src/HOL/Nat.thy
author blanchet
Wed Feb 12 08:37:28 2014 +0100 (2014-02-12)
changeset 55423 07dea66779f3
parent 55417 01fbfb60c33e
child 55424 9ab4129a76a3
permissions -rw-r--r--
for extraction -- use the exhaust rule that's registered with 'datatype_realizer.ML'
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(*  Title:      HOL/Nat.thy
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    Author:     Tobias Nipkow and Lawrence C Paulson and Markus Wenzel
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Type "nat" is a linear order, and a datatype; arithmetic operators + -
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and * (for div and mod, see theory Divides).
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*)
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header {* Natural numbers *}
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theory Nat
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imports Inductive Typedef Fun Fields
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begin
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ML_file "~~/src/Tools/rat.ML"
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ML_file "Tools/arith_data.ML"
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ML_file "~~/src/Provers/Arith/fast_lin_arith.ML"
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subsection {* Type @{text ind} *}
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typedecl ind
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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
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  -- {* the axiom of infinity in 2 parts *}
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  Suc_Rep_inject:       "Suc_Rep x = Suc_Rep y ==> x = y" and
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  Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"
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subsection {* Type nat *}
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text {* Type definition *}
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inductive Nat :: "ind \<Rightarrow> bool" where
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  Zero_RepI: "Nat Zero_Rep"
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| Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"
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typedef nat = "{n. Nat n}"
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  morphisms Rep_Nat Abs_Nat
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  using Nat.Zero_RepI by auto
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lemma Nat_Rep_Nat:
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  "Nat (Rep_Nat n)"
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  using Rep_Nat by simp
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lemma Nat_Abs_Nat_inverse:
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  "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
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  using Abs_Nat_inverse by simp
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lemma Nat_Abs_Nat_inject:
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  "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
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  using Abs_Nat_inject by simp
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instantiation nat :: zero
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begin
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definition Zero_nat_def:
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  "0 = Abs_Nat Zero_Rep"
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instance ..
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end
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definition Suc :: "nat \<Rightarrow> nat" where
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  "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"
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lemma Suc_not_Zero: "Suc m \<noteq> 0"
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  by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)
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lemma Zero_not_Suc: "0 \<noteq> Suc m"
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  by (rule not_sym, rule Suc_not_Zero not_sym)
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lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
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  by (rule iffI, rule Suc_Rep_inject) simp_all
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lemma nat_induct0:
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  fixes n
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  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
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  shows "P n"
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using assms
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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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done
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wrap_free_constructors ["0 \<Colon> nat", Suc] case_nat [=] [[], [pred]]
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  apply atomize_elim
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  apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
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 apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI
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   Suc_Rep_inject' Rep_Nat_inject)
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apply (simp only: Suc_not_Zero)
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done
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-- {* Avoid name clashes by prefixing the output of @{text rep_datatype} with @{text old}. *}
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setup {* Sign.mandatory_path "old" *}
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rep_datatype "0 \<Colon> nat" Suc
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  apply (erule nat_induct0, assumption)
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 apply (rule nat.inject)
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apply (rule nat.distinct(1))
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done
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setup {* Sign.parent_path *}
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-- {* But erase the prefix for properties that are not generated by @{text wrap_free_constructors}. *}
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setup {* Sign.mandatory_path "nat" *}
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declare
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  old.nat.inject[iff del]
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  old.nat.distinct(1)[simp del, induct_simp del]
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lemmas induct = old.nat.induct
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lemmas inducts = old.nat.inducts
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lemmas recs = old.nat.recs
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lemmas cases = nat.case
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lemmas simps = nat.inject nat.distinct nat.case old.nat.recs
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setup {* Sign.parent_path *}
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abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a" where
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  "rec_nat \<equiv> old.rec_nat"
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hide_const Nat.pred -- {* hide everything related to the selector *}
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hide_fact
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  nat.case_eq_if
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  nat.collapse
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  nat.expand
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  nat.sel
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  nat.sel_exhaust
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  nat.sel_split
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  nat.sel_split_asm
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lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
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  -- {* for backward compatibility -- names of variables differ *}
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  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
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by (rule old.nat.exhaust)
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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" 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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hide_fact
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  nat_exhaust
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  nat_induct0
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text {* Injectiveness and distinctness lemmas *}
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lemma inj_Suc[simp]: "inj_on Suc N"
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  by (simp add: inj_on_def)
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lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
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by (rule notE, rule Suc_not_Zero)
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lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
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by (rule Suc_neq_Zero, erule sym)
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lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
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by (rule inj_Suc [THEN injD])
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lemma n_not_Suc_n: "n \<noteq> Suc n"
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by (induct n) simp_all
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lemma Suc_n_not_n: "Suc n \<noteq> n"
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by (rule not_sym, rule n_not_Suc_n)
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text {* A special form of induction for reasoning
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  about @{term "m < n"} and @{term "m - n"} *}
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lemma diff_induct: "(!!x. P x 0) ==> (!!y. P 0 (Suc y)) ==>
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    (!!x y. P x y ==> P (Suc x) (Suc y)) ==> P m n"
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  apply (rule_tac x = m in spec)
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  apply (induct n)
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  prefer 2
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  apply (rule allI)
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  apply (induct_tac x, iprover+)
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  done
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subsection {* Arithmetic operators *}
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instantiation nat :: comm_monoid_diff
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begin
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primrec plus_nat where
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  add_0:      "0 + n = (n\<Colon>nat)"
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| add_Suc:  "Suc m + n = Suc (m + n)"
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lemma add_0_right [simp]: "m + 0 = (m::nat)"
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  by (induct m) simp_all
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lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
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  by (induct m) simp_all
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declare add_0 [code]
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lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
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  by simp
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primrec minus_nat where
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  diff_0 [code]: "m - 0 = (m\<Colon>nat)"
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| diff_Suc: "m - Suc n = (case m - n of 0 => 0 | Suc k => k)"
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declare diff_Suc [simp del]
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lemma diff_0_eq_0 [simp, code]: "0 - n = (0::nat)"
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  by (induct n) (simp_all add: diff_Suc)
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lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
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  by (induct n) (simp_all add: diff_Suc)
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instance proof
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  fix n m q :: nat
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  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
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  show "n + m = m + n" by (induct n) simp_all
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  show "0 + n = n" by simp
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  show "n - 0 = n" by simp
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  show "0 - n = 0" by simp
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  show "(q + n) - (q + m) = n - m" by (induct q) simp_all
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  show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
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qed
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end
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hide_fact (open) add_0 add_0_right diff_0
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition
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  One_nat_def [simp]: "1 = Suc 0"
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primrec times_nat where
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  mult_0:     "0 * n = (0\<Colon>nat)"
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| mult_Suc: "Suc m * n = n + (m * n)"
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lemma mult_0_right [simp]: "(m::nat) * 0 = 0"
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  by (induct m) simp_all
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lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
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  by (induct m) (simp_all add: add_left_commute)
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lemma add_mult_distrib: "(m + n) * k = (m * k) + ((n * k)::nat)"
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  by (induct m) (simp_all add: add_assoc)
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instance proof
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  fix n m q :: nat
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  show "0 \<noteq> (1::nat)" unfolding One_nat_def by simp
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  show "1 * n = n" unfolding One_nat_def by simp
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  show "n * m = m * n" by (induct n) simp_all
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  show "(n * m) * q = n * (m * q)" by (induct n) (simp_all add: add_mult_distrib)
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  show "(n + m) * q = n * q + m * q" by (rule add_mult_distrib)
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  assume "n + m = n + q" thus "m = q" by (induct n) simp_all
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qed
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end
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subsubsection {* Addition *}
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lemma nat_add_assoc: "(m + n) + k = m + ((n + k)::nat)"
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  by (rule add_assoc)
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lemma nat_add_commute: "m + n = n + (m::nat)"
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  by (rule add_commute)
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lemma nat_add_left_commute: "x + (y + z) = y + ((x + z)::nat)"
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  by (rule add_left_commute)
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lemma nat_add_left_cancel [simp]: "(k + m = k + n) = (m = (n::nat))"
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  by (rule add_left_cancel)
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lemma nat_add_right_cancel [simp]: "(m + k = n + k) = (m=(n::nat))"
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  by (rule add_right_cancel)
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text {* Reasoning about @{text "m + 0 = 0"}, etc. *}
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lemma add_is_0 [iff]:
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  fixes m n :: nat
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  shows "(m + n = 0) = (m = 0 & n = 0)"
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  by (cases m) simp_all
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lemma add_is_1:
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  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
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  by (cases m) simp_all
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lemma one_is_add:
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  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
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  by (rule trans, rule eq_commute, rule add_is_1)
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lemma add_eq_self_zero:
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  fixes m n :: nat
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  shows "m + n = m \<Longrightarrow> n = 0"
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  by (induct m) simp_all
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lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
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  apply (induct k)
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   apply simp
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  apply(drule comp_inj_on[OF _ inj_Suc])
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  apply (simp add:o_def)
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  done
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lemma Suc_eq_plus1: "Suc n = n + 1"
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  unfolding One_nat_def by simp
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lemma Suc_eq_plus1_left: "Suc n = 1 + n"
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  unfolding One_nat_def by simp
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subsubsection {* Difference *}
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lemma diff_self_eq_0 [simp]: "(m\<Colon>nat) - m = 0"
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  by (induct m) simp_all
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lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
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  by (induct i j rule: diff_induct) simp_all
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lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
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  by (simp add: diff_diff_left)
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lemma diff_commute: "(i::nat) - j - k = i - k - j"
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  by (simp add: diff_diff_left add_commute)
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lemma diff_add_inverse: "(n + m) - n = (m::nat)"
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  by (induct n) simp_all
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lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
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  by (simp add: diff_add_inverse add_commute [of m n])
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lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
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  by (induct k) simp_all
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lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
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  by (simp add: diff_cancel add_commute)
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lemma diff_add_0: "n - (n + m) = (0::nat)"
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  by (induct n) simp_all
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lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
huffman@30093
   340
  unfolding One_nat_def by simp
huffman@30093
   341
haftmann@26072
   342
text {* Difference distributes over multiplication *}
haftmann@26072
   343
haftmann@26072
   344
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
haftmann@26072
   345
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
haftmann@26072
   346
haftmann@26072
   347
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
haftmann@26072
   348
by (simp add: diff_mult_distrib mult_commute [of k])
haftmann@26072
   349
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
haftmann@26072
   350
haftmann@26072
   351
haftmann@26072
   352
subsubsection {* Multiplication *}
haftmann@26072
   353
haftmann@26072
   354
lemma nat_mult_assoc: "(m * n) * k = m * ((n * k)::nat)"
haftmann@26072
   355
  by (rule mult_assoc)
haftmann@26072
   356
haftmann@26072
   357
lemma nat_mult_commute: "m * n = n * (m::nat)"
haftmann@26072
   358
  by (rule mult_commute)
haftmann@26072
   359
haftmann@26072
   360
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
webertj@49962
   361
  by (rule distrib_left)
haftmann@26072
   362
haftmann@26072
   363
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   364
  by (induct m) auto
haftmann@26072
   365
haftmann@26072
   366
lemmas nat_distrib =
haftmann@26072
   367
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
haftmann@26072
   368
huffman@30079
   369
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   370
  apply (induct m)
haftmann@26072
   371
   apply simp
haftmann@26072
   372
  apply (induct n)
haftmann@26072
   373
   apply auto
haftmann@26072
   374
  done
haftmann@26072
   375
blanchet@54147
   376
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   377
  apply (rule trans)
nipkow@44890
   378
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
haftmann@26072
   379
  done
haftmann@26072
   380
huffman@30079
   381
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   382
  unfolding One_nat_def by (rule mult_eq_1_iff)
huffman@30079
   383
huffman@30079
   384
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   385
  unfolding One_nat_def by (rule one_eq_mult_iff)
huffman@30079
   386
haftmann@26072
   387
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   388
proof -
haftmann@26072
   389
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   390
  proof (induct n arbitrary: m)
haftmann@26072
   391
    case 0 then show "m = 0" by simp
haftmann@26072
   392
  next
haftmann@26072
   393
    case (Suc n) then show "m = Suc n"
haftmann@26072
   394
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   395
  qed
haftmann@26072
   396
  then show ?thesis by auto
haftmann@26072
   397
qed
haftmann@26072
   398
haftmann@26072
   399
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@26072
   400
  by (simp add: mult_commute)
haftmann@26072
   401
haftmann@26072
   402
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   403
  by (subst mult_cancel1) simp
haftmann@26072
   404
haftmann@24995
   405
haftmann@24995
   406
subsection {* Orders on @{typ nat} *}
haftmann@24995
   407
haftmann@26072
   408
subsubsection {* Operation definition *}
haftmann@24995
   409
haftmann@26072
   410
instantiation nat :: linorder
haftmann@25510
   411
begin
haftmann@25510
   412
haftmann@26072
   413
primrec less_eq_nat where
haftmann@26072
   414
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   415
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   416
haftmann@28514
   417
declare less_eq_nat.simps [simp del]
haftmann@26072
   418
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@54223
   419
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by simp
haftmann@26072
   420
haftmann@26072
   421
definition less_nat where
haftmann@28514
   422
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   423
haftmann@26072
   424
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   425
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   426
haftmann@26072
   427
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   428
  unfolding less_eq_Suc_le ..
haftmann@26072
   429
haftmann@26072
   430
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   431
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   432
haftmann@26072
   433
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   434
  by (simp add: less_eq_Suc_le)
haftmann@26072
   435
haftmann@26072
   436
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   437
  by simp
haftmann@26072
   438
haftmann@26072
   439
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   440
  by (simp add: less_eq_Suc_le)
haftmann@26072
   441
haftmann@26072
   442
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   443
  by (simp add: less_eq_Suc_le)
haftmann@26072
   444
haftmann@26072
   445
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   446
  by (induct m arbitrary: n)
haftmann@26072
   447
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   448
haftmann@26072
   449
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   450
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   451
haftmann@26072
   452
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   453
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   454
haftmann@26072
   455
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   456
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   457
wenzelm@26315
   458
instance
wenzelm@26315
   459
proof
haftmann@26072
   460
  fix n m :: nat
haftmann@27679
   461
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   462
  proof (induct n arbitrary: m)
haftmann@27679
   463
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   464
  next
haftmann@27679
   465
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   466
  qed
haftmann@26072
   467
next
haftmann@26072
   468
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   469
next
haftmann@26072
   470
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   471
  then show "n = m"
haftmann@26072
   472
    by (induct n arbitrary: m)
haftmann@26072
   473
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   474
next
haftmann@26072
   475
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   476
  then show "n \<le> q"
haftmann@26072
   477
  proof (induct n arbitrary: m q)
haftmann@26072
   478
    case 0 show ?case by simp
haftmann@26072
   479
  next
haftmann@26072
   480
    case (Suc n) then show ?case
haftmann@26072
   481
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   482
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   483
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   484
  qed
haftmann@26072
   485
next
haftmann@26072
   486
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   487
    by (induct n arbitrary: m)
haftmann@26072
   488
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   489
qed
haftmann@25510
   490
haftmann@25510
   491
end
berghofe@13449
   492
haftmann@52729
   493
instantiation nat :: order_bot
haftmann@29652
   494
begin
haftmann@29652
   495
haftmann@29652
   496
definition bot_nat :: nat where
haftmann@29652
   497
  "bot_nat = 0"
haftmann@29652
   498
haftmann@29652
   499
instance proof
haftmann@29652
   500
qed (simp add: bot_nat_def)
haftmann@29652
   501
haftmann@29652
   502
end
haftmann@29652
   503
hoelzl@51329
   504
instance nat :: no_top
haftmann@52289
   505
  by default (auto intro: less_Suc_eq_le [THEN iffD2])
haftmann@52289
   506
hoelzl@51329
   507
haftmann@26072
   508
subsubsection {* Introduction properties *}
berghofe@13449
   509
haftmann@26072
   510
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   511
  by (simp add: less_Suc_eq_le)
berghofe@13449
   512
haftmann@26072
   513
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   514
  by (simp add: less_Suc_eq_le)
berghofe@13449
   515
berghofe@13449
   516
berghofe@13449
   517
subsubsection {* Elimination properties *}
berghofe@13449
   518
berghofe@13449
   519
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   520
  by (rule order_less_irrefl)
berghofe@13449
   521
wenzelm@26335
   522
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   523
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   524
paulson@14267
   525
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   526
  by (rule less_imp_neq)
berghofe@13449
   527
wenzelm@26335
   528
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   529
  by (rule notE, rule less_not_refl)
berghofe@13449
   530
berghofe@13449
   531
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   532
  by (rule notE) (rule not_less0)
berghofe@13449
   533
berghofe@13449
   534
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   535
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   536
huffman@30079
   537
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   538
  by (simp add: less_Suc_eq)
berghofe@13449
   539
blanchet@54147
   540
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   541
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   542
berghofe@13449
   543
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   544
  by simp
berghofe@13449
   545
nipkow@14302
   546
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   547
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   548
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   549
paulson@14267
   550
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   551
  by (rule linorder_neq_iff)
berghofe@13449
   552
berghofe@13449
   553
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   554
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   555
  shows "P n m"
berghofe@13449
   556
  apply (rule less_linear [THEN disjE])
berghofe@13449
   557
  apply (erule_tac [2] disjE)
berghofe@13449
   558
  apply (erule lessCase)
berghofe@13449
   559
  apply (erule sym [THEN eqCase])
berghofe@13449
   560
  apply (erule major)
berghofe@13449
   561
  done
berghofe@13449
   562
berghofe@13449
   563
berghofe@13449
   564
subsubsection {* Inductive (?) properties *}
berghofe@13449
   565
paulson@14267
   566
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   567
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   568
haftmann@26072
   569
lemma lessE:
haftmann@26072
   570
  assumes major: "i < k"
haftmann@26072
   571
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   572
  shows P
haftmann@26072
   573
proof -
haftmann@26072
   574
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   575
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   576
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   577
    by (clarsimp simp add: less_le)
haftmann@26072
   578
  with p1 p2 show P by auto
haftmann@26072
   579
qed
haftmann@26072
   580
haftmann@26072
   581
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   582
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   583
  apply (rule major [THEN lessE])
haftmann@26072
   584
  apply (rule eq, blast)
haftmann@26072
   585
  apply (rule less, blast)
berghofe@13449
   586
  done
berghofe@13449
   587
berghofe@13449
   588
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   589
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   590
  apply (rule major [THEN lessE])
berghofe@13449
   591
  apply (erule lessI [THEN minor])
paulson@14208
   592
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   593
  done
berghofe@13449
   594
berghofe@13449
   595
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   596
  by simp
berghofe@13449
   597
berghofe@13449
   598
lemma less_trans_Suc:
berghofe@13449
   599
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   600
  apply (induct k, simp_all)
berghofe@13449
   601
  apply (insert le)
berghofe@13449
   602
  apply (simp add: less_Suc_eq)
berghofe@13449
   603
  apply (blast dest: Suc_lessD)
berghofe@13449
   604
  done
berghofe@13449
   605
berghofe@13449
   606
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   607
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   608
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   609
haftmann@26072
   610
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   611
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   612
haftmann@24995
   613
text {* Properties of "less than or equal" *}
berghofe@13449
   614
paulson@14267
   615
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   616
  unfolding less_Suc_eq_le .
berghofe@13449
   617
paulson@14267
   618
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   619
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   620
paulson@14267
   621
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   622
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   623
paulson@14267
   624
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   625
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   626
paulson@14267
   627
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   628
  unfolding Suc_le_eq .
berghofe@13449
   629
berghofe@13449
   630
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   631
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   632
  unfolding Suc_le_eq .
berghofe@13449
   633
wenzelm@26315
   634
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   635
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   636
paulson@14267
   637
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   638
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   639
berghofe@13449
   640
paulson@14267
   641
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   642
paulson@14267
   643
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   644
  unfolding le_less .
berghofe@13449
   645
paulson@14267
   646
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   647
  by (rule le_less)
berghofe@13449
   648
wenzelm@22718
   649
text {* Useful with @{text blast}. *}
paulson@14267
   650
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   651
  by auto
berghofe@13449
   652
paulson@14267
   653
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   654
  by simp
berghofe@13449
   655
paulson@14267
   656
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   657
  by (rule order_trans)
berghofe@13449
   658
nipkow@33657
   659
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   660
  by (rule antisym)
berghofe@13449
   661
paulson@14267
   662
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   663
  by (rule less_le)
berghofe@13449
   664
paulson@14267
   665
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   666
  unfolding less_le ..
berghofe@13449
   667
haftmann@26072
   668
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   669
  by (rule linear)
paulson@14341
   670
wenzelm@22718
   671
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   672
haftmann@26072
   673
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   674
  unfolding less_Suc_eq_le by auto
berghofe@13449
   675
haftmann@26072
   676
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   677
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   678
berghofe@13449
   679
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   680
paulson@14267
   681
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   682
by (cases n) simp_all
nipkow@25162
   683
nipkow@25162
   684
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   685
by (cases n) simp_all
berghofe@13449
   686
wenzelm@22718
   687
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   688
by (cases n) simp_all
berghofe@13449
   689
nipkow@25162
   690
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   691
by (cases n) simp_all
nipkow@25140
   692
berghofe@13449
   693
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   694
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   695
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   696
paulson@14267
   697
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   698
by (fast intro: not0_implies_Suc)
berghofe@13449
   699
blanchet@54147
   700
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   701
using neq0_conv by blast
berghofe@13449
   702
paulson@14267
   703
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   704
by (induct m') simp_all
berghofe@13449
   705
berghofe@13449
   706
text {* Useful in certain inductive arguments *}
paulson@14267
   707
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   708
by (cases m) simp_all
berghofe@13449
   709
berghofe@13449
   710
haftmann@26072
   711
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   712
haftmann@26072
   713
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   714
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   715
huffman@30128
   716
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   717
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   718
paulson@14331
   719
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   720
by (induct k) simp_all
berghofe@13449
   721
paulson@14331
   722
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   723
by (induct k) simp_all
berghofe@13449
   724
nipkow@25162
   725
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   726
by(auto dest:gr0_implies_Suc)
berghofe@13449
   727
paulson@14341
   728
text {* strict, in 1st argument *}
paulson@14341
   729
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   730
by (induct k) simp_all
paulson@14341
   731
paulson@14341
   732
text {* strict, in both arguments *}
paulson@14341
   733
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   734
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   735
  apply (induct j, simp_all)
paulson@14341
   736
  done
paulson@14341
   737
paulson@14341
   738
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   739
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   740
  apply (induct n)
paulson@14341
   741
  apply (simp_all add: order_le_less)
wenzelm@22718
   742
  apply (blast elim!: less_SucE
haftmann@35047
   743
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   744
  done
paulson@14341
   745
paulson@14341
   746
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   747
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   748
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   749
apply (induct_tac m)
nipkow@25134
   750
apply (simp_all add: add_less_mono)
nipkow@25134
   751
done
paulson@14341
   752
nipkow@14740
   753
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   754
instance nat :: linordered_semidom
paulson@14341
   755
proof
paulson@14348
   756
  show "0 < (1::nat)" by simp
haftmann@52289
   757
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
haftmann@52289
   758
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
paulson@14267
   759
qed
paulson@14267
   760
nipkow@30056
   761
instance nat :: no_zero_divisors
nipkow@30056
   762
proof
nipkow@30056
   763
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
nipkow@30056
   764
qed
nipkow@30056
   765
haftmann@44817
   766
haftmann@44817
   767
subsubsection {* @{term min} and @{term max} *}
haftmann@44817
   768
haftmann@44817
   769
lemma mono_Suc: "mono Suc"
haftmann@44817
   770
by (rule monoI) simp
haftmann@44817
   771
haftmann@44817
   772
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   773
by (rule min_absorb1) simp
haftmann@44817
   774
haftmann@44817
   775
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   776
by (rule min_absorb2) simp
haftmann@44817
   777
haftmann@44817
   778
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   779
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   780
haftmann@44817
   781
lemma min_Suc1:
haftmann@44817
   782
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   783
by (simp split: nat.split)
haftmann@44817
   784
haftmann@44817
   785
lemma min_Suc2:
haftmann@44817
   786
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   787
by (simp split: nat.split)
haftmann@44817
   788
haftmann@44817
   789
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   790
by (rule max_absorb2) simp
haftmann@44817
   791
haftmann@44817
   792
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   793
by (rule max_absorb1) simp
haftmann@44817
   794
haftmann@44817
   795
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   796
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   797
haftmann@44817
   798
lemma max_Suc1:
haftmann@44817
   799
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   800
by (simp split: nat.split)
haftmann@44817
   801
haftmann@44817
   802
lemma max_Suc2:
haftmann@44817
   803
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   804
by (simp split: nat.split)
paulson@14267
   805
haftmann@44817
   806
lemma nat_mult_min_left:
haftmann@44817
   807
  fixes m n q :: nat
haftmann@44817
   808
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   809
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   810
haftmann@44817
   811
lemma nat_mult_min_right:
haftmann@44817
   812
  fixes m n q :: nat
haftmann@44817
   813
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   814
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   815
haftmann@44817
   816
lemma nat_add_max_left:
haftmann@44817
   817
  fixes m n q :: nat
haftmann@44817
   818
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   819
  by (simp add: max_def)
haftmann@44817
   820
haftmann@44817
   821
lemma nat_add_max_right:
haftmann@44817
   822
  fixes m n q :: nat
haftmann@44817
   823
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   824
  by (simp add: max_def)
haftmann@44817
   825
haftmann@44817
   826
lemma nat_mult_max_left:
haftmann@44817
   827
  fixes m n q :: nat
haftmann@44817
   828
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   829
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   830
haftmann@44817
   831
lemma nat_mult_max_right:
haftmann@44817
   832
  fixes m n q :: nat
haftmann@44817
   833
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   834
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   835
paulson@14267
   836
krauss@26748
   837
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   838
krauss@26748
   839
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   840
haftmann@27823
   841
instance nat :: wellorder proof
haftmann@27823
   842
  fix P and n :: nat
haftmann@27823
   843
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   844
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   845
  proof (induct n)
haftmann@27823
   846
    case (0 n)
krauss@26748
   847
    have "P 0" by (rule step) auto
krauss@26748
   848
    thus ?case using 0 by auto
krauss@26748
   849
  next
haftmann@27823
   850
    case (Suc m n)
haftmann@27823
   851
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   852
    thus ?case
krauss@26748
   853
    proof
haftmann@27823
   854
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   855
    next
haftmann@27823
   856
      assume n: "n = Suc m"
haftmann@27823
   857
      show "P n"
haftmann@27823
   858
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   859
    qed
krauss@26748
   860
  qed
haftmann@27823
   861
  then show "P n" by auto
krauss@26748
   862
qed
krauss@26748
   863
haftmann@27823
   864
lemma Least_Suc:
haftmann@27823
   865
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   866
  apply (cases n, auto)
haftmann@27823
   867
  apply (frule LeastI)
haftmann@27823
   868
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   869
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   870
  apply (erule_tac [2] Least_le)
wenzelm@47988
   871
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   872
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   873
  apply (blast intro: order_antisym)
haftmann@27823
   874
  done
haftmann@27823
   875
haftmann@27823
   876
lemma Least_Suc2:
haftmann@27823
   877
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   878
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   879
  apply simp
haftmann@27823
   880
  done
haftmann@27823
   881
haftmann@27823
   882
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
   883
  apply (cases n)
haftmann@27823
   884
   apply blast
haftmann@27823
   885
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   886
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   887
  done
haftmann@27823
   888
haftmann@27823
   889
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
   890
  unfolding One_nat_def
haftmann@27823
   891
  apply (cases n)
haftmann@27823
   892
   apply blast
haftmann@27823
   893
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   894
  apply (erule exE)
haftmann@27823
   895
  apply (case_tac k)
haftmann@27823
   896
   apply simp
haftmann@27823
   897
  apply (rename_tac k1)
haftmann@27823
   898
  apply (rule_tac x=k1 in exI)
haftmann@27823
   899
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   900
  done
haftmann@27823
   901
krauss@26748
   902
lemma nat_less_induct:
krauss@26748
   903
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   904
  using assms less_induct by blast
krauss@26748
   905
krauss@26748
   906
lemma measure_induct_rule [case_names less]:
krauss@26748
   907
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   908
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   909
  shows "P a"
krauss@26748
   910
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   911
krauss@26748
   912
text {* old style induction rules: *}
krauss@26748
   913
lemma measure_induct:
krauss@26748
   914
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   915
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   916
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   917
krauss@26748
   918
lemma full_nat_induct:
krauss@26748
   919
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   920
  shows "P n"
krauss@26748
   921
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   922
paulson@19870
   923
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   924
lemma less_Suc_induct:
paulson@19870
   925
  assumes less:  "i < j"
paulson@19870
   926
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   927
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   928
  shows "P i j"
paulson@19870
   929
proof -
krauss@31714
   930
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   931
  have "P i (Suc (i + k))"
paulson@19870
   932
  proof (induct k)
wenzelm@22718
   933
    case 0
wenzelm@22718
   934
    show ?case by (simp add: step)
paulson@19870
   935
  next
paulson@19870
   936
    case (Suc k)
krauss@31714
   937
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
krauss@31714
   938
    hence "i < Suc (i + k)" by (simp add: add_commute)
krauss@31714
   939
    from trans[OF this lessI Suc step]
krauss@31714
   940
    show ?case by simp
paulson@19870
   941
  qed
wenzelm@22718
   942
  thus "P i j" by (simp add: j)
paulson@19870
   943
qed
paulson@19870
   944
krauss@26748
   945
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   946
Provided by Roelof Oosterhuis.
krauss@26748
   947
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   948
\begin{itemize}
krauss@26748
   949
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   950
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   951
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   952
\end{itemize} *}
krauss@26748
   953
krauss@26748
   954
text{* A compact version without explicit base case: *}
krauss@26748
   955
lemma infinite_descent:
krauss@26748
   956
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   957
by (induct n rule: less_induct) auto
krauss@26748
   958
krauss@26748
   959
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   960
  "\<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
   961
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   962
krauss@26748
   963
text {*
krauss@26748
   964
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   965
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   966
\begin{itemize}
krauss@26748
   967
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   968
\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
   969
\end{itemize}
krauss@26748
   970
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   971
krauss@26748
   972
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   973
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   974
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   975
  shows "P x"
krauss@26748
   976
proof -
krauss@26748
   977
  obtain n where "n = V x" by auto
krauss@26748
   978
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   979
  proof (induct n rule: infinite_descent0)
krauss@26748
   980
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   981
    with A0 show "P x" by auto
krauss@26748
   982
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   983
    case (smaller n)
krauss@26748
   984
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   985
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   986
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   987
    then show ?case by auto
krauss@26748
   988
  qed
krauss@26748
   989
  ultimately show "P x" by auto
krauss@26748
   990
qed
krauss@26748
   991
krauss@26748
   992
text{* Again, without explicit base case: *}
krauss@26748
   993
lemma infinite_descent_measure:
krauss@26748
   994
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
   995
proof -
krauss@26748
   996
  from assms obtain n where "n = V x" by auto
krauss@26748
   997
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
   998
  proof (induct n rule: infinite_descent, auto)
krauss@26748
   999
    fix x assume "\<not> P x"
krauss@26748
  1000
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
  1001
  qed
krauss@26748
  1002
  ultimately show "P x" by auto
krauss@26748
  1003
qed
krauss@26748
  1004
paulson@14267
  1005
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
  1006
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
  1007
lemma less_mono_imp_le_mono:
nipkow@24438
  1008
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
  1009
by (simp add: order_le_less) (blast)
nipkow@24438
  1010
paulson@14267
  1011
paulson@14267
  1012
text {* non-strict, in 1st argument *}
paulson@14267
  1013
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
  1014
by (rule add_right_mono)
paulson@14267
  1015
paulson@14267
  1016
text {* non-strict, in both arguments *}
paulson@14267
  1017
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
  1018
by (rule add_mono)
paulson@14267
  1019
paulson@14267
  1020
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
  1021
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
  1022
paulson@14267
  1023
lemma le_add1: "n \<le> ((n + m)::nat)"
nipkow@24438
  1024
by (simp add: add_commute, rule le_add2)
berghofe@13449
  1025
berghofe@13449
  1026
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
  1027
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
  1028
berghofe@13449
  1029
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
  1030
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
  1031
paulson@14267
  1032
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
  1033
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
  1034
paulson@14267
  1035
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
  1036
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
  1037
paulson@14267
  1038
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
  1039
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1040
berghofe@13449
  1041
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1042
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1043
berghofe@13449
  1044
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1045
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1046
berghofe@13449
  1047
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1048
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1049
apply (simp_all add: le_add1)
nipkow@24438
  1050
done
berghofe@13449
  1051
berghofe@13449
  1052
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1053
apply (rule notI)
wenzelm@26335
  1054
apply (drule add_lessD1)
wenzelm@26335
  1055
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1056
done
berghofe@13449
  1057
berghofe@13449
  1058
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
krauss@26748
  1059
by (simp add: add_commute)
berghofe@13449
  1060
paulson@14267
  1061
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1062
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1063
apply (simp_all add: le_add1)
nipkow@24438
  1064
done
berghofe@13449
  1065
paulson@14267
  1066
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
nipkow@24438
  1067
apply (simp add: add_commute)
nipkow@24438
  1068
apply (erule add_leD1)
nipkow@24438
  1069
done
berghofe@13449
  1070
paulson@14267
  1071
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1072
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1073
berghofe@13449
  1074
text {* needs @{text "!!k"} for @{text add_ac} to work *}
berghofe@13449
  1075
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1076
by (force simp del: add_Suc_right
berghofe@13449
  1077
    simp add: less_iff_Suc_add add_Suc_right [symmetric] add_ac)
berghofe@13449
  1078
berghofe@13449
  1079
haftmann@26072
  1080
subsubsection {* More results about difference *}
berghofe@13449
  1081
berghofe@13449
  1082
text {* Addition is the inverse of subtraction:
paulson@14267
  1083
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1084
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1085
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1086
paulson@14267
  1087
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1088
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1089
paulson@14267
  1090
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
krauss@26748
  1091
by (simp add: add_commute)
berghofe@13449
  1092
paulson@14267
  1093
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1094
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1095
berghofe@13449
  1096
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1097
apply (induct m n rule: diff_induct)
nipkow@24438
  1098
apply (erule_tac [3] less_SucE)
nipkow@24438
  1099
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1100
done
berghofe@13449
  1101
paulson@14267
  1102
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1103
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1104
haftmann@26072
  1105
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1106
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1107
haftmann@52289
  1108
instance nat :: ordered_cancel_comm_monoid_diff
haftmann@52289
  1109
proof
haftmann@52289
  1110
  show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
haftmann@52289
  1111
qed
haftmann@52289
  1112
berghofe@13449
  1113
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1114
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1115
berghofe@13449
  1116
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1117
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1118
paulson@14267
  1119
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1120
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1121
paulson@14267
  1122
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
nipkow@24438
  1123
by (simp add: add_commute diff_add_assoc)
berghofe@13449
  1124
paulson@14267
  1125
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1126
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1127
paulson@14267
  1128
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1129
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1130
paulson@14267
  1131
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1132
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1133
berghofe@13449
  1134
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1135
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1136
wenzelm@22718
  1137
lemma less_imp_add_positive:
wenzelm@22718
  1138
  assumes "i < j"
wenzelm@22718
  1139
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1140
proof
wenzelm@22718
  1141
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1142
    by (simp add: order_less_imp_le)
wenzelm@22718
  1143
qed
wenzelm@9436
  1144
haftmann@26072
  1145
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1146
lemma nat_minus_add_max:
haftmann@26072
  1147
  fixes n m :: nat
haftmann@26072
  1148
  shows "n - m + m = max n m"
haftmann@26072
  1149
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1150
haftmann@26072
  1151
lemma nat_diff_split:
haftmann@26072
  1152
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1153
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1154
by (cases "a < b")
haftmann@26072
  1155
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
haftmann@26072
  1156
    not_less le_less dest!: sym [of a] sym [of b] add_eq_self_zero)
berghofe@13449
  1157
haftmann@26072
  1158
lemma nat_diff_split_asm:
haftmann@26072
  1159
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1160
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1161
by (auto split: nat_diff_split)
berghofe@13449
  1162
huffman@47255
  1163
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1164
  by simp
huffman@47255
  1165
huffman@47255
  1166
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1167
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1168
huffman@47255
  1169
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1170
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1171
huffman@47255
  1172
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1173
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1174
huffman@47255
  1175
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1176
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1177
huffman@47255
  1178
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1179
  by (fact Let_def)
huffman@47255
  1180
berghofe@13449
  1181
haftmann@26072
  1182
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1183
paulson@14267
  1184
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1185
by (simp add: mult_right_mono)
berghofe@13449
  1186
paulson@14267
  1187
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1188
by (simp add: mult_left_mono)
berghofe@13449
  1189
paulson@14267
  1190
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1191
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1192
by (simp add: mult_mono)
berghofe@13449
  1193
berghofe@13449
  1194
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1195
by (simp add: mult_strict_right_mono)
berghofe@13449
  1196
paulson@14266
  1197
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1198
      there are no negative numbers.*}
paulson@14266
  1199
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1200
  apply (induct m)
wenzelm@22718
  1201
   apply simp
wenzelm@22718
  1202
  apply (case_tac n)
wenzelm@22718
  1203
   apply simp_all
berghofe@13449
  1204
  done
berghofe@13449
  1205
huffman@30079
  1206
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1207
  apply (induct m)
wenzelm@22718
  1208
   apply simp
wenzelm@22718
  1209
  apply (case_tac n)
wenzelm@22718
  1210
   apply simp_all
berghofe@13449
  1211
  done
berghofe@13449
  1212
paulson@14341
  1213
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1214
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1215
  apply (cases k, auto)
berghofe@13449
  1216
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1217
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1218
  done
berghofe@13449
  1219
berghofe@13449
  1220
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
nipkow@24438
  1221
by (simp add: mult_commute [of k])
berghofe@13449
  1222
paulson@14267
  1223
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1224
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1225
paulson@14267
  1226
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1227
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1228
berghofe@13449
  1229
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1230
by (subst mult_less_cancel1) simp
berghofe@13449
  1231
paulson@14267
  1232
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1233
by (subst mult_le_cancel1) simp
berghofe@13449
  1234
haftmann@26072
  1235
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1236
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1237
haftmann@26072
  1238
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1239
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1240
berghofe@13449
  1241
text {* Lemma for @{text gcd} *}
huffman@30128
  1242
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1243
  apply (drule sym)
berghofe@13449
  1244
  apply (rule disjCI)
berghofe@13449
  1245
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1246
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1247
    apply (auto)
berghofe@13449
  1248
  done
wenzelm@9436
  1249
haftmann@51263
  1250
lemma mono_times_nat:
haftmann@51263
  1251
  fixes n :: nat
haftmann@51263
  1252
  assumes "n > 0"
haftmann@51263
  1253
  shows "mono (times n)"
haftmann@51263
  1254
proof
haftmann@51263
  1255
  fix m q :: nat
haftmann@51263
  1256
  assume "m \<le> q"
haftmann@51263
  1257
  with assms show "n * m \<le> n * q" by simp
haftmann@51263
  1258
qed
haftmann@51263
  1259
haftmann@26072
  1260
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1261
haftmann@26072
  1262
instantiation nat :: distrib_lattice
haftmann@26072
  1263
begin
haftmann@24995
  1264
haftmann@26072
  1265
definition
haftmann@26072
  1266
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1267
haftmann@26072
  1268
definition
haftmann@26072
  1269
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1270
haftmann@26072
  1271
instance by intro_classes
haftmann@26072
  1272
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1273
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1274
haftmann@26072
  1275
end
haftmann@24995
  1276
haftmann@24995
  1277
haftmann@30954
  1278
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1279
haftmann@30971
  1280
text {*
haftmann@30971
  1281
  We use the same logical constant for the power operations on
haftmann@30971
  1282
  functions and relations, in order to share the same syntax.
haftmann@30971
  1283
*}
haftmann@30971
  1284
haftmann@45965
  1285
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1286
haftmann@45965
  1287
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1288
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1289
haftmann@30971
  1290
notation (latex output)
haftmann@30971
  1291
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1292
haftmann@30971
  1293
notation (HTML output)
haftmann@30971
  1294
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1295
haftmann@30971
  1296
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1297
haftmann@30971
  1298
overloading
haftmann@30971
  1299
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1300
begin
haftmann@30954
  1301
haftmann@30954
  1302
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1303
  "funpow 0 f = id"
haftmann@44325
  1304
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1305
haftmann@30971
  1306
end
haftmann@30971
  1307
haftmann@49723
  1308
lemma funpow_Suc_right:
haftmann@49723
  1309
  "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1310
proof (induct n)
haftmann@49723
  1311
  case 0 then show ?case by simp
haftmann@49723
  1312
next
haftmann@49723
  1313
  fix n
haftmann@49723
  1314
  assume "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1315
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
haftmann@49723
  1316
    by (simp add: o_assoc)
haftmann@49723
  1317
qed
haftmann@49723
  1318
haftmann@49723
  1319
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
haftmann@49723
  1320
haftmann@30971
  1321
text {* for code generation *}
haftmann@30971
  1322
haftmann@30971
  1323
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1324
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1325
haftmann@30971
  1326
lemma [code]:
haftmann@37430
  1327
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1328
  "funpow 0 f = id"
haftmann@37430
  1329
  by (simp_all add: funpow_code_def)
haftmann@30971
  1330
wenzelm@36176
  1331
hide_const (open) funpow
haftmann@30954
  1332
haftmann@30954
  1333
lemma funpow_add:
haftmann@30971
  1334
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1335
  by (induct m) simp_all
haftmann@30954
  1336
haftmann@37430
  1337
lemma funpow_mult:
haftmann@37430
  1338
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1339
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1340
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1341
haftmann@30954
  1342
lemma funpow_swap1:
haftmann@30971
  1343
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1344
proof -
haftmann@30971
  1345
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1346
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1347
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1348
  finally show ?thesis .
haftmann@30954
  1349
qed
haftmann@30954
  1350
haftmann@38621
  1351
lemma comp_funpow:
haftmann@38621
  1352
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1353
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1354
  by (induct n) simp_all
haftmann@30954
  1355
hoelzl@54496
  1356
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
hoelzl@54496
  1357
  by (induct n) simp_all
hoelzl@54496
  1358
hoelzl@54496
  1359
lemma id_funpow[simp]: "id ^^ n = id"
hoelzl@54496
  1360
  by (induct n) simp_all
haftmann@38621
  1361
nipkow@45833
  1362
subsection {* Kleene iteration *}
nipkow@45833
  1363
haftmann@52729
  1364
lemma Kleene_iter_lpfp:
haftmann@52729
  1365
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
nipkow@45833
  1366
proof(induction k)
nipkow@45833
  1367
  case 0 show ?case by simp
nipkow@45833
  1368
next
nipkow@45833
  1369
  case Suc
nipkow@45833
  1370
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1371
  show ?case by simp
nipkow@45833
  1372
qed
nipkow@45833
  1373
nipkow@45833
  1374
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1375
shows "lfp f = (f^^k) bot"
nipkow@45833
  1376
proof(rule antisym)
nipkow@45833
  1377
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1378
  proof(rule lfp_lowerbound)
nipkow@45833
  1379
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1380
  qed
nipkow@45833
  1381
next
nipkow@45833
  1382
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1383
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1384
qed
nipkow@45833
  1385
nipkow@45833
  1386
haftmann@38621
  1387
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1388
haftmann@24196
  1389
context semiring_1
haftmann@24196
  1390
begin
haftmann@24196
  1391
haftmann@38621
  1392
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1393
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1394
haftmann@38621
  1395
lemma of_nat_simps [simp]:
haftmann@38621
  1396
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1397
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1398
  by (simp_all add: of_nat_def)
haftmann@25193
  1399
haftmann@25193
  1400
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1401
  by (simp add: of_nat_def)
haftmann@25193
  1402
haftmann@25193
  1403
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@25193
  1404
  by (induct m) (simp_all add: add_ac)
haftmann@25193
  1405
haftmann@25193
  1406
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
webertj@49962
  1407
  by (induct m) (simp_all add: add_ac distrib_right)
haftmann@25193
  1408
haftmann@28514
  1409
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1410
  "of_nat_aux inc 0 i = i"
haftmann@44325
  1411
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1412
haftmann@30966
  1413
lemma of_nat_code:
haftmann@28514
  1414
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1415
proof (induct n)
haftmann@28514
  1416
  case 0 then show ?case by simp
haftmann@28514
  1417
next
haftmann@28514
  1418
  case (Suc n)
haftmann@28514
  1419
  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
  1420
    by (induct n) simp_all
haftmann@28514
  1421
  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
  1422
    by simp
haftmann@28514
  1423
  with Suc show ?case by (simp add: add_commute)
haftmann@28514
  1424
qed
haftmann@30966
  1425
haftmann@24196
  1426
end
haftmann@24196
  1427
bulwahn@45231
  1428
declare of_nat_code [code]
haftmann@30966
  1429
haftmann@26072
  1430
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1431
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1432
haftmann@26072
  1433
class semiring_char_0 = semiring_1 +
haftmann@38621
  1434
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1435
begin
haftmann@26072
  1436
haftmann@38621
  1437
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1438
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1439
haftmann@26072
  1440
text{*Special cases where either operand is zero*}
haftmann@26072
  1441
blanchet@54147
  1442
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1443
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1444
blanchet@54147
  1445
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1446
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1447
haftmann@26072
  1448
end
haftmann@26072
  1449
haftmann@35028
  1450
context linordered_semidom
haftmann@25193
  1451
begin
haftmann@25193
  1452
huffman@47489
  1453
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1454
  by (induct n) simp_all
haftmann@25193
  1455
huffman@47489
  1456
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1457
  by (simp add: not_less)
haftmann@25193
  1458
haftmann@25193
  1459
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
huffman@47489
  1460
  by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
haftmann@25193
  1461
haftmann@26072
  1462
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1463
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1464
huffman@47489
  1465
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1466
  by simp
huffman@47489
  1467
huffman@47489
  1468
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1469
  by simp
huffman@47489
  1470
haftmann@35028
  1471
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1472
haftmann@38621
  1473
subclass semiring_char_0 proof
haftmann@38621
  1474
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1475
haftmann@25193
  1476
text{*Special cases where either operand is zero*}
haftmann@25193
  1477
blanchet@54147
  1478
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1479
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1480
haftmann@26072
  1481
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1482
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1483
haftmann@26072
  1484
end
haftmann@26072
  1485
haftmann@26072
  1486
context ring_1
haftmann@26072
  1487
begin
haftmann@26072
  1488
haftmann@26072
  1489
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1490
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1491
haftmann@26072
  1492
end
haftmann@26072
  1493
haftmann@35028
  1494
context linordered_idom
haftmann@26072
  1495
begin
haftmann@26072
  1496
haftmann@26072
  1497
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1498
  unfolding abs_if by auto
haftmann@26072
  1499
haftmann@25193
  1500
end
haftmann@25193
  1501
haftmann@25193
  1502
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1503
  by (induct n) simp_all
haftmann@25193
  1504
haftmann@25193
  1505
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1506
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1507
haftmann@25193
  1508
haftmann@26149
  1509
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1510
haftmann@26072
  1511
context semiring_1
haftmann@25193
  1512
begin
haftmann@25193
  1513
haftmann@37767
  1514
definition Nats  :: "'a set" where
haftmann@37767
  1515
  "Nats = range of_nat"
haftmann@26072
  1516
haftmann@26072
  1517
notation (xsymbols)
haftmann@26072
  1518
  Nats  ("\<nat>")
haftmann@25193
  1519
haftmann@26072
  1520
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1521
  by (simp add: Nats_def)
haftmann@26072
  1522
haftmann@26072
  1523
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1524
apply (simp add: Nats_def)
haftmann@26072
  1525
apply (rule range_eqI)
haftmann@26072
  1526
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1527
done
haftmann@25193
  1528
haftmann@26072
  1529
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1530
apply (simp add: Nats_def)
haftmann@26072
  1531
apply (rule range_eqI)
haftmann@26072
  1532
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1533
done
haftmann@25193
  1534
haftmann@26072
  1535
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1536
apply (auto simp add: Nats_def)
haftmann@26072
  1537
apply (rule range_eqI)
haftmann@26072
  1538
apply (rule of_nat_add [symmetric])
haftmann@26072
  1539
done
haftmann@26072
  1540
haftmann@26072
  1541
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1542
apply (auto simp add: Nats_def)
haftmann@26072
  1543
apply (rule range_eqI)
haftmann@26072
  1544
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1545
done
haftmann@25193
  1546
huffman@35633
  1547
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1548
  assumes "x \<in> \<nat>"
huffman@35633
  1549
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1550
  unfolding Nats_def
huffman@35633
  1551
proof -
huffman@35633
  1552
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1553
  then obtain n where "x = of_nat n" ..
huffman@35633
  1554
  then show thesis ..
huffman@35633
  1555
qed
huffman@35633
  1556
huffman@35633
  1557
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1558
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1559
  by (rule Nats_cases) auto
huffman@35633
  1560
haftmann@25193
  1561
end
haftmann@25193
  1562
haftmann@25193
  1563
wenzelm@21243
  1564
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1565
haftmann@22845
  1566
lemma subst_equals:
haftmann@22845
  1567
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1568
  shows "u = s"
haftmann@22845
  1569
  using 2 1 by (rule trans)
haftmann@22845
  1570
haftmann@30686
  1571
setup Arith_Data.setup
haftmann@30686
  1572
wenzelm@48891
  1573
ML_file "Tools/nat_arith.ML"
huffman@48559
  1574
huffman@48559
  1575
simproc_setup nateq_cancel_sums
huffman@48559
  1576
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
wenzelm@54742
  1577
  {* fn phi => try o Nat_Arith.cancel_eq_conv *}
huffman@48559
  1578
huffman@48559
  1579
simproc_setup natless_cancel_sums
huffman@48559
  1580
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
wenzelm@54742
  1581
  {* fn phi => try o Nat_Arith.cancel_less_conv *}
huffman@48559
  1582
huffman@48559
  1583
simproc_setup natle_cancel_sums
huffman@48559
  1584
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
wenzelm@54742
  1585
  {* fn phi => try o Nat_Arith.cancel_le_conv *}
huffman@48559
  1586
huffman@48559
  1587
simproc_setup natdiff_cancel_sums
huffman@48559
  1588
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
wenzelm@54742
  1589
  {* fn phi => try o Nat_Arith.cancel_diff_conv *}
wenzelm@24091
  1590
wenzelm@48891
  1591
ML_file "Tools/lin_arith.ML"
haftmann@31100
  1592
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1593
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1594
wenzelm@43595
  1595
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@43595
  1596
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1597
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1598
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1599
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1600
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1601
solver all the time rather than add the additional check. *)
wenzelm@43595
  1602
wenzelm@43595
  1603
wenzelm@21243
  1604
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1605
nipkow@27625
  1606
context order
nipkow@27625
  1607
begin
nipkow@27625
  1608
nipkow@27625
  1609
lemma lift_Suc_mono_le:
haftmann@53986
  1610
  assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
krauss@27627
  1611
  shows "f n \<le> f n'"
krauss@27627
  1612
proof (cases "n < n'")
krauss@27627
  1613
  case True
haftmann@53986
  1614
  then show ?thesis
haftmann@53986
  1615
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
haftmann@53986
  1616
qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
nipkow@27625
  1617
nipkow@27625
  1618
lemma lift_Suc_mono_less:
haftmann@53986
  1619
  assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
krauss@27627
  1620
  shows "f n < f n'"
krauss@27627
  1621
using `n < n'`
haftmann@53986
  1622
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
nipkow@27625
  1623
nipkow@27789
  1624
lemma lift_Suc_mono_less_iff:
haftmann@53986
  1625
  "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
haftmann@53986
  1626
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
haftmann@53986
  1627
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
nipkow@27789
  1628
nipkow@27625
  1629
end
nipkow@27625
  1630
haftmann@53986
  1631
lemma mono_iff_le_Suc:
haftmann@53986
  1632
  "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1633
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1634
nipkow@27789
  1635
lemma mono_nat_linear_lb:
haftmann@53986
  1636
  fixes f :: "nat \<Rightarrow> nat"
haftmann@53986
  1637
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
haftmann@53986
  1638
  shows "f m + k \<le> f (m + k)"
haftmann@53986
  1639
proof (induct k)
haftmann@53986
  1640
  case 0 then show ?case by simp
haftmann@53986
  1641
next
haftmann@53986
  1642
  case (Suc k)
haftmann@53986
  1643
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
haftmann@53986
  1644
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
haftmann@53986
  1645
    by (simp add: Suc_le_eq)
haftmann@53986
  1646
  finally show ?case by simp
haftmann@53986
  1647
qed
nipkow@27789
  1648
nipkow@27789
  1649
wenzelm@21243
  1650
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1651
wenzelm@21243
  1652
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1653
by arith
wenzelm@21243
  1654
wenzelm@21243
  1655
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1656
by arith
wenzelm@21243
  1657
haftmann@51173
  1658
lemma less_diff_conv2:
haftmann@51173
  1659
  fixes j k i :: nat
haftmann@51173
  1660
  assumes "k \<le> j"
haftmann@51173
  1661
  shows "j - k < i \<longleftrightarrow> j < i + k"
haftmann@51173
  1662
  using assms by arith
haftmann@51173
  1663
wenzelm@21243
  1664
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1665
by arith
wenzelm@21243
  1666
wenzelm@21243
  1667
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
nipkow@24438
  1668
by arith
wenzelm@21243
  1669
wenzelm@21243
  1670
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1671
by arith
wenzelm@21243
  1672
wenzelm@21243
  1673
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
nipkow@24438
  1674
by arith
wenzelm@21243
  1675
wenzelm@21243
  1676
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1677
  second premise n\<le>m*)
wenzelm@21243
  1678
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1679
by arith
wenzelm@21243
  1680
haftmann@26072
  1681
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1682
wenzelm@21243
  1683
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1684
by (simp split add: nat_diff_split)
wenzelm@21243
  1685
wenzelm@36176
  1686
hide_fact (open) diff_diff_eq
haftmann@35064
  1687
wenzelm@21243
  1688
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1689
by (auto split add: nat_diff_split)
wenzelm@21243
  1690
wenzelm@21243
  1691
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1692
by (auto split add: nat_diff_split)
wenzelm@21243
  1693
wenzelm@21243
  1694
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1695
by (auto split add: nat_diff_split)
wenzelm@21243
  1696
wenzelm@21243
  1697
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1698
wenzelm@21243
  1699
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1700
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1701
by (simp split add: nat_diff_split)
wenzelm@21243
  1702
wenzelm@21243
  1703
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1704
by (simp split add: nat_diff_split)
wenzelm@21243
  1705
wenzelm@21243
  1706
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1707
by (simp split add: nat_diff_split)
wenzelm@21243
  1708
wenzelm@21243
  1709
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1710
by (simp split add: nat_diff_split)
wenzelm@21243
  1711
bulwahn@26143
  1712
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1713
by auto
bulwahn@26143
  1714
bulwahn@26143
  1715
lemma inj_on_diff_nat: 
bulwahn@26143
  1716
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1717
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1718
proof (rule inj_onI)
bulwahn@26143
  1719
  fix x y
bulwahn@26143
  1720
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1721
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1722
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1723
qed
bulwahn@26143
  1724
haftmann@26072
  1725
text{*Rewriting to pull differences out*}
haftmann@26072
  1726
haftmann@26072
  1727
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1728
by arith
haftmann@26072
  1729
haftmann@26072
  1730
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1731
by arith
haftmann@26072
  1732
haftmann@26072
  1733
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1734
by arith
haftmann@26072
  1735
noschinl@45933
  1736
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1737
by simp
noschinl@45933
  1738
bulwahn@46350
  1739
(*The others are
bulwahn@46350
  1740
      i - j - k = i - (j + k),
bulwahn@46350
  1741
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1742
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1743
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1744
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
bulwahn@46350
  1745
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1746
bulwahn@46350
  1747
text{*At present we prove no analogue of @{text not_less_Least} or @{text
bulwahn@46350
  1748
Least_Suc}, since there appears to be no need.*}
bulwahn@46350
  1749
wenzelm@21243
  1750
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1751
wenzelm@21243
  1752
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1753
by (cases m) auto
wenzelm@21243
  1754
wenzelm@21243
  1755
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1756
by (cases m) auto
wenzelm@21243
  1757
wenzelm@21243
  1758
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1759
by (cases m) auto
wenzelm@21243
  1760
krauss@23001
  1761
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1762
krauss@23001
  1763
lemma inc_induct[consumes 1, case_names base step]:
hoelzl@54411
  1764
  assumes less: "i \<le> j"
krauss@23001
  1765
  assumes base: "P j"
hoelzl@54411
  1766
  assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
krauss@23001
  1767
  shows "P i"
hoelzl@54411
  1768
  using less step
hoelzl@54411
  1769
proof (induct d\<equiv>"j - i" arbitrary: i)
krauss@23001
  1770
  case (0 i)
krauss@23001
  1771
  hence "i = j" by simp
krauss@23001
  1772
  with base show ?case by simp
krauss@23001
  1773
next
hoelzl@54411
  1774
  case (Suc d n)
hoelzl@54411
  1775
  hence "n \<le> n" "n < j" "P (Suc n)"
krauss@23001
  1776
    by simp_all
hoelzl@54411
  1777
  then show "P n" by fact
krauss@23001
  1778
qed
krauss@23001
  1779
krauss@23001
  1780
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1781
  assumes less: "i < j"
krauss@23001
  1782
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1783
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1784
  shows "P i"
krauss@23001
  1785
  using less
krauss@23001
  1786
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1787
  case (0 i)
krauss@23001
  1788
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1789
  with base show ?case by simp
krauss@23001
  1790
next
krauss@23001
  1791
  case (Suc d i)
krauss@23001
  1792
  hence "i < j" "P (Suc i)"
krauss@23001
  1793
    by simp_all
krauss@23001
  1794
  thus "P i" by (rule step)
krauss@23001
  1795
qed
krauss@23001
  1796
krauss@23001
  1797
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1798
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1799
krauss@23001
  1800
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1801
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1802
bulwahn@46351
  1803
text {* Further induction rule similar to @{thm inc_induct} *}
nipkow@27625
  1804
bulwahn@46351
  1805
lemma dec_induct[consumes 1, case_names base step]:
hoelzl@54411
  1806
  "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
bulwahn@46351
  1807
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
bulwahn@46351
  1808
 
haftmann@33274
  1809
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1810
haftmann@33274
  1811
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1812
unfolding dvd_def by simp
haftmann@33274
  1813
haftmann@33274
  1814
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1815
by (simp add: dvd_def)
haftmann@33274
  1816
haftmann@33274
  1817
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1818
by (simp add: dvd_def)
haftmann@33274
  1819
nipkow@33657
  1820
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1821
  unfolding dvd_def
huffman@35216
  1822
  by (force dest: mult_eq_self_implies_10 simp add: mult_assoc)
haftmann@33274
  1823
haftmann@33274
  1824
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1825
haftmann@33274
  1826
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1827
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1828
haftmann@33274
  1829
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1830
unfolding dvd_def
haftmann@33274
  1831
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1832
haftmann@33274
  1833
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1834
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1835
  apply (blast intro: dvd_add)
haftmann@33274
  1836
  done
haftmann@33274
  1837
haftmann@33274
  1838
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1839
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1840
haftmann@33274
  1841
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
haftmann@33274
  1842
  apply (rule iffI)
haftmann@33274
  1843
   apply (erule_tac [2] dvd_add)
haftmann@33274
  1844
   apply (rule_tac [2] dvd_refl)
haftmann@33274
  1845
  apply (subgoal_tac "n = (n+k) -k")
haftmann@33274
  1846
   prefer 2 apply simp
haftmann@33274
  1847
  apply (erule ssubst)
haftmann@33274
  1848
  apply (erule dvd_diff_nat)
haftmann@33274
  1849
  apply (rule dvd_refl)
haftmann@33274
  1850
  done
haftmann@33274
  1851
haftmann@33274
  1852
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1853
  unfolding dvd_def
haftmann@33274
  1854
  apply (erule exE)
haftmann@33274
  1855
  apply (simp add: mult_ac)
haftmann@33274
  1856
  done
haftmann@33274
  1857
haftmann@33274
  1858
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1859
  apply auto
haftmann@33274
  1860
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1861
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1862
  done
haftmann@33274
  1863
haftmann@33274
  1864
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@33274
  1865
  apply (subst mult_commute)
haftmann@33274
  1866
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1867
  done
haftmann@33274
  1868
haftmann@33274
  1869
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1870
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1871
haftmann@33274
  1872
lemma nat_dvd_not_less:
haftmann@33274
  1873
  fixes m n :: nat
haftmann@33274
  1874
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1875
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1876
haftmann@51173
  1877
lemma dvd_plusE:
haftmann@51173
  1878
  fixes m n q :: nat
haftmann@51173
  1879
  assumes "m dvd n + q" "m dvd n"
haftmann@51173
  1880
  obtains "m dvd q"
haftmann@51173
  1881
proof (cases "m = 0")
haftmann@51173
  1882
  case True with assms that show thesis by simp
haftmann@51173
  1883
next
haftmann@51173
  1884
  case False then have "m > 0" by simp
haftmann@51173
  1885
  from assms obtain r s where "n = m * r" and "n + q = m * s" by (blast elim: dvdE)
haftmann@51173
  1886
  then have *: "m * r + q = m * s" by simp
haftmann@51173
  1887
  show thesis proof (cases "r \<le> s")
haftmann@51173
  1888
    case False then have "s < r" by (simp add: not_le)
haftmann@51173
  1889
    with * have "m * r + q - m * s = m * s - m * s" by simp
haftmann@51173
  1890
    then have "m * r + q - m * s = 0" by simp
haftmann@53986
  1891
    with `m > 0` `s < r` have "m * r - m * s + q = 0" by (unfold less_le_not_le) auto
haftmann@51173
  1892
    then have "m * (r - s) + q = 0" by auto
haftmann@51173
  1893
    then have "m * (r - s) = 0" by simp
haftmann@51173
  1894
    then have "m = 0 \<or> r - s = 0" by simp
haftmann@53986
  1895
    with `s < r` have "m = 0" by (simp add: less_le_not_le)
haftmann@51173
  1896
    with `m > 0` show thesis by auto
haftmann@51173
  1897
  next
haftmann@51173
  1898
    case True with * have "m * r + q - m * r = m * s - m * r" by simp
haftmann@51173
  1899
    with `m > 0` `r \<le> s` have "m * r - m * r + q = m * s - m * r" by simp
haftmann@51173
  1900
    then have "q = m * (s - r)" by (simp add: diff_mult_distrib2)
haftmann@51173
  1901
    with assms that show thesis by (auto intro: dvdI)
haftmann@51173
  1902
  qed
haftmann@51173
  1903
qed
haftmann@51173
  1904
haftmann@51173
  1905
lemma dvd_plus_eq_right:
haftmann@51173
  1906
  fixes m n q :: nat
haftmann@51173
  1907
  assumes "m dvd n"
haftmann@51173
  1908
  shows "m dvd n + q \<longleftrightarrow> m dvd q"
haftmann@51173
  1909
  using assms by (auto elim: dvd_plusE)
haftmann@51173
  1910
haftmann@51173
  1911
lemma dvd_plus_eq_left:
haftmann@51173
  1912
  fixes m n q :: nat
haftmann@51173
  1913
  assumes "m dvd q"
haftmann@51173
  1914
  shows "m dvd n + q \<longleftrightarrow> m dvd n"
haftmann@51173
  1915
  using assms by (simp add: dvd_plus_eq_right add_commute [of n])
haftmann@51173
  1916
haftmann@54222
  1917
lemma less_eq_dvd_minus:
haftmann@51173
  1918
  fixes m n :: nat
haftmann@54222
  1919
  assumes "m \<le> n"
haftmann@54222
  1920
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
haftmann@51173
  1921
proof -
haftmann@54222
  1922
  from assms have "n = m + (n - m)" by simp
haftmann@51173
  1923
  then obtain q where "n = m + q" ..
haftmann@51173
  1924
  then show ?thesis by (simp add: dvd_reduce add_commute [of m])
haftmann@51173
  1925
qed
haftmann@51173
  1926
haftmann@51173
  1927
lemma dvd_minus_self:
haftmann@51173
  1928
  fixes m n :: nat
haftmann@51173
  1929
  shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
haftmann@51173
  1930
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
haftmann@51173
  1931
haftmann@51173
  1932
lemma dvd_minus_add:
haftmann@51173
  1933
  fixes m n q r :: nat
haftmann@51173
  1934
  assumes "q \<le> n" "q \<le> r * m"
haftmann@51173
  1935
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
haftmann@51173
  1936
proof -
haftmann@51173
  1937
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
haftmann@51173
  1938
    by (auto elim: dvd_plusE)
wenzelm@53374
  1939
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
wenzelm@53374
  1940
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
haftmann@51173
  1941
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add_commute)
haftmann@51173
  1942
  finally show ?thesis .
haftmann@51173
  1943
qed
haftmann@51173
  1944
haftmann@33274
  1945
haftmann@44817
  1946
subsection {* aliasses *}
haftmann@44817
  1947
haftmann@44817
  1948
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@44817
  1949
  by simp
haftmann@44817
  1950
 
haftmann@44817
  1951
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@44817
  1952
  by simp
haftmann@44817
  1953
haftmann@44817
  1954
haftmann@26072
  1955
subsection {* size of a datatype value *}
haftmann@25193
  1956
haftmann@29608
  1957
class size =
krauss@26748
  1958
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1959
haftmann@33364
  1960
haftmann@33364
  1961
subsection {* code module namespace *}
haftmann@33364
  1962
haftmann@52435
  1963
code_identifier
haftmann@52435
  1964
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1965
huffman@47108
  1966
hide_const (open) of_nat_aux
huffman@47108
  1967
haftmann@25193
  1968
end