(*  Title:      HOL/Nat.thy
    Author:     Tobias Nipkow
    Author:     Lawrence C Paulson
    Author:     Markus Wenzel

Type "nat" is a linear order, and a datatype; arithmetic operators + -
and * (for div and mod, see theory Divides).
*)

section \<open>Natural numbers\<close>

theory Nat
  imports Inductive Typedef Fun Rings
begin

named_theorems arith "arith facts -- only ground formulas"
ML_file "Tools/arith_data.ML"

subsection \<open>Type \<open>ind\<close>\<close>

typedecl ind

axiomatization Zero_Rep :: ind and Suc_Rep :: "ind \<Rightarrow> ind"
  \<comment> \<open>The axiom of infinity in 2 parts:\<close>
  where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
    and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"

subsection \<open>Type nat\<close>

text \<open>Type definition\<close>

inductive Nat :: "ind \<Rightarrow> bool"
  where
    Zero_RepI: "Nat Zero_Rep"
  | Suc_RepI: "Nat i \<Longrightarrow> Nat (Suc_Rep i)"

typedef nat = "{n. Nat n}"
  morphisms Rep_Nat Abs_Nat
  using Nat.Zero_RepI by auto

lemma Nat_Rep_Nat: "Nat (Rep_Nat n)"
  using Rep_Nat by simp

lemma Nat_Abs_Nat_inverse: "Nat n \<Longrightarrow> Rep_Nat (Abs_Nat n) = n"
  using Abs_Nat_inverse by simp

lemma Nat_Abs_Nat_inject: "Nat n \<Longrightarrow> Nat m \<Longrightarrow> Abs_Nat n = Abs_Nat m \<longleftrightarrow> n = m"
  using Abs_Nat_inject by simp

instantiation nat :: zero
begin

definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"

instance ..

end

definition Suc :: "nat \<Rightarrow> nat"
  where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"

lemma Suc_not_Zero: "Suc m \<noteq> 0"
  by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI
      Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)

lemma Zero_not_Suc: "0 \<noteq> Suc m"
  by (rule not_sym) (rule Suc_not_Zero)

lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y \<longleftrightarrow> x = y"
  by (rule iffI, rule Suc_Rep_inject) simp_all

lemma nat_induct0:
  assumes "P 0"
    and "\<And>n. P n \<Longrightarrow> P (Suc n)"
  shows "P n"
  using assms
  apply (unfold Zero_nat_def Suc_def)
  apply (rule Rep_Nat_inverse [THEN subst]) \<comment> \<open>types force good instantiation\<close>
  apply (erule Nat_Rep_Nat [THEN Nat.induct])
  apply (iprover elim: Nat_Abs_Nat_inverse [THEN subst])
  done

free_constructors case_nat for "0 :: nat" | Suc pred
  where "pred (0 :: nat) = (0 :: nat)"
    apply atomize_elim
    apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
   apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
  apply (simp only: Suc_not_Zero)
  done

\<comment> \<open>Avoid name clashes by prefixing the output of \<open>old_rep_datatype\<close> with \<open>old\<close>.\<close>
setup \<open>Sign.mandatory_path "old"\<close>

old_rep_datatype "0 :: nat" Suc
    apply (erule nat_induct0)
    apply assumption
   apply (rule nat.inject)
  apply (rule nat.distinct(1))
  done

setup \<open>Sign.parent_path\<close>

\<comment> \<open>But erase the prefix for properties that are not generated by \<open>free_constructors\<close>.\<close>
setup \<open>Sign.mandatory_path "nat"\<close>

declare old.nat.inject[iff del]
  and old.nat.distinct(1)[simp del, induct_simp del]

lemmas induct = old.nat.induct
lemmas inducts = old.nat.inducts
lemmas rec = old.nat.rec
lemmas simps = nat.inject nat.distinct nat.case nat.rec

setup \<open>Sign.parent_path\<close>

abbreviation rec_nat :: "'a \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
  where "rec_nat \<equiv> old.rec_nat"

declare nat.sel[code del]

hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>
hide_fact
  nat.case_eq_if
  nat.collapse
  nat.expand
  nat.sel
  nat.exhaust_sel
  nat.split_sel
  nat.split_sel_asm

lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
  "(y = 0 \<Longrightarrow> P) \<Longrightarrow> (\<And>nat. y = Suc nat \<Longrightarrow> P) \<Longrightarrow> P"
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
  by (rule old.nat.exhaust)

lemma nat_induct [case_names 0 Suc, induct type: nat]:
  fixes n
  assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
  shows "P n"
  \<comment> \<open>for backward compatibility -- names of variables differ\<close>
  using assms by (rule nat.induct)

hide_fact
  nat_exhaust
  nat_induct0

ML \<open>
val nat_basic_lfp_sugar =
  let
    val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global @{theory} @{type_name nat});
    val recx = Logic.varify_types_global @{term rec_nat};
    val C = body_type (fastype_of recx);
  in
    {T = HOLogic.natT, fp_res_index = 0, C = C, fun_arg_Tsss = [[], [[HOLogic.natT, C]]],
     ctr_sugar = ctr_sugar, recx = recx, rec_thms = @{thms nat.rec}}
  end;
\<close>

setup \<open>
let
  fun basic_lfp_sugars_of _ [@{typ nat}] _ _ ctxt =
      ([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*), [], false, ctxt)
    | basic_lfp_sugars_of bs arg_Ts callers callssss ctxt =
      BNF_LFP_Rec_Sugar.default_basic_lfp_sugars_of bs arg_Ts callers callssss ctxt;
in
  BNF_LFP_Rec_Sugar.register_lfp_rec_extension
    {nested_simps = [], is_new_datatype = K (K true), basic_lfp_sugars_of = basic_lfp_sugars_of,
     rewrite_nested_rec_call = NONE}
end
\<close>

text \<open>Injectiveness and distinctness lemmas\<close>

lemma inj_Suc[simp]: "inj_on Suc N"
  by (simp add: inj_on_def)

lemma Suc_neq_Zero: "Suc m = 0 \<Longrightarrow> R"
  by (rule notE) (rule Suc_not_Zero)

lemma Zero_neq_Suc: "0 = Suc m \<Longrightarrow> R"
  by (rule Suc_neq_Zero) (erule sym)

lemma Suc_inject: "Suc x = Suc y \<Longrightarrow> x = y"
  by (rule inj_Suc [THEN injD])

lemma n_not_Suc_n: "n \<noteq> Suc n"
  by (induct n) simp_all

lemma Suc_n_not_n: "Suc n \<noteq> n"
  by (rule not_sym) (rule n_not_Suc_n)

text \<open>A special form of induction for reasoning about @{term "m < n"} and @{term "m - n"}.\<close>
lemma diff_induct:
  assumes "\<And>x. P x 0"
    and "\<And>y. P 0 (Suc y)"
    and "\<And>x y. P x y \<Longrightarrow> P (Suc x) (Suc y)"
  shows "P m n"
proof (induct n arbitrary: m)
  case 0
  show ?case by (rule assms(1))
next
  case (Suc n)
  show ?case
  proof (induct m)
    case 0
    show ?case by (rule assms(2))
  next
    case (Suc m)
    from \<open>P m n\<close> show ?case by (rule assms(3))
  qed
qed

subsection \<open>Arithmetic operators\<close>

instantiation nat :: comm_monoid_diff
begin

primrec plus_nat
  where
    add_0: "0 + n = (n::nat)"
  | add_Suc: "Suc m + n = Suc (m + n)"

lemma add_0_right [simp]: "m + 0 = m"
  for m :: nat
  by (induct m) simp_all

lemma add_Suc_right [simp]: "m + Suc n = Suc (m + n)"
  by (induct m) simp_all

declare add_0 [code]

lemma add_Suc_shift [code]: "Suc m + n = m + Suc n"
  by simp

primrec minus_nat
  where
    diff_0 [code]: "m - 0 = (m::nat)"
  | diff_Suc: "m - Suc n = (case m - n of 0 \<Rightarrow> 0 | Suc k \<Rightarrow> k)"

declare diff_Suc [simp del]

lemma diff_0_eq_0 [simp, code]: "0 - n = 0"
  for n :: nat
  by (induct n) (simp_all add: diff_Suc)

lemma diff_Suc_Suc [simp, code]: "Suc m - Suc n = m - n"
  by (induct n) (simp_all add: diff_Suc)

instance
proof
  fix n m q :: nat
  show "(n + m) + q = n + (m + q)" by (induct n) simp_all
  show "n + m = m + n" by (induct n) simp_all
  show "m + n - m = n" by (induct m) simp_all
  show "n - m - q = n - (m + q)" by (induct q) (simp_all add: diff_Suc)
  show "0 + n = n" by simp
  show "0 - n = 0" by simp
qed

end

hide_fact (open) add_0 add_0_right diff_0

instantiation nat :: comm_semiring_1_cancel
begin

definition One_nat_def [simp]: "1 = Suc 0"

primrec times_nat
  where
    mult_0: "0 * n = (0::nat)"
  | mult_Suc: "Suc m * n = n + (m * n)"

lemma mult_0_right [simp]: "m * 0 = 0"
  for m :: nat
  by (induct m) simp_all

lemma mult_Suc_right [simp]: "m * Suc n = m + (m * n)"
  by (induct m) (simp_all add: add.left_commute)

lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
  for m n k :: nat
  by (induct m) (simp_all add: add.assoc)

instance
proof
  fix k n m q :: nat
  show "0 \<noteq> (1::nat)"
    by simp
  show "1 * n = n"
    by simp
  show "n * m = m * n"
    by (induct n) simp_all
  show "(n * m) * q = n * (m * q)"
    by (induct n) (simp_all add: add_mult_distrib)
  show "(n + m) * q = n * q + m * q"
    by (rule add_mult_distrib)
  show "k * (m - n) = (k * m) - (k * n)"
    by (induct m n rule: diff_induct) simp_all
qed

end

subsubsection \<open>Addition\<close>

text \<open>Reasoning about \<open>m + 0 = 0\<close>, etc.\<close>

lemma add_is_0 [iff]: "m + n = 0 \<longleftrightarrow> m = 0 \<and> n = 0"
  for m n :: nat
  by (cases m) simp_all

lemma add_is_1: "m + n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
  by (cases m) simp_all

lemma one_is_add: "Suc 0 = m + n \<longleftrightarrow> m = Suc 0 \<and> n = 0 | m = 0 \<and> n = Suc 0"
  by (rule trans, rule eq_commute, rule add_is_1)

lemma add_eq_self_zero: "m + n = m \<Longrightarrow> n = 0"
  for m n :: nat
  by (induct m) simp_all

lemma inj_on_add_nat [simp]: "inj_on (\<lambda>n. n + k) N"
  for k :: nat
proof (induct k)
  case 0
  then show ?case by simp
next
  case (Suc k)
  show ?case
    using comp_inj_on [OF Suc inj_Suc] by (simp add: o_def)
qed

lemma Suc_eq_plus1: "Suc n = n + 1"
  by simp

lemma Suc_eq_plus1_left: "Suc n = 1 + n"
  by simp

subsubsection \<open>Difference\<close>

lemma Suc_diff_diff [simp]: "(Suc m - n) - Suc k = m - n - k"
  by (simp add: diff_diff_add)

lemma diff_Suc_1 [simp]: "Suc n - 1 = n"
  by simp

subsubsection \<open>Multiplication\<close>

lemma mult_is_0 [simp]: "m * n = 0 \<longleftrightarrow> m = 0 \<or> n = 0" for m n :: nat
  by (induct m) auto

lemma mult_eq_1_iff [simp]: "m * n = Suc 0 \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
proof (induct m)
  case 0
  then show ?case by simp
next
  case (Suc m)
  then show ?case by (induct n) auto
qed

lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
  apply (rule trans)
   apply (rule_tac [2] mult_eq_1_iff)
  apply fastforce
  done

lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
  for m n :: nat
  unfolding One_nat_def by (rule mult_eq_1_iff)

lemma nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1"
  for m n :: nat
  unfolding One_nat_def by (rule one_eq_mult_iff)

lemma mult_cancel1 [simp]: "k * m = k * n \<longleftrightarrow> m = n \<or> k = 0"
  for k m n :: nat
proof -
  have "k \<noteq> 0 \<Longrightarrow> k * m = k * n \<Longrightarrow> m = n"
  proof (induct n arbitrary: m)
    case 0
    then show "m = 0" by simp
  next
    case (Suc n)
    then show "m = Suc n"
      by (cases m) (simp_all add: eq_commute [of 0])
  qed
  then show ?thesis by auto
qed

lemma mult_cancel2 [simp]: "m * k = n * k \<longleftrightarrow> m = n \<or> k = 0"
  for k m n :: nat
  by (simp add: mult.commute)

lemma Suc_mult_cancel1: "Suc k * m = Suc k * n \<longleftrightarrow> m = n"
  by (subst mult_cancel1) simp

subsection \<open>Orders on @{typ nat}\<close>

subsubsection \<open>Operation definition\<close>

instantiation nat :: linorder
begin

primrec less_eq_nat
  where
    "(0::nat) \<le> n \<longleftrightarrow> True"
  | "Suc m \<le> n \<longleftrightarrow> (case n of 0 \<Rightarrow> False | Suc n \<Rightarrow> m \<le> n)"

declare less_eq_nat.simps [simp del]

lemma le0 [iff]: "0 \<le> n" for
  n :: nat
  by (simp add: less_eq_nat.simps)

definition less_nat
  where less_eq_Suc_le: "n < m \<longleftrightarrow> Suc n \<le> m"

lemma Suc_le_mono [iff]: "Suc n \<le> Suc m \<longleftrightarrow> n \<le> m"
  by (simp add: less_eq_nat.simps(2))

lemma Suc_le_eq [code]: "Suc m \<le> n \<longleftrightarrow> m < n"
  unfolding less_eq_Suc_le ..

lemma le_0_eq [iff]: "n \<le> 0 \<longleftrightarrow> n = 0"
  for n :: nat
  by (induct n) (simp_all add: less_eq_nat.simps(2))

lemma not_less0 [iff]: "\<not> n < 0"
  for n :: nat
  by (simp add: less_eq_Suc_le)

lemma less_nat_zero_code [code]: "n < 0 \<longleftrightarrow> False"
  for n :: nat
  by simp

lemma Suc_less_eq [iff]: "Suc m < Suc n \<longleftrightarrow> m < n"
  by (simp add: less_eq_Suc_le)

lemma less_Suc_eq_le [code]: "m < Suc n \<longleftrightarrow> m \<le> n"
  by (simp add: less_eq_Suc_le)

lemma Suc_less_eq2: "Suc n < m \<longleftrightarrow> (\<exists>m'. m = Suc m' \<and> n < m')"
  by (cases m) auto

lemma le_SucI: "m \<le> n \<Longrightarrow> m \<le> Suc n"
  by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits)

lemma Suc_leD: "Suc m \<le> n \<Longrightarrow> m \<le> n"
  by (cases n) (auto intro: le_SucI)

lemma less_SucI: "m < n \<Longrightarrow> m < Suc n"
  by (simp add: less_eq_Suc_le) (erule Suc_leD)

lemma Suc_lessD: "Suc m < n \<Longrightarrow> m < n"
  by (simp add: less_eq_Suc_le) (erule Suc_leD)

instance
proof
  fix n m q :: nat
  show "n < m \<longleftrightarrow> n \<le> m \<and> \<not> m \<le> n"
  proof (induct n arbitrary: m)
    case 0
    then show ?case
      by (cases m) (simp_all add: less_eq_Suc_le)
  next
    case (Suc n)
    then show ?case
      by (cases m) (simp_all add: less_eq_Suc_le)
  qed
  show "n \<le> n"
    by (induct n) simp_all
  then show "n = m" if "n \<le> m" and "m \<le> n"
    using that by (induct n arbitrary: m)
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
  show "n \<le> q" if "n \<le> m" and "m \<le> q"
    using that
  proof (induct n arbitrary: m q)
    case 0
    show ?case by simp
  next
    case (Suc n)
    then show ?case
      by (simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits, clarify,
        simp_all (no_asm_use) add: less_eq_nat.simps(2) split: nat.splits)
  qed
  show "n \<le> m \<or> m \<le> n"
    by (induct n arbitrary: m)
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
qed

end

instantiation nat :: order_bot
begin

definition bot_nat :: nat
  where "bot_nat = 0"

instance
  by standard (simp add: bot_nat_def)

end

instance nat :: no_top
  by standard (auto intro: less_Suc_eq_le [THEN iffD2])

subsubsection \<open>Introduction properties\<close>

lemma lessI [iff]: "n < Suc n"
  by (simp add: less_Suc_eq_le)

lemma zero_less_Suc [iff]: "0 < Suc n"
  by (simp add: less_Suc_eq_le)

subsubsection \<open>Elimination properties\<close>

lemma less_not_refl: "\<not> n < n"
  for n :: nat
  by (rule order_less_irrefl)

lemma less_not_refl2: "n < m \<Longrightarrow> m \<noteq> n"
  for m n :: nat
  by (rule not_sym) (rule less_imp_neq)

lemma less_not_refl3: "s < t \<Longrightarrow> s \<noteq> t"
  for s t :: nat
  by (rule less_imp_neq)

lemma less_irrefl_nat: "n < n \<Longrightarrow> R"
  for n :: nat
  by (rule notE, rule less_not_refl)

lemma less_zeroE: "n < 0 \<Longrightarrow> R"
  for n :: nat
  by (rule notE) (rule not_less0)

lemma less_Suc_eq: "m < Suc n \<longleftrightarrow> m < n \<or> m = n"
  unfolding less_Suc_eq_le le_less ..

lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"
  by (simp add: less_Suc_eq)

lemma less_one [iff]: "n < 1 \<longleftrightarrow> n = 0"
  for n :: nat
  unfolding One_nat_def by (rule less_Suc0)

lemma Suc_mono: "m < n \<Longrightarrow> Suc m < Suc n"
  by simp

text \<open>"Less than" is antisymmetric, sort of.\<close>
lemma less_antisym: "\<not> n < m \<Longrightarrow> n < Suc m \<Longrightarrow> m = n"
  unfolding not_less less_Suc_eq_le by (rule antisym)

lemma nat_neq_iff: "m \<noteq> n \<longleftrightarrow> m < n \<or> n < m"
  for m n :: nat
  by (rule linorder_neq_iff)

subsubsection \<open>Inductive (?) properties\<close>

lemma Suc_lessI: "m < n \<Longrightarrow> Suc m \<noteq> n \<Longrightarrow> Suc m < n"
  unfolding less_eq_Suc_le [of m] le_less by simp

lemma lessE:
  assumes major: "i < k"
    and 1: "k = Suc i \<Longrightarrow> P"
    and 2: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
  shows P
proof -
  from major have "\<exists>j. i \<le> j \<and> k = Suc j"
    unfolding less_eq_Suc_le by (induct k) simp_all
  then have "(\<exists>j. i < j \<and> k = Suc j) \<or> k = Suc i"
    by (auto simp add: less_le)
  with 1 2 show P by auto
qed

lemma less_SucE:
  assumes major: "m < Suc n"
    and less: "m < n \<Longrightarrow> P"
    and eq: "m = n \<Longrightarrow> P"
  shows P
  apply (rule major [THEN lessE])
   apply (rule eq)
   apply blast
  apply (rule less)
  apply blast
  done

lemma Suc_lessE:
  assumes major: "Suc i < k"
    and minor: "\<And>j. i < j \<Longrightarrow> k = Suc j \<Longrightarrow> P"
  shows P
  apply (rule major [THEN lessE])
   apply (erule lessI [THEN minor])
  apply (erule Suc_lessD [THEN minor])
  apply assumption
  done

lemma Suc_less_SucD: "Suc m < Suc n \<Longrightarrow> m < n"
  by simp

lemma less_trans_Suc:
  assumes le: "i < j"
  shows "j < k \<Longrightarrow> Suc i < k"
proof (induct k)
  case 0
  then show ?case by simp
next
  case (Suc k)
  with le show ?case
    by simp (auto simp add: less_Suc_eq dest: Suc_lessD)
qed

text \<open>Can be used with \<open>less_Suc_eq\<close> to get @{prop "n = m \<or> n < m"}.\<close>
lemma not_less_eq: "\<not> m < n \<longleftrightarrow> n < Suc m"
  by (simp only: not_less less_Suc_eq_le)

lemma not_less_eq_eq: "\<not> m \<le> n \<longleftrightarrow> Suc n \<le> m"
  by (simp only: not_le Suc_le_eq)

text \<open>Properties of "less than or equal".\<close>

lemma le_imp_less_Suc: "m \<le> n \<Longrightarrow> m < Suc n"
  by (simp only: less_Suc_eq_le)

lemma Suc_n_not_le_n: "\<not> Suc n \<le> n"
  by (simp add: not_le less_Suc_eq_le)

lemma le_Suc_eq: "m \<le> Suc n \<longleftrightarrow> m \<le> n \<or> m = Suc n"
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)

lemma le_SucE: "m \<le> Suc n \<Longrightarrow> (m \<le> n \<Longrightarrow> R) \<Longrightarrow> (m = Suc n \<Longrightarrow> R) \<Longrightarrow> R"
  by (drule le_Suc_eq [THEN iffD1], iprover+)

lemma Suc_leI: "m < n \<Longrightarrow> Suc m \<le> n"
  by (simp only: Suc_le_eq)

text \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
lemma Suc_le_lessD: "Suc m \<le> n \<Longrightarrow> m < n"
  by (simp only: Suc_le_eq)

lemma less_imp_le_nat: "m < n \<Longrightarrow> m \<le> n" for m n :: nat
  unfolding less_eq_Suc_le by (rule Suc_leD)

text \<open>For instance, \<open>(Suc m < Suc n) = (Suc m \<le> n) = (m < n)\<close>\<close>
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq

text \<open>Equivalence of \<open>m \<le> n\<close> and \<open>m < n \<or> m = n\<close>\<close>

lemma less_or_eq_imp_le: "m < n \<or> m = n \<Longrightarrow> m \<le> n"
  for m n :: nat
  unfolding le_less .

lemma le_eq_less_or_eq: "m \<le> n \<longleftrightarrow> m < n \<or> m = n"
  for m n :: nat
  by (rule le_less)

text \<open>Useful with \<open>blast\<close>.\<close>
lemma eq_imp_le: "m = n \<Longrightarrow> m \<le> n"
  for m n :: nat
  by auto

lemma le_refl: "n \<le> n"
  for n :: nat
  by simp

lemma le_trans: "i \<le> j \<Longrightarrow> j \<le> k \<Longrightarrow> i \<le> k"
  for i j k :: nat
  by (rule order_trans)

lemma le_antisym: "m \<le> n \<Longrightarrow> n \<le> m \<Longrightarrow> m = n"
  for m n :: nat
  by (rule antisym)

lemma nat_less_le: "m < n \<longleftrightarrow> m \<le> n \<and> m \<noteq> n"
  for m n :: nat
  by (rule less_le)

lemma le_neq_implies_less: "m \<le> n \<Longrightarrow> m \<noteq> n \<Longrightarrow> m < n"
  for m n :: nat
  unfolding less_le ..

lemma nat_le_linear: "m \<le> n | n \<le> m"
  for m n :: nat
  by (rule linear)

lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

lemma le_less_Suc_eq: "m \<le> n \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
  unfolding less_Suc_eq_le by auto

lemma not_less_less_Suc_eq: "\<not> n < m \<Longrightarrow> n < Suc m \<longleftrightarrow> n = m"
  unfolding not_less by (rule le_less_Suc_eq)

lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

lemma not0_implies_Suc: "n \<noteq> 0 \<Longrightarrow> \<exists>m. n = Suc m"
  by (cases n) simp_all

lemma gr0_implies_Suc: "n > 0 \<Longrightarrow> \<exists>m. n = Suc m"
  by (cases n) simp_all

lemma gr_implies_not0: "m < n \<Longrightarrow> n \<noteq> 0"
  for m n :: nat
  by (cases n) simp_all

lemma neq0_conv[iff]: "n \<noteq> 0 \<longleftrightarrow> 0 < n"
  for n :: nat
  by (cases n) simp_all

text \<open>This theorem is useful with \<open>blast\<close>\<close>
lemma gr0I: "(n = 0 \<Longrightarrow> False) \<Longrightarrow> 0 < n"
  for n :: nat
  by (rule neq0_conv[THEN iffD1]) iprover

lemma gr0_conv_Suc: "0 < n \<longleftrightarrow> (\<exists>m. n = Suc m)"
  by (fast intro: not0_implies_Suc)

lemma not_gr0 [iff]: "\<not> 0 < n \<longleftrightarrow> n = 0"
  for n :: nat
  using neq0_conv by blast

lemma Suc_le_D: "Suc n \<le> m' \<Longrightarrow> \<exists>m. m' = Suc m"
  by (induct m') simp_all

text \<open>Useful in certain inductive arguments\<close>
lemma less_Suc_eq_0_disj: "m < Suc n \<longleftrightarrow> m = 0 \<or> (\<exists>j. m = Suc j \<and> j < n)"
  by (cases m) simp_all

subsubsection \<open>Monotonicity of Addition\<close>

lemma Suc_pred [simp]: "n > 0 \<Longrightarrow> Suc (n - Suc 0) = n"
  by (simp add: diff_Suc split: nat.split)

lemma Suc_diff_1 [simp]: "0 < n \<Longrightarrow> Suc (n - 1) = n"
  unfolding One_nat_def by (rule Suc_pred)

lemma nat_add_left_cancel_le [simp]: "k + m \<le> k + n \<longleftrightarrow> m \<le> n"
  for k m n :: nat
  by (induct k) simp_all

lemma nat_add_left_cancel_less [simp]: "k + m < k + n \<longleftrightarrow> m < n"
  for k m n :: nat
  by (induct k) simp_all

lemma add_gr_0 [iff]: "m + n > 0 \<longleftrightarrow> m > 0 \<or> n > 0"
  for m n :: nat
  by (auto dest: gr0_implies_Suc)

text \<open>strict, in 1st argument\<close>
lemma add_less_mono1: "i < j \<Longrightarrow> i + k < j + k"
  for i j k :: nat
  by (induct k) simp_all

text \<open>strict, in both arguments\<close>
lemma add_less_mono: "i < j \<Longrightarrow> k < l \<Longrightarrow> i + k < j + l"
  for i j k l :: nat
  apply (rule add_less_mono1 [THEN less_trans], assumption+)
  apply (induct j)
   apply simp_all
  done

text \<open>Deleted \<open>less_natE\<close>; use \<open>less_imp_Suc_add RS exE\<close>\<close>
lemma less_imp_Suc_add: "m < n \<Longrightarrow> \<exists>k. n = Suc (m + k)"
proof (induct n)
  case 0
  then show ?case by simp
next
  case Suc
  then show ?case
    by (simp add: order_le_less)
      (blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
qed

lemma le_Suc_ex: "k \<le> l \<Longrightarrow> (\<exists>n. l = k + n)"
  for k l :: nat
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)

text \<open>strict, in 1st argument; proof is by induction on \<open>k > 0\<close>\<close>
lemma mult_less_mono2:
  fixes i j :: nat
  assumes "i < j" and "0 < k"
  shows "k * i < k * j"
  using \<open>0 < k\<close>
proof (induct k)
  case 0
  then show ?case by simp
next
  case (Suc k)
  with \<open>i < j\<close> show ?case
    by (cases k) (simp_all add: add_less_mono)
qed

text \<open>Addition is the inverse of subtraction:
  if @{term "n \<le> m"} then @{term "n + (m - n) = m"}.\<close>
lemma add_diff_inverse_nat: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all

lemma nat_le_iff_add: "m \<le> n \<longleftrightarrow> (\<exists>k. n = m + k)"
  for m n :: nat
  using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)

text \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>

instance nat :: linordered_semidom
proof
  fix m n q :: nat
  show "0 < (1::nat)"
    by simp
  show "m \<le> n \<Longrightarrow> q + m \<le> q + n"
    by simp
  show "m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n"
    by (simp add: mult_less_mono2)
  show "m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0"
    by simp
  show "n \<le> m \<Longrightarrow> (m - n) + n = m"
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
qed

instance nat :: dioid
  by standard (rule nat_le_iff_add)

declare le0[simp del] \<comment> \<open>This is now @{thm zero_le}\<close>
declare le_0_eq[simp del] \<comment> \<open>This is now @{thm le_zero_eq}\<close>
declare not_less0[simp del] \<comment> \<open>This is now @{thm not_less_zero}\<close>
declare not_gr0[simp del] \<comment> \<open>This is now @{thm not_gr_zero}\<close>

instance nat :: ordered_cancel_comm_monoid_add ..
instance nat :: ordered_cancel_comm_monoid_diff ..

subsubsection \<open>@{term min} and @{term max}\<close>

lemma mono_Suc: "mono Suc"
  by (rule monoI) simp

lemma min_0L [simp]: "min 0 n = 0"
  for n :: nat
  by (rule min_absorb1) simp

lemma min_0R [simp]: "min n 0 = 0"
  for n :: nat
  by (rule min_absorb2) simp

lemma min_Suc_Suc [simp]: "min (Suc m) (Suc n) = Suc (min m n)"
  by (simp add: mono_Suc min_of_mono)

lemma min_Suc1: "min (Suc n) m = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min n m'))"
  by (simp split: nat.split)

lemma min_Suc2: "min m (Suc n) = (case m of 0 \<Rightarrow> 0 | Suc m' \<Rightarrow> Suc(min m' n))"
  by (simp split: nat.split)

lemma max_0L [simp]: "max 0 n = n"
  for n :: nat
  by (rule max_absorb2) simp

lemma max_0R [simp]: "max n 0 = n"
  for n :: nat
  by (rule max_absorb1) simp

lemma max_Suc_Suc [simp]: "max (Suc m) (Suc n) = Suc (max m n)"
  by (simp add: mono_Suc max_of_mono)

lemma max_Suc1: "max (Suc n) m = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max n m'))"
  by (simp split: nat.split)

lemma max_Suc2: "max m (Suc n) = (case m of 0 \<Rightarrow> Suc n | Suc m' \<Rightarrow> Suc (max m' n))"
  by (simp split: nat.split)

lemma nat_mult_min_left: "min m n * q = min (m * q) (n * q)"
  for m n q :: nat
  by (simp add: min_def not_le)
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

lemma nat_mult_min_right: "m * min n q = min (m * n) (m * q)"
  for m n q :: nat
  by (simp add: min_def not_le)
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

lemma nat_add_max_left: "max m n + q = max (m + q) (n + q)"
  for m n q :: nat
  by (simp add: max_def)

lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
  for m n q :: nat
  by (simp add: max_def)

lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
  for m n q :: nat
  by (simp add: max_def not_le)
    (auto dest: mult_right_le_imp_le mult_right_less_imp_less le_less_trans)

lemma nat_mult_max_right: "m * max n q = max (m * n) (m * q)"
  for m n q :: nat
  by (simp add: max_def not_le)
    (auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

subsubsection \<open>Additional theorems about @{term "op \<le>"}\<close>

text \<open>Complete induction, aka course-of-values induction\<close>

instance nat :: wellorder
proof
  fix P and n :: nat
  assume step: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat
  have "\<And>q. q \<le> n \<Longrightarrow> P q"
  proof (induct n)
    case (0 n)
    have "P 0" by (rule step) auto
    with 0 show ?case by auto
  next
    case (Suc m n)
    then have "n \<le> m \<or> n = Suc m"
      by (simp add: le_Suc_eq)
    then show ?case
    proof
      assume "n \<le> m"
      then show "P n" by (rule Suc(1))
    next
      assume n: "n = Suc m"
      show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
    qed
  qed
  then show "P n" by auto
qed

lemma Least_eq_0[simp]: "P 0 \<Longrightarrow> Least P = 0"
  for P :: "nat \<Rightarrow> bool"
  by (rule Least_equality[OF _ le0])

lemma Least_Suc: "P n \<Longrightarrow> \<not> P 0 \<Longrightarrow> (LEAST n. P n) = Suc (LEAST m. P (Suc m))"
  apply (cases n)
   apply auto
  apply (frule LeastI)
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in LeastI)
  apply (subgoal_tac " (LEAST x. P x) \<le> Suc (LEAST x. P (Suc x))")
   apply (erule_tac [2] Least_le)
  apply (cases "LEAST x. P x")
   apply auto
  apply (drule_tac P = "\<lambda>x. P (Suc x)" in Least_le)
  apply (blast intro: order_antisym)
  done

lemma Least_Suc2: "P n \<Longrightarrow> Q m \<Longrightarrow> \<not> P 0 \<Longrightarrow> \<forall>k. P (Suc k) = Q k \<Longrightarrow> Least P = Suc (Least Q)"
  by (erule (1) Least_Suc [THEN ssubst]) simp

lemma ex_least_nat_le: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> P k"
  for P :: "nat \<Rightarrow> bool"
  apply (cases n)
   apply blast
  apply (rule_tac x="LEAST k. P k" in exI)
  apply (blast intro: Least_le dest: not_less_Least intro: LeastI_ex)
  done

lemma ex_least_nat_less: "\<not> P 0 \<Longrightarrow> P n \<Longrightarrow> \<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (k + 1)"
  for P :: "nat \<Rightarrow> bool"
  apply (cases n)
   apply blast
  apply (frule (1) ex_least_nat_le)
  apply (erule exE)
  apply (case_tac k)
   apply simp
  apply (rename_tac k1)
  apply (rule_tac x=k1 in exI)
  apply (auto simp add: less_eq_Suc_le)
  done

lemma nat_less_induct:
  fixes P :: "nat \<Rightarrow> bool"
  assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
  shows "P n"
  using assms less_induct by blast

lemma measure_induct_rule [case_names less]:
  fixes f :: "'a \<Rightarrow> nat"
  assumes step: "\<And>x. (\<And>y. f y < f x \<Longrightarrow> P y) \<Longrightarrow> P x"
  shows "P a"
  by (induct m \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step)

text \<open>old style induction rules:\<close>
lemma measure_induct:
  fixes f :: "'a \<Rightarrow> nat"
  shows "(\<And>x. \<forall>y. f y < f x \<longrightarrow> P y \<Longrightarrow> P x) \<Longrightarrow> P a"
  by (rule measure_induct_rule [of f P a]) iprover

lemma full_nat_induct:
  assumes step: "\<And>n. (\<forall>m. Suc m \<le> n \<longrightarrow> P m) \<Longrightarrow> P n"
  shows "P n"
  by (rule less_induct) (auto intro: step simp:le_simps)

text\<open>An induction rule for establishing binary relations\<close>
lemma less_Suc_induct [consumes 1]:
  assumes less: "i < j"
    and step: "\<And>i. P i (Suc i)"
    and trans: "\<And>i j k. i < j \<Longrightarrow> j < k \<Longrightarrow> P i j \<Longrightarrow> P j k \<Longrightarrow> P i k"
  shows "P i j"
proof -
  from less obtain k where j: "j = Suc (i + k)"
    by (auto dest: less_imp_Suc_add)
  have "P i (Suc (i + k))"
  proof (induct k)
    case 0
    show ?case by (simp add: step)
  next
    case (Suc k)
    have "0 + i < Suc k + i" by (rule add_less_mono1) simp
    then have "i < Suc (i + k)" by (simp add: add.commute)
    from trans[OF this lessI Suc step]
    show ?case by simp
  qed
  then show "P i j" by (simp add: j)
qed

text \<open>
  The method of infinite descent, frequently used in number theory.
  Provided by Roelof Oosterhuis.
  \<open>P n\<close> is true for all natural numbers if
  \<^item> case ``0'': given \<open>n = 0\<close> prove \<open>P n\<close>
  \<^item> case ``smaller'': given \<open>n > 0\<close> and \<open>\<not> P n\<close> prove there exists
    a smaller natural number \<open>m\<close> such that \<open>\<not> P m\<close>.
\<close>

lemma infinite_descent: "(\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m) \<Longrightarrow> P n" for P :: "nat \<Rightarrow> bool"
  \<comment> \<open>compact version without explicit base case\<close>
  by (induct n rule: less_induct) auto

lemma infinite_descent0 [case_names 0 smaller]:
  fixes P :: "nat \<Rightarrow> bool"
  assumes "P 0"
    and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m"
  shows "P n"
  apply (rule infinite_descent)
  using assms
  apply (case_tac "n > 0")
   apply auto
  done

text \<open>
  Infinite descent using a mapping to \<open>nat\<close>:
  \<open>P x\<close> is true for all \<open>x \<in> D\<close> if there exists a \<open>V \<in> D \<Rightarrow> nat\<close> and
  \<^item> case ``0'': given \<open>V x = 0\<close> prove \<open>P x\<close>
  \<^item> ``smaller'': given \<open>V x > 0\<close> and \<open>\<not> P x\<close> prove
  there exists a \<open>y \<in> D\<close> such that \<open>V y < V x\<close> and \<open>\<not> P y\<close>.
\<close>
corollary infinite_descent0_measure [case_names 0 smaller]:
  fixes V :: "'a \<Rightarrow> nat"
  assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
    and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
  shows "P x"
proof -
  obtain n where "n = V x" by auto
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
  proof (induct n rule: infinite_descent0)
    case 0
    with 1 show "P x" by auto
  next
    case (smaller n)
    then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
    with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
    with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto
    then show ?case by auto
  qed
  ultimately show "P x" by auto
qed

text \<open>Again, without explicit base case:\<close>
lemma infinite_descent_measure:
  fixes V :: "'a \<Rightarrow> nat"
  assumes "\<And>x. \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
  shows "P x"
proof -
  from assms obtain n where "n = V x" by auto
  moreover have "\<And>x. V x = n \<Longrightarrow> P x"
  proof (induct n rule: infinite_descent, auto)
    show "\<exists>m < V x. \<exists>y. V y = m \<and> \<not> P y" if "\<not> P x" for x
      using assms and that by auto
  qed
  ultimately show "P x" by auto
qed

text \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close>
lemma less_mono_imp_le_mono:
  fixes f :: "nat \<Rightarrow> nat"
    and i j :: nat
  assumes "\<And>i j::nat. i < j \<Longrightarrow> f i < f j"
    and "i \<le> j"
  shows "f i \<le> f j"
  using assms by (auto simp add: order_le_less)

text \<open>non-strict, in 1st argument\<close>
lemma add_le_mono1: "i \<le> j \<Longrightarrow> i + k \<le> j + k"
  for i j k :: nat
  by (rule add_right_mono)

text \<open>non-strict, in both arguments\<close>
lemma add_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i + k \<le> j + l"
  for i j k l :: nat
  by (rule add_mono)

lemma le_add2: "n \<le> m + n"
  for m n :: nat
  by simp

lemma le_add1: "n \<le> n + m"
  for m n :: nat
  by simp

lemma less_add_Suc1: "i < Suc (i + m)"
  by (rule le_less_trans, rule le_add1, rule lessI)

lemma less_add_Suc2: "i < Suc (m + i)"
  by (rule le_less_trans, rule le_add2, rule lessI)

lemma less_iff_Suc_add: "m < n \<longleftrightarrow> (\<exists>k. n = Suc (m + k))"
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)

lemma trans_le_add1: "i \<le> j \<Longrightarrow> i \<le> j + m"
  for i j m :: nat
  by (rule le_trans, assumption, rule le_add1)

lemma trans_le_add2: "i \<le> j \<Longrightarrow> i \<le> m + j"
  for i j m :: nat
  by (rule le_trans, assumption, rule le_add2)

lemma trans_less_add1: "i < j \<Longrightarrow> i < j + m"
  for i j m :: nat
  by (rule less_le_trans, assumption, rule le_add1)

lemma trans_less_add2: "i < j \<Longrightarrow> i < m + j"
  for i j m :: nat
  by (rule less_le_trans, assumption, rule le_add2)

lemma add_lessD1: "i + j < k \<Longrightarrow> i < k"
  for i j k :: nat
  by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)

lemma not_add_less1 [iff]: "\<not> i + j < i"
  for i j :: nat
  apply (rule notI)
  apply (drule add_lessD1)
  apply (erule less_irrefl [THEN notE])
  done

lemma not_add_less2 [iff]: "\<not> j + i < i"
  for i j :: nat
  by (simp add: add.commute)

lemma add_leD1: "m + k \<le> n \<Longrightarrow> m \<le> n"
  for k m n :: nat
  by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)

lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
  for k m n :: nat
  apply (simp add: add.commute)
  apply (erule add_leD1)
  done

lemma add_leE: "m + k \<le> n \<Longrightarrow> (m \<le> n \<Longrightarrow> k \<le> n \<Longrightarrow> R) \<Longrightarrow> R"
  for k m n :: nat
  by (blast dest: add_leD1 add_leD2)

text \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
lemma less_add_eq_less: "\<And>k. k < l \<Longrightarrow> m + l = k + n \<Longrightarrow> m < n"
  for l m n :: nat
  by (force simp del: add_Suc_right simp add: less_iff_Suc_add add_Suc_right [symmetric] ac_simps)

subsubsection \<open>More results about difference\<close>

lemma Suc_diff_le: "n \<le> m \<Longrightarrow> Suc m - n = Suc (m - n)"
  by (induct m n rule: diff_induct) simp_all

lemma diff_less_Suc: "m - n < Suc m"
  apply (induct m n rule: diff_induct)
    apply (erule_tac [3] less_SucE)
     apply (simp_all add: less_Suc_eq)
  done

lemma diff_le_self [simp]: "m - n \<le> m"
  for m n :: nat
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)

lemma less_imp_diff_less: "j < k \<Longrightarrow> j - n < k"
  for j k n :: nat
  by (rule le_less_trans, rule diff_le_self)

lemma diff_Suc_less [simp]: "0 < n \<Longrightarrow> n - Suc i < n"
  by (cases n) (auto simp add: le_simps)

lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
  for i j k :: nat
  by (induct j k rule: diff_induct) simp_all

lemma add_diff_assoc [simp]: "k \<le> j \<Longrightarrow> i + (j - k) = i + j - k"
  for i j k :: nat
  by (fact diff_add_assoc [symmetric])

lemma diff_add_assoc2: "k \<le> j \<Longrightarrow> (j + i) - k = (j - k) + i"
  for i j k :: nat
  by (simp add: ac_simps)

lemma add_diff_assoc2 [simp]: "k \<le> j \<Longrightarrow> j - k + i = j + i - k"
  for i j k :: nat
  by (fact diff_add_assoc2 [symmetric])

lemma le_imp_diff_is_add: "i \<le> j \<Longrightarrow> (j - i = k) = (j = k + i)"
  for i j k :: nat
  by auto

lemma diff_is_0_eq [simp]: "m - n = 0 \<longleftrightarrow> m \<le> n"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all

lemma diff_is_0_eq' [simp]: "m \<le> n \<Longrightarrow> m - n = 0"
  for m n :: nat
  by (rule iffD2, rule diff_is_0_eq)

lemma zero_less_diff [simp]: "0 < n - m \<longleftrightarrow> m < n"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all

lemma less_imp_add_positive:
  assumes "i < j"
  shows "\<exists>k::nat. 0 < k \<and> i + k = j"
proof
  from assms show "0 < j - i \<and> i + (j - i) = j"
    by (simp add: order_less_imp_le)
qed

text \<open>a nice rewrite for bounded subtraction\<close>
lemma nat_minus_add_max: "n - m + m = max n m"
  for m n :: nat
  by (simp add: max_def not_le order_less_imp_le)

lemma nat_diff_split: "P (a - b) \<longleftrightarrow> (a < b \<longrightarrow> P 0) \<and> (\<forall>d. a = b + d \<longrightarrow> P d)"
  for a b :: nat
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close>\<close>
  by (cases "a < b") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])

lemma nat_diff_split_asm: "P (a - b) \<longleftrightarrow> \<not> (a < b \<and> \<not> P 0 \<or> (\<exists>d. a = b + d \<and> \<not> P d))"
  for a b :: nat
  \<comment> \<open>elimination of \<open>-\<close> on \<open>nat\<close> in assumptions\<close>
  by (auto split: nat_diff_split)

lemma Suc_pred': "0 < n \<Longrightarrow> n = Suc(n - 1)"
  by simp

lemma add_eq_if: "m + n = (if m = 0 then n else Suc ((m - 1) + n))"
  unfolding One_nat_def by (cases m) simp_all

lemma mult_eq_if: "m * n = (if m = 0 then 0 else n + ((m - 1) * n))"
  for m n :: nat
  by (cases m) simp_all

lemma Suc_diff_eq_diff_pred: "0 < n \<Longrightarrow> Suc m - n = m - (n - 1)"
  by (cases n) simp_all

lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n"
  by (cases m) simp_all

lemma Let_Suc [simp]: "Let (Suc n) f \<equiv> f (Suc n)"
  by (fact Let_def)

subsubsection \<open>Monotonicity of multiplication\<close>

lemma mult_le_mono1: "i \<le> j \<Longrightarrow> i * k \<le> j * k"
  for i j k :: nat
  by (simp add: mult_right_mono)

lemma mult_le_mono2: "i \<le> j \<Longrightarrow> k * i \<le> k * j"
  for i j k :: nat
  by (simp add: mult_left_mono)

text \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
lemma mult_le_mono: "i \<le> j \<Longrightarrow> k \<le> l \<Longrightarrow> i * k \<le> j * l"
  for i j k l :: nat
  by (simp add: mult_mono)

lemma mult_less_mono1: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> i * k < j * k"
  for i j k :: nat
  by (simp add: mult_strict_right_mono)

text \<open>Differs from the standard \<open>zero_less_mult_iff\<close> in that there are no negative numbers.\<close>
lemma nat_0_less_mult_iff [simp]: "0 < m * n \<longleftrightarrow> 0 < m \<and> 0 < n"
  for m n :: nat
proof (induct m)
  case 0
  then show ?case by simp
next
  case (Suc m)
  then show ?case by (cases n) simp_all
qed

lemma one_le_mult_iff [simp]: "Suc 0 \<le> m * n \<longleftrightarrow> Suc 0 \<le> m \<and> Suc 0 \<le> n"
proof (induct m)
  case 0
  then show ?case by simp
next
  case (Suc m)
  then show ?case by (cases n) simp_all
qed

lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n"
  for k m n :: nat
  apply (safe intro!: mult_less_mono1)
   apply (cases k)
    apply auto
  apply (simp add: linorder_not_le [symmetric])
  apply (blast intro: mult_le_mono1)
  done

lemma mult_less_cancel1 [simp]: "k * m < k * n \<longleftrightarrow> 0 < k \<and> m < n"
  for k m n :: nat
  by (simp add: mult.commute [of k])

lemma mult_le_cancel1 [simp]: "k * m \<le> k * n \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
  for k m n :: nat
  by (simp add: linorder_not_less [symmetric], auto)

lemma mult_le_cancel2 [simp]: "m * k \<le> n * k \<longleftrightarrow> (0 < k \<longrightarrow> m \<le> n)"
  for k m n :: nat
  by (simp add: linorder_not_less [symmetric], auto)

lemma Suc_mult_less_cancel1: "Suc k * m < Suc k * n \<longleftrightarrow> m < n"
  by (subst mult_less_cancel1) simp

lemma Suc_mult_le_cancel1: "Suc k * m \<le> Suc k * n \<longleftrightarrow> m \<le> n"
  by (subst mult_le_cancel1) simp

lemma le_square: "m \<le> m * m"
  for m :: nat
  by (cases m) (auto intro: le_add1)

lemma le_cube: "m \<le> m * (m * m)"
  for m :: nat
  by (cases m) (auto intro: le_add1)

text \<open>Lemma for \<open>gcd\<close>\<close>
lemma mult_eq_self_implies_10: "m = m * n \<Longrightarrow> n = 1 \<or> m = 0"
  for m n :: nat
  apply (drule sym)
  apply (rule disjCI)
  apply (rule linorder_cases)
    defer
    apply assumption
   apply (drule mult_less_mono2)
    apply auto
  done

lemma mono_times_nat:
  fixes n :: nat
  assumes "n > 0"
  shows "mono (times n)"
proof
  fix m q :: nat
  assume "m \<le> q"
  with assms show "n * m \<le> n * q" by simp
qed

text \<open>The lattice order on @{typ nat}.\<close>

instantiation nat :: distrib_lattice
begin

definition "(inf :: nat \<Rightarrow> nat \<Rightarrow> nat) = min"

definition "(sup :: nat \<Rightarrow> nat \<Rightarrow> nat) = max"

instance
  by intro_classes
    (auto simp add: inf_nat_def sup_nat_def max_def not_le min_def
      intro: order_less_imp_le antisym elim!: order_trans order_less_trans)

end

subsection \<open>Natural operation of natural numbers on functions\<close>

text \<open>
  We use the same logical constant for the power operations on
  functions and relations, in order to share the same syntax.
\<close>

consts compow :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"

abbreviation compower :: "'a \<Rightarrow> nat \<Rightarrow> 'a" (infixr "^^" 80)
  where "f ^^ n \<equiv> compow n f"

notation (latex output)
  compower ("(_\<^bsup>_\<^esup>)" [1000] 1000)

text \<open>\<open>f ^^ n = f \<circ> \<dots> \<circ> f\<close>, the \<open>n\<close>-fold composition of \<open>f\<close>\<close>

overloading
  funpow \<equiv> "compow :: nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a)"
begin

primrec funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
  where
    "funpow 0 f = id"
  | "funpow (Suc n) f = f \<circ> funpow n f"

end

lemma funpow_0 [simp]: "(f ^^ 0) x = x"
  by simp

lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n \<circ> f"
proof (induct n)
  case 0
  then show ?case by simp
next
  fix n
  assume "f ^^ Suc n = f ^^ n \<circ> f"
  then show "f ^^ Suc (Suc n) = f ^^ Suc n \<circ> f"
    by (simp add: o_assoc)
qed

lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

text \<open>For code generation.\<close>

definition funpow :: "nat \<Rightarrow> ('a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"
  where funpow_code_def [code_abbrev]: "funpow = compow"

lemma [code]:
  "funpow (Suc n) f = f \<circ> funpow n f"
  "funpow 0 f = id"
  by (simp_all add: funpow_code_def)

hide_const (open) funpow

lemma funpow_add: "f ^^ (m + n) = f ^^ m \<circ> f ^^ n"
  by (induct m) simp_all

lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)"
  for f :: "'a \<Rightarrow> 'a"
  by (induct n) (simp_all add: funpow_add)

lemma funpow_swap1: "f ((f ^^ n) x) = (f ^^ n) (f x)"
proof -
  have "f ((f ^^ n) x) = (f ^^ (n + 1)) x" by simp
  also have "\<dots>  = (f ^^ n \<circ> f ^^ 1) x" by (simp only: funpow_add)
  also have "\<dots> = (f ^^ n) (f x)" by simp
  finally show ?thesis .
qed

lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)"
  for f :: "'a \<Rightarrow> 'a"
  by (induct n) simp_all

lemma Suc_funpow[simp]: "Suc ^^ n = (op + n)"
  by (induct n) simp_all

lemma id_funpow[simp]: "id ^^ n = id"
  by (induct n) simp_all

lemma funpow_mono: "mono f \<Longrightarrow> A \<le> B \<Longrightarrow> (f ^^ n) A \<le> (f ^^ n) B"
  for f :: "'a \<Rightarrow> ('a::order)"
  by (induct n arbitrary: A B)
     (auto simp del: funpow.simps(2) simp add: funpow_Suc_right mono_def)

lemma funpow_mono2:
  assumes "mono f"
    and "i \<le> j"
    and "x \<le> y"
    and "x \<le> f x"
  shows "(f ^^ i) x \<le> (f ^^ j) y"
  using assms(2,3)
proof (induct j arbitrary: y)
  case 0
  then show ?case by simp
next
  case (Suc j)
  show ?case
  proof(cases "i = Suc j")
    case True
    with assms(1) Suc show ?thesis
      by (simp del: funpow.simps add: funpow_simps_right monoD funpow_mono)
  next
    case False
    with assms(1,4) Suc show ?thesis
      by (simp del: funpow.simps add: funpow_simps_right le_eq_less_or_eq less_Suc_eq_le)
        (simp add: Suc.hyps monoD order_subst1)
  qed
qed

subsection \<open>Kleene iteration\<close>

lemma Kleene_iter_lpfp:
  fixes f :: "'a::order_bot \<Rightarrow> 'a"
  assumes "mono f"
    and "f p \<le> p"
  shows "(f ^^ k) bot \<le> p"
proof (induct k)
  case 0
  show ?case by simp
next
  case Suc
  show ?case
    using monoD[OF assms(1) Suc] assms(2) by simp
qed

lemma lfp_Kleene_iter:
  assumes "mono f"
    and "(f ^^ Suc k) bot = (f ^^ k) bot"
  shows "lfp f = (f ^^ k) bot"
proof (rule antisym)
  show "lfp f \<le> (f ^^ k) bot"
  proof (rule lfp_lowerbound)
    show "f ((f ^^ k) bot) \<le> (f ^^ k) bot"
      using assms(2) by simp
  qed
  show "(f ^^ k) bot \<le> lfp f"
    using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
qed

lemma mono_pow: "mono f \<Longrightarrow> mono (f ^^ n)"
  for f :: "'a \<Rightarrow> 'a::complete_lattice"
  by (induct n) (auto simp: mono_def)

lemma lfp_funpow:
  assumes f: "mono f"
  shows "lfp (f ^^ Suc n) = lfp f"
proof (rule antisym)
  show "lfp f \<le> lfp (f ^^ Suc n)"
  proof (rule lfp_lowerbound)
    have "f (lfp (f ^^ Suc n)) = lfp (\<lambda>x. f ((f ^^ n) x))"
      unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
    then show "f (lfp (f ^^ Suc n)) \<le> lfp (f ^^ Suc n)"
      by (simp add: comp_def)
  qed
  have "(f ^^ n) (lfp f) = lfp f" for n
    by (induct n) (auto intro: f lfp_fixpoint)
  then show "lfp (f ^^ Suc n) \<le> lfp f"
    by (intro lfp_lowerbound) (simp del: funpow.simps)
qed

lemma gfp_funpow:
  assumes f: "mono f"
  shows "gfp (f ^^ Suc n) = gfp f"
proof (rule antisym)
  show "gfp f \<ge> gfp (f ^^ Suc n)"
  proof (rule gfp_upperbound)
    have "f (gfp (f ^^ Suc n)) = gfp (\<lambda>x. f ((f ^^ n) x))"
      unfolding funpow_Suc_right by (simp add: gfp_rolling f mono_pow comp_def)
    then show "f (gfp (f ^^ Suc n)) \<ge> gfp (f ^^ Suc n)"
      by (simp add: comp_def)
  qed
  have "(f ^^ n) (gfp f) = gfp f" for n
    by (induct n) (auto intro: f gfp_fixpoint)
  then show "gfp (f ^^ Suc n) \<ge> gfp f"
    by (intro gfp_upperbound) (simp del: funpow.simps)
qed

lemma Kleene_iter_gpfp:
  fixes f :: "'a::order_top \<Rightarrow> 'a"
  assumes "mono f"
    and "p \<le> f p"
  shows "p \<le> (f ^^ k) top"
proof (induct k)
  case 0
  show ?case by simp
next
  case Suc
  show ?case
    using monoD[OF assms(1) Suc] assms(2) by simp
qed

lemma gfp_Kleene_iter:
  assumes "mono f"
    and "(f ^^ Suc k) top = (f ^^ k) top"
  shows "gfp f = (f ^^ k) top"
    (is "?lhs = ?rhs")
proof (rule antisym)
  have "?rhs \<le> f ?rhs"
    using assms(2) by simp
  then show "?rhs \<le> ?lhs"
    by (rule gfp_upperbound)
  show "?lhs \<le> ?rhs"
    using Kleene_iter_gpfp[OF assms(1)] gfp_unfold[OF assms(1)] by simp
qed

subsection \<open>Embedding of the naturals into any \<open>semiring_1\<close>: @{term of_nat}\<close>

context semiring_1
begin

definition of_nat :: "nat \<Rightarrow> 'a"
  where "of_nat n = (plus 1 ^^ n) 0"

lemma of_nat_simps [simp]:
  shows of_nat_0: "of_nat 0 = 0"
    and of_nat_Suc: "of_nat (Suc m) = 1 + of_nat m"
  by (simp_all add: of_nat_def)

lemma of_nat_1 [simp]: "of_nat 1 = 1"
  by (simp add: of_nat_def)

lemma of_nat_add [simp]: "of_nat (m + n) = of_nat m + of_nat n"
  by (induct m) (simp_all add: ac_simps)

lemma of_nat_mult [simp]: "of_nat (m * n) = of_nat m * of_nat n"
  by (induct m) (simp_all add: ac_simps distrib_right)

lemma mult_of_nat_commute: "of_nat x * y = y * of_nat x"
  by (induct x) (simp_all add: algebra_simps)

primrec of_nat_aux :: "('a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> 'a"
  where
    "of_nat_aux inc 0 i = i"
  | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" \<comment> \<open>tail recursive\<close>

lemma of_nat_code: "of_nat n = of_nat_aux (\<lambda>i. i + 1) n 0"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "\<And>i. of_nat_aux (\<lambda>i. i + 1) n (i + 1) = of_nat_aux (\<lambda>i. i + 1) n i + 1"
    by (induct n) simp_all
  from this [of 0] have "of_nat_aux (\<lambda>i. i + 1) n 1 = of_nat_aux (\<lambda>i. i + 1) n 0 + 1"
    by simp
  with Suc show ?case
    by (simp add: add.commute)
qed

end

declare of_nat_code [code]

context ring_1
begin

lemma of_nat_diff: "n \<le> m \<Longrightarrow> of_nat (m - n) = of_nat m - of_nat n"
  by (simp add: algebra_simps of_nat_add [symmetric])

end

text \<open>Class for unital semirings with characteristic zero.
 Includes non-ordered rings like the complex numbers.\<close>

class semiring_char_0 = semiring_1 +
  assumes inj_of_nat: "inj of_nat"
begin

lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n \<longleftrightarrow> m = n"
  by (auto intro: inj_of_nat injD)

text \<open>Special cases where either operand is zero\<close>

lemma of_nat_0_eq_iff [simp]: "0 = of_nat n \<longleftrightarrow> 0 = n"
  by (fact of_nat_eq_iff [of 0 n, unfolded of_nat_0])

lemma of_nat_eq_0_iff [simp]: "of_nat m = 0 \<longleftrightarrow> m = 0"
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])

lemma of_nat_neq_0 [simp]: "of_nat (Suc n) \<noteq> 0"
  unfolding of_nat_eq_0_iff by simp

lemma of_nat_0_neq [simp]: "0 \<noteq> of_nat (Suc n)"
  unfolding of_nat_0_eq_iff by simp

end

class ring_char_0 = ring_1 + semiring_char_0

context linordered_semidom
begin

lemma of_nat_0_le_iff [simp]: "0 \<le> of_nat n"
  by (induct n) simp_all

lemma of_nat_less_0_iff [simp]: "\<not> of_nat m < 0"
  by (simp add: not_less)

lemma of_nat_less_iff [simp]: "of_nat m < of_nat n \<longleftrightarrow> m < n"
  by (induct m n rule: diff_induct) (simp_all add: add_pos_nonneg)

lemma of_nat_le_iff [simp]: "of_nat m \<le> of_nat n \<longleftrightarrow> m \<le> n"
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])

lemma less_imp_of_nat_less: "m < n \<Longrightarrow> of_nat m < of_nat n"
  by simp

lemma of_nat_less_imp_less: "of_nat m < of_nat n \<Longrightarrow> m < n"
  by simp

text \<open>Every \<open>linordered_semidom\<close> has characteristic zero.\<close>

subclass semiring_char_0
  by standard (auto intro!: injI simp add: eq_iff)

text \<open>Special cases where either operand is zero\<close>

lemma of_nat_le_0_iff [simp]: "of_nat m \<le> 0 \<longleftrightarrow> m = 0"
  by (rule of_nat_le_iff [of _ 0, simplified])

lemma of_nat_0_less_iff [simp]: "0 < of_nat n \<longleftrightarrow> 0 < n"
  by (rule of_nat_less_iff [of 0, simplified])

end

context linordered_idom
begin

lemma abs_of_nat [simp]: "\<bar>of_nat n\<bar> = of_nat n"
  unfolding abs_if by auto

end

lemma of_nat_id [simp]: "of_nat n = n"
  by (induct n) simp_all

lemma of_nat_eq_id [simp]: "of_nat = id"
  by (auto simp add: fun_eq_iff)

subsection \<open>The set of natural numbers\<close>

context semiring_1
begin

definition Nats :: "'a set"  ("\<nat>")
  where "\<nat> = range of_nat"

lemma of_nat_in_Nats [simp]: "of_nat n \<in> \<nat>"
  by (simp add: Nats_def)

lemma Nats_0 [simp]: "0 \<in> \<nat>"
  apply (simp add: Nats_def)
  apply (rule range_eqI)
  apply (rule of_nat_0 [symmetric])
  done

lemma Nats_1 [simp]: "1 \<in> \<nat>"
  apply (simp add: Nats_def)
  apply (rule range_eqI)
  apply (rule of_nat_1 [symmetric])
  done

lemma Nats_add [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a + b \<in> \<nat>"
  apply (auto simp add: Nats_def)
  apply (rule range_eqI)
  apply (rule of_nat_add [symmetric])
  done

lemma Nats_mult [simp]: "a \<in> \<nat> \<Longrightarrow> b \<in> \<nat> \<Longrightarrow> a * b \<in> \<nat>"
  apply (auto simp add: Nats_def)
  apply (rule range_eqI)
  apply (rule of_nat_mult [symmetric])
  done

lemma Nats_cases [cases set: Nats]:
  assumes "x \<in> \<nat>"
  obtains (of_nat) n where "x = of_nat n"
  unfolding Nats_def
proof -
  from \<open>x \<in> \<nat>\<close> have "x \<in> range of_nat" unfolding Nats_def .
  then obtain n where "x = of_nat n" ..
  then show thesis ..
qed

lemma Nats_induct [case_names of_nat, induct set: Nats]: "x \<in> \<nat> \<Longrightarrow> (\<And>n. P (of_nat n)) \<Longrightarrow> P x"
  by (rule Nats_cases) auto

end

subsection \<open>Further arithmetic facts concerning the natural numbers\<close>

lemma subst_equals:
  assumes "t = s" and "u = t"
  shows "u = s"
  using assms(2,1) by (rule trans)

ML_file "Tools/nat_arith.ML"

simproc_setup nateq_cancel_sums
  ("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
  \<open>fn phi => try o Nat_Arith.cancel_eq_conv\<close>

simproc_setup natless_cancel_sums
  ("(l::nat) + m < n" | "(l::nat) < m + n" | "Suc m < n" | "m < Suc n") =
  \<open>fn phi => try o Nat_Arith.cancel_less_conv\<close>

simproc_setup natle_cancel_sums
  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" | "Suc m \<le> n" | "m \<le> Suc n") =
  \<open>fn phi => try o Nat_Arith.cancel_le_conv\<close>

simproc_setup natdiff_cancel_sums
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
  \<open>fn phi => try o Nat_Arith.cancel_diff_conv\<close>

context order
begin

lemma lift_Suc_mono_le:
  assumes mono: "\<And>n. f n \<le> f (Suc n)"
    and "n \<le> n'"
  shows "f n \<le> f n'"
proof (cases "n < n'")
  case True
  then show ?thesis
    by (induct n n' rule: less_Suc_induct) (auto intro: mono)
next
  case False
  with \<open>n \<le> n'\<close> show ?thesis by auto
qed

lemma lift_Suc_antimono_le:
  assumes mono: "\<And>n. f n \<ge> f (Suc n)"
    and "n \<le> n'"
  shows "f n \<ge> f n'"
proof (cases "n < n'")
  case True
  then show ?thesis
    by (induct n n' rule: less_Suc_induct) (auto intro: mono)
next
  case False
  with \<open>n \<le> n'\<close> show ?thesis by auto
qed

lemma lift_Suc_mono_less:
  assumes mono: "\<And>n. f n < f (Suc n)"
    and "n < n'"
  shows "f n < f n'"
  using \<open>n < n'\<close> by (induct n n' rule: less_Suc_induct) (auto intro: mono)

lemma lift_Suc_mono_less_iff: "(\<And>n. f n < f (Suc n)) \<Longrightarrow> f n < f m \<longleftrightarrow> n < m"
  by (blast intro: less_asym' lift_Suc_mono_less [of f]
    dest: linorder_not_less[THEN iffD1] le_eq_less_or_eq [THEN iffD1])

end

lemma mono_iff_le_Suc: "mono f \<longleftrightarrow> (\<forall>n. f n \<le> f (Suc n))"
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])

lemma antimono_iff_le_Suc: "antimono f \<longleftrightarrow> (\<forall>n. f (Suc n) \<le> f n)"
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])

lemma mono_nat_linear_lb:
  fixes f :: "nat \<Rightarrow> nat"
  assumes "\<And>m n. m < n \<Longrightarrow> f m < f n"
  shows "f m + k \<le> f (m + k)"
proof (induct k)
  case 0
  then show ?case by simp
next
  case (Suc k)
  then have "Suc (f m + k) \<le> Suc (f (m + k))" by simp
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) \<le> f (Suc (m + k))"
    by (simp add: Suc_le_eq)
  finally show ?case by simp
qed

text \<open>Subtraction laws, mostly by Clemens Ballarin\<close>

lemma diff_less_mono:
  fixes a b c :: nat
  assumes "a < b" and "c \<le> a"
  shows "a - c < b - c"
proof -
  from assms obtain d e where "b = c + (d + e)" and "a = c + e" and "d > 0"
    by (auto dest!: le_Suc_ex less_imp_Suc_add simp add: ac_simps)
  then show ?thesis by simp
qed

lemma less_diff_conv: "i < j - k \<longleftrightarrow> i + k < j"
  for i j k :: nat
  by (cases "k \<le> j") (auto simp add: not_le dest: less_imp_Suc_add le_Suc_ex)

lemma less_diff_conv2: "k \<le> j \<Longrightarrow> j - k < i \<longleftrightarrow> j < i + k"
  for j k i :: nat
  by (auto dest: le_Suc_ex)

lemma le_diff_conv: "j - k \<le> i \<longleftrightarrow> j \<le> i + k"
  for j k i :: nat
  by (cases "k \<le> j") (auto simp add: not_le dest!: less_imp_Suc_add le_Suc_ex)

lemma diff_diff_cancel [simp]: "i \<le> n \<Longrightarrow> n - (n - i) = i"
  for i n :: nat
  by (auto dest: le_Suc_ex)

lemma diff_less [simp]: "0 < n \<Longrightarrow> 0 < m \<Longrightarrow> m - n < m"
  for i n :: nat
  by (auto dest: less_imp_Suc_add)

text \<open>Simplification of relational expressions involving subtraction\<close>

lemma diff_diff_eq: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k - (n - k) = m - n"
  for m n k :: nat
  by (auto dest!: le_Suc_ex)

hide_fact (open) diff_diff_eq

lemma eq_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k = n - k \<longleftrightarrow> m = n"
  for m n k :: nat
  by (auto dest: le_Suc_ex)

lemma less_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k < n - k \<longleftrightarrow> m < n"
  for m n k :: nat
  by (auto dest!: le_Suc_ex)

lemma le_diff_iff: "k \<le> m \<Longrightarrow> k \<le> n \<Longrightarrow> m - k \<le> n - k \<longleftrightarrow> m \<le> n"
  for m n k :: nat
  by (auto dest!: le_Suc_ex)

lemma le_diff_iff': "a \<le> c \<Longrightarrow> b \<le> c \<Longrightarrow> c - a \<le> c - b \<longleftrightarrow> b \<le> a"
  for a b c :: nat
  by (force dest: le_Suc_ex)

text \<open>(Anti)Monotonicity of subtraction -- by Stephan Merz\<close>

lemma diff_le_mono: "m \<le> n \<Longrightarrow> m - l \<le> n - l"
  for m n l :: nat
  by (auto dest: less_imp_le less_imp_Suc_add split: nat_diff_split)

lemma diff_le_mono2: "m \<le> n \<Longrightarrow> l - n \<le> l - m"
  for m n l :: nat
  by (auto dest: less_imp_le le_Suc_ex less_imp_Suc_add less_le_trans split: nat_diff_split)

lemma diff_less_mono2: "m < n \<Longrightarrow> m < l \<Longrightarrow> l - n < l - m"
  for m n l :: nat
  by (auto dest: less_imp_Suc_add split: nat_diff_split)

lemma diffs0_imp_equal: "m - n = 0 \<Longrightarrow> n - m = 0 \<Longrightarrow> m = n"
  for m n :: nat
  by (simp split: nat_diff_split)

lemma min_diff: "min (m - i) (n - i) = min m n - i"
  for m n i :: nat
  by (cases m n rule: le_cases)
    (auto simp add: not_le min.absorb1 min.absorb2 min.absorb_iff1 [symmetric] diff_le_mono)

lemma inj_on_diff_nat:
  fixes k :: nat
  assumes "\<forall>n \<in> N. k \<le> n"
  shows "inj_on (\<lambda>n. n - k) N"
proof (rule inj_onI)
  fix x y
  assume a: "x \<in> N" "y \<in> N" "x - k = y - k"
  with assms have "x - k + k = y - k + k" by auto
  with a assms show "x = y" by (auto simp add: eq_diff_iff)
qed

text \<open>Rewriting to pull differences out\<close>

lemma diff_diff_right [simp]: "k \<le> j \<Longrightarrow> i - (j - k) = i + k - j"
  for i j k :: nat
  by (fact diff_diff_right)

lemma diff_Suc_diff_eq1 [simp]:
  assumes "k \<le> j"
  shows "i - Suc (j - k) = i + k - Suc j"
proof -
  from assms have *: "Suc (j - k) = Suc j - k"
    by (simp add: Suc_diff_le)
  from assms have "k \<le> Suc j"
    by (rule order_trans) simp
  with diff_diff_right [of k "Suc j" i] * show ?thesis
    by simp
qed

lemma diff_Suc_diff_eq2 [simp]:
  assumes "k \<le> j"
  shows "Suc (j - k) - i = Suc j - (k + i)"
proof -
  from assms obtain n where "j = k + n"
    by (auto dest: le_Suc_ex)
  moreover have "Suc n - i = (k + Suc n) - (k + i)"
    using add_diff_cancel_left [of k "Suc n" i] by simp
  ultimately show ?thesis by simp
qed

lemma Suc_diff_Suc:
  assumes "n < m"
  shows "Suc (m - Suc n) = m - n"
proof -
  from assms obtain q where "m = n + Suc q"
    by (auto dest: less_imp_Suc_add)
  moreover define r where "r = Suc q"
  ultimately have "Suc (m - Suc n) = r" and "m = n + r"
    by simp_all
  then show ?thesis by simp
qed

lemma one_less_mult: "Suc 0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> Suc 0 < m * n"
  using less_1_mult [of n m] by (simp add: ac_simps)

lemma n_less_m_mult_n: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < m * n"
  using mult_strict_right_mono [of 1 m n] by simp

lemma n_less_n_mult_m: "0 < n \<Longrightarrow> Suc 0 < m \<Longrightarrow> n < n * m"
  using mult_strict_left_mono [of 1 m n] by simp

text \<open>Specialized induction principles that work "backwards":\<close>

lemma inc_induct [consumes 1, case_names base step]:
  assumes less: "i \<le> j"
    and base: "P j"
    and step: "\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P (Suc n) \<Longrightarrow> P n"
  shows "P i"
  using less step
proof (induct "j - i" arbitrary: i)
  case (0 i)
  then have "i = j" by simp
  with base show ?case by simp
next
  case (Suc d n)
  from Suc.hyps have "n \<noteq> j" by auto
  with Suc have "n < j" by (simp add: less_le)
  from \<open>Suc d = j - n\<close> have "d + 1 = j - n" by simp
  then have "d + 1 - 1 = j - n - 1" by simp
  then have "d = j - n - 1" by simp
  then have "d = j - (n + 1)" by (simp add: diff_diff_eq)
  then have "d = j - Suc n" by simp
  moreover from \<open>n < j\<close> have "Suc n \<le> j" by (simp add: Suc_le_eq)
  ultimately have "P (Suc n)"
  proof (rule Suc.hyps)
    fix q
    assume "Suc n \<le> q"
    then have "n \<le> q" by (simp add: Suc_le_eq less_imp_le)
    moreover assume "q < j"
    moreover assume "P (Suc q)"
    ultimately show "P q" by (rule Suc.prems)
  qed
  with order_refl \<open>n < j\<close> show "P n" by (rule Suc.prems)
qed

lemma strict_inc_induct [consumes 1, case_names base step]:
  assumes less: "i < j"
    and base: "\<And>i. j = Suc i \<Longrightarrow> P i"
    and step: "\<And>i. i < j \<Longrightarrow> P (Suc i) \<Longrightarrow> P i"
  shows "P i"
using less proof (induct "j - i - 1" arbitrary: i)
  case (0 i)
  from \<open>i < j\<close> obtain n where "j = i + n" and "n > 0"
    by (auto dest!: less_imp_Suc_add)
  with 0 have "j = Suc i"
    by (auto intro: order_antisym simp add: Suc_le_eq)
  with base show ?case by simp
next
  case (Suc d i)
  from \<open>Suc d = j - i - 1\<close> have *: "Suc d = j - Suc i"
    by (simp add: diff_diff_add)
  then have "Suc d - 1 = j - Suc i - 1" by simp
  then have "d = j - Suc i - 1" by simp
  moreover from * have "j - Suc i \<noteq> 0" by auto
  then have "Suc i < j" by (simp add: not_le)
  ultimately have "P (Suc i)" by (rule Suc.hyps)
  with \<open>i < j\<close> show "P i" by (rule step)
qed

lemma zero_induct_lemma: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P (k - i)"
  using inc_induct[of "k - i" k P, simplified] by blast

lemma zero_induct: "P k \<Longrightarrow> (\<And>n. P (Suc n) \<Longrightarrow> P n) \<Longrightarrow> P 0"
  using inc_induct[of 0 k P] by blast

text \<open>Further induction rule similar to @{thm inc_induct}.\<close>

lemma dec_induct [consumes 1, case_names base step]:
  "i \<le> j \<Longrightarrow> P i \<Longrightarrow> (\<And>n. i \<le> n \<Longrightarrow> n < j \<Longrightarrow> P n \<Longrightarrow> P (Suc n)) \<Longrightarrow> P j"
proof (induct j arbitrary: i)
  case 0
  then show ?case by simp
next
  case (Suc j)
  from Suc.prems consider "i \<le> j" | "i = Suc j"
    by (auto simp add: le_Suc_eq)
  then show ?case
  proof cases
    case 1
    moreover have "j < Suc j" by simp
    moreover have "P j" using \<open>i \<le> j\<close> \<open>P i\<close>
    proof (rule Suc.hyps)
      fix q
      assume "i \<le> q"
      moreover assume "q < j" then have "q < Suc j"
        by (simp add: less_Suc_eq)
      moreover assume "P q"
      ultimately show "P (Suc q)" by (rule Suc.prems)
    qed
    ultimately show "P (Suc j)" by (rule Suc.prems)
  next
    case 2
    with \<open>P i\<close> show "P (Suc j)" by simp
  qed
qed

subsection \<open>Monotonicity of \<open>funpow\<close>\<close>

lemma funpow_increasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ n) \<top> \<le> (f ^^ m) \<top>"
  for f :: "'a::{lattice,order_top} \<Rightarrow> 'a"
  by (induct rule: inc_induct)
    (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
      intro: order_trans[OF _ funpow_mono])

lemma funpow_decreasing: "m \<le> n \<Longrightarrow> mono f \<Longrightarrow> (f ^^ m) \<bottom> \<le> (f ^^ n) \<bottom>"
  for f :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
  by (induct rule: dec_induct)
    (auto simp del: funpow.simps(2) simp add: funpow_Suc_right
      intro: order_trans[OF _ funpow_mono])

lemma mono_funpow: "mono Q \<Longrightarrow> mono (\<lambda>i. (Q ^^ i) \<bottom>)"
  for Q :: "'a::{lattice,order_bot} \<Rightarrow> 'a"
  by (auto intro!: funpow_decreasing simp: mono_def)

lemma antimono_funpow: "mono Q \<Longrightarrow> antimono (\<lambda>i. (Q ^^ i) \<top>)"
  for Q :: "'a::{lattice,order_top} \<Rightarrow> 'a"
  by (auto intro!: funpow_increasing simp: antimono_def)

subsection \<open>The divides relation on @{typ nat}\<close>

lemma dvd_1_left [iff]: "Suc 0 dvd k"
  by (simp add: dvd_def)

lemma dvd_1_iff_1 [simp]: "m dvd Suc 0 \<longleftrightarrow> m = Suc 0"
  by (simp add: dvd_def)

lemma nat_dvd_1_iff_1 [simp]: "m dvd 1 \<longleftrightarrow> m = 1"
  for m :: nat
  by (simp add: dvd_def)

lemma dvd_antisym: "m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
  for m n :: nat
  unfolding dvd_def by (force dest: mult_eq_self_implies_10 simp add: mult.assoc)

lemma dvd_diff_nat [simp]: "k dvd m \<Longrightarrow> k dvd n \<Longrightarrow> k dvd (m - n)"
  for k m n :: nat
  unfolding dvd_def by (blast intro: right_diff_distrib' [symmetric])

lemma dvd_diffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd m"
  for k m n :: nat
  apply (erule linorder_not_less [THEN iffD2, THEN add_diff_inverse, THEN subst])
  apply (blast intro: dvd_add)
  done

lemma dvd_diffD1: "k dvd m - n \<Longrightarrow> k dvd m \<Longrightarrow> n \<le> m \<Longrightarrow> k dvd n"
  for k m n :: nat
  by (drule_tac m = m in dvd_diff_nat) auto

lemma dvd_mult_cancel:
  fixes m n k :: nat
  assumes "k * m dvd k * n" and "0 < k"
  shows "m dvd n"
proof -
  from assms(1) obtain q where "k * n = (k * m) * q" ..
  then have "k * n = k * (m * q)" by (simp add: ac_simps)
  with \<open>0 < k\<close> have "n = m * q" by (auto simp add: mult_left_cancel)
  then show ?thesis ..
qed

lemma dvd_mult_cancel1: "0 < m \<Longrightarrow> m * n dvd m \<longleftrightarrow> n = 1"
  for m n :: nat
  apply auto
  apply (subgoal_tac "m * n dvd m * 1")
   apply (drule dvd_mult_cancel)
    apply auto
  done

lemma dvd_mult_cancel2: "0 < m \<Longrightarrow> n * m dvd m \<longleftrightarrow> n = 1"
  for m n :: nat
  using dvd_mult_cancel1 [of m n] by (simp add: ac_simps)

lemma dvd_imp_le: "k dvd n \<Longrightarrow> 0 < n \<Longrightarrow> k \<le> n"
  for k n :: nat
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

lemma nat_dvd_not_less: "0 < m \<Longrightarrow> m < n \<Longrightarrow> \<not> n dvd m"
  for m n :: nat
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)

lemma less_eq_dvd_minus:
  fixes m n :: nat
  assumes "m \<le> n"
  shows "m dvd n \<longleftrightarrow> m dvd n - m"
proof -
  from assms have "n = m + (n - m)" by simp
  then obtain q where "n = m + q" ..
  then show ?thesis by (simp add: add.commute [of m])
qed

lemma dvd_minus_self: "m dvd n - m \<longleftrightarrow> n < m \<or> m dvd n"
  for m n :: nat
  by (cases "n < m") (auto elim!: dvdE simp add: not_less le_imp_diff_is_add dest: less_imp_le)

lemma dvd_minus_add:
  fixes m n q r :: nat
  assumes "q \<le> n" "q \<le> r * m"
  shows "m dvd n - q \<longleftrightarrow> m dvd n + (r * m - q)"
proof -
  have "m dvd n - q \<longleftrightarrow> m dvd r * m + (n - q)"
    using dvd_add_times_triv_left_iff [of m r] by simp
  also from assms have "\<dots> \<longleftrightarrow> m dvd r * m + n - q" by simp
  also from assms have "\<dots> \<longleftrightarrow> m dvd (r * m - q) + n" by simp
  also have "\<dots> \<longleftrightarrow> m dvd n + (r * m - q)" by (simp add: add.commute)
  finally show ?thesis .
qed

subsection \<open>Aliasses\<close>

lemma nat_mult_1: "1 * n = n"
  for n :: nat
  by (fact mult_1_left)

lemma nat_mult_1_right: "n * 1 = n"
  for n :: nat
  by (fact mult_1_right)

lemma nat_add_left_cancel: "k + m = k + n \<longleftrightarrow> m = n"
  for k m n :: nat
  by (fact add_left_cancel)

lemma nat_add_right_cancel: "m + k = n + k \<longleftrightarrow> m = n"
  for k m n :: nat
  by (fact add_right_cancel)

lemma diff_mult_distrib: "(m - n) * k = (m * k) - (n * k)"
  for k m n :: nat
  by (fact left_diff_distrib')

lemma diff_mult_distrib2: "k * (m - n) = (k * m) - (k * n)"
  for k m n :: nat
  by (fact right_diff_distrib')

lemma le_add_diff: "k \<le> n \<Longrightarrow> m \<le> n + m - k"
  for k m n :: nat
  by (fact le_add_diff)  (* FIXME delete *)

lemma le_diff_conv2: "k \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> j)"
  for i j k :: nat
  by (fact le_diff_conv2) (* FIXME delete *)

lemma diff_self_eq_0 [simp]: "m - m = 0"
  for m :: nat
  by (fact diff_cancel)

lemma diff_diff_left [simp]: "i - j - k = i - (j + k)"
  for i j k :: nat
  by (fact diff_diff_add)

lemma diff_commute: "i - j - k = i - k - j"
  for i j k :: nat
  by (fact diff_right_commute)

lemma diff_add_inverse: "(n + m) - n = m"
  for m n :: nat
  by (fact add_diff_cancel_left')

lemma diff_add_inverse2: "(m + n) - n = m"
  for m n :: nat
  by (fact add_diff_cancel_right')

lemma diff_cancel: "(k + m) - (k + n) = m - n"
  for k m n :: nat
  by (fact add_diff_cancel_left)

lemma diff_cancel2: "(m + k) - (n + k) = m - n"
  for k m n :: nat
  by (fact add_diff_cancel_right)

lemma diff_add_0: "n - (n + m) = 0"
  for m n :: nat
  by (fact diff_add_zero)

lemma add_mult_distrib2: "k * (m + n) = (k * m) + (k * n)"
  for k m n :: nat
  by (fact distrib_left)

lemmas nat_distrib =
  add_mult_distrib distrib_left diff_mult_distrib diff_mult_distrib2

subsection \<open>Size of a datatype value\<close>

class size =
  fixes size :: "'a \<Rightarrow> nat" \<comment> \<open>see further theory \<open>Wellfounded\<close>\<close>

instantiation nat :: size
begin

definition size_nat where [simp, code]: "size (n::nat) = n"

instance ..

end

subsection \<open>Code module namespace\<close>

code_identifier
  code_module Nat \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith

hide_const (open) of_nat_aux

end