src/HOL/Nat.thy
author haftmann
Thu Aug 18 13:55:26 2011 +0200 (2011-08-18)
changeset 44278 1220ecb81e8f
parent 43595 7ae4a23b5be6
child 44325 84696670feb1
permissions -rw-r--r--
observe distinction between sets and predicates more properly
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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
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  Zero_Rep :: ind and
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  Suc_Rep :: "ind => ind"
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where
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  -- {* the axiom of infinity in 2 parts *}
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  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"
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where
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    Zero_RepI: "Nat Zero_Rep"
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  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
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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
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where
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  add_0:      "0 + n = (n\<Colon>nat)"
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  | add_Suc:  "Suc m + n = Suc (m + n)"
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lemma add_0_right [simp]: "m + 0 = (m::nat)"
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  by (induct m) simp_all
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
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  by (induct m) simp_all
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declare add_0 [code]
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
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  by simp
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primrec minus_nat
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where
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  diff_0 [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
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where
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  mult_0:     "0 * n = (0\<Colon>nat)"
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  | mult_Suc: "Suc m * n = n + (m * n)"
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lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
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  by (induct m) simp_all
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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
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  by (induct m) (simp_all add: add_left_commute)
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lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
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  by (induct m) (simp_all add: add_assoc)
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instance proof
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  fix n m q :: nat
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  show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
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  show "1 * n = n" unfolding One_nat_def by simp
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  show "n * m = m * n" by (induct n) simp_all
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  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
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  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
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  assume "n + m = n + q" thus "m = q" by (induct n) simp_all
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qed
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end
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subsubsection {* Addition *}
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lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
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  by (rule add_assoc)
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lemma nat_add_commute: "m + n = n + (m::nat)"
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  by (rule add_commute)
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lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
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  by (rule add_left_commute)
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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)
huffman@30079
   338
haftmann@26072
   339
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   340
proof -
haftmann@26072
   341
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   342
  proof (induct n arbitrary: m)
haftmann@26072
   343
    case 0 then show "m = 0" by simp
haftmann@26072
   344
  next
haftmann@26072
   345
    case (Suc n) then show "m = Suc n"
haftmann@26072
   346
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   347
  qed
haftmann@26072
   348
  then show ?thesis by auto
haftmann@26072
   349
qed
haftmann@26072
   350
haftmann@26072
   351
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@26072
   352
  by (simp add: mult_commute)
haftmann@26072
   353
haftmann@26072
   354
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   355
  by (subst mult_cancel1) simp
haftmann@26072
   356
haftmann@24995
   357
haftmann@24995
   358
subsection {* Orders on @{typ nat} *}
haftmann@24995
   359
haftmann@26072
   360
subsubsection {* Operation definition *}
haftmann@24995
   361
haftmann@26072
   362
instantiation nat :: linorder
haftmann@25510
   363
begin
haftmann@25510
   364
haftmann@26072
   365
primrec less_eq_nat where
haftmann@26072
   366
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@26072
   367
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   368
haftmann@28514
   369
declare less_eq_nat.simps [simp del]
haftmann@26072
   370
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by (simp add: less_eq_nat.simps)
haftmann@26072
   371
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@26072
   372
haftmann@26072
   373
definition less_nat where
haftmann@28514
   374
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   375
haftmann@26072
   376
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   377
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   378
haftmann@26072
   379
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   380
  unfolding less_eq_Suc_le ..
haftmann@26072
   381
haftmann@26072
   382
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   383
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   384
haftmann@26072
   385
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   386
  by (simp add: less_eq_Suc_le)
haftmann@26072
   387
haftmann@26072
   388
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   389
  by simp
haftmann@26072
   390
haftmann@26072
   391
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   392
  by (simp add: less_eq_Suc_le)
haftmann@26072
   393
haftmann@26072
   394
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   395
  by (simp add: less_eq_Suc_le)
haftmann@26072
   396
haftmann@26072
   397
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   398
  by (induct m arbitrary: n)
haftmann@26072
   399
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   400
haftmann@26072
   401
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   402
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   403
haftmann@26072
   404
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   405
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   406
haftmann@26072
   407
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   408
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   409
wenzelm@26315
   410
instance
wenzelm@26315
   411
proof
haftmann@26072
   412
  fix n m :: nat
haftmann@27679
   413
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   414
  proof (induct n arbitrary: m)
haftmann@27679
   415
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   416
  next
haftmann@27679
   417
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   418
  qed
haftmann@26072
   419
next
haftmann@26072
   420
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   421
next
haftmann@26072
   422
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   423
  then show "n = m"
haftmann@26072
   424
    by (induct n arbitrary: m)
haftmann@26072
   425
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   426
next
haftmann@26072
   427
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   428
  then show "n \<le> q"
haftmann@26072
   429
  proof (induct n arbitrary: m q)
haftmann@26072
   430
    case 0 show ?case by simp
haftmann@26072
   431
  next
haftmann@26072
   432
    case (Suc n) then show ?case
haftmann@26072
   433
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   434
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   435
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   436
  qed
haftmann@26072
   437
next
haftmann@26072
   438
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   439
    by (induct n arbitrary: m)
haftmann@26072
   440
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   441
qed
haftmann@25510
   442
haftmann@25510
   443
end
berghofe@13449
   444
haftmann@29652
   445
instantiation nat :: bot
haftmann@29652
   446
begin
haftmann@29652
   447
haftmann@29652
   448
definition bot_nat :: nat where
haftmann@29652
   449
  "bot_nat = 0"
haftmann@29652
   450
haftmann@29652
   451
instance proof
haftmann@29652
   452
qed (simp add: bot_nat_def)
haftmann@29652
   453
haftmann@29652
   454
end
haftmann@29652
   455
haftmann@26072
   456
subsubsection {* Introduction properties *}
berghofe@13449
   457
haftmann@26072
   458
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   459
  by (simp add: less_Suc_eq_le)
berghofe@13449
   460
haftmann@26072
   461
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   462
  by (simp add: less_Suc_eq_le)
berghofe@13449
   463
berghofe@13449
   464
berghofe@13449
   465
subsubsection {* Elimination properties *}
berghofe@13449
   466
berghofe@13449
   467
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   468
  by (rule order_less_irrefl)
berghofe@13449
   469
wenzelm@26335
   470
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   471
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   472
paulson@14267
   473
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   474
  by (rule less_imp_neq)
berghofe@13449
   475
wenzelm@26335
   476
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   477
  by (rule notE, rule less_not_refl)
berghofe@13449
   478
berghofe@13449
   479
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   480
  by (rule notE) (rule not_less0)
berghofe@13449
   481
berghofe@13449
   482
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   483
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   484
huffman@30079
   485
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   486
  by (simp add: less_Suc_eq)
berghofe@13449
   487
blanchet@35828
   488
lemma less_one [iff, no_atp]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   489
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   490
berghofe@13449
   491
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   492
  by simp
berghofe@13449
   493
nipkow@14302
   494
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   495
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   496
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   497
paulson@14267
   498
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   499
  by (rule linorder_neq_iff)
berghofe@13449
   500
berghofe@13449
   501
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   502
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   503
  shows "P n m"
berghofe@13449
   504
  apply (rule less_linear [THEN disjE])
berghofe@13449
   505
  apply (erule_tac [2] disjE)
berghofe@13449
   506
  apply (erule lessCase)
berghofe@13449
   507
  apply (erule sym [THEN eqCase])
berghofe@13449
   508
  apply (erule major)
berghofe@13449
   509
  done
berghofe@13449
   510
berghofe@13449
   511
berghofe@13449
   512
subsubsection {* Inductive (?) properties *}
berghofe@13449
   513
paulson@14267
   514
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   515
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   516
haftmann@26072
   517
lemma lessE:
haftmann@26072
   518
  assumes major: "i < k"
haftmann@26072
   519
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   520
  shows P
haftmann@26072
   521
proof -
haftmann@26072
   522
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   523
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   524
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   525
    by (clarsimp simp add: less_le)
haftmann@26072
   526
  with p1 p2 show P by auto
haftmann@26072
   527
qed
haftmann@26072
   528
haftmann@26072
   529
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   530
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   531
  apply (rule major [THEN lessE])
haftmann@26072
   532
  apply (rule eq, blast)
haftmann@26072
   533
  apply (rule less, blast)
berghofe@13449
   534
  done
berghofe@13449
   535
berghofe@13449
   536
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   537
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   538
  apply (rule major [THEN lessE])
berghofe@13449
   539
  apply (erule lessI [THEN minor])
paulson@14208
   540
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   541
  done
berghofe@13449
   542
berghofe@13449
   543
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   544
  by simp
berghofe@13449
   545
berghofe@13449
   546
lemma less_trans_Suc:
berghofe@13449
   547
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   548
  apply (induct k, simp_all)
berghofe@13449
   549
  apply (insert le)
berghofe@13449
   550
  apply (simp add: less_Suc_eq)
berghofe@13449
   551
  apply (blast dest: Suc_lessD)
berghofe@13449
   552
  done
berghofe@13449
   553
berghofe@13449
   554
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   555
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   556
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   557
haftmann@26072
   558
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   559
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   560
haftmann@24995
   561
text {* Properties of "less than or equal" *}
berghofe@13449
   562
paulson@14267
   563
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   564
  unfolding less_Suc_eq_le .
berghofe@13449
   565
paulson@14267
   566
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   567
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   568
paulson@14267
   569
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   570
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   571
paulson@14267
   572
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   573
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   574
paulson@14267
   575
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   576
  unfolding Suc_le_eq .
berghofe@13449
   577
berghofe@13449
   578
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   579
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   580
  unfolding Suc_le_eq .
berghofe@13449
   581
wenzelm@26315
   582
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   583
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   584
paulson@14267
   585
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   586
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   587
berghofe@13449
   588
paulson@14267
   589
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   590
paulson@14267
   591
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   592
  unfolding le_less .
berghofe@13449
   593
paulson@14267
   594
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   595
  by (rule le_less)
berghofe@13449
   596
wenzelm@22718
   597
text {* Useful with @{text blast}. *}
paulson@14267
   598
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   599
  by auto
berghofe@13449
   600
paulson@14267
   601
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   602
  by simp
berghofe@13449
   603
paulson@14267
   604
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   605
  by (rule order_trans)
berghofe@13449
   606
nipkow@33657
   607
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   608
  by (rule antisym)
berghofe@13449
   609
paulson@14267
   610
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   611
  by (rule less_le)
berghofe@13449
   612
paulson@14267
   613
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   614
  unfolding less_le ..
berghofe@13449
   615
haftmann@26072
   616
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   617
  by (rule linear)
paulson@14341
   618
wenzelm@22718
   619
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   620
haftmann@26072
   621
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   622
  unfolding less_Suc_eq_le by auto
berghofe@13449
   623
haftmann@26072
   624
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   625
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   626
berghofe@13449
   627
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   628
wenzelm@22718
   629
text {* These two rules ease the use of primitive recursion.
paulson@14341
   630
NOTE USE OF @{text "=="} *}
berghofe@13449
   631
lemma def_nat_rec_0: "(!!n. f n == nat_rec c h n) ==> f 0 = c"
nipkow@25162
   632
by simp
berghofe@13449
   633
berghofe@13449
   634
lemma def_nat_rec_Suc: "(!!n. f n == nat_rec c h n) ==> f (Suc n) = h n (f n)"
nipkow@25162
   635
by simp
berghofe@13449
   636
paulson@14267
   637
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   638
by (cases n) simp_all
nipkow@25162
   639
nipkow@25162
   640
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   641
by (cases n) simp_all
berghofe@13449
   642
wenzelm@22718
   643
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   644
by (cases n) simp_all
berghofe@13449
   645
nipkow@25162
   646
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   647
by (cases n) simp_all
nipkow@25140
   648
berghofe@13449
   649
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   650
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   651
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   652
paulson@14267
   653
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   654
by (fast intro: not0_implies_Suc)
berghofe@13449
   655
blanchet@35828
   656
lemma not_gr0 [iff,no_atp]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   657
using neq0_conv by blast
berghofe@13449
   658
paulson@14267
   659
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   660
by (induct m') simp_all
berghofe@13449
   661
berghofe@13449
   662
text {* Useful in certain inductive arguments *}
paulson@14267
   663
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   664
by (cases m) simp_all
berghofe@13449
   665
berghofe@13449
   666
haftmann@26072
   667
subsubsection {* @{term min} and @{term max} *}
berghofe@13449
   668
haftmann@25076
   669
lemma mono_Suc: "mono Suc"
nipkow@25162
   670
by (rule monoI) simp
haftmann@25076
   671
berghofe@13449
   672
lemma min_0L [simp]: "min 0 n = (0::nat)"
nipkow@25162
   673
by (rule min_leastL) simp
berghofe@13449
   674
berghofe@13449
   675
lemma min_0R [simp]: "min n 0 = (0::nat)"
nipkow@25162
   676
by (rule min_leastR) simp
berghofe@13449
   677
berghofe@13449
   678
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
nipkow@25162
   679
by (simp add: mono_Suc min_of_mono)
berghofe@13449
   680
paulson@22191
   681
lemma min_Suc1:
paulson@22191
   682
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
nipkow@25162
   683
by (simp split: nat.split)
paulson@22191
   684
paulson@22191
   685
lemma min_Suc2:
paulson@22191
   686
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
nipkow@25162
   687
by (simp split: nat.split)
paulson@22191
   688
berghofe@13449
   689
lemma max_0L [simp]: "max 0 n = (n::nat)"
nipkow@25162
   690
by (rule max_leastL) simp
berghofe@13449
   691
berghofe@13449
   692
lemma max_0R [simp]: "max n 0 = (n::nat)"
nipkow@25162
   693
by (rule max_leastR) simp
berghofe@13449
   694
berghofe@13449
   695
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
nipkow@25162
   696
by (simp add: mono_Suc max_of_mono)
berghofe@13449
   697
paulson@22191
   698
lemma max_Suc1:
paulson@22191
   699
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
nipkow@25162
   700
by (simp split: nat.split)
paulson@22191
   701
paulson@22191
   702
lemma max_Suc2:
paulson@22191
   703
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
nipkow@25162
   704
by (simp split: nat.split)
paulson@22191
   705
berghofe@13449
   706
haftmann@26072
   707
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   708
haftmann@26072
   709
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   710
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   711
huffman@30128
   712
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   713
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   714
paulson@14331
   715
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   716
by (induct k) simp_all
berghofe@13449
   717
paulson@14331
   718
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   719
by (induct k) simp_all
berghofe@13449
   720
nipkow@25162
   721
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   722
by(auto dest:gr0_implies_Suc)
berghofe@13449
   723
paulson@14341
   724
text {* strict, in 1st argument *}
paulson@14341
   725
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   726
by (induct k) simp_all
paulson@14341
   727
paulson@14341
   728
text {* strict, in both arguments *}
paulson@14341
   729
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   730
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   731
  apply (induct j, simp_all)
paulson@14341
   732
  done
paulson@14341
   733
paulson@14341
   734
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   735
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   736
  apply (induct n)
paulson@14341
   737
  apply (simp_all add: order_le_less)
wenzelm@22718
   738
  apply (blast elim!: less_SucE
haftmann@35047
   739
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   740
  done
paulson@14341
   741
paulson@14341
   742
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   743
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   744
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   745
apply (induct_tac m)
nipkow@25134
   746
apply (simp_all add: add_less_mono)
nipkow@25134
   747
done
paulson@14341
   748
nipkow@14740
   749
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   750
instance nat :: linordered_semidom
paulson@14341
   751
proof
paulson@14341
   752
  fix i j k :: nat
paulson@14348
   753
  show "0 < (1::nat)" by simp
paulson@14267
   754
  show "i \<le> j ==> k + i \<le> k + j" by simp
paulson@14267
   755
  show "i < j ==> 0 < k ==> k * i < k * j" by (simp add: mult_less_mono2)
paulson@14267
   756
qed
paulson@14267
   757
nipkow@30056
   758
instance nat :: no_zero_divisors
nipkow@30056
   759
proof
nipkow@30056
   760
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
nipkow@30056
   761
qed
nipkow@30056
   762
paulson@14267
   763
lemma nat_mult_1: "(1::nat) * n = n"
nipkow@25162
   764
by simp
paulson@14267
   765
paulson@14267
   766
lemma nat_mult_1_right: "n * (1::nat) = n"
nipkow@25162
   767
by simp
paulson@14267
   768
paulson@14267
   769
krauss@26748
   770
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   771
krauss@26748
   772
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   773
haftmann@27823
   774
instance nat :: wellorder proof
haftmann@27823
   775
  fix P and n :: nat
haftmann@27823
   776
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   777
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   778
  proof (induct n)
haftmann@27823
   779
    case (0 n)
krauss@26748
   780
    have "P 0" by (rule step) auto
krauss@26748
   781
    thus ?case using 0 by auto
krauss@26748
   782
  next
haftmann@27823
   783
    case (Suc m n)
haftmann@27823
   784
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   785
    thus ?case
krauss@26748
   786
    proof
haftmann@27823
   787
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   788
    next
haftmann@27823
   789
      assume n: "n = Suc m"
haftmann@27823
   790
      show "P n"
haftmann@27823
   791
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   792
    qed
krauss@26748
   793
  qed
haftmann@27823
   794
  then show "P n" by auto
krauss@26748
   795
qed
krauss@26748
   796
haftmann@27823
   797
lemma Least_Suc:
haftmann@27823
   798
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
haftmann@27823
   799
  apply (case_tac "n", auto)
haftmann@27823
   800
  apply (frule LeastI)
haftmann@27823
   801
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   802
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   803
  apply (erule_tac [2] Least_le)
haftmann@27823
   804
  apply (case_tac "LEAST x. P x", auto)
haftmann@27823
   805
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   806
  apply (blast intro: order_antisym)
haftmann@27823
   807
  done
haftmann@27823
   808
haftmann@27823
   809
lemma Least_Suc2:
haftmann@27823
   810
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   811
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   812
  apply simp
haftmann@27823
   813
  done
haftmann@27823
   814
haftmann@27823
   815
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
   816
  apply (cases n)
haftmann@27823
   817
   apply blast
haftmann@27823
   818
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   819
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   820
  done
haftmann@27823
   821
haftmann@27823
   822
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
   823
  unfolding One_nat_def
haftmann@27823
   824
  apply (cases n)
haftmann@27823
   825
   apply blast
haftmann@27823
   826
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   827
  apply (erule exE)
haftmann@27823
   828
  apply (case_tac k)
haftmann@27823
   829
   apply simp
haftmann@27823
   830
  apply (rename_tac k1)
haftmann@27823
   831
  apply (rule_tac x=k1 in exI)
haftmann@27823
   832
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   833
  done
haftmann@27823
   834
krauss@26748
   835
lemma nat_less_induct:
krauss@26748
   836
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   837
  using assms less_induct by blast
krauss@26748
   838
krauss@26748
   839
lemma measure_induct_rule [case_names less]:
krauss@26748
   840
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   841
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   842
  shows "P a"
krauss@26748
   843
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   844
krauss@26748
   845
text {* old style induction rules: *}
krauss@26748
   846
lemma measure_induct:
krauss@26748
   847
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   848
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   849
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   850
krauss@26748
   851
lemma full_nat_induct:
krauss@26748
   852
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   853
  shows "P n"
krauss@26748
   854
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   855
paulson@19870
   856
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   857
lemma less_Suc_induct:
paulson@19870
   858
  assumes less:  "i < j"
paulson@19870
   859
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   860
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   861
  shows "P i j"
paulson@19870
   862
proof -
krauss@31714
   863
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   864
  have "P i (Suc (i + k))"
paulson@19870
   865
  proof (induct k)
wenzelm@22718
   866
    case 0
wenzelm@22718
   867
    show ?case by (simp add: step)
paulson@19870
   868
  next
paulson@19870
   869
    case (Suc k)
krauss@31714
   870
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
krauss@31714
   871
    hence "i < Suc (i + k)" by (simp add: add_commute)
krauss@31714
   872
    from trans[OF this lessI Suc step]
krauss@31714
   873
    show ?case by simp
paulson@19870
   874
  qed
wenzelm@22718
   875
  thus "P i j" by (simp add: j)
paulson@19870
   876
qed
paulson@19870
   877
krauss@26748
   878
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   879
Provided by Roelof Oosterhuis.
krauss@26748
   880
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   881
\begin{itemize}
krauss@26748
   882
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   883
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   884
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   885
\end{itemize} *}
krauss@26748
   886
krauss@26748
   887
text{* A compact version without explicit base case: *}
krauss@26748
   888
lemma infinite_descent:
krauss@26748
   889
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
krauss@26748
   890
by (induct n rule: less_induct, auto)
krauss@26748
   891
krauss@26748
   892
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   893
  "\<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
   894
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   895
krauss@26748
   896
text {*
krauss@26748
   897
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   898
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   899
\begin{itemize}
krauss@26748
   900
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   901
\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
   902
\end{itemize}
krauss@26748
   903
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   904
krauss@26748
   905
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   906
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   907
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   908
  shows "P x"
krauss@26748
   909
proof -
krauss@26748
   910
  obtain n where "n = V x" by auto
krauss@26748
   911
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   912
  proof (induct n rule: infinite_descent0)
krauss@26748
   913
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   914
    with A0 show "P x" by auto
krauss@26748
   915
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   916
    case (smaller n)
krauss@26748
   917
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   918
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   919
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   920
    then show ?case by auto
krauss@26748
   921
  qed
krauss@26748
   922
  ultimately show "P x" by auto
krauss@26748
   923
qed
krauss@26748
   924
krauss@26748
   925
text{* Again, without explicit base case: *}
krauss@26748
   926
lemma infinite_descent_measure:
krauss@26748
   927
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   928
proof -
krauss@26748
   929
  from assms obtain n where "n = V x" by auto
krauss@26748
   930
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   931
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   932
    fix x assume "\<not> P x"
krauss@26748
   933
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
   934
  qed
krauss@26748
   935
  ultimately show "P x" by auto
krauss@26748
   936
qed
krauss@26748
   937
paulson@14267
   938
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
   939
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
   940
lemma less_mono_imp_le_mono:
nipkow@24438
   941
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
   942
by (simp add: order_le_less) (blast)
nipkow@24438
   943
paulson@14267
   944
paulson@14267
   945
text {* non-strict, in 1st argument *}
paulson@14267
   946
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
   947
by (rule add_right_mono)
paulson@14267
   948
paulson@14267
   949
text {* non-strict, in both arguments *}
paulson@14267
   950
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
   951
by (rule add_mono)
paulson@14267
   952
paulson@14267
   953
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
   954
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
   955
paulson@14267
   956
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
   957
by (simp add: add_commute, rule le_add2)
berghofe@13449
   958
berghofe@13449
   959
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
   960
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
   961
berghofe@13449
   962
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
   963
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
   964
paulson@14267
   965
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
   966
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
   967
paulson@14267
   968
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
   969
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
   970
paulson@14267
   971
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
   972
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
   973
berghofe@13449
   974
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
   975
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
   976
berghofe@13449
   977
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
   978
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
   979
berghofe@13449
   980
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
   981
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
   982
apply (simp_all add: le_add1)
nipkow@24438
   983
done
berghofe@13449
   984
berghofe@13449
   985
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
   986
apply (rule notI)
wenzelm@26335
   987
apply (drule add_lessD1)
wenzelm@26335
   988
apply (erule less_irrefl [THEN notE])
nipkow@24438
   989
done
berghofe@13449
   990
berghofe@13449
   991
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
   992
by (simp add: add_commute)
berghofe@13449
   993
paulson@14267
   994
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
   995
apply (rule order_trans [of _ "m+k"])
nipkow@24438
   996
apply (simp_all add: le_add1)
nipkow@24438
   997
done
berghofe@13449
   998
paulson@14267
   999
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
  1000
apply (simp add: add_commute)
nipkow@24438
  1001
apply (erule add_leD1)
nipkow@24438
  1002
done
berghofe@13449
  1003
paulson@14267
  1004
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1005
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1006
berghofe@13449
  1007
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
  1008
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1009
by (force simp del: add_Suc_right
berghofe@13449
  1010
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
  1011
berghofe@13449
  1012
haftmann@26072
  1013
subsubsection {* More results about difference *}
berghofe@13449
  1014
berghofe@13449
  1015
text {* Addition is the inverse of subtraction:
paulson@14267
  1016
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1017
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1018
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1019
paulson@14267
  1020
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1021
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1022
paulson@14267
  1023
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
  1024
by (simp add: add_commute)
berghofe@13449
  1025
paulson@14267
  1026
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1027
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1028
berghofe@13449
  1029
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1030
apply (induct m n rule: diff_induct)
nipkow@24438
  1031
apply (erule_tac [3] less_SucE)
nipkow@24438
  1032
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1033
done
berghofe@13449
  1034
paulson@14267
  1035
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1036
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1037
haftmann@26072
  1038
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1039
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1040
berghofe@13449
  1041
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1042
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1043
berghofe@13449
  1044
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1045
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1046
paulson@14267
  1047
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1048
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1049
paulson@14267
  1050
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1051
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1052
paulson@14267
  1053
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1054
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1055
paulson@14267
  1056
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1057
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1058
paulson@14267
  1059
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1060
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1061
berghofe@13449
  1062
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1063
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1064
wenzelm@22718
  1065
lemma less_imp_add_positive:
wenzelm@22718
  1066
  assumes "i < j"
wenzelm@22718
  1067
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1068
proof
wenzelm@22718
  1069
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1070
    by (simp add: order_less_imp_le)
wenzelm@22718
  1071
qed
wenzelm@9436
  1072
haftmann@26072
  1073
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1074
lemma nat_minus_add_max:
haftmann@26072
  1075
  fixes n m :: nat
haftmann@26072
  1076
  shows "n - m + m = max n m"
haftmann@26072
  1077
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1078
haftmann@26072
  1079
lemma nat_diff_split:
haftmann@26072
  1080
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1081
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1082
by (cases "a < b")
haftmann@26072
  1083
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1084
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1085
haftmann@26072
  1086
lemma nat_diff_split_asm:
haftmann@26072
  1087
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1088
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1089
by (auto split: nat_diff_split)
berghofe@13449
  1090
berghofe@13449
  1091
haftmann@26072
  1092
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1093
paulson@14267
  1094
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1095
by (simp add: mult_right_mono)
berghofe@13449
  1096
paulson@14267
  1097
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1098
by (simp add: mult_left_mono)
berghofe@13449
  1099
paulson@14267
  1100
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1101
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1102
by (simp add: mult_mono)
berghofe@13449
  1103
berghofe@13449
  1104
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1105
by (simp add: mult_strict_right_mono)
berghofe@13449
  1106
paulson@14266
  1107
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1108
      there are no negative numbers.*}
paulson@14266
  1109
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1110
  apply (induct m)
wenzelm@22718
  1111
   apply simp
wenzelm@22718
  1112
  apply (case_tac n)
wenzelm@22718
  1113
   apply simp_all
berghofe@13449
  1114
  done
berghofe@13449
  1115
huffman@30079
  1116
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1117
  apply (induct m)
wenzelm@22718
  1118
   apply simp
wenzelm@22718
  1119
  apply (case_tac n)
wenzelm@22718
  1120
   apply simp_all
berghofe@13449
  1121
  done
berghofe@13449
  1122
paulson@14341
  1123
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1124
  apply (safe intro!: mult_less_mono1)
paulson@14208
  1125
  apply (case_tac k, auto)
berghofe@13449
  1126
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1127
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1128
  done
berghofe@13449
  1129
berghofe@13449
  1130
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1131
by (simp add: mult_commute [of k])
berghofe@13449
  1132
paulson@14267
  1133
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1134
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1135
paulson@14267
  1136
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1137
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1138
berghofe@13449
  1139
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1140
by (subst mult_less_cancel1) simp
berghofe@13449
  1141
paulson@14267
  1142
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1143
by (subst mult_le_cancel1) simp
berghofe@13449
  1144
haftmann@26072
  1145
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1146
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1147
haftmann@26072
  1148
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1149
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1150
berghofe@13449
  1151
text {* Lemma for @{text gcd} *}
huffman@30128
  1152
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1153
  apply (drule sym)
berghofe@13449
  1154
  apply (rule disjCI)
berghofe@13449
  1155
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1156
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1157
    apply (auto)
berghofe@13449
  1158
  done
wenzelm@9436
  1159
haftmann@26072
  1160
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1161
haftmann@26072
  1162
instantiation nat :: distrib_lattice
haftmann@26072
  1163
begin
haftmann@24995
  1164
haftmann@26072
  1165
definition
haftmann@26072
  1166
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1167
haftmann@26072
  1168
definition
haftmann@26072
  1169
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1170
haftmann@26072
  1171
instance by intro_classes
haftmann@26072
  1172
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1173
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1174
haftmann@26072
  1175
end
haftmann@24995
  1176
haftmann@24995
  1177
haftmann@30954
  1178
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1179
haftmann@30971
  1180
text {*
haftmann@30971
  1181
  We use the same logical constant for the power operations on
haftmann@30971
  1182
  functions and relations, in order to share the same syntax.
haftmann@30971
  1183
*}
haftmann@30971
  1184
haftmann@30971
  1185
consts compow :: "nat \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a \<Rightarrow> 'b)"
haftmann@30971
  1186
haftmann@30971
  1187
abbreviation compower :: "('a \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'b" (infixr "^^" 80) where
haftmann@30971
  1188
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1189
haftmann@30971
  1190
notation (latex output)
haftmann@30971
  1191
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1192
haftmann@30971
  1193
notation (HTML output)
haftmann@30971
  1194
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1195
haftmann@30971
  1196
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1197
haftmann@30971
  1198
overloading
haftmann@30971
  1199
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1200
begin
haftmann@30954
  1201
haftmann@30954
  1202
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@30954
  1203
    "funpow 0 f = id"
haftmann@30954
  1204
  | "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1205
haftmann@30971
  1206
end
haftmann@30971
  1207
haftmann@30971
  1208
text {* for code generation *}
haftmann@30971
  1209
haftmann@30971
  1210
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@31998
  1211
  funpow_code_def [code_post]: "funpow = compow"
haftmann@30954
  1212
haftmann@31998
  1213
lemmas [code_unfold] = funpow_code_def [symmetric]
haftmann@30954
  1214
haftmann@30971
  1215
lemma [code]:
haftmann@37430
  1216
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1217
  "funpow 0 f = id"
haftmann@37430
  1218
  by (simp_all add: funpow_code_def)
haftmann@30971
  1219
wenzelm@36176
  1220
hide_const (open) funpow
haftmann@30954
  1221
haftmann@30954
  1222
lemma funpow_add:
haftmann@30971
  1223
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1224
  by (induct m) simp_all
haftmann@30954
  1225
haftmann@37430
  1226
lemma funpow_mult:
haftmann@37430
  1227
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1228
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1229
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1230
haftmann@30954
  1231
lemma funpow_swap1:
haftmann@30971
  1232
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1233
proof -
haftmann@30971
  1234
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1235
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1236
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1237
  finally show ?thesis .
haftmann@30954
  1238
qed
haftmann@30954
  1239
haftmann@38621
  1240
lemma comp_funpow:
haftmann@38621
  1241
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1242
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1243
  by (induct n) simp_all
haftmann@30954
  1244
haftmann@38621
  1245
haftmann@38621
  1246
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1247
haftmann@24196
  1248
context semiring_1
haftmann@24196
  1249
begin
haftmann@24196
  1250
haftmann@38621
  1251
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1252
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1253
haftmann@38621
  1254
lemma of_nat_simps [simp]:
haftmann@38621
  1255
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1256
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1257
  by (simp_all add: of_nat_def)
haftmann@25193
  1258
haftmann@25193
  1259
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1260
  by (simp add: of_nat_def)
haftmann@25193
  1261
haftmann@25193
  1262
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1263
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1264
haftmann@25193
  1265
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@25193
  1266
  by (induct m) (simp_all add: add_ac left_distrib)
haftmann@25193
  1267
haftmann@28514
  1268
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1269
  "of_nat_aux inc 0 i = i"
haftmann@28514
  1270
  | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1271
haftmann@30966
  1272
lemma of_nat_code:
haftmann@28514
  1273
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1274
proof (induct n)
haftmann@28514
  1275
  case 0 then show ?case by simp
haftmann@28514
  1276
next
haftmann@28514
  1277
  case (Suc n)
haftmann@28514
  1278
  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
  1279
    by (induct n) simp_all
haftmann@28514
  1280
  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
  1281
    by simp
haftmann@28514
  1282
  with Suc show ?case by (simp add: add_commute)
haftmann@28514
  1283
qed
haftmann@30966
  1284
haftmann@24196
  1285
end
haftmann@24196
  1286
haftmann@31998
  1287
declare of_nat_code [code, code_unfold, code_inline del]
haftmann@30966
  1288
haftmann@26072
  1289
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1290
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1291
haftmann@26072
  1292
class semiring_char_0 = semiring_1 +
haftmann@38621
  1293
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1294
begin
haftmann@26072
  1295
haftmann@38621
  1296
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1297
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1298
haftmann@26072
  1299
text{*Special cases where either operand is zero*}
haftmann@26072
  1300
blanchet@35828
  1301
lemma of_nat_0_eq_iff [simp, no_atp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1302
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1303
blanchet@35828
  1304
lemma of_nat_eq_0_iff [simp, no_atp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1305
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1306
haftmann@26072
  1307
end
haftmann@26072
  1308
haftmann@35028
  1309
context linordered_semidom
haftmann@25193
  1310
begin
haftmann@25193
  1311
haftmann@25193
  1312
lemma zero_le_imp_of_nat: "0 \<le> of_nat m"
huffman@36977
  1313
  by (induct m) simp_all
haftmann@25193
  1314
haftmann@25193
  1315
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
haftmann@25193
  1316
  apply (induct m n rule: diff_induct, simp_all)
huffman@36977
  1317
  apply (rule add_pos_nonneg [OF zero_less_one zero_le_imp_of_nat])
haftmann@25193
  1318
  done
haftmann@25193
  1319
haftmann@25193
  1320
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
haftmann@25193
  1321
  apply (induct m n rule: diff_induct, simp_all)
haftmann@25193
  1322
  apply (insert zero_le_imp_of_nat)
haftmann@25193
  1323
  apply (force simp add: not_less [symmetric])
haftmann@25193
  1324
  done
haftmann@25193
  1325
haftmann@25193
  1326
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
haftmann@25193
  1327
  by (blast intro: of_nat_less_imp_less less_imp_of_nat_less)
haftmann@25193
  1328
haftmann@26072
  1329
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1330
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1331
haftmann@35028
  1332
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1333
haftmann@38621
  1334
subclass semiring_char_0 proof
haftmann@38621
  1335
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1336
haftmann@25193
  1337
text{*Special cases where either operand is zero*}
haftmann@25193
  1338
haftmann@25193
  1339
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
haftmann@25193
  1340
  by (rule of_nat_le_iff [of 0, simplified])
haftmann@25193
  1341
blanchet@35828
  1342
lemma of_nat_le_0_iff [simp, no_atp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1343
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1344
haftmann@26072
  1345
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1346
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1347
haftmann@26072
  1348
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
haftmann@26072
  1349
  by (rule of_nat_less_iff [of _ 0, simplified])
haftmann@26072
  1350
haftmann@26072
  1351
end
haftmann@26072
  1352
haftmann@26072
  1353
context ring_1
haftmann@26072
  1354
begin
haftmann@26072
  1355
haftmann@26072
  1356
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1357
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1358
haftmann@26072
  1359
end
haftmann@26072
  1360
haftmann@35028
  1361
context linordered_idom
haftmann@26072
  1362
begin
haftmann@26072
  1363
haftmann@26072
  1364
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1365
  unfolding abs_if by auto
haftmann@26072
  1366
haftmann@25193
  1367
end
haftmann@25193
  1368
haftmann@25193
  1369
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1370
  by (induct n) simp_all
haftmann@25193
  1371
haftmann@25193
  1372
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1373
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1374
haftmann@25193
  1375
haftmann@26149
  1376
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1377
haftmann@26072
  1378
context semiring_1
haftmann@25193
  1379
begin
haftmann@25193
  1380
haftmann@37767
  1381
definition Nats  :: "'a set" where
haftmann@37767
  1382
  "Nats = range of_nat"
haftmann@26072
  1383
haftmann@26072
  1384
notation (xsymbols)
haftmann@26072
  1385
  Nats  ("\<nat>")
haftmann@25193
  1386
haftmann@26072
  1387
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1388
  by (simp add: Nats_def)
haftmann@26072
  1389
haftmann@26072
  1390
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1391
apply (simp add: Nats_def)
haftmann@26072
  1392
apply (rule range_eqI)
haftmann@26072
  1393
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1394
done
haftmann@25193
  1395
haftmann@26072
  1396
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1397
apply (simp add: Nats_def)
haftmann@26072
  1398
apply (rule range_eqI)
haftmann@26072
  1399
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1400
done
haftmann@25193
  1401
haftmann@26072
  1402
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1403
apply (auto simp add: Nats_def)
haftmann@26072
  1404
apply (rule range_eqI)
haftmann@26072
  1405
apply (rule of_nat_add [symmetric])
haftmann@26072
  1406
done
haftmann@26072
  1407
haftmann@26072
  1408
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1409
apply (auto simp add: Nats_def)
haftmann@26072
  1410
apply (rule range_eqI)
haftmann@26072
  1411
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1412
done
haftmann@25193
  1413
huffman@35633
  1414
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1415
  assumes "x \<in> \<nat>"
huffman@35633
  1416
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1417
  unfolding Nats_def
huffman@35633
  1418
proof -
huffman@35633
  1419
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1420
  then obtain n where "x = of_nat n" ..
huffman@35633
  1421
  then show thesis ..
huffman@35633
  1422
qed
huffman@35633
  1423
huffman@35633
  1424
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1425
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1426
  by (rule Nats_cases) auto
huffman@35633
  1427
haftmann@25193
  1428
end
haftmann@25193
  1429
haftmann@25193
  1430
wenzelm@21243
  1431
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1432
haftmann@22845
  1433
lemma subst_equals:
haftmann@22845
  1434
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1435
  shows "u = s"
haftmann@22845
  1436
  using 2 1 by (rule trans)
haftmann@22845
  1437
haftmann@30686
  1438
setup Arith_Data.setup
haftmann@30686
  1439
haftmann@30496
  1440
use "Tools/nat_arith.ML"
haftmann@30496
  1441
declaration {* K Nat_Arith.setup *}
wenzelm@24091
  1442
wenzelm@24091
  1443
use "Tools/lin_arith.ML"
haftmann@31100
  1444
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1445
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1446
wenzelm@43595
  1447
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@43595
  1448
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1449
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1450
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1451
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1452
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1453
solver all the time rather than add the additional check. *)
wenzelm@43595
  1454
wenzelm@43595
  1455
wenzelm@21243
  1456
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1457
nipkow@27625
  1458
context order
nipkow@27625
  1459
begin
nipkow@27625
  1460
nipkow@27625
  1461
lemma lift_Suc_mono_le:
krauss@27627
  1462
  assumes mono: "!!n. f n \<le> f(Suc n)" and "n\<le>n'"
krauss@27627
  1463
  shows "f n \<le> f n'"
krauss@27627
  1464
proof (cases "n < n'")
krauss@27627
  1465
  case True
krauss@27627
  1466
  thus ?thesis
krauss@27627
  1467
    by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
krauss@27627
  1468
qed (insert `n \<le> n'`, auto) -- {*trivial for @{prop "n = n'"} *}
nipkow@27625
  1469
nipkow@27625
  1470
lemma lift_Suc_mono_less:
krauss@27627
  1471
  assumes mono: "!!n. f n < f(Suc n)" and "n < n'"
krauss@27627
  1472
  shows "f n < f n'"
krauss@27627
  1473
using `n < n'`
krauss@27627
  1474
by (induct n n' rule: less_Suc_induct[consumes 1]) (auto intro: mono)
nipkow@27625
  1475
nipkow@27789
  1476
lemma lift_Suc_mono_less_iff:
nipkow@27789
  1477
  "(!!n. f n < f(Suc n)) \<Longrightarrow> f(n) < f(m) \<longleftrightarrow> n<m"
nipkow@27789
  1478
by(blast intro: less_asym' lift_Suc_mono_less[of f]
nipkow@27789
  1479
         dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq[THEN iffD1])
nipkow@27789
  1480
nipkow@27625
  1481
end
nipkow@27625
  1482
nipkow@29879
  1483
lemma mono_iff_le_Suc: "mono f = (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1484
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1485
nipkow@27789
  1486
lemma mono_nat_linear_lb:
nipkow@27789
  1487
  "(!!m n::nat. m<n \<Longrightarrow> f m < f n) \<Longrightarrow> f(m)+k \<le> f(m+k)"
nipkow@27789
  1488
apply(induct_tac k)
nipkow@27789
  1489
 apply simp
nipkow@27789
  1490
apply(erule_tac x="m+n" in meta_allE)
huffman@30079
  1491
apply(erule_tac x="Suc(m+n)" in meta_allE)
nipkow@27789
  1492
apply simp
nipkow@27789
  1493
done
nipkow@27789
  1494
nipkow@27789
  1495
wenzelm@21243
  1496
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1497
wenzelm@21243
  1498
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1499
by arith
wenzelm@21243
  1500
wenzelm@21243
  1501
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1502
by arith
wenzelm@21243
  1503
wenzelm@21243
  1504
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1505
by arith
wenzelm@21243
  1506
wenzelm@21243
  1507
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1508
by arith
wenzelm@21243
  1509
wenzelm@21243
  1510
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1511
by arith
wenzelm@21243
  1512
wenzelm@21243
  1513
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1514
by arith
wenzelm@21243
  1515
wenzelm@21243
  1516
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1517
  second premise n\<le>m*)
wenzelm@21243
  1518
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1519
by arith
wenzelm@21243
  1520
haftmann@26072
  1521
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1522
wenzelm@21243
  1523
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1524
by (simp split add: nat_diff_split)
wenzelm@21243
  1525
wenzelm@36176
  1526
hide_fact (open) diff_diff_eq
haftmann@35064
  1527
wenzelm@21243
  1528
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1529
by (auto split add: nat_diff_split)
wenzelm@21243
  1530
wenzelm@21243
  1531
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1532
by (auto split add: nat_diff_split)
wenzelm@21243
  1533
wenzelm@21243
  1534
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1535
by (auto split add: nat_diff_split)
wenzelm@21243
  1536
wenzelm@21243
  1537
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1538
wenzelm@21243
  1539
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1540
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1541
by (simp split add: nat_diff_split)
wenzelm@21243
  1542
wenzelm@21243
  1543
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1544
by (simp split add: nat_diff_split)
wenzelm@21243
  1545
wenzelm@21243
  1546
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1547
by (simp split add: nat_diff_split)
wenzelm@21243
  1548
wenzelm@21243
  1549
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1550
by (simp split add: nat_diff_split)
wenzelm@21243
  1551
bulwahn@26143
  1552
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1553
by auto
bulwahn@26143
  1554
bulwahn@26143
  1555
lemma inj_on_diff_nat: 
bulwahn@26143
  1556
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1557
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1558
proof (rule inj_onI)
bulwahn@26143
  1559
  fix x y
bulwahn@26143
  1560
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1561
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1562
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1563
qed
bulwahn@26143
  1564
haftmann@26072
  1565
text{*Rewriting to pull differences out*}
haftmann@26072
  1566
haftmann@26072
  1567
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1568
by arith
haftmann@26072
  1569
haftmann@26072
  1570
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1571
by arith
haftmann@26072
  1572
haftmann@26072
  1573
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1574
by arith
haftmann@26072
  1575
wenzelm@21243
  1576
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1577
wenzelm@21243
  1578
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1579
by (cases m) auto
wenzelm@21243
  1580
wenzelm@21243
  1581
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1582
by (cases m) auto
wenzelm@21243
  1583
wenzelm@21243
  1584
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1585
by (cases m) auto
wenzelm@21243
  1586
krauss@23001
  1587
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1588
krauss@23001
  1589
lemma inc_induct[consumes 1, case_names base step]:
krauss@23001
  1590
  assumes less: "i <= j"
krauss@23001
  1591
  assumes base: "P j"
krauss@23001
  1592
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1593
  shows "P i"
krauss@23001
  1594
  using less
krauss@23001
  1595
proof (induct d=="j - i" arbitrary: i)
krauss@23001
  1596
  case (0 i)
krauss@23001
  1597
  hence "i = j" by simp
krauss@23001
  1598
  with base show ?case by simp
krauss@23001
  1599
next
krauss@23001
  1600
  case (Suc d i)
krauss@23001
  1601
  hence "i < j" "P (Suc i)"
krauss@23001
  1602
    by simp_all
krauss@23001
  1603
  thus "P i" by (rule step)
krauss@23001
  1604
qed
krauss@23001
  1605
krauss@23001
  1606
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1607
  assumes less: "i < j"
krauss@23001
  1608
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1609
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1610
  shows "P i"
krauss@23001
  1611
  using less
krauss@23001
  1612
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1613
  case (0 i)
krauss@23001
  1614
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1615
  with base show ?case by simp
krauss@23001
  1616
next
krauss@23001
  1617
  case (Suc d i)
krauss@23001
  1618
  hence "i < j" "P (Suc i)"
krauss@23001
  1619
    by simp_all
krauss@23001
  1620
  thus "P i" by (rule step)
krauss@23001
  1621
qed
krauss@23001
  1622
krauss@23001
  1623
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1624
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1625
krauss@23001
  1626
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1627
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1628
wenzelm@21243
  1629
(*The others are
wenzelm@21243
  1630
      i - j - k = i - (j + k),
wenzelm@21243
  1631
      k \<le> j ==> j - k + i = j + i - k,
wenzelm@21243
  1632
      k \<le> j ==> i + (j - k) = i + j - k *)
wenzelm@21243
  1633
lemmas add_diff_assoc = diff_add_assoc [symmetric]
wenzelm@21243
  1634
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
haftmann@26072
  1635
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
wenzelm@21243
  1636
wenzelm@21243
  1637
text{*At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@21243
  1638
Least_Suc}, since there appears to be no need.*}
wenzelm@21243
  1639
nipkow@27625
  1640
haftmann@33274
  1641
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1642
haftmann@33274
  1643
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1644
unfolding dvd_def by simp
haftmann@33274
  1645
haftmann@33274
  1646
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1647
by (simp add: dvd_def)
haftmann@33274
  1648
haftmann@33274
  1649
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1650
by (simp add: dvd_def)
haftmann@33274
  1651
nipkow@33657
  1652
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1653
  unfolding dvd_def
huffman@35216
  1654
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
haftmann@33274
  1655
haftmann@33274
  1656
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1657
haftmann@33274
  1658
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1659
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1660
haftmann@33274
  1661
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1662
unfolding dvd_def
haftmann@33274
  1663
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1664
haftmann@33274
  1665
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1666
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1667
  apply (blast intro: dvd_add)
haftmann@33274
  1668
  done
haftmann@33274
  1669
haftmann@33274
  1670
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1671
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1672
haftmann@33274
  1673
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
haftmann@33274
  1674
  apply (rule iffI)
haftmann@33274
  1675
   apply (erule_tac [2] dvd_add)
haftmann@33274
  1676
   apply (rule_tac [2] dvd_refl)
haftmann@33274
  1677
  apply (subgoal_tac "n = (n+k) -k")
haftmann@33274
  1678
   prefer 2 apply simp
haftmann@33274
  1679
  apply (erule ssubst)
haftmann@33274
  1680
  apply (erule dvd_diff_nat)
haftmann@33274
  1681
  apply (rule dvd_refl)
haftmann@33274
  1682
  done
haftmann@33274
  1683
haftmann@33274
  1684
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1685
  unfolding dvd_def
haftmann@33274
  1686
  apply (erule exE)
haftmann@33274
  1687
  apply (simp add: mult_ac)
haftmann@33274
  1688
  done
haftmann@33274
  1689
haftmann@33274
  1690
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1691
  apply auto
haftmann@33274
  1692
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1693
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1694
  done
haftmann@33274
  1695
haftmann@33274
  1696
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@33274
  1697
  apply (subst mult_commute)
haftmann@33274
  1698
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1699
  done
haftmann@33274
  1700
haftmann@33274
  1701
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1702
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1703
haftmann@33274
  1704
lemma nat_dvd_not_less:
haftmann@33274
  1705
  fixes m n :: nat
haftmann@33274
  1706
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1707
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1708
haftmann@33274
  1709
haftmann@26072
  1710
subsection {* size of a datatype value *}
haftmann@25193
  1711
haftmann@29608
  1712
class size =
krauss@26748
  1713
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1714
haftmann@33364
  1715
haftmann@33364
  1716
subsection {* code module namespace *}
haftmann@33364
  1717
haftmann@33364
  1718
code_modulename SML
haftmann@33364
  1719
  Nat Arith
haftmann@33364
  1720
haftmann@33364
  1721
code_modulename OCaml
haftmann@33364
  1722
  Nat Arith
haftmann@33364
  1723
haftmann@33364
  1724
code_modulename Haskell
haftmann@33364
  1725
  Nat Arith
haftmann@33364
  1726
haftmann@25193
  1727
end