src/HOL/Nat.thy
author haftmann
Wed Sep 07 23:07:16 2011 +0200 (2011-09-07)
changeset 44817 b63e445c8f6d
parent 44325 84696670feb1
child 44848 f4d0b060c7ca
permissions -rw-r--r--
lemmas about +, *, min, max on nat
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(*  Title:      HOL/Nat.thy
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div and mod, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Inductive Typedef Fun Fields
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uses
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  "~~/src/Tools/rat.ML"
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  "~~/src/Provers/Arith/cancel_sums.ML"
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  "Tools/arith_data.ML"
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  ("Tools/nat_arith.ML")
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  "~~/src/Provers/Arith/fast_lin_arith.ML"
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  ("Tools/lin_arith.ML")
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begin
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
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  -- {* the axiom of infinity in 2 parts *}
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  Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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inductive Nat :: "ind \<Rightarrow> bool" where
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  Zero_RepI: "Nat Zero_Rep"
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| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
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typedef (open Nat) nat = "{n. Nat n}"
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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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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, fastsimp)
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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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lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
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proof -
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  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   335
  proof (induct n arbitrary: m)
haftmann@26072
   336
    case 0 then show "m = 0" by simp
haftmann@26072
   337
  next
haftmann@26072
   338
    case (Suc n) then show "m = Suc n"
haftmann@26072
   339
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   340
  qed
haftmann@26072
   341
  then show ?thesis by auto
haftmann@26072
   342
qed
haftmann@26072
   343
haftmann@26072
   344
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@26072
   345
  by (simp add: mult_commute)
haftmann@26072
   346
haftmann@26072
   347
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   348
  by (subst mult_cancel1) simp
haftmann@26072
   349
haftmann@24995
   350
haftmann@24995
   351
subsection {* Orders on @{typ nat} *}
haftmann@24995
   352
haftmann@26072
   353
subsubsection {* Operation definition *}
haftmann@24995
   354
haftmann@26072
   355
instantiation nat :: linorder
haftmann@25510
   356
begin
haftmann@25510
   357
haftmann@26072
   358
primrec less_eq_nat where
haftmann@26072
   359
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   360
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   361
haftmann@28514
   362
declare less_eq_nat.simps [simp del]
haftmann@26072
   363
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
haftmann@26072
   364
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@26072
   365
haftmann@26072
   366
definition less_nat where
haftmann@28514
   367
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   368
haftmann@26072
   369
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   370
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   371
haftmann@26072
   372
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   373
  unfolding less_eq_Suc_le ..
haftmann@26072
   374
haftmann@26072
   375
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   376
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   377
haftmann@26072
   378
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   379
  by (simp add: less_eq_Suc_le)
haftmann@26072
   380
haftmann@26072
   381
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   382
  by simp
haftmann@26072
   383
haftmann@26072
   384
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   385
  by (simp add: less_eq_Suc_le)
haftmann@26072
   386
haftmann@26072
   387
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   388
  by (simp add: less_eq_Suc_le)
haftmann@26072
   389
haftmann@26072
   390
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   391
  by (induct m arbitrary: n)
haftmann@26072
   392
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   393
haftmann@26072
   394
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   395
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   396
haftmann@26072
   397
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   398
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   399
haftmann@26072
   400
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   401
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   402
wenzelm@26315
   403
instance
wenzelm@26315
   404
proof
haftmann@26072
   405
  fix n m :: nat
haftmann@27679
   406
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   407
  proof (induct n arbitrary: m)
haftmann@27679
   408
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   409
  next
haftmann@27679
   410
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   411
  qed
haftmann@26072
   412
next
haftmann@26072
   413
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   414
next
haftmann@26072
   415
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   416
  then show "n = m"
haftmann@26072
   417
    by (induct n arbitrary: m)
haftmann@26072
   418
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   419
next
haftmann@26072
   420
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   421
  then show "n \<le> q"
haftmann@26072
   422
  proof (induct n arbitrary: m q)
haftmann@26072
   423
    case 0 show ?case by simp
haftmann@26072
   424
  next
haftmann@26072
   425
    case (Suc n) then show ?case
haftmann@26072
   426
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   427
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   428
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   429
  qed
haftmann@26072
   430
next
haftmann@26072
   431
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   432
    by (induct n arbitrary: m)
haftmann@26072
   433
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   434
qed
haftmann@25510
   435
haftmann@25510
   436
end
berghofe@13449
   437
haftmann@29652
   438
instantiation nat :: bot
haftmann@29652
   439
begin
haftmann@29652
   440
haftmann@29652
   441
definition bot_nat :: nat where
haftmann@29652
   442
  "bot_nat = 0"
haftmann@29652
   443
haftmann@29652
   444
instance proof
haftmann@29652
   445
qed (simp add: bot_nat_def)
haftmann@29652
   446
haftmann@29652
   447
end
haftmann@29652
   448
haftmann@26072
   449
subsubsection {* Introduction properties *}
berghofe@13449
   450
haftmann@26072
   451
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   452
  by (simp add: less_Suc_eq_le)
berghofe@13449
   453
haftmann@26072
   454
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   455
  by (simp add: less_Suc_eq_le)
berghofe@13449
   456
berghofe@13449
   457
berghofe@13449
   458
subsubsection {* Elimination properties *}
berghofe@13449
   459
berghofe@13449
   460
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   461
  by (rule order_less_irrefl)
berghofe@13449
   462
wenzelm@26335
   463
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   464
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   465
paulson@14267
   466
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   467
  by (rule less_imp_neq)
berghofe@13449
   468
wenzelm@26335
   469
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   470
  by (rule notE, rule less_not_refl)
berghofe@13449
   471
berghofe@13449
   472
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   473
  by (rule notE) (rule not_less0)
berghofe@13449
   474
berghofe@13449
   475
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   476
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   477
huffman@30079
   478
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   479
  by (simp add: less_Suc_eq)
berghofe@13449
   480
blanchet@35828
   481
lemma less_one [iff, no_atp]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   482
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   483
berghofe@13449
   484
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   485
  by simp
berghofe@13449
   486
nipkow@14302
   487
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   488
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   489
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   490
paulson@14267
   491
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   492
  by (rule linorder_neq_iff)
berghofe@13449
   493
berghofe@13449
   494
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   495
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   496
  shows "P n m"
berghofe@13449
   497
  apply (rule less_linear [THEN disjE])
berghofe@13449
   498
  apply (erule_tac [2] disjE)
berghofe@13449
   499
  apply (erule lessCase)
berghofe@13449
   500
  apply (erule sym [THEN eqCase])
berghofe@13449
   501
  apply (erule major)
berghofe@13449
   502
  done
berghofe@13449
   503
berghofe@13449
   504
berghofe@13449
   505
subsubsection {* Inductive (?) properties *}
berghofe@13449
   506
paulson@14267
   507
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   508
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   509
haftmann@26072
   510
lemma lessE:
haftmann@26072
   511
  assumes major: "i < k"
haftmann@26072
   512
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   513
  shows P
haftmann@26072
   514
proof -
haftmann@26072
   515
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   516
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   517
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   518
    by (clarsimp simp add: less_le)
haftmann@26072
   519
  with p1 p2 show P by auto
haftmann@26072
   520
qed
haftmann@26072
   521
haftmann@26072
   522
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   523
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   524
  apply (rule major [THEN lessE])
haftmann@26072
   525
  apply (rule eq, blast)
haftmann@26072
   526
  apply (rule less, blast)
berghofe@13449
   527
  done
berghofe@13449
   528
berghofe@13449
   529
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   530
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   531
  apply (rule major [THEN lessE])
berghofe@13449
   532
  apply (erule lessI [THEN minor])
paulson@14208
   533
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   534
  done
berghofe@13449
   535
berghofe@13449
   536
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   537
  by simp
berghofe@13449
   538
berghofe@13449
   539
lemma less_trans_Suc:
berghofe@13449
   540
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   541
  apply (induct k, simp_all)
berghofe@13449
   542
  apply (insert le)
berghofe@13449
   543
  apply (simp add: less_Suc_eq)
berghofe@13449
   544
  apply (blast dest: Suc_lessD)
berghofe@13449
   545
  done
berghofe@13449
   546
berghofe@13449
   547
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   548
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   549
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   550
haftmann@26072
   551
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   552
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   553
haftmann@24995
   554
text {* Properties of "less than or equal" *}
berghofe@13449
   555
paulson@14267
   556
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   557
  unfolding less_Suc_eq_le .
berghofe@13449
   558
paulson@14267
   559
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   560
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   561
paulson@14267
   562
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   563
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   564
paulson@14267
   565
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   566
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   567
paulson@14267
   568
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   569
  unfolding Suc_le_eq .
berghofe@13449
   570
berghofe@13449
   571
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   572
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   573
  unfolding Suc_le_eq .
berghofe@13449
   574
wenzelm@26315
   575
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   576
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   577
paulson@14267
   578
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   579
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   580
berghofe@13449
   581
paulson@14267
   582
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   583
paulson@14267
   584
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   585
  unfolding le_less .
berghofe@13449
   586
paulson@14267
   587
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   588
  by (rule le_less)
berghofe@13449
   589
wenzelm@22718
   590
text {* Useful with @{text blast}. *}
paulson@14267
   591
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   592
  by auto
berghofe@13449
   593
paulson@14267
   594
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   595
  by simp
berghofe@13449
   596
paulson@14267
   597
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   598
  by (rule order_trans)
berghofe@13449
   599
nipkow@33657
   600
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   601
  by (rule antisym)
berghofe@13449
   602
paulson@14267
   603
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   604
  by (rule less_le)
berghofe@13449
   605
paulson@14267
   606
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   607
  unfolding less_le ..
berghofe@13449
   608
haftmann@26072
   609
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   610
  by (rule linear)
paulson@14341
   611
wenzelm@22718
   612
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   613
haftmann@26072
   614
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   615
  unfolding less_Suc_eq_le by auto
berghofe@13449
   616
haftmann@26072
   617
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   618
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   619
berghofe@13449
   620
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   621
wenzelm@22718
   622
text {* These two rules ease the use of primitive recursion.
paulson@14341
   623
NOTE USE OF @{text "=="} *}
berghofe@13449
   624
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   625
by simp
berghofe@13449
   626
berghofe@13449
   627
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   628
by simp
berghofe@13449
   629
paulson@14267
   630
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   631
by (cases n) simp_all
nipkow@25162
   632
nipkow@25162
   633
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   634
by (cases n) simp_all
berghofe@13449
   635
wenzelm@22718
   636
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   637
by (cases n) simp_all
berghofe@13449
   638
nipkow@25162
   639
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   640
by (cases n) simp_all
nipkow@25140
   641
berghofe@13449
   642
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   643
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   644
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   645
paulson@14267
   646
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   647
by (fast intro: not0_implies_Suc)
berghofe@13449
   648
blanchet@35828
   649
lemma not_gr0 [iff,no_atp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   650
using neq0_conv by blast
berghofe@13449
   651
paulson@14267
   652
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   653
by (induct m') simp_all
berghofe@13449
   654
berghofe@13449
   655
text {* Useful in certain inductive arguments *}
paulson@14267
   656
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   657
by (cases m) simp_all
berghofe@13449
   658
berghofe@13449
   659
haftmann@26072
   660
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   661
haftmann@26072
   662
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   663
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   664
huffman@30128
   665
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   666
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   667
paulson@14331
   668
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   669
by (induct k) simp_all
berghofe@13449
   670
paulson@14331
   671
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   672
by (induct k) simp_all
berghofe@13449
   673
nipkow@25162
   674
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   675
by(auto dest:gr0_implies_Suc)
berghofe@13449
   676
paulson@14341
   677
text {* strict, in 1st argument *}
paulson@14341
   678
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   679
by (induct k) simp_all
paulson@14341
   680
paulson@14341
   681
text {* strict, in both arguments *}
paulson@14341
   682
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   683
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   684
  apply (induct j, simp_all)
paulson@14341
   685
  done
paulson@14341
   686
paulson@14341
   687
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   688
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   689
  apply (induct n)
paulson@14341
   690
  apply (simp_all add: order_le_less)
wenzelm@22718
   691
  apply (blast elim!: less_SucE
haftmann@35047
   692
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   693
  done
paulson@14341
   694
paulson@14341
   695
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   696
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   697
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   698
apply (induct_tac m)
nipkow@25134
   699
apply (simp_all add: add_less_mono)
nipkow@25134
   700
done
paulson@14341
   701
nipkow@14740
   702
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   703
instance nat :: linordered_semidom
paulson@14341
   704
proof
paulson@14341
   705
  fix i j k :: nat
paulson@14348
   706
  show "0 < (1::nat)" by simp
paulson@14267
   707
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   708
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   709
qed
paulson@14267
   710
nipkow@30056
   711
instance nat :: no_zero_divisors
nipkow@30056
   712
proof
nipkow@30056
   713
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
nipkow@30056
   714
qed
nipkow@30056
   715
haftmann@44817
   716
haftmann@44817
   717
subsubsection {* @{term min} and @{term max} *}
haftmann@44817
   718
haftmann@44817
   719
lemma mono_Suc: "mono Suc"
haftmann@44817
   720
by (rule monoI) simp
haftmann@44817
   721
haftmann@44817
   722
lemma min_0L [simp]: "min 0 n = (0::nat)"
haftmann@44817
   723
by (rule min_leastL) simp
haftmann@44817
   724
haftmann@44817
   725
lemma min_0R [simp]: "min n 0 = (0::nat)"
haftmann@44817
   726
by (rule min_leastR) simp
haftmann@44817
   727
haftmann@44817
   728
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   729
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   730
haftmann@44817
   731
lemma min_Suc1:
haftmann@44817
   732
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   733
by (simp split: nat.split)
haftmann@44817
   734
haftmann@44817
   735
lemma min_Suc2:
haftmann@44817
   736
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   737
by (simp split: nat.split)
haftmann@44817
   738
haftmann@44817
   739
lemma max_0L [simp]: "max 0 n = (n::nat)"
haftmann@44817
   740
by (rule max_leastL) simp
haftmann@44817
   741
haftmann@44817
   742
lemma max_0R [simp]: "max n 0 = (n::nat)"
haftmann@44817
   743
by (rule max_leastR) simp
haftmann@44817
   744
haftmann@44817
   745
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   746
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   747
haftmann@44817
   748
lemma max_Suc1:
haftmann@44817
   749
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   750
by (simp split: nat.split)
haftmann@44817
   751
haftmann@44817
   752
lemma max_Suc2:
haftmann@44817
   753
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   754
by (simp split: nat.split)
paulson@14267
   755
haftmann@44817
   756
lemma nat_add_min_left:
haftmann@44817
   757
  fixes m n q :: nat
haftmann@44817
   758
  shows "min m n + q = min (m + q) (n + q)"
haftmann@44817
   759
  by (simp add: min_def)
haftmann@44817
   760
haftmann@44817
   761
lemma nat_add_min_right:
haftmann@44817
   762
  fixes m n q :: nat
haftmann@44817
   763
  shows "m + min n q = min (m + n) (m + q)"
haftmann@44817
   764
  by (simp add: min_def)
haftmann@44817
   765
haftmann@44817
   766
lemma nat_mult_min_left:
haftmann@44817
   767
  fixes m n q :: nat
haftmann@44817
   768
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   769
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   770
haftmann@44817
   771
lemma nat_mult_min_right:
haftmann@44817
   772
  fixes m n q :: nat
haftmann@44817
   773
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   774
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   775
haftmann@44817
   776
lemma nat_add_max_left:
haftmann@44817
   777
  fixes m n q :: nat
haftmann@44817
   778
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   779
  by (simp add: max_def)
haftmann@44817
   780
haftmann@44817
   781
lemma nat_add_max_right:
haftmann@44817
   782
  fixes m n q :: nat
haftmann@44817
   783
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   784
  by (simp add: max_def)
haftmann@44817
   785
haftmann@44817
   786
lemma nat_mult_max_left:
haftmann@44817
   787
  fixes m n q :: nat
haftmann@44817
   788
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   789
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   790
haftmann@44817
   791
lemma nat_mult_max_right:
haftmann@44817
   792
  fixes m n q :: nat
haftmann@44817
   793
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   794
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   795
paulson@14267
   796
krauss@26748
   797
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   798
krauss@26748
   799
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   800
haftmann@27823
   801
instance nat :: wellorder proof
haftmann@27823
   802
  fix P and n :: nat
haftmann@27823
   803
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   804
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   805
  proof (induct n)
haftmann@27823
   806
    case (0 n)
krauss@26748
   807
    have "P 0" by (rule step) auto
krauss@26748
   808
    thus ?case using 0 by auto
krauss@26748
   809
  next
haftmann@27823
   810
    case (Suc m n)
haftmann@27823
   811
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   812
    thus ?case
krauss@26748
   813
    proof
haftmann@27823
   814
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   815
    next
haftmann@27823
   816
      assume n: "n = Suc m"
haftmann@27823
   817
      show "P n"
haftmann@27823
   818
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   819
    qed
krauss@26748
   820
  qed
haftmann@27823
   821
  then show "P n" by auto
krauss@26748
   822
qed
krauss@26748
   823
haftmann@27823
   824
lemma Least_Suc:
haftmann@27823
   825
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
haftmann@27823
   826
  apply (case_tac "n", auto)
haftmann@27823
   827
  apply (frule LeastI)
haftmann@27823
   828
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   829
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   830
  apply (erule_tac [2] Least_le)
haftmann@27823
   831
  apply (case_tac "LEAST x. P x", auto)
haftmann@27823
   832
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   833
  apply (blast intro: order_antisym)
haftmann@27823
   834
  done
haftmann@27823
   835
haftmann@27823
   836
lemma Least_Suc2:
haftmann@27823
   837
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   838
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   839
  apply simp
haftmann@27823
   840
  done
haftmann@27823
   841
haftmann@27823
   842
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
   843
  apply (cases n)
haftmann@27823
   844
   apply blast
haftmann@27823
   845
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   846
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   847
  done
haftmann@27823
   848
haftmann@27823
   849
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
   850
  unfolding One_nat_def
haftmann@27823
   851
  apply (cases n)
haftmann@27823
   852
   apply blast
haftmann@27823
   853
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   854
  apply (erule exE)
haftmann@27823
   855
  apply (case_tac k)
haftmann@27823
   856
   apply simp
haftmann@27823
   857
  apply (rename_tac k1)
haftmann@27823
   858
  apply (rule_tac x=k1 in exI)
haftmann@27823
   859
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   860
  done
haftmann@27823
   861
krauss@26748
   862
lemma nat_less_induct:
krauss@26748
   863
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   864
  using assms less_induct by blast
krauss@26748
   865
krauss@26748
   866
lemma measure_induct_rule [case_names less]:
krauss@26748
   867
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   868
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   869
  shows "P a"
krauss@26748
   870
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   871
krauss@26748
   872
text {* old style induction rules: *}
krauss@26748
   873
lemma measure_induct:
krauss@26748
   874
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   875
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   876
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   877
krauss@26748
   878
lemma full_nat_induct:
krauss@26748
   879
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   880
  shows "P n"
krauss@26748
   881
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   882
paulson@19870
   883
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   884
lemma less_Suc_induct:
paulson@19870
   885
  assumes less:  "i < j"
paulson@19870
   886
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   887
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   888
  shows "P i j"
paulson@19870
   889
proof -
krauss@31714
   890
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   891
  have "P i (Suc (i + k))"
paulson@19870
   892
  proof (induct k)
wenzelm@22718
   893
    case 0
wenzelm@22718
   894
    show ?case by (simp add: step)
paulson@19870
   895
  next
paulson@19870
   896
    case (Suc k)
krauss@31714
   897
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
krauss@31714
   898
    hence "i < Suc (i + k)" by (simp add: add_commute)
krauss@31714
   899
    from trans[OF this lessI Suc step]
krauss@31714
   900
    show ?case by simp
paulson@19870
   901
  qed
wenzelm@22718
   902
  thus "P i j" by (simp add: j)
paulson@19870
   903
qed
paulson@19870
   904
krauss@26748
   905
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   906
Provided by Roelof Oosterhuis.
krauss@26748
   907
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   908
\begin{itemize}
krauss@26748
   909
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   910
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   911
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   912
\end{itemize} *}
krauss@26748
   913
krauss@26748
   914
text{* A compact version without explicit base case: *}
krauss@26748
   915
lemma infinite_descent:
krauss@26748
   916
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
krauss@26748
   917
by (induct n rule: less_induct, auto)
krauss@26748
   918
krauss@26748
   919
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   920
  "\<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
   921
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   922
krauss@26748
   923
text {*
krauss@26748
   924
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   925
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   926
\begin{itemize}
krauss@26748
   927
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   928
\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
   929
\end{itemize}
krauss@26748
   930
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   931
krauss@26748
   932
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   933
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   934
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   935
  shows "P x"
krauss@26748
   936
proof -
krauss@26748
   937
  obtain n where "n = V x" by auto
krauss@26748
   938
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   939
  proof (induct n rule: infinite_descent0)
krauss@26748
   940
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   941
    with A0 show "P x" by auto
krauss@26748
   942
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   943
    case (smaller n)
krauss@26748
   944
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   945
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   946
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   947
    then show ?case by auto
krauss@26748
   948
  qed
krauss@26748
   949
  ultimately show "P x" by auto
krauss@26748
   950
qed
krauss@26748
   951
krauss@26748
   952
text{* Again, without explicit base case: *}
krauss@26748
   953
lemma infinite_descent_measure:
krauss@26748
   954
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   955
proof -
krauss@26748
   956
  from assms obtain n where "n = V x" by auto
krauss@26748
   957
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   958
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   959
    fix x assume "\<not> P x"
krauss@26748
   960
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
   961
  qed
krauss@26748
   962
  ultimately show "P x" by auto
krauss@26748
   963
qed
krauss@26748
   964
paulson@14267
   965
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   966
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   967
lemma less_mono_imp_le_mono:
nipkow@24438
   968
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   969
by (simp add: order_le_less) (blast)
nipkow@24438
   970
paulson@14267
   971
paulson@14267
   972
text {* non-strict, in 1st argument *}
paulson@14267
   973
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   974
by (rule add_right_mono)
paulson@14267
   975
paulson@14267
   976
text {* non-strict, in both arguments *}
paulson@14267
   977
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   978
by (rule add_mono)
paulson@14267
   979
paulson@14267
   980
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   981
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   982
paulson@14267
   983
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   984
by (simp add: add_commute, rule le_add2)
berghofe@13449
   985
berghofe@13449
   986
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   987
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   988
berghofe@13449
   989
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   990
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   991
paulson@14267
   992
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   993
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   994
paulson@14267
   995
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   996
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   997
paulson@14267
   998
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   999
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1000
berghofe@13449
  1001
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1002
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1003
berghofe@13449
  1004
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1005
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1006
berghofe@13449
  1007
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1008
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1009
apply (simp_all add: le_add1)
nipkow@24438
  1010
done
berghofe@13449
  1011
berghofe@13449
  1012
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1013
apply (rule notI)
wenzelm@26335
  1014
apply (drule add_lessD1)
wenzelm@26335
  1015
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1016
done
berghofe@13449
  1017
berghofe@13449
  1018
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
  1019
by (simp add: add_commute)
berghofe@13449
  1020
paulson@14267
  1021
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1022
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1023
apply (simp_all add: le_add1)
nipkow@24438
  1024
done
berghofe@13449
  1025
paulson@14267
  1026
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
  1027
apply (simp add: add_commute)
nipkow@24438
  1028
apply (erule add_leD1)
nipkow@24438
  1029
done
berghofe@13449
  1030
paulson@14267
  1031
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1032
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1033
berghofe@13449
  1034
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
  1035
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1036
by (force simp del: add_Suc_right
berghofe@13449
  1037
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
  1038
berghofe@13449
  1039
haftmann@26072
  1040
subsubsection {* More results about difference *}
berghofe@13449
  1041
berghofe@13449
  1042
text {* Addition is the inverse of subtraction:
paulson@14267
  1043
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1044
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1045
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1046
paulson@14267
  1047
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1048
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1049
paulson@14267
  1050
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
  1051
by (simp add: add_commute)
berghofe@13449
  1052
paulson@14267
  1053
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1054
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1055
berghofe@13449
  1056
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1057
apply (induct m n rule: diff_induct)
nipkow@24438
  1058
apply (erule_tac [3] less_SucE)
nipkow@24438
  1059
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1060
done
berghofe@13449
  1061
paulson@14267
  1062
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1063
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1064
haftmann@26072
  1065
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1066
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1067
berghofe@13449
  1068
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1069
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1070
berghofe@13449
  1071
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1072
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1073
paulson@14267
  1074
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1075
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1076
paulson@14267
  1077
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1078
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1079
paulson@14267
  1080
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1081
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1082
paulson@14267
  1083
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1084
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1085
paulson@14267
  1086
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1087
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1088
berghofe@13449
  1089
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1090
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1091
wenzelm@22718
  1092
lemma less_imp_add_positive:
wenzelm@22718
  1093
  assumes "i < j"
wenzelm@22718
  1094
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1095
proof
wenzelm@22718
  1096
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1097
    by (simp add: order_less_imp_le)
wenzelm@22718
  1098
qed
wenzelm@9436
  1099
haftmann@26072
  1100
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1101
lemma nat_minus_add_max:
haftmann@26072
  1102
  fixes n m :: nat
haftmann@26072
  1103
  shows "n - m + m = max n m"
haftmann@26072
  1104
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1105
haftmann@26072
  1106
lemma nat_diff_split:
haftmann@26072
  1107
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1108
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1109
by (cases "a < b")
haftmann@26072
  1110
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1111
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1112
haftmann@26072
  1113
lemma nat_diff_split_asm:
haftmann@26072
  1114
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1115
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1116
by (auto split: nat_diff_split)
berghofe@13449
  1117
berghofe@13449
  1118
haftmann@26072
  1119
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1120
paulson@14267
  1121
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1122
by (simp add: mult_right_mono)
berghofe@13449
  1123
paulson@14267
  1124
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1125
by (simp add: mult_left_mono)
berghofe@13449
  1126
paulson@14267
  1127
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1128
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1129
by (simp add: mult_mono)
berghofe@13449
  1130
berghofe@13449
  1131
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1132
by (simp add: mult_strict_right_mono)
berghofe@13449
  1133
paulson@14266
  1134
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1135
      there are no negative numbers.*}
paulson@14266
  1136
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1137
  apply (induct m)
wenzelm@22718
  1138
   apply simp
wenzelm@22718
  1139
  apply (case_tac n)
wenzelm@22718
  1140
   apply simp_all
berghofe@13449
  1141
  done
berghofe@13449
  1142
huffman@30079
  1143
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1144
  apply (induct m)
wenzelm@22718
  1145
   apply simp
wenzelm@22718
  1146
  apply (case_tac n)
wenzelm@22718
  1147
   apply simp_all
berghofe@13449
  1148
  done
berghofe@13449
  1149
paulson@14341
  1150
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1151
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1152
  apply (case_tac k, auto)
berghofe@13449
  1153
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1154
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1155
  done
berghofe@13449
  1156
berghofe@13449
  1157
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1158
by (simp add: mult_commute [of k])
berghofe@13449
  1159
paulson@14267
  1160
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1161
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1162
paulson@14267
  1163
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1164
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1165
berghofe@13449
  1166
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1167
by (subst mult_less_cancel1) simp
berghofe@13449
  1168
paulson@14267
  1169
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1170
by (subst mult_le_cancel1) simp
berghofe@13449
  1171
haftmann@26072
  1172
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1173
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1174
haftmann@26072
  1175
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1176
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1177
berghofe@13449
  1178
text {* Lemma for @{text gcd} *}
huffman@30128
  1179
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1180
  apply (drule sym)
berghofe@13449
  1181
  apply (rule disjCI)
berghofe@13449
  1182
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1183
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1184
    apply (auto)
berghofe@13449
  1185
  done
wenzelm@9436
  1186
haftmann@26072
  1187
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1188
haftmann@26072
  1189
instantiation nat :: distrib_lattice
haftmann@26072
  1190
begin
haftmann@24995
  1191
haftmann@26072
  1192
definition
haftmann@26072
  1193
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1194
haftmann@26072
  1195
definition
haftmann@26072
  1196
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1197
haftmann@26072
  1198
instance by intro_classes
haftmann@26072
  1199
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1200
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1201
haftmann@26072
  1202
end
haftmann@24995
  1203
haftmann@24995
  1204
haftmann@30954
  1205
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1206
haftmann@30971
  1207
text {*
haftmann@30971
  1208
  We use the same logical constant for the power operations on
haftmann@30971
  1209
  functions and relations, in order to share the same syntax.
haftmann@30971
  1210
*}
haftmann@30971
  1211
haftmann@30971
  1212
consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@30971
  1213
haftmann@30971
  1214
abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
haftmann@30971
  1215
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1216
haftmann@30971
  1217
notation (latex output)
haftmann@30971
  1218
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1219
haftmann@30971
  1220
notation (HTML output)
haftmann@30971
  1221
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1222
haftmann@30971
  1223
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1224
haftmann@30971
  1225
overloading
haftmann@30971
  1226
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1227
begin
haftmann@30954
  1228
haftmann@30954
  1229
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1230
  "funpow 0 f = id"
haftmann@44325
  1231
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1232
haftmann@30971
  1233
end
haftmann@30971
  1234
haftmann@30971
  1235
text {* for code generation *}
haftmann@30971
  1236
haftmann@30971
  1237
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@31998
  1238
  funpow_code_def [code_post]: "funpow = compow"
haftmann@30954
  1239
haftmann@31998
  1240
lemmas [code_unfold] = funpow_code_def [symmetric]
haftmann@30954
  1241
haftmann@30971
  1242
lemma [code]:
haftmann@37430
  1243
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1244
  "funpow 0 f = id"
haftmann@37430
  1245
  by (simp_all add: funpow_code_def)
haftmann@30971
  1246
wenzelm@36176
  1247
hide_const (open) funpow
haftmann@30954
  1248
haftmann@30954
  1249
lemma funpow_add:
haftmann@30971
  1250
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1251
  by (induct m) simp_all
haftmann@30954
  1252
haftmann@37430
  1253
lemma funpow_mult:
haftmann@37430
  1254
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1255
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1256
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1257
haftmann@30954
  1258
lemma funpow_swap1:
haftmann@30971
  1259
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1260
proof -
haftmann@30971
  1261
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1262
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1263
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1264
  finally show ?thesis .
haftmann@30954
  1265
qed
haftmann@30954
  1266
haftmann@38621
  1267
lemma comp_funpow:
haftmann@38621
  1268
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1269
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1270
  by (induct n) simp_all
haftmann@30954
  1271
haftmann@38621
  1272
haftmann@38621
  1273
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1274
haftmann@24196
  1275
context semiring_1
haftmann@24196
  1276
begin
haftmann@24196
  1277
haftmann@38621
  1278
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1279
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1280
haftmann@38621
  1281
lemma of_nat_simps [simp]:
haftmann@38621
  1282
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1283
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1284
  by (simp_all add: of_nat_def)
haftmann@25193
  1285
haftmann@25193
  1286
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1287
  by (simp add: of_nat_def)
haftmann@25193
  1288
haftmann@25193
  1289
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1290
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1291
haftmann@25193
  1292
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1293
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1294
haftmann@28514
  1295
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1296
  "of_nat_aux inc 0 i = i"
haftmann@44325
  1297
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1298
haftmann@30966
  1299
lemma of_nat_code:
haftmann@28514
  1300
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1301
proof (induct n)
haftmann@28514
  1302
  case 0 then show ?case by simp
haftmann@28514
  1303
next
haftmann@28514
  1304
  case (Suc n)
haftmann@28514
  1305
  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
  1306
    by (induct n) simp_all
haftmann@28514
  1307
  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
  1308
    by simp
haftmann@28514
  1309
  with Suc show ?case by (simp add: add_commute)
haftmann@28514
  1310
qed
haftmann@30966
  1311
haftmann@24196
  1312
end
haftmann@24196
  1313
haftmann@31998
  1314
declare of_nat_code [code, code_unfold, code_inline del]
haftmann@30966
  1315
haftmann@26072
  1316
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1317
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1318
haftmann@26072
  1319
class semiring_char_0 = semiring_1 +
haftmann@38621
  1320
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1321
begin
haftmann@26072
  1322
haftmann@38621
  1323
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1324
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1325
haftmann@26072
  1326
text{*Special cases where either operand is zero*}
haftmann@26072
  1327
blanchet@35828
  1328
lemma of_nat_0_eq_iff [simp, no_atp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1329
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1330
blanchet@35828
  1331
lemma of_nat_eq_0_iff [simp, no_atp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1332
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1333
haftmann@26072
  1334
end
haftmann@26072
  1335
haftmann@35028
  1336
context linordered_semidom
haftmann@25193
  1337
begin
haftmann@25193
  1338
haftmann@25193
  1339
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
huffman@36977
  1340
  by (induct m) simp_all
haftmann@25193
  1341
haftmann@25193
  1342
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1343
  apply (induct m n rule: diff_induct, simp_all)
huffman@36977
  1344
  apply (rule add_pos_nonneg [OF zero_less_one zero_le_imp_of_nat])
haftmann@25193
  1345
  done
haftmann@25193
  1346
haftmann@25193
  1347
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1348
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1349
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1350
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1351
  done
haftmann@25193
  1352
haftmann@25193
  1353
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1354
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1355
haftmann@26072
  1356
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1357
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1358
haftmann@35028
  1359
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1360
haftmann@38621
  1361
subclass semiring_char_0 proof
haftmann@38621
  1362
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1363
haftmann@25193
  1364
text{*Special cases where either operand is zero*}
haftmann@25193
  1365
haftmann@25193
  1366
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1367
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1368
blanchet@35828
  1369
lemma of_nat_le_0_iff [simp, no_atp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1370
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1371
haftmann@26072
  1372
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1373
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1374
haftmann@26072
  1375
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1376
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1377
haftmann@26072
  1378
end
haftmann@26072
  1379
haftmann@26072
  1380
context ring_1
haftmann@26072
  1381
begin
haftmann@26072
  1382
haftmann@26072
  1383
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1384
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1385
haftmann@26072
  1386
end
haftmann@26072
  1387
haftmann@35028
  1388
context linordered_idom
haftmann@26072
  1389
begin
haftmann@26072
  1390
haftmann@26072
  1391
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1392
  unfolding abs_if by auto
haftmann@26072
  1393
haftmann@25193
  1394
end
haftmann@25193
  1395
haftmann@25193
  1396
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1397
  by (induct n) simp_all
haftmann@25193
  1398
haftmann@25193
  1399
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1400
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1401
haftmann@25193
  1402
haftmann@26149
  1403
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1404
haftmann@26072
  1405
context semiring_1
haftmann@25193
  1406
begin
haftmann@25193
  1407
haftmann@37767
  1408
definition Nats  :: "'a set" where
haftmann@37767
  1409
  "Nats = range of_nat"
haftmann@26072
  1410
haftmann@26072
  1411
notation (xsymbols)
haftmann@26072
  1412
  Nats  ("\<nat>")
haftmann@25193
  1413
haftmann@26072
  1414
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1415
  by (simp add: Nats_def)
haftmann@26072
  1416
haftmann@26072
  1417
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1418
apply (simp add: Nats_def)
haftmann@26072
  1419
apply (rule range_eqI)
haftmann@26072
  1420
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1421
done
haftmann@25193
  1422
haftmann@26072
  1423
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1424
apply (simp add: Nats_def)
haftmann@26072
  1425
apply (rule range_eqI)
haftmann@26072
  1426
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1427
done
haftmann@25193
  1428
haftmann@26072
  1429
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1430
apply (auto simp add: Nats_def)
haftmann@26072
  1431
apply (rule range_eqI)
haftmann@26072
  1432
apply (rule of_nat_add [symmetric])
haftmann@26072
  1433
done
haftmann@26072
  1434
haftmann@26072
  1435
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1436
apply (auto simp add: Nats_def)
haftmann@26072
  1437
apply (rule range_eqI)
haftmann@26072
  1438
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1439
done
haftmann@25193
  1440
huffman@35633
  1441
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1442
  assumes "x \<in> \<nat>"
huffman@35633
  1443
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1444
  unfolding Nats_def
huffman@35633
  1445
proof -
huffman@35633
  1446
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1447
  then obtain n where "x = of_nat n" ..
huffman@35633
  1448
  then show thesis ..
huffman@35633
  1449
qed
huffman@35633
  1450
huffman@35633
  1451
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1452
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1453
  by (rule Nats_cases) auto
huffman@35633
  1454
haftmann@25193
  1455
end
haftmann@25193
  1456
haftmann@25193
  1457
wenzelm@21243
  1458
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1459
haftmann@22845
  1460
lemma subst_equals:
haftmann@22845
  1461
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1462
  shows "u = s"
haftmann@22845
  1463
  using 2 1 by (rule trans)
haftmann@22845
  1464
haftmann@30686
  1465
setup Arith_Data.setup
haftmann@30686
  1466
haftmann@30496
  1467
use "Tools/nat_arith.ML"
haftmann@30496
  1468
declaration {* K Nat_Arith.setup *}
wenzelm@24091
  1469
wenzelm@24091
  1470
use "Tools/lin_arith.ML"
haftmann@31100
  1471
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1472
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1473
wenzelm@43595
  1474
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@43595
  1475
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1476
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1477
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1478
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1479
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1480
solver all the time rather than add the additional check. *)
wenzelm@43595
  1481
wenzelm@43595
  1482
wenzelm@21243
  1483
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1484
nipkow@27625
  1485
context order
nipkow@27625
  1486
begin
nipkow@27625
  1487
nipkow@27625
  1488
lemma lift_Suc_mono_le:
krauss@27627
  1489
  assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
krauss@27627
  1490
  shows "f n \<le> f n'"
krauss@27627
  1491
proof (cases "n < n'")
krauss@27627
  1492
  case True
krauss@27627
  1493
  thus ?thesis
krauss@27627
  1494
    by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
krauss@27627
  1495
qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}
nipkow@27625
  1496
nipkow@27625
  1497
lemma lift_Suc_mono_less:
krauss@27627
  1498
  assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
krauss@27627
  1499
  shows "f n < f n'"
krauss@27627
  1500
using `n < n'`
krauss@27627
  1501
by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
nipkow@27625
  1502
nipkow@27789
  1503
lemma lift_Suc_mono_less_iff:
nipkow@27789
  1504
  "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
nipkow@27789
  1505
by(blast intro: less_asym' lift_Suc_mono_less[of f]
nipkow@27789
  1506
         dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
nipkow@27789
  1507
nipkow@27625
  1508
end
nipkow@27625
  1509
nipkow@29879
  1510
lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1511
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1512
nipkow@27789
  1513
lemma mono_nat_linear_lb:
nipkow@27789
  1514
  "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
nipkow@27789
  1515
apply(induct_tac k)
nipkow@27789
  1516
 apply simp
nipkow@27789
  1517
apply(erule_tac x="m+n" in meta_allE)
huffman@30079
  1518
apply(erule_tac x="Suc(m+n)" in meta_allE)
nipkow@27789
  1519
apply simp
nipkow@27789
  1520
done
nipkow@27789
  1521
nipkow@27789
  1522
wenzelm@21243
  1523
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1524
wenzelm@21243
  1525
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1526
by arith
wenzelm@21243
  1527
wenzelm@21243
  1528
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1529
by arith
wenzelm@21243
  1530
wenzelm@21243
  1531
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1532
by arith
wenzelm@21243
  1533
wenzelm@21243
  1534
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1535
by arith
wenzelm@21243
  1536
wenzelm@21243
  1537
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1538
by arith
wenzelm@21243
  1539
wenzelm@21243
  1540
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1541
by arith
wenzelm@21243
  1542
wenzelm@21243
  1543
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1544
  second premise n\<le>m*)
wenzelm@21243
  1545
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1546
by arith
wenzelm@21243
  1547
haftmann@26072
  1548
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1549
wenzelm@21243
  1550
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1551
by (simp split add: nat_diff_split)
wenzelm@21243
  1552
wenzelm@36176
  1553
hide_fact (open) diff_diff_eq
haftmann@35064
  1554
wenzelm@21243
  1555
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1556
by (auto split add: nat_diff_split)
wenzelm@21243
  1557
wenzelm@21243
  1558
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1559
by (auto split add: nat_diff_split)
wenzelm@21243
  1560
wenzelm@21243
  1561
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1562
by (auto split add: nat_diff_split)
wenzelm@21243
  1563
wenzelm@21243
  1564
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1565
wenzelm@21243
  1566
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1567
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1568
by (simp split add: nat_diff_split)
wenzelm@21243
  1569
wenzelm@21243
  1570
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1571
by (simp split add: nat_diff_split)
wenzelm@21243
  1572
wenzelm@21243
  1573
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1574
by (simp split add: nat_diff_split)
wenzelm@21243
  1575
wenzelm@21243
  1576
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1577
by (simp split add: nat_diff_split)
wenzelm@21243
  1578
bulwahn@26143
  1579
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1580
by auto
bulwahn@26143
  1581
bulwahn@26143
  1582
lemma inj_on_diff_nat: 
bulwahn@26143
  1583
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1584
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1585
proof (rule inj_onI)
bulwahn@26143
  1586
  fix x y
bulwahn@26143
  1587
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1588
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1589
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1590
qed
bulwahn@26143
  1591
haftmann@26072
  1592
text{*Rewriting to pull differences out*}
haftmann@26072
  1593
haftmann@26072
  1594
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1595
by arith
haftmann@26072
  1596
haftmann@26072
  1597
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1598
by arith
haftmann@26072
  1599
haftmann@26072
  1600
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1601
by arith
haftmann@26072
  1602
wenzelm@21243
  1603
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1604
wenzelm@21243
  1605
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1606
by (cases m) auto
wenzelm@21243
  1607
wenzelm@21243
  1608
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1609
by (cases m) auto
wenzelm@21243
  1610
wenzelm@21243
  1611
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1612
by (cases m) auto
wenzelm@21243
  1613
krauss@23001
  1614
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1615
krauss@23001
  1616
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1617
  assumes less: "i <= j"
krauss@23001
  1618
  assumes base: "P j"
krauss@23001
  1619
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1620
  shows "P i"
krauss@23001
  1621
  using less
krauss@23001
  1622
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1623
  case (0 i)
krauss@23001
  1624
  hence "i = j" by simp
krauss@23001
  1625
  with base show ?case by simp
krauss@23001
  1626
next
krauss@23001
  1627
  case (Suc d i)
krauss@23001
  1628
  hence "i < j" "P (Suc i)"
krauss@23001
  1629
    by simp_all
krauss@23001
  1630
  thus "P i" by (rule step)
krauss@23001
  1631
qed
krauss@23001
  1632
krauss@23001
  1633
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1634
  assumes less: "i < j"
krauss@23001
  1635
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1636
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1637
  shows "P i"
krauss@23001
  1638
  using less
krauss@23001
  1639
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1640
  case (0 i)
krauss@23001
  1641
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1642
  with base show ?case by simp
krauss@23001
  1643
next
krauss@23001
  1644
  case (Suc d i)
krauss@23001
  1645
  hence "i < j" "P (Suc i)"
krauss@23001
  1646
    by simp_all
krauss@23001
  1647
  thus "P i" by (rule step)
krauss@23001
  1648
qed
krauss@23001
  1649
krauss@23001
  1650
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1651
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1652
krauss@23001
  1653
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1654
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1655
wenzelm@21243
  1656
(*The others are
wenzelm@21243
  1657
      i - j - k = i - (j + k),
wenzelm@21243
  1658
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1659
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1660
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1661
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1662
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1663
wenzelm@21243
  1664
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1665
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1666
nipkow@27625
  1667
haftmann@33274
  1668
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1669
haftmann@33274
  1670
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1671
unfolding dvd_def by simp
haftmann@33274
  1672
haftmann@33274
  1673
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1674
by (simp add: dvd_def)
haftmann@33274
  1675
haftmann@33274
  1676
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1677
by (simp add: dvd_def)
haftmann@33274
  1678
nipkow@33657
  1679
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1680
  unfolding dvd_def
huffman@35216
  1681
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
haftmann@33274
  1682
haftmann@33274
  1683
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1684
haftmann@33274
  1685
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1686
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1687
haftmann@33274
  1688
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1689
unfolding dvd_def
haftmann@33274
  1690
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1691
haftmann@33274
  1692
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1693
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1694
  apply (blast intro: dvd_add)
haftmann@33274
  1695
  done
haftmann@33274
  1696
haftmann@33274
  1697
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1698
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1699
haftmann@33274
  1700
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
haftmann@33274
  1701
  apply (rule iffI)
haftmann@33274
  1702
   apply (erule_tac [2] dvd_add)
haftmann@33274
  1703
   apply (rule_tac [2] dvd_refl)
haftmann@33274
  1704
  apply (subgoal_tac "n = (n+k) -k")
haftmann@33274
  1705
   prefer 2 apply simp
haftmann@33274
  1706
  apply (erule ssubst)
haftmann@33274
  1707
  apply (erule dvd_diff_nat)
haftmann@33274
  1708
  apply (rule dvd_refl)
haftmann@33274
  1709
  done
haftmann@33274
  1710
haftmann@33274
  1711
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1712
  unfolding dvd_def
haftmann@33274
  1713
  apply (erule exE)
haftmann@33274
  1714
  apply (simp add: mult_ac)
haftmann@33274
  1715
  done
haftmann@33274
  1716
haftmann@33274
  1717
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1718
  apply auto
haftmann@33274
  1719
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1720
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1721
  done
haftmann@33274
  1722
haftmann@33274
  1723
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@33274
  1724
  apply (subst mult_commute)
haftmann@33274
  1725
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1726
  done
haftmann@33274
  1727
haftmann@33274
  1728
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1729
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1730
haftmann@33274
  1731
lemma nat_dvd_not_less:
haftmann@33274
  1732
  fixes m n :: nat
haftmann@33274
  1733
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1734
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1735
haftmann@33274
  1736
haftmann@44817
  1737
subsection {* aliasses *}
haftmann@44817
  1738
haftmann@44817
  1739
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@44817
  1740
  by simp
haftmann@44817
  1741
 
haftmann@44817
  1742
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@44817
  1743
  by simp
haftmann@44817
  1744
haftmann@44817
  1745
haftmann@26072
  1746
subsection {* size of a datatype value *}
haftmann@25193
  1747
haftmann@29608
  1748
class size =
krauss@26748
  1749
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1750
haftmann@33364
  1751
haftmann@33364
  1752
subsection {* code module namespace *}
haftmann@33364
  1753
haftmann@33364
  1754
code_modulename SML
haftmann@33364
  1755
  Nat Arith
haftmann@33364
  1756
haftmann@33364
  1757
code_modulename OCaml
haftmann@33364
  1758
  Nat Arith
haftmann@33364
  1759
haftmann@33364
  1760
code_modulename Haskell
haftmann@33364
  1761
  Nat Arith
haftmann@33364
  1762
haftmann@25193
  1763
end