src/HOL/Nat.thy
author blanchet
Fri Sep 05 00:41:01 2014 +0200 (2014-09-05)
changeset 58189 9d714be4f028
parent 57983 6edc3529bb4e
child 58306 117ba6cbe414
permissions -rw-r--r--
added 'plugins' option to control which hooks are enabled
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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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named_theorems arith "arith facts -- only ground formulas"
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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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free_constructors case_nat for
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    "0 \<Colon> nat"
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  | Suc pred
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where
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  "pred (0 \<Colon> nat) = (0 \<Colon> nat)"
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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 Suc_Rep_inject'
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     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 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 rec = old.nat.rec
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lemmas simps = nat.inject nat.distinct nat.case nat.rec
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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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declare nat.sel[code del]
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hide_const (open) 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.exhaust_sel
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  nat.split_sel
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  nat.split_sel_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_left_cancel:
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  fixes k m n :: nat
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  shows "k + m = k + n \<longleftrightarrow> m = n"
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  by (fact add_left_cancel)
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lemma nat_add_right_cancel:
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  fixes k m n :: nat
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  shows "m + k = n + k \<longleftrightarrow> m = n"
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  by (fact 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 (fact diff_cancel)
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lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
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  by (fact diff_diff_add)
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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 (fact diff_right_commute)
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lemma diff_add_inverse: "(n + m) - n = (m::nat)"
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  by (fact add_diff_cancel_left')
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lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
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  by (fact add_diff_cancel_right')
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lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
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  by (fact comm_monoid_diff_class.add_diff_cancel_left)
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lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
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  by (fact add_diff_cancel_right)
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lemma diff_add_0: "n - (n + m) = (0::nat)"
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  by (fact diff_add_zero)
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lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
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  unfolding One_nat_def by simp
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text {* Difference distributes over multiplication *}
haftmann@26072
   345
haftmann@26072
   346
lemma diff_mult_distrib: "((m::nat) - n) * k = (m * k) - (n * k)"
haftmann@26072
   347
by (induct m n rule: diff_induct) (simp_all add: diff_cancel)
haftmann@26072
   348
haftmann@26072
   349
lemma diff_mult_distrib2: "k * ((m::nat) - n) = (k * m) - (k * n)"
haftmann@57512
   350
by (simp add: diff_mult_distrib mult.commute [of k])
haftmann@26072
   351
  -- {* NOT added as rewrites, since sometimes they are used from right-to-left *}
haftmann@26072
   352
haftmann@26072
   353
haftmann@26072
   354
subsubsection {* Multiplication *}
haftmann@26072
   355
haftmann@26072
   356
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
haftmann@57512
   357
  by (fact distrib_left)
haftmann@26072
   358
haftmann@26072
   359
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   360
  by (induct m) auto
haftmann@26072
   361
haftmann@26072
   362
lemmas nat_distrib =
haftmann@26072
   363
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
haftmann@26072
   364
huffman@30079
   365
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   366
  apply (induct m)
haftmann@26072
   367
   apply simp
haftmann@26072
   368
  apply (induct n)
haftmann@26072
   369
   apply auto
haftmann@26072
   370
  done
haftmann@26072
   371
blanchet@54147
   372
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   373
  apply (rule trans)
nipkow@44890
   374
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
haftmann@26072
   375
  done
haftmann@26072
   376
huffman@30079
   377
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   378
  unfolding One_nat_def by (rule mult_eq_1_iff)
huffman@30079
   379
huffman@30079
   380
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   381
  unfolding One_nat_def by (rule one_eq_mult_iff)
huffman@30079
   382
haftmann@26072
   383
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   384
proof -
haftmann@26072
   385
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   386
  proof (induct n arbitrary: m)
haftmann@26072
   387
    case 0 then show "m = 0" by simp
haftmann@26072
   388
  next
haftmann@26072
   389
    case (Suc n) then show "m = Suc n"
haftmann@26072
   390
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   391
  qed
haftmann@26072
   392
  then show ?thesis by auto
haftmann@26072
   393
qed
haftmann@26072
   394
haftmann@26072
   395
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@57512
   396
  by (simp add: mult.commute)
haftmann@26072
   397
haftmann@26072
   398
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   399
  by (subst mult_cancel1) simp
haftmann@26072
   400
haftmann@24995
   401
haftmann@24995
   402
subsection {* Orders on @{typ nat} *}
haftmann@24995
   403
haftmann@26072
   404
subsubsection {* Operation definition *}
haftmann@24995
   405
haftmann@26072
   406
instantiation nat :: linorder
haftmann@25510
   407
begin
haftmann@25510
   408
blanchet@55575
   409
primrec less_eq_nat where
haftmann@26072
   410
  "(0\<Colon>nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   411
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   412
haftmann@28514
   413
declare less_eq_nat.simps [simp del]
haftmann@26072
   414
lemma le0 [iff]: "0 \<le> (n\<Colon>nat)" by (simp add: less_eq_nat.simps)
haftmann@54223
   415
lemma [code]: "(0\<Colon>nat) \<le> n \<longleftrightarrow> True" by simp
haftmann@26072
   416
haftmann@26072
   417
definition less_nat where
haftmann@28514
   418
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   419
haftmann@26072
   420
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   421
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   422
haftmann@26072
   423
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   424
  unfolding less_eq_Suc_le ..
haftmann@26072
   425
haftmann@26072
   426
lemma le_0_eq [iff]: "(n\<Colon>nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   427
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   428
haftmann@26072
   429
lemma not_less0 [iff]: "\<not> n < (0\<Colon>nat)"
haftmann@26072
   430
  by (simp add: less_eq_Suc_le)
haftmann@26072
   431
haftmann@26072
   432
lemma less_nat_zero_code [code]: "n < (0\<Colon>nat) \<longleftrightarrow> False"
haftmann@26072
   433
  by simp
haftmann@26072
   434
haftmann@26072
   435
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   436
  by (simp add: less_eq_Suc_le)
haftmann@26072
   437
haftmann@26072
   438
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   439
  by (simp add: less_eq_Suc_le)
haftmann@26072
   440
hoelzl@56194
   441
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
hoelzl@56194
   442
  by (cases m) auto
hoelzl@56194
   443
haftmann@26072
   444
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   445
  by (induct m arbitrary: n)
haftmann@26072
   446
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   447
haftmann@26072
   448
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   449
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   450
haftmann@26072
   451
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   452
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   453
haftmann@26072
   454
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   455
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   456
wenzelm@26315
   457
instance
wenzelm@26315
   458
proof
haftmann@26072
   459
  fix n m :: nat
haftmann@27679
   460
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n" 
haftmann@26072
   461
  proof (induct n arbitrary: m)
haftmann@27679
   462
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   463
  next
haftmann@27679
   464
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   465
  qed
haftmann@26072
   466
next
haftmann@26072
   467
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   468
next
haftmann@26072
   469
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   470
  then show "n = m"
haftmann@26072
   471
    by (induct n arbitrary: m)
haftmann@26072
   472
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   473
next
haftmann@26072
   474
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   475
  then show "n \<le> q"
haftmann@26072
   476
  proof (induct n arbitrary: m q)
haftmann@26072
   477
    case 0 show ?case by simp
haftmann@26072
   478
  next
haftmann@26072
   479
    case (Suc n) then show ?case
haftmann@26072
   480
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   481
        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)
haftmann@26072
   483
  qed
haftmann@26072
   484
next
haftmann@26072
   485
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   486
    by (induct n arbitrary: m)
haftmann@26072
   487
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   488
qed
haftmann@25510
   489
haftmann@25510
   490
end
berghofe@13449
   491
haftmann@52729
   492
instantiation nat :: order_bot
haftmann@29652
   493
begin
haftmann@29652
   494
haftmann@29652
   495
definition bot_nat :: nat where
haftmann@29652
   496
  "bot_nat = 0"
haftmann@29652
   497
haftmann@29652
   498
instance proof
haftmann@29652
   499
qed (simp add: bot_nat_def)
haftmann@29652
   500
haftmann@29652
   501
end
haftmann@29652
   502
hoelzl@51329
   503
instance nat :: no_top
haftmann@52289
   504
  by default (auto intro: less_Suc_eq_le [THEN iffD2])
haftmann@52289
   505
hoelzl@51329
   506
haftmann@26072
   507
subsubsection {* Introduction properties *}
berghofe@13449
   508
haftmann@26072
   509
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   510
  by (simp add: less_Suc_eq_le)
berghofe@13449
   511
haftmann@26072
   512
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   513
  by (simp add: less_Suc_eq_le)
berghofe@13449
   514
berghofe@13449
   515
berghofe@13449
   516
subsubsection {* Elimination properties *}
berghofe@13449
   517
berghofe@13449
   518
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   519
  by (rule order_less_irrefl)
berghofe@13449
   520
wenzelm@26335
   521
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
wenzelm@26335
   522
  by (rule not_sym) (rule less_imp_neq) 
berghofe@13449
   523
paulson@14267
   524
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   525
  by (rule less_imp_neq)
berghofe@13449
   526
wenzelm@26335
   527
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   528
  by (rule notE, rule less_not_refl)
berghofe@13449
   529
berghofe@13449
   530
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   531
  by (rule notE) (rule not_less0)
berghofe@13449
   532
berghofe@13449
   533
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   534
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   535
huffman@30079
   536
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   537
  by (simp add: less_Suc_eq)
berghofe@13449
   538
blanchet@54147
   539
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   540
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   541
berghofe@13449
   542
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   543
  by simp
berghofe@13449
   544
nipkow@14302
   545
text {* "Less than" is antisymmetric, sort of *}
nipkow@14302
   546
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   547
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   548
paulson@14267
   549
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   550
  by (rule linorder_neq_iff)
berghofe@13449
   551
berghofe@13449
   552
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   553
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   554
  shows "P n m"
berghofe@13449
   555
  apply (rule less_linear [THEN disjE])
berghofe@13449
   556
  apply (erule_tac [2] disjE)
berghofe@13449
   557
  apply (erule lessCase)
berghofe@13449
   558
  apply (erule sym [THEN eqCase])
berghofe@13449
   559
  apply (erule major)
berghofe@13449
   560
  done
berghofe@13449
   561
berghofe@13449
   562
berghofe@13449
   563
subsubsection {* Inductive (?) properties *}
berghofe@13449
   564
paulson@14267
   565
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
haftmann@26072
   566
  unfolding less_eq_Suc_le [of m] le_less by simp 
berghofe@13449
   567
haftmann@26072
   568
lemma lessE:
haftmann@26072
   569
  assumes major: "i < k"
haftmann@26072
   570
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   571
  shows P
haftmann@26072
   572
proof -
haftmann@26072
   573
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   574
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   575
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   576
    by (clarsimp simp add: less_le)
haftmann@26072
   577
  with p1 p2 show P by auto
haftmann@26072
   578
qed
haftmann@26072
   579
haftmann@26072
   580
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   581
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   582
  apply (rule major [THEN lessE])
haftmann@26072
   583
  apply (rule eq, blast)
haftmann@26072
   584
  apply (rule less, blast)
berghofe@13449
   585
  done
berghofe@13449
   586
berghofe@13449
   587
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   588
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   589
  apply (rule major [THEN lessE])
berghofe@13449
   590
  apply (erule lessI [THEN minor])
paulson@14208
   591
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   592
  done
berghofe@13449
   593
berghofe@13449
   594
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   595
  by simp
berghofe@13449
   596
berghofe@13449
   597
lemma less_trans_Suc:
berghofe@13449
   598
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   599
  apply (induct k, simp_all)
berghofe@13449
   600
  apply (insert le)
berghofe@13449
   601
  apply (simp add: less_Suc_eq)
berghofe@13449
   602
  apply (blast dest: Suc_lessD)
berghofe@13449
   603
  done
berghofe@13449
   604
berghofe@13449
   605
text {* Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"} *}
haftmann@26072
   606
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   607
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   608
haftmann@26072
   609
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   610
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   611
haftmann@24995
   612
text {* Properties of "less than or equal" *}
berghofe@13449
   613
paulson@14267
   614
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   615
  unfolding less_Suc_eq_le .
berghofe@13449
   616
paulson@14267
   617
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   618
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   619
paulson@14267
   620
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   621
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   622
paulson@14267
   623
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   624
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   625
paulson@14267
   626
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   627
  unfolding Suc_le_eq .
berghofe@13449
   628
berghofe@13449
   629
text {* Stronger version of @{text Suc_leD} *}
paulson@14267
   630
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   631
  unfolding Suc_le_eq .
berghofe@13449
   632
wenzelm@26315
   633
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   634
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   635
paulson@14267
   636
text {* For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"} *}
wenzelm@26315
   637
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   638
berghofe@13449
   639
paulson@14267
   640
text {* Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"} *}
berghofe@13449
   641
paulson@14267
   642
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   643
  unfolding le_less .
berghofe@13449
   644
paulson@14267
   645
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   646
  by (rule le_less)
berghofe@13449
   647
wenzelm@22718
   648
text {* Useful with @{text blast}. *}
paulson@14267
   649
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   650
  by auto
berghofe@13449
   651
paulson@14267
   652
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   653
  by simp
berghofe@13449
   654
paulson@14267
   655
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   656
  by (rule order_trans)
berghofe@13449
   657
nipkow@33657
   658
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   659
  by (rule antisym)
berghofe@13449
   660
paulson@14267
   661
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   662
  by (rule less_le)
berghofe@13449
   663
paulson@14267
   664
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   665
  unfolding less_le ..
berghofe@13449
   666
haftmann@26072
   667
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   668
  by (rule linear)
paulson@14341
   669
wenzelm@22718
   670
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   671
haftmann@26072
   672
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   673
  unfolding less_Suc_eq_le by auto
berghofe@13449
   674
haftmann@26072
   675
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   676
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   677
berghofe@13449
   678
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   679
paulson@14267
   680
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   681
by (cases n) simp_all
nipkow@25162
   682
nipkow@25162
   683
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   684
by (cases n) simp_all
berghofe@13449
   685
wenzelm@22718
   686
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   687
by (cases n) simp_all
berghofe@13449
   688
nipkow@25162
   689
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   690
by (cases n) simp_all
nipkow@25140
   691
berghofe@13449
   692
text {* This theorem is useful with @{text blast} *}
berghofe@13449
   693
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   694
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   695
paulson@14267
   696
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   697
by (fast intro: not0_implies_Suc)
berghofe@13449
   698
blanchet@54147
   699
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   700
using neq0_conv by blast
berghofe@13449
   701
paulson@14267
   702
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   703
by (induct m') simp_all
berghofe@13449
   704
berghofe@13449
   705
text {* Useful in certain inductive arguments *}
paulson@14267
   706
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   707
by (cases m) simp_all
berghofe@13449
   708
berghofe@13449
   709
haftmann@26072
   710
subsubsection {* Monotonicity of Addition *}
berghofe@13449
   711
haftmann@26072
   712
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   713
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   714
huffman@30128
   715
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   716
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   717
paulson@14331
   718
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   719
by (induct k) simp_all
berghofe@13449
   720
paulson@14331
   721
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   722
by (induct k) simp_all
berghofe@13449
   723
nipkow@25162
   724
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   725
by(auto dest:gr0_implies_Suc)
berghofe@13449
   726
paulson@14341
   727
text {* strict, in 1st argument *}
paulson@14341
   728
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   729
by (induct k) simp_all
paulson@14341
   730
paulson@14341
   731
text {* strict, in both arguments *}
paulson@14341
   732
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   733
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   734
  apply (induct j, simp_all)
paulson@14341
   735
  done
paulson@14341
   736
paulson@14341
   737
text {* Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"} *}
paulson@14341
   738
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   739
  apply (induct n)
paulson@14341
   740
  apply (simp_all add: order_le_less)
wenzelm@22718
   741
  apply (blast elim!: less_SucE
haftmann@35047
   742
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   743
  done
paulson@14341
   744
hoelzl@56194
   745
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
hoelzl@56194
   746
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
hoelzl@56194
   747
paulson@14341
   748
text {* strict, in 1st argument; proof is by induction on @{text "k > 0"} *}
nipkow@25134
   749
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   750
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   751
apply (induct_tac m)
nipkow@25134
   752
apply (simp_all add: add_less_mono)
nipkow@25134
   753
done
paulson@14341
   754
nipkow@14740
   755
text{*The naturals form an ordered @{text comm_semiring_1_cancel}*}
haftmann@35028
   756
instance nat :: linordered_semidom
paulson@14341
   757
proof
paulson@14348
   758
  show "0 < (1::nat)" by simp
haftmann@52289
   759
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
haftmann@52289
   760
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
paulson@14267
   761
qed
paulson@14267
   762
nipkow@30056
   763
instance nat :: no_zero_divisors
nipkow@30056
   764
proof
nipkow@30056
   765
  fix a::nat and b::nat show "a ~= 0 \<Longrightarrow> b ~= 0 \<Longrightarrow> a * b ~= 0" by auto
nipkow@30056
   766
qed
nipkow@30056
   767
haftmann@44817
   768
haftmann@44817
   769
subsubsection {* @{term min} and @{term max} *}
haftmann@44817
   770
haftmann@44817
   771
lemma mono_Suc: "mono Suc"
haftmann@44817
   772
by (rule monoI) simp
haftmann@44817
   773
haftmann@44817
   774
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   775
by (rule min_absorb1) simp
haftmann@44817
   776
haftmann@44817
   777
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   778
by (rule min_absorb2) simp
haftmann@44817
   779
haftmann@44817
   780
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   781
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   782
haftmann@44817
   783
lemma min_Suc1:
haftmann@44817
   784
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   785
by (simp split: nat.split)
haftmann@44817
   786
haftmann@44817
   787
lemma min_Suc2:
haftmann@44817
   788
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   789
by (simp split: nat.split)
haftmann@44817
   790
haftmann@44817
   791
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   792
by (rule max_absorb2) simp
haftmann@44817
   793
haftmann@44817
   794
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   795
by (rule max_absorb1) simp
haftmann@44817
   796
haftmann@44817
   797
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   798
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   799
haftmann@44817
   800
lemma max_Suc1:
haftmann@44817
   801
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   802
by (simp split: nat.split)
haftmann@44817
   803
haftmann@44817
   804
lemma max_Suc2:
haftmann@44817
   805
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   806
by (simp split: nat.split)
paulson@14267
   807
haftmann@44817
   808
lemma nat_mult_min_left:
haftmann@44817
   809
  fixes m n q :: nat
haftmann@44817
   810
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   811
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   812
haftmann@44817
   813
lemma nat_mult_min_right:
haftmann@44817
   814
  fixes m n q :: nat
haftmann@44817
   815
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   816
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   817
haftmann@44817
   818
lemma nat_add_max_left:
haftmann@44817
   819
  fixes m n q :: nat
haftmann@44817
   820
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   821
  by (simp add: max_def)
haftmann@44817
   822
haftmann@44817
   823
lemma nat_add_max_right:
haftmann@44817
   824
  fixes m n q :: nat
haftmann@44817
   825
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   826
  by (simp add: max_def)
haftmann@44817
   827
haftmann@44817
   828
lemma nat_mult_max_left:
haftmann@44817
   829
  fixes m n q :: nat
haftmann@44817
   830
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   831
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   832
haftmann@44817
   833
lemma nat_mult_max_right:
haftmann@44817
   834
  fixes m n q :: nat
haftmann@44817
   835
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   836
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   837
paulson@14267
   838
krauss@26748
   839
subsubsection {* Additional theorems about @{term "op \<le>"} *}
krauss@26748
   840
krauss@26748
   841
text {* Complete induction, aka course-of-values induction *}
krauss@26748
   842
haftmann@27823
   843
instance nat :: wellorder proof
haftmann@27823
   844
  fix P and n :: nat
haftmann@27823
   845
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   846
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   847
  proof (induct n)
haftmann@27823
   848
    case (0 n)
krauss@26748
   849
    have "P 0" by (rule step) auto
krauss@26748
   850
    thus ?case using 0 by auto
krauss@26748
   851
  next
haftmann@27823
   852
    case (Suc m n)
haftmann@27823
   853
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   854
    thus ?case
krauss@26748
   855
    proof
haftmann@27823
   856
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   857
    next
haftmann@27823
   858
      assume n: "n = Suc m"
haftmann@27823
   859
      show "P n"
haftmann@27823
   860
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   861
    qed
krauss@26748
   862
  qed
haftmann@27823
   863
  then show "P n" by auto
krauss@26748
   864
qed
krauss@26748
   865
nipkow@57015
   866
nipkow@57015
   867
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
nipkow@57015
   868
by (rule Least_equality[OF _ le0])
nipkow@57015
   869
haftmann@27823
   870
lemma Least_Suc:
haftmann@27823
   871
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   872
  apply (cases n, auto)
haftmann@27823
   873
  apply (frule LeastI)
haftmann@27823
   874
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   875
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   876
  apply (erule_tac [2] Least_le)
wenzelm@47988
   877
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   878
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   879
  apply (blast intro: order_antisym)
haftmann@27823
   880
  done
haftmann@27823
   881
haftmann@27823
   882
lemma Least_Suc2:
haftmann@27823
   883
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   884
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   885
  apply simp
haftmann@27823
   886
  done
haftmann@27823
   887
haftmann@27823
   888
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
   889
  apply (cases n)
haftmann@27823
   890
   apply blast
haftmann@27823
   891
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   892
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   893
  done
haftmann@27823
   894
haftmann@27823
   895
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
   896
  unfolding One_nat_def
haftmann@27823
   897
  apply (cases n)
haftmann@27823
   898
   apply blast
haftmann@27823
   899
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   900
  apply (erule exE)
haftmann@27823
   901
  apply (case_tac k)
haftmann@27823
   902
   apply simp
haftmann@27823
   903
  apply (rename_tac k1)
haftmann@27823
   904
  apply (rule_tac x=k1 in exI)
haftmann@27823
   905
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   906
  done
haftmann@27823
   907
krauss@26748
   908
lemma nat_less_induct:
krauss@26748
   909
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   910
  using assms less_induct by blast
krauss@26748
   911
krauss@26748
   912
lemma measure_induct_rule [case_names less]:
krauss@26748
   913
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   914
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   915
  shows "P a"
krauss@26748
   916
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   917
krauss@26748
   918
text {* old style induction rules: *}
krauss@26748
   919
lemma measure_induct:
krauss@26748
   920
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   921
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   922
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   923
krauss@26748
   924
lemma full_nat_induct:
krauss@26748
   925
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   926
  shows "P n"
krauss@26748
   927
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   928
paulson@19870
   929
text{*An induction rule for estabilishing binary relations*}
wenzelm@22718
   930
lemma less_Suc_induct:
paulson@19870
   931
  assumes less:  "i < j"
paulson@19870
   932
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   933
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   934
  shows "P i j"
paulson@19870
   935
proof -
krauss@31714
   936
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   937
  have "P i (Suc (i + k))"
paulson@19870
   938
  proof (induct k)
wenzelm@22718
   939
    case 0
wenzelm@22718
   940
    show ?case by (simp add: step)
paulson@19870
   941
  next
paulson@19870
   942
    case (Suc k)
krauss@31714
   943
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
haftmann@57512
   944
    hence "i < Suc (i + k)" by (simp add: add.commute)
krauss@31714
   945
    from trans[OF this lessI Suc step]
krauss@31714
   946
    show ?case by simp
paulson@19870
   947
  qed
wenzelm@22718
   948
  thus "P i j" by (simp add: j)
paulson@19870
   949
qed
paulson@19870
   950
krauss@26748
   951
text {* The method of infinite descent, frequently used in number theory.
krauss@26748
   952
Provided by Roelof Oosterhuis.
krauss@26748
   953
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   954
\begin{itemize}
krauss@26748
   955
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   956
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   957
        a smaller integer $m$ such that $\neg P(m)$.
krauss@26748
   958
\end{itemize} *}
krauss@26748
   959
krauss@26748
   960
text{* A compact version without explicit base case: *}
krauss@26748
   961
lemma infinite_descent:
krauss@26748
   962
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   963
by (induct n rule: less_induct) auto
krauss@26748
   964
krauss@26748
   965
lemma infinite_descent0[case_names 0 smaller]: 
krauss@26748
   966
  "\<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
   967
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
   968
krauss@26748
   969
text {*
krauss@26748
   970
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
   971
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
   972
\begin{itemize}
krauss@26748
   973
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
   974
\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
   975
\end{itemize}
krauss@26748
   976
NB: the proof also shows how to use the previous lemma. *}
krauss@26748
   977
krauss@26748
   978
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
   979
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
   980
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
   981
  shows "P x"
krauss@26748
   982
proof -
krauss@26748
   983
  obtain n where "n = V x" by auto
krauss@26748
   984
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
   985
  proof (induct n rule: infinite_descent0)
krauss@26748
   986
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
   987
    with A0 show "P x" by auto
krauss@26748
   988
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
   989
    case (smaller n)
krauss@26748
   990
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
   991
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
   992
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
   993
    then show ?case by auto
krauss@26748
   994
  qed
krauss@26748
   995
  ultimately show "P x" by auto
krauss@26748
   996
qed
krauss@26748
   997
krauss@26748
   998
text{* Again, without explicit base case: *}
krauss@26748
   999
lemma infinite_descent_measure:
krauss@26748
  1000
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
  1001
proof -
krauss@26748
  1002
  from assms obtain n where "n = V x" by auto
krauss@26748
  1003
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
  1004
  proof (induct n rule: infinite_descent, auto)
krauss@26748
  1005
    fix x assume "\<not> P x"
krauss@26748
  1006
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
  1007
  qed
krauss@26748
  1008
  ultimately show "P x" by auto
krauss@26748
  1009
qed
krauss@26748
  1010
paulson@14267
  1011
text {* A [clumsy] way of lifting @{text "<"}
paulson@14267
  1012
  monotonicity to @{text "\<le>"} monotonicity *}
paulson@14267
  1013
lemma less_mono_imp_le_mono:
nipkow@24438
  1014
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
  1015
by (simp add: order_le_less) (blast)
nipkow@24438
  1016
paulson@14267
  1017
paulson@14267
  1018
text {* non-strict, in 1st argument *}
paulson@14267
  1019
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
  1020
by (rule add_right_mono)
paulson@14267
  1021
paulson@14267
  1022
text {* non-strict, in both arguments *}
paulson@14267
  1023
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
  1024
by (rule add_mono)
paulson@14267
  1025
paulson@14267
  1026
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
  1027
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
  1028
paulson@14267
  1029
lemma le_add1: "n \<le> ((n + m)::nat)"
haftmann@57512
  1030
by (simp add: add.commute, rule le_add2)
berghofe@13449
  1031
berghofe@13449
  1032
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
  1033
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
  1034
berghofe@13449
  1035
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
  1036
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
  1037
paulson@14267
  1038
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
  1039
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
  1040
paulson@14267
  1041
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
  1042
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
  1043
paulson@14267
  1044
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
  1045
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1046
berghofe@13449
  1047
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1048
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1049
berghofe@13449
  1050
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1051
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1052
berghofe@13449
  1053
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1054
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1055
apply (simp_all add: le_add1)
nipkow@24438
  1056
done
berghofe@13449
  1057
berghofe@13449
  1058
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1059
apply (rule notI)
wenzelm@26335
  1060
apply (drule add_lessD1)
wenzelm@26335
  1061
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1062
done
berghofe@13449
  1063
berghofe@13449
  1064
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
haftmann@57512
  1065
by (simp add: add.commute)
berghofe@13449
  1066
paulson@14267
  1067
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1068
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1069
apply (simp_all add: le_add1)
nipkow@24438
  1070
done
berghofe@13449
  1071
paulson@14267
  1072
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
haftmann@57512
  1073
apply (simp add: add.commute)
nipkow@24438
  1074
apply (erule add_leD1)
nipkow@24438
  1075
done
berghofe@13449
  1076
paulson@14267
  1077
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1078
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1079
haftmann@57514
  1080
text {* needs @{text "!!k"} for @{text ac_simps} to work *}
berghofe@13449
  1081
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1082
by (force simp del: add_Suc_right
haftmann@57514
  1083
    simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
berghofe@13449
  1084
berghofe@13449
  1085
haftmann@26072
  1086
subsubsection {* More results about difference *}
berghofe@13449
  1087
berghofe@13449
  1088
text {* Addition is the inverse of subtraction:
paulson@14267
  1089
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}. *}
berghofe@13449
  1090
lemma add_diff_inverse: "~  m < n ==> n + (m - n) = (m::nat)"
nipkow@24438
  1091
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1092
paulson@14267
  1093
lemma le_add_diff_inverse [simp]: "n \<le> m ==> n + (m - n) = (m::nat)"
nipkow@24438
  1094
by (simp add: add_diff_inverse linorder_not_less)
berghofe@13449
  1095
paulson@14267
  1096
lemma le_add_diff_inverse2 [simp]: "n \<le> m ==> (m - n) + n = (m::nat)"
haftmann@57512
  1097
by (simp add: add.commute)
berghofe@13449
  1098
paulson@14267
  1099
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1100
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1101
berghofe@13449
  1102
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1103
apply (induct m n rule: diff_induct)
nipkow@24438
  1104
apply (erule_tac [3] less_SucE)
nipkow@24438
  1105
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1106
done
berghofe@13449
  1107
paulson@14267
  1108
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1109
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1110
haftmann@26072
  1111
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1112
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1113
haftmann@52289
  1114
instance nat :: ordered_cancel_comm_monoid_diff
haftmann@52289
  1115
proof
haftmann@52289
  1116
  show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
haftmann@52289
  1117
qed
haftmann@52289
  1118
berghofe@13449
  1119
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1120
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1121
berghofe@13449
  1122
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1123
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1124
paulson@14267
  1125
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1126
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1127
paulson@14267
  1128
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
haftmann@57512
  1129
by (simp add: add.commute diff_add_assoc)
berghofe@13449
  1130
paulson@14267
  1131
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1132
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1133
paulson@14267
  1134
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1135
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1136
paulson@14267
  1137
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1138
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1139
berghofe@13449
  1140
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1141
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1142
wenzelm@22718
  1143
lemma less_imp_add_positive:
wenzelm@22718
  1144
  assumes "i < j"
wenzelm@22718
  1145
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1146
proof
wenzelm@22718
  1147
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1148
    by (simp add: order_less_imp_le)
wenzelm@22718
  1149
qed
wenzelm@9436
  1150
haftmann@26072
  1151
text {* a nice rewrite for bounded subtraction *}
haftmann@26072
  1152
lemma nat_minus_add_max:
haftmann@26072
  1153
  fixes n m :: nat
haftmann@26072
  1154
  shows "n - m + m = max n m"
haftmann@26072
  1155
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1156
haftmann@26072
  1157
lemma nat_diff_split:
haftmann@26072
  1158
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
haftmann@26072
  1159
    -- {* elimination of @{text -} on @{text nat} *}
haftmann@26072
  1160
by (cases "a < b")
haftmann@26072
  1161
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
thomas@57492
  1162
    not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])
berghofe@13449
  1163
haftmann@26072
  1164
lemma nat_diff_split_asm:
haftmann@26072
  1165
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
haftmann@26072
  1166
    -- {* elimination of @{text -} on @{text nat} in assumptions *}
haftmann@26072
  1167
by (auto split: nat_diff_split)
berghofe@13449
  1168
huffman@47255
  1169
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1170
  by simp
huffman@47255
  1171
huffman@47255
  1172
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1173
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1174
huffman@47255
  1175
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1176
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1177
huffman@47255
  1178
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1179
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1180
huffman@47255
  1181
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1182
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1183
huffman@47255
  1184
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1185
  by (fact Let_def)
huffman@47255
  1186
berghofe@13449
  1187
haftmann@26072
  1188
subsubsection {* Monotonicity of Multiplication *}
berghofe@13449
  1189
paulson@14267
  1190
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1191
by (simp add: mult_right_mono)
berghofe@13449
  1192
paulson@14267
  1193
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1194
by (simp add: mult_left_mono)
berghofe@13449
  1195
paulson@14267
  1196
text {* @{text "\<le>"} monotonicity, BOTH arguments *}
paulson@14267
  1197
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1198
by (simp add: mult_mono)
berghofe@13449
  1199
berghofe@13449
  1200
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1201
by (simp add: mult_strict_right_mono)
berghofe@13449
  1202
paulson@14266
  1203
text{*Differs from the standard @{text zero_less_mult_iff} in that
paulson@14266
  1204
      there are no negative numbers.*}
paulson@14266
  1205
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1206
  apply (induct m)
wenzelm@22718
  1207
   apply simp
wenzelm@22718
  1208
  apply (case_tac n)
wenzelm@22718
  1209
   apply simp_all
berghofe@13449
  1210
  done
berghofe@13449
  1211
huffman@30079
  1212
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1213
  apply (induct m)
wenzelm@22718
  1214
   apply simp
wenzelm@22718
  1215
  apply (case_tac n)
wenzelm@22718
  1216
   apply simp_all
berghofe@13449
  1217
  done
berghofe@13449
  1218
paulson@14341
  1219
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1220
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1221
  apply (cases k, auto)
berghofe@13449
  1222
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1223
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1224
  done
berghofe@13449
  1225
berghofe@13449
  1226
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
haftmann@57512
  1227
by (simp add: mult.commute [of k])
berghofe@13449
  1228
paulson@14267
  1229
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1230
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1231
paulson@14267
  1232
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1233
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1234
berghofe@13449
  1235
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1236
by (subst mult_less_cancel1) simp
berghofe@13449
  1237
paulson@14267
  1238
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1239
by (subst mult_le_cancel1) simp
berghofe@13449
  1240
haftmann@26072
  1241
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1242
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1243
haftmann@26072
  1244
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1245
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1246
berghofe@13449
  1247
text {* Lemma for @{text gcd} *}
huffman@30128
  1248
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1249
  apply (drule sym)
berghofe@13449
  1250
  apply (rule disjCI)
berghofe@13449
  1251
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1252
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1253
    apply (auto)
berghofe@13449
  1254
  done
wenzelm@9436
  1255
haftmann@51263
  1256
lemma mono_times_nat:
haftmann@51263
  1257
  fixes n :: nat
haftmann@51263
  1258
  assumes "n > 0"
haftmann@51263
  1259
  shows "mono (times n)"
haftmann@51263
  1260
proof
haftmann@51263
  1261
  fix m q :: nat
haftmann@51263
  1262
  assume "m \<le> q"
haftmann@51263
  1263
  with assms show "n * m \<le> n * q" by simp
haftmann@51263
  1264
qed
haftmann@51263
  1265
haftmann@26072
  1266
text {* the lattice order on @{typ nat} *}
haftmann@24995
  1267
haftmann@26072
  1268
instantiation nat :: distrib_lattice
haftmann@26072
  1269
begin
haftmann@24995
  1270
haftmann@26072
  1271
definition
haftmann@26072
  1272
  "(inf \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1273
haftmann@26072
  1274
definition
haftmann@26072
  1275
  "(sup \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1276
haftmann@26072
  1277
instance by intro_classes
haftmann@26072
  1278
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1279
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1280
haftmann@26072
  1281
end
haftmann@24995
  1282
haftmann@24995
  1283
haftmann@30954
  1284
subsection {* Natural operation of natural numbers on functions *}
haftmann@30954
  1285
haftmann@30971
  1286
text {*
haftmann@30971
  1287
  We use the same logical constant for the power operations on
haftmann@30971
  1288
  functions and relations, in order to share the same syntax.
haftmann@30971
  1289
*}
haftmann@30971
  1290
haftmann@45965
  1291
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1292
haftmann@45965
  1293
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1294
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1295
haftmann@30971
  1296
notation (latex output)
haftmann@30971
  1297
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1298
haftmann@30971
  1299
notation (HTML output)
haftmann@30971
  1300
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1301
haftmann@30971
  1302
text {* @{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f} *}
haftmann@30971
  1303
haftmann@30971
  1304
overloading
haftmann@30971
  1305
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1306
begin
haftmann@30954
  1307
blanchet@55575
  1308
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1309
  "funpow 0 f = id"
haftmann@44325
  1310
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1311
haftmann@30971
  1312
end
haftmann@30971
  1313
haftmann@49723
  1314
lemma funpow_Suc_right:
haftmann@49723
  1315
  "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1316
proof (induct n)
haftmann@49723
  1317
  case 0 then show ?case by simp
haftmann@49723
  1318
next
haftmann@49723
  1319
  fix n
haftmann@49723
  1320
  assume "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1321
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
haftmann@49723
  1322
    by (simp add: o_assoc)
haftmann@49723
  1323
qed
haftmann@49723
  1324
haftmann@49723
  1325
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
haftmann@49723
  1326
haftmann@30971
  1327
text {* for code generation *}
haftmann@30971
  1328
haftmann@30971
  1329
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1330
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1331
haftmann@30971
  1332
lemma [code]:
haftmann@37430
  1333
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1334
  "funpow 0 f = id"
haftmann@37430
  1335
  by (simp_all add: funpow_code_def)
haftmann@30971
  1336
wenzelm@36176
  1337
hide_const (open) funpow
haftmann@30954
  1338
haftmann@30954
  1339
lemma funpow_add:
haftmann@30971
  1340
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1341
  by (induct m) simp_all
haftmann@30954
  1342
haftmann@37430
  1343
lemma funpow_mult:
haftmann@37430
  1344
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1345
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1346
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1347
haftmann@30954
  1348
lemma funpow_swap1:
haftmann@30971
  1349
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1350
proof -
haftmann@30971
  1351
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1352
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1353
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1354
  finally show ?thesis .
haftmann@30954
  1355
qed
haftmann@30954
  1356
haftmann@38621
  1357
lemma comp_funpow:
haftmann@38621
  1358
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1359
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1360
  by (induct n) simp_all
haftmann@30954
  1361
hoelzl@54496
  1362
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
hoelzl@54496
  1363
  by (induct n) simp_all
hoelzl@54496
  1364
hoelzl@54496
  1365
lemma id_funpow[simp]: "id ^^ n = id"
hoelzl@54496
  1366
  by (induct n) simp_all
haftmann@38621
  1367
nipkow@45833
  1368
subsection {* Kleene iteration *}
nipkow@45833
  1369
haftmann@52729
  1370
lemma Kleene_iter_lpfp:
haftmann@52729
  1371
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
nipkow@45833
  1372
proof(induction k)
nipkow@45833
  1373
  case 0 show ?case by simp
nipkow@45833
  1374
next
nipkow@45833
  1375
  case Suc
nipkow@45833
  1376
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1377
  show ?case by simp
nipkow@45833
  1378
qed
nipkow@45833
  1379
nipkow@45833
  1380
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1381
shows "lfp f = (f^^k) bot"
nipkow@45833
  1382
proof(rule antisym)
nipkow@45833
  1383
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1384
  proof(rule lfp_lowerbound)
nipkow@45833
  1385
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1386
  qed
nipkow@45833
  1387
next
nipkow@45833
  1388
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1389
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1390
qed
nipkow@45833
  1391
nipkow@45833
  1392
haftmann@38621
  1393
subsection {* Embedding of the Naturals into any @{text semiring_1}: @{term of_nat} *}
haftmann@24196
  1394
haftmann@24196
  1395
context semiring_1
haftmann@24196
  1396
begin
haftmann@24196
  1397
haftmann@38621
  1398
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1399
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1400
haftmann@38621
  1401
lemma of_nat_simps [simp]:
haftmann@38621
  1402
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1403
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1404
  by (simp_all add: of_nat_def)
haftmann@25193
  1405
haftmann@25193
  1406
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1407
  by (simp add: of_nat_def)
haftmann@25193
  1408
haftmann@25193
  1409
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@57514
  1410
  by (induct m) (simp_all add: ac_simps)
haftmann@25193
  1411
haftmann@25193
  1412
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@57514
  1413
  by (induct m) (simp_all add: ac_simps distrib_right)
haftmann@25193
  1414
blanchet@55575
  1415
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1416
  "of_nat_aux inc 0 i = i"
haftmann@44325
  1417
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- {* tail recursive *}
haftmann@25928
  1418
haftmann@30966
  1419
lemma of_nat_code:
haftmann@28514
  1420
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1421
proof (induct n)
haftmann@28514
  1422
  case 0 then show ?case by simp
haftmann@28514
  1423
next
haftmann@28514
  1424
  case (Suc n)
haftmann@28514
  1425
  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
  1426
    by (induct n) simp_all
haftmann@28514
  1427
  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
  1428
    by simp
haftmann@57512
  1429
  with Suc show ?case by (simp add: add.commute)
haftmann@28514
  1430
qed
haftmann@30966
  1431
haftmann@24196
  1432
end
haftmann@24196
  1433
bulwahn@45231
  1434
declare of_nat_code [code]
haftmann@30966
  1435
haftmann@26072
  1436
text{*Class for unital semirings with characteristic zero.
haftmann@26072
  1437
 Includes non-ordered rings like the complex numbers.*}
haftmann@26072
  1438
haftmann@26072
  1439
class semiring_char_0 = semiring_1 +
haftmann@38621
  1440
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1441
begin
haftmann@26072
  1442
haftmann@38621
  1443
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1444
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1445
haftmann@26072
  1446
text{*Special cases where either operand is zero*}
haftmann@26072
  1447
blanchet@54147
  1448
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1449
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1450
blanchet@54147
  1451
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1452
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1453
haftmann@26072
  1454
end
haftmann@26072
  1455
haftmann@35028
  1456
context linordered_semidom
haftmann@25193
  1457
begin
haftmann@25193
  1458
huffman@47489
  1459
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1460
  by (induct n) simp_all
haftmann@25193
  1461
huffman@47489
  1462
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1463
  by (simp add: not_less)
haftmann@25193
  1464
haftmann@25193
  1465
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
huffman@47489
  1466
  by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
haftmann@25193
  1467
haftmann@26072
  1468
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1469
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1470
huffman@47489
  1471
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1472
  by simp
huffman@47489
  1473
huffman@47489
  1474
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1475
  by simp
huffman@47489
  1476
haftmann@35028
  1477
text{*Every @{text linordered_semidom} has characteristic zero.*}
haftmann@25193
  1478
haftmann@38621
  1479
subclass semiring_char_0 proof
haftmann@38621
  1480
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1481
haftmann@25193
  1482
text{*Special cases where either operand is zero*}
haftmann@25193
  1483
blanchet@54147
  1484
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1485
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1486
haftmann@26072
  1487
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1488
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1489
haftmann@26072
  1490
end
haftmann@26072
  1491
haftmann@26072
  1492
context ring_1
haftmann@26072
  1493
begin
haftmann@26072
  1494
haftmann@26072
  1495
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1496
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1497
haftmann@26072
  1498
end
haftmann@26072
  1499
haftmann@35028
  1500
context linordered_idom
haftmann@26072
  1501
begin
haftmann@26072
  1502
haftmann@26072
  1503
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1504
  unfolding abs_if by auto
haftmann@26072
  1505
haftmann@25193
  1506
end
haftmann@25193
  1507
haftmann@25193
  1508
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1509
  by (induct n) simp_all
haftmann@25193
  1510
haftmann@25193
  1511
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1512
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1513
haftmann@25193
  1514
haftmann@26149
  1515
subsection {* The Set of Natural Numbers *}
haftmann@25193
  1516
haftmann@26072
  1517
context semiring_1
haftmann@25193
  1518
begin
haftmann@25193
  1519
haftmann@37767
  1520
definition Nats  :: "'a set" where
haftmann@37767
  1521
  "Nats = range of_nat"
haftmann@26072
  1522
haftmann@26072
  1523
notation (xsymbols)
haftmann@26072
  1524
  Nats  ("\<nat>")
haftmann@25193
  1525
haftmann@26072
  1526
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1527
  by (simp add: Nats_def)
haftmann@26072
  1528
haftmann@26072
  1529
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1530
apply (simp add: Nats_def)
haftmann@26072
  1531
apply (rule range_eqI)
haftmann@26072
  1532
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1533
done
haftmann@25193
  1534
haftmann@26072
  1535
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1536
apply (simp add: Nats_def)
haftmann@26072
  1537
apply (rule range_eqI)
haftmann@26072
  1538
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1539
done
haftmann@25193
  1540
haftmann@26072
  1541
lemma Nats_add [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_add [symmetric])
haftmann@26072
  1545
done
haftmann@26072
  1546
haftmann@26072
  1547
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1548
apply (auto simp add: Nats_def)
haftmann@26072
  1549
apply (rule range_eqI)
haftmann@26072
  1550
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1551
done
haftmann@25193
  1552
huffman@35633
  1553
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1554
  assumes "x \<in> \<nat>"
huffman@35633
  1555
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1556
  unfolding Nats_def
huffman@35633
  1557
proof -
huffman@35633
  1558
  from `x \<in> \<nat>` have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1559
  then obtain n where "x = of_nat n" ..
huffman@35633
  1560
  then show thesis ..
huffman@35633
  1561
qed
huffman@35633
  1562
huffman@35633
  1563
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1564
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1565
  by (rule Nats_cases) auto
huffman@35633
  1566
haftmann@25193
  1567
end
haftmann@25193
  1568
haftmann@25193
  1569
wenzelm@21243
  1570
subsection {* Further Arithmetic Facts Concerning the Natural Numbers *}
wenzelm@21243
  1571
haftmann@22845
  1572
lemma subst_equals:
haftmann@22845
  1573
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1574
  shows "u = s"
haftmann@22845
  1575
  using 2 1 by (rule trans)
haftmann@22845
  1576
haftmann@30686
  1577
setup Arith_Data.setup
haftmann@30686
  1578
wenzelm@48891
  1579
ML_file "Tools/nat_arith.ML"
huffman@48559
  1580
huffman@48559
  1581
simproc_setup nateq_cancel_sums
huffman@48559
  1582
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
wenzelm@54742
  1583
  {* fn phi => try o Nat_Arith.cancel_eq_conv *}
huffman@48559
  1584
huffman@48559
  1585
simproc_setup natless_cancel_sums
huffman@48559
  1586
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
wenzelm@54742
  1587
  {* fn phi => try o Nat_Arith.cancel_less_conv *}
huffman@48559
  1588
huffman@48559
  1589
simproc_setup natle_cancel_sums
huffman@48559
  1590
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
wenzelm@54742
  1591
  {* fn phi => try o Nat_Arith.cancel_le_conv *}
huffman@48559
  1592
huffman@48559
  1593
simproc_setup natdiff_cancel_sums
huffman@48559
  1594
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
wenzelm@54742
  1595
  {* fn phi => try o Nat_Arith.cancel_diff_conv *}
wenzelm@24091
  1596
wenzelm@48891
  1597
ML_file "Tools/lin_arith.ML"
haftmann@31100
  1598
setup {* Lin_Arith.global_setup *}
haftmann@30686
  1599
declaration {* K Lin_Arith.setup *}
wenzelm@24091
  1600
wenzelm@43595
  1601
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) <= n" | "(m::nat) = n") =
wenzelm@43595
  1602
  {* fn _ => fn ss => fn ct => Lin_Arith.simproc ss (term_of ct) *}
wenzelm@43595
  1603
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1604
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1605
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1606
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1607
solver all the time rather than add the additional check. *)
wenzelm@43595
  1608
wenzelm@43595
  1609
wenzelm@21243
  1610
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1611
nipkow@27625
  1612
context order
nipkow@27625
  1613
begin
nipkow@27625
  1614
nipkow@27625
  1615
lemma lift_Suc_mono_le:
haftmann@53986
  1616
  assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
krauss@27627
  1617
  shows "f n \<le> f n'"
krauss@27627
  1618
proof (cases "n < n'")
krauss@27627
  1619
  case True
haftmann@53986
  1620
  then show ?thesis
haftmann@53986
  1621
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
haftmann@53986
  1622
qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
nipkow@27625
  1623
hoelzl@56020
  1624
lemma lift_Suc_antimono_le:
hoelzl@56020
  1625
  assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
hoelzl@56020
  1626
  shows "f n \<ge> f n'"
hoelzl@56020
  1627
proof (cases "n < n'")
hoelzl@56020
  1628
  case True
hoelzl@56020
  1629
  then show ?thesis
hoelzl@56020
  1630
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
hoelzl@56020
  1631
qed (insert `n \<le> n'`, auto) -- {* trivial for @{prop "n = n'"} *}
hoelzl@56020
  1632
nipkow@27625
  1633
lemma lift_Suc_mono_less:
haftmann@53986
  1634
  assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
krauss@27627
  1635
  shows "f n < f n'"
krauss@27627
  1636
using `n < n'`
haftmann@53986
  1637
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
nipkow@27625
  1638
nipkow@27789
  1639
lemma lift_Suc_mono_less_iff:
haftmann@53986
  1640
  "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
haftmann@53986
  1641
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
haftmann@53986
  1642
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
nipkow@27789
  1643
nipkow@27625
  1644
end
nipkow@27625
  1645
haftmann@53986
  1646
lemma mono_iff_le_Suc:
haftmann@53986
  1647
  "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1648
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1649
hoelzl@56020
  1650
lemma antimono_iff_le_Suc:
hoelzl@56020
  1651
  "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@56020
  1652
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
hoelzl@56020
  1653
nipkow@27789
  1654
lemma mono_nat_linear_lb:
haftmann@53986
  1655
  fixes f :: "nat \<Rightarrow> nat"
haftmann@53986
  1656
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
haftmann@53986
  1657
  shows "f m + k \<le> f (m + k)"
haftmann@53986
  1658
proof (induct k)
haftmann@53986
  1659
  case 0 then show ?case by simp
haftmann@53986
  1660
next
haftmann@53986
  1661
  case (Suc k)
haftmann@53986
  1662
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
haftmann@53986
  1663
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
haftmann@53986
  1664
    by (simp add: Suc_le_eq)
haftmann@53986
  1665
  finally show ?case by simp
haftmann@53986
  1666
qed
nipkow@27789
  1667
nipkow@27789
  1668
wenzelm@21243
  1669
text{*Subtraction laws, mostly by Clemens Ballarin*}
wenzelm@21243
  1670
wenzelm@21243
  1671
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1672
by arith
wenzelm@21243
  1673
wenzelm@21243
  1674
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1675
by arith
wenzelm@21243
  1676
haftmann@51173
  1677
lemma less_diff_conv2:
haftmann@51173
  1678
  fixes j k i :: nat
haftmann@51173
  1679
  assumes "k \<le> j"
haftmann@51173
  1680
  shows "j - k < i \<longleftrightarrow> j < i + k"
haftmann@51173
  1681
  using assms by arith
haftmann@51173
  1682
wenzelm@21243
  1683
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1684
by arith
wenzelm@21243
  1685
wenzelm@21243
  1686
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
haftmann@57512
  1687
  by (fact le_diff_conv2) -- {* FIXME delete *}
wenzelm@21243
  1688
wenzelm@21243
  1689
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1690
by arith
wenzelm@21243
  1691
wenzelm@21243
  1692
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
haftmann@57512
  1693
  by (fact le_add_diff) -- {* FIXME delete *}
wenzelm@21243
  1694
wenzelm@21243
  1695
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1696
  second premise n\<le>m*)
wenzelm@21243
  1697
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1698
by arith
wenzelm@21243
  1699
haftmann@26072
  1700
text {* Simplification of relational expressions involving subtraction *}
wenzelm@21243
  1701
wenzelm@21243
  1702
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1703
by (simp split add: nat_diff_split)
wenzelm@21243
  1704
wenzelm@36176
  1705
hide_fact (open) diff_diff_eq
haftmann@35064
  1706
wenzelm@21243
  1707
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1708
by (auto split add: nat_diff_split)
wenzelm@21243
  1709
wenzelm@21243
  1710
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1711
by (auto split add: nat_diff_split)
wenzelm@21243
  1712
wenzelm@21243
  1713
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1714
by (auto split add: nat_diff_split)
wenzelm@21243
  1715
wenzelm@21243
  1716
text{*(Anti)Monotonicity of subtraction -- by Stephan Merz*}
wenzelm@21243
  1717
wenzelm@21243
  1718
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1719
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1720
by (simp split add: nat_diff_split)
wenzelm@21243
  1721
wenzelm@21243
  1722
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1723
by (simp split add: nat_diff_split)
wenzelm@21243
  1724
wenzelm@21243
  1725
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1726
by (simp split add: nat_diff_split)
wenzelm@21243
  1727
wenzelm@21243
  1728
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1729
by (simp split add: nat_diff_split)
wenzelm@21243
  1730
bulwahn@26143
  1731
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1732
by auto
bulwahn@26143
  1733
bulwahn@26143
  1734
lemma inj_on_diff_nat: 
bulwahn@26143
  1735
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1736
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1737
proof (rule inj_onI)
bulwahn@26143
  1738
  fix x y
bulwahn@26143
  1739
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1740
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1741
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1742
qed
bulwahn@26143
  1743
haftmann@26072
  1744
text{*Rewriting to pull differences out*}
haftmann@26072
  1745
haftmann@26072
  1746
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1747
by arith
haftmann@26072
  1748
haftmann@26072
  1749
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1750
by arith
haftmann@26072
  1751
haftmann@26072
  1752
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1753
by arith
haftmann@26072
  1754
noschinl@45933
  1755
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1756
by simp
noschinl@45933
  1757
bulwahn@46350
  1758
(*The others are
bulwahn@46350
  1759
      i - j - k = i - (j + k),
bulwahn@46350
  1760
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1761
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1762
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1763
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
bulwahn@46350
  1764
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1765
bulwahn@46350
  1766
text{*At present we prove no analogue of @{text not_less_Least} or @{text
bulwahn@46350
  1767
Least_Suc}, since there appears to be no need.*}
bulwahn@46350
  1768
wenzelm@21243
  1769
text{*Lemmas for ex/Factorization*}
wenzelm@21243
  1770
wenzelm@21243
  1771
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1772
by (cases m) auto
wenzelm@21243
  1773
wenzelm@21243
  1774
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1775
by (cases m) auto
wenzelm@21243
  1776
wenzelm@21243
  1777
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1778
by (cases m) auto
wenzelm@21243
  1779
krauss@23001
  1780
text {* Specialized induction principles that work "backwards": *}
krauss@23001
  1781
krauss@23001
  1782
lemma inc_induct[consumes 1, case_names base step]:
hoelzl@54411
  1783
  assumes less: "i \<le> j"
krauss@23001
  1784
  assumes base: "P j"
hoelzl@54411
  1785
  assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
krauss@23001
  1786
  shows "P i"
hoelzl@54411
  1787
  using less step
hoelzl@54411
  1788
proof (induct d\<equiv>"j - i" arbitrary: i)
krauss@23001
  1789
  case (0 i)
krauss@23001
  1790
  hence "i = j" by simp
krauss@23001
  1791
  with base show ?case by simp
krauss@23001
  1792
next
hoelzl@54411
  1793
  case (Suc d n)
hoelzl@54411
  1794
  hence "n \<le> n" "n < j" "P (Suc n)"
krauss@23001
  1795
    by simp_all
hoelzl@54411
  1796
  then show "P n" by fact
krauss@23001
  1797
qed
krauss@23001
  1798
krauss@23001
  1799
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1800
  assumes less: "i < j"
krauss@23001
  1801
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1802
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1803
  shows "P i"
krauss@23001
  1804
  using less
krauss@23001
  1805
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1806
  case (0 i)
krauss@23001
  1807
  with `i < j` have "j = Suc i" by simp
krauss@23001
  1808
  with base show ?case by simp
krauss@23001
  1809
next
krauss@23001
  1810
  case (Suc d i)
krauss@23001
  1811
  hence "i < j" "P (Suc i)"
krauss@23001
  1812
    by simp_all
krauss@23001
  1813
  thus "P i" by (rule step)
krauss@23001
  1814
qed
krauss@23001
  1815
krauss@23001
  1816
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1817
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1818
krauss@23001
  1819
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1820
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1821
bulwahn@46351
  1822
text {* Further induction rule similar to @{thm inc_induct} *}
nipkow@27625
  1823
bulwahn@46351
  1824
lemma dec_induct[consumes 1, case_names base step]:
hoelzl@54411
  1825
  "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
  1826
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
bulwahn@46351
  1827
 
haftmann@33274
  1828
subsection {* The divides relation on @{typ nat} *}
haftmann@33274
  1829
haftmann@33274
  1830
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1831
unfolding dvd_def by simp
haftmann@33274
  1832
haftmann@33274
  1833
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1834
by (simp add: dvd_def)
haftmann@33274
  1835
haftmann@33274
  1836
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1837
by (simp add: dvd_def)
haftmann@33274
  1838
nipkow@33657
  1839
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1840
  unfolding dvd_def
haftmann@57512
  1841
  by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
haftmann@33274
  1842
haftmann@33274
  1843
text {* @{term "op dvd"} is a partial order *}
haftmann@33274
  1844
haftmann@33274
  1845
interpretation dvd: order "op dvd" "\<lambda>n m \<Colon> nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1846
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1847
haftmann@33274
  1848
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1849
unfolding dvd_def
haftmann@33274
  1850
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1851
haftmann@33274
  1852
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1853
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1854
  apply (blast intro: dvd_add)
haftmann@33274
  1855
  done
haftmann@33274
  1856
haftmann@33274
  1857
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1858
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1859
haftmann@33274
  1860
lemma dvd_reduce: "(k dvd n + k) = (k dvd (n::nat))"
haftmann@33274
  1861
  apply (rule iffI)
haftmann@33274
  1862
   apply (erule_tac [2] dvd_add)
haftmann@33274
  1863
   apply (rule_tac [2] dvd_refl)
haftmann@33274
  1864
  apply (subgoal_tac "n = (n+k) -k")
haftmann@33274
  1865
   prefer 2 apply simp
haftmann@33274
  1866
  apply (erule ssubst)
haftmann@33274
  1867
  apply (erule dvd_diff_nat)
haftmann@33274
  1868
  apply (rule dvd_refl)
haftmann@33274
  1869
  done
haftmann@33274
  1870
haftmann@33274
  1871
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1872
  unfolding dvd_def
haftmann@33274
  1873
  apply (erule exE)
haftmann@57514
  1874
  apply (simp add: ac_simps)
haftmann@33274
  1875
  done
haftmann@33274
  1876
haftmann@33274
  1877
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1878
  apply auto
haftmann@33274
  1879
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1880
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1881
  done
haftmann@33274
  1882
haftmann@33274
  1883
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@57512
  1884
  apply (subst mult.commute)
haftmann@33274
  1885
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1886
  done
haftmann@33274
  1887
haftmann@33274
  1888
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1889
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1890
haftmann@33274
  1891
lemma nat_dvd_not_less:
haftmann@33274
  1892
  fixes m n :: nat
haftmann@33274
  1893
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1894
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1895
haftmann@51173
  1896
lemma dvd_plusE:
haftmann@51173
  1897
  fixes m n q :: nat
haftmann@51173
  1898
  assumes "m dvd n + q" "m dvd n"
haftmann@51173
  1899
  obtains "m dvd q"
haftmann@51173
  1900
proof (cases "m = 0")
haftmann@51173
  1901
  case True with assms that show thesis by simp
haftmann@51173
  1902
next
haftmann@51173
  1903
  case False then have "m > 0" by simp
haftmann@51173
  1904
  from assms obtain r s where "n = m * r" and "n + q = m * s" by (blast elim: dvdE)
haftmann@51173
  1905
  then have *: "m * r + q = m * s" by simp
haftmann@51173
  1906
  show thesis proof (cases "r \<le> s")
haftmann@51173
  1907
    case False then have "s < r" by (simp add: not_le)
haftmann@51173
  1908
    with * have "m * r + q - m * s = m * s - m * s" by simp
haftmann@51173
  1909
    then have "m * r + q - m * s = 0" by simp
haftmann@53986
  1910
    with `m > 0` `s < r` have "m * r - m * s + q = 0" by (unfold less_le_not_le) auto
haftmann@51173
  1911
    then have "m * (r - s) + q = 0" by auto
haftmann@51173
  1912
    then have "m * (r - s) = 0" by simp
haftmann@51173
  1913
    then have "m = 0 \<or> r - s = 0" by simp
haftmann@53986
  1914
    with `s < r` have "m = 0" by (simp add: less_le_not_le)
haftmann@51173
  1915
    with `m > 0` show thesis by auto
haftmann@51173
  1916
  next
haftmann@51173
  1917
    case True with * have "m * r + q - m * r = m * s - m * r" by simp
haftmann@51173
  1918
    with `m > 0` `r \<le> s` have "m * r - m * r + q = m * s - m * r" by simp
haftmann@51173
  1919
    then have "q = m * (s - r)" by (simp add: diff_mult_distrib2)
haftmann@51173
  1920
    with assms that show thesis by (auto intro: dvdI)
haftmann@51173
  1921
  qed
haftmann@51173
  1922
qed
haftmann@51173
  1923
haftmann@51173
  1924
lemma dvd_plus_eq_right:
haftmann@51173
  1925
  fixes m n q :: nat
haftmann@51173
  1926
  assumes "m dvd n"
haftmann@51173
  1927
  shows "m dvd n + q \<longleftrightarrow> m dvd q"
haftmann@51173
  1928
  using assms by (auto elim: dvd_plusE)
haftmann@51173
  1929
haftmann@51173
  1930
lemma dvd_plus_eq_left:
haftmann@51173
  1931
  fixes m n q :: nat
haftmann@51173
  1932
  assumes "m dvd q"
haftmann@51173
  1933
  shows "m dvd n + q \<longleftrightarrow> m dvd n"
haftmann@57512
  1934
  using assms by (simp add: dvd_plus_eq_right add.commute [of n])
haftmann@51173
  1935
haftmann@54222
  1936
lemma less_eq_dvd_minus:
haftmann@51173
  1937
  fixes m n :: nat
haftmann@54222
  1938
  assumes "m \<le> n"
haftmann@54222
  1939
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
haftmann@51173
  1940
proof -
haftmann@54222
  1941
  from assms have "n = m + (n - m)" by simp
haftmann@51173
  1942
  then obtain q where "n = m + q" ..
haftmann@57512
  1943
  then show ?thesis by (simp add: dvd_reduce add.commute [of m])
haftmann@51173
  1944
qed
haftmann@51173
  1945
haftmann@51173
  1946
lemma dvd_minus_self:
haftmann@51173
  1947
  fixes m n :: nat
haftmann@51173
  1948
  shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
haftmann@51173
  1949
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
haftmann@51173
  1950
haftmann@51173
  1951
lemma dvd_minus_add:
haftmann@51173
  1952
  fixes m n q r :: nat
haftmann@51173
  1953
  assumes "q \<le> n" "q \<le> r * m"
haftmann@51173
  1954
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
haftmann@51173
  1955
proof -
haftmann@51173
  1956
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
haftmann@51173
  1957
    by (auto elim: dvd_plusE)
wenzelm@53374
  1958
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
wenzelm@53374
  1959
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
haftmann@57512
  1960
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
haftmann@51173
  1961
  finally show ?thesis .
haftmann@51173
  1962
qed
haftmann@51173
  1963
haftmann@33274
  1964
blanchet@55424
  1965
subsection {* aliases *}
haftmann@44817
  1966
haftmann@44817
  1967
lemma nat_mult_1: "(1::nat) * n = n"
blanchet@55424
  1968
  by (rule mult_1_left)
haftmann@44817
  1969
 
haftmann@44817
  1970
lemma nat_mult_1_right: "n * (1::nat) = n"
blanchet@55424
  1971
  by (rule mult_1_right)
haftmann@44817
  1972
haftmann@44817
  1973
haftmann@26072
  1974
subsection {* size of a datatype value *}
haftmann@25193
  1975
haftmann@29608
  1976
class size =
krauss@26748
  1977
  fixes size :: "'a \<Rightarrow> nat" -- {* see further theory @{text Wellfounded} *}
haftmann@23852
  1978
haftmann@33364
  1979
haftmann@33364
  1980
subsection {* code module namespace *}
haftmann@33364
  1981
haftmann@52435
  1982
code_identifier
haftmann@52435
  1983
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  1984
huffman@47108
  1985
hide_const (open) of_nat_aux
huffman@47108
  1986
haftmann@25193
  1987
end