src/HOL/Nat.thy
author wenzelm
Wed Aug 22 22:55:41 2012 +0200 (2012-08-22)
changeset 48891 c0eafbd55de3
parent 48560 e0875d956a6b
child 49388 1ffd5a055acf
permissions -rw-r--r--
prefer ML_file over old uses;
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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 :: "{minus, comm_monoid_add}"
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begin
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primrec plus_nat where
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  add_0:      "0 + n = (n\<Colon>nat)"
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| add_Suc:  "Suc m + n = Suc (m + n)"
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lemma add_0_right [simp]: "m + 0 = (m::nat)"
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  by (induct m) simp_all
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
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  by (induct m) simp_all
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declare add_0 [code]
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
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  by simp
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primrec minus_nat where
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  diff_0 [code]: "m - 0 = (m\<Colon>nat)"
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| diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
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declare diff_Suc [simp del]
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lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
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  by (induct n) (simp_all add: diff_Suc)
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lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
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  by (induct n) (simp_all add: diff_Suc)
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instance proof
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  fix n m q :: nat
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  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
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  show "n + m = m + n" by (induct n) simp_all
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  show "0 + n = n" by simp
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qed
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end
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hide_fact (open) add_0 add_0_right diff_0
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition
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  One_nat_def [simp]: "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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qed
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end
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subsubsection {* Addition *}
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lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
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  by (rule add_assoc)
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lemma nat_add_commute: "m + n = n + (m::nat)"
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  by (rule add_commute)
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lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
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  by (rule add_left_commute)
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lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
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  by (rule add_left_cancel)
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lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
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  by (rule add_right_cancel)
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text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
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lemma add_is_0 [iff]:
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  fixes m n :: nat
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  shows "(m + n = 0) = (m = 0 & n = 0)"
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  by (cases m) simp_all
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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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  by (cases m) simp_all
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lemma one_is_add:
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  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
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  by (rule trans, rule eq_commute, rule add_is_1)
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lemma add_eq_self_zero:
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  fixes m n :: nat
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  shows "m + n = m \<Longrightarrow> n = 0"
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  by (induct m) simp_all
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lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
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  apply (induct k)
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   apply simp
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  apply(drule comp_inj_on[OF _ inj_Suc])
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  apply (simp add:o_def)
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  done
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lemma Suc_eq_plus1: "Suc n = n + 1"
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  unfolding One_nat_def by simp
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lemma Suc_eq_plus1_left: "Suc n = 1 + n"
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  unfolding One_nat_def by simp
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subsubsection {* Difference *}
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lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
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  by (induct m) simp_all
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lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
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  by (induct i j rule: diff_induct) simp_all
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lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
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  by (simp add: diff_diff_left)
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lemma diff_commute: "(i::nat) - j - k = i - k - j"
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  by (simp add: diff_diff_left add_commute)
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lemma diff_add_inverse: "(n + m) - n = (m::nat)"
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  by (induct n) simp_all
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lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
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  by (simp add: diff_add_inverse add_commute [of m n])
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lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
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  by (induct k) simp_all
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lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
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  by (simp add: diff_cancel add_commute)
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lemma diff_add_0: "n - (n + m) = (0::nat)"
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  by (induct n) simp_all
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lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
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  unfolding One_nat_def by simp
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text {* Difference distributes over multiplication *}
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lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
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by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
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lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
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by (simp add: diff_mult_distrib mult_commute [of k])
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  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
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subsubsection {* Multiplication *}
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lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
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  by (rule mult_assoc)
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lemma nat_mult_commute: "m * n = n * (m::nat)"
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  by (rule mult_commute)
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lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
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  by (rule right_distrib)
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lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
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  by (induct m) auto
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lemmas nat_distrib =
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  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
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lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
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  apply (induct m)
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   apply simp
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  apply (induct n)
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   apply auto
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  done
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lemma one_eq_mult_iff [simp,no_atp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
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  apply (rule trans)
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  apply (rule_tac [2] mult_eq_1_iff, fastforce)
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  done
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lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
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  unfolding One_nat_def by (rule mult_eq_1_iff)
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lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
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  unfolding One_nat_def by (rule one_eq_mult_iff)
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   336
haftmann@26072
   337
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   338
proof -
haftmann@26072
   339
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   340
  proof (induct n arbitrary: m)
haftmann@26072
   341
    case 0 then show "m = 0" by simp
haftmann@26072
   342
  next
haftmann@26072
   343
    case (Suc n) then show "m = Suc n"
haftmann@26072
   344
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   345
  qed
haftmann@26072
   346
  then show ?thesis by auto
haftmann@26072
   347
qed
haftmann@26072
   348
haftmann@26072
   349
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@26072
   350
  by (simp add: mult_commute)
haftmann@26072
   351
haftmann@26072
   352
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   353
  by (subst mult_cancel1) simp
haftmann@26072
   354
haftmann@24995
   355
haftmann@24995
   356
subsection {* Orders on @{typ nat} *}
haftmann@24995
   357
haftmann@26072
   358
subsubsection {* Operation definition *}
haftmann@24995
   359
haftmann@26072
   360
instantiation nat :: linorder
haftmann@25510
   361
begin
haftmann@25510
   362
haftmann@26072
   363
primrec less_eq_nat where
haftmann@26072
   364
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   365
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   366
haftmann@28514
   367
declare less_eq_nat.simps [simp del]
haftmann@26072
   368
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
haftmann@26072
   369
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@26072
   370
haftmann@26072
   371
definition less_nat where
haftmann@28514
   372
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   373
haftmann@26072
   374
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   375
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   376
haftmann@26072
   377
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   378
  unfolding less_eq_Suc_le ..
haftmann@26072
   379
haftmann@26072
   380
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   381
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   382
haftmann@26072
   383
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   384
  by (simp add: less_eq_Suc_le)
haftmann@26072
   385
haftmann@26072
   386
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   387
  by simp
haftmann@26072
   388
haftmann@26072
   389
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   390
  by (simp add: less_eq_Suc_le)
haftmann@26072
   391
haftmann@26072
   392
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   393
  by (simp add: less_eq_Suc_le)
haftmann@26072
   394
haftmann@26072
   395
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   396
  by (induct m arbitrary: n)
haftmann@26072
   397
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   398
haftmann@26072
   399
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   400
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   401
haftmann@26072
   402
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   403
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   404
haftmann@26072
   405
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   406
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   407
wenzelm@26315
   408
instance
wenzelm@26315
   409
proof
haftmann@26072
   410
  fix n m :: nat
haftmann@27679
   411
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   412
  proof (induct n arbitrary: m)
haftmann@27679
   413
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   414
  next
haftmann@27679
   415
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   416
  qed
haftmann@26072
   417
next
haftmann@26072
   418
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   419
next
haftmann@26072
   420
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   421
  then show "n = m"
haftmann@26072
   422
    by (induct n arbitrary: m)
haftmann@26072
   423
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   424
next
haftmann@26072
   425
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   426
  then show "n \<le> q"
haftmann@26072
   427
  proof (induct n arbitrary: m q)
haftmann@26072
   428
    case 0 show ?case by simp
haftmann@26072
   429
  next
haftmann@26072
   430
    case (Suc n) then show ?case
haftmann@26072
   431
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   432
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   433
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   434
  qed
haftmann@26072
   435
next
haftmann@26072
   436
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   437
    by (induct n arbitrary: m)
haftmann@26072
   438
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   439
qed
haftmann@25510
   440
haftmann@25510
   441
end
berghofe@13449
   442
haftmann@29652
   443
instantiation nat :: bot
haftmann@29652
   444
begin
haftmann@29652
   445
haftmann@29652
   446
definition bot_nat :: nat where
haftmann@29652
   447
  "bot_nat = 0"
haftmann@29652
   448
haftmann@29652
   449
instance proof
haftmann@29652
   450
qed (simp add: bot_nat_def)
haftmann@29652
   451
haftmann@29652
   452
end
haftmann@29652
   453
haftmann@26072
   454
subsubsection {* Introduction properties *}
berghofe@13449
   455
haftmann@26072
   456
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   457
  by (simp add: less_Suc_eq_le)
berghofe@13449
   458
haftmann@26072
   459
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   460
  by (simp add: less_Suc_eq_le)
berghofe@13449
   461
berghofe@13449
   462
berghofe@13449
   463
subsubsection {* Elimination properties *}
berghofe@13449
   464
berghofe@13449
   465
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   466
  by (rule order_less_irrefl)
berghofe@13449
   467
wenzelm@26335
   468
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   469
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   470
paulson@14267
   471
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   472
  by (rule less_imp_neq)
berghofe@13449
   473
wenzelm@26335
   474
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   475
  by (rule notE, rule less_not_refl)
berghofe@13449
   476
berghofe@13449
   477
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   478
  by (rule notE) (rule not_less0)
berghofe@13449
   479
berghofe@13449
   480
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   481
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   482
huffman@30079
   483
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   484
  by (simp add: less_Suc_eq)
berghofe@13449
   485
blanchet@35828
   486
lemma less_one [iff, no_atp]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   487
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   488
berghofe@13449
   489
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   490
  by simp
berghofe@13449
   491
nipkow@14302
   492
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   493
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   494
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   495
paulson@14267
   496
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   497
  by (rule linorder_neq_iff)
berghofe@13449
   498
berghofe@13449
   499
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   500
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   501
  shows "P n m"
berghofe@13449
   502
  apply (rule less_linear [THEN disjE])
berghofe@13449
   503
  apply (erule_tac [2] disjE)
berghofe@13449
   504
  apply (erule lessCase)
berghofe@13449
   505
  apply (erule sym [THEN eqCase])
berghofe@13449
   506
  apply (erule major)
berghofe@13449
   507
  done
berghofe@13449
   508
berghofe@13449
   509
berghofe@13449
   510
subsubsection {* Inductive (?) properties *}
berghofe@13449
   511
paulson@14267
   512
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   513
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   514
haftmann@26072
   515
lemma lessE:
haftmann@26072
   516
  assumes major: "i < k"
haftmann@26072
   517
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   518
  shows P
haftmann@26072
   519
proof -
haftmann@26072
   520
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   521
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   522
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   523
    by (clarsimp simp add: less_le)
haftmann@26072
   524
  with p1 p2 show P by auto
haftmann@26072
   525
qed
haftmann@26072
   526
haftmann@26072
   527
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   528
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   529
  apply (rule major [THEN lessE])
haftmann@26072
   530
  apply (rule eq, blast)
haftmann@26072
   531
  apply (rule less, blast)
berghofe@13449
   532
  done
berghofe@13449
   533
berghofe@13449
   534
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   535
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   536
  apply (rule major [THEN lessE])
berghofe@13449
   537
  apply (erule lessI [THEN minor])
paulson@14208
   538
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   539
  done
berghofe@13449
   540
berghofe@13449
   541
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   542
  by simp
berghofe@13449
   543
berghofe@13449
   544
lemma less_trans_Suc:
berghofe@13449
   545
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   546
  apply (induct k, simp_all)
berghofe@13449
   547
  apply (insert le)
berghofe@13449
   548
  apply (simp add: less_Suc_eq)
berghofe@13449
   549
  apply (blast dest: Suc_lessD)
berghofe@13449
   550
  done
berghofe@13449
   551
berghofe@13449
   552
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   553
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   554
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   555
haftmann@26072
   556
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   557
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   558
haftmann@24995
   559
text {* Properties of "less than or equal" *}
berghofe@13449
   560
paulson@14267
   561
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   562
  unfolding less_Suc_eq_le .
berghofe@13449
   563
paulson@14267
   564
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   565
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   566
paulson@14267
   567
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   568
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   569
paulson@14267
   570
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   571
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   572
paulson@14267
   573
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   574
  unfolding Suc_le_eq .
berghofe@13449
   575
berghofe@13449
   576
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   577
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   578
  unfolding Suc_le_eq .
berghofe@13449
   579
wenzelm@26315
   580
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   581
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   582
paulson@14267
   583
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   584
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   585
berghofe@13449
   586
paulson@14267
   587
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   588
paulson@14267
   589
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   590
  unfolding le_less .
berghofe@13449
   591
paulson@14267
   592
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   593
  by (rule le_less)
berghofe@13449
   594
wenzelm@22718
   595
text {* Useful with @{text blast}. *}
paulson@14267
   596
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   597
  by auto
berghofe@13449
   598
paulson@14267
   599
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   600
  by simp
berghofe@13449
   601
paulson@14267
   602
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   603
  by (rule order_trans)
berghofe@13449
   604
nipkow@33657
   605
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   606
  by (rule antisym)
berghofe@13449
   607
paulson@14267
   608
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   609
  by (rule less_le)
berghofe@13449
   610
paulson@14267
   611
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   612
  unfolding less_le ..
berghofe@13449
   613
haftmann@26072
   614
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   615
  by (rule linear)
paulson@14341
   616
wenzelm@22718
   617
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   618
haftmann@26072
   619
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   620
  unfolding less_Suc_eq_le by auto
berghofe@13449
   621
haftmann@26072
   622
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   623
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   624
berghofe@13449
   625
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   626
wenzelm@22718
   627
text {* These two rules ease the use of primitive recursion.
paulson@14341
   628
NOTE USE OF @{text "=="} *}
berghofe@13449
   629
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   630
by simp
berghofe@13449
   631
berghofe@13449
   632
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   633
by simp
berghofe@13449
   634
paulson@14267
   635
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   636
by (cases n) simp_all
nipkow@25162
   637
nipkow@25162
   638
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   639
by (cases n) simp_all
berghofe@13449
   640
wenzelm@22718
   641
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   642
by (cases n) simp_all
berghofe@13449
   643
nipkow@25162
   644
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   645
by (cases n) simp_all
nipkow@25140
   646
berghofe@13449
   647
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   648
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   649
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   650
paulson@14267
   651
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   652
by (fast intro: not0_implies_Suc)
berghofe@13449
   653
blanchet@35828
   654
lemma not_gr0 [iff,no_atp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   655
using neq0_conv by blast
berghofe@13449
   656
paulson@14267
   657
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   658
by (induct m') simp_all
berghofe@13449
   659
berghofe@13449
   660
text {* Useful in certain inductive arguments *}
paulson@14267
   661
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   662
by (cases m) simp_all
berghofe@13449
   663
berghofe@13449
   664
haftmann@26072
   665
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   666
haftmann@26072
   667
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   668
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   669
huffman@30128
   670
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   671
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   672
paulson@14331
   673
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   674
by (induct k) simp_all
berghofe@13449
   675
paulson@14331
   676
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   677
by (induct k) simp_all
berghofe@13449
   678
nipkow@25162
   679
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   680
by(auto dest:gr0_implies_Suc)
berghofe@13449
   681
paulson@14341
   682
text {* strict, in 1st argument *}
paulson@14341
   683
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   684
by (induct k) simp_all
paulson@14341
   685
paulson@14341
   686
text {* strict, in both arguments *}
paulson@14341
   687
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   688
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   689
  apply (induct j, simp_all)
paulson@14341
   690
  done
paulson@14341
   691
paulson@14341
   692
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   693
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   694
  apply (induct n)
paulson@14341
   695
  apply (simp_all add: order_le_less)
wenzelm@22718
   696
  apply (blast elim!: less_SucE
haftmann@35047
   697
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   698
  done
paulson@14341
   699
paulson@14341
   700
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   701
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   702
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   703
apply (induct_tac m)
nipkow@25134
   704
apply (simp_all add: add_less_mono)
nipkow@25134
   705
done
paulson@14341
   706
nipkow@14740
   707
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   708
instance nat :: linordered_semidom
paulson@14341
   709
proof
paulson@14341
   710
  fix i j k :: nat
paulson@14348
   711
  show "0 < (1::nat)" by simp
paulson@14267
   712
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   713
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   714
qed
paulson@14267
   715
nipkow@30056
   716
instance nat :: no_zero_divisors
nipkow@30056
   717
proof
nipkow@30056
   718
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
nipkow@30056
   719
qed
nipkow@30056
   720
haftmann@44817
   721
haftmann@44817
   722
subsubsection {* @{term min} and @{term max} *}
haftmann@44817
   723
haftmann@44817
   724
lemma mono_Suc: "mono Suc"
haftmann@44817
   725
by (rule monoI) simp
haftmann@44817
   726
haftmann@44817
   727
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   728
by (rule min_absorb1) simp
haftmann@44817
   729
haftmann@44817
   730
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   731
by (rule min_absorb2) simp
haftmann@44817
   732
haftmann@44817
   733
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   734
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   735
haftmann@44817
   736
lemma min_Suc1:
haftmann@44817
   737
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   738
by (simp split: nat.split)
haftmann@44817
   739
haftmann@44817
   740
lemma min_Suc2:
haftmann@44817
   741
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   742
by (simp split: nat.split)
haftmann@44817
   743
haftmann@44817
   744
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   745
by (rule max_absorb2) simp
haftmann@44817
   746
haftmann@44817
   747
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   748
by (rule max_absorb1) simp
haftmann@44817
   749
haftmann@44817
   750
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   751
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   752
haftmann@44817
   753
lemma max_Suc1:
haftmann@44817
   754
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   755
by (simp split: nat.split)
haftmann@44817
   756
haftmann@44817
   757
lemma max_Suc2:
haftmann@44817
   758
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   759
by (simp split: nat.split)
paulson@14267
   760
haftmann@44817
   761
lemma nat_mult_min_left:
haftmann@44817
   762
  fixes m n q :: nat
haftmann@44817
   763
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   764
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   765
haftmann@44817
   766
lemma nat_mult_min_right:
haftmann@44817
   767
  fixes m n q :: nat
haftmann@44817
   768
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   769
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   770
haftmann@44817
   771
lemma nat_add_max_left:
haftmann@44817
   772
  fixes m n q :: nat
haftmann@44817
   773
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   774
  by (simp add: max_def)
haftmann@44817
   775
haftmann@44817
   776
lemma nat_add_max_right:
haftmann@44817
   777
  fixes m n q :: nat
haftmann@44817
   778
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   779
  by (simp add: max_def)
haftmann@44817
   780
haftmann@44817
   781
lemma nat_mult_max_left:
haftmann@44817
   782
  fixes m n q :: nat
haftmann@44817
   783
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   784
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   785
haftmann@44817
   786
lemma nat_mult_max_right:
haftmann@44817
   787
  fixes m n q :: nat
haftmann@44817
   788
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   789
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   790
paulson@14267
   791
krauss@26748
   792
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   793
krauss@26748
   794
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   795
haftmann@27823
   796
instance nat :: wellorder proof
haftmann@27823
   797
  fix P and n :: nat
haftmann@27823
   798
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   799
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   800
  proof (induct n)
haftmann@27823
   801
    case (0 n)
krauss@26748
   802
    have "P 0" by (rule step) auto
krauss@26748
   803
    thus ?case using 0 by auto
krauss@26748
   804
  next
haftmann@27823
   805
    case (Suc m n)
haftmann@27823
   806
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   807
    thus ?case
krauss@26748
   808
    proof
haftmann@27823
   809
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   810
    next
haftmann@27823
   811
      assume n: "n = Suc m"
haftmann@27823
   812
      show "P n"
haftmann@27823
   813
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   814
    qed
krauss@26748
   815
  qed
haftmann@27823
   816
  then show "P n" by auto
krauss@26748
   817
qed
krauss@26748
   818
haftmann@27823
   819
lemma Least_Suc:
haftmann@27823
   820
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   821
  apply (cases n, auto)
haftmann@27823
   822
  apply (frule LeastI)
haftmann@27823
   823
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   824
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   825
  apply (erule_tac [2] Least_le)
wenzelm@47988
   826
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   827
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   828
  apply (blast intro: order_antisym)
haftmann@27823
   829
  done
haftmann@27823
   830
haftmann@27823
   831
lemma Least_Suc2:
haftmann@27823
   832
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   833
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   834
  apply simp
haftmann@27823
   835
  done
haftmann@27823
   836
haftmann@27823
   837
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)"
haftmann@27823
   838
  apply (cases n)
haftmann@27823
   839
   apply blast
haftmann@27823
   840
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   841
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   842
  done
haftmann@27823
   843
haftmann@27823
   844
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)"
huffman@30079
   845
  unfolding One_nat_def
haftmann@27823
   846
  apply (cases n)
haftmann@27823
   847
   apply blast
haftmann@27823
   848
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   849
  apply (erule exE)
haftmann@27823
   850
  apply (case_tac k)
haftmann@27823
   851
   apply simp
haftmann@27823
   852
  apply (rename_tac k1)
haftmann@27823
   853
  apply (rule_tac x=k1 in exI)
haftmann@27823
   854
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   855
  done
haftmann@27823
   856
krauss@26748
   857
lemma nat_less_induct:
krauss@26748
   858
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   859
  using assms less_induct by blast
krauss@26748
   860
krauss@26748
   861
lemma measure_induct_rule [case_names less]:
krauss@26748
   862
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   863
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   864
  shows "P a"
krauss@26748
   865
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   866
krauss@26748
   867
text {* old style induction rules: *}
krauss@26748
   868
lemma measure_induct:
krauss@26748
   869
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   870
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   871
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   872
krauss@26748
   873
lemma full_nat_induct:
krauss@26748
   874
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   875
  shows "P n"
krauss@26748
   876
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   877
paulson@19870
   878
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   879
lemma less_Suc_induct:
paulson@19870
   880
  assumes less:  "i < j"
paulson@19870
   881
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   882
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   883
  shows "P i j"
paulson@19870
   884
proof -
krauss@31714
   885
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   886
  have "P i (Suc (i + k))"
paulson@19870
   887
  proof (induct k)
wenzelm@22718
   888
    case 0
wenzelm@22718
   889
    show ?case by (simp add: step)
paulson@19870
   890
  next
paulson@19870
   891
    case (Suc k)
krauss@31714
   892
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
krauss@31714
   893
    hence "i < Suc (i + k)" by (simp add: add_commute)
krauss@31714
   894
    from trans[OF this lessI Suc step]
krauss@31714
   895
    show ?case by simp
paulson@19870
   896
  qed
wenzelm@22718
   897
  thus "P i j" by (simp add: j)
paulson@19870
   898
qed
paulson@19870
   899
krauss@26748
   900
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   901
Provided by Roelof Oosterhuis.
krauss@26748
   902
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   903
\begin{itemize}
krauss@26748
   904
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   905
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   906
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   907
\end{itemize} *}
krauss@26748
   908
krauss@26748
   909
text{* A compact version without explicit base case: *}
krauss@26748
   910
lemma infinite_descent:
krauss@26748
   911
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   912
by (induct n rule: less_induct) auto
krauss@26748
   913
krauss@26748
   914
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   915
  "\<lbrakk> P 0; !!n. n>0 \<Longrightarrow> \<not> P n \<Longrightarrow> (\<exists>m::nat. m < n \<and> \<not>P m) \<rbrakk> \<Longrightarrow> P n"
krauss@26748
   916
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   917
krauss@26748
   918
text {*
krauss@26748
   919
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   920
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   921
\begin{itemize}
krauss@26748
   922
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   923
\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 $~\neg P(y)$.
krauss@26748
   924
\end{itemize}
krauss@26748
   925
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   926
krauss@26748
   927
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   928
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   929
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   930
  shows "P x"
krauss@26748
   931
proof -
krauss@26748
   932
  obtain n where "n = V x" by auto
krauss@26748
   933
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   934
  proof (induct n rule: infinite_descent0)
krauss@26748
   935
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   936
    with A0 show "P x" by auto
krauss@26748
   937
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   938
    case (smaller n)
krauss@26748
   939
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   940
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   941
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   942
    then show ?case by auto
krauss@26748
   943
  qed
krauss@26748
   944
  ultimately show "P x" by auto
krauss@26748
   945
qed
krauss@26748
   946
krauss@26748
   947
text{* Again, without explicit base case: *}
krauss@26748
   948
lemma infinite_descent_measure:
krauss@26748
   949
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   950
proof -
krauss@26748
   951
  from assms obtain n where "n = V x" by auto
krauss@26748
   952
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   953
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   954
    fix x assume "\<not> P x"
krauss@26748
   955
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
   956
  qed
krauss@26748
   957
  ultimately show "P x" by auto
krauss@26748
   958
qed
krauss@26748
   959
paulson@14267
   960
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   961
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   962
lemma less_mono_imp_le_mono:
nipkow@24438
   963
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   964
by (simp add: order_le_less) (blast)
nipkow@24438
   965
paulson@14267
   966
paulson@14267
   967
text {* non-strict, in 1st argument *}
paulson@14267
   968
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   969
by (rule add_right_mono)
paulson@14267
   970
paulson@14267
   971
text {* non-strict, in both arguments *}
paulson@14267
   972
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   973
by (rule add_mono)
paulson@14267
   974
paulson@14267
   975
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   976
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   977
paulson@14267
   978
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   979
by (simp add: add_commute, rule le_add2)
berghofe@13449
   980
berghofe@13449
   981
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   982
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   983
berghofe@13449
   984
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   985
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   986
paulson@14267
   987
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   988
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   989
paulson@14267
   990
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   991
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   992
paulson@14267
   993
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   994
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   995
berghofe@13449
   996
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   997
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   998
berghofe@13449
   999
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1000
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1001
berghofe@13449
  1002
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1003
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1004
apply (simp_all add: le_add1)
nipkow@24438
  1005
done
berghofe@13449
  1006
berghofe@13449
  1007
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1008
apply (rule notI)
wenzelm@26335
  1009
apply (drule add_lessD1)
wenzelm@26335
  1010
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1011
done
berghofe@13449
  1012
berghofe@13449
  1013
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
  1014
by (simp add: add_commute)
berghofe@13449
  1015
paulson@14267
  1016
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1017
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1018
apply (simp_all add: le_add1)
nipkow@24438
  1019
done
berghofe@13449
  1020
paulson@14267
  1021
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
  1022
apply (simp add: add_commute)
nipkow@24438
  1023
apply (erule add_leD1)
nipkow@24438
  1024
done
berghofe@13449
  1025
paulson@14267
  1026
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1027
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1028
berghofe@13449
  1029
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
  1030
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1031
by (force simp del: add_Suc_right
berghofe@13449
  1032
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
  1033
berghofe@13449
  1034
haftmann@26072
  1035
subsubsection {* More results about difference *}
berghofe@13449
  1036
berghofe@13449
  1037
text {* Addition is the inverse of subtraction:
paulson@14267
  1038
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1039
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1040
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1041
paulson@14267
  1042
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1043
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1044
paulson@14267
  1045
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
  1046
by (simp add: add_commute)
berghofe@13449
  1047
paulson@14267
  1048
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1049
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1050
berghofe@13449
  1051
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1052
apply (induct m n rule: diff_induct)
nipkow@24438
  1053
apply (erule_tac [3] less_SucE)
nipkow@24438
  1054
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1055
done
berghofe@13449
  1056
paulson@14267
  1057
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1058
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1059
haftmann@26072
  1060
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1061
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1062
berghofe@13449
  1063
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1064
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1065
berghofe@13449
  1066
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1067
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1068
paulson@14267
  1069
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1070
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1071
paulson@14267
  1072
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1073
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1074
paulson@14267
  1075
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1076
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1077
paulson@14267
  1078
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1079
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1080
paulson@14267
  1081
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1082
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1083
berghofe@13449
  1084
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1085
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1086
wenzelm@22718
  1087
lemma less_imp_add_positive:
wenzelm@22718
  1088
  assumes "i < j"
wenzelm@22718
  1089
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1090
proof
wenzelm@22718
  1091
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1092
    by (simp add: order_less_imp_le)
wenzelm@22718
  1093
qed
wenzelm@9436
  1094
haftmann@26072
  1095
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1096
lemma nat_minus_add_max:
haftmann@26072
  1097
  fixes n m :: nat
haftmann@26072
  1098
  shows "n - m + m = max n m"
haftmann@26072
  1099
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1100
haftmann@26072
  1101
lemma nat_diff_split:
haftmann@26072
  1102
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1103
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1104
by (cases "a < b")
haftmann@26072
  1105
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1106
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1107
haftmann@26072
  1108
lemma nat_diff_split_asm:
haftmann@26072
  1109
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1110
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1111
by (auto split: nat_diff_split)
berghofe@13449
  1112
huffman@47255
  1113
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1114
  by simp
huffman@47255
  1115
huffman@47255
  1116
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1117
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1118
huffman@47255
  1119
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1120
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1121
huffman@47255
  1122
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1123
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1124
huffman@47255
  1125
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1126
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1127
huffman@47255
  1128
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1129
  by (fact Let_def)
huffman@47255
  1130
berghofe@13449
  1131
haftmann@26072
  1132
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1133
paulson@14267
  1134
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1135
by (simp add: mult_right_mono)
berghofe@13449
  1136
paulson@14267
  1137
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1138
by (simp add: mult_left_mono)
berghofe@13449
  1139
paulson@14267
  1140
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1141
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1142
by (simp add: mult_mono)
berghofe@13449
  1143
berghofe@13449
  1144
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1145
by (simp add: mult_strict_right_mono)
berghofe@13449
  1146
paulson@14266
  1147
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1148
      there are no negative numbers.*}
paulson@14266
  1149
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1150
  apply (induct m)
wenzelm@22718
  1151
   apply simp
wenzelm@22718
  1152
  apply (case_tac n)
wenzelm@22718
  1153
   apply simp_all
berghofe@13449
  1154
  done
berghofe@13449
  1155
huffman@30079
  1156
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1157
  apply (induct m)
wenzelm@22718
  1158
   apply simp
wenzelm@22718
  1159
  apply (case_tac n)
wenzelm@22718
  1160
   apply simp_all
berghofe@13449
  1161
  done
berghofe@13449
  1162
paulson@14341
  1163
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1164
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1165
  apply (cases k, auto)
berghofe@13449
  1166
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1167
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1168
  done
berghofe@13449
  1169
berghofe@13449
  1170
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1171
by (simp add: mult_commute [of k])
berghofe@13449
  1172
paulson@14267
  1173
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1174
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1175
paulson@14267
  1176
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1177
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1178
berghofe@13449
  1179
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1180
by (subst mult_less_cancel1) simp
berghofe@13449
  1181
paulson@14267
  1182
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1183
by (subst mult_le_cancel1) simp
berghofe@13449
  1184
haftmann@26072
  1185
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1186
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1187
haftmann@26072
  1188
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1189
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1190
berghofe@13449
  1191
text {* Lemma for @{text gcd} *}
huffman@30128
  1192
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1193
  apply (drule sym)
berghofe@13449
  1194
  apply (rule disjCI)
berghofe@13449
  1195
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1196
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1197
    apply (auto)
berghofe@13449
  1198
  done
wenzelm@9436
  1199
haftmann@26072
  1200
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1201
haftmann@26072
  1202
instantiation nat :: distrib_lattice
haftmann@26072
  1203
begin
haftmann@24995
  1204
haftmann@26072
  1205
definition
haftmann@26072
  1206
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1207
haftmann@26072
  1208
definition
haftmann@26072
  1209
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1210
haftmann@26072
  1211
instance by intro_classes
haftmann@26072
  1212
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1213
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1214
haftmann@26072
  1215
end
haftmann@24995
  1216
haftmann@24995
  1217
haftmann@30954
  1218
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1219
haftmann@30971
  1220
text {*
haftmann@30971
  1221
  We use the same logical constant for the power operations on
haftmann@30971
  1222
  functions and relations, in order to share the same syntax.
haftmann@30971
  1223
*}
haftmann@30971
  1224
haftmann@45965
  1225
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1226
haftmann@45965
  1227
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1228
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1229
haftmann@30971
  1230
notation (latex output)
haftmann@30971
  1231
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1232
haftmann@30971
  1233
notation (HTML output)
haftmann@30971
  1234
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1235
haftmann@30971
  1236
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1237
haftmann@30971
  1238
overloading
haftmann@30971
  1239
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1240
begin
haftmann@30954
  1241
haftmann@30954
  1242
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1243
  "funpow 0 f = id"
haftmann@44325
  1244
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1245
haftmann@30971
  1246
end
haftmann@30971
  1247
haftmann@30971
  1248
text {* for code generation *}
haftmann@30971
  1249
haftmann@30971
  1250
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1251
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1252
haftmann@30971
  1253
lemma [code]:
haftmann@37430
  1254
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1255
  "funpow 0 f = id"
haftmann@37430
  1256
  by (simp_all add: funpow_code_def)
haftmann@30971
  1257
wenzelm@36176
  1258
hide_const (open) funpow
haftmann@30954
  1259
haftmann@30954
  1260
lemma funpow_add:
haftmann@30971
  1261
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1262
  by (induct m) simp_all
haftmann@30954
  1263
haftmann@37430
  1264
lemma funpow_mult:
haftmann@37430
  1265
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1266
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1267
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1268
haftmann@30954
  1269
lemma funpow_swap1:
haftmann@30971
  1270
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1271
proof -
haftmann@30971
  1272
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1273
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1274
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1275
  finally show ?thesis .
haftmann@30954
  1276
qed
haftmann@30954
  1277
haftmann@38621
  1278
lemma comp_funpow:
haftmann@38621
  1279
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1280
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1281
  by (induct n) simp_all
haftmann@30954
  1282
haftmann@38621
  1283
nipkow@45833
  1284
subsection {* Kleene iteration *}
nipkow@45833
  1285
nipkow@45833
  1286
lemma Kleene_iter_lpfp: assumes "mono f" and "f p \<le> p" shows "(f^^k) bot \<le> p"
nipkow@45833
  1287
proof(induction k)
nipkow@45833
  1288
  case 0 show ?case by simp
nipkow@45833
  1289
next
nipkow@45833
  1290
  case Suc
nipkow@45833
  1291
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1292
  show ?case by simp
nipkow@45833
  1293
qed
nipkow@45833
  1294
nipkow@45833
  1295
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1296
shows "lfp f = (f^^k) bot"
nipkow@45833
  1297
proof(rule antisym)
nipkow@45833
  1298
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1299
  proof(rule lfp_lowerbound)
nipkow@45833
  1300
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1301
  qed
nipkow@45833
  1302
next
nipkow@45833
  1303
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1304
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1305
qed
nipkow@45833
  1306
nipkow@45833
  1307
haftmann@38621
  1308
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1309
haftmann@24196
  1310
context semiring_1
haftmann@24196
  1311
begin
haftmann@24196
  1312
haftmann@38621
  1313
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1314
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1315
haftmann@38621
  1316
lemma of_nat_simps [simp]:
haftmann@38621
  1317
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1318
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1319
  by (simp_all add: of_nat_def)
haftmann@25193
  1320
haftmann@25193
  1321
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1322
  by (simp add: of_nat_def)
haftmann@25193
  1323
haftmann@25193
  1324
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1325
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1326
haftmann@25193
  1327
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1328
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1329
haftmann@28514
  1330
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1331
  "of_nat_aux inc 0 i = i"
haftmann@44325
  1332
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1333
haftmann@30966
  1334
lemma of_nat_code:
haftmann@28514
  1335
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1336
proof (induct n)
haftmann@28514
  1337
  case 0 then show ?case by simp
haftmann@28514
  1338
next
haftmann@28514
  1339
  case (Suc n)
haftmann@28514
  1340
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
haftmann@28514
  1341
    by (induct n) simp_all
haftmann@28514
  1342
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
haftmann@28514
  1343
    by simp
haftmann@28514
  1344
  with Suc show ?case by (simp add: add_commute)
haftmann@28514
  1345
qed
haftmann@30966
  1346
haftmann@24196
  1347
end
haftmann@24196
  1348
bulwahn@45231
  1349
declare of_nat_code [code]
haftmann@30966
  1350
haftmann@26072
  1351
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1352
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1353
haftmann@26072
  1354
class semiring_char_0 = semiring_1 +
haftmann@38621
  1355
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1356
begin
haftmann@26072
  1357
haftmann@38621
  1358
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1359
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1360
haftmann@26072
  1361
text{*Special cases where either operand is zero*}
haftmann@26072
  1362
blanchet@35828
  1363
lemma of_nat_0_eq_iff [simp, no_atp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1364
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1365
blanchet@35828
  1366
lemma of_nat_eq_0_iff [simp, no_atp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1367
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1368
haftmann@26072
  1369
end
haftmann@26072
  1370
haftmann@35028
  1371
context linordered_semidom
haftmann@25193
  1372
begin
haftmann@25193
  1373
huffman@47489
  1374
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1375
  by (induct n) simp_all
haftmann@25193
  1376
huffman@47489
  1377
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1378
  by (simp add: not_less)
haftmann@25193
  1379
haftmann@25193
  1380
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
huffman@47489
  1381
  by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
haftmann@25193
  1382
haftmann@26072
  1383
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1384
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1385
huffman@47489
  1386
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1387
  by simp
huffman@47489
  1388
huffman@47489
  1389
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1390
  by simp
huffman@47489
  1391
haftmann@35028
  1392
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1393
haftmann@38621
  1394
subclass semiring_char_0 proof
haftmann@38621
  1395
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1396
haftmann@25193
  1397
text{*Special cases where either operand is zero*}
haftmann@25193
  1398
blanchet@35828
  1399
lemma of_nat_le_0_iff [simp, no_atp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1400
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1401
haftmann@26072
  1402
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1403
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1404
haftmann@26072
  1405
end
haftmann@26072
  1406
haftmann@26072
  1407
context ring_1
haftmann@26072
  1408
begin
haftmann@26072
  1409
haftmann@26072
  1410
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1411
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1412
haftmann@26072
  1413
end
haftmann@26072
  1414
haftmann@35028
  1415
context linordered_idom
haftmann@26072
  1416
begin
haftmann@26072
  1417
haftmann@26072
  1418
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1419
  unfolding abs_if by auto
haftmann@26072
  1420
haftmann@25193
  1421
end
haftmann@25193
  1422
haftmann@25193
  1423
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1424
  by (induct n) simp_all
haftmann@25193
  1425
haftmann@25193
  1426
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1427
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1428
haftmann@25193
  1429
haftmann@26149
  1430
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1431
haftmann@26072
  1432
context semiring_1
haftmann@25193
  1433
begin
haftmann@25193
  1434
haftmann@37767
  1435
definition Nats  :: "'a set" where
haftmann@37767
  1436
  "Nats = range of_nat"
haftmann@26072
  1437
haftmann@26072
  1438
notation (xsymbols)
haftmann@26072
  1439
  Nats  ("\<nat>")
haftmann@25193
  1440
haftmann@26072
  1441
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1442
  by (simp add: Nats_def)
haftmann@26072
  1443
haftmann@26072
  1444
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1445
apply (simp add: Nats_def)
haftmann@26072
  1446
apply (rule range_eqI)
haftmann@26072
  1447
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1448
done
haftmann@25193
  1449
haftmann@26072
  1450
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1451
apply (simp add: Nats_def)
haftmann@26072
  1452
apply (rule range_eqI)
haftmann@26072
  1453
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1454
done
haftmann@25193
  1455
haftmann@26072
  1456
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1457
apply (auto simp add: Nats_def)
haftmann@26072
  1458
apply (rule range_eqI)
haftmann@26072
  1459
apply (rule of_nat_add [symmetric])
haftmann@26072
  1460
done
haftmann@26072
  1461
haftmann@26072
  1462
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1463
apply (auto simp add: Nats_def)
haftmann@26072
  1464
apply (rule range_eqI)
haftmann@26072
  1465
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1466
done
haftmann@25193
  1467
huffman@35633
  1468
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1469
  assumes "x \<in> \<nat>"
huffman@35633
  1470
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1471
  unfolding Nats_def
huffman@35633
  1472
proof -
huffman@35633
  1473
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1474
  then obtain n where "x = of_nat n" ..
huffman@35633
  1475
  then show thesis ..
huffman@35633
  1476
qed
huffman@35633
  1477
huffman@35633
  1478
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1479
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1480
  by (rule Nats_cases) auto
huffman@35633
  1481
haftmann@25193
  1482
end
haftmann@25193
  1483
haftmann@25193
  1484
wenzelm@21243
  1485
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1486
haftmann@22845
  1487
lemma subst_equals:
haftmann@22845
  1488
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1489
  shows "u = s"
haftmann@22845
  1490
  using 2 1 by (rule trans)
haftmann@22845
  1491
haftmann@30686
  1492
setup Arith_Data.setup
haftmann@30686
  1493
wenzelm@48891
  1494
ML_file "Tools/nat_arith.ML"
huffman@48559
  1495
huffman@48559
  1496
simproc_setup nateq_cancel_sums
huffman@48559
  1497
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
huffman@48560
  1498
  {* fn phi => fn ss => try Nat_Arith.cancel_eq_conv *}
huffman@48559
  1499
huffman@48559
  1500
simproc_setup natless_cancel_sums
huffman@48559
  1501
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
huffman@48560
  1502
  {* fn phi => fn ss => try Nat_Arith.cancel_less_conv *}
huffman@48559
  1503
huffman@48559
  1504
simproc_setup natle_cancel_sums
huffman@48559
  1505
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
huffman@48560
  1506
  {* fn phi => fn ss => try Nat_Arith.cancel_le_conv *}
huffman@48559
  1507
huffman@48559
  1508
simproc_setup natdiff_cancel_sums
huffman@48559
  1509
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
huffman@48560
  1510
  {* fn phi => fn ss => try Nat_Arith.cancel_diff_conv *}
wenzelm@24091
  1511
wenzelm@48891
  1512
ML_file "Tools/lin_arith.ML"
haftmann@31100
  1513
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1514
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1515
wenzelm@43595
  1516
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@43595
  1517
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1518
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1519
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1520
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1521
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1522
solver all the time rather than add the additional check. *)
wenzelm@43595
  1523
wenzelm@43595
  1524
wenzelm@21243
  1525
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1526
nipkow@27625
  1527
context order
nipkow@27625
  1528
begin
nipkow@27625
  1529
nipkow@27625
  1530
lemma lift_Suc_mono_le:
krauss@27627
  1531
  assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
krauss@27627
  1532
  shows "f n \<le> f n'"
krauss@27627
  1533
proof (cases "n < n'")
krauss@27627
  1534
  case True
krauss@27627
  1535
  thus ?thesis
krauss@27627
  1536
    by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
krauss@27627
  1537
qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}
nipkow@27625
  1538
nipkow@27625
  1539
lemma lift_Suc_mono_less:
krauss@27627
  1540
  assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
krauss@27627
  1541
  shows "f n < f n'"
krauss@27627
  1542
using `n < n'`
krauss@27627
  1543
by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
nipkow@27625
  1544
nipkow@27789
  1545
lemma lift_Suc_mono_less_iff:
nipkow@27789
  1546
  "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
nipkow@27789
  1547
by(blast intro: less_asym' lift_Suc_mono_less[of f]
nipkow@27789
  1548
         dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
nipkow@27789
  1549
nipkow@27625
  1550
end
nipkow@27625
  1551
nipkow@29879
  1552
lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1553
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1554
nipkow@27789
  1555
lemma mono_nat_linear_lb:
nipkow@27789
  1556
  "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
nipkow@27789
  1557
apply(induct_tac k)
nipkow@27789
  1558
 apply simp
nipkow@27789
  1559
apply(erule_tac x="m+n" in meta_allE)
huffman@30079
  1560
apply(erule_tac x="Suc(m+n)" in meta_allE)
nipkow@27789
  1561
apply simp
nipkow@27789
  1562
done
nipkow@27789
  1563
nipkow@27789
  1564
wenzelm@21243
  1565
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1566
wenzelm@21243
  1567
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1568
by arith
wenzelm@21243
  1569
wenzelm@21243
  1570
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1571
by arith
wenzelm@21243
  1572
wenzelm@21243
  1573
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1574
by arith
wenzelm@21243
  1575
wenzelm@21243
  1576
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1577
by arith
wenzelm@21243
  1578
wenzelm@21243
  1579
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1580
by arith
wenzelm@21243
  1581
wenzelm@21243
  1582
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1583
by arith
wenzelm@21243
  1584
wenzelm@21243
  1585
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1586
  second premise n\<le>m*)
wenzelm@21243
  1587
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1588
by arith
wenzelm@21243
  1589
haftmann@26072
  1590
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1591
wenzelm@21243
  1592
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1593
by (simp split add: nat_diff_split)
wenzelm@21243
  1594
wenzelm@36176
  1595
hide_fact (open) diff_diff_eq
haftmann@35064
  1596
wenzelm@21243
  1597
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1598
by (auto split add: nat_diff_split)
wenzelm@21243
  1599
wenzelm@21243
  1600
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1601
by (auto split add: nat_diff_split)
wenzelm@21243
  1602
wenzelm@21243
  1603
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1604
by (auto split add: nat_diff_split)
wenzelm@21243
  1605
wenzelm@21243
  1606
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1607
wenzelm@21243
  1608
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1609
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1610
by (simp split add: nat_diff_split)
wenzelm@21243
  1611
wenzelm@21243
  1612
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1613
by (simp split add: nat_diff_split)
wenzelm@21243
  1614
wenzelm@21243
  1615
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1616
by (simp split add: nat_diff_split)
wenzelm@21243
  1617
wenzelm@21243
  1618
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1619
by (simp split add: nat_diff_split)
wenzelm@21243
  1620
bulwahn@26143
  1621
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1622
by auto
bulwahn@26143
  1623
bulwahn@26143
  1624
lemma inj_on_diff_nat: 
bulwahn@26143
  1625
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1626
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1627
proof (rule inj_onI)
bulwahn@26143
  1628
  fix x y
bulwahn@26143
  1629
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1630
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1631
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1632
qed
bulwahn@26143
  1633
haftmann@26072
  1634
text{*Rewriting to pull differences out*}
haftmann@26072
  1635
haftmann@26072
  1636
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1637
by arith
haftmann@26072
  1638
haftmann@26072
  1639
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1640
by arith
haftmann@26072
  1641
haftmann@26072
  1642
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1643
by arith
haftmann@26072
  1644
noschinl@45933
  1645
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1646
by simp
noschinl@45933
  1647
bulwahn@46350
  1648
(*The others are
bulwahn@46350
  1649
      i - j - k = i - (j + k),
bulwahn@46350
  1650
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1651
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1652
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1653
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
bulwahn@46350
  1654
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1655
bulwahn@46350
  1656
text{*At present we prove no analogue of @{text not_less_Least} or @{text
bulwahn@46350
  1657
Least_Suc}, since there appears to be no need.*}
bulwahn@46350
  1658
wenzelm@21243
  1659
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1660
wenzelm@21243
  1661
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1662
by (cases m) auto
wenzelm@21243
  1663
wenzelm@21243
  1664
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1665
by (cases m) auto
wenzelm@21243
  1666
wenzelm@21243
  1667
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1668
by (cases m) auto
wenzelm@21243
  1669
krauss@23001
  1670
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1671
krauss@23001
  1672
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1673
  assumes less: "i <= j"
krauss@23001
  1674
  assumes base: "P j"
krauss@23001
  1675
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1676
  shows "P i"
krauss@23001
  1677
  using less
krauss@23001
  1678
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1679
  case (0 i)
krauss@23001
  1680
  hence "i = j" by simp
krauss@23001
  1681
  with base show ?case by simp
krauss@23001
  1682
next
krauss@23001
  1683
  case (Suc d i)
krauss@23001
  1684
  hence "i < j" "P (Suc i)"
krauss@23001
  1685
    by simp_all
krauss@23001
  1686
  thus "P i" by (rule step)
krauss@23001
  1687
qed
krauss@23001
  1688
krauss@23001
  1689
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1690
  assumes less: "i < j"
krauss@23001
  1691
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1692
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1693
  shows "P i"
krauss@23001
  1694
  using less
krauss@23001
  1695
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1696
  case (0 i)
krauss@23001
  1697
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1698
  with base show ?case by simp
krauss@23001
  1699
next
krauss@23001
  1700
  case (Suc d i)
krauss@23001
  1701
  hence "i < j" "P (Suc i)"
krauss@23001
  1702
    by simp_all
krauss@23001
  1703
  thus "P i" by (rule step)
krauss@23001
  1704
qed
krauss@23001
  1705
krauss@23001
  1706
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1707
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1708
krauss@23001
  1709
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1710
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1711
bulwahn@46351
  1712
text {* Further induction rule similar to @{thm inc_induct} *}
nipkow@27625
  1713
bulwahn@46351
  1714
lemma dec_induct[consumes 1, case_names base step]:
bulwahn@46351
  1715
  "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
bulwahn@46351
  1716
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
bulwahn@46351
  1717
bulwahn@46351
  1718
 
haftmann@33274
  1719
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1720
haftmann@33274
  1721
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1722
unfolding dvd_def by simp
haftmann@33274
  1723
haftmann@33274
  1724
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1725
by (simp add: dvd_def)
haftmann@33274
  1726
haftmann@33274
  1727
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1728
by (simp add: dvd_def)
haftmann@33274
  1729
nipkow@33657
  1730
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1731
  unfolding dvd_def
huffman@35216
  1732
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
haftmann@33274
  1733
haftmann@33274
  1734
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1735
haftmann@33274
  1736
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1737
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1738
haftmann@33274
  1739
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1740
unfolding dvd_def
haftmann@33274
  1741
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1742
haftmann@33274
  1743
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1744
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1745
  apply (blast intro: dvd_add)
haftmann@33274
  1746
  done
haftmann@33274
  1747
haftmann@33274
  1748
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1749
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1750
haftmann@33274
  1751
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
haftmann@33274
  1752
  apply (rule iffI)
haftmann@33274
  1753
   apply (erule_tac [2] dvd_add)
haftmann@33274
  1754
   apply (rule_tac [2] dvd_refl)
haftmann@33274
  1755
  apply (subgoal_tac "n = (n+k) -k")
haftmann@33274
  1756
   prefer 2 apply simp
haftmann@33274
  1757
  apply (erule ssubst)
haftmann@33274
  1758
  apply (erule dvd_diff_nat)
haftmann@33274
  1759
  apply (rule dvd_refl)
haftmann@33274
  1760
  done
haftmann@33274
  1761
haftmann@33274
  1762
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1763
  unfolding dvd_def
haftmann@33274
  1764
  apply (erule exE)
haftmann@33274
  1765
  apply (simp add: mult_ac)
haftmann@33274
  1766
  done
haftmann@33274
  1767
haftmann@33274
  1768
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1769
  apply auto
haftmann@33274
  1770
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1771
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1772
  done
haftmann@33274
  1773
haftmann@33274
  1774
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@33274
  1775
  apply (subst mult_commute)
haftmann@33274
  1776
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1777
  done
haftmann@33274
  1778
haftmann@33274
  1779
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1780
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1781
haftmann@33274
  1782
lemma nat_dvd_not_less:
haftmann@33274
  1783
  fixes m n :: nat
haftmann@33274
  1784
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1785
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1786
haftmann@33274
  1787
haftmann@44817
  1788
subsection {* aliasses *}
haftmann@44817
  1789
haftmann@44817
  1790
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@44817
  1791
  by simp
haftmann@44817
  1792
 
haftmann@44817
  1793
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@44817
  1794
  by simp
haftmann@44817
  1795
haftmann@44817
  1796
haftmann@26072
  1797
subsection {* size of a datatype value *}
haftmann@25193
  1798
haftmann@29608
  1799
class size =
krauss@26748
  1800
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1801
haftmann@33364
  1802
haftmann@33364
  1803
subsection {* code module namespace *}
haftmann@33364
  1804
haftmann@33364
  1805
code_modulename SML
haftmann@33364
  1806
  Nat Arith
haftmann@33364
  1807
haftmann@33364
  1808
code_modulename OCaml
haftmann@33364
  1809
  Nat Arith
haftmann@33364
  1810
haftmann@33364
  1811
code_modulename Haskell
haftmann@33364
  1812
  Nat Arith
haftmann@33364
  1813
huffman@47108
  1814
hide_const (open) of_nat_aux
huffman@47108
  1815
haftmann@25193
  1816
end