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(* Title: HOL/Nat.thy 
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Author: Tobias Nipkow and Lawrence C Paulson and Markus Wenzel 
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Type "nat" is a linear order, and a datatype; arithmetic operators +  
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and * (for div, mod and dvd, see theory Divides). 
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*) 
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header {* Natural numbers *} 
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theory Nat 
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imports Inductive Ring_and_Field 
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uses 
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"~~/src/Tools/rat.ML" 

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

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

15131  18 
begin 
13449  19 

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subsection {* Type @{text ind} *} 

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typedecl ind 

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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" 
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13449  32 

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subsection {* Type nat *} 

34 

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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)" 
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global 

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typedef (open Nat) 

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nat = Nat 
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by (rule exI, unfold mem_def, rule Nat.Zero_RepI) 
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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)))" 
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local 

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instantiation nat :: zero 
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begin 

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definition Zero_nat_def [code 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 Suc_not_Zero: "Suc m \<noteq> 0" 
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apply (simp add: Zero_nat_def Suc_def Abs_Nat_inject [unfolded mem_def] 
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Rep_Nat [unfolded mem_def] Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def]) 

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done 

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lemma Zero_not_Suc: "0 \<noteq> Suc m" 
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by (rule not_sym, rule Suc_not_Zero not_sym) 
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rep_datatype "0 \<Colon> nat" Suc 
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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 [unfolded mem_def, THEN Nat.induct]) 

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apply (iprover elim: Abs_Nat_inverse [unfolded mem_def, THEN subst]) 

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apply (simp_all add: Abs_Nat_inject [unfolded mem_def] Rep_Nat [unfolded mem_def] 

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Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep [unfolded mem_def] 

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Suc_Rep_not_Zero_Rep [unfolded mem_def, symmetric] 

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inj_Suc_Rep [THEN inj_eq] Rep_Nat_inject) 

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done 

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lemma nat_induct [case_names 0 Suc, induct type: nat]: 
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 {* for backward compatibility  naming of variables differs *} 
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fixes n 
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assumes "P 0" 
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and "\<And>n. P n \<Longrightarrow> P (Suc n)" 
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shows "P n" 
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using assms by (rule nat.induct) 
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declare nat.exhaust [case_names 0 Suc, cases type: nat] 

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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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text {* Injectiveness and distinctness lemmas *} 

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lemma inj_Suc[simp]: "inj_on Suc N" 
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by (simp add: inj_on_def) 
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R" 
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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" 
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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 
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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) 
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text {* A special form of induction for reasoning 

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

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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" 
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apply (rule_tac x = m in spec) 
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apply (induct n) 
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prefer 2 
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apply (rule allI) 

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apply (induct_tac x, iprover+) 
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done 
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24995  132 

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subsection {* Arithmetic operators *} 

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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)" 
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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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28514  149 
declare add_0 [code] 
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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] 
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declare diff_0 [code] 

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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)" unfolding One_nat_def by simp 
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show "1 * n = n" unfolding One_nat_def 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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219 

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220 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
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221 
by (rule add_left_cancel) 
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222 

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223 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
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224 
by (rule add_right_cancel) 
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225 

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226 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 
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227 

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228 
lemma add_is_0 [iff]: 
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229 
fixes m n :: nat 
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230 
shows "(m + n = 0) = (m = 0 & n = 0)" 
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231 
by (cases m) simp_all 
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232 

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233 
lemma add_is_1: 
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234 
"(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 
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235 
by (cases m) simp_all 
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236 

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237 
lemma one_is_add: 
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238 
"(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 
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239 
by (rule trans, rule eq_commute, rule add_is_1) 
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240 

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241 
lemma add_eq_self_zero: 
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242 
fixes m n :: nat 
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243 
shows "m + n = m \<Longrightarrow> n = 0" 
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244 
by (induct m) simp_all 
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245 

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246 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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247 
apply (induct k) 
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248 
apply simp 
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249 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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250 
apply (simp add:o_def) 
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251 
done 
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252 

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253 

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254 
subsubsection {* Difference *} 
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255 

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256 
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat)  m = 0" 
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257 
by (induct m) simp_all 
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258 

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259 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 
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260 
by (induct i j rule: diff_induct) simp_all 
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261 

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262 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 
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263 
by (simp add: diff_diff_left) 
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264 

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265 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 
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266 
by (simp add: diff_diff_left add_commute) 
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267 

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268 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 
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269 
by (induct n) simp_all 
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270 

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271 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 
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272 
by (simp add: diff_add_inverse add_commute [of m n]) 
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273 

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274 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 
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275 
by (induct k) simp_all 
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276 

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277 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 
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278 
by (simp add: diff_cancel add_commute) 
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279 

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280 
lemma diff_add_0: "n  (n + m) = (0::nat)" 
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281 
by (induct n) simp_all 
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282 

30240  283 
lemma diff_Suc_1 [simp]: "Suc n  1 = n" 
284 
unfolding One_nat_def by simp 

285 

26072
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286 
text {* Difference distributes over multiplication *} 
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287 

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288 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 
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289 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 
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290 

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291 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 
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292 
by (simp add: diff_mult_distrib mult_commute [of k]) 
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293 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 
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294 

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295 

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296 
subsubsection {* Multiplication *} 
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297 

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298 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
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299 
by (rule mult_assoc) 
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300 

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301 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
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302 
by (rule mult_commute) 
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303 

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304 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
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305 
by (rule right_distrib) 
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306 

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307 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
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308 
by (induct m) auto 
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309 

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310 
lemmas nat_distrib = 
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311 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 
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312 

30240  313 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)" 
26072
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314 
apply (induct m) 
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315 
apply simp 
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316 
apply (induct n) 
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317 
apply auto 
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318 
done 
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319 

30240  320 
lemma one_eq_mult_iff [simp,noatp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)" 
26072
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321 
apply (rule trans) 
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322 
apply (rule_tac [2] mult_eq_1_iff, fastsimp) 
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323 
done 
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324 

30240  325 
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1" 
326 
unfolding One_nat_def by (rule mult_eq_1_iff) 

327 

328 
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1" 

329 
unfolding One_nat_def by (rule one_eq_mult_iff) 

330 

26072
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331 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 
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332 
proof  
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333 
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n" 
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334 
proof (induct n arbitrary: m) 
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335 
case 0 then show "m = 0" by simp 
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336 
next 
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337 
case (Suc n) then show "m = Suc n" 
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338 
by (cases m) (simp_all add: eq_commute [of "0"]) 
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339 
qed 
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340 
then show ?thesis by auto 
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341 
qed 
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342 

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343 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
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344 
by (simp add: mult_commute) 
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345 

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346 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 
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347 
by (subst mult_cancel1) simp 
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348 

24995  349 

350 
subsection {* Orders on @{typ nat} *} 

351 

26072
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352 
subsubsection {* Operation definition *} 
24995  353 

26072
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354 
instantiation nat :: linorder 
25510  355 
begin 
356 

26072
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357 
primrec less_eq_nat where 
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358 
"(0\<Colon>nat) \<le> n \<longleftrightarrow> True" 
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359 
 "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False  Suc n \<Rightarrow> m \<le> n)" 
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360 

28514  361 
declare less_eq_nat.simps [simp del] 
26072
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362 
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps) 
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363 
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps) 
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364 

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365 
definition less_nat where 
28514  366 
less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

367 

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

368 
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

369 
by (simp add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

370 

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

371 
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

372 
unfolding less_eq_Suc_le .. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

373 

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

374 
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

375 
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

376 

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

377 
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

378 
by (simp add: less_eq_Suc_le) 
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 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

381 
by simp 
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 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

384 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

385 

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

386 
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

387 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

388 

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

389 
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

390 
by (induct m arbitrary: n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

391 
(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

392 

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

393 
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

394 
by (cases n) (auto intro: le_SucI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

395 

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

396 
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

397 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
24995  398 

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

399 
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

400 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
25510  401 

26315
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset

402 
instance 
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset

403 
proof 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

404 
fix n m :: nat 
27679  405 
show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

406 
proof (induct n arbitrary: m) 
27679  407 
case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

408 
next 
27679  409 
case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le) 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

410 
qed 
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 :: 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

413 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

414 
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

415 
then show "n = m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

416 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

417 
(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

418 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

419 
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

420 
then show "n \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

421 
proof (induct n arbitrary: m q) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

422 
case 0 show ?case by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

423 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

424 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

425 
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

426 
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

427 
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

428 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

429 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

430 
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

431 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

432 
(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

433 
qed 
25510  434 

435 
end 

13449  436 

29652  437 
instantiation nat :: bot 
438 
begin 

439 

440 
definition bot_nat :: nat where 

441 
"bot_nat = 0" 

442 

443 
instance proof 

444 
qed (simp add: bot_nat_def) 

445 

446 
end 

447 

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

448 
subsubsection {* Introduction properties *} 
13449  449 

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

450 
lemma lessI [iff]: "n < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

451 
by (simp add: less_Suc_eq_le) 
13449  452 

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

453 
lemma zero_less_Suc [iff]: "0 < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

454 
by (simp add: less_Suc_eq_le) 
13449  455 

456 

457 
subsubsection {* Elimination properties *} 

458 

459 
lemma less_not_refl: "~ n < (n::nat)" 

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

460 
by (rule order_less_irrefl) 
13449  461 

26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

462 
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)" 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

463 
by (rule not_sym) (rule less_imp_neq) 
13449  464 

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

465 
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

466 
by (rule less_imp_neq) 
13449  467 

26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

468 
lemma less_irrefl_nat: "(n::nat) < n ==> R" 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

469 
by (rule notE, rule less_not_refl) 
13449  470 

471 
lemma less_zeroE: "(n::nat) < 0 ==> R" 

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

472 
by (rule notE) (rule not_less0) 
13449  473 

474 
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

475 
unfolding less_Suc_eq_le le_less .. 
13449  476 

30240  477 
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

478 
by (simp add: less_Suc_eq) 
13449  479 

30240  480 
lemma less_one [iff, noatp]: "(n < (1::nat)) = (n = 0)" 
481 
unfolding One_nat_def by (rule less_Suc0) 

13449  482 

483 
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

484 
by simp 
13449  485 

14302  486 
text {* "Less than" is antisymmetric, sort of *} 
487 
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

488 
unfolding not_less less_Suc_eq_le by (rule antisym) 
14302  489 

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

490 
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

491 
by (rule linorder_neq_iff) 
13449  492 

493 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

494 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

495 
shows "P n m" 

496 
apply (rule less_linear [THEN disjE]) 

497 
apply (erule_tac [2] disjE) 

498 
apply (erule lessCase) 

499 
apply (erule sym [THEN eqCase]) 

500 
apply (erule major) 

501 
done 

502 

503 

504 
subsubsection {* Inductive (?) properties *} 

505 

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

506 
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

507 
unfolding less_eq_Suc_le [of m] le_less by simp 
13449  508 

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

509 
lemma lessE: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

510 
assumes major: "i < k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

511 
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

512 
shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

513 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

514 
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

515 
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

516 
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

517 
by (clarsimp simp add: less_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

518 
with p1 p2 show P by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

519 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

520 

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

521 
lemma less_SucE: assumes major: "m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

522 
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

523 
apply (rule major [THEN lessE]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

524 
apply (rule eq, blast) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

525 
apply (rule less, blast) 
13449  526 
done 
527 

528 
lemma Suc_lessE: assumes major: "Suc i < k" 

529 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

530 
apply (rule major [THEN lessE]) 

531 
apply (erule lessI [THEN minor]) 

14208  532 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  533 
done 
534 

535 
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

536 
by simp 
13449  537 

538 
lemma less_trans_Suc: 

539 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  540 
apply (induct k, simp_all) 
13449  541 
apply (insert le) 
542 
apply (simp add: less_Suc_eq) 

543 
apply (blast dest: Suc_lessD) 

544 
done 

545 

546 
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:
25928
diff
changeset

547 
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

548 
unfolding not_less less_Suc_eq_le .. 
13449  549 

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

550 
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:
25928
diff
changeset

551 
unfolding not_le Suc_le_eq .. 
21243  552 

24995  553 
text {* Properties of "less than or equal" *} 
13449  554 

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

555 
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

556 
unfolding less_Suc_eq_le . 
13449  557 

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

558 
lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

559 
unfolding not_le less_Suc_eq_le .. 
13449  560 

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

561 
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:
25928
diff
changeset

562 
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) 
13449  563 

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

564 
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:
25928
diff
changeset

565 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  566 

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

567 
lemma Suc_leI: "m < n ==> Suc(m) \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

568 
unfolding Suc_le_eq . 
13449  569 

570 
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:
14266
diff
changeset

571 
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

572 
unfolding Suc_le_eq . 
13449  573 

26315
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset

574 
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

575 
unfolding less_eq_Suc_le by (rule Suc_leD) 
13449  576 

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

577 
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *} 
26315
cb3badaa192e
removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents:
26300
diff
changeset

578 
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq 
13449  579 

580 

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

581 
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  582 

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

583 
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:
25928
diff
changeset

584 
unfolding le_less . 
13449  585 

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

586 
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

587 
by (rule le_less) 
13449  588 

22718  589 
text {* Useful with @{text blast}. *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

590 
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

591 
by auto 
13449  592 

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

593 
lemma le_refl: "n \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

594 
by simp 
13449  595 

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

596 
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

597 
by (rule order_trans) 
13449  598 

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

599 
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

600 
by (rule antisym) 
13449  601 

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

602 
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:
25928
diff
changeset

603 
by (rule less_le) 
13449  604 

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

605 
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:
25928
diff
changeset

606 
unfolding less_le .. 
13449  607 

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

608 
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

609 
by (rule linear) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

610 

22718  611 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  612 

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

613 
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:
25928
diff
changeset

614 
unfolding less_Suc_eq_le by auto 
13449  615 

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

616 
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:
25928
diff
changeset

617 
unfolding not_less by (rule le_less_Suc_eq) 
13449  618 

619 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

620 

22718  621 
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

622 
NOTE USE OF @{text "=="} *} 
13449  623 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
25162  624 
by simp 
13449  625 

626 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

25162  627 
by simp 
13449  628 

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

629 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
25162  630 
by (cases n) simp_all 
631 

632 
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m" 

633 
by (cases n) simp_all 

13449  634 

22718  635 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
25162  636 
by (cases n) simp_all 
13449  637 

25162  638 
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
639 
by (cases n) simp_all 

25140  640 

13449  641 
text {* This theorem is useful with @{text blast} *} 
642 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

25162  643 
by (rule neq0_conv[THEN iffD1], iprover) 
13449  644 

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

645 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
25162  646 
by (fast intro: not0_implies_Suc) 
13449  647 

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

648 
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

649 
using neq0_conv by blast 
13449  650 

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

651 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
25162  652 
by (induct m') simp_all 
13449  653 

654 
text {* Useful in certain inductive arguments *} 

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

655 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
25162  656 
by (cases m) simp_all 
13449  657 

658 

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

659 
subsubsection {* @{term min} and @{term max} *} 
13449  660 

25076  661 
lemma mono_Suc: "mono Suc" 
25162  662 
by (rule monoI) simp 
25076  663 

13449  664 
lemma min_0L [simp]: "min 0 n = (0::nat)" 
25162  665 
by (rule min_leastL) simp 
13449  666 

667 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

25162  668 
by (rule min_leastR) simp 
13449  669 

670 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

25162  671 
by (simp add: mono_Suc min_of_mono) 
13449  672 

22191  673 
lemma min_Suc1: 
674 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

25162  675 
by (simp split: nat.split) 
22191  676 

677 
lemma min_Suc2: 

678 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

25162  679 
by (simp split: nat.split) 
22191  680 

13449  681 
lemma max_0L [simp]: "max 0 n = (n::nat)" 
25162  682 
by (rule max_leastL) simp 
13449  683 

684 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

25162  685 
by (rule max_leastR) simp 
13449  686 

687 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

25162  688 
by (simp add: mono_Suc max_of_mono) 
13449  689 

22191  690 
lemma max_Suc1: 
691 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

25162  692 
by (simp split: nat.split) 
22191  693 

694 
lemma max_Suc2: 

695 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

25162  696 
by (simp split: nat.split) 
22191  697 

13449  698 

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

699 
subsubsection {* Monotonicity of Addition *} 
13449  700 

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

701 
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

702 
by (simp add: diff_Suc split: nat.split) 
13449  703 

30240  704 
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n  1) = n" 
705 
unfolding One_nat_def by (rule Suc_pred) 

706 

14331  707 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
25162  708 
by (induct k) simp_all 
13449  709 

14331  710 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
25162  711 
by (induct k) simp_all 
13449  712 

25162  713 
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0  n>0)" 
714 
by(auto dest:gr0_implies_Suc) 

13449  715 

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

716 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

717 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
25162  718 
by (induct k) simp_all 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

719 

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

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

721 
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

722 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  723 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

724 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

725 

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

726 
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

727 
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

728 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

729 
apply (simp_all add: order_le_less) 
22718  730 
apply (blast elim!: less_SucE 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

731 
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

732 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

733 

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

734 
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

735 
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

736 
apply(auto simp: gr0_conv_Suc) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

737 
apply (induct_tac m) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

738 
apply (simp_all add: add_less_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

739 
done 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

740 

14740  741 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
14738  742 
instance nat :: ordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

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

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

745 
show "0 < (1::nat)" by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

746 
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

747 
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

748 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

749 

30240  750 
instance nat :: no_zero_divisors 
751 
proof 

752 
fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto 

753 
qed 

754 

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

755 
lemma nat_mult_1: "(1::nat) * n = n" 
25162  756 
by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

757 

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

758 
lemma nat_mult_1_right: "n * (1::nat) = n" 
25162  759 
by simp 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

760 

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

761 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

762 
subsubsection {* Additional theorems about @{term "op \<le>"} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

763 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

764 
text {* Complete induction, aka courseofvalues induction *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

765 

27823  766 
instance nat :: wellorder proof 
767 
fix P and n :: nat 

768 
assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" 

769 
have "\<And>q. q \<le> n \<Longrightarrow> P q" 

770 
proof (induct n) 

771 
case (0 n) 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

772 
have "P 0" by (rule step) auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

773 
thus ?case using 0 by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

774 
next 
27823  775 
case (Suc m n) 
776 
then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq) 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

777 
thus ?case 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

778 
proof 
27823  779 
assume "n \<le> m" thus "P n" by (rule Suc(1)) 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

780 
next 
27823  781 
assume n: "n = Suc m" 
782 
show "P n" 

783 
by (rule step) (rule Suc(1), simp add: n le_simps) 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

784 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

785 
qed 
27823  786 
then show "P n" by auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

787 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

788 

27823  789 
lemma Least_Suc: 
790 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 

791 
apply (case_tac "n", auto) 

792 
apply (frule LeastI) 

793 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

794 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 

795 
apply (erule_tac [2] Least_le) 

796 
apply (case_tac "LEAST x. P x", auto) 

797 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 

798 
apply (blast intro: order_antisym) 

799 
done 

800 

801 
lemma Least_Suc2: 

802 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 

803 
apply (erule (1) Least_Suc [THEN ssubst]) 

804 
apply simp 

805 
done 

806 

807 
lemma ex_least_nat_le: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not>P i) & P(k)" 

808 
apply (cases n) 

809 
apply blast 

810 
apply (rule_tac x="LEAST k. P(k)" in exI) 

811 
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex) 

812 
done 

813 

814 
lemma ex_least_nat_less: "\<not>P(0) \<Longrightarrow> P(n::nat) \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not>P i) & P(k+1)" 

30240  815 
unfolding One_nat_def 
27823  816 
apply (cases n) 
817 
apply blast 

818 
apply (frule (1) ex_least_nat_le) 

819 
apply (erule exE) 

820 
apply (case_tac k) 

821 
apply simp 

822 
apply (rename_tac k1) 

823 
apply (rule_tac x=k1 in exI) 

824 
apply (auto simp add: less_eq_Suc_le) 

825 
done 

826 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

827 
lemma nat_less_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

828 
assumes "!!n. \<forall>m::nat. m < n > P m ==> P n" shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

829 
using assms less_induct by blast 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

830 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

831 
lemma measure_induct_rule [case_names less]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

832 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

833 
assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

834 
shows "P a" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

835 
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

836 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

837 
text {* old style induction rules: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

838 
lemma measure_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

839 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

840 
shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

841 
by (rule measure_induct_rule [of f P a]) iprover 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

842 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

843 
lemma full_nat_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

844 
assumes step: "(!!n. (ALL m. Suc m <= n > P m) ==> P n)" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

845 
shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

846 
by (rule less_induct) (auto intro: step simp:le_simps) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

847 

19870  848 
text{*An induction rule for estabilishing binary relations*} 
22718  849 
lemma less_Suc_induct: 
19870  850 
assumes less: "i < j" 
851 
and step: "!!i. P i (Suc i)" 

852 
and trans: "!!i j k. P i j ==> P j k ==> P i k" 

853 
shows "P i j" 

854 
proof  

22718  855 
from less obtain k where j: "j = Suc(i+k)" by (auto dest: less_imp_Suc_add) 
856 
have "P i (Suc (i + k))" 

19870  857 
proof (induct k) 
22718  858 
case 0 
859 
show ?case by (simp add: step) 

19870  860 
next 
861 
case (Suc k) 

22718  862 
thus ?case by (auto intro: assms) 
19870  863 
qed 
22718  864 
thus "P i j" by (simp add: j) 
19870  865 
qed 
866 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

867 
text {* The method of infinite descent, frequently used in number theory. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

868 
Provided by Roelof Oosterhuis. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

869 
$P(n)$ is true for all $n\in\mathbb{N}$ if 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

870 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

871 
\item case ``0'': given $n=0$ prove $P(n)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

872 
\item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

873 
a smaller integer $m$ such that $\neg P(m)$. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

874 
\end{itemize} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

875 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

876 
text{* A compact version without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

877 
lemma infinite_descent: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

878 
"\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

879 
by (induct n rule: less_induct, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

880 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

881 
lemma infinite_descent0[case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

882 
"\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

883 
by (rule infinite_descent) (case_tac "n>0", auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

884 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

885 
text {* 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

886 
Infinite descent using a mapping to $\mathbb{N}$: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

887 
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

888 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

889 
\item case ``0'': given $V(x)=0$ prove $P(x)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

890 
\item case ``smaller'': given $V(x)>0$ and $\neg P(x)$ prove there exists a $y \in D$ such that $V(y)<V(x)$ and $~\neg P(y)$. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

891 
\end{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

892 
NB: the proof also shows how to use the previous lemma. *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

893 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

894 
corollary infinite_descent0_measure [case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

895 
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

896 
and A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

897 
shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

898 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

899 
obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

900 
moreover have "\<And>x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

901 
proof (induct n rule: infinite_descent0) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

902 
case 0  "i.e. $V(x) = 0$" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

903 
with A0 show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

904 
next  "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

905 
case (smaller n) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

906 
then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

907 
with A1 obtain y where "V y < V x \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

908 
with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

909 
then show ?case by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

910 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

911 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

912 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

913 

4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

914 
text{* Again, without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

915 
lemma infinite_descent_measure: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

916 
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

917 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

918 
from assms obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

919 
moreover have "!!x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

920 
proof (induct n rule: infinite_descent, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

921 
fix x assume "\<not> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

922 
with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

923 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

924 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

925 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

926 

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

927 
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

928 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

929 
lemma less_mono_imp_le_mono: 
24438  930 
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)" 
931 
by (simp add: order_le_less) (blast) 

932 

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

933 

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

934 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

935 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
24438  936 
by (rule add_right_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

937 

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

938 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

939 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
24438  940 
by (rule add_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

941 

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

942 
lemma le_add2: "n \<le> ((m + n)::nat)" 
24438  943 
by (insert add_right_mono [of 0 m n], simp) 
13449  944 

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

945 
lemma le_add1: "n \<le> ((n + m)::nat)" 
24438  946 
by (simp add: add_commute, rule le_add2) 
13449  947 

948 
lemma less_add_Suc1: "i < Suc (i + m)" 

24438  949 
by (rule le_less_trans, rule le_add1, rule lessI) 
13449  950 

951 
lemma less_add_Suc2: "i < Suc (m + i)" 

24438  952 
by (rule le_less_trans, rule le_add2, rule lessI) 
13449  953 

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

954 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
24438  955 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  956 

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

957 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
24438  958 
by (rule le_trans, assumption, rule le_add1) 
13449  959 

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

960 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
24438  961 
by (rule le_trans, assumption, rule le_add2) 
13449  962 

963 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

24438  964 
by (rule less_le_trans, assumption, rule le_add1) 
13449  965 

966 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

24438  967 
by (rule less_le_trans, assumption, rule le_add2) 
13449  968 

969 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

24438  970 
apply (rule le_less_trans [of _ "i+j"]) 
971 
apply (simp_all add: le_add1) 

972 
done 

13449  973 

974 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

24438  975 
apply (rule notI) 
26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

976 
apply (drule add_lessD1) 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

977 
apply (erule less_irrefl [THEN notE]) 
24438  978 
done 
13449  979 

980 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

981 
by (simp add: add_commute) 
13449  982 

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

983 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
24438  984 
apply (rule order_trans [of _ "m+k"]) 
985 
apply (simp_all add: le_add1) 

986 
done 

13449  987 

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

988 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
24438  989 
apply (simp add: add_commute) 
990 
apply (erule add_leD1) 

991 
done 

13449  992 

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

993 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
24438  994 
by (blast dest: add_leD1 add_leD2) 
13449  995 

996 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

997 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

24438  998 
by (force simp del: add_Suc_right 
13449  999 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 
1000 

1001 

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

1002 
subsubsection {* More results about difference *} 
13449  1003 

1004 
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

1005 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  1006 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
24438  1007 
by (induct m n rule: diff_induct) simp_all 
13449  1008 

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

1009 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
24438  1010 
by (simp add: add_diff_inverse linorder_not_less) 
13449  1011 

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

1012 
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m  n) + n = (m::nat)" 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

1013 
by (simp add: add_commute) 
13449  1014 

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

1015 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
24438  1016 
by (induct m n rule: diff_induct) simp_all 
13449  1017 

1018 
lemma diff_less_Suc: "m  n < Suc m" 

24438  1019 
apply (induct m n rule: diff_induct) 
1020 
apply (erule_tac [3] less_SucE) 

1021 
apply (simp_all add: less_Suc_eq) 

1022 
done 

13449  1023 

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

1024 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
24438  1025 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
13449  1026 

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

1027 
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

1028 
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

1029 

13449  1030 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 
24438  1031 
by (rule le_less_trans, rule diff_le_self) 
13449  1032 

1033 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

24438  1034 
by (cases n) (auto simp add: le_simps) 
13449  1035 

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

1036 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
24438  1037 
by (induct j k rule: diff_induct) simp_all 
13449  1038 

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

1039 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
24438  1040 
by (simp add: add_commute diff_add_assoc) 
13449  1041 

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

1042 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
24438  1043 
by (auto simp add: diff_add_inverse2) 
13449  1044 

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

1045 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
24438  1046 
by (induct m n rule: diff_induct) simp_all 
13449  1047 

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

1048 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
24438  1049 
by (rule iffD2, rule diff_is_0_eq) 
13449  1050 

1051 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

24438  1052 
by (induct m n rule: diff_induct) simp_all 
13449  1053 

22718  1054 
lemma less_imp_add_positive: 
1055 
assumes "i < j" 

1056 
shows "\<exists>k::nat. 0 < k & i + k = j" 

1057 
proof 

1058 
from assms show "0 < j  i & i + (j  i) = j" 

23476  1059 
by (simp add: order_less_imp_le) 
22718  1060 
qed 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1061 

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

1062 
text {* a nice rewrite for bounded subtraction *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1063 
lemma nat_minus_add_max: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1064 
fixes n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1065 
shows "n  m + m = max n m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1066 
by (simp add: max_def not_le order_less_imp_le) 
13449  1067 

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

1068 
lemma nat_diff_split: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1069 
"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

1070 
 {* elimination of @{text } on @{text nat} *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1071 
by (cases "a < b") 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1072 
(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

1073 
not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero) 
13449  1074 

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

1075 
lemma nat_diff_split_asm: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1076 
"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

1077 
 {* elimination of @{text } on @{text nat} in assumptions *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1078 
by (auto split: nat_diff_split) 
13449  1079 

1080 

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

1081 
subsubsection {* Monotonicity of Multiplication *} 
13449  1082 

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

1083 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
24438  1084 
by (simp add: mult_right_mono) 
13449  1085 

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

1086 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
24438  1087 
by (simp add: mult_left_mono) 
13449  1088 

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

1089 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1090 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
24438  1091 
by (simp add: mult_mono) 
13449  1092 

1093 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

24438  1094 
by (simp add: mult_strict_right_mono) 
13449  1095 

14266  1096 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1097 
there are no negative numbers.*} 

1098 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1099 
apply (induct m) 
22718  1100 
apply simp 
1101 
apply (case_tac n) 

1102 
apply simp_all 

13449  1103 
done 
1104 

30240  1105 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)" 
13449  1106 
apply (induct m) 
22718  1107 
apply simp 
1108 
apply (case_tac n) 

1109 
apply simp_all 

13449  1110 
done 
1111 

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

1112 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1113 
apply (safe intro!: mult_less_mono1) 
14208  1114 
apply (case_tac k, auto) 
13449  1115 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1116 
apply (blast intro: mult_le_mono1) 

1117 
done 

1118 

1119 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

24438  1120 
by (simp add: mult_commute [of k]) 
13449  1121 

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

1122 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
24438  1123 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1124 

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

1125 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
24438  1126 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1127 

1128 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

24438  1129 
by (subst mult_less_cancel1) simp 
13449  1130 

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

1131 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
24438  1132 
by (subst mult_le_cancel1) simp 
13449  1133 

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

1134 
lemma le_square: "m \<le> m * (m::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1135 
by (cases m) (auto intro: le_add1) 
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 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1138 
by (cases m) (auto intro: le_add1) 
13449  1139 

1140 
text {* Lemma for @{text gcd} *} 

1141 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 

1142 
apply (drule sym) 

1143 
apply (rule disjCI) 

1144 
apply (rule nat_less_cases, erule_tac [2] _) 

25157  1145 
apply (drule_tac [2] mult_less_mono2) 
25162  1146 
apply (auto) 
13449  1147 
done 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1148 

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

1149 
text {* the lattice order on @{typ nat} *} 
24995  1150 

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

1151 
instantiation nat :: distrib_lattice 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1152 
begin 
24995  1153 

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

1154 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1155 
"(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min" 
24995  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 
"(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max" 
24995  1159 

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

1160 
instance by intro_classes 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1161 
(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

1162 
intro: order_less_imp_le antisym elim!: order_trans order_less_trans) 
24995  1163 

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

1164 
end 
24995  1165 

1166 

25193  1167 
subsection {* Embedding of the Naturals into any 
1168 
@{text semiring_1}: @{term of_nat} *} 

24196  1169 

1170 
context semiring_1 

1171 
begin 

1172 

25559  1173 
primrec 
1174 
of_nat :: "nat \<Rightarrow> 'a" 

1175 
where 

1176 
of_nat_0: "of_nat 0 = 0" 

1177 
 of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m" 

25193  1178 

1179 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 

30240  1180 
unfolding One_nat_def by simp 
25193  1181 

1182 
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n" 

1183 
by (induct m) (simp_all add: add_ac) 

1184 

1185 
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n" 

1186 
by (induct m) (simp_all add: add_ac left_distrib) 

1187 

28514  1188 
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where 
1189 
"of_nat_aux inc 0 i = i" 

1190 
 "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)"  {* tail recursive *} 

25928  1191 

28514  1192 
lemma of_nat_code [code, code unfold, code inline del]: 
1193 
"of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0" 

1194 
proof (induct n) 

1195 
case 0 then show ?case by simp 

1196 
next 

1197 
case (Suc n) 

1198 
have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1" 

1199 
by (induct n) simp_all 

1200 
from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1" 

1201 
by simp 

1202 
with Suc show ?case by (simp add: add_commute) 

1203 
qed 

1204 

24196  1205 
end 
1206 

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

1207 
text{*Class for unital semirings with characteristic zero. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1208 
Includes nonordered rings like the complex numbers.*} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1209 

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

1210 
class semiring_char_0 = semiring_1 + 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1211 
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

1212 
begin 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1213 

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

1214 
text{*Special cases where either operand is zero*} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1215 

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

1216 
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

1217 
by (rule of_nat_eq_iff [of 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1218 

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

1219 
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

1220 
by (rule of_nat_eq_iff [of _ 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1221 

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

1222 
lemma inj_of_nat: "inj of_nat" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1223 
by (simp add: inj_on_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1224 

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

1225 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1226 

25193  1227 
context ordered_semidom 
1228 
begin 

1229 

1230 
lemma zero_le_imp_of_nat: "0 \<le> of_nat m" 

1231 
apply (induct m, simp_all) 

1232 
apply (erule order_trans) 

1233 
apply (rule ord_le_eq_trans [OF _ add_commute]) 

1234 
apply (rule less_add_one [THEN less_imp_le]) 

1235 
done 

1236 

1237 
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n" 

1238 
apply (induct m n rule: diff_induct, simp_all) 

1239 
apply (insert add_less_le_mono [OF zero_less_one zero_le_imp_of_nat], force) 

1240 
done 

1241 

1242 
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n" 

1243 
apply (induct m n rule: diff_induct, simp_all) 

1244 
apply (insert zero_le_imp_of_nat) 

1245 
apply (force simp add: not_less [symmetric]) 

1246 
done 

1247 

1248 
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n" 

1249 
by (blast intro: of_nat_less_imp_less less_imp_of_nat_less) 

1250 

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

1251 
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

1252 
by (simp add: not_less [symmetric] linorder_not_less [symmetric]) 
25193  1253 

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

1254 
text{*Every @{text ordered_semidom} has characteristic zero.*} 
25193  1255 

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

1256 
subclass semiring_char_0 
28823  1257 
proof qed (simp add: eq_iff order_eq_iff) 
25193  1258 

1259 
text{*Special cases where either operand is zero*} 

1260 

1261 
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n" 

1262 
by (rule of_nat_le_iff [of 0, simplified]) 

1263 

1264 
lemma of_nat_le_0_iff [simp, noatp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0" 

1265 
by (rule of_nat_le_iff [of _ 0, simplified]) 

1266 

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

1267 
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

1268 
by (rule of_nat_less_iff [of 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1269 

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

1270 
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

1271 
by (rule of_nat_less_iff [of _ 0, simplified]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1272 

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

1273 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1274 

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

1275 
context ring_1 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1276 
begin 
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 of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m  n) = of_nat m  of_nat n" 
29667  1279 
by (simp add: algebra_simps of_nat_add [symmetric]) 
26072
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 
end 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1282 

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

1283 
context ordered_idom 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1284 
begin 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1285 

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

1286 
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

1287 
unfolding abs_if by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1288 

25193  1289 
end 
1290 

1291 
lemma of_nat_id [simp]: "of_nat n = n" 

30240  1292 
by (induct n) (auto simp add: One_nat_def) 
25193  1293 

1294 
lemma of_nat_eq_id [simp]: "of_nat = id" 

1295 
by (auto simp add: expand_fun_eq) 

1296 

1297 

26149  1298 
subsection {* The Set of Natural Numbers *} 
25193  1299 

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

1300 
context semiring_1 
25193  1301 
begin 
1302 

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

1303 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1304 
Nats :: "'a set" where 
28562  1305 
[code del]: "Nats = range of_nat" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1306 

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

1307 
notation (xsymbols) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1308 
Nats ("\<nat>") 
25193  1309 

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

1310 
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

1311 
by (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1312 

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

1313 
lemma Nats_0 [simp]: "0 \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1314 
apply (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1315 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1316 
apply (rule of_nat_0 [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1317 
done 
25193  1318 

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

1319 
lemma Nats_1 [simp]: "1 \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1320 
apply (simp add: Nats_def) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1321 
apply (rule range_eqI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1322 
apply (rule of_nat_1 [symmetric]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1323 
done 
25193  1324 

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

1325 
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>" 
f65a7fa2da6c
<= and < on nat no long 