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permissions  rwrr 
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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 and mod, 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 Typedef Fun Fields 
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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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("Tools/nat_arith.ML") 
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"~~/src/Provers/Arith/fast_lin_arith.ML" 
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("Tools/lin_arith.ML") 

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

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typedecl ind 

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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where 
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 {* the axiom of infinity in 2 parts *} 
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Suc_Rep_inject: "Suc_Rep x = Suc_Rep y ==> x = y" and 
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Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep" 
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subsection {* Type nat *} 
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text {* Type definition *} 

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inductive Nat :: "ind \<Rightarrow> bool" 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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typedef (open) nat = "{n. Nat n}" 
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morphisms Rep_Nat Abs_Nat 

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using Nat.Zero_RepI by auto 
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lemma Nat_Rep_Nat: 
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"Nat (Rep_Nat n)" 
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using Rep_Nat by simp 
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lemma Nat_Abs_Nat_inverse: 
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"Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n" 
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using Abs_Nat_inverse by simp 
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lemma Nat_Abs_Nat_inject: 
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"Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m" 
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using Abs_Nat_inject by simp 
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instantiation nat :: zero 
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begin 

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definition Zero_nat_def: 
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"0 = Abs_Nat Zero_Rep" 
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instance .. 

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end 

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definition Suc :: "nat \<Rightarrow> nat" where 
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"Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))" 
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lemma Suc_not_Zero: "Suc m \<noteq> 0" 
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by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat) 
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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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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y" 
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by (rule iffI, rule Suc_Rep_inject) simp_all 
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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 Nat_Rep_Nat [THEN Nat.induct]) 
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apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst]) 
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apply (simp_all add: Nat_Abs_Nat_inject Nat_Rep_Nat 
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Suc_RepI Zero_RepI Suc_Rep_not_Zero_Rep 
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Suc_Rep_not_Zero_Rep [symmetric] 
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Suc_Rep_inject' 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  names of variables differ *} 
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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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137 
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 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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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 where 
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diff_0 [code]: "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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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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hide_fact (open) add_0 add_0_right diff_0 
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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, code_post]: "1 = Suc 0" 
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primrec times_nat 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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206 
assume "n + m = n + q" thus "m = q" by (induct n) simp_all 
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207 
qed 
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208 

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209 
end 
24995  210 

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211 
subsubsection {* Addition *} 
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212 

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213 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
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214 
by (rule add_assoc) 
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215 

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216 
lemma nat_add_commute: "m + n = n + (m::nat)" 
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217 
by (rule add_commute) 
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218 

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219 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
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220 
by (rule add_left_commute) 
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221 

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222 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
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223 
by (rule add_left_cancel) 
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224 

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225 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
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226 
by (rule add_right_cancel) 
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227 

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228 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 
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229 

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230 
lemma add_is_0 [iff]: 
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231 
fixes m n :: nat 
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232 
shows "(m + n = 0) = (m = 0 & n = 0)" 
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233 
by (cases m) simp_all 
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234 

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235 
lemma add_is_1: 
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"(m+n= Suc 0) = (m= Suc 0 & n=0  m=0 & n= Suc 0)" 
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237 
by (cases m) simp_all 
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238 

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239 
lemma one_is_add: 
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240 
"(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 
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241 
by (rule trans, rule eq_commute, rule add_is_1) 
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242 

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243 
lemma add_eq_self_zero: 
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244 
fixes m n :: nat 
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245 
shows "m + n = m \<Longrightarrow> n = 0" 
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246 
by (induct m) simp_all 
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247 

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248 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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249 
apply (induct k) 
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250 
apply simp 
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251 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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252 
apply (simp add:o_def) 
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253 
done 
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254 

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255 

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256 
subsubsection {* Difference *} 
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257 

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258 
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat)  m = 0" 
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259 
by (induct m) simp_all 
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260 

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261 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 
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262 
by (induct i j rule: diff_induct) simp_all 
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263 

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264 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 
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265 
by (simp add: diff_diff_left) 
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266 

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267 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 
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268 
by (simp add: diff_diff_left add_commute) 
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269 

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

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273 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 
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274 
by (simp add: diff_add_inverse add_commute [of m n]) 
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275 

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

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279 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 
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280 
by (simp add: diff_cancel add_commute) 
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281 

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282 
lemma diff_add_0: "n  (n + m) = (0::nat)" 
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283 
by (induct n) simp_all 
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284 

30093  285 
lemma diff_Suc_1 [simp]: "Suc n  1 = n" 
286 
unfolding One_nat_def by simp 

287 

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288 
text {* Difference distributes over multiplication *} 
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289 

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290 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 
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291 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 
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292 

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293 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 
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294 
by (simp add: diff_mult_distrib mult_commute [of k]) 
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295 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 
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296 

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297 

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298 
subsubsection {* Multiplication *} 
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299 

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300 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
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301 
by (rule mult_assoc) 
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302 

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303 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
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304 
by (rule mult_commute) 
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305 

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306 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
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307 
by (rule right_distrib) 
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308 

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309 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
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310 
by (induct m) auto 
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311 

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312 
lemmas nat_distrib = 
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313 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 
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314 

30079
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315 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)" 
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316 
apply (induct m) 
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317 
apply simp 
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318 
apply (induct n) 
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319 
apply auto 
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320 
done 
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321 

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322 
lemma one_eq_mult_iff [simp,no_atp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)" 
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323 
apply (rule trans) 
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324 
apply (rule_tac [2] mult_eq_1_iff, fastforce) 
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325 
done 
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326 

30079
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327 
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1" 
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328 
unfolding One_nat_def by (rule mult_eq_1_iff) 
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329 

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330 
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1" 
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331 
unfolding One_nat_def by (rule one_eq_mult_iff) 
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332 

26072
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333 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 
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334 
proof  
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335 
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n" 
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336 
proof (induct n arbitrary: m) 
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337 
case 0 then show "m = 0" by simp 
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338 
next 
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339 
case (Suc n) then show "m = Suc n" 
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parents:
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diff
changeset

340 
by (cases m) (simp_all add: eq_commute [of "0"]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

341 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

342 
then show ?thesis by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

343 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

344 

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

345 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
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parents:
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diff
changeset

346 
by (simp add: mult_commute) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

347 

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

348 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
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parents:
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diff
changeset

349 
by (subst mult_cancel1) simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

350 

24995  351 

352 
subsection {* Orders on @{typ nat} *} 

353 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
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parents:
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354 
subsubsection {* Operation definition *} 
24995  355 

26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
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356 
instantiation nat :: linorder 
25510  357 
begin 
358 

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

359 
primrec less_eq_nat where 
f65a7fa2da6c
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haftmann
parents:
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diff
changeset

360 
"(0\<Colon>nat) \<le> n \<longleftrightarrow> True" 
44325  361 
 "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False  Suc n \<Rightarrow> m \<le> n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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changeset

362 

28514  363 
declare less_eq_nat.simps [simp del] 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

364 
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

365 
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

366 

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

367 
definition less_nat where 
28514  368 
less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

369 

f65a7fa2da6c
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parents:
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diff
changeset

370 
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:
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diff
changeset

371 
by (simp add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

372 

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

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

374 
unfolding less_eq_Suc_le .. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

375 

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

376 
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:
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diff
changeset

377 
by (induct n) (simp_all add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

378 

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

379 
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

380 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

381 

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

382 
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

383 
by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

384 

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

385 
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

386 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

387 

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

388 
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

389 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

390 

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

391 
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

392 
by (induct m arbitrary: n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

393 
(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

394 

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

395 
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

396 
by (cases n) (auto intro: le_SucI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

397 

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

398 
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

399 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
24995  400 

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

401 
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

402 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
25510  403 

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

404 
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

405 
proof 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

406 
fix n m :: nat 
27679  407 
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

408 
proof (induct n arbitrary: m) 
27679  409 
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

410 
next 
27679  411 
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

412 
qed 
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 :: 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

415 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

416 
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

417 
then show "n = m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

418 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

419 
(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

420 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

421 
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

422 
then show "n \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

423 
proof (induct n arbitrary: m q) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

424 
case 0 show ?case by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

425 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

426 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

427 
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

428 
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

429 
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

430 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

431 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

432 
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

433 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

434 
(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

435 
qed 
25510  436 

437 
end 

13449  438 

29652  439 
instantiation nat :: bot 
440 
begin 

441 

442 
definition bot_nat :: nat where 

443 
"bot_nat = 0" 

444 

445 
instance proof 

446 
qed (simp add: bot_nat_def) 

447 

448 
end 

449 

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

450 
subsubsection {* Introduction properties *} 
13449  451 

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

452 
lemma lessI [iff]: "n < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

453 
by (simp add: less_Suc_eq_le) 
13449  454 

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

455 
lemma zero_less_Suc [iff]: "0 < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

456 
by (simp add: less_Suc_eq_le) 
13449  457 

458 

459 
subsubsection {* Elimination properties *} 

460 

461 
lemma less_not_refl: "~ n < (n::nat)" 

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

462 
by (rule order_less_irrefl) 
13449  463 

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

464 
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

465 
by (rule not_sym) (rule less_imp_neq) 
13449  466 

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

467 
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

468 
by (rule less_imp_neq) 
13449  469 

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

470 
lemma less_irrefl_nat: "(n::nat) < n ==> R" 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

471 
by (rule notE, rule less_not_refl) 
13449  472 

473 
lemma less_zeroE: "(n::nat) < 0 ==> R" 

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

474 
by (rule notE) (rule not_less0) 
13449  475 

476 
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

477 
unfolding less_Suc_eq_le le_less .. 
13449  478 

30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset

479 
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

480 
by (simp add: less_Suc_eq) 
13449  481 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35633
diff
changeset

482 
lemma less_one [iff, no_atp]: "(n < (1::nat)) = (n = 0)" 
30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset

483 
unfolding One_nat_def by (rule less_Suc0) 
13449  484 

485 
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

486 
by simp 
13449  487 

14302  488 
text {* "Less than" is antisymmetric, sort of *} 
489 
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

490 
unfolding not_less less_Suc_eq_le by (rule antisym) 
14302  491 

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

492 
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

493 
by (rule linorder_neq_iff) 
13449  494 

495 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

496 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

497 
shows "P n m" 

498 
apply (rule less_linear [THEN disjE]) 

499 
apply (erule_tac [2] disjE) 

500 
apply (erule lessCase) 

501 
apply (erule sym [THEN eqCase]) 

502 
apply (erule major) 

503 
done 

504 

505 

506 
subsubsection {* Inductive (?) properties *} 

507 

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

508 
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

509 
unfolding less_eq_Suc_le [of m] le_less by simp 
13449  510 

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

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

512 
assumes major: "i < k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

513 
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

514 
shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

515 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

516 
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

517 
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

518 
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

519 
by (clarsimp simp add: less_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

520 
with p1 p2 show P by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

521 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

522 

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

523 
lemma less_SucE: assumes major: "m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

524 
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

525 
apply (rule major [THEN lessE]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

526 
apply (rule eq, blast) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

527 
apply (rule less, blast) 
13449  528 
done 
529 

530 
lemma Suc_lessE: assumes major: "Suc i < k" 

531 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

532 
apply (rule major [THEN lessE]) 

533 
apply (erule lessI [THEN minor]) 

14208  534 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  535 
done 
536 

537 
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

538 
by simp 
13449  539 

540 
lemma less_trans_Suc: 

541 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  542 
apply (induct k, simp_all) 
13449  543 
apply (insert le) 
544 
apply (simp add: less_Suc_eq) 

545 
apply (blast dest: Suc_lessD) 

546 
done 

547 

548 
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

549 
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

550 
unfolding not_less less_Suc_eq_le .. 
13449  551 

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

552 
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

553 
unfolding not_le Suc_le_eq .. 
21243  554 

24995  555 
text {* Properties of "less than or equal" *} 
13449  556 

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

557 
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

558 
unfolding less_Suc_eq_le . 
13449  559 

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

560 
lemma Suc_n_not_le_n: "~ Suc n \<le> n" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

561 
unfolding not_le less_Suc_eq_le .. 
13449  562 

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

563 
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n  m = Suc n)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
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diff
changeset

564 
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) 
13449  565 

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

566 
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

567 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  568 

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

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

570 
unfolding Suc_le_eq . 
13449  571 

572 
text {* Stronger version of @{text Suc_leD} *} 

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

573 
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

574 
unfolding Suc_le_eq . 
13449  575 

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

576 
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

577 
unfolding less_eq_Suc_le by (rule Suc_leD) 
13449  578 

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

579 
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

580 
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq 
13449  581 

582 

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

583 
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  584 

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

585 
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

586 
unfolding le_less . 
13449  587 

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

588 
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

589 
by (rule le_less) 
13449  590 

22718  591 
text {* Useful with @{text blast}. *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

592 
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

593 
by auto 
13449  594 

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

595 
lemma le_refl: "n \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

596 
by simp 
13449  597 

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

598 
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

599 
by (rule order_trans) 
13449  600 

33657  601 
lemma le_antisym: "[ 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

602 
by (rule antisym) 
13449  603 

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

604 
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

605 
by (rule less_le) 
13449  606 

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

607 
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

608 
unfolding less_le .. 
13449  609 

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

610 
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

611 
by (rule linear) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

612 

22718  613 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  614 

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

615 
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

616 
unfolding less_Suc_eq_le by auto 
13449  617 

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

618 
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

619 
unfolding not_less by (rule le_less_Suc_eq) 
13449  620 

621 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

622 

22718  623 
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

624 
NOTE USE OF @{text "=="} *} 
13449  625 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
25162  626 
by simp 
13449  627 

628 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

25162  629 
by simp 
13449  630 

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

631 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
25162  632 
by (cases n) simp_all 
633 

634 
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m" 

635 
by (cases n) simp_all 

13449  636 

22718  637 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
25162  638 
by (cases n) simp_all 
13449  639 

25162  640 
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
641 
by (cases n) simp_all 

25140  642 

13449  643 
text {* This theorem is useful with @{text blast} *} 
644 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

25162  645 
by (rule neq0_conv[THEN iffD1], iprover) 
13449  646 

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

647 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
25162  648 
by (fast intro: not0_implies_Suc) 
13449  649 

35828
46cfc4b8112e
now use "Named_Thms" for "noatp", and renamed "noatp" to "no_atp"
blanchet
parents:
35633
diff
changeset

650 
lemma not_gr0 [iff,no_atp]: "!!n::nat. (~ (0 < n)) = (n = 0)" 
25134
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

651 
using neq0_conv by blast 
13449  652 

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

653 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
25162  654 
by (induct m') simp_all 
13449  655 

656 
text {* Useful in certain inductive arguments *} 

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

657 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
25162  658 
by (cases m) simp_all 
13449  659 

660 

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

661 
subsubsection {* Monotonicity of Addition *} 
13449  662 

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

663 
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

664 
by (simp add: diff_Suc split: nat.split) 
13449  665 

30128
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

666 
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n  1) = n" 
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

667 
unfolding One_nat_def by (rule Suc_pred) 
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

668 

14331  669 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
25162  670 
by (induct k) simp_all 
13449  671 

14331  672 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
25162  673 
by (induct k) simp_all 
13449  674 

25162  675 
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0  n>0)" 
676 
by(auto dest:gr0_implies_Suc) 

13449  677 

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

678 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

679 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
25162  680 
by (induct k) simp_all 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

681 

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

682 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

683 
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

684 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  685 
apply (induct j, simp_all) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

686 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

687 

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

688 
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

689 
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

690 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

691 
apply (simp_all add: order_le_less) 
22718  692 
apply (blast elim!: less_SucE 
35047
1b2bae06c796
hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents:
35028
diff
changeset

693 
intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric]) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

694 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

695 

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

696 
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

697 
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

698 
apply(auto simp: gr0_conv_Suc) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

699 
apply (induct_tac m) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

700 
apply (simp_all add: add_less_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

701 
done 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

702 

14740  703 
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*} 
35028
108662d50512
more consistent naming of type classes involving orderings (and lattices)  c.f. NEWS
haftmann
parents:
34208
diff
changeset

704 
instance nat :: linordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

705 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

706 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

707 
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

708 
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

709 
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

710 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

711 

30056  712 
instance nat :: no_zero_divisors 
713 
proof 

714 
fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto 

715 
qed 

716 

44817  717 

718 
subsubsection {* @{term min} and @{term max} *} 

719 

720 
lemma mono_Suc: "mono Suc" 

721 
by (rule monoI) simp 

722 

723 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

45931  724 
by (rule min_absorb1) simp 
44817  725 

726 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

45931  727 
by (rule min_absorb2) simp 
44817  728 

729 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

730 
by (simp add: mono_Suc min_of_mono) 

731 

732 
lemma min_Suc1: 

733 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

734 
by (simp split: nat.split) 

735 

736 
lemma min_Suc2: 

737 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

738 
by (simp split: nat.split) 

739 

740 
lemma max_0L [simp]: "max 0 n = (n::nat)" 

45931  741 
by (rule max_absorb2) simp 
44817  742 

743 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

45931  744 
by (rule max_absorb1) simp 
44817  745 

746 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

747 
by (simp add: mono_Suc max_of_mono) 

748 

749 
lemma max_Suc1: 

750 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

751 
by (simp split: nat.split) 

752 

753 
lemma max_Suc2: 

754 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

755 
by (simp split: nat.split) 

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

756 

44817  757 
lemma nat_mult_min_left: 
758 
fixes m n q :: nat 

759 
shows "min m n * q = min (m * q) (n * q)" 

760 
by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

761 

762 
lemma nat_mult_min_right: 

763 
fixes m n q :: nat 

764 
shows "m * min n q = min (m * n) (m * q)" 

765 
by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) 

766 

767 
lemma nat_add_max_left: 

768 
fixes m n q :: nat 

769 
shows "max m n + q = max (m + q) (n + q)" 

770 
by (simp add: max_def) 

771 

772 
lemma nat_add_max_right: 

773 
fixes m n q :: nat 

774 
shows "m + max n q = max (m + n) (m + q)" 

775 
by (simp add: max_def) 

776 

777 
lemma nat_mult_max_left: 

778 
fixes m n q :: nat 

779 
shows "max m n * q = max (m * q) (n * q)" 

780 
by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

781 

782 
lemma nat_mult_max_right: 

783 
fixes m n q :: nat 

784 
shows "m * max n q = max (m * n) (m * q)" 

785 
by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) 

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

786 

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

787 

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

788 
subsubsection {* Additional theorems about @{term "op \<le>"} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

789 

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

790 
text {* Complete induction, aka courseofvalues induction *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

791 

27823  792 
instance nat :: wellorder proof 
793 
fix P and n :: nat 

794 
assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" 

795 
have "\<And>q. q \<le> n \<Longrightarrow> P q" 

796 
proof (induct n) 

797 
case (0 n) 

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

798 
have "P 0" by (rule step) auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

799 
thus ?case using 0 by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

800 
next 
27823  801 
case (Suc m n) 
802 
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

803 
thus ?case 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

804 
proof 
27823  805 
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

806 
next 
27823  807 
assume n: "n = Suc m" 
808 
show "P n" 

809 
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

810 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

811 
qed 
27823  812 
then show "P n" by auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

813 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

814 

27823  815 
lemma Least_Suc: 
816 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 

817 
apply (case_tac "n", auto) 

818 
apply (frule LeastI) 

819 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

820 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 

821 
apply (erule_tac [2] Least_le) 

822 
apply (case_tac "LEAST x. P x", auto) 

823 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 

824 
apply (blast intro: order_antisym) 

825 
done 

826 

827 
lemma Least_Suc2: 

828 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 

829 
apply (erule (1) Least_Suc [THEN ssubst]) 

830 
apply simp 

831 
done 

832 

833 
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)" 

834 
apply (cases n) 

835 
apply blast 

836 
apply (rule_tac x="LEAST k. P(k)" in exI) 

837 
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex) 

838 
done 

839 

840 
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)" 

30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset

841 
unfolding One_nat_def 
27823  842 
apply (cases n) 
843 
apply blast 

844 
apply (frule (1) ex_least_nat_le) 

845 
apply (erule exE) 

846 
apply (case_tac k) 

847 
apply simp 

848 
apply (rename_tac k1) 

849 
apply (rule_tac x=k1 in exI) 

850 
apply (auto simp add: less_eq_Suc_le) 

851 
done 

852 

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

853 
lemma nat_less_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

854 
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

855 
using assms less_induct by blast 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

856 

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

857 
lemma measure_induct_rule [case_names less]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

858 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

859 
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

860 
shows "P a" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

861 
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

862 

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

863 
text {* old style induction rules: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

864 
lemma measure_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

865 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

866 
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

867 
by (rule measure_induct_rule [of f P a]) iprover 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

868 

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

869 
lemma full_nat_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

870 
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

871 
shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

872 
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

873 

19870  874 
text{*An induction rule for estabilishing binary relations*} 
22718  875 
lemma less_Suc_induct: 
19870  876 
assumes less: "i < j" 
877 
and step: "!!i. P i (Suc i)" 

31714  878 
and trans: "!!i j k. i < j ==> j < k ==> P i j ==> P j k ==> P i k" 
19870  879 
shows "P i j" 
880 
proof  

31714  881 
from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add) 
22718  882 
have "P i (Suc (i + k))" 
19870  883 
proof (induct k) 
22718  884 
case 0 
885 
show ?case by (simp add: step) 

19870  886 
next 
887 
case (Suc k) 

31714  888 
have "0 + i < Suc k + i" by (rule add_less_mono1) simp 
889 
hence "i < Suc (i + k)" by (simp add: add_commute) 

890 
from trans[OF this lessI Suc step] 

891 
show ?case by simp 

19870  892 
qed 
22718  893 
thus "P i j" by (simp add: j) 
19870  894 
qed 
895 

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

896 
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

897 
Provided by Roelof Oosterhuis. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

898 
$P(n)$ is true for all $n\in\mathbb{N}$ if 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

899 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

900 
\item case ``0'': given $n=0$ prove $P(n)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

901 
\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

902 
a smaller integer $m$ such that $\neg P(m)$. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

903 
\end{itemize} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

904 

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

905 
text{* A compact version without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

906 
lemma infinite_descent: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

907 
"\<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

908 
by (induct n rule: less_induct, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

909 

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

910 
lemma infinite_descent0[case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

911 
"\<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

912 
by (rule infinite_descent) (case_tac "n>0", auto) 
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 {* 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

915 
Infinite descent using a mapping to $\mathbb{N}$: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

916 
$P(x)$ 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

917 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

918 
\item case ``0'': given $V(x)=0$ prove $P(x)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

919 
\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

920 
\end{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

921 
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

922 

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

923 
corollary infinite_descent0_measure [case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

924 
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

925 
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

926 
shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

927 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

928 
obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

929 
moreover have "\<And>x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

930 
proof (induct n rule: infinite_descent0) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

931 
case 0  "i.e. $V(x) = 0$" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

932 
with A0 show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

933 
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

934 
case (smaller n) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

935 
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

936 
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

937 
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

938 
then show ?case by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

939 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

940 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

941 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

942 

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

943 
text{* Again, without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

944 
lemma infinite_descent_measure: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

945 
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

946 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

947 
from assms obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

948 
moreover have "!!x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

949 
proof (induct n rule: infinite_descent, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

950 
fix x assume "\<not> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

951 
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

952 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

953 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

954 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

955 

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

956 
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

957 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

958 
lemma less_mono_imp_le_mono: 
24438  959 
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)" 
960 
by (simp add: order_le_less) (blast) 

961 

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

962 

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

963 
text {* nonstrict, in 1st argument *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

964 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
24438  965 
by (rule add_right_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

966 

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

967 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

968 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
24438  969 
by (rule add_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

970 

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

971 
lemma le_add2: "n \<le> ((m + n)::nat)" 
24438  972 
by (insert add_right_mono [of 0 m n], simp) 
13449  973 

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

974 
lemma le_add1: "n \<le> ((n + m)::nat)" 
24438  975 
by (simp add: add_commute, rule le_add2) 
13449  976 

977 
lemma less_add_Suc1: "i < Suc (i + m)" 

24438  978 
by (rule le_less_trans, rule le_add1, rule lessI) 
13449  979 

980 
lemma less_add_Suc2: "i < Suc (m + i)" 

24438  981 
by (rule le_less_trans, rule le_add2, rule lessI) 
13449  982 

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

983 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
24438  984 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  985 

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

986 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
24438  987 
by (rule le_trans, assumption, rule le_add1) 
13449  988 

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

989 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
24438  990 
by (rule le_trans, assumption, rule le_add2) 
13449  991 

992 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

24438  993 
by (rule less_le_trans, assumption, rule le_add1) 
13449  994 

995 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

24438  996 
by (rule less_le_trans, assumption, rule le_add2) 
13449  997 

998 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

24438  999 
apply (rule le_less_trans [of _ "i+j"]) 
1000 
apply (simp_all add: le_add1) 

1001 
done 

13449  1002 

1003 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

24438  1004 
apply (rule notI) 
26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1005 
apply (drule add_lessD1) 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1006 
apply (erule less_irrefl [THEN notE]) 
24438  1007 
done 
13449  1008 

1009 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

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

1010 
by (simp add: add_commute) 
13449  1011 

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

1012 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
24438  1013 
apply (rule order_trans [of _ "m+k"]) 
1014 
apply (simp_all add: le_add1) 

1015 
done 

13449  1016 

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

1017 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
24438  1018 
apply (simp add: add_commute) 
1019 
apply (erule add_leD1) 

1020 
done 

13449  1021 

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

1022 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
24438  1023 
by (blast dest: add_leD1 add_leD2) 
13449  1024 

1025 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

1026 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

24438  1027 
by (force simp del: add_Suc_right 
13449  1028 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 
1029 

1030 

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

1031 
subsubsection {* More results about difference *} 
13449  1032 

1033 
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

1034 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  1035 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
24438  1036 
by (induct m n rule: diff_induct) simp_all 
13449  1037 

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

1038 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
24438  1039 
by (simp add: add_diff_inverse linorder_not_less) 
13449  1040 

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

1041 
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

1042 
by (simp add: add_commute) 
13449  1043 

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

1044 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
24438  1045 
by (induct m n rule: diff_induct) simp_all 
13449  1046 

1047 
lemma diff_less_Suc: "m  n < Suc m" 

24438  1048 
apply (induct m n rule: diff_induct) 
1049 
apply (erule_tac [3] less_SucE) 

1050 
apply (simp_all add: less_Suc_eq) 

1051 
done 

13449  1052 

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

1053 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
24438  1054 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
13449  1055 

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

1056 
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

1057 
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

1058 

13449  1059 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 
24438  1060 
by (rule le_less_trans, rule diff_le_self) 
13449  1061 

1062 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

24438  1063 
by (cases n) (auto simp add: le_simps) 
13449  1064 

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

1065 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
24438  1066 
by (induct j k rule: diff_induct) simp_all 
13449  1067 

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

1068 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
24438  1069 
by (simp add: add_commute diff_add_assoc) 
13449  1070 

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

1071 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
24438  1072 
by (auto simp add: diff_add_inverse2) 
13449  1073 

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

1074 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
24438  1075 
by (induct m n rule: diff_induct) simp_all 
13449  1076 

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

1077 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
24438  1078 
by (rule iffD2, rule diff_is_0_eq) 
13449  1079 

1080 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

24438  1081 
by (induct m n rule: diff_induct) simp_all 
13449  1082 

22718  1083 
lemma less_imp_add_positive: 
1084 
assumes "i < j" 

1085 
shows "\<exists>k::nat. 0 < k & i + k = j" 

1086 
proof 

1087 
from assms show "0 < j  i & i + (j  i) = j" 

23476  1088 
by (simp add: order_less_imp_le) 
22718  1089 
qed 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1090 

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

1091 
text {* a nice rewrite for bounded subtraction *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1092 
lemma nat_minus_add_max: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1093 
fixes n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1094 
shows "n  m + m = max n m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1095 
by (simp add: max_def not_le order_less_imp_le) 
13449  1096 

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

1097 
lemma nat_diff_split: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1098 
"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

1099 
 {* elimination of @{text } on @{text nat} *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1100 
by (cases "a < b") 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1101 
(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

1102 
not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero) 
13449  1103 

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

1104 
lemma nat_diff_split_asm: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1105 
"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

1106 
 {* elimination of @{text } on @{text nat} in assumptions *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1107 
by (auto split: nat_diff_split) 
13449  1108 

1109 

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

1110 
subsubsection {* Monotonicity of Multiplication *} 
13449  1111 

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

1112 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
24438  1113 
by (simp add: mult_right_mono) 
13449  1114 

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

1115 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
24438  1116 
by (simp add: mult_left_mono) 
13449  1117 

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

1118 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1119 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
24438  1120 
by (simp add: mult_mono) 
13449  1121 

1122 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

24438  1123 
by (simp add: mult_strict_right_mono) 
13449  1124 

14266  1125 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1126 
there are no negative numbers.*} 

1127 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1128 
apply (induct m) 
22718  1129 
apply simp 
1130 
apply (case_tac n) 

1131 
apply simp_all 

13449  1132 
done 
1133 

30079
293b896b9c25
make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents:
30056
diff
changeset

1134 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)" 
13449  1135 
apply (induct m) 
22718  1136 
apply simp 
1137 
apply (case_tac n) 

1138 
apply simp_all 

13449  1139 
done 
1140 

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

1141 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1142 
apply (safe intro!: mult_less_mono1) 
14208  1143 
apply (case_tac k, auto) 
13449  1144 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1145 
apply (blast intro: mult_le_mono1) 

1146 
done 

1147 

1148 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

24438  1149 
by (simp add: mult_commute [of k]) 
13449  1150 

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

1151 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
24438  1152 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1153 

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

1154 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
24438  1155 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1156 

1157 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

24438  1158 
by (subst mult_less_cancel1) simp 
13449  1159 

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

1160 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
24438  1161 
by (subst mult_le_cancel1) simp 
13449  1162 

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

1163 
lemma le_square: "m \<le> m * (m::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1164 
by (cases m) (auto intro: le_add1) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1165 

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

1166 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1167 
by (cases m) (auto intro: le_add1) 
13449  1168 

1169 
text {* Lemma for @{text gcd} *} 

30128
365ee7319b86
revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents:
30093
diff
changeset

1170 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 
13449  1171 
apply (drule sym) 
1172 
apply (rule disjCI) 

1173 
apply (rule nat_less_cases, erule_tac [2] _) 

25157  1174 
apply (drule_tac [2] mult_less_mono2) 
25162  1175 
apply (auto) 
13449  1176 
done 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1177 

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

1178 
text {* the lattice order on @{typ nat} *} 
24995  1179 

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

1180 
instantiation nat :: distrib_lattice 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1181 
begin 
24995  1182 

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

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

1184 
"(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min" 
24995  1185 

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

1186 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1187 
"(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max" 
24995  1188 

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

1189 
instance by intro_classes 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1190 
(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

1191 
intro: order_less_imp_le antisym elim!: order_trans order_less_trans) 
24995  1192 

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

1193 
end 
24995  1194 

1195 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1196 
subsection {* Natural operation of natural numbers on functions *} 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1197 

30971  1198 
text {* 
1199 
We use the same logical constant for the power operations on 

1200 
functions and relations, in order to share the same syntax. 

1201 
*} 

1202 

45965
2af982715e5c
generalized type signature to permit overloading on `set`
haftmann
parents:
45933
diff
changeset

1203 
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" 
30971  1204 

45965
2af982715e5c
generalized type signature to permit overloading on `set`
haftmann
parents:
45933
diff
changeset

1205 
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where 
30971  1206 
"f ^^ n \<equiv> compow n f" 
1207 

1208 
notation (latex output) 

1209 
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000) 

1210 

1211 
notation (HTML output) 

1212 
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000) 

1213 

1214 
text {* @{text "f ^^ n = f o ... o f"}, the nfold composition of @{text f} *} 

1215 

1216 
overloading 

1217 
funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" 

1218 
begin 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1219 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1220 
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where 
44325  1221 
"funpow 0 f = id" 
1222 
 "funpow (Suc n) f = f o funpow n f" 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1223 

30971  1224 
end 
1225 

1226 
text {* for code generation *} 

1227 

1228 
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where 

46028
9f113cdf3d66
attribute code_abbrev superseedes code_unfold_post
haftmann
parents:
45965
diff
changeset

1229 
funpow_code_def [code_abbrev]: "funpow = compow" 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1230 

30971  1231 
lemma [code]: 
37430  1232 
"funpow (Suc n) f = f o funpow n f" 
30971  1233 
"funpow 0 f = id" 
37430  1234 
by (simp_all add: funpow_code_def) 
30971  1235 

36176
3fe7e97ccca8
replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact'  frees some popular keywords;
wenzelm
parents:
35828
diff
changeset

1236 
hide_const (open) funpow 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1237 

cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1238 
lemma funpow_add: 
30971  1239 
"f ^^ (m + n) = f ^^ m \<circ> f ^^ n" 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1240 
by (induct m) simp_all 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1241 

37430  1242 
lemma funpow_mult: 
1243 
fixes f :: "'a \<Rightarrow> 'a" 

1244 
shows "(f ^^ m) ^^ n = f ^^ (m * n)" 

1245 
by (induct n) (simp_all add: funpow_add) 

1246 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1247 
lemma funpow_swap1: 
30971  1248 
"f ((f ^^ n) x) = (f ^^ n) (f x)" 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1249 
proof  
30971  1250 
have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp 
1251 
also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add) 

1252 
also have "\<dots> = (f ^^ n) (f x)" by simp 

30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1253 
finally show ?thesis . 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1254 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1255 

38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1256 
lemma comp_funpow: 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1257 
fixes f :: "'a \<Rightarrow> 'a" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1258 
shows "comp f ^^ n = comp (f ^^ n)" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1259 
by (induct n) simp_all 
30954
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1260 

38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1261 

45833  1262 
subsection {* Kleene iteration *} 
1263 

1264 
lemma Kleene_iter_lpfp: assumes "mono f" and "f p \<le> p" shows "(f^^k) bot \<le> p" 

1265 
proof(induction k) 

1266 
case 0 show ?case by simp 

1267 
next 

1268 
case Suc 

1269 
from monoD[OF assms(1) Suc] assms(2) 

1270 
show ?case by simp 

1271 
qed 

1272 

1273 
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot" 

1274 
shows "lfp f = (f^^k) bot" 

1275 
proof(rule antisym) 

1276 
show "lfp f \<le> (f^^k) bot" 

1277 
proof(rule lfp_lowerbound) 

1278 
show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp 

1279 
qed 

1280 
next 

1281 
show "(f^^k) bot \<le> lfp f" 

1282 
using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp 

1283 
qed 

1284 

1285 

38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1286 
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *} 
24196  1287 

1288 
context semiring_1 

1289 
begin 

1290 

38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1291 
definition of_nat :: "nat \<Rightarrow> 'a" where 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1292 
"of_nat n = (plus 1 ^^ n) 0" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1293 

d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1294 
lemma of_nat_simps [simp]: 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1295 
shows of_nat_0: "of_nat 0 = 0" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1296 
and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1297 
by (simp_all add: of_nat_def) 
25193  1298 

1299 
lemma of_nat_1 [simp]: "of_nat 1 = 1" 

38621
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1300 
by (simp add: of_nat_def) 
25193  1301 

1302 
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n" 

1303 
by (induct m) (simp_all add: add_ac) 

1304 

1305 
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n" 

1306 
by (induct m) (simp_all add: add_ac left_distrib) 

1307 

28514  1308 
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where 
1309 
"of_nat_aux inc 0 i = i" 

44325  1310 
 "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)"  {* tail recursive *} 
25928  1311 

30966  1312 
lemma of_nat_code: 
28514  1313 
"of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0" 
1314 
proof (induct n) 

1315 
case 0 then show ?case by simp 

1316 
next 

1317 
case (Suc n) 

1318 
have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1" 

da83a614c454
tuned of_nat code generation
ha 