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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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begin 
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ML_file "~~/src/Tools/rat.ML" 
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ML_file "Tools/arith_data.ML" 

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ML_file "~~/src/Provers/Arith/fast_lin_arith.ML" 

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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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subsection {* Arithmetic operators *} 

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instantiation nat :: comm_monoid_diff 
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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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show "n  0 = n" by simp 
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show "0  n = 0" by simp 

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show "(q + n)  (q + m) = n  m" by (induct q) simp_all 

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show "n  m  q = n  (m + q)" by (induct q) (simp_all add: diff_Suc) 

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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]: "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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assume "n + m = n + q" thus "m = q" by (induct n) simp_all 
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209 
qed 
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210 

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211 
end 
24995  212 

26072
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213 
subsubsection {* Addition *} 
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214 

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215 
lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)" 
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216 
by (rule add_assoc) 
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217 

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218 
lemma nat_add_commute: "m + n = n + (m::nat)" 
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219 
by (rule add_commute) 
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220 

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221 
lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)" 
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222 
by (rule add_left_commute) 
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223 

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224 
lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))" 
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225 
by (rule add_left_cancel) 
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226 

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227 
lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))" 
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228 
by (rule add_right_cancel) 
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229 

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230 
text {* Reasoning about @{text "m + 0 = 0"}, etc. *} 
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231 

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232 
lemma add_is_0 [iff]: 
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233 
fixes m n :: nat 
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234 
shows "(m + n = 0) = (m = 0 & n = 0)" 
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235 
by (cases m) simp_all 
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236 

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237 
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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239 
by (cases m) simp_all 
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240 

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241 
lemma one_is_add: 
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242 
"(Suc 0 = m + n) = (m = Suc 0 & n = 0  m = 0 & n = Suc 0)" 
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243 
by (rule trans, rule eq_commute, rule add_is_1) 
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244 

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245 
lemma add_eq_self_zero: 
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246 
fixes m n :: nat 
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247 
shows "m + n = m \<Longrightarrow> n = 0" 
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248 
by (induct m) simp_all 
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249 

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250 
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N" 
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251 
apply (induct k) 
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252 
apply simp 
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253 
apply(drule comp_inj_on[OF _ inj_Suc]) 
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254 
apply (simp add:o_def) 
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255 
done 
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256 

47208  257 
lemma Suc_eq_plus1: "Suc n = n + 1" 
258 
unfolding One_nat_def by simp 

259 

260 
lemma Suc_eq_plus1_left: "Suc n = 1 + n" 

261 
unfolding One_nat_def by simp 

262 

26072
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263 

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264 
subsubsection {* Difference *} 
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265 

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266 
lemma diff_self_eq_0 [simp]: "(m\<Colon>nat)  m = 0" 
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267 
by (induct m) simp_all 
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268 

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269 
lemma diff_diff_left: "(i::nat)  j  k = i  (j + k)" 
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270 
by (induct i j rule: diff_induct) simp_all 
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271 

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272 
lemma Suc_diff_diff [simp]: "(Suc m  n)  Suc k = m  n  k" 
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273 
by (simp add: diff_diff_left) 
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274 

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275 
lemma diff_commute: "(i::nat)  j  k = i  k  j" 
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276 
by (simp add: diff_diff_left add_commute) 
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277 

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278 
lemma diff_add_inverse: "(n + m)  n = (m::nat)" 
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279 
by (induct n) simp_all 
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280 

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281 
lemma diff_add_inverse2: "(m + n)  n = (m::nat)" 
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282 
by (simp add: diff_add_inverse add_commute [of m n]) 
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283 

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284 
lemma diff_cancel: "(k + m)  (k + n) = m  (n::nat)" 
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285 
by (induct k) simp_all 
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286 

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287 
lemma diff_cancel2: "(m + k)  (n + k) = m  (n::nat)" 
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288 
by (simp add: diff_cancel add_commute) 
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289 

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290 
lemma diff_add_0: "n  (n + m) = (0::nat)" 
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291 
by (induct n) simp_all 
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292 

30093  293 
lemma diff_Suc_1 [simp]: "Suc n  1 = n" 
294 
unfolding One_nat_def by simp 

295 

26072
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296 
text {* Difference distributes over multiplication *} 
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297 

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298 
lemma diff_mult_distrib: "((m::nat)  n) * k = (m * k)  (n * k)" 
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299 
by (induct m n rule: diff_induct) (simp_all add: diff_cancel) 
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300 

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301 
lemma diff_mult_distrib2: "k * ((m::nat)  n) = (k * m)  (k * n)" 
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302 
by (simp add: diff_mult_distrib mult_commute [of k]) 
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303 
 {* NOT added as rewrites, since sometimes they are used from righttoleft *} 
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304 

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305 

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306 
subsubsection {* Multiplication *} 
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307 

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308 
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)" 
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309 
by (rule mult_assoc) 
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310 

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311 
lemma nat_mult_commute: "m * n = n * (m::nat)" 
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312 
by (rule mult_commute) 
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313 

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314 
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)" 
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315 
by (rule right_distrib) 
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316 

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317 
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0  n=0)" 
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318 
by (induct m) auto 
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319 

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320 
lemmas nat_distrib = 
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321 
add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2 
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322 

30079
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323 
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)" 
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324 
apply (induct m) 
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325 
apply simp 
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326 
apply (induct n) 
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327 
apply auto 
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328 
done 
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329 

35828
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330 
lemma one_eq_mult_iff [simp,no_atp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)" 
26072
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331 
apply (rule trans) 
44890
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332 
apply (rule_tac [2] mult_eq_1_iff, fastforce) 
26072
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333 
done 
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334 

30079
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335 
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1" 
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336 
unfolding One_nat_def by (rule mult_eq_1_iff) 
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337 

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338 
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1" 
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339 
unfolding One_nat_def by (rule one_eq_mult_iff) 
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340 

26072
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341 
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n  (k = (0::nat)))" 
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342 
proof  
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343 
have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n" 
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344 
proof (induct n arbitrary: m) 
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345 
case 0 then show "m = 0" by simp 
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346 
next 
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347 
case (Suc n) then show "m = Suc n" 
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changeset

348 
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

349 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

350 
then show ?thesis by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

351 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

352 

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

353 
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n  (k = (0::nat)))" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

354 
by (simp add: mult_commute) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

355 

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

356 
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

357 
by (subst mult_cancel1) simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

358 

24995  359 

360 
subsection {* Orders on @{typ nat} *} 

361 

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

362 
subsubsection {* Operation definition *} 
24995  363 

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

364 
instantiation nat :: linorder 
25510  365 
begin 
366 

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

367 
primrec less_eq_nat where 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

368 
"(0\<Colon>nat) \<le> n \<longleftrightarrow> True" 
44325  369 
 "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:
25928
diff
changeset

370 

28514  371 
declare less_eq_nat.simps [simp del] 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

372 
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

373 
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:
25928
diff
changeset

374 

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

375 
definition less_nat where 
28514  376 
less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

377 

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

378 
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

379 
by (simp add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

380 

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

381 
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

382 
unfolding less_eq_Suc_le .. 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

383 

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

384 
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

385 
by (induct n) (simp_all add: less_eq_nat.simps(2)) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

386 

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

387 
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

388 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

389 

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

390 
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

391 
by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

392 

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

393 
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

394 
by (simp add: less_eq_Suc_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

395 

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

396 
lemma less_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

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

398 

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

399 
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

400 
by (induct m arbitrary: n) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

401 
(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

402 

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

403 
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

404 
by (cases n) (auto intro: le_SucI) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

405 

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

406 
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

407 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
24995  408 

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

409 
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

410 
by (simp add: less_eq_Suc_le) (erule Suc_leD) 
25510  411 

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

412 
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

413 
proof 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

414 
fix n m :: nat 
27679  415 
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

416 
proof (induct n arbitrary: m) 
27679  417 
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

418 
next 
27679  419 
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

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

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

422 
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

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

424 
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

425 
then show "n = m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

426 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

427 
(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

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

429 
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

430 
then show "n \<le> q" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

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

432 
case 0 show ?case by simp 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

433 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

434 
case (Suc n) then show ?case 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

435 
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

436 
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

437 
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

438 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

439 
next 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

440 
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

441 
by (induct n arbitrary: m) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

442 
(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

443 
qed 
25510  444 

445 
end 

13449  446 

29652  447 
instantiation nat :: bot 
448 
begin 

449 

450 
definition bot_nat :: nat where 

451 
"bot_nat = 0" 

452 

453 
instance proof 

454 
qed (simp add: bot_nat_def) 

455 

456 
end 

457 

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

458 
subsubsection {* Introduction properties *} 
13449  459 

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

460 
lemma lessI [iff]: "n < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

461 
by (simp add: less_Suc_eq_le) 
13449  462 

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

463 
lemma zero_less_Suc [iff]: "0 < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

464 
by (simp add: less_Suc_eq_le) 
13449  465 

466 

467 
subsubsection {* Elimination properties *} 

468 

469 
lemma less_not_refl: "~ n < (n::nat)" 

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

470 
by (rule order_less_irrefl) 
13449  471 

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

472 
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

473 
by (rule not_sym) (rule less_imp_neq) 
13449  474 

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

475 
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

476 
by (rule less_imp_neq) 
13449  477 

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

478 
lemma less_irrefl_nat: "(n::nat) < n ==> R" 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

479 
by (rule notE, rule less_not_refl) 
13449  480 

481 
lemma less_zeroE: "(n::nat) < 0 ==> R" 

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

482 
by (rule notE) (rule not_less0) 
13449  483 

484 
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

485 
unfolding less_Suc_eq_le le_less .. 
13449  486 

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

487 
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

488 
by (simp add: less_Suc_eq) 
13449  489 

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

490 
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

491 
unfolding One_nat_def by (rule less_Suc0) 
13449  492 

493 
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

494 
by simp 
13449  495 

14302  496 
text {* "Less than" is antisymmetric, sort of *} 
497 
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

498 
unfolding not_less less_Suc_eq_le by (rule antisym) 
14302  499 

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

500 
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

501 
by (rule linorder_neq_iff) 
13449  502 

503 
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m" 

504 
and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m" 

505 
shows "P n m" 

506 
apply (rule less_linear [THEN disjE]) 

507 
apply (erule_tac [2] disjE) 

508 
apply (erule lessCase) 

509 
apply (erule sym [THEN eqCase]) 

510 
apply (erule major) 

511 
done 

512 

513 

514 
subsubsection {* Inductive (?) properties *} 

515 

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

516 
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

517 
unfolding less_eq_Suc_le [of m] le_less by simp 
13449  518 

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

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

520 
assumes major: "i < k" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

521 
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

522 
shows P 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

523 
proof  
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

524 
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

525 
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

526 
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

527 
by (clarsimp simp add: less_le) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

528 
with p1 p2 show P by auto 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

529 
qed 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

530 

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

531 
lemma less_SucE: assumes major: "m < Suc n" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

532 
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

533 
apply (rule major [THEN lessE]) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

534 
apply (rule eq, blast) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

535 
apply (rule less, blast) 
13449  536 
done 
537 

538 
lemma Suc_lessE: assumes major: "Suc i < k" 

539 
and minor: "!!j. i < j ==> k = Suc j ==> P" shows P 

540 
apply (rule major [THEN lessE]) 

541 
apply (erule lessI [THEN minor]) 

14208  542 
apply (erule Suc_lessD [THEN minor], assumption) 
13449  543 
done 
544 

545 
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

546 
by simp 
13449  547 

548 
lemma less_trans_Suc: 

549 
assumes le: "i < j" shows "j < k ==> Suc i < k" 

14208  550 
apply (induct k, simp_all) 
13449  551 
apply (insert le) 
552 
apply (simp add: less_Suc_eq) 

553 
apply (blast dest: Suc_lessD) 

554 
done 

555 

556 
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

557 
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

558 
unfolding not_less less_Suc_eq_le .. 
13449  559 

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

560 
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

561 
unfolding not_le Suc_le_eq .. 
21243  562 

24995  563 
text {* Properties of "less than or equal" *} 
13449  564 

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

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

566 
unfolding less_Suc_eq_le . 
13449  567 

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

568 
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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changeset

569 
unfolding not_le less_Suc_eq_le .. 
13449  570 

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

571 
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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changeset

572 
by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq) 
13449  573 

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

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

575 
by (drule le_Suc_eq [THEN iffD1], iprover+) 
13449  576 

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

577 
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

578 
unfolding Suc_le_eq . 
13449  579 

580 
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

581 
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

582 
unfolding Suc_le_eq . 
13449  583 

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

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

585 
unfolding less_eq_Suc_le by (rule Suc_leD) 
13449  586 

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

587 
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

588 
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq 
13449  589 

590 

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

591 
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n  m = n"} *} 
13449  592 

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

593 
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

594 
unfolding le_less . 
13449  595 

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

596 
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

597 
by (rule le_less) 
13449  598 

22718  599 
text {* Useful with @{text blast}. *} 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

600 
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

601 
by auto 
13449  602 

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

603 
lemma le_refl: "n \<le> (n::nat)" 
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

604 
by simp 
13449  605 

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

606 
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

607 
by (rule order_trans) 
13449  608 

33657  609 
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

610 
by (rule antisym) 
13449  611 

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

612 
lemma 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

613 
by (rule less_le) 
13449  614 

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

615 
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

616 
unfolding less_le .. 
13449  617 

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

618 
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

619 
by (rule linear) 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

620 

22718  621 
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat] 
15921  622 

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

623 
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

624 
unfolding less_Suc_eq_le by auto 
13449  625 

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

626 
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

627 
unfolding not_less by (rule le_less_Suc_eq) 
13449  628 

629 
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq 

630 

22718  631 
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

632 
NOTE USE OF @{text "=="} *} 
13449  633 
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c" 
25162  634 
by simp 
13449  635 

636 
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)" 

25162  637 
by simp 
13449  638 

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

639 
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m" 
25162  640 
by (cases n) simp_all 
641 

642 
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m" 

643 
by (cases n) simp_all 

13449  644 

22718  645 
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0" 
25162  646 
by (cases n) simp_all 
13449  647 

25162  648 
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)" 
649 
by (cases n) simp_all 

25140  650 

13449  651 
text {* This theorem is useful with @{text blast} *} 
652 
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n" 

25162  653 
by (rule neq0_conv[THEN iffD1], iprover) 
13449  654 

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

655 
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)" 
25162  656 
by (fast intro: not0_implies_Suc) 
13449  657 

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

658 
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

659 
using neq0_conv by blast 
13449  660 

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

661 
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)" 
25162  662 
by (induct m') simp_all 
13449  663 

664 
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

665 
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0  (\<exists>j. m = Suc j & j < n))" 
25162  666 
by (cases m) simp_all 
13449  667 

668 

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

669 
subsubsection {* Monotonicity of Addition *} 
13449  670 

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

671 
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

672 
by (simp add: diff_Suc split: nat.split) 
13449  673 

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

674 
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

675 
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

676 

14331  677 
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))" 
25162  678 
by (induct k) simp_all 
13449  679 

14331  680 
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))" 
25162  681 
by (induct k) simp_all 
13449  682 

25162  683 
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0  n>0)" 
684 
by(auto dest:gr0_implies_Suc) 

13449  685 

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

686 
text {* strict, in 1st argument *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

687 
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)" 
25162  688 
by (induct k) simp_all 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

689 

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

690 
text {* strict, in both arguments *} 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

691 
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

692 
apply (rule add_less_mono1 [THEN less_trans], assumption+) 
15251  693 
apply (induct j, simp_all) 
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 {* 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

697 
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

698 
apply (induct n) 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

699 
apply (simp_all add: order_le_less) 
22718  700 
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

701 
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

702 
done 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

703 

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

704 
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

705 
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

706 
apply(auto simp: gr0_conv_Suc) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

707 
apply (induct_tac m) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

708 
apply (simp_all add: add_less_mono) 
3d4953e88449
Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents:
25111
diff
changeset

709 
done 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

710 

14740  711 
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

712 
instance nat :: linordered_semidom 
14341
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

713 
proof 
a09441bd4f1e
Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents:
14331
diff
changeset

714 
fix i j k :: nat 
14348
744c868ee0b7
Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents:
14341
diff
changeset

715 
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

716 
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

717 
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

718 
qed 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

719 

30056  720 
instance nat :: no_zero_divisors 
721 
proof 

722 
fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto 

723 
qed 

724 

44817  725 

726 
subsubsection {* @{term min} and @{term max} *} 

727 

728 
lemma mono_Suc: "mono Suc" 

729 
by (rule monoI) simp 

730 

731 
lemma min_0L [simp]: "min 0 n = (0::nat)" 

45931  732 
by (rule min_absorb1) simp 
44817  733 

734 
lemma min_0R [simp]: "min n 0 = (0::nat)" 

45931  735 
by (rule min_absorb2) simp 
44817  736 

737 
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)" 

738 
by (simp add: mono_Suc min_of_mono) 

739 

740 
lemma min_Suc1: 

741 
"min (Suc n) m = (case m of 0 => 0  Suc m' => Suc(min n m'))" 

742 
by (simp split: nat.split) 

743 

744 
lemma min_Suc2: 

745 
"min m (Suc n) = (case m of 0 => 0  Suc m' => Suc(min m' n))" 

746 
by (simp split: nat.split) 

747 

748 
lemma max_0L [simp]: "max 0 n = (n::nat)" 

45931  749 
by (rule max_absorb2) simp 
44817  750 

751 
lemma max_0R [simp]: "max n 0 = (n::nat)" 

45931  752 
by (rule max_absorb1) simp 
44817  753 

754 
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)" 

755 
by (simp add: mono_Suc max_of_mono) 

756 

757 
lemma max_Suc1: 

758 
"max (Suc n) m = (case m of 0 => Suc n  Suc m' => Suc(max n m'))" 

759 
by (simp split: nat.split) 

760 

761 
lemma max_Suc2: 

762 
"max m (Suc n) = (case m of 0 => Suc n  Suc m' => Suc(max m' n))" 

763 
by (simp split: nat.split) 

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

764 

44817  765 
lemma nat_mult_min_left: 
766 
fixes m n q :: nat 

767 
shows "min m n * q = min (m * q) (n * q)" 

768 
by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

769 

770 
lemma nat_mult_min_right: 

771 
fixes m n q :: nat 

772 
shows "m * min n q = min (m * n) (m * q)" 

773 
by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans) 

774 

775 
lemma nat_add_max_left: 

776 
fixes m n q :: nat 

777 
shows "max m n + q = max (m + q) (n + q)" 

778 
by (simp add: max_def) 

779 

780 
lemma nat_add_max_right: 

781 
fixes m n q :: nat 

782 
shows "m + max n q = max (m + n) (m + q)" 

783 
by (simp add: max_def) 

784 

785 
lemma nat_mult_max_left: 

786 
fixes m n q :: nat 

787 
shows "max m n * q = max (m * q) (n * q)" 

788 
by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans) 

789 

790 
lemma nat_mult_max_right: 

791 
fixes m n q :: nat 

792 
shows "m * max n q = max (m * n) (m * q)" 

793 
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

794 

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

795 

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

796 
subsubsection {* Additional theorems about @{term "op \<le>"} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

797 

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

798 
text {* Complete induction, aka courseofvalues induction *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

799 

27823  800 
instance nat :: wellorder proof 
801 
fix P and n :: nat 

802 
assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" 

803 
have "\<And>q. q \<le> n \<Longrightarrow> P q" 

804 
proof (induct n) 

805 
case (0 n) 

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

806 
have "P 0" by (rule step) auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

807 
thus ?case using 0 by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

808 
next 
27823  809 
case (Suc m n) 
810 
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

811 
thus ?case 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

812 
proof 
27823  813 
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

814 
next 
27823  815 
assume n: "n = Suc m" 
816 
show "P n" 

817 
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

818 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

819 
qed 
27823  820 
then show "P n" by auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

821 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

822 

27823  823 
lemma Least_Suc: 
824 
"[ P n; ~ P 0 ] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))" 

47988  825 
apply (cases n, auto) 
27823  826 
apply (frule LeastI) 
827 
apply (drule_tac P = "%x. P (Suc x) " in LeastI) 

828 
apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))") 

829 
apply (erule_tac [2] Least_le) 

47988  830 
apply (cases "LEAST x. P x", auto) 
27823  831 
apply (drule_tac P = "%x. P (Suc x) " in Least_le) 
832 
apply (blast intro: order_antisym) 

833 
done 

834 

835 
lemma Least_Suc2: 

836 
"[P n; Q m; ~P 0; !k. P (Suc k) = Q k] ==> Least P = Suc (Least Q)" 

837 
apply (erule (1) Least_Suc [THEN ssubst]) 

838 
apply simp 

839 
done 

840 

841 
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)" 

842 
apply (cases n) 

843 
apply blast 

844 
apply (rule_tac x="LEAST k. P(k)" in exI) 

845 
apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex) 

846 
done 

847 

848 
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

849 
unfolding One_nat_def 
27823  850 
apply (cases n) 
851 
apply blast 

852 
apply (frule (1) ex_least_nat_le) 

853 
apply (erule exE) 

854 
apply (case_tac k) 

855 
apply simp 

856 
apply (rename_tac k1) 

857 
apply (rule_tac x=k1 in exI) 

858 
apply (auto simp add: less_eq_Suc_le) 

859 
done 

860 

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

861 
lemma nat_less_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

862 
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

863 
using assms less_induct by blast 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

864 

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

865 
lemma measure_induct_rule [case_names less]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

866 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

867 
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

868 
shows "P a" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

869 
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

870 

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

871 
text {* old style induction rules: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

872 
lemma measure_induct: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

873 
fixes f :: "'a \<Rightarrow> nat" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

874 
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

875 
by (rule measure_induct_rule [of f P a]) iprover 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

876 

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

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

878 
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

879 
shows "P n" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

880 
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

881 

19870  882 
text{*An induction rule for estabilishing binary relations*} 
22718  883 
lemma less_Suc_induct: 
19870  884 
assumes less: "i < j" 
885 
and step: "!!i. P i (Suc i)" 

31714  886 
and trans: "!!i j k. i < j ==> j < k ==> P i j ==> P j k ==> P i k" 
19870  887 
shows "P i j" 
888 
proof  

31714  889 
from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add) 
22718  890 
have "P i (Suc (i + k))" 
19870  891 
proof (induct k) 
22718  892 
case 0 
893 
show ?case by (simp add: step) 

19870  894 
next 
895 
case (Suc k) 

31714  896 
have "0 + i < Suc k + i" by (rule add_less_mono1) simp 
897 
hence "i < Suc (i + k)" by (simp add: add_commute) 

898 
from trans[OF this lessI Suc step] 

899 
show ?case by simp 

19870  900 
qed 
22718  901 
thus "P i j" by (simp add: j) 
19870  902 
qed 
903 

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

904 
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

905 
Provided by Roelof Oosterhuis. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

906 
$P(n)$ is true for all $n\in\mathbb{N}$ if 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

907 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

908 
\item case ``0'': given $n=0$ prove $P(n)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

909 
\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

910 
a smaller integer $m$ such that $\neg P(m)$. 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

911 
\end{itemize} *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

912 

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

913 
text{* A compact version without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

914 
lemma infinite_descent: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

915 
"\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m \<rbrakk> \<Longrightarrow> P n" 
47988  916 
by (induct n rule: less_induct) auto 
26748
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

917 

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

918 
lemma infinite_descent0[case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

919 
"\<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

920 
by (rule infinite_descent) (case_tac "n>0", auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

921 

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

922 
text {* 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

923 
Infinite descent using a mapping to $\mathbb{N}$: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

924 
$P(x)$ 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

925 
\begin{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

926 
\item case ``0'': given $V(x)=0$ prove $P(x)$, 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

927 
\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

928 
\end{itemize} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

929 
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

930 

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

931 
corollary infinite_descent0_measure [case_names 0 smaller]: 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

932 
assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

933 
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

934 
shows "P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

935 
proof  
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

936 
obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

937 
moreover have "\<And>x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

938 
proof (induct n rule: infinite_descent0) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

939 
case 0  "i.e. $V(x) = 0$" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

940 
with A0 show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

941 
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

942 
case (smaller n) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

943 
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

944 
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

945 
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

946 
then show ?case by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

947 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

948 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

949 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

950 

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

951 
text{* Again, without explicit base case: *} 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

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

953 
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

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

955 
from assms obtain n where "n = V x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

956 
moreover have "!!x. V x = n \<Longrightarrow> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

957 
proof (induct n rule: infinite_descent, auto) 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

958 
fix x assume "\<not> P x" 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

959 
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

960 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

961 
ultimately show "P x" by auto 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

962 
qed 
4d51ddd6aa5c
Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents:
26335
diff
changeset

963 

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

964 
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

965 
monotonicity to @{text "\<le>"} monotonicity *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

966 
lemma less_mono_imp_le_mono: 
24438  967 
"\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)" 
968 
by (simp add: order_le_less) (blast) 

969 

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

972 
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)" 
24438  973 
by (rule add_right_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

974 

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

975 
text {* nonstrict, in both arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

976 
lemma add_le_mono: "[ i \<le> j; k \<le> l ] ==> i + k \<le> j + (l::nat)" 
24438  977 
by (rule add_mono) 
14267
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

978 

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

979 
lemma le_add2: "n \<le> ((m + n)::nat)" 
24438  980 
by (insert add_right_mono [of 0 m n], simp) 
13449  981 

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

982 
lemma le_add1: "n \<le> ((n + m)::nat)" 
24438  983 
by (simp add: add_commute, rule le_add2) 
13449  984 

985 
lemma less_add_Suc1: "i < Suc (i + m)" 

24438  986 
by (rule le_less_trans, rule le_add1, rule lessI) 
13449  987 

988 
lemma less_add_Suc2: "i < Suc (m + i)" 

24438  989 
by (rule le_less_trans, rule le_add2, rule lessI) 
13449  990 

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

991 
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))" 
24438  992 
by (iprover intro!: less_add_Suc1 less_imp_Suc_add) 
13449  993 

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

994 
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m" 
24438  995 
by (rule le_trans, assumption, rule le_add1) 
13449  996 

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

997 
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j" 
24438  998 
by (rule le_trans, assumption, rule le_add2) 
13449  999 

1000 
lemma trans_less_add1: "(i::nat) < j ==> i < j + m" 

24438  1001 
by (rule less_le_trans, assumption, rule le_add1) 
13449  1002 

1003 
lemma trans_less_add2: "(i::nat) < j ==> i < m + j" 

24438  1004 
by (rule less_le_trans, assumption, rule le_add2) 
13449  1005 

1006 
lemma add_lessD1: "i + j < (k::nat) ==> i < k" 

24438  1007 
apply (rule le_less_trans [of _ "i+j"]) 
1008 
apply (simp_all add: le_add1) 

1009 
done 

13449  1010 

1011 
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))" 

24438  1012 
apply (rule notI) 
26335
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1013 
apply (drule add_lessD1) 
961bbcc9d85b
removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents:
26315
diff
changeset

1014 
apply (erule less_irrefl [THEN notE]) 
24438  1015 
done 
13449  1016 

1017 
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))" 

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

1018 
by (simp add: add_commute) 
13449  1019 

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

1020 
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)" 
24438  1021 
apply (rule order_trans [of _ "m+k"]) 
1022 
apply (simp_all add: le_add1) 

1023 
done 

13449  1024 

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

1025 
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)" 
24438  1026 
apply (simp add: add_commute) 
1027 
apply (erule add_leD1) 

1028 
done 

13449  1029 

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

1030 
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R" 
24438  1031 
by (blast dest: add_leD1 add_leD2) 
13449  1032 

1033 
text {* needs @{text "!!k"} for @{text add_ac} to work *} 

1034 
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n" 

24438  1035 
by (force simp del: add_Suc_right 
13449  1036 
simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac) 
1037 

1038 

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

1039 
subsubsection {* More results about difference *} 
13449  1040 

1041 
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

1042 
if @{term "n \<le> m"} then @{term "n + (m  n) = m"}. *} 
13449  1043 
lemma add_diff_inverse: "~ m < n ==> n + (m  n) = (m::nat)" 
24438  1044 
by (induct m n rule: diff_induct) simp_all 
13449  1045 

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

1046 
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m  n) = (m::nat)" 
24438  1047 
by (simp add: add_diff_inverse linorder_not_less) 
13449  1048 

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

1049 
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

1050 
by (simp add: add_commute) 
13449  1051 

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

1052 
lemma Suc_diff_le: "n \<le> m ==> Suc m  n = Suc (m  n)" 
24438  1053 
by (induct m n rule: diff_induct) simp_all 
13449  1054 

1055 
lemma diff_less_Suc: "m  n < Suc m" 

24438  1056 
apply (induct m n rule: diff_induct) 
1057 
apply (erule_tac [3] less_SucE) 

1058 
apply (simp_all add: less_Suc_eq) 

1059 
done 

13449  1060 

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

1061 
lemma diff_le_self [simp]: "m  n \<le> (m::nat)" 
24438  1062 
by (induct m n rule: diff_induct) (simp_all add: le_SucI) 
13449  1063 

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

1064 
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

1065 
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

1066 

13449  1067 
lemma less_imp_diff_less: "(j::nat) < k ==> j  n < k" 
24438  1068 
by (rule le_less_trans, rule diff_le_self) 
13449  1069 

1070 
lemma diff_Suc_less [simp]: "0<n ==> n  Suc i < n" 

24438  1071 
by (cases n) (auto simp add: le_simps) 
13449  1072 

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

1073 
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j)  k = i + (j  k)" 
24438  1074 
by (induct j k rule: diff_induct) simp_all 
13449  1075 

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

1076 
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i)  k = (j  k) + i" 
24438  1077 
by (simp add: add_commute diff_add_assoc) 
13449  1078 

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

1079 
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j  i = k) = (j = k + i)" 
24438  1080 
by (auto simp add: diff_add_inverse2) 
13449  1081 

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

1082 
lemma diff_is_0_eq [simp]: "((m::nat)  n = 0) = (m \<le> n)" 
24438  1083 
by (induct m n rule: diff_induct) simp_all 
13449  1084 

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

1085 
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat)  n = 0" 
24438  1086 
by (rule iffD2, rule diff_is_0_eq) 
13449  1087 

1088 
lemma zero_less_diff [simp]: "(0 < n  (m::nat)) = (m < n)" 

24438  1089 
by (induct m n rule: diff_induct) simp_all 
13449  1090 

22718  1091 
lemma less_imp_add_positive: 
1092 
assumes "i < j" 

1093 
shows "\<exists>k::nat. 0 < k & i + k = j" 

1094 
proof 

1095 
from assms show "0 < j  i & i + (j  i) = j" 

23476  1096 
by (simp add: order_less_imp_le) 
22718  1097 
qed 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1098 

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

1099 
text {* a nice rewrite for bounded subtraction *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1100 
lemma nat_minus_add_max: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1101 
fixes n m :: nat 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1102 
shows "n  m + m = max n m" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1103 
by (simp add: max_def not_le order_less_imp_le) 
13449  1104 

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

1105 
lemma nat_diff_split: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1106 
"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

1107 
 {* elimination of @{text } on @{text nat} *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1108 
by (cases "a < b") 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1109 
(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

1110 
not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero) 
13449  1111 

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

1112 
lemma nat_diff_split_asm: 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1113 
"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

1114 
 {* elimination of @{text } on @{text nat} in assumptions *} 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1115 
by (auto split: nat_diff_split) 
13449  1116 

47255
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1117 
lemma Suc_pred': "0 < n ==> n = Suc(n  1)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1118 
by simp 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1119 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1120 
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m  1) + n))" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1121 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1122 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1123 
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m  1) * n))" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1124 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1125 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1126 
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m  n = m  (n  1)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1127 
unfolding One_nat_def by (cases n) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1128 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1129 
lemma diff_Suc_eq_diff_pred: "m  Suc n = (m  1)  n" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1130 
unfolding One_nat_def by (cases m) simp_all 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1131 

30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1132 
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)" 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1133 
by (fact Let_def) 
30a1692557b0
removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents:
47208
diff
changeset

1134 

13449  1135 

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

1136 
subsubsection {* Monotonicity of Multiplication *} 
13449  1137 

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

1138 
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k" 
24438  1139 
by (simp add: mult_right_mono) 
13449  1140 

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

1141 
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j" 
24438  1142 
by (simp add: mult_left_mono) 
13449  1143 

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

1144 
text {* @{text "\<le>"} monotonicity, BOTH arguments *} 
b963e9cee2a0
More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents:
14266
diff
changeset

1145 
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l" 
24438  1146 
by (simp add: mult_mono) 
13449  1147 

1148 
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k" 

24438  1149 
by (simp add: mult_strict_right_mono) 
13449  1150 

14266  1151 
text{*Differs from the standard @{text zero_less_mult_iff} in that 
1152 
there are no negative numbers.*} 

1153 
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)" 

13449  1154 
apply (induct m) 
22718  1155 
apply simp 
1156 
apply (case_tac n) 

1157 
apply simp_all 

13449  1158 
done 
1159 

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

1160 
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)" 
13449  1161 
apply (induct m) 
22718  1162 
apply simp 
1163 
apply (case_tac n) 

1164 
apply simp_all 

13449  1165 
done 
1166 

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

1167 
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)" 
13449  1168 
apply (safe intro!: mult_less_mono1) 
47988  1169 
apply (cases k, auto) 
13449  1170 
apply (simp del: le_0_eq add: linorder_not_le [symmetric]) 
1171 
apply (blast intro: mult_le_mono1) 

1172 
done 

1173 

1174 
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)" 

24438  1175 
by (simp add: mult_commute [of k]) 
13449  1176 

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

1177 
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k > m \<le> n)" 
24438  1178 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1179 

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

1180 
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k > m \<le> n)" 
24438  1181 
by (simp add: linorder_not_less [symmetric], auto) 
13449  1182 

1183 
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)" 

24438  1184 
by (subst mult_less_cancel1) simp 
13449  1185 

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

1186 
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)" 
24438  1187 
by (subst mult_le_cancel1) simp 
13449  1188 

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

1189 
lemma le_square: "m \<le> m * (m::nat)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1190 
by (cases m) (auto intro: le_add1) 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1191 

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

1192 
lemma le_cube: "(m::nat) \<le> m * (m * m)" 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1193 
by (cases m) (auto intro: le_add1) 
13449  1194 

1195 
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

1196 
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1  m = 0" 
13449  1197 
apply (drule sym) 
1198 
apply (rule disjCI) 

1199 
apply (rule nat_less_cases, erule_tac [2] _) 

25157  1200 
apply (drule_tac [2] mult_less_mono2) 
25162  1201 
apply (auto) 
13449  1202 
done 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
7702
diff
changeset

1203 

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

1204 
text {* the lattice order on @{typ nat} *} 
24995  1205 

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

1206 
instantiation nat :: distrib_lattice 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1207 
begin 
24995  1208 

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

1209 
definition 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1210 
"(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min" 
24995  1211 

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

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

1213 
"(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max" 
24995  1214 

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

1215 
instance by intro_classes 
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25928
diff
changeset

1216 
(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

1217 
intro: order_less_imp_le antisym elim!: order_trans order_less_trans) 
24995  1218 

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

1219 
end 
24995  1220 

1221 

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

1222 
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

1223 

30971  1224 
text {* 
1225 
We use the same logical constant for the power operations on 

1226 
functions and relations, in order to share the same syntax. 

1227 
*} 

1228 

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

1229 
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" 
30971  1230 

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

1231 
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where 
30971  1232 
"f ^^ n \<equiv> compow n f" 
1233 

1234 
notation (latex output) 

1235 
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000) 

1236 

1237 
notation (HTML output) 

1238 
compower ("(_\<^bsup>_\<^esup>)" [1000] 1000) 

1239 

1240 
text {* @{text "f ^^ n = f o ... o f"}, the nfold composition of @{text f} *} 

1241 

1242 
overloading 

1243 
funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)" 

1244 
begin 

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

1245 

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

1246 
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where 
44325  1247 
"funpow 0 f = id" 
1248 
 "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

1249 

30971  1250 
end 
1251 

49723
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1252 
lemma funpow_Suc_right: 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1253 
"f ^^ Suc n = f ^^ n \<circ> f" 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1254 
proof (induct n) 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1255 
case 0 then show ?case by simp 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1256 
next 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1257 
fix n 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1258 
assume "f ^^ Suc n = f ^^ n \<circ> f" 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1259 
then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f" 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1260 
by (simp add: o_assoc) 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1261 
qed 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1262 

bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1263 
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right 
bbc2942ba09f
alternative simplification of ^^ to the righthand side;
haftmann
parents:
49388
diff
changeset

1264 

30971  1265 
text {* for code generation *} 
1266 

1267 
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

1268 
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

1269 

30971  1270 
lemma [code]: 
37430  1271 
"funpow (Suc n) f = f o funpow n f" 
30971  1272 
"funpow 0 f = id" 
37430  1273 
by (simp_all add: funpow_code_def) 
30971  1274 

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

1275 
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

1276 

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

1277 
lemma funpow_add: 
30971  1278 
"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

1279 
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

1280 

37430  1281 
lemma funpow_mult: 
1282 
fixes f :: "'a \<Rightarrow> 'a" 

1283 
shows "(f ^^ m) ^^ n = f ^^ (m * n)" 

1284 
by (induct n) (simp_all add: funpow_add) 

1285 

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

1286 
lemma funpow_swap1: 
30971  1287 
"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

1288 
proof  
30971  1289 
have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp 
1290 
also have "\<dots> = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add) 

1291 
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

1292 
finally show ?thesis . 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1293 
qed 
cf50e67bc1d1
power operation on functions in theory Nat; power operation on relations in theory Transitive_Closure
haftmann
parents:
30686
diff
changeset

1294 

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

1295 
lemma comp_funpow: 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1296 
fixes f :: "'a \<Rightarrow> 'a" 
d6cb7e625d75
more concise characterization of of_nat operation and class semiring_char_0
haftmann
parents:
37767
diff
changeset

1297 
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

1298 
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

1299 

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

1300 

45833  1301 
subsection {* Kleene iteration *} 
1302 

1303 
lemma Kleene_iter_lpfp: assumes "mono f" and "f p \<le> p" shows "(f^^k) bot \<le> p" 

1304 
proof(induction k) 

1305 
case 0 show ?case by simp 

1306 
next 

1307 
case Suc 

1308 
from monoD[OF assms(1) Suc] assms(2) 

1309 
show ?case by simp 

1310 
qed 

1311 

1312 
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot" 

1313 
shows "lfp f = (f^^k) bot" 

1314 
proof(rule antisym) 

1315 
show "lfp f \<le> (f^^k) bot" 

1316 
proof(rule lfp_lowerbound) 

1317 
show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp 
