src/HOL/Nat.thy
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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 \<open>ind\<close>\<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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  \<comment> \<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]) \<comment> \<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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\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<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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\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<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 \<comment> \<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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  \<comment> \<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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  \<comment> \<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)"
39793
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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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3fe7e97ccca8 replaced generic 'hide' command by more conventional 'hide_class', 'hide_type', 'hide_const', 'hide_fact' -- frees some popular keywords;
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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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24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
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   285
next
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   286
  fix k m n :: nat
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   287
  show "k * ((m::nat) - n) = (k * m) - (k * n)"
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   288
    by (induct m n rule: diff_induct) simp_all
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   289
qed
25571
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   290
c9e39eafc7a0 instantiation target rather than legacy instance
haftmann
parents: 25563
diff changeset
   291
end
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   292
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   293
text \<open>Difference distributes over multiplication\<close>
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   294
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   295
lemma diff_mult_distrib:
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   296
  "((m::nat) - n) * k = (m * k) - (n * k)"
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   297
  by (fact left_diff_distrib')
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   298
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   299
lemma diff_mult_distrib2:
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   300
  "k * ((m::nat) - n) = (k * m) - (k * n)"
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   301
  by (fact right_diff_distrib')
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   302
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   303
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   304
subsubsection \<open>Addition\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   305
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   306
lemma nat_add_left_cancel:
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   307
  fixes k m n :: nat
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   308
  shows "k + m = k + n \<longleftrightarrow> m = n"
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   309
  by (fact add_left_cancel)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   310
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   311
lemma nat_add_right_cancel:
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   312
  fixes k m n :: nat
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   313
  shows "m + k = n + k \<longleftrightarrow> m = n"
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   314
  by (fact add_right_cancel)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   315
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   316
text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   317
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   318
lemma add_is_0 [iff]:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   319
  fixes m n :: nat
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   320
  shows "(m + n = 0) = (m = 0 & n = 0)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   321
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   322
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   323
lemma add_is_1:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   324
  "(m+n= Suc 0) = (m= Suc 0 & n=0 | m=0 & n= Suc 0)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   325
  by (cases m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   326
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   327
lemma one_is_add:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   328
  "(Suc 0 = m + n) = (m = Suc 0 & n = 0 | m = 0 & n = Suc 0)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   329
  by (rule trans, rule eq_commute, rule add_is_1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   330
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   331
lemma add_eq_self_zero:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   332
  fixes m n :: nat
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   333
  shows "m + n = m \<Longrightarrow> n = 0"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   334
  by (induct m) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   335
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   336
lemma inj_on_add_nat[simp]: "inj_on (%n::nat. n+k) N"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   337
  apply (induct k)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   338
   apply simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   339
  apply(drule comp_inj_on[OF _ inj_Suc])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   340
  apply (simp add:o_def)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   341
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   342
47208
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   343
lemma Suc_eq_plus1: "Suc n = n + 1"
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   344
  unfolding One_nat_def by simp
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   345
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   346
lemma Suc_eq_plus1_left: "Suc n = 1 + n"
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   347
  unfolding One_nat_def by simp
9a91b163bb71 move lemmas from Nat_Numeral.thy to Nat.thy
huffman
parents: 47108
diff changeset
   348
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   349
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   350
subsubsection \<open>Difference\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   351
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   352
lemma diff_self_eq_0 [simp]: "(m::nat) - m = 0"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   353
  by (fact diff_cancel)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   354
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   355
lemma diff_diff_left: "(i::nat) - j - k = i - (j + k)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   356
  by (fact diff_diff_add)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   357
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   358
lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   359
  by (simp add: diff_diff_left)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   360
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   361
lemma diff_commute: "(i::nat) - j - k = i - k - j"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   362
  by (fact diff_right_commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   363
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   364
lemma diff_add_inverse: "(n + m) - n = (m::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   365
  by (fact add_diff_cancel_left')
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   366
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   367
lemma diff_add_inverse2: "(m + n) - n = (m::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   368
  by (fact add_diff_cancel_right')
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   369
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   370
lemma diff_cancel: "(k + m) - (k + n) = m - (n::nat)"
59815
cce82e360c2f explicit commutative additive inverse operation;
haftmann
parents: 59582
diff changeset
   371
  by (fact add_diff_cancel_left)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   372
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   373
lemma diff_cancel2: "(m + k) - (n + k) = m - (n::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   374
  by (fact add_diff_cancel_right)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   375
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   376
lemma diff_add_0: "n - (n + m) = (0::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   377
  by (fact diff_add_zero)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   378
30093
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   379
lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   380
  unfolding One_nat_def by simp
ecb557b021b2 add lemma diff_Suc_1
huffman
parents: 30079
diff changeset
   381
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   382
subsubsection \<open>Multiplication\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   383
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   384
lemma add_mult_distrib2: "k * (m + n) = (k * m) + ((k * n)::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   385
  by (fact distrib_left)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   386
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   387
lemma mult_is_0 [simp]: "((m::nat) * n = 0) = (m=0 | n=0)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   388
  by (induct m) auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   389
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   390
lemmas nat_distrib =
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   391
  add_mult_distrib add_mult_distrib2 diff_mult_distrib diff_mult_distrib2
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   392
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   393
lemma mult_eq_1_iff [simp]: "(m * n = Suc 0) = (m = Suc 0 & n = Suc 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   394
  apply (induct m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   395
   apply simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   396
  apply (induct n)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   397
   apply auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   398
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   399
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53986
diff changeset
   400
lemma one_eq_mult_iff [simp]: "(Suc 0 = m * n) = (m = Suc 0 & n = Suc 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   401
  apply (rule trans)
44890
22f665a2e91c new fastforce replacing fastsimp - less confusing name
nipkow
parents: 44848
diff changeset
   402
  apply (rule_tac [2] mult_eq_1_iff, fastforce)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   403
  done
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   404
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   405
lemma nat_mult_eq_1_iff [simp]: "m * n = (1::nat) \<longleftrightarrow> m = 1 \<and> n = 1"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   406
  unfolding One_nat_def by (rule mult_eq_1_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   407
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   408
lemma nat_1_eq_mult_iff [simp]: "(1::nat) = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   409
  unfolding One_nat_def by (rule one_eq_mult_iff)
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   410
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   411
lemma mult_cancel1 [simp]: "(k * m = k * n) = (m = n | (k = (0::nat)))"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   412
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   413
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   414
  proof (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   415
    case 0 then show "m = 0" by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   416
  next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   417
    case (Suc n) then show "m = Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   418
      by (cases m) (simp_all add: eq_commute [of "0"])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   419
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   420
  then show ?thesis by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   421
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   422
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   423
lemma mult_cancel2 [simp]: "(m * k = n * k) = (m = n | (k = (0::nat)))"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   424
  by (simp add: mult.commute)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   425
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   426
lemma Suc_mult_cancel1: "(Suc k * m = Suc k * n) = (m = n)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   427
  by (subst mult_cancel1) simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   428
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   429
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   430
subsection \<open>Orders on @{typ nat}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   431
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   432
subsubsection \<open>Operation definition\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   433
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   434
instantiation nat :: linorder
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   435
begin
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   436
55575
a5e33e18fb5c moved 'primrec' up (for real this time) and removed temporary 'old_primrec'
blanchet
parents: 55534
diff changeset
   437
primrec less_eq_nat where
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   438
  "(0::nat) \<le> n \<longleftrightarrow> True"
44325
84696670feb1 more uniform formatting of specifications
haftmann
parents: 44278
diff changeset
   439
| "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   440
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   441
declare less_eq_nat.simps [simp del]
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   442
lemma le0 [iff]: "0 \<le> (n::nat)" by (simp add: less_eq_nat.simps)
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   443
lemma [code]: "(0::nat) \<le> n \<longleftrightarrow> True" by simp
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   444
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   445
definition less_nat where
28514
da83a614c454 tuned of_nat code generation
haftmann
parents: 27823
diff changeset
   446
  less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   447
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   448
lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   449
  by (simp add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   450
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   451
lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   452
  unfolding less_eq_Suc_le ..
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   453
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   454
lemma le_0_eq [iff]: "(n::nat) \<le> 0 \<longleftrightarrow> n = 0"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   455
  by (induct n) (simp_all add: less_eq_nat.simps(2))
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   456
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   457
lemma not_less0 [iff]: "\<not> n < (0::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   458
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   459
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
   460
lemma less_nat_zero_code [code]: "n < (0::nat) \<longleftrightarrow> False"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   461
  by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   462
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   463
lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   464
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   465
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   466
lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   467
  by (simp add: less_eq_Suc_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   468
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   469
lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   470
  by (cases m) auto
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   471
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   472
lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   473
  by (induct m arbitrary: n)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   474
    (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   475
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   476
lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   477
  by (cases n) (auto intro: le_SucI)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   478
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   479
lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   480
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
   481
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   482
lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   483
  by (simp add: less_eq_Suc_le) (erule Suc_leD)
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   484
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   485
instance
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   486
proof
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   487
  fix n m :: nat
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   488
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   489
  proof (induct n arbitrary: m)
27679
haftmann
parents: 27627
diff changeset
   490
    case 0 then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   491
  next
27679
haftmann
parents: 27627
diff changeset
   492
    case (Suc n) then show ?case by (cases m) (simp_all add: less_eq_Suc_le)
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   493
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   494
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   495
  fix n :: nat show "n \<le> n" by (induct n) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   496
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   497
  fix n m :: nat assume "n \<le> m" and "m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   498
  then show "n = m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   499
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   500
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   501
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   502
  fix n m q :: nat assume "n \<le> m" and "m \<le> q"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   503
  then show "n \<le> q"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   504
  proof (induct n arbitrary: m q)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   505
    case 0 show ?case by simp
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   506
  next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   507
    case (Suc n) then show ?case
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   508
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   509
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   510
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   511
  qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   512
next
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   513
  fix n m :: nat show "n \<le> m \<or> m \<le> n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   514
    by (induct n arbitrary: m)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   515
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   516
qed
25510
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   517
38c15efe603b adjustions to due to instance target
haftmann
parents: 25502
diff changeset
   518
end
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   519
52729
412c9e0381a1 factored syntactic type classes for bot and top (by Alessandro Coglio)
haftmann
parents: 52435
diff changeset
   520
instantiation nat :: order_bot
29652
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   521
begin
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   522
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   523
definition bot_nat :: nat where
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   524
  "bot_nat = 0"
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   525
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   526
instance proof
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   527
qed (simp add: bot_nat_def)
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   528
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   529
end
f4c6e546b7fe nat is a bot instance
haftmann
parents: 29608
diff changeset
   530
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   531
instance nat :: no_top
61169
4de9ff3ea29a tuned proofs -- less legacy;
wenzelm
parents: 61144
diff changeset
   532
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   533
51329
4a3c453f99a1 split dense into inner_dense_order and no_top/no_bot
hoelzl
parents: 51263
diff changeset
   534
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   535
subsubsection \<open>Introduction properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   536
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   537
lemma lessI [iff]: "n < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   538
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   539
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   540
lemma zero_less_Suc [iff]: "0 < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   541
  by (simp add: less_Suc_eq_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   542
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   543
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   544
subsubsection \<open>Elimination properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   545
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   546
lemma less_not_refl: "~ n < (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   547
  by (rule order_less_irrefl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   548
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   549
lemma less_not_refl2: "n < m ==> m \<noteq> (n::nat)"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   550
  by (rule not_sym) (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   551
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   552
lemma less_not_refl3: "(s::nat) < t ==> s \<noteq> t"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   553
  by (rule less_imp_neq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   554
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   555
lemma less_irrefl_nat: "(n::nat) < n ==> R"
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
   556
  by (rule notE, rule less_not_refl)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   557
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   558
lemma less_zeroE: "(n::nat) < 0 ==> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   559
  by (rule notE) (rule not_less0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   560
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   561
lemma less_Suc_eq: "(m < Suc n) = (m < n | m = n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   562
  unfolding less_Suc_eq_le le_less ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   563
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   564
lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   565
  by (simp add: less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   566
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53986
diff changeset
   567
lemma less_one [iff]: "(n < (1::nat)) = (n = 0)"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   568
  unfolding One_nat_def by (rule less_Suc0)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   569
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   570
lemma Suc_mono: "m < n ==> Suc m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   571
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   572
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   573
text \<open>"Less than" is antisymmetric, sort of\<close>
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   574
lemma less_antisym: "\<lbrakk> \<not> n < m; n < Suc m \<rbrakk> \<Longrightarrow> m = n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   575
  unfolding not_less less_Suc_eq_le by (rule antisym)
14302
6c24235e8d5d *** empty log message ***
nipkow
parents: 14267
diff changeset
   576
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   577
lemma nat_neq_iff: "((m::nat) \<noteq> n) = (m < n | n < m)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   578
  by (rule linorder_neq_iff)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   579
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   580
lemma nat_less_cases: assumes major: "(m::nat) < n ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   581
  and eqCase: "m = n ==> P n m" and lessCase: "n<m ==> P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   582
  shows "P n m"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   583
  apply (rule less_linear [THEN disjE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   584
  apply (erule_tac [2] disjE)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   585
  apply (erule lessCase)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   586
  apply (erule sym [THEN eqCase])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   587
  apply (erule major)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   588
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   589
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   590
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   591
subsubsection \<open>Inductive (?) properties\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   592
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   593
lemma Suc_lessI: "m < n ==> Suc m \<noteq> n ==> Suc m < n"
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   594
  unfolding less_eq_Suc_le [of m] le_less by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   595
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   596
lemma lessE:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   597
  assumes major: "i < k"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   598
  and p1: "k = Suc i ==> P" and p2: "!!j. i < j ==> k = Suc j ==> P"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   599
  shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   600
proof -
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   601
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   602
    unfolding less_eq_Suc_le by (induct k) simp_all
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   603
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   604
    by (clarsimp simp add: less_le)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   605
  with p1 p2 show P by auto
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   606
qed
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   607
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   608
lemma less_SucE: assumes major: "m < Suc n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   609
  and less: "m < n ==> P" and eq: "m = n ==> P" shows P
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   610
  apply (rule major [THEN lessE])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   611
  apply (rule eq, blast)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   612
  apply (rule less, blast)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   613
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   614
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   615
lemma Suc_lessE: assumes major: "Suc i < k"
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   616
  and minor: "!!j. i < j ==> k = Suc j ==> P" shows P
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   617
  apply (rule major [THEN lessE])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   618
  apply (erule lessI [THEN minor])
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   619
  apply (erule Suc_lessD [THEN minor], assumption)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   620
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   621
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   622
lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   623
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   624
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   625
lemma less_trans_Suc:
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   626
  assumes le: "i < j" shows "j < k ==> Suc i < k"
14208
144f45277d5a misc tidying
paulson
parents: 14193
diff changeset
   627
  apply (induct k, simp_all)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   628
  apply (insert le)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   629
  apply (simp add: less_Suc_eq)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   630
  apply (blast dest: Suc_lessD)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   631
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   632
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   633
text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{term "n = m | n < m"}\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   634
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   635
  unfolding not_less less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   636
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   637
lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   638
  unfolding not_le Suc_le_eq ..
21243
afffe1f72143 removed theory NatArith (now part of Nat);
wenzelm
parents: 21191
diff changeset
   639
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   640
text \<open>Properties of "less than or equal"\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   641
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   642
lemma le_imp_less_Suc: "m \<le> n ==> m < Suc n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   643
  unfolding less_Suc_eq_le .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   644
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   645
lemma Suc_n_not_le_n: "~ Suc n \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   646
  unfolding not_le less_Suc_eq_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   647
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   648
lemma le_Suc_eq: "(m \<le> Suc n) = (m \<le> n | m = Suc n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   649
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   650
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   651
lemma le_SucE: "m \<le> Suc n ==> (m \<le> n ==> R) ==> (m = Suc n ==> R) ==> R"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   652
  by (drule le_Suc_eq [THEN iffD1], iprover+)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   653
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   654
lemma Suc_leI: "m < n ==> Suc(m) \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   655
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   656
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   657
text \<open>Stronger version of \<open>Suc_leD\<close>\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   658
lemma Suc_le_lessD: "Suc m \<le> n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   659
  unfolding Suc_le_eq .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   660
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   661
lemma less_imp_le_nat: "m < n ==> m \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   662
  unfolding less_eq_Suc_le by (rule Suc_leD)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   663
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   664
text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
26315
cb3badaa192e removed redundant less_trans, less_linear, le_imp_less_or_eq, le_less_trans, less_le_trans (cf. Orderings.thy);
wenzelm
parents: 26300
diff changeset
   665
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   666
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   667
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   668
text \<open>Equivalence of @{term "m \<le> n"} and @{term "m < n | m = n"}\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   669
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   670
lemma less_or_eq_imp_le: "m < n | m = n ==> m \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   671
  unfolding le_less .
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   672
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   673
lemma le_eq_less_or_eq: "(m \<le> (n::nat)) = (m < n | m=n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   674
  by (rule le_less)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   675
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   676
text \<open>Useful with \<open>blast\<close>.\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   677
lemma eq_imp_le: "(m::nat) = n ==> m \<le> n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   678
  by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   679
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   680
lemma le_refl: "n \<le> (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   681
  by simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   682
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   683
lemma le_trans: "[| i \<le> j; j \<le> k |] ==> i \<le> (k::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   684
  by (rule order_trans)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   685
33657
a4179bf442d1 renamed lemmas "anti_sym" -> "antisym"
nipkow
parents: 33364
diff changeset
   686
lemma le_antisym: "[| m \<le> n; n \<le> m |] ==> m = (n::nat)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   687
  by (rule antisym)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   688
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   689
lemma nat_less_le: "((m::nat) < n) = (m \<le> n & m \<noteq> n)"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   690
  by (rule less_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   691
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   692
lemma le_neq_implies_less: "(m::nat) \<le> n ==> m \<noteq> n ==> m < n"
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   693
  unfolding less_le ..
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   694
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   695
lemma nat_le_linear: "(m::nat) \<le> n | n \<le> m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   696
  by (rule linear)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   697
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   698
lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]
15921
b6e345548913 Fixing a problem with lin.arith.
nipkow
parents: 15539
diff changeset
   699
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   700
lemma le_less_Suc_eq: "m \<le> n ==> (n < Suc m) = (n = m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   701
  unfolding less_Suc_eq_le by auto
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   702
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   703
lemma not_less_less_Suc_eq: "~ n < m ==> (n < Suc m) = (n = m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   704
  unfolding not_less by (rule le_less_Suc_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   705
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   706
lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   707
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   708
lemma not0_implies_Suc: "n \<noteq> 0 ==> \<exists>m. n = Suc m"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   709
by (cases n) simp_all
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   710
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   711
lemma gr0_implies_Suc: "n > 0 ==> \<exists>m. n = Suc m"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   712
by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   713
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   714
lemma gr_implies_not0: fixes n :: nat shows "m<n ==> n \<noteq> 0"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   715
by (cases n) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   716
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   717
lemma neq0_conv[iff]: fixes n :: nat shows "(n \<noteq> 0) = (0 < n)"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   718
by (cases n) simp_all
25140
273772abbea2 More changes from >0 to ~=0::nat
nipkow
parents: 25134
diff changeset
   719
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   720
text \<open>This theorem is useful with \<open>blast\<close>\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   721
lemma gr0I: "((n::nat) = 0 ==> False) ==> 0 < n"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   722
by (rule neq0_conv[THEN iffD1], iprover)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   723
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   724
lemma gr0_conv_Suc: "(0 < n) = (\<exists>m. n = Suc m)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   725
by (fast intro: not0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   726
54147
97a8ff4e4ac9 killed most "no_atp", to make Sledgehammer more complete
blanchet
parents: 53986
diff changeset
   727
lemma not_gr0 [iff]: "!!n::nat. (~ (0 < n)) = (n = 0)"
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   728
using neq0_conv by blast
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   729
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   730
lemma Suc_le_D: "(Suc n \<le> m') ==> (? m. m' = Suc m)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   731
by (induct m') simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   732
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   733
text \<open>Useful in certain inductive arguments\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   734
lemma less_Suc_eq_0_disj: "(m < Suc n) = (m = 0 | (\<exists>j. m = Suc j & j < n))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   735
by (cases m) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   736
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   737
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   738
subsubsection \<open>Monotonicity of Addition\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   739
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   740
lemma Suc_pred [simp]: "n>0 ==> Suc (n - Suc 0) = n"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
   741
by (simp add: diff_Suc split: nat.split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   742
30128
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   743
lemma Suc_diff_1 [simp]: "0 < n ==> Suc (n - 1) = n"
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   744
unfolding One_nat_def by (rule Suc_pred)
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
   745
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   746
lemma nat_add_left_cancel_le [simp]: "(k + m \<le> k + n) = (m\<le>(n::nat))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   747
by (induct k) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   748
14331
8dbbb7cf3637 re-organized numeric lemmas
paulson
parents: 14302
diff changeset
   749
lemma nat_add_left_cancel_less [simp]: "(k + m < k + n) = (m<(n::nat))"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   750
by (induct k) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   751
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   752
lemma add_gr_0 [iff]: "!!m::nat. (m + n > 0) = (m>0 | n>0)"
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   753
by(auto dest:gr0_implies_Suc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
   754
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   755
text \<open>strict, in 1st argument\<close>
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   756
lemma add_less_mono1: "i < j ==> i + k < j + (k::nat)"
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
   757
by (induct k) simp_all
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   758
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   759
text \<open>strict, in both arguments\<close>
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   760
lemma add_less_mono: "[|i < j; k < l|] ==> i + k < j + (l::nat)"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   761
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
15251
bb6f072c8d10 converted some induct_tac to induct
paulson
parents: 15140
diff changeset
   762
  apply (induct j, simp_all)
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   763
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   764
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   765
text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   766
lemma less_imp_Suc_add: "m < n ==> (\<exists>k. n = Suc (m + k))"
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   767
  apply (induct n)
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   768
  apply (simp_all add: order_le_less)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   769
  apply (blast elim!: less_SucE
35047
1b2bae06c796 hide fact Nat.add_0_right; make add_0_right from Groups priority
haftmann
parents: 35028
diff changeset
   770
               intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   771
  done
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   772
56194
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   773
lemma le_Suc_ex: "(k::nat) \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   774
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)
9ffbb4004c81 fix HOL-NSA; move lemmas
hoelzl
parents: 56020
diff changeset
   775
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   776
text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
25134
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   777
lemma mult_less_mono2: "(i::nat) < j ==> 0<k ==> k * i < k * j"
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   778
apply(auto simp: gr0_conv_Suc)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   779
apply (induct_tac m)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   780
apply (simp_all add: add_less_mono)
3d4953e88449 Eliminated most of the neq0_conv occurrences. As a result, many
nipkow
parents: 25111
diff changeset
   781
done
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   782
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   783
text \<open>Addition is the inverse of subtraction:
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   784
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   785
lemma add_diff_inverse_nat: "~  m < n ==> n + (m - n) = (m::nat)"
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   786
by (induct m n rule: diff_induct) simp_all
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   787
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   788
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
   789
text\<open>The naturals form an ordered \<open>semidom\<close>\<close>
35028
108662d50512 more consistent naming of type classes involving orderings (and lattices) -- c.f. NEWS
haftmann
parents: 34208
diff changeset
   790
instance nat :: linordered_semidom
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
   791
proof
14348
744c868ee0b7 Defining the type class "ringpower" and deleting superseded theorems for
paulson
parents: 14341
diff changeset
   792
  show "0 < (1::nat)" by simp
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   793
  show "\<And>m n q :: nat. m \<le> n \<Longrightarrow> q + m \<le> q + n" by simp
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
   794
  show "\<And>m n q :: nat. m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n" by (simp add: mult_less_mono2)
59833
ab828c2c5d67 clarified no_zero_devisors: makes only sense in a semiring;
haftmann
parents: 59816
diff changeset
   795
  show "\<And>m n :: nat. m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0" by simp
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   796
  show "\<And>m n :: nat. n \<le> m ==> (m - n) + n = (m::nat)"
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   797
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   798
qed 
30056
0a35bee25c20 added lemmas
nipkow
parents: 29879
diff changeset
   799
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   800
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   801
subsubsection \<open>@{term min} and @{term max}\<close>
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   802
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   803
lemma mono_Suc: "mono Suc"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   804
by (rule monoI) simp
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   805
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   806
lemma min_0L [simp]: "min 0 n = (0::nat)"
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45833
diff changeset
   807
by (rule min_absorb1) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   808
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   809
lemma min_0R [simp]: "min n 0 = (0::nat)"
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45833
diff changeset
   810
by (rule min_absorb2) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   811
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   812
lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   813
by (simp add: mono_Suc min_of_mono)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   814
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   815
lemma min_Suc1:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   816
   "min (Suc n) m = (case m of 0 => 0 | Suc m' => Suc(min n m'))"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   817
by (simp split: nat.split)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   818
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   819
lemma min_Suc2:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   820
   "min m (Suc n) = (case m of 0 => 0 | Suc m' => Suc(min m' n))"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   821
by (simp split: nat.split)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   822
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   823
lemma max_0L [simp]: "max 0 n = (n::nat)"
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45833
diff changeset
   824
by (rule max_absorb2) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   825
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   826
lemma max_0R [simp]: "max n 0 = (n::nat)"
45931
99cf6e470816 weaken preconditions on lemmas
noschinl
parents: 45833
diff changeset
   827
by (rule max_absorb1) simp
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   828
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   829
lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc(max m n)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   830
by (simp add: mono_Suc max_of_mono)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   831
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   832
lemma max_Suc1:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   833
   "max (Suc n) m = (case m of 0 => Suc n | Suc m' => Suc(max n m'))"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   834
by (simp split: nat.split)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   835
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   836
lemma max_Suc2:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   837
   "max m (Suc n) = (case m of 0 => Suc n | Suc m' => Suc(max m' n))"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   838
by (simp split: nat.split)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   839
44817
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   840
lemma nat_mult_min_left:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   841
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   842
  shows "min m n * q = min (m * q) (n * q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   843
  by (simp add: min_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   844
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   845
lemma nat_mult_min_right:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   846
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   847
  shows "m * min n q = min (m * n) (m * q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   848
  by (simp add: min_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   849
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   850
lemma nat_add_max_left:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   851
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   852
  shows "max m n + q = max (m + q) (n + q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   853
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   854
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   855
lemma nat_add_max_right:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   856
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   857
  shows "m + max n q = max (m + n) (m + q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   858
  by (simp add: max_def)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   859
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   860
lemma nat_mult_max_left:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   861
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   862
  shows "max m n * q = max (m * q) (n * q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   863
  by (simp add: max_def not_le) (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   864
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   865
lemma nat_mult_max_right:
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   866
  fixes m n q :: nat
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   867
  shows "m * max n q = max (m * n) (m * q)"
b63e445c8f6d lemmas about +, *, min, max on nat
haftmann
parents: 44325
diff changeset
   868
  by (simp add: max_def not_le) (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   869
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   870
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   871
subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   872
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   873
text \<open>Complete induction, aka course-of-values induction\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   874
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   875
instance nat :: wellorder proof
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   876
  fix P and n :: nat
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   877
  assume step: "\<And>n::nat. (\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   878
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   879
  proof (induct n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   880
    case (0 n)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   881
    have "P 0" by (rule step) auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   882
    thus ?case using 0 by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   883
  next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   884
    case (Suc m n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   885
    then have "n \<le> m \<or> n = Suc m" by (simp add: le_Suc_eq)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   886
    thus ?case
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   887
    proof
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   888
      assume "n \<le> m" thus "P n" by (rule Suc(1))
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   889
    next
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   890
      assume n: "n = Suc m"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   891
      show "P n"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   892
        by (rule step) (rule Suc(1), simp add: n le_simps)
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   893
    qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   894
  qed
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   895
  then show "P n" by auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   896
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   897
57015
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   898
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   899
lemma Least_eq_0[simp]: "P(0::nat) \<Longrightarrow> Least P = 0"
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   900
by (rule Least_equality[OF _ le0])
842bb6d36263 added lemma
nipkow
parents: 56194
diff changeset
   901
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   902
lemma Least_Suc:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   903
     "[| P n; ~ P 0 |] ==> (LEAST n. P n) = Suc (LEAST m. P(Suc m))"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
   904
  apply (cases n, auto)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   905
  apply (frule LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   906
  apply (drule_tac P = "%x. P (Suc x) " in LeastI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   907
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   908
  apply (erule_tac [2] Least_le)
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
   909
  apply (cases "LEAST x. P x", auto)
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   910
  apply (drule_tac P = "%x. P (Suc x) " in Least_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   911
  apply (blast intro: order_antisym)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   912
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   913
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   914
lemma Least_Suc2:
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   915
   "[|P n; Q m; ~P 0; !k. P (Suc k) = Q k|] ==> Least P = Suc (Least Q)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   916
  apply (erule (1) Least_Suc [THEN ssubst])
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   917
  apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   918
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   919
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   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)"
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   921
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   922
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   923
  apply (rule_tac x="LEAST k. P(k)" in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   924
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   925
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   926
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   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)"
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
   928
  unfolding One_nat_def
27823
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   929
  apply (cases n)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   930
   apply blast
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   931
  apply (frule (1) ex_least_nat_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   932
  apply (erule exE)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   933
  apply (case_tac k)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   934
   apply simp
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   935
  apply (rename_tac k1)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   936
  apply (rule_tac x=k1 in exI)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   937
  apply (auto simp add: less_eq_Suc_le)
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   938
  done
52971512d1a2 moved class wellorder to theory Orderings
haftmann
parents: 27789
diff changeset
   939
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   940
lemma nat_less_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   941
  assumes "!!n. \<forall>m::nat. m < n --> P m ==> P n" shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   942
  using assms less_induct by blast
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   943
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   944
lemma measure_induct_rule [case_names less]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   945
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   946
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   947
  shows "P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   948
by (induct m\<equiv>"f a" arbitrary: a rule: less_induct) (auto intro: step)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   949
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   950
text \<open>old style induction rules:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   951
lemma measure_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   952
  fixes f :: "'a \<Rightarrow> nat"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   953
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   954
  by (rule measure_induct_rule [of f P a]) iprover
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   955
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   956
lemma full_nat_induct:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   957
  assumes step: "(!!n. (ALL m. Suc m <= n --> P m) ==> P n)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   958
  shows "P n"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   959
  by (rule less_induct) (auto intro: step simp:le_simps)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
   960
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   961
text\<open>An induction rule for estabilishing binary relations\<close>
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   962
lemma less_Suc_induct:
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   963
  assumes less:  "i < j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   964
     and  step:  "!!i. P i (Suc i)"
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   965
     and  trans: "!!i j k. i < j ==> j < k ==>  P i j ==> P j k ==> P i k"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   966
  shows "P i j"
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   967
proof -
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   968
  from less obtain k where j: "j = Suc (i + k)" by (auto dest: less_imp_Suc_add)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   969
  have "P i (Suc (i + k))"
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   970
  proof (induct k)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   971
    case 0
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   972
    show ?case by (simp add: step)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   973
  next
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   974
    case (Suc k)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   975
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
   976
    hence "i < Suc (i + k)" by (simp add: add.commute)
31714
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   977
    from trans[OF this lessI Suc step]
347e9d18f372 generalized less_Suc_induct
krauss
parents: 31155
diff changeset
   978
    show ?case by simp
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   979
  qed
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
   980
  thus "P i j" by (simp add: j)
19870
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   981
qed
ef037d1b32d1 new results
paulson
parents: 19573
diff changeset
   982
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   983
text \<open>The method of infinite descent, frequently used in number theory.
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   984
Provided by Roelof Oosterhuis.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   985
$P(n)$ is true for all $n\in\mathbb{N}$ if
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   986
\begin{itemize}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   987
  \item case ``0'': given $n=0$ prove $P(n)$,
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   988
  \item case ``smaller'': given $n>0$ and $\neg P(n)$ prove there exists
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   989
        a smaller integer $m$ such that $\neg P(m)$.
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   990
\end{itemize}\<close>
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   991
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
   992
text\<open>A compact version without explicit base case:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   993
lemma infinite_descent:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   994
  "\<lbrakk> !!n::nat. \<not> P n \<Longrightarrow>  \<exists>m<n. \<not>  P m \<rbrakk> \<Longrightarrow>  P n"
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
   995
by (induct n rule: less_induct) auto
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   996
60562
24af00b010cf Amalgamation of the class comm_semiring_1_diff_distrib into comm_semiring_1_cancel. Moving axiom le_add_diff_inverse2 from semiring_numeral_div to linordered_semidom.
paulson <lp15@cam.ac.uk>
parents: 60427
diff changeset
   997
lemma infinite_descent0[case_names 0 smaller]:
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   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"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
   999
by (rule infinite_descent) (case_tac "n>0", auto)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1000
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1001
text \<open>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1002
Infinite descent using a mapping to $\mathbb{N}$:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1003
$P(x)$ is true for all $x\in D$ if there exists a $V: D \to \mathbb{N}$ and
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1004
\begin{itemize}
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1005
\item case ``0'': given $V(x)=0$ prove $P(x)$,
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  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 $~\neg P(y)$.
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1007
\end{itemize}
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1008
NB: the proof also shows how to use the previous lemma.\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1009
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1010
corollary infinite_descent0_measure [case_names 0 smaller]:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1011
  assumes A0: "!!x. V x = (0::nat) \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1012
    and   A1: "!!x. V x > 0 \<Longrightarrow> \<not>P x \<Longrightarrow> (\<exists>y. V y < V x \<and> \<not>P y)"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1013
  shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1014
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1015
  obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1016
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1017
  proof (induct n rule: infinite_descent0)
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1018
    case 0 \<comment> "i.e. $V(x) = 0$"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1019
    with A0 show "P x" by auto
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1020
  next \<comment> "now $n>0$ and $P(x)$ does not hold for some $x$ with $V(x)=n$"
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1021
    case (smaller n)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1022
    then obtain x where vxn: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1023
    with A1 obtain y where "V y < V x \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1024
    with vxn obtain m where "m = V y \<and> m<n \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1025
    then show ?case by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1026
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1027
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1028
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1029
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1030
text\<open>Again, without explicit base case:\<close>
26748
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1031
lemma infinite_descent_measure:
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1032
assumes "!!x. \<not> P x \<Longrightarrow> \<exists>y. (V::'a\<Rightarrow>nat) y < V x \<and> \<not> P y" shows "P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1033
proof -
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1034
  from assms obtain n where "n = V x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1035
  moreover have "!!x. V x = n \<Longrightarrow> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1036
  proof (induct n rule: infinite_descent, auto)
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1037
    fix x assume "\<not> P x"
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1038
    with assms show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1039
  qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1040
  ultimately show "P x" by auto
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1041
qed
4d51ddd6aa5c Merged theories about wellfoundedness into one: Wellfounded.thy
krauss
parents: 26335
diff changeset
  1042
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1043
text \<open>A [clumsy] way of lifting \<open><\<close>
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1044
  monotonicity to \<open>\<le>\<close> monotonicity\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1045
lemma less_mono_imp_le_mono:
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1046
  "\<lbrakk> !!i j::nat. i < j \<Longrightarrow> f i < f j; i \<le> j \<rbrakk> \<Longrightarrow> f i \<le> ((f j)::nat)"
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1047
by (simp add: order_le_less) (blast)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1048
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1049
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1050
text \<open>non-strict, in 1st argument\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1051
lemma add_le_mono1: "i \<le> j ==> i + k \<le> j + (k::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1052
by (rule add_right_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1053
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1054
text \<open>non-strict, in both arguments\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1055
lemma add_le_mono: "[| i \<le> j;  k \<le> l |] ==> i + k \<le> j + (l::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1056
by (rule add_mono)
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1057
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1058
lemma le_add2: "n \<le> ((m + n)::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1059
by (insert add_right_mono [of 0 m n], simp)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1060
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1061
lemma le_add1: "n \<le> ((n + m)::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1062
by (simp add: add.commute, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1063
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1064
lemma less_add_Suc1: "i < Suc (i + m)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1065
by (rule le_less_trans, rule le_add1, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1066
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1067
lemma less_add_Suc2: "i < Suc (m + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1068
by (rule le_less_trans, rule le_add2, rule lessI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1069
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1070
lemma less_iff_Suc_add: "(m < n) = (\<exists>k. n = Suc (m + k))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1071
by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1072
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1073
lemma trans_le_add1: "(i::nat) \<le> j ==> i \<le> j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1074
by (rule le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1075
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1076
lemma trans_le_add2: "(i::nat) \<le> j ==> i \<le> m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1077
by (rule le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1078
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1079
lemma trans_less_add1: "(i::nat) < j ==> i < j + m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1080
by (rule less_le_trans, assumption, rule le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1081
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1082
lemma trans_less_add2: "(i::nat) < j ==> i < m + j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1083
by (rule less_le_trans, assumption, rule le_add2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1084
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1085
lemma add_lessD1: "i + j < (k::nat) ==> i < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1086
apply (rule le_less_trans [of _ "i+j"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1087
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1088
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1089
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1090
lemma not_add_less1 [iff]: "~ (i + j < (i::nat))"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1091
apply (rule notI)
26335
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
  1092
apply (drule add_lessD1)
961bbcc9d85b removed redundant Nat.less_not_sym, Nat.less_asym;
wenzelm
parents: 26315
diff changeset
  1093
apply (erule less_irrefl [THEN notE])
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1094
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1095
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1096
lemma not_add_less2 [iff]: "~ (j + i < (i::nat))"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1097
by (simp add: add.commute)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1098
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1099
lemma add_leD1: "m + k \<le> n ==> m \<le> (n::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1100
apply (rule order_trans [of _ "m+k"])
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1101
apply (simp_all add: le_add1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1102
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1103
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1104
lemma add_leD2: "m + k \<le> n ==> k \<le> (n::nat)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1105
apply (simp add: add.commute)
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1106
apply (erule add_leD1)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1107
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1108
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1109
lemma add_leE: "(m::nat) + k \<le> n ==> (m \<le> n ==> k \<le> n ==> R) ==> R"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1110
by (blast dest: add_leD1 add_leD2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1111
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1112
text \<open>needs \<open>!!k\<close> for \<open>ac_simps\<close> to work\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1113
lemma less_add_eq_less: "!!k::nat. k < l ==> m + l = k + n ==> m < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1114
by (force simp del: add_Suc_right
57514
bdc2c6b40bf2 prefer ac_simps collections over separate name bindings for add and mult
haftmann
parents: 57512
diff changeset
  1115
    simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1116
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1117
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1118
subsubsection \<open>More results about difference\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1119
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1120
lemma Suc_diff_le: "n \<le> m ==> Suc m - n = Suc (m - n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1121
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1122
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1123
lemma diff_less_Suc: "m - n < Suc m"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1124
apply (induct m n rule: diff_induct)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1125
apply (erule_tac [3] less_SucE)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1126
apply (simp_all add: less_Suc_eq)
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1127
done
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1128
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1129
lemma diff_le_self [simp]: "m - n \<le> (m::nat)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1130
by (induct m n rule: diff_induct) (simp_all add: le_SucI)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1131
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1132
lemma le_iff_add: "(m::nat) \<le> n = (\<exists>k. n = m + k)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1133
  by (auto simp: le_add1 dest!: le_add_diff_inverse sym [of _ n])
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1134
52289
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
  1135
instance nat :: ordered_cancel_comm_monoid_diff
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
  1136
proof
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
  1137
  show "\<And>m n :: nat. m \<le> n \<longleftrightarrow> (\<exists>q. n = m + q)" by (fact le_iff_add)
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
  1138
qed
83ce5d2841e7 type class for confined subtraction
haftmann
parents: 51329
diff changeset
  1139
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1140
lemma less_imp_diff_less: "(j::nat) < k ==> j - n < k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1141
by (rule le_less_trans, rule diff_le_self)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1142
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1143
lemma diff_Suc_less [simp]: "0<n ==> n - Suc i < n"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1144
by (cases n) (auto simp add: le_simps)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1145
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1146
lemma diff_add_assoc: "k \<le> (j::nat) ==> (i + j) - k = i + (j - k)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1147
by (induct j k rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1148
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1149
lemma diff_add_assoc2: "k \<le> (j::nat) ==> (j + i) - k = (j - k) + i"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1150
by (simp add: add.commute diff_add_assoc)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1151
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1152
lemma le_imp_diff_is_add: "i \<le> (j::nat) ==> (j - i = k) = (j = k + i)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1153
by (auto simp add: diff_add_inverse2)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1154
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1155
lemma diff_is_0_eq [simp]: "((m::nat) - n = 0) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1156
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1157
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1158
lemma diff_is_0_eq' [simp]: "m \<le> n ==> (m::nat) - n = 0"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1159
by (rule iffD2, rule diff_is_0_eq)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1160
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1161
lemma zero_less_diff [simp]: "(0 < n - (m::nat)) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1162
by (induct m n rule: diff_induct) simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1163
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1164
lemma less_imp_add_positive:
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1165
  assumes "i < j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1166
  shows "\<exists>k::nat. 0 < k & i + k = j"
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1167
proof
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1168
  from assms show "0 < j - i & i + (j - i) = j"
23476
839db6346cc8 fix looping simp rule
huffman
parents: 23438
diff changeset
  1169
    by (simp add: order_less_imp_le)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1170
qed
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1171
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1172
text \<open>a nice rewrite for bounded subtraction\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1173
lemma nat_minus_add_max:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1174
  fixes n m :: nat
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1175
  shows "n - m + m = max n m"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1176
    by (simp add: max_def not_le order_less_imp_le)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1177
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1178
lemma nat_diff_split:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1179
  "P(a - b::nat) = ((a<b --> P 0) & (ALL d. a = b + d --> P d))"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1180
    \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1181
by (cases "a < b")
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1182
  (auto simp add: diff_is_0_eq [THEN iffD2] diff_add_inverse
57492
74bf65a1910a Hypsubst preserves equality hypotheses
Thomas Sewell <thomas.sewell@nicta.com.au>
parents: 57200
diff changeset
  1183
    not_less le_less dest!: add_eq_self_zero add_eq_self_zero[OF sym])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1184
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1185
lemma nat_diff_split_asm:
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1186
  "P(a - b::nat) = (~ (a < b & ~ P 0 | (EX d. a = b + d & ~ P d)))"
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1187
    \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1188
by (auto split: nat_diff_split)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1189
47255
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1190
lemma Suc_pred': "0 < n ==> n = Suc(n - 1)"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1191
  by simp
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1192
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1193
lemma add_eq_if: "(m::nat) + n = (if m=0 then n else Suc ((m - 1) + n))"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1194
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1195
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1196
lemma mult_eq_if: "(m::nat) * n = (if m=0 then 0 else n + ((m - 1) * n))"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1197
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1198
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1199
lemma Suc_diff_eq_diff_pred: "0 < n ==> Suc m - n = m - (n - 1)"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1200
  unfolding One_nat_def by (cases n) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1201
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1202
lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1203
  unfolding One_nat_def by (cases m) simp_all
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1204
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1205
lemma Let_Suc [simp]: "Let (Suc n) f == f (Suc n)"
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1206
  by (fact Let_def)
30a1692557b0 removed Nat_Numeral.thy, moving all theorems elsewhere
huffman
parents: 47208
diff changeset
  1207
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1208
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1209
subsubsection \<open>Monotonicity of multiplication\<close>
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1210
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1211
lemma mult_le_mono1: "i \<le> (j::nat) ==> i * k \<le> j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1212
by (simp add: mult_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1213
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1214
lemma mult_le_mono2: "i \<le> (j::nat) ==> k * i \<le> k * j"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1215
by (simp add: mult_left_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1216
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1217
text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1218
lemma mult_le_mono: "i \<le> (j::nat) ==> k \<le> l ==> i * k \<le> j * l"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1219
by (simp add: mult_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1220
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1221
lemma mult_less_mono1: "(i::nat) < j ==> 0 < k ==> i * k < j * k"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1222
by (simp add: mult_strict_right_mono)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1223
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1224
text\<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1225
      there are no negative numbers.\<close>
14266
08b34c902618 conversion of integers to use Ring_and_Field;
paulson
parents: 14265
diff changeset
  1226
lemma nat_0_less_mult_iff [simp]: "(0 < (m::nat) * n) = (0 < m & 0 < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1227
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1228
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1229
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1230
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1231
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1232
30079
293b896b9c25 make proofs work whether or not One_nat_def is a simp rule; replace 1 with Suc 0 in the rhs of some simp rules
huffman
parents: 30056
diff changeset
  1233
lemma one_le_mult_iff [simp]: "(Suc 0 \<le> m * n) = (Suc 0 \<le> m & Suc 0 \<le> n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1234
  apply (induct m)
22718
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1235
   apply simp
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1236
  apply (case_tac n)
936f7580937d tuned proofs;
wenzelm
parents: 22483
diff changeset
  1237
   apply simp_all
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1238
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1239
14341
a09441bd4f1e Ring_and_Field now requires axiom add_left_imp_eq for semirings.
paulson
parents: 14331
diff changeset
  1240
lemma mult_less_cancel2 [simp]: "((m::nat) * k < n * k) = (0 < k & m < n)"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1241
  apply (safe intro!: mult_less_mono1)
47988
e4b69e10b990 tuned proofs;
wenzelm
parents: 47489
diff changeset
  1242
  apply (cases k, auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1243
  apply (simp del: le_0_eq add: linorder_not_le [symmetric])
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1244
  apply (blast intro: mult_le_mono1)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1245
  done
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1246
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1247
lemma mult_less_cancel1 [simp]: "(k * (m::nat) < k * n) = (0 < k & m < n)"
57512
cc97b347b301 reduced name variants for assoc and commute on plus and mult
haftmann
parents: 57492
diff changeset
  1248
by (simp add: mult.commute [of k])
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1249
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1250
lemma mult_le_cancel1 [simp]: "(k * (m::nat) \<le> k * n) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1251
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1252
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1253
lemma mult_le_cancel2 [simp]: "((m::nat) * k \<le> n * k) = (0 < k --> m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1254
by (simp add: linorder_not_less [symmetric], auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1255
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1256
lemma Suc_mult_less_cancel1: "(Suc k * m < Suc k * n) = (m < n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1257
by (subst mult_less_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1258
14267
b963e9cee2a0 More refinements to Ring_and_Field and numerics. Conversion of Divides_lemmas
paulson
parents: 14266
diff changeset
  1259
lemma Suc_mult_le_cancel1: "(Suc k * m \<le> Suc k * n) = (m \<le> n)"
24438
2d8058804a76 Added infinite_descent
nipkow
parents: 24286
diff changeset
  1260
by (subst mult_le_cancel1) simp
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1261
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1262
lemma le_square: "m \<le> m * (m::nat)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1263
  by (cases m) (auto intro: le_add1)
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1264
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1265
lemma le_cube: "(m::nat) \<le> m * (m * m)"
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1266
  by (cases m) (auto intro: le_add1)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1267
61799
4cf66f21b764 isabelle update_cartouches -c -t;
wenzelm
parents: 61649
diff changeset
  1268
text \<open>Lemma for \<open>gcd\<close>\<close>
30128
365ee7319b86 revert some Suc 0 lemmas back to their original forms; added some simp rules for (1::nat)
huffman
parents: 30093
diff changeset
  1269
lemma mult_eq_self_implies_10: "(m::nat) = m * n ==> n = 1 | m = 0"
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1270
  apply (drule sym)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1271
  apply (rule disjCI)
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1272
  apply (rule nat_less_cases, erule_tac [2] _)
25157
8b80535cd017 random tidying of proofs
paulson
parents: 25145
diff changeset
  1273
   apply (drule_tac [2] mult_less_mono2)
25162
ad4d5365d9d8 went back to >0
nipkow
parents: 25157
diff changeset
  1274
    apply (auto)
13449
43c9ec498291 - Converted to new theory format
berghofe
parents: 12338
diff changeset
  1275
  done
9436
62bb04ab4b01 rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents: 7702
diff changeset
  1276
51263
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1277
lemma mono_times_nat:
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1278
  fixes n :: nat
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1279
  assumes "n > 0"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1280
  shows "mono (times n)"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1281
proof
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1282
  fix m q :: nat
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1283
  assume "m \<le> q"
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1284
  with assms show "n * m \<le> n * q" by simp
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1285
qed
31e786e0e6a7 turned example into library for comparing growth of functions
haftmann
parents: 51173
diff changeset
  1286
60758
d8d85a8172b5 isabelle update_cartouches;
wenzelm
parents: 60636
diff changeset
  1287
text \<open>the lattice order on @{typ nat}\<close>
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1288
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1289
instantiation nat :: distrib_lattice
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1290
begin
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1291
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1292
definition
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1293
  "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1294
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1295
definition
61076
bdc1e2f0a86a eliminated \<Colon>;
wenzelm
parents: 61070
diff changeset
  1296
  "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"
24995
c26e0166e568 refined; moved class power to theory Power
haftmann
parents: 24729
diff changeset
  1297
26072
f65a7fa2da6c <= and < on nat no longer depend on wellfounded relations
haftmann
parents: 25928
diff changeset
  1298
instance by intro_classes