src/HOL/Nat.thy
author wenzelm
Sun Sep 13 22:56:52 2015 +0200 (2015-09-13)
changeset 61169 4de9ff3ea29a
parent 61144 5e94dfead1c2
child 61378 3e04c9ca001a
permissions -rw-r--r--
tuned proofs -- less legacy;
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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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section \<open>Natural numbers\<close>
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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 \<open>Type @{text ind}\<close>
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typedecl ind
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axiomatization Zero_Rep :: ind and Suc_Rep :: "ind => ind" where
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  -- \<open>the axiom of infinity in 2 parts\<close>
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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 \<open>Type nat\<close>
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text \<open>Type definition\<close>
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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]) -- \<open>types force good instantiation\<close>
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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 :: nat"
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  | Suc pred
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where
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  "pred (0 :: nat) = (0 :: 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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-- \<open>Avoid name clashes by prefixing the output of @{text old_rep_datatype} with @{text old}.\<close>
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setup \<open>Sign.mandatory_path "old"\<close>
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old_rep_datatype "0 :: 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 \<open>Sign.parent_path\<close>
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-- \<open>But erase the prefix for properties that are not generated by @{text free_constructors}.\<close>
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setup \<open>Sign.mandatory_path "nat"\<close>
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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 \<open>Sign.parent_path\<close>
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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 -- \<open>hide everything related to the selector\<close>
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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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  -- \<open>for backward compatibility -- names of variables differ\<close>
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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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  -- \<open>for backward compatibility -- names of variables differ\<close>
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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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ML \<open>
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val nat_basic_lfp_sugar =
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  let
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    val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
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    val recx = Logic.varify_types_global @{term rec_nat};
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    val C = body_type (fastype_of recx);
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  in
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    {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
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     ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
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  end;
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\<close>
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setup \<open>
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let
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  fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
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      ([], [0], [nat_basic_lfp_sugar], [], [], TrueI (*dummy*), [], false, ctxt)
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    | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
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      BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
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in
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  BNF_LFP_Rec_Sugar.register_lfp_rec_extension
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    {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
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     rewrite_nested_rec_call = NONE}
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end
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\<close>
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text \<open>Injectiveness and distinctness lemmas\<close>
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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 \<open>A special form of induction for reasoning
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  about @{term "m < n"} and @{term "m - n"}\<close>
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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 \<open>Arithmetic operators\<close>
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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::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::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 "m + n - m = n" by (induct m) 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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  show "0 + n = n" by simp
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  show "0 - n = 0" by simp
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qed
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end
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hide_fact (open) add_0 add_0_right diff_0
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instantiation nat :: comm_semiring_1_cancel
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begin
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definition
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  One_nat_def [simp]: "1 = Suc 0"
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primrec times_nat where
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  mult_0: "0 * n = (0::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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next
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  fix k m n :: nat
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  show "k * ((m::nat) - n) = (k * m) - (k * n)"
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    by (induct m n rule: diff_induct) simp_all
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qed
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end
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text \<open>Difference distributes over multiplication\<close>
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lemma diff_mult_distrib:
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  "((m::nat) - n) * k = (m * k) - (n * k)"
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  by (fact left_diff_distrib')
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lemma diff_mult_distrib2:
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  "k * ((m::nat) - n) = (k * m) - (k * n)"
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  by (fact right_diff_distrib')
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subsubsection \<open>Addition\<close>
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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 \<open>Reasoning about @{text "m + 0 = 0"}, etc.\<close>
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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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   340
  apply (simp add:o_def)
haftmann@26072
   341
  done
haftmann@26072
   342
huffman@47208
   343
lemma Suc_eq_plus1: "Suc n = n + 1"
huffman@47208
   344
  unfolding One_nat_def by simp
huffman@47208
   345
huffman@47208
   346
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
huffman@47208
   347
  unfolding One_nat_def by simp
huffman@47208
   348
haftmann@26072
   349
wenzelm@60758
   350
subsubsection \<open>Difference\<close>
haftmann@26072
   351
wenzelm@61076
   352
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
haftmann@57512
   353
  by (fact diff_cancel)
haftmann@26072
   354
haftmann@26072
   355
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
haftmann@57512
   356
  by (fact diff_diff_add)
haftmann@26072
   357
haftmann@26072
   358
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
haftmann@26072
   359
  by (simp add: diff_diff_left)
haftmann@26072
   360
haftmann@26072
   361
lemma diff_commute: "(i::nat) - j - k = i - k - j"
haftmann@57512
   362
  by (fact diff_right_commute)
haftmann@26072
   363
haftmann@26072
   364
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
haftmann@57512
   365
  by (fact add_diff_cancel_left')
haftmann@26072
   366
haftmann@26072
   367
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
haftmann@57512
   368
  by (fact add_diff_cancel_right')
haftmann@26072
   369
haftmann@26072
   370
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
haftmann@59815
   371
  by (fact add_diff_cancel_left)
haftmann@26072
   372
haftmann@26072
   373
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
haftmann@57512
   374
  by (fact add_diff_cancel_right)
haftmann@26072
   375
haftmann@26072
   376
lemma diff_add_0: "n - (n + m) = (0::nat)"
haftmann@57512
   377
  by (fact diff_add_zero)
haftmann@26072
   378
huffman@30093
   379
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
huffman@30093
   380
  unfolding One_nat_def by simp
huffman@30093
   381
wenzelm@60758
   382
subsubsection \<open>Multiplication\<close>
haftmann@26072
   383
haftmann@26072
   384
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
haftmann@57512
   385
  by (fact distrib_left)
haftmann@26072
   386
haftmann@26072
   387
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
haftmann@26072
   388
  by (induct m) auto
haftmann@26072
   389
haftmann@26072
   390
lemmas nat_distrib =
haftmann@26072
   391
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
haftmann@26072
   392
huffman@30079
   393
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   394
  apply (induct m)
haftmann@26072
   395
   apply simp
haftmann@26072
   396
  apply (induct n)
haftmann@26072
   397
   apply auto
haftmann@26072
   398
  done
haftmann@26072
   399
blanchet@54147
   400
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
haftmann@26072
   401
  apply (rule trans)
nipkow@44890
   402
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
haftmann@26072
   403
  done
haftmann@26072
   404
huffman@30079
   405
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   406
  unfolding One_nat_def by (rule mult_eq_1_iff)
huffman@30079
   407
huffman@30079
   408
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
huffman@30079
   409
  unfolding One_nat_def by (rule one_eq_mult_iff)
huffman@30079
   410
haftmann@26072
   411
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
haftmann@26072
   412
proof -
haftmann@26072
   413
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
haftmann@26072
   414
  proof (induct n arbitrary: m)
haftmann@26072
   415
    case 0 then show "m = 0" by simp
haftmann@26072
   416
  next
haftmann@26072
   417
    case (Suc n) then show "m = Suc n"
haftmann@26072
   418
      by (cases m) (simp_all add: eq_commute [of "0"])
haftmann@26072
   419
  qed
haftmann@26072
   420
  then show ?thesis by auto
haftmann@26072
   421
qed
haftmann@26072
   422
haftmann@26072
   423
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
haftmann@57512
   424
  by (simp add: mult.commute)
haftmann@26072
   425
haftmann@26072
   426
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
haftmann@26072
   427
  by (subst mult_cancel1) simp
haftmann@26072
   428
haftmann@24995
   429
wenzelm@60758
   430
subsection \<open>Orders on @{typ nat}\<close>
wenzelm@60758
   431
wenzelm@60758
   432
subsubsection \<open>Operation definition\<close>
haftmann@24995
   433
haftmann@26072
   434
instantiation nat :: linorder
haftmann@25510
   435
begin
haftmann@25510
   436
blanchet@55575
   437
primrec less_eq_nat where
wenzelm@61076
   438
  "(0::nat) \<le> n \<longleftrightarrow> True"
haftmann@44325
   439
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
haftmann@26072
   440
haftmann@28514
   441
declare less_eq_nat.simps [simp del]
wenzelm@61076
   442
lemma le0 [iff]: "0 \<le> (n::nat)" by (simp add: less_eq_nat.simps)
wenzelm@61076
   443
lemma [code]: "(0::nat) \<le> n \<longleftrightarrow> True" by simp
haftmann@26072
   444
haftmann@26072
   445
definition less_nat where
haftmann@28514
   446
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   447
haftmann@26072
   448
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
haftmann@26072
   449
  by (simp add: less_eq_nat.simps(2))
haftmann@26072
   450
haftmann@26072
   451
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
haftmann@26072
   452
  unfolding less_eq_Suc_le ..
haftmann@26072
   453
wenzelm@61076
   454
lemma le_0_eq [iff]: "(n::nat) \<le> 0 \<longleftrightarrow> n = 0"
haftmann@26072
   455
  by (induct n) (simp_all add: less_eq_nat.simps(2))
haftmann@26072
   456
wenzelm@61076
   457
lemma not_less0 [iff]: "\<not> n < (0::nat)"
haftmann@26072
   458
  by (simp add: less_eq_Suc_le)
haftmann@26072
   459
wenzelm@61076
   460
lemma less_nat_zero_code [code]: "n < (0::nat) \<longleftrightarrow> False"
haftmann@26072
   461
  by simp
haftmann@26072
   462
haftmann@26072
   463
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
haftmann@26072
   464
  by (simp add: less_eq_Suc_le)
haftmann@26072
   465
haftmann@26072
   466
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
haftmann@26072
   467
  by (simp add: less_eq_Suc_le)
haftmann@26072
   468
hoelzl@56194
   469
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
hoelzl@56194
   470
  by (cases m) auto
hoelzl@56194
   471
haftmann@26072
   472
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
haftmann@26072
   473
  by (induct m arbitrary: n)
haftmann@26072
   474
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   475
haftmann@26072
   476
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
haftmann@26072
   477
  by (cases n) (auto intro: le_SucI)
haftmann@26072
   478
haftmann@26072
   479
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
haftmann@26072
   480
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@24995
   481
haftmann@26072
   482
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
haftmann@26072
   483
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
haftmann@25510
   484
wenzelm@26315
   485
instance
wenzelm@26315
   486
proof
haftmann@26072
   487
  fix n m :: nat
lp15@60562
   488
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
haftmann@26072
   489
  proof (induct n arbitrary: m)
haftmann@27679
   490
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   491
  next
haftmann@27679
   492
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
haftmann@26072
   493
  qed
haftmann@26072
   494
next
haftmann@26072
   495
  fix n :: nat show "n \<le> n" by (induct n) simp_all
haftmann@26072
   496
next
haftmann@26072
   497
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
haftmann@26072
   498
  then show "n = m"
haftmann@26072
   499
    by (induct n arbitrary: m)
haftmann@26072
   500
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   501
next
haftmann@26072
   502
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
haftmann@26072
   503
  then show "n \<le> q"
haftmann@26072
   504
  proof (induct n arbitrary: m q)
haftmann@26072
   505
    case 0 show ?case by simp
haftmann@26072
   506
  next
haftmann@26072
   507
    case (Suc n) then show ?case
haftmann@26072
   508
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   509
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
haftmann@26072
   510
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   511
  qed
haftmann@26072
   512
next
haftmann@26072
   513
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
haftmann@26072
   514
    by (induct n arbitrary: m)
haftmann@26072
   515
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
haftmann@26072
   516
qed
haftmann@25510
   517
haftmann@25510
   518
end
berghofe@13449
   519
haftmann@52729
   520
instantiation nat :: order_bot
haftmann@29652
   521
begin
haftmann@29652
   522
haftmann@29652
   523
definition bot_nat :: nat where
haftmann@29652
   524
  "bot_nat = 0"
haftmann@29652
   525
haftmann@29652
   526
instance proof
haftmann@29652
   527
qed (simp add: bot_nat_def)
haftmann@29652
   528
haftmann@29652
   529
end
haftmann@29652
   530
hoelzl@51329
   531
instance nat :: no_top
wenzelm@61169
   532
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
haftmann@52289
   533
hoelzl@51329
   534
wenzelm@60758
   535
subsubsection \<open>Introduction properties\<close>
berghofe@13449
   536
haftmann@26072
   537
lemma lessI [iff]: "n < Suc n"
haftmann@26072
   538
  by (simp add: less_Suc_eq_le)
berghofe@13449
   539
haftmann@26072
   540
lemma zero_less_Suc [iff]: "0 < Suc n"
haftmann@26072
   541
  by (simp add: less_Suc_eq_le)
berghofe@13449
   542
berghofe@13449
   543
wenzelm@60758
   544
subsubsection \<open>Elimination properties\<close>
berghofe@13449
   545
berghofe@13449
   546
lemma less_not_refl: "~ n < (n::nat)"
haftmann@26072
   547
  by (rule order_less_irrefl)
berghofe@13449
   548
wenzelm@26335
   549
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
lp15@60562
   550
  by (rule not_sym) (rule less_imp_neq)
berghofe@13449
   551
paulson@14267
   552
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
haftmann@26072
   553
  by (rule less_imp_neq)
berghofe@13449
   554
wenzelm@26335
   555
lemma less_irrefl_nat: "(n::nat) < n ==> R"
wenzelm@26335
   556
  by (rule notE, rule less_not_refl)
berghofe@13449
   557
berghofe@13449
   558
lemma less_zeroE: "(n::nat) < 0 ==> R"
haftmann@26072
   559
  by (rule notE) (rule not_less0)
berghofe@13449
   560
berghofe@13449
   561
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
haftmann@26072
   562
  unfolding less_Suc_eq_le le_less ..
berghofe@13449
   563
huffman@30079
   564
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
haftmann@26072
   565
  by (simp add: less_Suc_eq)
berghofe@13449
   566
blanchet@54147
   567
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
huffman@30079
   568
  unfolding One_nat_def by (rule less_Suc0)
berghofe@13449
   569
berghofe@13449
   570
lemma Suc_mono: "m < n ==> Suc m < Suc n"
haftmann@26072
   571
  by simp
berghofe@13449
   572
wenzelm@60758
   573
text \<open>"Less than" is antisymmetric, sort of\<close>
nipkow@14302
   574
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
haftmann@26072
   575
  unfolding not_less less_Suc_eq_le by (rule antisym)
nipkow@14302
   576
paulson@14267
   577
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
haftmann@26072
   578
  by (rule linorder_neq_iff)
berghofe@13449
   579
berghofe@13449
   580
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
berghofe@13449
   581
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
berghofe@13449
   582
  shows "P n m"
berghofe@13449
   583
  apply (rule less_linear [THEN disjE])
berghofe@13449
   584
  apply (erule_tac [2] disjE)
berghofe@13449
   585
  apply (erule lessCase)
berghofe@13449
   586
  apply (erule sym [THEN eqCase])
berghofe@13449
   587
  apply (erule major)
berghofe@13449
   588
  done
berghofe@13449
   589
berghofe@13449
   590
wenzelm@60758
   591
subsubsection \<open>Inductive (?) properties\<close>
berghofe@13449
   592
paulson@14267
   593
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
lp15@60562
   594
  unfolding less_eq_Suc_le [of m] le_less by simp
berghofe@13449
   595
haftmann@26072
   596
lemma lessE:
haftmann@26072
   597
  assumes major: "i < k"
haftmann@26072
   598
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
haftmann@26072
   599
  shows P
haftmann@26072
   600
proof -
haftmann@26072
   601
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
haftmann@26072
   602
    unfolding less_eq_Suc_le by (induct k) simp_all
haftmann@26072
   603
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
haftmann@26072
   604
    by (clarsimp simp add: less_le)
haftmann@26072
   605
  with p1 p2 show P by auto
haftmann@26072
   606
qed
haftmann@26072
   607
haftmann@26072
   608
lemma less_SucE: assumes major: "m < Suc n"
haftmann@26072
   609
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
haftmann@26072
   610
  apply (rule major [THEN lessE])
haftmann@26072
   611
  apply (rule eq, blast)
haftmann@26072
   612
  apply (rule less, blast)
berghofe@13449
   613
  done
berghofe@13449
   614
berghofe@13449
   615
lemma Suc_lessE: assumes major: "Suc i < k"
berghofe@13449
   616
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
berghofe@13449
   617
  apply (rule major [THEN lessE])
berghofe@13449
   618
  apply (erule lessI [THEN minor])
paulson@14208
   619
  apply (erule Suc_lessD [THEN minor], assumption)
berghofe@13449
   620
  done
berghofe@13449
   621
berghofe@13449
   622
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
haftmann@26072
   623
  by simp
berghofe@13449
   624
berghofe@13449
   625
lemma less_trans_Suc:
berghofe@13449
   626
  assumes le: "i < j" shows "j < k ==> Suc i < k"
paulson@14208
   627
  apply (induct k, simp_all)
berghofe@13449
   628
  apply (insert le)
berghofe@13449
   629
  apply (simp add: less_Suc_eq)
berghofe@13449
   630
  apply (blast dest: Suc_lessD)
berghofe@13449
   631
  done
berghofe@13449
   632
wenzelm@60758
   633
text \<open>Can be used with @{text less_Suc_eq} to get @{term "n = m | n < m"}\<close>
haftmann@26072
   634
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
haftmann@26072
   635
  unfolding not_less less_Suc_eq_le ..
berghofe@13449
   636
haftmann@26072
   637
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
haftmann@26072
   638
  unfolding not_le Suc_le_eq ..
wenzelm@21243
   639
wenzelm@60758
   640
text \<open>Properties of "less than or equal"\<close>
berghofe@13449
   641
paulson@14267
   642
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
haftmann@26072
   643
  unfolding less_Suc_eq_le .
berghofe@13449
   644
paulson@14267
   645
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
haftmann@26072
   646
  unfolding not_le less_Suc_eq_le ..
berghofe@13449
   647
paulson@14267
   648
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
haftmann@26072
   649
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
berghofe@13449
   650
paulson@14267
   651
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
haftmann@26072
   652
  by (drule le_Suc_eq [THEN iffD1], iprover+)
berghofe@13449
   653
paulson@14267
   654
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
haftmann@26072
   655
  unfolding Suc_le_eq .
berghofe@13449
   656
wenzelm@60758
   657
text \<open>Stronger version of @{text Suc_leD}\<close>
paulson@14267
   658
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
haftmann@26072
   659
  unfolding Suc_le_eq .
berghofe@13449
   660
wenzelm@26315
   661
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
haftmann@26072
   662
  unfolding less_eq_Suc_le by (rule Suc_leD)
berghofe@13449
   663
wenzelm@60758
   664
text \<open>For instance, @{text "(Suc m < Suc n) = (Suc m \<le> n) = (m < n)"}\<close>
wenzelm@26315
   665
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
berghofe@13449
   666
berghofe@13449
   667
wenzelm@60758
   668
text \<open>Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"}\<close>
berghofe@13449
   669
paulson@14267
   670
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
haftmann@26072
   671
  unfolding le_less .
berghofe@13449
   672
paulson@14267
   673
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
haftmann@26072
   674
  by (rule le_less)
berghofe@13449
   675
wenzelm@60758
   676
text \<open>Useful with @{text blast}.\<close>
paulson@14267
   677
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
haftmann@26072
   678
  by auto
berghofe@13449
   679
paulson@14267
   680
lemma le_refl: "n \<le> (n::nat)"
haftmann@26072
   681
  by simp
berghofe@13449
   682
paulson@14267
   683
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
haftmann@26072
   684
  by (rule order_trans)
berghofe@13449
   685
nipkow@33657
   686
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
haftmann@26072
   687
  by (rule antisym)
berghofe@13449
   688
paulson@14267
   689
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
haftmann@26072
   690
  by (rule less_le)
berghofe@13449
   691
paulson@14267
   692
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
haftmann@26072
   693
  unfolding less_le ..
berghofe@13449
   694
haftmann@26072
   695
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
haftmann@26072
   696
  by (rule linear)
paulson@14341
   697
wenzelm@22718
   698
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
nipkow@15921
   699
haftmann@26072
   700
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
haftmann@26072
   701
  unfolding less_Suc_eq_le by auto
berghofe@13449
   702
haftmann@26072
   703
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
haftmann@26072
   704
  unfolding not_less by (rule le_less_Suc_eq)
berghofe@13449
   705
berghofe@13449
   706
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
berghofe@13449
   707
paulson@14267
   708
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   709
by (cases n) simp_all
nipkow@25162
   710
nipkow@25162
   711
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
nipkow@25162
   712
by (cases n) simp_all
berghofe@13449
   713
wenzelm@22718
   714
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
nipkow@25162
   715
by (cases n) simp_all
berghofe@13449
   716
nipkow@25162
   717
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
nipkow@25162
   718
by (cases n) simp_all
nipkow@25140
   719
wenzelm@60758
   720
text \<open>This theorem is useful with @{text blast}\<close>
berghofe@13449
   721
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
nipkow@25162
   722
by (rule neq0_conv[THEN iffD1], iprover)
berghofe@13449
   723
paulson@14267
   724
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
nipkow@25162
   725
by (fast intro: not0_implies_Suc)
berghofe@13449
   726
blanchet@54147
   727
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
nipkow@25134
   728
using neq0_conv by blast
berghofe@13449
   729
paulson@14267
   730
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
nipkow@25162
   731
by (induct m') simp_all
berghofe@13449
   732
wenzelm@60758
   733
text \<open>Useful in certain inductive arguments\<close>
paulson@14267
   734
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
nipkow@25162
   735
by (cases m) simp_all
berghofe@13449
   736
berghofe@13449
   737
wenzelm@60758
   738
subsubsection \<open>Monotonicity of Addition\<close>
berghofe@13449
   739
haftmann@26072
   740
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
haftmann@26072
   741
by (simp add: diff_Suc split: nat.split)
berghofe@13449
   742
huffman@30128
   743
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
huffman@30128
   744
unfolding One_nat_def by (rule Suc_pred)
huffman@30128
   745
paulson@14331
   746
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
nipkow@25162
   747
by (induct k) simp_all
berghofe@13449
   748
paulson@14331
   749
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
nipkow@25162
   750
by (induct k) simp_all
berghofe@13449
   751
nipkow@25162
   752
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
nipkow@25162
   753
by(auto dest:gr0_implies_Suc)
berghofe@13449
   754
wenzelm@60758
   755
text \<open>strict, in 1st argument\<close>
paulson@14341
   756
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
nipkow@25162
   757
by (induct k) simp_all
paulson@14341
   758
wenzelm@60758
   759
text \<open>strict, in both arguments\<close>
paulson@14341
   760
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
paulson@14341
   761
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
paulson@15251
   762
  apply (induct j, simp_all)
paulson@14341
   763
  done
paulson@14341
   764
wenzelm@60758
   765
text \<open>Deleted @{text less_natE}; use @{text "less_imp_Suc_add RS exE"}\<close>
paulson@14341
   766
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
paulson@14341
   767
  apply (induct n)
paulson@14341
   768
  apply (simp_all add: order_le_less)
wenzelm@22718
   769
  apply (blast elim!: less_SucE
haftmann@35047
   770
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
paulson@14341
   771
  done
paulson@14341
   772
hoelzl@56194
   773
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
hoelzl@56194
   774
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
hoelzl@56194
   775
wenzelm@60758
   776
text \<open>strict, in 1st argument; proof is by induction on @{text "k > 0"}\<close>
nipkow@25134
   777
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
nipkow@25134
   778
apply(auto simp: gr0_conv_Suc)
nipkow@25134
   779
apply (induct_tac m)
nipkow@25134
   780
apply (simp_all add: add_less_mono)
nipkow@25134
   781
done
paulson@14341
   782
wenzelm@60758
   783
text \<open>Addition is the inverse of subtraction:
wenzelm@60758
   784
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
lp15@60562
   785
lemma add_diff_inverse_nat: "~  m < n ==> n + (m - n) = (m::nat)"
lp15@60562
   786
by (induct m n rule: diff_induct) simp_all
lp15@60562
   787
lp15@60562
   788
wenzelm@60758
   789
text\<open>The naturals form an ordered @{text semidom}\<close>
haftmann@35028
   790
instance nat :: linordered_semidom
paulson@14341
   791
proof
paulson@14348
   792
  show "0 < (1::nat)" by simp
haftmann@52289
   793
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
haftmann@52289
   794
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
haftmann@59833
   795
  show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
lp15@60562
   796
  show "\<And>m n :: nat. n \<le> m ==> (m - n) + n = (m::nat)"
lp15@60562
   797
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
lp15@60562
   798
qed 
nipkow@30056
   799
haftmann@44817
   800
wenzelm@60758
   801
subsubsection \<open>@{term min} and @{term max}\<close>
haftmann@44817
   802
haftmann@44817
   803
lemma mono_Suc: "mono Suc"
haftmann@44817
   804
by (rule monoI) simp
haftmann@44817
   805
haftmann@44817
   806
lemma min_0L [simp]: "min 0 n = (0::nat)"
noschinl@45931
   807
by (rule min_absorb1) simp
haftmann@44817
   808
haftmann@44817
   809
lemma min_0R [simp]: "min n 0 = (0::nat)"
noschinl@45931
   810
by (rule min_absorb2) simp
haftmann@44817
   811
haftmann@44817
   812
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
haftmann@44817
   813
by (simp add: mono_Suc min_of_mono)
haftmann@44817
   814
haftmann@44817
   815
lemma min_Suc1:
haftmann@44817
   816
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
haftmann@44817
   817
by (simp split: nat.split)
haftmann@44817
   818
haftmann@44817
   819
lemma min_Suc2:
haftmann@44817
   820
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
haftmann@44817
   821
by (simp split: nat.split)
haftmann@44817
   822
haftmann@44817
   823
lemma max_0L [simp]: "max 0 n = (n::nat)"
noschinl@45931
   824
by (rule max_absorb2) simp
haftmann@44817
   825
haftmann@44817
   826
lemma max_0R [simp]: "max n 0 = (n::nat)"
noschinl@45931
   827
by (rule max_absorb1) simp
haftmann@44817
   828
haftmann@44817
   829
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
haftmann@44817
   830
by (simp add: mono_Suc max_of_mono)
haftmann@44817
   831
haftmann@44817
   832
lemma max_Suc1:
haftmann@44817
   833
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
haftmann@44817
   834
by (simp split: nat.split)
haftmann@44817
   835
haftmann@44817
   836
lemma max_Suc2:
haftmann@44817
   837
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
haftmann@44817
   838
by (simp split: nat.split)
paulson@14267
   839
haftmann@44817
   840
lemma nat_mult_min_left:
haftmann@44817
   841
  fixes m n q :: nat
haftmann@44817
   842
  shows "min m n * q = min (m * q) (n * q)"
haftmann@44817
   843
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   844
haftmann@44817
   845
lemma nat_mult_min_right:
haftmann@44817
   846
  fixes m n q :: nat
haftmann@44817
   847
  shows "m * min n q = min (m * n) (m * q)"
haftmann@44817
   848
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
haftmann@44817
   849
haftmann@44817
   850
lemma nat_add_max_left:
haftmann@44817
   851
  fixes m n q :: nat
haftmann@44817
   852
  shows "max m n + q = max (m + q) (n + q)"
haftmann@44817
   853
  by (simp add: max_def)
haftmann@44817
   854
haftmann@44817
   855
lemma nat_add_max_right:
haftmann@44817
   856
  fixes m n q :: nat
haftmann@44817
   857
  shows "m + max n q = max (m + n) (m + q)"
haftmann@44817
   858
  by (simp add: max_def)
haftmann@44817
   859
haftmann@44817
   860
lemma nat_mult_max_left:
haftmann@44817
   861
  fixes m n q :: nat
haftmann@44817
   862
  shows "max m n * q = max (m * q) (n * q)"
haftmann@44817
   863
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
haftmann@44817
   864
haftmann@44817
   865
lemma nat_mult_max_right:
haftmann@44817
   866
  fixes m n q :: nat
haftmann@44817
   867
  shows "m * max n q = max (m * n) (m * q)"
haftmann@44817
   868
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
paulson@14267
   869
paulson@14267
   870
wenzelm@60758
   871
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
wenzelm@60758
   872
wenzelm@60758
   873
text \<open>Complete induction, aka course-of-values induction\<close>
krauss@26748
   874
haftmann@27823
   875
instance nat :: wellorder proof
haftmann@27823
   876
  fix P and n :: nat
haftmann@27823
   877
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
haftmann@27823
   878
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
haftmann@27823
   879
  proof (induct n)
haftmann@27823
   880
    case (0 n)
krauss@26748
   881
    have "P 0" by (rule step) auto
krauss@26748
   882
    thus ?case using 0 by auto
krauss@26748
   883
  next
haftmann@27823
   884
    case (Suc m n)
haftmann@27823
   885
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
krauss@26748
   886
    thus ?case
krauss@26748
   887
    proof
haftmann@27823
   888
      assume "n \<le> m" thus "P n" by (rule Suc(1))
krauss@26748
   889
    next
haftmann@27823
   890
      assume n: "n = Suc m"
haftmann@27823
   891
      show "P n"
haftmann@27823
   892
        by (rule step) (rule Suc(1), simp add: n le_simps)
krauss@26748
   893
    qed
krauss@26748
   894
  qed
haftmann@27823
   895
  then show "P n" by auto
krauss@26748
   896
qed
krauss@26748
   897
nipkow@57015
   898
nipkow@57015
   899
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
nipkow@57015
   900
by (rule Least_equality[OF _ le0])
nipkow@57015
   901
haftmann@27823
   902
lemma Least_Suc:
haftmann@27823
   903
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
wenzelm@47988
   904
  apply (cases n, auto)
haftmann@27823
   905
  apply (frule LeastI)
haftmann@27823
   906
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
haftmann@27823
   907
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
haftmann@27823
   908
  apply (erule_tac [2] Least_le)
wenzelm@47988
   909
  apply (cases "LEAST x. P x", auto)
haftmann@27823
   910
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
haftmann@27823
   911
  apply (blast intro: order_antisym)
haftmann@27823
   912
  done
haftmann@27823
   913
haftmann@27823
   914
lemma Least_Suc2:
haftmann@27823
   915
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
haftmann@27823
   916
  apply (erule (1) Least_Suc [THEN ssubst])
haftmann@27823
   917
  apply simp
haftmann@27823
   918
  done
haftmann@27823
   919
haftmann@27823
   920
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
   921
  apply (cases n)
haftmann@27823
   922
   apply blast
haftmann@27823
   923
  apply (rule_tac x="LEAST k. P(k)" in exI)
haftmann@27823
   924
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
haftmann@27823
   925
  done
haftmann@27823
   926
haftmann@27823
   927
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
   928
  unfolding One_nat_def
haftmann@27823
   929
  apply (cases n)
haftmann@27823
   930
   apply blast
haftmann@27823
   931
  apply (frule (1) ex_least_nat_le)
haftmann@27823
   932
  apply (erule exE)
haftmann@27823
   933
  apply (case_tac k)
haftmann@27823
   934
   apply simp
haftmann@27823
   935
  apply (rename_tac k1)
haftmann@27823
   936
  apply (rule_tac x=k1 in exI)
haftmann@27823
   937
  apply (auto simp add: less_eq_Suc_le)
haftmann@27823
   938
  done
haftmann@27823
   939
krauss@26748
   940
lemma nat_less_induct:
krauss@26748
   941
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
krauss@26748
   942
  using assms less_induct by blast
krauss@26748
   943
krauss@26748
   944
lemma measure_induct_rule [case_names less]:
krauss@26748
   945
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   946
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
krauss@26748
   947
  shows "P a"
krauss@26748
   948
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
krauss@26748
   949
wenzelm@60758
   950
text \<open>old style induction rules:\<close>
krauss@26748
   951
lemma measure_induct:
krauss@26748
   952
  fixes f :: "'a \<Rightarrow> nat"
krauss@26748
   953
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
krauss@26748
   954
  by (rule measure_induct_rule [of f P a]) iprover
krauss@26748
   955
krauss@26748
   956
lemma full_nat_induct:
krauss@26748
   957
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
krauss@26748
   958
  shows "P n"
krauss@26748
   959
  by (rule less_induct) (auto intro: step simp:le_simps)
paulson@14267
   960
wenzelm@60758
   961
text\<open>An induction rule for estabilishing binary relations\<close>
wenzelm@22718
   962
lemma less_Suc_induct:
paulson@19870
   963
  assumes less:  "i < j"
paulson@19870
   964
     and  step:  "!!i. P i (Suc i)"
krauss@31714
   965
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
paulson@19870
   966
  shows "P i j"
paulson@19870
   967
proof -
krauss@31714
   968
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
wenzelm@22718
   969
  have "P i (Suc (i + k))"
paulson@19870
   970
  proof (induct k)
wenzelm@22718
   971
    case 0
wenzelm@22718
   972
    show ?case by (simp add: step)
paulson@19870
   973
  next
paulson@19870
   974
    case (Suc k)
krauss@31714
   975
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
haftmann@57512
   976
    hence "i < Suc (i + k)" by (simp add: add.commute)
krauss@31714
   977
    from trans[OF this lessI Suc step]
krauss@31714
   978
    show ?case by simp
paulson@19870
   979
  qed
wenzelm@22718
   980
  thus "P i j" by (simp add: j)
paulson@19870
   981
qed
paulson@19870
   982
wenzelm@60758
   983
text \<open>The method of infinite descent, frequently used in number theory.
krauss@26748
   984
Provided by Roelof Oosterhuis.
krauss@26748
   985
$P(n)$ is true for all $n\in\mathbb{N}$ if
krauss@26748
   986
\begin{itemize}
krauss@26748
   987
  \item case ``0'': given $n=0$ prove $P(n)$,
krauss@26748
   988
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
krauss@26748
   989
        a smaller integer $m$ such that $\neg P(m)$.
wenzelm@60758
   990
\end{itemize}\<close>
wenzelm@60758
   991
wenzelm@60758
   992
text\<open>A compact version without explicit base case:\<close>
krauss@26748
   993
lemma infinite_descent:
krauss@26748
   994
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
wenzelm@47988
   995
by (induct n rule: less_induct) auto
krauss@26748
   996
lp15@60562
   997
lemma infinite_descent0[case_names 0 smaller]:
krauss@26748
   998
  "\<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
   999
by (rule infinite_descent) (case_tac "n>0", auto)
krauss@26748
  1000
wenzelm@60758
  1001
text \<open>
krauss@26748
  1002
Infinite descent using a mapping to $\mathbb{N}$:
krauss@26748
  1003
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
krauss@26748
  1004
\begin{itemize}
krauss@26748
  1005
\item case ``0'': given $V(x)=0$ prove $P(x)$,
krauss@26748
  1006
\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
  1007
\end{itemize}
wenzelm@60758
  1008
NB: the proof also shows how to use the previous lemma.\<close>
krauss@26748
  1009
krauss@26748
  1010
corollary infinite_descent0_measure [case_names 0 smaller]:
krauss@26748
  1011
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
krauss@26748
  1012
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
krauss@26748
  1013
  shows "P x"
krauss@26748
  1014
proof -
krauss@26748
  1015
  obtain n where "n = V x" by auto
krauss@26748
  1016
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
krauss@26748
  1017
  proof (induct n rule: infinite_descent0)
krauss@26748
  1018
    case 0 -- "i.e. $V(x) = 0$"
krauss@26748
  1019
    with A0 show "P x" by auto
krauss@26748
  1020
  next -- "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
krauss@26748
  1021
    case (smaller n)
krauss@26748
  1022
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
krauss@26748
  1023
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
krauss@26748
  1024
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
krauss@26748
  1025
    then show ?case by auto
krauss@26748
  1026
  qed
krauss@26748
  1027
  ultimately show "P x" by auto
krauss@26748
  1028
qed
krauss@26748
  1029
wenzelm@60758
  1030
text\<open>Again, without explicit base case:\<close>
krauss@26748
  1031
lemma infinite_descent_measure:
krauss@26748
  1032
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
krauss@26748
  1033
proof -
krauss@26748
  1034
  from assms obtain n where "n = V x" by auto
krauss@26748
  1035
  moreover have "!!x. V x = n \<Longrightarrow> P x"
krauss@26748
  1036
  proof (induct n rule: infinite_descent, auto)
krauss@26748
  1037
    fix x assume "\<not> P x"
krauss@26748
  1038
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
krauss@26748
  1039
  qed
krauss@26748
  1040
  ultimately show "P x" by auto
krauss@26748
  1041
qed
krauss@26748
  1042
wenzelm@60758
  1043
text \<open>A [clumsy] way of lifting @{text "<"}
wenzelm@60758
  1044
  monotonicity to @{text "\<le>"} monotonicity\<close>
paulson@14267
  1045
lemma less_mono_imp_le_mono:
nipkow@24438
  1046
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
nipkow@24438
  1047
by (simp add: order_le_less) (blast)
nipkow@24438
  1048
paulson@14267
  1049
wenzelm@60758
  1050
text \<open>non-strict, in 1st argument\<close>
paulson@14267
  1051
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
nipkow@24438
  1052
by (rule add_right_mono)
paulson@14267
  1053
wenzelm@60758
  1054
text \<open>non-strict, in both arguments\<close>
paulson@14267
  1055
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
nipkow@24438
  1056
by (rule add_mono)
paulson@14267
  1057
paulson@14267
  1058
lemma le_add2: "n \<le> ((m + n)::nat)"
nipkow@24438
  1059
by (insert add_right_mono [of 0 m n], simp)
berghofe@13449
  1060
paulson@14267
  1061
lemma le_add1: "n \<le> ((n + m)::nat)"
haftmann@57512
  1062
by (simp add: add.commute, rule le_add2)
berghofe@13449
  1063
berghofe@13449
  1064
lemma less_add_Suc1: "i < Suc (i + m)"
nipkow@24438
  1065
by (rule le_less_trans, rule le_add1, rule lessI)
berghofe@13449
  1066
berghofe@13449
  1067
lemma less_add_Suc2: "i < Suc (m + i)"
nipkow@24438
  1068
by (rule le_less_trans, rule le_add2, rule lessI)
berghofe@13449
  1069
paulson@14267
  1070
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
nipkow@24438
  1071
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
berghofe@13449
  1072
paulson@14267
  1073
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
nipkow@24438
  1074
by (rule le_trans, assumption, rule le_add1)
berghofe@13449
  1075
paulson@14267
  1076
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
nipkow@24438
  1077
by (rule le_trans, assumption, rule le_add2)
berghofe@13449
  1078
berghofe@13449
  1079
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
nipkow@24438
  1080
by (rule less_le_trans, assumption, rule le_add1)
berghofe@13449
  1081
berghofe@13449
  1082
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
nipkow@24438
  1083
by (rule less_le_trans, assumption, rule le_add2)
berghofe@13449
  1084
berghofe@13449
  1085
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
nipkow@24438
  1086
apply (rule le_less_trans [of _ "i+j"])
nipkow@24438
  1087
apply (simp_all add: le_add1)
nipkow@24438
  1088
done
berghofe@13449
  1089
berghofe@13449
  1090
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
nipkow@24438
  1091
apply (rule notI)
wenzelm@26335
  1092
apply (drule add_lessD1)
wenzelm@26335
  1093
apply (erule less_irrefl [THEN notE])
nipkow@24438
  1094
done
berghofe@13449
  1095
berghofe@13449
  1096
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
haftmann@57512
  1097
by (simp add: add.commute)
berghofe@13449
  1098
paulson@14267
  1099
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
nipkow@24438
  1100
apply (rule order_trans [of _ "m+k"])
nipkow@24438
  1101
apply (simp_all add: le_add1)
nipkow@24438
  1102
done
berghofe@13449
  1103
paulson@14267
  1104
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
haftmann@57512
  1105
apply (simp add: add.commute)
nipkow@24438
  1106
apply (erule add_leD1)
nipkow@24438
  1107
done
berghofe@13449
  1108
paulson@14267
  1109
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
nipkow@24438
  1110
by (blast dest: add_leD1 add_leD2)
berghofe@13449
  1111
wenzelm@60758
  1112
text \<open>needs @{text "!!k"} for @{text ac_simps} to work\<close>
berghofe@13449
  1113
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
nipkow@24438
  1114
by (force simp del: add_Suc_right
haftmann@57514
  1115
    simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
berghofe@13449
  1116
berghofe@13449
  1117
wenzelm@60758
  1118
subsubsection \<open>More results about difference\<close>
berghofe@13449
  1119
paulson@14267
  1120
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
nipkow@24438
  1121
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1122
berghofe@13449
  1123
lemma diff_less_Suc: "m - n < Suc m"
nipkow@24438
  1124
apply (induct m n rule: diff_induct)
nipkow@24438
  1125
apply (erule_tac [3] less_SucE)
nipkow@24438
  1126
apply (simp_all add: less_Suc_eq)
nipkow@24438
  1127
done
berghofe@13449
  1128
paulson@14267
  1129
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
nipkow@24438
  1130
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
berghofe@13449
  1131
haftmann@26072
  1132
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
haftmann@26072
  1133
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
haftmann@26072
  1134
haftmann@52289
  1135
instance nat :: ordered_cancel_comm_monoid_diff
haftmann@52289
  1136
proof
haftmann@52289
  1137
  show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
haftmann@52289
  1138
qed
haftmann@52289
  1139
berghofe@13449
  1140
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
nipkow@24438
  1141
by (rule le_less_trans, rule diff_le_self)
berghofe@13449
  1142
berghofe@13449
  1143
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
nipkow@24438
  1144
by (cases n) (auto simp add: le_simps)
berghofe@13449
  1145
paulson@14267
  1146
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
nipkow@24438
  1147
by (induct j k rule: diff_induct) simp_all
berghofe@13449
  1148
paulson@14267
  1149
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
haftmann@57512
  1150
by (simp add: add.commute diff_add_assoc)
berghofe@13449
  1151
paulson@14267
  1152
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
nipkow@24438
  1153
by (auto simp add: diff_add_inverse2)
berghofe@13449
  1154
paulson@14267
  1155
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
nipkow@24438
  1156
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1157
paulson@14267
  1158
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
nipkow@24438
  1159
by (rule iffD2, rule diff_is_0_eq)
berghofe@13449
  1160
berghofe@13449
  1161
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
nipkow@24438
  1162
by (induct m n rule: diff_induct) simp_all
berghofe@13449
  1163
wenzelm@22718
  1164
lemma less_imp_add_positive:
wenzelm@22718
  1165
  assumes "i < j"
wenzelm@22718
  1166
  shows "\<exists>k::nat. 0 < k & i + k = j"
wenzelm@22718
  1167
proof
wenzelm@22718
  1168
  from assms show "0 < j - i & i + (j - i) = j"
huffman@23476
  1169
    by (simp add: order_less_imp_le)
wenzelm@22718
  1170
qed
wenzelm@9436
  1171
wenzelm@60758
  1172
text \<open>a nice rewrite for bounded subtraction\<close>
haftmann@26072
  1173
lemma nat_minus_add_max:
haftmann@26072
  1174
  fixes n m :: nat
haftmann@26072
  1175
  shows "n - m + m = max n m"
haftmann@26072
  1176
    by (simp add: max_def not_le order_less_imp_le)
berghofe@13449
  1177
haftmann@26072
  1178
lemma nat_diff_split:
haftmann@26072
  1179
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
wenzelm@60758
  1180
    -- \<open>elimination of @{text -} on @{text nat}\<close>
haftmann@26072
  1181
by (cases "a < b")
haftmann@26072
  1182
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
thomas@57492
  1183
    not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])
berghofe@13449
  1184
haftmann@26072
  1185
lemma nat_diff_split_asm:
haftmann@26072
  1186
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
wenzelm@60758
  1187
    -- \<open>elimination of @{text -} on @{text nat} in assumptions\<close>
haftmann@26072
  1188
by (auto split: nat_diff_split)
berghofe@13449
  1189
huffman@47255
  1190
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
huffman@47255
  1191
  by simp
huffman@47255
  1192
huffman@47255
  1193
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
huffman@47255
  1194
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1195
huffman@47255
  1196
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
huffman@47255
  1197
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1198
huffman@47255
  1199
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
huffman@47255
  1200
  unfolding One_nat_def by (cases n) simp_all
huffman@47255
  1201
huffman@47255
  1202
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
huffman@47255
  1203
  unfolding One_nat_def by (cases m) simp_all
huffman@47255
  1204
huffman@47255
  1205
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
huffman@47255
  1206
  by (fact Let_def)
huffman@47255
  1207
berghofe@13449
  1208
wenzelm@60758
  1209
subsubsection \<open>Monotonicity of multiplication\<close>
berghofe@13449
  1210
paulson@14267
  1211
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
nipkow@24438
  1212
by (simp add: mult_right_mono)
berghofe@13449
  1213
paulson@14267
  1214
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
nipkow@24438
  1215
by (simp add: mult_left_mono)
berghofe@13449
  1216
wenzelm@60758
  1217
text \<open>@{text "\<le>"} monotonicity, BOTH arguments\<close>
paulson@14267
  1218
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
nipkow@24438
  1219
by (simp add: mult_mono)
berghofe@13449
  1220
berghofe@13449
  1221
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
nipkow@24438
  1222
by (simp add: mult_strict_right_mono)
berghofe@13449
  1223
wenzelm@60758
  1224
text\<open>Differs from the standard @{text zero_less_mult_iff} in that
wenzelm@60758
  1225
      there are no negative numbers.\<close>
paulson@14266
  1226
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
berghofe@13449
  1227
  apply (induct m)
wenzelm@22718
  1228
   apply simp
wenzelm@22718
  1229
  apply (case_tac n)
wenzelm@22718
  1230
   apply simp_all
berghofe@13449
  1231
  done
berghofe@13449
  1232
huffman@30079
  1233
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
berghofe@13449
  1234
  apply (induct m)
wenzelm@22718
  1235
   apply simp
wenzelm@22718
  1236
  apply (case_tac n)
wenzelm@22718
  1237
   apply simp_all
berghofe@13449
  1238
  done
berghofe@13449
  1239
paulson@14341
  1240
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
berghofe@13449
  1241
  apply (safe intro!: mult_less_mono1)
wenzelm@47988
  1242
  apply (cases k, auto)
berghofe@13449
  1243
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
berghofe@13449
  1244
  apply (blast intro: mult_le_mono1)
berghofe@13449
  1245
  done
berghofe@13449
  1246
berghofe@13449
  1247
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
haftmann@57512
  1248
by (simp add: mult.commute [of k])
berghofe@13449
  1249
paulson@14267
  1250
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
nipkow@24438
  1251
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1252
paulson@14267
  1253
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
nipkow@24438
  1254
by (simp add: linorder_not_less [symmetric], auto)
berghofe@13449
  1255
berghofe@13449
  1256
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
nipkow@24438
  1257
by (subst mult_less_cancel1) simp
berghofe@13449
  1258
paulson@14267
  1259
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
nipkow@24438
  1260
by (subst mult_le_cancel1) simp
berghofe@13449
  1261
haftmann@26072
  1262
lemma le_square: "m \<le> m * (m::nat)"
haftmann@26072
  1263
  by (cases m) (auto intro: le_add1)
haftmann@26072
  1264
haftmann@26072
  1265
lemma le_cube: "(m::nat) \<le> m * (m * m)"
haftmann@26072
  1266
  by (cases m) (auto intro: le_add1)
berghofe@13449
  1267
wenzelm@60758
  1268
text \<open>Lemma for @{text gcd}\<close>
huffman@30128
  1269
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
berghofe@13449
  1270
  apply (drule sym)
berghofe@13449
  1271
  apply (rule disjCI)
berghofe@13449
  1272
  apply (rule nat_less_cases, erule_tac [2] _)
paulson@25157
  1273
   apply (drule_tac [2] mult_less_mono2)
nipkow@25162
  1274
    apply (auto)
berghofe@13449
  1275
  done
wenzelm@9436
  1276
haftmann@51263
  1277
lemma mono_times_nat:
haftmann@51263
  1278
  fixes n :: nat
haftmann@51263
  1279
  assumes "n > 0"
haftmann@51263
  1280
  shows "mono (times n)"
haftmann@51263
  1281
proof
haftmann@51263
  1282
  fix m q :: nat
haftmann@51263
  1283
  assume "m \<le> q"
haftmann@51263
  1284
  with assms show "n * m \<le> n * q" by simp
haftmann@51263
  1285
qed
haftmann@51263
  1286
wenzelm@60758
  1287
text \<open>the lattice order on @{typ nat}\<close>
haftmann@24995
  1288
haftmann@26072
  1289
instantiation nat :: distrib_lattice
haftmann@26072
  1290
begin
haftmann@24995
  1291
haftmann@26072
  1292
definition
wenzelm@61076
  1293
  "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
haftmann@24995
  1294
haftmann@26072
  1295
definition
wenzelm@61076
  1296
  "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
haftmann@24995
  1297
haftmann@26072
  1298
instance by intro_classes
haftmann@26072
  1299
  (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
haftmann@26072
  1300
    intro: order_less_imp_le antisym elim!: order_trans order_less_trans)
haftmann@24995
  1301
haftmann@26072
  1302
end
haftmann@24995
  1303
haftmann@24995
  1304
wenzelm@60758
  1305
subsection \<open>Natural operation of natural numbers on functions\<close>
wenzelm@60758
  1306
wenzelm@60758
  1307
text \<open>
haftmann@30971
  1308
  We use the same logical constant for the power operations on
haftmann@30971
  1309
  functions and relations, in order to share the same syntax.
wenzelm@60758
  1310
\<close>
haftmann@30971
  1311
haftmann@45965
  1312
consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"
haftmann@30971
  1313
haftmann@45965
  1314
abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80) where
haftmann@30971
  1315
  "f ^^ n \<equiv> compow n f"
haftmann@30971
  1316
haftmann@30971
  1317
notation (latex output)
haftmann@30971
  1318
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1319
haftmann@30971
  1320
notation (HTML output)
haftmann@30971
  1321
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)
haftmann@30971
  1322
wenzelm@60758
  1323
text \<open>@{text "f ^^ n = f o ... o f"}, the n-fold composition of @{text f}\<close>
haftmann@30971
  1324
haftmann@30971
  1325
overloading
haftmann@30971
  1326
  funpow == "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
haftmann@30971
  1327
begin
haftmann@30954
  1328
blanchet@55575
  1329
primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@44325
  1330
  "funpow 0 f = id"
haftmann@44325
  1331
| "funpow (Suc n) f = f o funpow n f"
haftmann@30954
  1332
haftmann@30971
  1333
end
haftmann@30971
  1334
haftmann@49723
  1335
lemma funpow_Suc_right:
haftmann@49723
  1336
  "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1337
proof (induct n)
haftmann@49723
  1338
  case 0 then show ?case by simp
haftmann@49723
  1339
next
haftmann@49723
  1340
  fix n
haftmann@49723
  1341
  assume "f ^^ Suc n = f ^^ n \<circ> f"
haftmann@49723
  1342
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
haftmann@49723
  1343
    by (simp add: o_assoc)
haftmann@49723
  1344
qed
haftmann@49723
  1345
haftmann@49723
  1346
lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
haftmann@49723
  1347
wenzelm@60758
  1348
text \<open>for code generation\<close>
haftmann@30971
  1349
haftmann@30971
  1350
definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@46028
  1351
  funpow_code_def [code_abbrev]: "funpow = compow"
haftmann@30954
  1352
haftmann@30971
  1353
lemma [code]:
haftmann@37430
  1354
  "funpow (Suc n) f = f o funpow n f"
haftmann@30971
  1355
  "funpow 0 f = id"
haftmann@37430
  1356
  by (simp_all add: funpow_code_def)
haftmann@30971
  1357
wenzelm@36176
  1358
hide_const (open) funpow
haftmann@30954
  1359
haftmann@30954
  1360
lemma funpow_add:
haftmann@30971
  1361
  "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
haftmann@30954
  1362
  by (induct m) simp_all
haftmann@30954
  1363
haftmann@37430
  1364
lemma funpow_mult:
haftmann@37430
  1365
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@37430
  1366
  shows "(f ^^ m) ^^ n = f ^^ (m * n)"
haftmann@37430
  1367
  by (induct n) (simp_all add: funpow_add)
haftmann@37430
  1368
haftmann@30954
  1369
lemma funpow_swap1:
haftmann@30971
  1370
  "f ((f ^^ n) x) = (f ^^ n) (f x)"
haftmann@30954
  1371
proof -
haftmann@30971
  1372
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
haftmann@30971
  1373
  also have "\<dots>  = (f ^^ n o f ^^ 1) x" by (simp only: funpow_add)
haftmann@30971
  1374
  also have "\<dots> = (f ^^ n) (f x)" by simp
haftmann@30954
  1375
  finally show ?thesis .
haftmann@30954
  1376
qed
haftmann@30954
  1377
haftmann@38621
  1378
lemma comp_funpow:
haftmann@38621
  1379
  fixes f :: "'a \<Rightarrow> 'a"
haftmann@38621
  1380
  shows "comp f ^^ n = comp (f ^^ n)"
haftmann@38621
  1381
  by (induct n) simp_all
haftmann@30954
  1382
hoelzl@54496
  1383
lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
hoelzl@54496
  1384
  by (induct n) simp_all
hoelzl@54496
  1385
hoelzl@54496
  1386
lemma id_funpow[simp]: "id ^^ n = id"
hoelzl@54496
  1387
  by (induct n) simp_all
haftmann@38621
  1388
hoelzl@59000
  1389
lemma funpow_mono:
hoelzl@59000
  1390
  fixes f :: "'a \<Rightarrow> ('a::lattice)"
hoelzl@59000
  1391
  shows "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
hoelzl@59000
  1392
  by (induct n arbitrary: A B)
hoelzl@59000
  1393
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)
hoelzl@59000
  1394
wenzelm@60758
  1395
subsection \<open>Kleene iteration\<close>
nipkow@45833
  1396
haftmann@52729
  1397
lemma Kleene_iter_lpfp:
haftmann@52729
  1398
assumes "mono f" and "f p \<le> p" shows "(f^^k) (bot::'a::order_bot) \<le> p"
nipkow@45833
  1399
proof(induction k)
nipkow@45833
  1400
  case 0 show ?case by simp
nipkow@45833
  1401
next
nipkow@45833
  1402
  case Suc
nipkow@45833
  1403
  from monoD[OF assms(1) Suc] assms(2)
nipkow@45833
  1404
  show ?case by simp
nipkow@45833
  1405
qed
nipkow@45833
  1406
nipkow@45833
  1407
lemma lfp_Kleene_iter: assumes "mono f" and "(f^^Suc k) bot = (f^^k) bot"
nipkow@45833
  1408
shows "lfp f = (f^^k) bot"
nipkow@45833
  1409
proof(rule antisym)
nipkow@45833
  1410
  show "lfp f \<le> (f^^k) bot"
nipkow@45833
  1411
  proof(rule lfp_lowerbound)
nipkow@45833
  1412
    show "f ((f^^k) bot) \<le> (f^^k) bot" using assms(2) by simp
nipkow@45833
  1413
  qed
nipkow@45833
  1414
next
nipkow@45833
  1415
  show "(f^^k) bot \<le> lfp f"
nipkow@45833
  1416
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
nipkow@45833
  1417
qed
nipkow@45833
  1418
hoelzl@60636
  1419
lemma mono_pow:
hoelzl@60636
  1420
  fixes f :: "'a \<Rightarrow> 'a::complete_lattice"
hoelzl@60636
  1421
  shows "mono f \<Longrightarrow> mono (f ^^ n)"
hoelzl@60636
  1422
  by (induction n) (auto simp: mono_def)
hoelzl@60636
  1423
hoelzl@60636
  1424
lemma lfp_funpow:
hoelzl@60636
  1425
  assumes f: "mono f" shows "lfp (f ^^ Suc n) = lfp f"
hoelzl@60636
  1426
proof (rule antisym)
hoelzl@60636
  1427
  show "lfp f \<le> lfp (f ^^ Suc n)"
hoelzl@60636
  1428
  proof (rule lfp_lowerbound)
hoelzl@60636
  1429
    have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
hoelzl@60636
  1430
      unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
hoelzl@60636
  1431
    then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
hoelzl@60636
  1432
      by (simp add: comp_def)
hoelzl@60636
  1433
  qed
hoelzl@60636
  1434
  have "(f^^n) (lfp f) = lfp f" for n
hoelzl@60636
  1435
    by (induction n) (auto intro: f lfp_unfold[symmetric])
hoelzl@60636
  1436
  then show "lfp (f^^Suc n) \<le> lfp f"
hoelzl@60636
  1437
    by (intro lfp_lowerbound) (simp del: funpow.simps)
hoelzl@60636
  1438
qed
hoelzl@60636
  1439
hoelzl@60636
  1440
lemma gfp_funpow:
hoelzl@60636
  1441
  assumes f: "mono f" shows "gfp (f ^^ Suc n) = gfp f"
hoelzl@60636
  1442
proof (rule antisym)
hoelzl@60636
  1443
  show "gfp f \<ge> gfp (f ^^ Suc n)"
hoelzl@60636
  1444
  proof (rule gfp_upperbound)
hoelzl@60636
  1445
    have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
hoelzl@60636
  1446
      unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
hoelzl@60636
  1447
    then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
hoelzl@60636
  1448
      by (simp add: comp_def)
hoelzl@60636
  1449
  qed
hoelzl@60636
  1450
  have "(f^^n) (gfp f) = gfp f" for n
hoelzl@60636
  1451
    by (induction n) (auto intro: f gfp_unfold[symmetric])
hoelzl@60636
  1452
  then show "gfp (f^^Suc n) \<ge> gfp f"
hoelzl@60636
  1453
    by (intro gfp_upperbound) (simp del: funpow.simps)
hoelzl@60636
  1454
qed
nipkow@45833
  1455
wenzelm@60758
  1456
subsection \<open>Embedding of the naturals into any @{text semiring_1}: @{term of_nat}\<close>
haftmann@24196
  1457
haftmann@24196
  1458
context semiring_1
haftmann@24196
  1459
begin
haftmann@24196
  1460
haftmann@38621
  1461
definition of_nat :: "nat \<Rightarrow> 'a" where
haftmann@38621
  1462
  "of_nat n = (plus 1 ^^ n) 0"
haftmann@38621
  1463
haftmann@38621
  1464
lemma of_nat_simps [simp]:
haftmann@38621
  1465
  shows of_nat_0: "of_nat 0 = 0"
haftmann@38621
  1466
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
haftmann@38621
  1467
  by (simp_all add: of_nat_def)
haftmann@25193
  1468
haftmann@25193
  1469
lemma of_nat_1 [simp]: "of_nat 1 = 1"
haftmann@38621
  1470
  by (simp add: of_nat_def)
haftmann@25193
  1471
haftmann@25193
  1472
lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
haftmann@57514
  1473
  by (induct m) (simp_all add: ac_simps)
haftmann@25193
  1474
haftmann@25193
  1475
lemma of_nat_mult: "of_nat (m * n) = of_nat m * of_nat n"
haftmann@57514
  1476
  by (induct m) (simp_all add: ac_simps distrib_right)
haftmann@25193
  1477
blanchet@55575
  1478
primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a" where
haftmann@28514
  1479
  "of_nat_aux inc 0 i = i"
wenzelm@60758
  1480
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" -- \<open>tail recursive\<close>
haftmann@25928
  1481
haftmann@30966
  1482
lemma of_nat_code:
haftmann@28514
  1483
  "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
haftmann@28514
  1484
proof (induct n)
haftmann@28514
  1485
  case 0 then show ?case by simp
haftmann@28514
  1486
next
haftmann@28514
  1487
  case (Suc n)
haftmann@28514
  1488
  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
  1489
    by (induct n) simp_all
haftmann@28514
  1490
  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
  1491
    by simp
haftmann@57512
  1492
  with Suc show ?case by (simp add: add.commute)
haftmann@28514
  1493
qed
haftmann@30966
  1494
haftmann@24196
  1495
end
haftmann@24196
  1496
bulwahn@45231
  1497
declare of_nat_code [code]
haftmann@30966
  1498
wenzelm@60758
  1499
text\<open>Class for unital semirings with characteristic zero.
wenzelm@60758
  1500
 Includes non-ordered rings like the complex numbers.\<close>
haftmann@26072
  1501
haftmann@26072
  1502
class semiring_char_0 = semiring_1 +
haftmann@38621
  1503
  assumes inj_of_nat: "inj of_nat"
haftmann@26072
  1504
begin
haftmann@26072
  1505
haftmann@38621
  1506
lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
haftmann@38621
  1507
  by (auto intro: inj_of_nat injD)
haftmann@38621
  1508
wenzelm@60758
  1509
text\<open>Special cases where either operand is zero\<close>
haftmann@26072
  1510
blanchet@54147
  1511
lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
haftmann@38621
  1512
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])
haftmann@26072
  1513
blanchet@54147
  1514
lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
haftmann@38621
  1515
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
haftmann@26072
  1516
haftmann@60353
  1517
lemma of_nat_neq_0 [simp]:
haftmann@60353
  1518
  "of_nat (Suc n) \<noteq> 0"
haftmann@60353
  1519
  unfolding of_nat_eq_0_iff by simp
haftmann@60353
  1520
haftmann@60353
  1521
lemma of_nat_0_neq [simp]:
haftmann@60353
  1522
  "0 \<noteq> of_nat (Suc n)"
lp15@60562
  1523
  unfolding of_nat_0_eq_iff by simp
lp15@60562
  1524
haftmann@26072
  1525
end
haftmann@26072
  1526
haftmann@35028
  1527
context linordered_semidom
haftmann@25193
  1528
begin
haftmann@25193
  1529
huffman@47489
  1530
lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
huffman@47489
  1531
  by (induct n) simp_all
haftmann@25193
  1532
huffman@47489
  1533
lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
huffman@47489
  1534
  by (simp add: not_less)
haftmann@25193
  1535
haftmann@25193
  1536
lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
huffman@47489
  1537
  by (induct m n rule: diff_induct, simp_all add: add_pos_nonneg)
haftmann@25193
  1538
haftmann@26072
  1539
lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
haftmann@26072
  1540
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
haftmann@25193
  1541
huffman@47489
  1542
lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
huffman@47489
  1543
  by simp
huffman@47489
  1544
huffman@47489
  1545
lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
huffman@47489
  1546
  by simp
huffman@47489
  1547
wenzelm@60758
  1548
text\<open>Every @{text linordered_semidom} has characteristic zero.\<close>
haftmann@25193
  1549
haftmann@38621
  1550
subclass semiring_char_0 proof
haftmann@38621
  1551
qed (auto intro!: injI simp add: eq_iff)
haftmann@25193
  1552
wenzelm@60758
  1553
text\<open>Special cases where either operand is zero\<close>
haftmann@25193
  1554
blanchet@54147
  1555
lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
haftmann@25193
  1556
  by (rule of_nat_le_iff [of _ 0, simplified])
haftmann@25193
  1557
haftmann@26072
  1558
lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
haftmann@26072
  1559
  by (rule of_nat_less_iff [of 0, simplified])
haftmann@26072
  1560
haftmann@26072
  1561
end
haftmann@26072
  1562
haftmann@26072
  1563
context ring_1
haftmann@26072
  1564
begin
haftmann@26072
  1565
haftmann@26072
  1566
lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
nipkow@29667
  1567
by (simp add: algebra_simps of_nat_add [symmetric])
haftmann@26072
  1568
haftmann@26072
  1569
end
haftmann@26072
  1570
haftmann@35028
  1571
context linordered_idom
haftmann@26072
  1572
begin
haftmann@26072
  1573
haftmann@26072
  1574
lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
haftmann@26072
  1575
  unfolding abs_if by auto
haftmann@26072
  1576
haftmann@25193
  1577
end
haftmann@25193
  1578
haftmann@25193
  1579
lemma of_nat_id [simp]: "of_nat n = n"
huffman@35216
  1580
  by (induct n) simp_all
haftmann@25193
  1581
haftmann@25193
  1582
lemma of_nat_eq_id [simp]: "of_nat = id"
nipkow@39302
  1583
  by (auto simp add: fun_eq_iff)
haftmann@25193
  1584
haftmann@25193
  1585
wenzelm@60758
  1586
subsection \<open>The set of natural numbers\<close>
haftmann@25193
  1587
haftmann@26072
  1588
context semiring_1
haftmann@25193
  1589
begin
haftmann@25193
  1590
wenzelm@61070
  1591
definition Nats :: "'a set"  ("\<nat>")
wenzelm@61070
  1592
  where "\<nat> = range of_nat"
haftmann@25193
  1593
haftmann@26072
  1594
lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
haftmann@26072
  1595
  by (simp add: Nats_def)
haftmann@26072
  1596
haftmann@26072
  1597
lemma Nats_0 [simp]: "0 \<in> \<nat>"
haftmann@26072
  1598
apply (simp add: Nats_def)
haftmann@26072
  1599
apply (rule range_eqI)
haftmann@26072
  1600
apply (rule of_nat_0 [symmetric])
haftmann@26072
  1601
done
haftmann@25193
  1602
haftmann@26072
  1603
lemma Nats_1 [simp]: "1 \<in> \<nat>"
haftmann@26072
  1604
apply (simp add: Nats_def)
haftmann@26072
  1605
apply (rule range_eqI)
haftmann@26072
  1606
apply (rule of_nat_1 [symmetric])
haftmann@26072
  1607
done
haftmann@25193
  1608
haftmann@26072
  1609
lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
haftmann@26072
  1610
apply (auto simp add: Nats_def)
haftmann@26072
  1611
apply (rule range_eqI)
haftmann@26072
  1612
apply (rule of_nat_add [symmetric])
haftmann@26072
  1613
done
haftmann@26072
  1614
haftmann@26072
  1615
lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
haftmann@26072
  1616
apply (auto simp add: Nats_def)
haftmann@26072
  1617
apply (rule range_eqI)
haftmann@26072
  1618
apply (rule of_nat_mult [symmetric])
haftmann@26072
  1619
done
haftmann@25193
  1620
huffman@35633
  1621
lemma Nats_cases [cases set: Nats]:
huffman@35633
  1622
  assumes "x \<in> \<nat>"
huffman@35633
  1623
  obtains (of_nat) n where "x = of_nat n"
huffman@35633
  1624
  unfolding Nats_def
huffman@35633
  1625
proof -
wenzelm@60758
  1626
  from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def .
huffman@35633
  1627
  then obtain n where "x = of_nat n" ..
huffman@35633
  1628
  then show thesis ..
huffman@35633
  1629
qed
huffman@35633
  1630
huffman@35633
  1631
lemma Nats_induct [case_names of_nat, induct set: Nats]:
huffman@35633
  1632
  "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
huffman@35633
  1633
  by (rule Nats_cases) auto
huffman@35633
  1634
haftmann@25193
  1635
end
haftmann@25193
  1636
haftmann@25193
  1637
wenzelm@60758
  1638
subsection \<open>Further arithmetic facts concerning the natural numbers\<close>
wenzelm@21243
  1639
haftmann@22845
  1640
lemma subst_equals:
haftmann@22845
  1641
  assumes 1: "t = s" and 2: "u = t"
haftmann@22845
  1642
  shows "u = s"
haftmann@22845
  1643
  using 2 1 by (rule trans)
haftmann@22845
  1644
wenzelm@48891
  1645
ML_file "Tools/nat_arith.ML"
huffman@48559
  1646
huffman@48559
  1647
simproc_setup nateq_cancel_sums
huffman@48559
  1648
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
wenzelm@60758
  1649
  \<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close>
huffman@48559
  1650
huffman@48559
  1651
simproc_setup natless_cancel_sums
huffman@48559
  1652
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
wenzelm@60758
  1653
  \<open>fn phi => try o Nat_Arith.cancel_less_conv\<close>
huffman@48559
  1654
huffman@48559
  1655
simproc_setup natle_cancel_sums
huffman@48559
  1656
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
wenzelm@60758
  1657
  \<open>fn phi => try o Nat_Arith.cancel_le_conv\<close>
huffman@48559
  1658
huffman@48559
  1659
simproc_setup natdiff_cancel_sums
huffman@48559
  1660
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
wenzelm@60758
  1661
  \<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close>
wenzelm@24091
  1662
wenzelm@48891
  1663
ML_file "Tools/lin_arith.ML"
wenzelm@60758
  1664
setup \<open>Lin_Arith.global_setup\<close>
wenzelm@60758
  1665
declaration \<open>K Lin_Arith.setup\<close>
wenzelm@24091
  1666
wenzelm@61144
  1667
simproc_setup fast_arith_nat ("(m::nat) < n" | "(m::nat) \<le> n" | "(m::nat) = n") =
wenzelm@61144
  1668
  \<open>K Lin_Arith.simproc\<close>
wenzelm@43595
  1669
(* Because of this simproc, the arithmetic solver is really only
wenzelm@43595
  1670
useful to detect inconsistencies among the premises for subgoals which are
wenzelm@43595
  1671
*not* themselves (in)equalities, because the latter activate
wenzelm@43595
  1672
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the
wenzelm@43595
  1673
solver all the time rather than add the additional check. *)
wenzelm@43595
  1674
wenzelm@43595
  1675
wenzelm@21243
  1676
lemmas [arith_split] = nat_diff_split split_min split_max
wenzelm@21243
  1677
nipkow@27625
  1678
context order
nipkow@27625
  1679
begin
nipkow@27625
  1680
nipkow@27625
  1681
lemma lift_Suc_mono_le:
haftmann@53986
  1682
  assumes mono: "\<And>n. f n \<le> f (Suc n)" and "n \<le> n'"
krauss@27627
  1683
  shows "f n \<le> f n'"
krauss@27627
  1684
proof (cases "n < n'")
krauss@27627
  1685
  case True
haftmann@53986
  1686
  then show ?thesis
haftmann@53986
  1687
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
wenzelm@60758
  1688
qed (insert \<open>n \<le> n'\<close>, auto) -- \<open>trivial for @{prop "n = n'"}\<close>
nipkow@27625
  1689
hoelzl@56020
  1690
lemma lift_Suc_antimono_le:
hoelzl@56020
  1691
  assumes mono: "\<And>n. f n \<ge> f (Suc n)" and "n \<le> n'"
hoelzl@56020
  1692
  shows "f n \<ge> f n'"
hoelzl@56020
  1693
proof (cases "n < n'")
hoelzl@56020
  1694
  case True
hoelzl@56020
  1695
  then show ?thesis
hoelzl@56020
  1696
    by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
wenzelm@60758
  1697
qed (insert \<open>n \<le> n'\<close>, auto) -- \<open>trivial for @{prop "n = n'"}\<close>
hoelzl@56020
  1698
nipkow@27625
  1699
lemma lift_Suc_mono_less:
haftmann@53986
  1700
  assumes mono: "\<And>n. f n < f (Suc n)" and "n < n'"
krauss@27627
  1701
  shows "f n < f n'"
wenzelm@60758
  1702
using \<open>n < n'\<close>
haftmann@53986
  1703
by (induct n n' rule: less_Suc_induct [consumes 1]) (auto intro: mono)
nipkow@27625
  1704
nipkow@27789
  1705
lemma lift_Suc_mono_less_iff:
haftmann@53986
  1706
  "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
haftmann@53986
  1707
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
haftmann@53986
  1708
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])
nipkow@27789
  1709
nipkow@27625
  1710
end
nipkow@27625
  1711
haftmann@53986
  1712
lemma mono_iff_le_Suc:
haftmann@53986
  1713
  "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
haftmann@37387
  1714
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
nipkow@27625
  1715
hoelzl@56020
  1716
lemma antimono_iff_le_Suc:
hoelzl@56020
  1717
  "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
hoelzl@56020
  1718
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
hoelzl@56020
  1719
nipkow@27789
  1720
lemma mono_nat_linear_lb:
haftmann@53986
  1721
  fixes f :: "nat \<Rightarrow> nat"
haftmann@53986
  1722
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
haftmann@53986
  1723
  shows "f m + k \<le> f (m + k)"
haftmann@53986
  1724
proof (induct k)
haftmann@53986
  1725
  case 0 then show ?case by simp
haftmann@53986
  1726
next
haftmann@53986
  1727
  case (Suc k)
haftmann@53986
  1728
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
haftmann@53986
  1729
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
haftmann@53986
  1730
    by (simp add: Suc_le_eq)
haftmann@53986
  1731
  finally show ?case by simp
haftmann@53986
  1732
qed
nipkow@27789
  1733
nipkow@27789
  1734
wenzelm@60758
  1735
text\<open>Subtraction laws, mostly by Clemens Ballarin\<close>
wenzelm@21243
  1736
wenzelm@21243
  1737
lemma diff_less_mono: "[| a < (b::nat); c \<le> a |] ==> a-c < b-c"
nipkow@24438
  1738
by arith
wenzelm@21243
  1739
wenzelm@21243
  1740
lemma less_diff_conv: "(i < j-k) = (i+k < (j::nat))"
nipkow@24438
  1741
by arith
wenzelm@21243
  1742
haftmann@51173
  1743
lemma less_diff_conv2:
haftmann@51173
  1744
  fixes j k i :: nat
haftmann@51173
  1745
  assumes "k \<le> j"
haftmann@51173
  1746
  shows "j - k < i \<longleftrightarrow> j < i + k"
haftmann@51173
  1747
  using assms by arith
haftmann@51173
  1748
wenzelm@21243
  1749
lemma le_diff_conv: "(j-k \<le> (i::nat)) = (j \<le> i+k)"
nipkow@24438
  1750
by arith
wenzelm@21243
  1751
wenzelm@21243
  1752
lemma le_diff_conv2: "k \<le> j ==> (i \<le> j-k) = (i+k \<le> (j::nat))"
wenzelm@60758
  1753
  by (fact le_diff_conv2) -- \<open>FIXME delete\<close>
wenzelm@21243
  1754
wenzelm@21243
  1755
lemma diff_diff_cancel [simp]: "i \<le> (n::nat) ==> n - (n - i) = i"
nipkow@24438
  1756
by arith
wenzelm@21243
  1757
wenzelm@21243
  1758
lemma le_add_diff: "k \<le> (n::nat) ==> m \<le> n + m - k"
wenzelm@60758
  1759
  by (fact le_add_diff) -- \<open>FIXME delete\<close>
wenzelm@21243
  1760
wenzelm@21243
  1761
(*Replaces the previous diff_less and le_diff_less, which had the stronger
wenzelm@21243
  1762
  second premise n\<le>m*)
wenzelm@21243
  1763
lemma diff_less[simp]: "!!m::nat. [| 0<n; 0<m |] ==> m - n < m"
nipkow@24438
  1764
by arith
wenzelm@21243
  1765
wenzelm@60758
  1766
text \<open>Simplification of relational expressions involving subtraction\<close>
wenzelm@21243
  1767
wenzelm@21243
  1768
lemma diff_diff_eq: "[| k \<le> m;  k \<le> (n::nat) |] ==> ((m-k) - (n-k)) = (m-n)"
nipkow@24438
  1769
by (simp split add: nat_diff_split)
wenzelm@21243
  1770
wenzelm@36176
  1771
hide_fact (open) diff_diff_eq
haftmann@35064
  1772
wenzelm@21243
  1773
lemma eq_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k = n-k) = (m=n)"
nipkow@24438
  1774
by (auto split add: nat_diff_split)
wenzelm@21243
  1775
wenzelm@21243
  1776
lemma less_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k < n-k) = (m<n)"
nipkow@24438
  1777
by (auto split add: nat_diff_split)
wenzelm@21243
  1778
wenzelm@21243
  1779
lemma le_diff_iff: "[| k \<le> m;  k \<le> (n::nat) |] ==> (m-k \<le> n-k) = (m\<le>n)"
nipkow@24438
  1780
by (auto split add: nat_diff_split)
wenzelm@21243
  1781
wenzelm@60758
  1782
text\<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>
wenzelm@21243
  1783
wenzelm@21243
  1784
(* Monotonicity of subtraction in first argument *)
wenzelm@21243
  1785
lemma diff_le_mono: "m \<le> (n::nat) ==> (m-l) \<le> (n-l)"
nipkow@24438
  1786
by (simp split add: nat_diff_split)
wenzelm@21243
  1787
wenzelm@21243
  1788
lemma diff_le_mono2: "m \<le> (n::nat) ==> (l-n) \<le> (l-m)"
nipkow@24438
  1789
by (simp split add: nat_diff_split)
wenzelm@21243
  1790
wenzelm@21243
  1791
lemma diff_less_mono2: "[| m < (n::nat); m<l |] ==> (l-n) < (l-m)"
nipkow@24438
  1792
by (simp split add: nat_diff_split)
wenzelm@21243
  1793
wenzelm@21243
  1794
lemma diffs0_imp_equal: "!!m::nat. [| m-n = 0; n-m = 0 |] ==>  m=n"
nipkow@24438
  1795
by (simp split add: nat_diff_split)
wenzelm@21243
  1796
bulwahn@26143
  1797
lemma min_diff: "min (m - (i::nat)) (n - i) = min m n - i"
nipkow@32437
  1798
by auto
bulwahn@26143
  1799
lp15@60562
  1800
lemma inj_on_diff_nat:
bulwahn@26143
  1801
  assumes k_le_n: "\<forall>n \<in> N. k \<le> (n::nat)"
bulwahn@26143
  1802
  shows "inj_on (\<lambda>n. n - k) N"
bulwahn@26143
  1803
proof (rule inj_onI)
bulwahn@26143
  1804
  fix x y
bulwahn@26143
  1805
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
bulwahn@26143
  1806
  with k_le_n have "x - k + k = y - k + k" by auto
bulwahn@26143
  1807
  with a k_le_n show "x = y" by auto
bulwahn@26143
  1808
qed
bulwahn@26143
  1809
wenzelm@60758
  1810
text\<open>Rewriting to pull differences out\<close>
haftmann@26072
  1811
haftmann@26072
  1812
lemma diff_diff_right [simp]: "k\<le>j --> i - (j - k) = i + (k::nat) - j"
haftmann@26072
  1813
by arith
haftmann@26072
  1814
haftmann@26072
  1815
lemma diff_Suc_diff_eq1 [simp]: "k \<le> j ==> m - Suc (j - k) = m + k - Suc j"
haftmann@26072
  1816
by arith
haftmann@26072
  1817
haftmann@26072
  1818
lemma diff_Suc_diff_eq2 [simp]: "k \<le> j ==> Suc (j - k) - m = Suc j - (k + m)"
haftmann@26072
  1819
by arith
haftmann@26072
  1820
noschinl@45933
  1821
lemma Suc_diff_Suc: "n < m \<Longrightarrow> Suc (m - Suc n) = m - n"
noschinl@45933
  1822
by simp
noschinl@45933
  1823
bulwahn@46350
  1824
(*The others are
bulwahn@46350
  1825
      i - j - k = i - (j + k),
bulwahn@46350
  1826
      k \<le> j ==> j - k + i = j + i - k,
bulwahn@46350
  1827
      k \<le> j ==> i + (j - k) = i + j - k *)
bulwahn@46350
  1828
lemmas add_diff_assoc = diff_add_assoc [symmetric]
bulwahn@46350
  1829
lemmas add_diff_assoc2 = diff_add_assoc2[symmetric]
bulwahn@46350
  1830
declare diff_diff_left [simp]  add_diff_assoc [simp] add_diff_assoc2[simp]
bulwahn@46350
  1831
wenzelm@60758
  1832
text\<open>At present we prove no analogue of @{text not_less_Least} or @{text
wenzelm@60758
  1833
Least_Suc}, since there appears to be no need.\<close>
wenzelm@60758
  1834
wenzelm@60758
  1835
text\<open>Lemmas for ex/Factorization\<close>
wenzelm@21243
  1836
wenzelm@21243
  1837
lemma one_less_mult: "[| Suc 0 < n; Suc 0 < m |] ==> Suc 0 < m*n"
nipkow@24438
  1838
by (cases m) auto
wenzelm@21243
  1839
wenzelm@21243
  1840
lemma n_less_m_mult_n: "[| Suc 0 < n; Suc 0 < m |] ==> n<m*n"
nipkow@24438
  1841
by (cases m) auto
wenzelm@21243
  1842
wenzelm@21243
  1843
lemma n_less_n_mult_m: "[| Suc 0 < n; Suc 0 < m |] ==> n<n*m"
nipkow@24438
  1844
by (cases m) auto
wenzelm@21243
  1845
wenzelm@60758
  1846
text \<open>Specialized induction principles that work "backwards":\<close>
krauss@23001
  1847
krauss@23001
  1848
lemma inc_induct[consumes 1, case_names base step]:
hoelzl@54411
  1849
  assumes less: "i \<le> j"
krauss@23001
  1850
  assumes base: "P j"
hoelzl@54411
  1851
  assumes step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
krauss@23001
  1852
  shows "P i"
hoelzl@54411
  1853
  using less step
hoelzl@54411
  1854
proof (induct d\<equiv>"j - i" arbitrary: i)
krauss@23001
  1855
  case (0 i)
krauss@23001
  1856
  hence "i = j" by simp
krauss@23001
  1857
  with base show ?case by simp
krauss@23001
  1858
next
hoelzl@54411
  1859
  case (Suc d n)
hoelzl@54411
  1860
  hence "n \<le> n" "n < j" "P (Suc n)"
krauss@23001
  1861
    by simp_all
hoelzl@54411
  1862
  then show "P n" by fact
krauss@23001
  1863
qed
krauss@23001
  1864
krauss@23001
  1865
lemma strict_inc_induct[consumes 1, case_names base step]:
krauss@23001
  1866
  assumes less: "i < j"
krauss@23001
  1867
  assumes base: "!!i. j = Suc i ==> P i"
krauss@23001
  1868
  assumes step: "!!i. [| i < j; P (Suc i) |] ==> P i"
krauss@23001
  1869
  shows "P i"
krauss@23001
  1870
  using less
krauss@23001
  1871
proof (induct d=="j - i - 1" arbitrary: i)
krauss@23001
  1872
  case (0 i)
wenzelm@60758
  1873
  with \<open>i < j\<close> have "j = Suc i" by simp
krauss@23001
  1874
  with base show ?case by simp
krauss@23001
  1875
next
krauss@23001
  1876
  case (Suc d i)
krauss@23001
  1877
  hence "i < j" "P (Suc i)"
krauss@23001
  1878
    by simp_all
krauss@23001
  1879
  thus "P i" by (rule step)
krauss@23001
  1880
qed
krauss@23001
  1881
krauss@23001
  1882
lemma zero_induct_lemma: "P k ==> (!!n. P (Suc n) ==> P n) ==> P (k - i)"
krauss@23001
  1883
  using inc_induct[of "k - i" k P, simplified] by blast
krauss@23001
  1884
krauss@23001
  1885
lemma zero_induct: "P k ==> (!!n. P (Suc n) ==> P n) ==> P 0"
krauss@23001
  1886
  using inc_induct[of 0 k P] by blast
wenzelm@21243
  1887
wenzelm@60758
  1888
text \<open>Further induction rule similar to @{thm inc_induct}\<close>
nipkow@27625
  1889
bulwahn@46351
  1890
lemma dec_induct[consumes 1, case_names base step]:
hoelzl@54411
  1891
  "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
  1892
  by (induct j arbitrary: i) (auto simp: le_Suc_eq)
hoelzl@59000
  1893
hoelzl@59000
  1894
subsection \<open> Monotonicity of funpow \<close>
hoelzl@59000
  1895
hoelzl@59000
  1896
lemma funpow_increasing:
hoelzl@59000
  1897
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_top})"
hoelzl@59000
  1898
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
hoelzl@59000
  1899
  by (induct rule: inc_induct)
hoelzl@59000
  1900
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1901
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1902
hoelzl@59000
  1903
lemma funpow_decreasing:
hoelzl@59000
  1904
  fixes f :: "'a \<Rightarrow> ('a::{lattice, order_bot})"
hoelzl@59000
  1905
  shows "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
hoelzl@59000
  1906
  by (induct rule: dec_induct)
hoelzl@59000
  1907
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
hoelzl@59000
  1908
           intro: order_trans[OF _ funpow_mono])
hoelzl@59000
  1909
hoelzl@59000
  1910
lemma mono_funpow:
hoelzl@60427
  1911
  fixes Q :: "'a::{lattice, order_bot} \<Rightarrow> 'a"
hoelzl@59000
  1912
  shows "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
hoelzl@59000
  1913
  by (auto intro!: funpow_decreasing simp: mono_def)
blanchet@58377
  1914
hoelzl@60175
  1915
lemma antimono_funpow:
hoelzl@60427
  1916
  fixes Q :: "'a::{lattice, order_top} \<Rightarrow> 'a"
hoelzl@60175
  1917
  shows "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
hoelzl@60175
  1918
  by (auto intro!: funpow_increasing simp: antimono_def)
hoelzl@60175
  1919
wenzelm@60758
  1920
subsection \<open>The divides relation on @{typ nat}\<close>
haftmann@33274
  1921
haftmann@33274
  1922
lemma dvd_1_left [iff]: "Suc 0 dvd k"
haftmann@33274
  1923
unfolding dvd_def by simp
haftmann@33274
  1924
haftmann@33274
  1925
lemma dvd_1_iff_1 [simp]: "(m dvd Suc 0) = (m = Suc 0)"
haftmann@33274
  1926
by (simp add: dvd_def)
haftmann@33274
  1927
haftmann@33274
  1928
lemma nat_dvd_1_iff_1 [simp]: "m dvd (1::nat) \<longleftrightarrow> m = 1"
haftmann@33274
  1929
by (simp add: dvd_def)
haftmann@33274
  1930
nipkow@33657
  1931
lemma dvd_antisym: "[| m dvd n; n dvd m |] ==> m = (n::nat)"
haftmann@33274
  1932
  unfolding dvd_def
haftmann@57512
  1933
  by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)
haftmann@33274
  1934
wenzelm@60758
  1935
text \<open>@{term "op dvd"} is a partial order\<close>
haftmann@33274
  1936
wenzelm@61076
  1937
interpretation dvd: order "op dvd" "\<lambda>n m :: nat. n dvd m \<and> \<not> m dvd n"
nipkow@33657
  1938
  proof qed (auto intro: dvd_refl dvd_trans dvd_antisym)
haftmann@33274
  1939
haftmann@33274
  1940
lemma dvd_diff_nat[simp]: "[| k dvd m; k dvd n |] ==> k dvd (m-n :: nat)"
haftmann@33274
  1941
unfolding dvd_def
haftmann@33274
  1942
by (blast intro: diff_mult_distrib2 [symmetric])
haftmann@33274
  1943
haftmann@33274
  1944
lemma dvd_diffD: "[| k dvd m-n; k dvd n; n\<le>m |] ==> k dvd (m::nat)"
haftmann@33274
  1945
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
haftmann@33274
  1946
  apply (blast intro: dvd_add)
haftmann@33274
  1947
  done
haftmann@33274
  1948
haftmann@33274
  1949
lemma dvd_diffD1: "[| k dvd m-n; k dvd m; n\<le>m |] ==> k dvd (n::nat)"
haftmann@33274
  1950
by (drule_tac m = m in dvd_diff_nat, auto)
haftmann@33274
  1951
haftmann@33274
  1952
lemma dvd_mult_cancel: "!!k::nat. [| k*m dvd k*n; 0<k |] ==> m dvd n"
haftmann@33274
  1953
  unfolding dvd_def
haftmann@33274
  1954
  apply (erule exE)
haftmann@57514
  1955
  apply (simp add: ac_simps)
haftmann@33274
  1956
  done
haftmann@33274
  1957
haftmann@33274
  1958
lemma dvd_mult_cancel1: "0<m ==> (m*n dvd m) = (n = (1::nat))"
haftmann@33274
  1959
  apply auto
haftmann@33274
  1960
   apply (subgoal_tac "m*n dvd m*1")
haftmann@33274
  1961
   apply (drule dvd_mult_cancel, auto)
haftmann@33274
  1962
  done
haftmann@33274
  1963
haftmann@33274
  1964
lemma dvd_mult_cancel2: "0<m ==> (n*m dvd m) = (n = (1::nat))"
haftmann@57512
  1965
  apply (subst mult.commute)
haftmann@33274
  1966
  apply (erule dvd_mult_cancel1)
haftmann@33274
  1967
  done
haftmann@33274
  1968
haftmann@33274
  1969
lemma dvd_imp_le: "[| k dvd n; 0 < n |] ==> k \<le> (n::nat)"
haftmann@33274
  1970
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1971
haftmann@33274
  1972
lemma nat_dvd_not_less:
haftmann@33274
  1973
  fixes m n :: nat
haftmann@33274
  1974
  shows "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
haftmann@33274
  1975
by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
haftmann@33274
  1976
haftmann@54222
  1977
lemma less_eq_dvd_minus:
haftmann@51173
  1978
  fixes m n :: nat
haftmann@54222
  1979
  assumes "m \<le> n"
haftmann@54222
  1980
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
haftmann@51173
  1981
proof -
haftmann@54222
  1982
  from assms have "n = m + (n - m)" by simp
haftmann@51173
  1983
  then obtain q where "n = m + q" ..
haftmann@58647
  1984
  then show ?thesis by (simp add: add.commute [of m])
haftmann@51173
  1985
qed
haftmann@51173
  1986
haftmann@51173
  1987
lemma dvd_minus_self:
haftmann@51173
  1988
  fixes m n :: nat
haftmann@51173
  1989
  shows "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
haftmann@51173
  1990
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add)
haftmann@51173
  1991
haftmann@51173
  1992
lemma dvd_minus_add:
haftmann@51173
  1993
  fixes m n q r :: nat
haftmann@51173
  1994
  assumes "q \<le> n" "q \<le> r * m"
haftmann@51173
  1995
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
haftmann@51173
  1996
proof -
haftmann@51173
  1997
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
haftmann@58649
  1998
    using dvd_add_times_triv_left_iff [of m r] by simp
wenzelm@53374
  1999
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
wenzelm@53374
  2000
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
haftmann@57512
  2001
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
haftmann@51173
  2002
  finally show ?thesis .
haftmann@51173
  2003
qed
haftmann@51173
  2004
haftmann@33274
  2005
wenzelm@60758
  2006
subsection \<open>Aliases\<close>
haftmann@44817
  2007
haftmann@44817
  2008
lemma nat_mult_1: "(1::nat) * n = n"
haftmann@58647
  2009
  by (fact mult_1_left)
lp15@60562
  2010
haftmann@44817
  2011
lemma nat_mult_1_right: "n * (1::nat) = n"
haftmann@58647
  2012
  by (fact mult_1_right)
haftmann@58647
  2013
haftmann@44817
  2014
wenzelm@60758
  2015
subsection \<open>Size of a datatype value\<close>
haftmann@25193
  2016
haftmann@29608
  2017
class size =
wenzelm@60758
  2018
  fixes size :: "'a \<Rightarrow> nat" -- \<open>see further theory @{text Wellfounded}\<close>
haftmann@23852
  2019
blanchet@58377
  2020
instantiation nat :: size
blanchet@58377
  2021
begin
blanchet@58377
  2022
blanchet@58377
  2023
definition size_nat where
wenzelm@61076
  2024
  [simp, code]: "size (n::nat) = n"
blanchet@58377
  2025
blanchet@58377
  2026
instance ..
blanchet@58377
  2027
blanchet@58377
  2028
end
blanchet@58377
  2029
blanchet@58377
  2030
wenzelm@60758
  2031
subsection \<open>Code module namespace\<close>
haftmann@33364
  2032
haftmann@52435
  2033
code_identifier
haftmann@52435
  2034
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
haftmann@33364
  2035
huffman@47108
  2036
hide_const (open) of_nat_aux
huffman@47108
  2037
haftmann@25193
  2038
end