklisting is now in theory data, attribute noatp
paulson
parents:
24196
diff
changeset

655 
lemma not_gr0 [iff,noatp]: "!!n::nat. (~ (0 < n)) = (n = 0)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

656 
using neq0_conv by blast 
13449  657 

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

658 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
25162  659 
by (induct m') simp_all 
13449  660 

661 
text {* Useful in certain inductive arguments *} 

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

662 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
25162  663 
by (cases m) simp_all 
13449  664 

665 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

666 
subsubsection {* @{term min} and @{term max} *} 
13449  667 

25076  668 
lemma mono_Suc: "mono Suc" 
25162  669 
by (rule monoI) simp 
25076  670 

13449  671 
lemma min_0L [simp]: "min 0 n = (0::nat)" 
25162  672 
by (rule min_leastL) simp 
13449  673 

674 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

25162  675 
by (rule min_leastR) simp 
13449  676 

677 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

25162  678 
by (simp add: mono_Suc min_of_mono) 
13449  679 

22191  680 
lemma min_Suc1: 
681 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

25162  682 
by (simp split: nat.split) 
22191  683 

684 
lemma min_Suc2: 

685 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

25162  686 
by (simp split: nat.split) 
22191  687 

13449  688 
lemma max_0L [simp]: "max 0 n = (n::nat)" 
25162  689 
by (rule max_leastL) simp 
13449  690 

691 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

25162  692 
by (rule max_leastR) simp 
13449  693 

694 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

25162  695 
by (simp add: mono_Suc max_of_mono) 
13449  696 

22191  697 
lemma max_Suc1: 
698 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

25162  699 
by (simp split: nat.split) 
22191  700 

701 
lemma max_Suc2: 

702 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

25162  703 
by (simp split: nat.split) 
22191  704 

13449  705 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

706 
subsubsection {* Monotonicity of Addition *} 
13449  707 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

708 
lemma Suc_pred [simp]: "n>0 ==> Suc (n  Suc 0) = n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

709 
by (simp add: diff_Suc split: nat.split) 
13449  710 

14331  711 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
25162  712 
by (induct k) simp_all 
13449  713 

14331  714 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
25162  715 
by (induct k) simp_all 
13449  716 

25162  717 
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0  n>0)" 
718 
by(auto dest:gr0_implies_Suc) 

13449  719 

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

720 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

721 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
25162  722 
by (induct k) simp_all 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

723 

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

724 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

725 
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

726 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  727 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

728 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

729 

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

730 
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

731 
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

732 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

733 
apply (simp_all add: order_le_less) 
22718  734 
apply (blast elim!: less_SucE 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

735 
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

736 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

737 

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

738 
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *} 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

739 
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j" 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

740 
apply(auto simp: gr0_conv_Suc) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

741 
apply (induct_tac m) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

742 
apply (simp_all add: add_less_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

743 
done 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

744 

14740  745 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
14738  746 
instance nat :: ordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

747 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

748 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

749 
show "0 < (1::nat)" by simp 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

750 
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

751 
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

752 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

753 

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

754 
lemma nat_mult_1: "(1::nat) * n = n" 
25162  755 
by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

756 

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

757 
lemma nat_mult_1_right: "n * (1::nat) = n" 
25162  758 
by simp 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

759 

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

760 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

761 
subsubsection {* Additional theorems about "less than" *} 
14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

762 

19870  763 
text{*An induction rule for estabilishing binary relations*} 
22718  764 
lemma less_Suc_induct: 
19870  765 
assumes less: "i < j" 
766 
and step: "!!i. P i (Suc i)" 

767 
and trans: "!!i j k. P i j ==> P j k ==> P i k" 

768 
shows "P i j" 

769 
proof  

22718  770 
from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 
771 
have "P i (Suc (i + k))" 

19870  772 
proof (induct k) 
22718  773 
case 0 
774 
show ?case by (simp add: step) 

19870  775 
next 
776 
case (Suc k) 

22718  777 
thus ?case by (auto intro: assms) 
19870  778 
qed 
22718  779 
thus "P i j" by (simp add: j) 
19870  780 
qed 
781 

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

782 
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

783 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

784 
lemma less_mono_imp_le_mono: 
24438  785 
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)" 
786 
by (simp add: order_le_less) (blast) 

787 

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

788 

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

789 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

790 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
24438  791 
by (rule add_right_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

792 

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

793 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

794 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
24438  795 
by (rule add_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

796 

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

797 
lemma le_add2: "n \<le> ((m + n)::nat)" 
24438  798 
by (insert add_right_mono [of 0 m n], simp) 
13449  799 

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

800 
lemma le_add1: "n \<le> ((n + m)::nat)" 
24438  801 
by (simp add: add_commute, rule le_add2) 
13449  802 

803 
lemma less_add_Suc1: "i < Suc (i + m)" 

24438  804 
by (rule le_less_trans, rule le_add1, rule lessI) 
13449  805 

806 
lemma less_add_Suc2: "i < Suc (m + i)" 

24438  807 
by (rule le_less_trans, rule le_add2, rule lessI) 
13449  808 

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

809 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
24438  810 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  811 

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

812 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
24438  813 
by (rule le_trans, assumption, rule le_add1) 
13449  814 

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

815 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
24438  816 
by (rule le_trans, assumption, rule le_add2) 
13449  817 

818 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

24438  819 
by (rule less_le_trans, assumption, rule le_add1) 
13449  820 

821 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

24438  822 
by (rule less_le_trans, assumption, rule le_add2) 
13449  823 

824 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

24438  825 
apply (rule le_less_trans [of _ "i+j"]) 
826 
apply (simp_all add: le_add1) 

827 
done 

13449  828 

829 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

24438  830 
apply (rule notI) 
831 
apply (erule add_lessD1 [THEN less_irrefl]) 

832 
done 

13449  833 

834 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

24438  835 
by (simp add: add_commute not_add_less1) 
13449  836 

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

837 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
24438  838 
apply (rule order_trans [of _ "m+k"]) 
839 
apply (simp_all add: le_add1) 

840 
done 

13449  841 

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

842 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
24438  843 
apply (simp add: add_commute) 
844 
apply (erule add_leD1) 

845 
done 

13449  846 

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

847 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
24438  848 
by (blast dest: add_leD1 add_leD2) 
13449  849 

850 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

851 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

24438  852 
by (force simp del: add_Suc_right 
13449  853 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 
854 

855 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

856 
subsubsection {* More results about difference *} 
13449  857 

858 
lemma diff_self_eq_0 [simp]: "(m::nat)  m = 0" 

24438  859 
by (induct m) simp_all 
13449  860 

861 
text {* Addition is the inverse of subtraction: 

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

862 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  863 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
24438  864 
by (induct m n rule: diff_induct) simp_all 
13449  865 

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

866 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
24438  867 
by (simp add: add_diff_inverse linorder_not_less) 
13449  868 

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

869 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
24438  870 
by (simp add: le_add_diff_inverse add_commute) 
13449  871 

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

872 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
24438  873 
by (induct m n rule: diff_induct) simp_all 
13449  874 

875 
lemma diff_less_Suc: "m  n < Suc m" 

24438  876 
apply (induct m n rule: diff_induct) 
877 
apply (erule_tac [3] less_SucE) 

878 
apply (simp_all add: less_Suc_eq) 

879 
done 

13449  880 

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

881 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
24438  882 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
13449  883 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

884 
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

885 
by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

886 

13449  887 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 
24438  888 
by (rule le_less_trans, rule diff_le_self) 
13449  889 

890 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

24438  891 
by (cases n) (auto simp add: le_simps) 
13449  892 

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

893 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
24438  894 
by (induct j k rule: diff_induct) simp_all 
13449  895 

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

896 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
24438  897 
by (simp add: add_commute diff_add_assoc) 
13449  898 

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

899 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
24438  900 
by (auto simp add: diff_add_inverse2) 
13449  901 

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

902 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
24438  903 
by (induct m n rule: diff_induct) simp_all 
13449  904 

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

905 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
24438  906 
by (rule iffD2, rule diff_is_0_eq) 
13449  907 

908 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

24438  909 
by (induct m n rule: diff_induct) simp_all 
13449  910 

22718  911 
lemma less_imp_add_positive: 
912 
assumes "i < j" 

913 
shows "\<exists>k::nat. 0 < k & i + k = j" 

914 
proof 

915 
from assms show "0 < j  i & i + (j  i) = j" 

23476  916 
by (simp add: order_less_imp_le) 
22718  917 
qed 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

918 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

919 
text {* a nice rewrite for bounded subtraction *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

920 
lemma nat_minus_add_max: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

921 
fixes n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

922 
shows "n  m + m = max n m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

923 
by (simp add: max_def not_le order_less_imp_le) 
13449  924 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

925 
lemma nat_diff_split: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

926 
"P(a  b::nat) = ((a<b > P 0) & (ALL d. a = b + d > P d))" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

927 
 {* elimination of @{text } on @{text nat} *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

928 
by (cases "a < b") 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

929 
(auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

930 
not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero) 
13449  931 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

932 
lemma nat_diff_split_asm: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

933 
"P(a  b::nat) = (~ (a < b & ~ P 0  (EX d. a = b + d & ~ P d)))" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

934 
 {* elimination of @{text } on @{text nat} in assumptions *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

935 
by (auto split: nat_diff_split) 
13449  936 

937 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

938 
subsubsection {* Monotonicity of Multiplication *} 
13449  939 

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

940 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
24438  941 
by (simp add: mult_right_mono) 
13449  942 

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

943 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
24438  944 
by (simp add: mult_left_mono) 
13449  945 

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

946 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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14266
diff
changeset

947 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
24438  948 
by (simp add: mult_mono) 
13449  949 

950 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

24438  951 
by (simp add: mult_strict_right_mono) 
13449  952 

14266  953 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
954 
there are no negative numbers.*} 

955 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  956 
apply (induct m) 
22718  957 
apply simp 
958 
apply (case_tac n) 

959 
apply simp_all 

13449  960 
done 
961 

14267
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More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
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diff
changeset

962 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (1 \<le> m & 1 \<le> n)" 
13449  963 
apply (induct m) 
22718  964 
apply simp 
965 
apply (case_tac n) 

966 
apply simp_all 

13449  967 
done 
968 

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

969 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  970 
apply (safe intro!: mult_less_mono1) 
14208  971 
apply (case_tac k, auto) 
13449  972 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
973 
apply (blast intro: mult_le_mono1) 

974 
done 

975 

976 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

24438  977 
by (simp add: mult_commute [of k]) 
13449  978 

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

979 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
24438  980 
by (simp add: linorder_not_less [symmetric], auto) 
13449  981 

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

982 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
24438  983 
by (simp add: linorder_not_less [symmetric], auto) 
13449  984 

985 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

24438  986 
by (subst mult_less_cancel1) simp 
13449  987 

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

988 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
24438  989 
by (subst mult_le_cancel1) simp 
13449  990 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

991 
lemma le_square: "m \<le> m * (m::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

992 
by (cases m) (auto intro: le_add1) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

993 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

994 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

995 
by (cases m) (auto intro: le_add1) 
13449  996 

997 
text {* Lemma for @{text gcd} *} 

998 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

999 
apply (drule sym) 

1000 
apply (rule disjCI) 

1001 
apply (rule nat_less_cases, erule_tac [2] _) 

25157  1002 
apply (drule_tac [2] mult_less_mono2) 
25162  1003 
apply (auto) 
13449  1004 
done 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1005 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1006 
text {* the lattice order on @{typ nat} *} 
24995  1007 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1008 
instantiation nat :: distrib_lattice 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1009 
begin 
24995  1010 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1011 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1012 
"(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min" 
24995  1013 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1014 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1015 
"(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max" 
24995  1016 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1017 
instance by intro_classes 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1018 
(auto simp add: inf_nat_def sup_nat_def max_def not_le min_def 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1019 
intro: order_less_imp_le antisym elim!: order_trans order_less_trans) 
24995  1020 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1021 
end 
24995  1022 

1023 

25193  1024 
subsection {* Embedding of the Naturals into any 
1025 
@{text semiring_1}: @{term of_nat} *} 

24196  1026 

1027 
context semiring_1 

1028 
begin 

1029 

25559  1030 
primrec 
1031 
of_nat :: "nat \<Rightarrow> 'a" 

1032 
where 

1033 
of_nat_0: "of_nat 0 = 0" 

1034 
 of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m" 

25193  1035 

1036 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 

1037 
by simp 

1038 

1039 
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n" 

1040 
by (induct m) (simp_all add: add_ac) 

1041 

1042 
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n" 

1043 
by (induct m) (simp_all add: add_ac left_distrib) 

1044 

25928  1045 
definition 
1046 
of_nat_aux :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" 

1047 
where 

1048 
[code func del]: "of_nat_aux n i = of_nat n + i" 

1049 

1050 
lemma of_nat_aux_code [code]: 

1051 
"of_nat_aux 0 i = i" 

1052 
"of_nat_aux (Suc n) i = of_nat_aux n (i + 1)"  {* tail recursive *} 

1053 
by (simp_all add: of_nat_aux_def add_ac) 

1054 

1055 
lemma of_nat_code [code]: 

1056 
"of_nat n = of_nat_aux n 0" 

1057 
by (simp add: of_nat_aux_def) 

1058 

24196  1059 
end 
1060 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1061 
text{*Class for unital semirings with characteristic zero. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1062 
Includes nonordered rings like the complex numbers.*} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1063 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1064 
class semiring_char_0 = semiring_1 + 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1065 
assumes of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1066 
begin 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1067 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1068 
text{*Special cases where either operand is zero*} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1069 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1070 
lemma of_nat_0_eq_iff [simp, noatp]: "0 = of_nat n \<longleftrightarrow> 0 = n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1071 
by (rule of_nat_eq_iff [of 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1072 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1073 
lemma of_nat_eq_0_iff [simp, noatp]: "of_nat m = 0 \<longleftrightarrow> m = 0" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1074 
by (rule of_nat_eq_iff [of _ 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1075 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1076 
lemma inj_of_nat: "inj of_nat" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1077 
by (simp add: inj_on_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1078 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1079 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1080 

25193  1081 
context ordered_semidom 
1082 
begin 

1083 

1084 
lemma zero_le_imp_of_nat: "0 \<le> of_nat m" 

1085 
apply (induct m, simp_all) 

1086 
apply (erule order_trans) 

1087 
apply (rule ord_le_eq_trans [OF _ add_commute]) 

1088 
apply (rule less_add_one [THEN less_imp_le]) 

1089 
done 

1090 

1091 
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n" 

1092 
apply (induct m n rule: diff_induct, simp_all) 

1093 
apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force) 

1094 
done 

1095 

1096 
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n" 

1097 
apply (induct m n rule: diff_induct, simp_all) 

1098 
apply (insert zero_le_imp_of_nat) 

1099 
apply (force simp add: not_less [symmetric]) 

1100 
done 

1101 

1102 
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n" 

1103 
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) 

1104 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1105 
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1106 
by (simp add: not_less [symmetric] linorder_not_less [symmetric]) 
25193  1107 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1108 
text{*Every @{text ordered_semidom} has characteristic zero.*} 
25193  1109 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1110 
subclass semiring_char_0 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1111 
by unfold_locales (simp add: eq_iff order_eq_iff) 
25193  1112 

1113 
text{*Special cases where either operand is zero*} 

1114 

1115 
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n" 

1116 
by (rule of_nat_le_iff [of 0, simplified]) 

1117 

1118 
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0" 

1119 
by (rule of_nat_le_iff [of _ 0, simplified]) 

1120 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1121 
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1122 
by (rule of_nat_less_iff [of 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1123 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1124 
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1125 
by (rule of_nat_less_iff [of _ 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1126 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1127 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1128 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1129 
context ring_1 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1130 
begin 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1131 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1132 
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m  n) = of_nat m  of_nat n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1133 
by (simp add: compare_rls of_nat_add [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1134 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1135 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1136 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1137 
context ordered_idom 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1138 
begin 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1139 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1140 
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1141 
unfolding abs_if by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1142 

25193  1143 
end 
1144 

1145 
lemma of_nat_id [simp]: "of_nat n = n" 

1146 
by (induct n) auto 

1147 

1148 
lemma of_nat_eq_id [simp]: "of_nat = id" 

1149 
by (auto simp add: expand_fun_eq) 

1150 

1151 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1152 
subsection {*The Set of Natural Numbers*} 
25193  1153 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1154 
context semiring_1 
25193  1155 
begin 
1156 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1157 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1158 
Nats :: "'a set" where 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1159 
"Nats = range of_nat" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1160 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1161 
notation (xsymbols) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1162 
Nats ("\<nat>") 
25193  1163 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1164 
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1165 
by (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1166 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1167 
lemma Nats_0 [simp]: "0 \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1168 
apply (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1169 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1170 
apply (rule of_nat_0 [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1171 
done 
25193  1172 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1173 
lemma Nats_1 [simp]: "1 \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1174 
apply (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1175 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1176 
apply (rule of_nat_1 [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1177 
done 
25193  1178 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1179 
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1180 
apply (auto simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1181 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1182 
apply (rule of_nat_add [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1183 
done 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1184 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1185 
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1186 
apply (auto simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1187 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1188 
apply (rule of_nat_mult [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1189 
done 
25193  1190 

1191 
end 

1192 

1193 

21243  1194 
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *} 
1195 

22845  1196 
lemma subst_equals: 
1197 
assumes 1: "t = s" and 2: "u = t" 

1198 
shows "u = s" 

1199 
using 2 1 by (rule trans) 

1200 

21243  1201 
use "arith_data.ML" 
26101  1202 
declaration {* K ArithData.setup *} 
24091  1203 

1204 
use "Tools/lin_arith.ML" 

1205 
declaration {* K LinArith.setup *} 

1206 

21243  1207 
lemmas [arith_split] = nat_diff_split split_min split_max 
1208 

1209 
text{*Subtraction laws, mostly by Clemens Ballarin*} 

1210 

1211 
lemma diff_less_mono: "[ a < (b::nat); c \<le> a ] ==> ac < bc" 

24438  1212 
by arith 
21243  1213 

1214 
lemma less_diff_conv: "(i < jk) = (i+k < (j::nat))" 

24438  1215 
by arith 
21243  1216 

1217 
lemma le_diff_conv: "(jk \<le> (i::nat)) = (j \<le> i+k)" 

24438  1218 
by arith 
21243  1219 

1220 
lemma le_diff_conv2: "k \<le> j ==> (i \<le> jk) = (i+k \<le> (j::nat))" 

24438  1221 
by arith 
21243  1222 

1223 
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n  (n  i) = i" 

24438  1224 
by arith 
21243  1225 

1226 
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m  k" 

24438  1227 
by arith 
21243  1228 

1229 
(*Replaces the previous diff_less and le_diff_less, which had the stronger 

1230 
second premise n\<le>m*) 

1231 
lemma diff_less[simp]: "!!m::nat. [ 0<n; 0<m ] ==> m  n < m" 

24438  1232 
by arith 
21243  1233 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1234 
text {* Simplification of relational expressions involving subtraction *} 
21243  1235 

1236 
lemma diff_diff_eq: "[ k \<le> m; k \<le> (n::nat) ] ==> ((mk)  (nk)) = (mn)" 

24438  1237 
by (simp split add: nat_diff_split) 
21243  1238 

1239 
lemma eq_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk = nk) = (m=n)" 

24438  1240 
by (auto split add: nat_diff_split) 
21243  1241 

1242 
lemma less_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk < nk) = (m<n)" 

24438  1243 
by (auto split add: nat_diff_split) 
21243  1244 

1245 
lemma le_diff_iff: "[ k \<le> m; k \<le> (n::nat) ] ==> (mk \<le> nk) = (m\<le>n)" 

24438  1246 
by (auto split add: nat_diff_split) 
21243  1247 

1248 
text{*(Anti)Monotonicity of subtraction  by Stephan Merz*} 

1249 

1250 
(* Monotonicity of subtraction in first argument *) 

1251 
lemma diff_le_mono: "m \<le> (n::nat) ==> (ml) \<le> (nl)" 

24438  1252 
by (simp split add: nat_diff_split) 
21243  1253 

1254 
lemma diff_le_mono2: "m \<le> (n::nat) ==> (ln) \<le> (lm)" 

24438  1255 
by (simp split add: nat_diff_split) 
21243  1256 

1257 
lemma diff_less_mono2: "[ m < (n::nat); m<l ] ==> (ln) < (lm)" 

24438  1258 
by (simp split add: nat_diff_split) 
21243  1259 

1260 
lemma diffs0_imp_equal: "!!m::nat. [ mn = 0; nm = 0 ] ==> m=n" 

24438  1261 
by (simp split add: nat_diff_split) 
21243  1262 

26143
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1263 
lemma min_diff: "min (m  (i::nat)) (n  i) = min m n  i" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1264 
unfolding min_def by auto 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1265 

314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1266 
lemma inj_on_diff_nat: 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1267 
assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1268 
shows "inj_on (\<lambda>n. n  k) N" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1269 
proof (rule inj_onI) 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1270 
fix x y 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1271 
assume a: "x \<in> N" "y \<in> N" "x  k = y  k" 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1272 
with k_le_n have "x  k + k = y  k + k" by auto 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1273 
with a k_le_n show "x = y" by auto 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1274 
qed 
314c0bcb7df7
Added useful general lemmas from the work with the HeapMonad
bulwahn
parents:
26101
diff
changeset

1275 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1276 
text{*Rewriting to pull differences out*} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1277 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1278 
lemma diff_diff_right [simp]: "k\<le>j > i  (j  k) = i + (k::nat)  j" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1279 
by arith 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1280 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1281 
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m  Suc (j  k) = m + k  Suc j" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1282 
by arith 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1283 

f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1284 
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j  k)  m = Suc j  (k + m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1285 
by arith 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1286 

21243  1287 
text{*Lemmas for ex/Factorization*} 
1288 

1289 
lemma one_less_mult: "[ Suc 0 < n; Suc 0 < m ] ==> Suc 0 < m*n" 

24438  1290 
by (cases m) auto 
21243  1291 

1292 
lemma n_less_m_mult_n: "[ Suc 0 < n; Suc 0 < m ] ==> n<m*n" 

24438  1293 
by (cases m) auto 
21243  1294 

1295 
lemma n_less_n_mult_m: "[ Suc 0 < n; Suc 0 < m ] ==> n<n*m" 

24438  1296 
by (cases m) auto 
21243  1297 

23001
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1298 
text {* Specialized induction principles that work "backwards": *} 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1299 

3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1300 
lemma inc_induct[consumes 1, case_names base step]: 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1301 
assumes less: "i <= j" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1302 
assumes base: "P j" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1303 
assumes step: "!!i. [ i < j; P (Suc i) ] ==> P i" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1304 
shows "P i" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1305 
using less 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1306 
proof (induct d=="j  i" arbitrary: i) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1307 
case (0 i) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1308 
hence "i = j" by simp 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1309 
with base show ?case by simp 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1310 
next 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1311 
case (Suc d i) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1312 
hence "i < j" "P (Suc i)" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1313 
by simp_all 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1314 
thus "P i" by (rule step) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1315 
qed 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1316 

3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1317 
lemma strict_inc_induct[consumes 1, case_names base step]: 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1318 
assumes less: "i < j" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1319 
assumes base: "!!i. j = Suc i ==> P i" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1320 
assumes step: "!!i. [ i < j; P (Suc i) ] ==> P i" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1321 
shows "P i" 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1322 
using less 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1323 
proof (induct d=="j  i  1" arbitrary: i) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset

1324 
case (0 i) 
3608f0362a91
added induction principles for induction "backwards": P (Suc n) ==> P n
krauss
parents:
22920
diff
changeset
