SSL Nat.thy

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

section ‹Natural numbers›

theory Nat
imports Inductive Typedef Fun Rings
begin

subsection ‹Type ‹ind›\›

typedecl ind

axiomatization Zero_Rep :: ind and Suc_Rep :: "ind ==> ind"
  🍋 ‹The axiom of infinity in 2 parts:›
  where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y ==> x = y"
    and Suc_Rep_not_Zero_Rep: "Suc_Rep x ≠ Zero_Rep"


subsection ‹Type nat›

text ‹Type definition›

inductive Nat :: "ind ==> bool"
  where
    Zero_RepI: "Nat Zero_Rep"
  | Suc_RepI: "Nat i ==> 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 ==> Rep_Nat (Abs_Nat n) = n"
  using Abs_Nat_inverse by simp

lemma Nat_Abs_Nat_inject: "Nat n ==> Nat m ==> Abs_Nat n = Abs_Nat m ⟷ 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 ==> nat"
  where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"

lemma Suc_not_Zero: "Suc m ≠ 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 ≠ Suc m"
  by (rule not_sym) (rule Suc_not_Zero)

lemma Suc_Rep_inject': "Suc_Rep x = Suc_Rep y ⟷ x = y"
  by (rule iffI, rule Suc_Rep_inject) simp_all

lemma nat_induct0:
  assumes "P 0" and "∧n. P n ==> P (Suc n)"
  shows "P n"
proof -
  have "P (Abs_Nat (Rep_Nat n))"
    using assms unfolding Zero_nat_def Suc_def
    by (iprover intro:  Nat_Rep_Nat [THEN Nat.induct] elim: Nat_Abs_Nat_inverse [THEN subst])
  then show ?thesis
    by (simp add: Rep_Nat_inverse)
qed

free_constructors case_nat for "0 :: nat" | Suc pred
  where "pred (0 :: nat) = (0 :: nat)"
proof atomize_elim
  fix n
  show "n = 0 ∨ (∃m. n = Suc m)"
    by (induction n rule: nat_induct0) auto
next
  fix n m
  show "(Suc n = Suc m) = (n = m)"
    by (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
next
  fix n
  show "0 ≠ Suc n"
    by (simp add: Suc_not_Zero)
qed


🍋 ‹Avoid name clashes by prefixing the output of ‹old_rep_datatype› with ‹old›.›
setup ‹Sign.mandatory_path "old"›

old_rep_datatype "0 :: nat" Suc
  by (erule nat_induct0) auto

setup ‹Sign.parent_path›

🍋 ‹But erase the prefix for properties that are not generated by ‹free_constructors›.›
setup ‹Sign.mandatory_path "nat"›

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 ‹Sign.parent_path›

abbreviation rec_nat :: "'a ==> (nat ==> 'a ==> 'a) ==> nat ==> 'a"
  where "rec_nat ≡ old.rec_nat"

declare nat.sel[code del]

hide_const (open) Nat.pred 🍋 ‹hide everything related to the selector›

lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
  "(y = 0 ==> P) ==> (∧nat. y = Suc nat ==> P) ==> P"
  🍋 ‹for backward compatibility -- names of variables differ›
  by (rule old.nat.exhaust)

lemma nat_induct [case_names 0 Suc, induct type: nat]:
  fixes n
  assumes "P 0" and "∧n. P n ==> P (Suc n)"
  shows "P n"
  🍋 ‹for backward compatibility -- names of variables differ›
  using assms by (rule nat.induct)

hide_fact
  nat_exhaust
  nat_induct0

ML ‹
 val nat_basic_lfp_sugar =
  let
  val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global 🍋 🍋‹nat›);
  val recx = Logic.varify_types_global 🍋‹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;
 ›

setup ‹
 let
  fun basic_lfp_sugars_of _ [🍋‹nat›] _ _ ctxt =
      ([], [0], [nat_basic_lfp_sugar], [], [], [], TrueI (*dummy*)
    | 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 = [], special_endgame_tac = K (K (K (K no_tac))), is_new_datatype = K (K true),
     basic_lfp_sugars_of = basic_lfp_sugars_of, rewrite_nested_rec_call = NONE}
end
›

text ‹Injectiveness and distinctness lemmas›

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

lemma bij_betw_Suc [simp]:
  "bij_betw Suc M N ⟷ Suc ` M = N"
  by (simp add: bij_betw_def)

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

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

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

lemma n_not_Suc_n: "n ≠ Suc n"
  by (induct n) simp_all

lemma Suc_n_not_n: "Suc n ≠ n"
  by (rule not_sym) (rule n_not_Suc_n)

text ‹A special form of induction for reasoning about 🍋‹m 🚫› and 🍋‹m - n›.›
lemma diff_induct:
  assumes "∧x. P x 0"
    and "∧y. P 0 (Suc y)"
    and "∧x y. P x y ==> 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 ‹P m n› show ?case by (rule assms(3))
  qed
qed


subsection ‹Arithmetic operators›

instantiation nat :: comm_monoid_diff
begin

primrec plus_nat
  where
    add_0 [code]: "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

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 ==> 0 | Suc k ==> 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 ≠ (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 ‹Addition›

text ‹Reasoning about ‹m + 0 = 0›, etc.›

lemma add_is_0 [iff]: "m + n = 0 ⟷ m = 0 ∧ n = 0"
  for m n :: nat
  by (cases m) simp_all

lemma add_is_1: "m + n = Suc 0 ⟷ m = Suc 0 ∧ n = 0 ∨ m = 0 ∧ n = Suc 0"
  by (cases m) simp_all

lemma one_is_add: "Suc 0 = m + n ⟷ m = Suc 0 ∧ n = 0 ∨ m = 0 ∧ n = Suc 0"
  by (rule trans, rule eq_commute, rule add_is_1)

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

lemma plus_1_eq_Suc:
  "plus 1 = Suc"
  by (simp add: fun_eq_iff)

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

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


subsubsection ‹Difference›

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

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

lemma diff_Suc_1' [simp]: "Suc n - Suc 0 = n"
  by simp


subsubsection ‹Multiplication›

lemma mult_is_0 [simp]: "m * n = 0 ⟷ m = 0 ∨ n = 0" for m n :: nat
  by (induct m) auto

lemma mult_eq_1_iff [simp]: "m * n = Suc 0 ⟷ m = Suc 0 ∧ 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 ⟷ m = Suc 0 ∧ n = Suc 0"
  by (simp add: eq_commute flip: mult_eq_1_iff)

lemma nat_mult_eq_1_iff [simp]: "m * n = 1 ⟷ m = 1 ∧ n = 1" 
  and nat_1_eq_mult_iff [simp]: "1 = m * n ⟷ m = 1 ∧ n = 1" for m n :: nat
  by auto

lemma mult_cancel1 [simp]: "k * m = k * n ⟷ m = n ∨ k = 0"
  for k m n :: nat
proof -
  have "k ≠ 0 ==> k * m = k * n ==> 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 ⟷ m = n ∨ k = 0"
  for k m n :: nat
  by (simp add: mult.commute)

lemma Suc_mult_cancel1: "Suc k * m = Suc k * n ⟷ m = n"
  by (subst mult_cancel1) simp


subsection ‹Orders on 🍋‹nat›\
subsubsection ‹Operation definition›

instantiation nat :: linorder
begin

primrec less_eq_nat
  where
    "(0::nat) ≤ n ⟷ True"
  | "Suc m ≤ n ⟷ (case n of 0 ==> False | Suc n ==> m ≤ n)"

declare less_eq_nat.simps [simp del]

lemma le0 [iff]: "0 ≤ n" for
  n :: nat
  by (simp add: less_eq_nat.simps)

lemma [code]: "0 ≤ n ⟷ True"
  for n :: nat
  by simp

definition less_nat
  where less_eq_Suc_le: "n < m ⟷ Suc n ≤ m"

lemma Suc_le_mono [iff]: "Suc n ≤ Suc m ⟷ n ≤ m"
  by (simp add: less_eq_nat.simps(2))

lemma Suc_le_eq [code]: "Suc m ≤ n ⟷ m < n"
  unfolding less_eq_Suc_le ..

lemma le_0_eq [iff]: "n ≤ 0 ⟷ n = 0"
  for n :: nat
  by (induct n) (simp_all add: less_eq_nat.simps(2))

lemma not_less0 [iff]: "¬ n < 0"
  for n :: nat
  by (simp add: less_eq_Suc_le)

lemma less_nat_zero_code [code]: "n < 0 ⟷ False"
  for n :: nat
  by simp

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

lemma less_Suc_eq_le [code]: "m < Suc n ⟷ m ≤ n"
  by (simp add: less_eq_Suc_le)

lemma Suc_less_eq2: "Suc n < m ⟷ (∃m'. m = Suc m' ∧ n < m')"
  by (cases m) auto

lemma le_SucI: "m ≤ n ==> m ≤ Suc n"
  by (induct m arbitrary: n) (simp_all add: less_eq_nat.simps(2) split: nat.splits)

lemma Suc_leD: "Suc m ≤ n ==> m ≤ n"
  by (cases n) (auto intro: le_SucI)

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

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

instance
proof
  fix n m q :: nat
  show "n < m ⟷ n ≤ m ∧ ¬ m ≤ 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 ≤ n"
    by (induct n) simp_all
  then show "n = m" if "n ≤ m" and "m ≤ n"
    using that by (induct n arbitrary: m)
      (simp_all add: less_eq_nat.simps(2) split: nat.splits)
  show "n ≤ q" if "n ≤ m" and "m ≤ 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 ≤ m ∨ m ≤ 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 ‹Introduction properties›

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 ‹Elimination properties›

lemma less_not_refl: "¬ n < n"
  for n :: nat
  by (rule order_less_irrefl)

lemma less_not_refl2: "n < m ==> m ≠ n"
  for m n :: nat
  by (rule not_sym) (rule less_imp_neq)

lemma less_not_refl3: "s < t ==> s ≠ t"
  for s t :: nat
  by (rule less_imp_neq)

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

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

lemma less_Suc_eq: "m < Suc n ⟷ m < n ∨ 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 ⟷ n = 0"
  for n :: nat
  unfolding One_nat_def by (rule less_Suc0)

lemma Suc_mono: "m < n ==> Suc m < Suc n"
  by simp

text ‹"Less than" is antisymmetric, sort of.›
lemma less_antisym: "¬ n < m ==> n < Suc m ==> m = n"
  unfolding not_less less_Suc_eq_le by (rule antisym)

lemma nat_neq_iff: "m ≠ n ⟷ m < n ∨ n < m"
  for m n :: nat
  by (rule linorder_neq_iff)


subsubsection ‹Inductive (?) properties›

lemma Suc_lessI: "m < n ==> Suc m ≠ n ==> 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 ==> P"
    and 2: "∧j. i < j ==> k = Suc j ==> P"
  shows P
proof -
  from major have "∃j. i ≤ j ∧ k = Suc j"
    unfolding less_eq_Suc_le by (induct k) simp_all
  then have "(∃j. i < j ∧ k = Suc j) ∨ 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 ==> P"
    and eq: "m = n ==> P"
  shows P
proof (rule major [THEN lessE])
  show "Suc n = Suc m ==> P"
    using eq by blast
  show "∧j. [m < j; Suc n = Suc j] ==> P"
    by (blast intro: less)
qed

lemma Suc_lessE:
  assumes major: "Suc i < k"
    and minor: "∧j. i < j ==> k = Suc j ==> P"
  shows P
proof (rule major [THEN lessE])
  show "k = Suc (Suc i) ==> P"
    using lessI minor by iprover
  show "∧j. [Suc i < j; k = Suc j] ==> P"
    using Suc_lessD minor by iprover
qed

lemma Suc_less_SucD: "Suc m < Suc n ==> m < n"
  by simp

lemma less_trans_Suc:
  assumes le: "i < j"
  shows "j < k ==> 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 ‹Can be used with ‹less_Suc_eq› to get 🍋‹n = m ∨ n 🚫›.›
lemma not_less_eq: "¬ m < n ⟷ n < Suc m"
  by (simp only: not_less less_Suc_eq_le)

lemma not_less_eq_eq: "¬ m ≤ n ⟷ Suc n ≤ m"
  by (simp only: not_le Suc_le_eq)

text ‹Properties of "less than or equal".›

lemma le_imp_less_Suc: "m ≤ n ==> m < Suc n"
  by (simp only: less_Suc_eq_le)

lemma Suc_n_not_le_n: "¬ Suc n ≤ n"
  by (simp add: not_le less_Suc_eq_le)

lemma le_Suc_eq: "m ≤ Suc n ⟷ m ≤ n ∨ m = Suc n"
  by (simp add: less_Suc_eq_le [symmetric] less_Suc_eq)

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

lemma Suc_leI: "m < n ==> Suc m ≤ n"
  by (simp only: Suc_le_eq)

text ‹Stronger version of ‹Suc_leD›.›
lemma Suc_le_lessD: "Suc m ≤ n ==> m < n"
  by (simp only: Suc_le_eq)

lemma less_imp_le_nat: "m < n ==> m ≤ n" for m n :: nat
  unfolding less_eq_Suc_le by (rule Suc_leD)

text ‹For instance, ‹(Suc m 🚫 n) = (Suc m ≤ n) = (m 🚫)›\›
lemmas le_simps = less_imp_le_nat less_Suc_eq_le Suc_le_eq


text ‹Equivalence of ‹m ≤ n› and ‹m 🚫 ∨ m = n›\›

lemma less_or_eq_imp_le: "m < n ∨ m = n ==> m ≤ n"
  for m n :: nat
  unfolding le_less .

lemma le_eq_less_or_eq: "m ≤ n ⟷ m < n ∨ m = n"
  for m n :: nat
  by (rule le_less)

text ‹Useful with ‹blast›.›
lemma eq_imp_le: "m = n ==> m ≤ n"
  for m n :: nat
  by auto

lemma le_refl: "n ≤ n"
  for n :: nat
  by simp

lemma le_trans: "i ≤ j ==> j ≤ k ==> i ≤ k"
  for i j k :: nat
  by (rule order_trans)

lemma le_antisym: "m ≤ n ==> n ≤ m ==> m = n"
  for m n :: nat
  by (rule antisym)

lemma nat_less_le: "m < n ⟷ m ≤ n ∧ m ≠ n"
  for m n :: nat
  by (rule less_le)

lemma le_neq_implies_less: "m ≤ n ==> m ≠ n ==> m < n"
  for m n :: nat
  unfolding less_le ..

lemma nat_le_linear: "m ≤ n ∨ n ≤ m"
  for m n :: nat
  by (rule linear)

lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

lemma le_less_Suc_eq: "m ≤ n ==> n < Suc m ⟷ n = m"
  unfolding less_Suc_eq_le by auto

lemma not_less_less_Suc_eq: "¬ n < m ==> n < Suc m ⟷ 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 ≠ 0 ==> ∃m. n = Suc m"
  by (cases n) simp_all

lemma gr0_implies_Suc: "n > 0 ==> ∃m. n = Suc m"
  by (cases n) simp_all

lemma gr_implies_not0: "m < n ==> n ≠ 0"
  for m n :: nat
  by (cases n) simp_all

lemma neq0_conv[iff]: "n ≠ 0 ⟷ 0 < n"
  for n :: nat
  by (cases n) simp_all

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

lemma gr0_conv_Suc: "0 < n ⟷ (∃m. n = Suc m)"
  by (fast intro: not0_implies_Suc)

lemma not_gr0 [iff]: "¬ 0 < n ⟷ n = 0"
  for n :: nat
  using neq0_conv by blast

lemma Suc_le_D: "Suc n ≤ m' ==> ∃m. m' = Suc m"
  by (induct m') simp_all

text ‹Useful in certain inductive arguments›
lemma less_Suc_eq_0_disj: "m < Suc n ⟷ m = 0 ∨ (∃j. m = Suc j ∧ j < n)"
  by (cases m) simp_all

lemma All_less_Suc: "(∀i < Suc n. P i) = (P n ∧ (∀i < n. P i))"
  by (auto simp: less_Suc_eq)

lemma All_less_Suc2: "(∀i < Suc n. P i) = (P 0 ∧ (∀i < n. P(Suc i)))"
  by (auto simp: less_Suc_eq_0_disj)

lemma Ex_less_Suc: "(∃i < Suc n. P i) = (P n ∨ (∃i < n. P i))"
  by (auto simp: less_Suc_eq)

lemma Ex_less_Suc2: "(∃i < Suc n. P i) = (P 0 ∨ (∃i < n. P(Suc i)))"
  by (auto simp: less_Suc_eq_0_disj)

text ‹@{term mono} (non-strict) doesn't imply increasing, as the function could be constant›
lemma strict_mono_imp_increasing:
  fixes n::nat
  assumes "strict_mono f" shows "f n ≥ n"
proof (induction n)
  case 0
  then show ?case
    by auto
next
  case (Suc n)
  then show ?case
    unfolding not_less_eq_eq [symmetric]
    using Suc_n_not_le_n assms order_trans strict_mono_less_eq by blast
qed

subsubsection ‹Monotonicity of Addition›

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

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

lemma nat_add_left_cancel_le [simp]: "k + m ≤ k + n ⟷ m ≤ n"
  for k m n :: nat
  by (induct k) simp_all

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

lemma add_gr_0 [iff]: "m + n > 0 ⟷ m > 0 ∨ n > 0"
  for m n :: nat
  by (auto dest: gr0_implies_Suc)

text ‹strict, in 1st argument›
lemma add_less_mono1: "i < j ==> i + k < j + k"
  for i j k :: nat
  by (induct k) simp_all

text ‹strict, in both arguments›
lemma add_less_mono: 
  fixes i j k l :: nat
  assumes "i < j" "k < l" shows "i + k < j + l"
proof -
  have "i + k < j + k"
    by (simp add: add_less_mono1 assms)
  also have "... < j + l"
    using ‹i 🚫› by (induction j) (auto simp: assms)
  finally show ?thesis .
qed

lemma less_imp_Suc_add: "m < n ==> ∃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 ≤ l ==> (∃n. l = k + n)"
  for k l :: nat
  by (auto simp: less_Suc_eq_le[symmetric] dest: less_imp_Suc_add)

lemma less_natE:
  assumes ‹m 🚫›
  obtains q where ‹n = Suc (m + q)›
  using assms by (auto dest: less_imp_Suc_add intro: that)

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

text ‹Addition is the inverse of subtraction:
  if 🍋‹n ≤ m› then 🍋‹n + (m - n) = m›.›
lemma add_diff_inverse_nat: "¬ m < n ==> n + (m - n) = m"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all

lemma nat_le_iff_add: "m ≤ n ⟷ (∃k. n = m + k)"
  for m n :: nat
  using nat_add_left_cancel_le[of m 0] by (auto dest: le_Suc_ex)

text ‹The naturals form an ordered ‹semidom› and a ‹dioid›.›

instance nat :: discrete_linordered_semidom
proof
  fix m n q :: nat
  show ‹0 🚫1::nat)›
    by simp
  show ‹m ≤ n ==> q + m ≤ q + n›
    by simp
  show ‹m 🚫 ==> 0 🚫 ==> q * m 🚫 * n›
    by (simp add: mult_less_mono2)
  show ‹m ≠ 0 ==> n ≠ 0 ==> m * n ≠ 0›
    by simp
  show ‹n ≤ m ==> (m - n) + n = m›
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
  show ‹m 🚫 ⟷ m + 1 ≤ n›
    by (simp add: Suc_le_eq)
qed

instance nat :: dioid
  by standard (rule nat_le_iff_add)

declare le0[simp del] 🍋 ‹This is now @{thm zero_le}›
declare le_0_eq[simp del] 🍋 ‹This is now @{thm le_zero_eq}›
declare not_less0[simp del] 🍋 ‹This is now @{thm not_less_zero}›
declare not_gr0[simp del] 🍋 ‹This is now @{thm not_gr_zero}›

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


subsubsection ‹🍋‹min› and 🍋‹max›\›

global_interpretation bot_nat_0: ordering_top ‹(≥)› ‹(>)› ‹0::nat›
  by standard simp

global_interpretation max_nat: semilattice_neutr_order max ‹0::nat› ‹(≥)› ‹(>)›
  by standard (simp add: max_def)

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 ==> 0 | Suc m' ==> Suc(min n m'))"
  by (simp split: nat.split)

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

lemma max_0L [simp]: "max 0 n = n"
  for n :: nat
  by (fact max_nat.left_neutral)

lemma max_0R [simp]: "max n 0 = n"
  for n :: nat
  by (fact max_nat.right_neutral)

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 ==> Suc n | Suc m' ==> Suc (max n m'))"
  by (simp split: nat.split)

lemma max_Suc2: "max m (Suc n) = (case m of 0 ==> Suc n | Suc m' ==> 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 ‹Additional theorems about 🍋‹(≤)›\›

text ‹Complete induction, aka course-of-values induction›

instance nat :: wellorder
proof
  fix P and n :: nat
  assume step: "(∧m. m < n ==> P m) ==> P n" for n :: nat
  have "∧q. q ≤ n ==> 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 ≤ m ∨ n = Suc m"
      by (simp add: le_Suc_eq)
    then show ?case
    proof
      assume "n ≤ 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 ==> Least P = 0"
  for P :: "nat ==> bool"
  by (rule Least_equality[OF _ le0])

lemma Least_Suc:
  assumes "P n" "¬ P 0" 
  shows "(LEAST n. P n) = Suc (LEAST m. P (Suc m))"
proof (cases n)
  case (Suc m)
  show ?thesis
  proof (rule antisym)
    show "(LEAST x. P x) ≤ Suc (LEAST x. P (Suc x))"
      using assms Suc by (force intro: LeastI Least_le)
    have 🍋: "P (LEAST x. P x)"
      by (blast intro: LeastI assms)
    show "Suc (LEAST m. P (Suc m)) ≤ (LEAST n. P n)"
    proof (cases "(LEAST n. P n)")
      case 0
      then show ?thesis
        using 🍋 by (simp add: assms)
    next
      case Suc
      with 🍋 show ?thesis
        by (auto simp: Least_le)
    qed
  qed
qed (use assms in auto)

lemma Least_Suc2: "P n ==> Q m ==> ¬ P 0 ==> ∀k. P (Suc k) = Q k ==> Least P = Suc (Least Q)"
  by (erule (1) Least_Suc [THEN ssubst]) simp

lemma ex_least_nat_le:
  fixes P :: "nat ==> bool"
  assumes "P n"
  shows "∃k≤n. (∀i¬ P i) ∧ P k"
proof (cases n)
  case (Suc m)
  with assms show ?thesis
    by (blast intro: Least_le LeastI_ex dest: not_less_Least)
qed (use assms in auto)

lemma ex_least_nat_less:
  fixes P :: "nat ==> bool"
  assumes "P n" "¬ P 0" 
  shows "∃k∀i≤k. ¬ P i) ∧ P (Suc k)"
proof (cases n)
  case (Suc m)
  then obtain k where k: "k ≤ n" "∀i¬ P i" "P k"
    using ex_least_nat_le ‹P n› by blast
  show ?thesis 
    by (cases k) (use assms k less_eq_Suc_le in auto)
qed (use assms in auto)


lemma nat_less_induct:
  fixes P :: "nat ==> bool"
  assumes "∧n. ∀m. m < n ⟶ P m ==> P n"
  shows "P n"
  using assms less_induct by blast

lemma measure_induct_rule [case_names less]:
  fixes f :: "'a ==> 'b::wellorder"
  assumes step: "∧x. (∧y. f y < f x ==> P y) ==> P x"
  shows "P a"
  by (induct m ≡ "f a" arbitrary: a rule: less_induct) (auto intro: step)

text ‹old style induction rules:›
lemma measure_induct:
  fixes f :: "'a ==> 'b::wellorder"
  shows "(∧x. ∀y. f y < f x ⟶ P y ==> P x) ==> P a"
  by (rule measure_induct_rule [of f P a]) iprover

lemma full_nat_induct:
  assumes step: "∧n. (∀m. Suc m ≤ n ⟶ P m) ==> P n"
  shows "P n"
  by (rule less_induct) (auto intro: step simp:le_simps)

text‹An induction rule for establishing binary relations›
lemma less_Suc_induct [consumes 1]:
  assumes less: "i < j"
    and step: "∧i. P i (Suc i)"
    and trans: "∧i j k. i < j ==> j < k ==> P i j ==> P j k ==> 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 ‹
  The method of infinite descent, frequently used in number theory.
  Provided by Roelof Oosterhuis.
  ‹P n› is true for all natural numbers if
  🪙 case ``0'': given ‹n = 0› prove ‹P n›
  🪙 case ``smaller'': given ‹n > 0› and ‹¬ P n› prove there exists
  a smaller natural number ‹m› such that ‹¬ P m›.
 ›

lemma infinite_descent: "(∧n. ¬ P n ==> ∃m¬ P m) ==> P n" for P :: "nat ==> bool"
  🍋 ‹compact version without explicit base case›
  by (induct n rule: less_induct) auto

lemma infinite_descent0 [case_names 0 smaller]:
  fixes P :: "nat ==> bool"
  assumes "P 0"
    and "∧n. n > 0 ==> ¬ P n ==> ∃m. m < n ∧ ¬ P m"
  shows "P n"
proof (rule infinite_descent)
  fix n
  show "¬ P n ==> ∃m¬ P m"
    using assms by (cases "n > 0") auto
qed

text ‹
  Infinite descent using a mapping to ‹nat›:
  ‹P x› is true for all ‹x ∈ D› if there exists a ‹V ∈ D ==> nat› and
  🪙 case ``0'': given ‹V x = 0› prove ‹P x›
  🪙 ``smaller'': given ‹V x > 0› and ‹¬ P x› prove
  there exists a ‹y ∈ D› such that ‹V y 🚫 x› and ‹¬ P y›.
 ›
corollary infinite_descent0_measure [case_names 0 smaller]:
  fixes V :: "'a ==> nat"
  assumes 1: "∧x. V x = 0 ==> P x"
    and 2: "∧x. V x > 0 ==> ¬ P x ==> ∃y. V y < V x ∧ ¬ P y"
  shows "P x"
proof -
  obtain n where "n = V x" by auto
  moreover have "∧x. V x = n ==> 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 ∧ ¬ P x" by auto
    with 2 obtain y where "V y < V x ∧ ¬ P y" by auto
    with * obtain m where "m = V y ∧ m < n ∧ ¬ P y" by auto
    then show ?case by auto
  qed
  ultimately show "P x" by auto
qed

text ‹Again, without explicit base case:›
lemma infinite_descent_measure:
  fixes V :: "'a ==> nat"
  assumes "∧x. ¬ P x ==> ∃y. V y < V x ∧ ¬ P y"
  shows "P x"
proof -
  from assms obtain n where "n = V x" by auto
  moreover have "∧x. V x = n ==> P x"
  proof -
    have "∃m < V x. ∃y. V y = m ∧ ¬ P y" if "¬ P x" for x
      using assms and that by auto
    then show "∧x. V x = n ==> P x"
      by (induct n rule: infinite_descent, auto)
  qed
  ultimately show "P x" by auto
qed

text ‹A (clumsy) way of lifting ‹🚫close> monotonicity to ‹≤› monotonicity›
 lemma less_mono_imp_le_mono:
  fixes f :: "nat ==> nat"
  and i j :: nat
  assumes "∧i j::nat. i 🚫 ==> f i 🚫 j"
  and "i ≤ j"
  shows "f i ≤ f j"
  using assms by (auto simp add: order_le_less)
 
 
 text ‹non-strict, in 1st argument›
 lemma add_le_mono1: "i ≤ j ==> i + k ≤ j + k"
  for i j k :: nat
  by (rule add_right_mono)
 
 text ‹non-strict, in both arguments›
 lemma add_le_mono: "i ≤ j ==> k ≤ l ==> i + k ≤ j + l"
  for i j k l :: nat
  by (rule add_mono)
 
 lemma le_add2: "n ≤ m + n"
  for m n :: nat
  by simp
 
 lemma le_add1: "n ≤ n + m"
  for m n :: nat
  by simp
 
 lemma less_add_Suc1: "i 🚫 (i + m)"
  by (rule le_less_trans, rule le_add1, rule lessI)
 
 lemma less_add_Suc2: "i 🚫 (m + i)"
  by (rule le_less_trans, rule le_add2, rule lessI)
 
 lemma less_iff_Suc_add: "m 🚫 ⟷ (∃k. n = Suc (m + k))"
  by (iprover intro!: less_add_Suc1 less_imp_Suc_add)
 
 lemma trans_le_add1: "i ≤ j ==> i ≤ j + m"
  for i j m :: nat
  by (rule le_trans, assumption, rule le_add1)
 
 lemma trans_le_add2: "i ≤ j ==> i ≤ m + j"
  for i j m :: nat
  by (rule le_trans, assumption, rule le_add2)
 
 lemma trans_less_add1: "i 🚫 ==> i 🚫 + m"
  for i j m :: nat
  by (rule less_le_trans, assumption, rule le_add1)
 
 lemma trans_less_add2: "i 🚫 ==> i 🚫 + j"
  for i j m :: nat
  by (rule less_le_trans, assumption, rule le_add2)
 
 lemma add_lessD1: "i + j 🚫 ==> i 🚫"
  for i j k :: nat
  by (rule le_less_trans [of _ "i+j"]) (simp_all add: le_add1)
 
 lemma not_add_less1 [iff]: "¬ i + j 🚫"
  for i j :: nat
  by simp
 
 lemma not_add_less2 [iff]: "¬ j + i 🚫"
  for i j :: nat
  by simp
 
 lemma add_leD1: "m + k ≤ n ==> m ≤ n"
  for k m n :: nat
  by (rule order_trans [of _ "m + k"]) (simp_all add: le_add1)
 
 lemma add_leD2: "m + k ≤ n ==> k ≤ n"
  for k m n :: nat
  by (force simp add: add.commute dest: add_leD1)
 
 lemma add_leE: "m + k ≤ n ==> (m ≤ n ==> k ≤ n ==> R) ==> R"
  for k m n :: nat
  by (blast dest: add_leD1 add_leD2)
 
 text ‹needs ‹∧k› for ‹ac_simps› to work›
 lemma less_add_eq_less: "∧k. k 🚫 ==> m + l = k + n ==> m 🚫"
  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 ‹More results about difference›
 
 lemma Suc_diff_le: "n ≤ m ==> Suc m - n = Suc (m - n)"
  by (induct m n rule: diff_induct) simp_all
 
 lemma diff_less_Suc: "m - n 🚫 m"
  by (induct m n rule: diff_induct) (auto simp: less_Suc_eq)
 
 lemma diff_le_self [simp]: "m - n ≤ m"
  for m n :: nat
  by (induct m n rule: diff_induct) (simp_all add: le_SucI)
 
 lemma less_imp_diff_less: "j 🚫 ==> j - n 🚫"
  for j k n :: nat
  by (rule le_less_trans, rule diff_le_self)
 
 lemma diff_Suc_less [simp]: "0 🚫 ==> n - Suc i 🚫"
  by (cases n) (auto simp add: le_simps)
 
 lemma diff_add_assoc: "k ≤ j ==> (i + j) - k = i + (j - k)"
  for i j k :: nat
  by (fact ordered_cancel_comm_monoid_diff_class.diff_add_assoc)
 
 lemma add_diff_assoc [simp]: "k ≤ j ==> i + (j - k) = i + j - k"
  for i j k :: nat
  by (fact ordered_cancel_comm_monoid_diff_class.add_diff_assoc)
 
 lemma diff_add_assoc2: "k ≤ j ==> (j + i) - k = (j - k) + i"
  for i j k :: nat
  by (fact ordered_cancel_comm_monoid_diff_class.diff_add_assoc2)
 
 lemma add_diff_assoc2 [simp]: "k ≤ j ==> j - k + i = j + i - k"
  for i j k :: nat
  by (fact ordered_cancel_comm_monoid_diff_class.add_diff_assoc2)
 
 lemma le_imp_diff_is_add: "i ≤ j ==> (j - i = k) = (j = k + i)"
  for i j k :: nat
  by auto
 
 lemma diff_is_0_eq [simp]: "m - n = 0 ⟷ m ≤ n"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all
 
 lemma diff_is_0_eq' [simp]: "m ≤ n ==> m - n = 0"
  for m n :: nat
  by simp
 
 lemma zero_less_diff [simp]: "0 🚫 - m ⟷ m 🚫"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all
 
 lemma less_imp_add_positive:
  assumes "i 🚫"
  shows "∃k::nat. 0 🚫 ∧ i + k = j"
 proof
  from assms show "0 🚫 - i ∧ i + (j - i) = j"
  by (simp add: order_less_imp_le)
 qed
 
 text ‹a nice rewrite for bounded subtraction›
 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) ⟷ (a 🚫 ⟶ P 0) ∧ (∀d. a = b + d ⟶ P d)"
  for a b :: nat
  🍋 ‹elimination of ‹-› on ‹nat›\›
  by (cases "a 🚫") (auto simp add: not_less le_less dest!: add_eq_self_zero [OF sym])
 
 lemma nat_diff_split_asm: "P (a - b) ⟷ ¬ (a 🚫 ∧ ¬ P 0 ∨ (∃d. a = b + d ∧ ¬ P d))"
  for a b :: nat
  🍋 ‹elimination of ‹-› on ‹nat› in assumptions›
  by (auto split: nat_diff_split)
 
 lemmas nat_diff_splits = nat_diff_split nat_diff_split_asm
 
 lemma Suc_pred': "0 🚫 ==> 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 🚫 ==> 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 ≡ f (Suc n)"
  by (fact Let_def)
 
 
 subsubsection ‹Monotonicity of multiplication›
 
 lemma mult_le_mono1: "i ≤ j ==> i * k ≤ j * k"
  for i j k :: nat
  by (simp add: mult_right_mono)
 
 lemma mult_le_mono2: "i ≤ j ==> k * i ≤ k * j"
  for i j k :: nat
  by (simp add: mult_left_mono)
 
 text ‹‹≤› monotonicity, BOTH arguments›
 lemma mult_le_mono: "i ≤ j ==> k ≤ l ==> i * k ≤ j * l"
  for i j k l :: nat
  by (simp add: mult_mono)
 
 lemma mult_less_mono1: "i 🚫 ==> 0 🚫 ==> i * k 🚫 * k"
  for i j k :: nat
  by (simp add: mult_strict_right_mono)
 
 text ‹Differs from the standard ‹zero_less_mult_iff› in that there are no negative numbers.›
 lemma nat_0_less_mult_iff [simp]: "0 🚫 * n ⟷ 0 🚫 ∧ 0 🚫"
  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 ≤ m * n ⟷ Suc 0 ≤ m ∧ Suc 0 ≤ 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 🚫 * k ⟷ 0 🚫 ∧ m 🚫"
  for k m n :: nat
 proof (intro iffI conjI)
  assume m: "m * k 🚫 * k"
  then show "0 🚫"
  by (cases k) auto
  show "m 🚫"
  proof (cases k)
  case 0
  then show ?thesis
  using m by auto
  next
  case (Suc k')
  then show ?thesis
  using m
  by (simp flip: linorder_not_le) (blast intro: add_mono mult_le_mono1)
  qed
 next
  assume "0 🚫 ∧ m 🚫"
  then show "m * k 🚫 * k"
  by (blast intro: mult_less_mono1)
 qed
 
 lemma mult_less_cancel1 [simp]: "k * m 🚫 * n ⟷ 0 🚫 ∧ m 🚫"
  for k m n :: nat
  by (simp add: mult.commute [of k])
 
 lemma mult_le_cancel1 [simp]: "k * m ≤ k * n ⟷ (0 🚫 ⟶ m ≤ n)"
  for k m n :: nat
  by (simp add: linorder_not_less [symmetric], auto)
 
 lemma mult_le_cancel2 [simp]: "m * k ≤ n * k ⟷ (0 🚫 ⟶ m ≤ n)"
  for k m n :: nat
  by (simp add: linorder_not_less [symmetric], auto)
 
 lemma Suc_mult_less_cancel1: "Suc k * m 🚫 k * n ⟷ m 🚫"
  by (subst mult_less_cancel1) simp
 
 lemma Suc_mult_le_cancel1: "Suc k * m ≤ Suc k * n ⟷ m ≤ n"
  by (subst mult_le_cancel1) simp
 
 lemma le_square: "m ≤ m * m"
  for m :: nat
  by (cases m) (auto intro: le_add1)
 
 lemma le_cube: "m ≤ m * (m * m)"
  for m :: nat
  by (cases m) (auto intro: le_add1)
 
 text ‹Lemma for ‹gcd›\›
 lemma mult_eq_self_implies_10:
  fixes m n :: nat
  assumes "m = m * n" shows "n = 1 ∨ m = 0"
 proof (rule disjCI)
  assume "m ≠ 0"
  show "n = 1"
  proof (cases n "1::nat" rule: linorder_cases)
  case greater
  show ?thesis
  using assms mult_less_mono2 [OF greater, of m] ‹m ≠ 0› by auto
  qed (use assms ‹m ≠ 0› in auto)
 qed
 
 lemma mono_times_nat:
  fixes n :: nat
  assumes "n > 0"
  shows "mono (times n)"
 proof
  fix m q :: nat
  assume "m ≤ q"
  with assms show "n * m ≤ n * q" by simp
 qed
 
 text ‹The lattice order on 🍋‹nat›.›
 
 instantiation nat :: distrib_lattice
 begin
 
 definition "(inf :: nat ==> nat ==> nat) = min"
 
 definition "(sup :: nat ==> nat ==> 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 ‹Natural operation of natural numbers on functions›
 
 text ‹
  We use the same logical constant for the power operations on
  functions and relations, in order to share the same syntax.
 ›
 
 consts compow :: "nat ==> 'a ==> 'a"
 
 abbreviation compower :: "'a ==> nat ==> 'a" (infixr ‹^^› 80)
  where "f ^^ n ≡ compow n f"
 
 notation (latex output)
  compower (‹(_🪙_🪙)› [1000] 1000)
 
 text ‹‹f ^^ n = f ∘ … ∘ f›, the ‹n›-fold composition of ‹f›\›
 
 overloading
  funpow ≡ "compow :: nat ==> ('a ==> 'a) ==> ('a ==> 'a)"
 begin
 
 primrec funpow :: "nat ==> ('a ==> 'a) ==> 'a ==> 'a"
  where
  "funpow 0 f = id"
  | "funpow (Suc n) f = f ∘ funpow n f"
 
 end
 
 lemma funpow_0 [simp]: "(f ^^ 0) x = x"
  by simp
 
 lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n ∘ f"
 proof (induct n)
  case 0
  then show ?case by simp
 next
  fix n
  assume "f ^^ Suc n = f ^^ n ∘ f"
  then show "f ^^ Suc (Suc n) = f ^^ Suc n ∘ f"
  by (simp add: o_assoc)
 qed
 
 lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right
 
 text ‹For code generation.›
 
 context
 begin
 
 qualified definition funpow :: "nat ==> ('a ==> 'a) ==> 'a ==> 'a"
  where funpow_code_def [code_abbrev]: "funpow = compow"
 
 lemma [code]:
  "funpow 0 f = id"
  "funpow (Suc n) f = f ∘ funpow n f"
  by (simp_all add: funpow_code_def)
 
 end
 
 lemma funpow_add: "f ^^ (m + n) = f ^^ m ∘ f ^^ n"
  by (induct m) simp_all
 
 lemma funpow_mult: "(f ^^ m) ^^ n = f ^^ (m * n)"
  for f :: "'a ==> '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 "… = (f ^^ n ∘ f ^^ 1) x" by (simp only: funpow_add)
  also have "… = (f ^^ n) (f x)" by simp
  finally show ?thesis .
 qed
 
 lemma comp_funpow: "comp f ^^ n = comp (f ^^ n)"
  for f :: "'a ==> 'a"
  by (induct n) simp_all
 
 lemma Suc_funpow[simp]: "Suc ^^ n = ((+) n)"
  by (induct n) simp_all
 
 lemma id_funpow[simp]: "id ^^ n = id"
  by (induct n) simp_all
 
 lemma funpow_mono: "mono f ==> A ≤ B ==> (f ^^ n) A ≤ (f ^^ n) B"
  for f :: "'a ==> ('a::order)"
  by (induct n) (auto simp: mono_def)
 
 lemma funpow_mono2:
  assumes "mono f"
  and "i ≤ j"
  and "x ≤ y"
  and "x ≤ f x"
  shows "(f ^^ i) x ≤ (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
 
 lemma inj_fn[simp]:
  fixes f::"'a ==> 'a"
  assumes "inj f"
  shows "inj (f^^n)"
 proof (induction n)
  case Suc thus ?case using inj_compose[OF assms Suc.IH] by (simp del: comp_apply)
 qed simp
 
 lemma surj_fn[simp]:
  fixes f::"'a ==> 'a"
  assumes "surj f"
  shows "surj (f^^n)"
 proof (induction n)
  case Suc thus ?case by (simp add: comp_surj[OF Suc.IH assms] del: comp_apply)
 qed simp
 
 lemma bij_fn[simp]:
  fixes f::"'a ==> 'a"
  assumes "bij f"
  shows "bij (f^^n)"
 by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]])
 
 lemma bij_betw_funpow: 🍋‹contributor ‹Lars Noschinski›\›
  assumes "bij_betw f S S" shows "bij_betw (f ^^ n) S S"
 proof (induct n)
  case 0 then show ?case by (auto simp: id_def[symmetric])
 next
  case (Suc n)
  then show ?case unfolding funpow.simps using assms by (rule bij_betw_trans)
 qed
 
 
 subsection ‹Kleene iteration›
 
 lemma Kleene_iter_lpfp:
  fixes f :: "'a::order_bot ==> 'a"
  assumes "mono f"
  and "f p ≤ p"
  shows "(f ^^ k) bot ≤ 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 ≤ (f ^^ k) bot"
  proof (rule lfp_lowerbound)
  show "f ((f ^^ k) bot) ≤ (f ^^ k) bot"
  using assms(2) by simp
  qed
  show "(f ^^ k) bot ≤ lfp f"
  using Kleene_iter_lpfp[OF assms(1)] lfp_unfold[OF assms(1)] by simp
 qed
 
 lemma mono_pow: "mono f ==> mono (f ^^ n)"
  for f :: "'a ==> 'a::order"
  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 ≤ lfp (f ^^ Suc n)"
  proof (rule lfp_lowerbound)
  have "f (lfp (f ^^ Suc n)) = lfp (λx. 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)) ≤ 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) ≤ 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 ≥ gfp (f ^^ Suc n)"
  proof (rule gfp_upperbound)
  have "f (gfp (f ^^ Suc n)) = gfp (λx. 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)) ≥ 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) ≥ gfp f"
  by (intro gfp_upperbound) (simp del: funpow.simps)
 qed
 
 lemma Kleene_iter_gpfp:
  fixes f :: "'a::order_top ==> 'a"
  assumes "mono f"
  and "p ≤ f p"
  shows "p ≤ (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 ≤ f ?rhs"
  using assms(2) by simp
  then show "?rhs ≤ ?lhs"
  by (rule gfp_upperbound)
  show "?lhs ≤ ?rhs"
  using Kleene_iter_gpfp[OF assms(1)] gfp_unfold[OF assms(1)] by simp
 qed
 
 
 subsection ‹Embedding of the naturals into any ‹semiring_1›: 🍋‹of_nat›\›
 
 context semiring_1
 begin
 
 definition of_nat :: "nat ==> '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 ==> 'a) ==> nat ==> 'a ==> 'a"
  where
  "of_nat_aux inc 0 i = i"
  | "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" 🍋 ‹tail recursive›
 
 lemma of_nat_code: "of_nat n = of_nat_aux (λi. i + 1) n 0"
 proof (induct n)
  case 0
  then show ?case by simp
 next
  case (Suc n)
  have "∧i. of_nat_aux (λi. i + 1) n (i + 1) = of_nat_aux (λi. i + 1) n i + 1"
  by (induct n) simp_all
  from this [of 0] have "of_nat_aux (λi. i + 1) n 1 = of_nat_aux (λi. i + 1) n 0 + 1"
  by simp
  with Suc show ?case
  by (simp add: add.commute)
 qed
 
 lemma of_nat_of_bool [simp]:
  "of_nat (of_bool P) = of_bool P"
  by auto
 
 end
 
 declare of_nat_code [code]
 
 context semiring_1_cancel
 begin
 
 lemma of_nat_diff [simp]:
  ‹of_nat (m - n) = of_nat m - of_nat n› if ‹n ≤ m›
 proof -
  from that obtain q where ‹m = n + q›
  by (blast dest: le_Suc_ex)
  then show ?thesis
  by simp
 qed
 
 lemma of_nat_diff_if: ‹of_nat (m - n) = (if n≤m then of_nat m - of_nat n else 0)›
  by (simp add: not_le less_imp_le)
 
 end
 
 text ‹Class for unital semirings with characteristic zero.
  Includes non-ordered rings like the complex numbers.›
 
 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 ⟷ m = n"
  by (auto intro: inj_of_nat injD)
 
 text ‹Special cases where either operand is zero›
 
 lemma of_nat_0_eq_iff [simp]: "0 = of_nat n ⟷ 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 ⟷ m = 0"
  by (fact of_nat_eq_iff [of m 0, unfolded of_nat_0])
 
 lemma of_nat_1_eq_iff [simp]: "1 = of_nat n ⟷ n=1"
  using of_nat_eq_iff by fastforce
 
 lemma of_nat_eq_1_iff [simp]: "of_nat n = 1 ⟷ n=1"
  using of_nat_eq_iff by fastforce
 
 lemma of_nat_neq_0 [simp]: "of_nat (Suc n) ≠ 0"
  unfolding of_nat_eq_0_iff by simp
 
 lemma of_nat_0_neq [simp]: "0 ≠ of_nat (Suc n)"
  unfolding of_nat_0_eq_iff by simp
 
 end
 
 class ring_char_0 = ring_1 + semiring_char_0
 
 lemma (in ordered_semiring_1) of_nat_0_le_iff [simp]: "0 ≤ of_nat n"
  by (induct n) simp_all
 
 context linordered_nonzero_semiring
 begin
 
 lemma of_nat_less_0_iff [simp]: "¬ of_nat m 🚫"
  by (simp add: not_less)
 
 lemma of_nat_mono[simp]: "i ≤ j ==> of_nat i ≤ of_nat j"
  by (auto simp: le_iff_add intro!: add_increasing2)
 
 lemma of_nat_less_iff [simp]: "of_nat m 🚫 n ⟷ m 🚫"
 proof(induct m n rule: diff_induct)
  case (1 m) then show ?case
  by auto
 next
  case (2 n) then show ?case
  by (simp add: add_pos_nonneg)
 next
  case (3 m n)
  then show ?case
  by (auto simp: add_commute [of 1] add_mono1 not_less add_right_mono leD)
 qed
 
 lemma of_nat_le_iff [simp]: "of_nat m ≤ of_nat n ⟷ m ≤ n"
  by (simp add: not_less [symmetric] linorder_not_less [symmetric])
 
 lemma less_imp_of_nat_less: "m 🚫 ==> of_nat m 🚫 n"
  by simp
 
 lemma of_nat_less_imp_less: "of_nat m 🚫 n ==> m 🚫"
  by simp
 
 text ‹Every ‹linordered_nonzero_semiring› has characteristic zero.›
 
 subclass semiring_char_0
  by standard (auto intro!: injI simp add: order.eq_iff)
 
 text ‹Special cases where either operand is zero›
 
 lemma of_nat_le_0_iff [simp]: "of_nat m ≤ 0 ⟷ m = 0"
  by (rule of_nat_le_iff [of _ 0, simplified])
 
 lemma of_nat_0_less_iff [simp]: "0 🚫 n ⟷ 0 🚫"
  by (rule of_nat_less_iff [of 0, simplified])
 
 end
 
 context linordered_nonzero_semiring
 begin
 
 lemma of_nat_max: "of_nat (max x y) = max (of_nat x) (of_nat y)"
  by (auto simp: max_def ord_class.max_def)
 
 lemma of_nat_min: "of_nat (min x y) = min (of_nat x) (of_nat y)"
  by (auto simp: min_def ord_class.min_def)
 
 end
 
 context linordered_semidom
 begin
 
 subclass linordered_nonzero_semiring ..
 
 subclass semiring_char_0 ..
 
 end
 
 context linordered_idom
 begin
 
 lemma abs_of_nat [simp]:
  "∣of_nat n∣ = of_nat n"
  by (simp add: abs_if)
 
 lemma sgn_of_nat [simp]:
  "sgn (of_nat n) = of_bool (n > 0)"
  by simp
 
 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 ‹The set of natural numbers›
 
 context semiring_1
 begin
 
 definition Nats :: "'a set" (‹ℕ›)
  where "ℕ = range of_nat"
 
 lemma of_nat_in_Nats [simp]: "of_nat n ∈ ℕ"
  by (simp add: Nats_def)
 
 lemma Nats_0 [simp]: "0 ∈ ℕ"
  using of_nat_0 [symmetric] unfolding Nats_def
  by (rule range_eqI)
 
 lemma Nats_1 [simp]: "1 ∈ ℕ"
  using of_nat_1 [symmetric] unfolding Nats_def
  by (rule range_eqI)
 
 lemma Nats_add [simp]: "a ∈ ℕ ==> b ∈ ℕ ==> a + b ∈ ℕ"
  unfolding Nats_def using of_nat_add [symmetric]
  by (blast intro: range_eqI)
 
 lemma Nats_mult [simp]: "a ∈ ℕ ==> b ∈ ℕ ==> a * b ∈ ℕ"
  unfolding Nats_def using of_nat_mult [symmetric]
  by (blast intro: range_eqI)
 
 lemma Nats_cases [cases set: Nats]:
  assumes "x ∈ ℕ"
  obtains (of_nat) n where "x = of_nat n"
  unfolding Nats_def
 proof -
  from ‹x ∈ ℕ› have "x ∈ 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 ∈ ℕ ==> (∧n. P (of_nat n)) ==> P x"
  by (rule Nats_cases) auto
 
 lemma Nats_nonempty [simp]: "ℕ ≠ {}"
  unfolding Nats_def by auto
 
 end
 
 lemma Nats_diff [simp]:
  fixes a:: "'a::linordered_idom"
  assumes "a ∈ ℕ" "b ∈ ℕ" "b ≤ a" shows "a - b ∈ ℕ"
 proof -
  obtain i where i: "a = of_nat i"
  using Nats_cases assms by blast
  obtain j where j: "b = of_nat j"
  using Nats_cases assms by blast
  have "j ≤ i"
  using ‹b ≤ a› i j of_nat_le_iff by blast
  then have *: "of_nat i - of_nat j = (of_nat (i-j) :: 'a)"
  by (simp add: of_nat_diff)
  then show ?thesis
  by (simp add: * i j)
 qed
 
 
 subsection ‹Further arithmetic facts concerning the natural numbers›
 
 lemma subst_equals:
  assumes "t = s" and "u = t"
  shows "u = s"
  using assms(2,1) by (rule trans)
 
 locale nat_arith
 begin
 
 lemma add1: "(A::'a::comm_monoid_add) ≡ k + a ==> A + b ≡ k + (a + b)"
  by (simp only: ac_simps)
 
 lemma add2: "(B::'a::comm_monoid_add) ≡ k + b ==> a + B ≡ k + (a + b)"
  by (simp only: ac_simps)
 
 lemma suc1: "A == k + a ==> Suc A ≡ k + Suc a"
  by (simp only: add_Suc_right)
 
 lemma rule0: "(a::'a::comm_monoid_add) ≡ a + 0"
  by (simp only: add_0_right)
 
 end
 
 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") =
  ‹K (try o Nat_Arith.cancel_eq_conv)›
 
 simproc_setup natless_cancel_sums
  ("(l::nat) + m 🚫" | "(l::nat) 🚫 + n" | "Suc m 🚫" | "m 🚫 n") =
  ‹K (try o Nat_Arith.cancel_less_conv)›
 
 simproc_setup natle_cancel_sums
  ("(l::nat) + m ≤ n" | "(l::nat) ≤ m + n" | "Suc m ≤ n" | "m ≤ Suc n") =
  ‹K (try o Nat_Arith.cancel_le_conv)›
 
 simproc_setup natdiff_cancel_sums
  ("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
  ‹K (try o Nat_Arith.cancel_diff_conv)›
 
 context preorder
 begin
 
 lemma lift_Suc_mono_le:
  assumes mono: "∧n. f n ≤ f (Suc n)"
  and "n ≤ n'"
  shows "f n ≤ f n'"
 proof (cases "n 🚫'")
  case True
  then show ?thesis
  by (induct n n' rule: less_Suc_induct) (auto intro: mono order.trans)
 next
  case False
  with ‹n ≤ n'› show ?thesis by auto
 qed
 
 lemma lift_Suc_antimono_le:
  assumes mono: "∧n. f n ≥ f (Suc n)"
  and "n ≤ n'"
  shows "f n ≥ f n'"
 proof (cases "n 🚫'")
  case True
  then show ?thesis
  by (induct n n' rule: less_Suc_induct) (auto intro: mono order.trans)
 next
  case False
  with ‹n ≤ n'› show ?thesis by auto
 qed
 
 lemma lift_Suc_mono_less:
  assumes mono: "∧n. f n 🚫 (Suc n)"
  and "n 🚫'"
  shows "f n 🚫 n'"
  using ‹n 🚫'› by (induct n n' rule: less_Suc_induct) (auto intro: mono order.strict_trans)
 
 lemma lift_Suc_mono_less_iff: "(∧n. f n 🚫 (Suc n)) ==> f n 🚫 m ⟷ n 🚫"
  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 ⟷ (∀n. f n ≤ f (Suc n))"
  unfolding mono_def by (auto intro: lift_Suc_mono_le [of f])
 
 lemma antimono_iff_le_Suc: "antimono f ⟷ (∀n. f (Suc n) ≤ f n)"
  unfolding antimono_def by (auto intro: lift_Suc_antimono_le [of f])
 
 lemma strict_mono_Suc_iff: "strict_mono f ⟷ (∀n. f n 🚫 (Suc n))"
 proof (intro iffI strict_monoI)
  assume *: "∀n. f n 🚫 (Suc n)"
  fix m n :: nat assume "m 🚫"
  thus "f m 🚫 n"
  by (induction rule: less_Suc_induct) (use * in auto)
 qed (auto simp: strict_mono_def)
 
 lemma strict_mono_add: "strict_mono (λn::'a::linordered_semidom. n + k)"
  by (auto simp: strict_mono_def)
 
 lemma mono_nat_linear_lb:
  fixes f :: "nat ==> nat"
  assumes "∧m n. m 🚫 ==> f m 🚫 n"
  shows "f m + k ≤ f (m + k)"
 proof (induct k)
  case 0
  then show ?case by simp
 next
  case (Suc k)
  then have "Suc (f m + k) ≤ Suc (f (m + k))" by simp
  also from assms [of "m + k" "Suc (m + k)"] have "Suc (f (m + k)) ≤ f (Suc (m + k))"
  by (simp add: Suc_le_eq)
  finally show ?case by simp
 qed
 
 lemma bex_const1_if_mono_below_diag: fixes f :: "nat ==> nat" assumes "mono f"
 shows "f n 🚫 ==> ∃i🚫 f(Suc i) = f i"
 proof(induction n)
  case 0
  then show ?case by simp
 next
  case (Suc n)
  have *: "f n ≤ f(Suc n)" using assms[simplified mono_iff_le_Suc] by blast
  from Suc.prems[simplified less_Suc_eq]
  show ?case
  proof
  assume "f(Suc n) 🚫"
  from order.strict_trans1[OF * this]
  show ?thesis using Suc.IH less_SucI by blast
  next
  assume "f(Suc n) = n"
  from order.strict_trans1[OF * Suc.prems, simplified less_Suc_eq]
  show ?case
  proof
  assume "f n 🚫"
  thus ?thesis using Suc.IH less_SucI by blast
  next
  assume "f n = n"
  with ‹f(Suc n) = n› show ?thesis by auto
  qed
  qed
 qed
 
 lemma bex_const1_if_mono_below_diag_Suc:
 fixes f :: "nat ==> nat" assumes "mono f" "f(Suc m) ≤ m"
 shows "∃i≤m. f (Suc i) = f i"
 using bex_const1_if_mono_below_diag[OF assms(1), of "Suc m"] assms(2) less_Suc_eq_le by blast
 
 
 text ‹Subtraction laws, mostly by Clemens Ballarin›
 
 lemma diff_less_mono:
  fixes a b c :: nat
  assumes "a 🚫" and "c ≤ a"
  shows "a - c 🚫 - 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 🚫 - k ⟷ i + k 🚫"
  for i j k :: nat
  by (cases "k ≤ j") (auto simp add: not_le dest: less_imp_Suc_add le_Suc_ex)
 
 lemma less_diff_conv2: "k ≤ j ==> j - k 🚫 ⟷ j 🚫 + k"
  for j k i :: nat
  by (auto dest: le_Suc_ex)
 
 lemma le_diff_conv: "j - k ≤ i ⟷ j ≤ i + k"
  for j k i :: nat
  by (cases "k ≤ j") (auto simp add: not_le dest!: less_imp_Suc_add le_Suc_ex)
 
 lemma diff_diff_cancel [simp]: "i ≤ n ==> n - (n - i) = i"
  for i n :: nat
  by (auto dest: le_Suc_ex)
 
 lemma diff_less [simp]: "0 🚫 ==> 0 🚫 ==> m - n 🚫"
  for i n :: nat
  by (auto dest: less_imp_Suc_add)
 
 text ‹Simplification of relational expressions involving subtraction›
 
 lemma diff_diff_eq: "k ≤ m ==> k ≤ n ==> 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 ≤ m ==> k ≤ n ==> m - k = n - k ⟷ m = n"
  for m n k :: nat
  by (auto dest: le_Suc_ex)
 
 lemma less_diff_iff: "k ≤ m ==> k ≤ n ==> m - k 🚫 - k ⟷ m 🚫"
  for m n k :: nat
  by (auto dest!: le_Suc_ex)
 
 lemma le_diff_iff: "k ≤ m ==> k ≤ n ==> m - k ≤ n - k ⟷ m ≤ n"
  for m n k :: nat
  by (auto dest!: le_Suc_ex)
 
 lemma le_diff_iff': "a ≤ c ==> b ≤ c ==> c - a ≤ c - b ⟷ b ≤ a"
  for a b c :: nat
  by (force dest: le_Suc_ex)
 
 
 text ‹(Anti)Monotonicity of subtraction -- by Stephan Merz›
 
 lemma diff_le_mono: "m ≤ n ==> m - l ≤ 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 ≤ n ==> l - n ≤ 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 🚫 ==> m 🚫 ==> l - n 🚫 - m"
  for m n l :: nat
  by (auto dest: less_imp_Suc_add split: nat_diff_split)
 
 lemma diffs0_imp_equal: "m - n = 0 ==> n - m = 0 ==> 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 "∧n. n ∈ N ==> k ≤ n"
  shows "inj_on (λn. n - k) N"
 proof (rule inj_onI)
  fix x y
  assume a: "x ∈ N" "y ∈ 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 ‹Rewriting to pull differences out›
 
 lemma diff_diff_right [simp]: "k ≤ j ==> i - (j - k) = i + k - j"
  for i j k :: nat
  by (fact diff_diff_right)
 
 lemma diff_Suc_diff_eq1 [simp]:
  assumes "k ≤ 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 ≤ 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 ≤ 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 🚫"
  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 🚫 ==> Suc 0 🚫 ==> Suc 0 🚫 * n"
  using less_1_mult [of n m] by (simp add: ac_simps)
 
 lemma n_less_m_mult_n: "0 🚫 ==> Suc 0 🚫 ==> n 🚫 * n"
  using mult_strict_right_mono [of 1 m n] by simp
 
 lemma n_less_n_mult_m: "0 🚫 ==> Suc 0 🚫 ==> n 🚫 * m"
  using mult_strict_left_mono [of 1 m n] by simp
 
 
 text ‹Induction starting beyond zero›
 
 lemma nat_induct_at_least [consumes 1, case_names base Suc]:
  "P n" if "n ≥ m" "P m" "∧n. n ≥ m ==> P n ==> P (Suc n)"
 proof -
  define q where "q = n - m"
  with ‹n ≥ m› have "n = m + q"
  by simp
  moreover have "P (m + q)"
  by (induction q) (use that in simp_all)
  ultimately show "P n"
  by simp
 qed
 
 lemma nat_induct_non_zero [consumes 1, case_names 1 Suc]:
  "P n" if "n > 0" "P 1" "∧n. n > 0 ==> P n ==> P (Suc n)"
 proof -
  from ‹n > 0› have "n ≥ 1"
  by (cases n) simp_all
  moreover note ‹P 1›
  moreover have "∧n. n ≥ 1 ==> P n ==> P (Suc n)"
  using ‹∧n. n > 0 ==> P n ==> P (Suc n)›
  by (simp add: Suc_le_eq)
  ultimately show "P n"
  by (rule nat_induct_at_least)
 qed
 
 
 text ‹Specialized induction principles that work "backwards":›
 
 lemma inc_induct [consumes 1, case_names base step]:
  assumes less: "i ≤ j"
  and base: "P j"
  and step: "∧n. i ≤ n ==> n 🚫 ==> P (Suc n) ==> 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 ≠ j" by auto
  with Suc have "n 🚫" by (simp add: less_le)
  from ‹Suc d = j - n› 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 ‹n 🚫› have "Suc n ≤ j" by (simp add: Suc_le_eq)
  ultimately have "P (Suc n)"
  proof (rule Suc.hyps)
  fix q
  assume "Suc n ≤ q"
  then have "n ≤ q" by (simp add: Suc_le_eq less_imp_le)
  moreover assume "q 🚫"
  moreover assume "P (Suc q)"
  ultimately show "P q" by (rule Suc.prems)
  qed
  with order_refl ‹n 🚫› show "P n" by (rule Suc.prems)
 qed
 
 lemma strict_inc_induct [consumes 1, case_names base step]:
  assumes less: "i 🚫"
  and base: "∧i. j = Suc i ==> P i"
  and step: "∧i. Suc i 🚫 ==> P (Suc i) ==> P i"
  shows "P i"
 using less proof (induct "j - i - 1" arbitrary: i)
  case (0 i)
  from ‹i 🚫› 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 ‹Suc d = j - i - 1› 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 ≠ 0" by auto
  then have "Suc i 🚫" by (simp add: not_le)
  ultimately have "P (Suc i)" by (rule Suc.hyps)
  with ‹Suc i 🚫› show "P i" by (rule step)
 qed
 
 lemma zero_induct_lemma: "P k ==> (∧n. P (Suc n) ==> P n) ==> P (k - i)"
  using inc_induct[of "k - i" k P, simplified] by blast
 
 lemma zero_induct: "P k ==> (∧n. P (Suc n) ==> P n) ==> P 0"
  using inc_induct[of 0 k P] by blast
 
 text ‹Further induction rule similar to @{thm inc_induct}.›
 
 lemma dec_induct [consumes 1, case_names base step]:
  "i ≤ j ==> P i ==> (∧n. i ≤ n ==> n 🚫 ==> P n ==> P (Suc n)) ==> P j"
 proof (induct j arbitrary: i)
  case 0
  then show ?case by simp
 next
  case (Suc j)
  from Suc.prems consider "i ≤ j" | "i = Suc j"
  by (auto simp add: le_Suc_eq)
  then show ?case
  proof cases
  case 1
  moreover have "j 🚫 j" by simp
  moreover have "P j" using ‹i ≤ j› ‹P i›
  proof (rule Suc.hyps)
  fix q
  assume "i ≤ q"
  moreover assume "q 🚫" then have "q 🚫 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 ‹P i› show "P (Suc j)" by simp
  qed
 qed
 
 lemma transitive_stepwise_le:
  assumes "m ≤ n" "∧x. R x x" "∧x y z. R x y ==> R y z ==> R x z" and "∧n. R n (Suc n)"
  shows "R m n"
 using ‹m ≤ n›
  by (induction rule: dec_induct) (use assms in blast)+
 
 
 subsubsection ‹Greatest operator›
 
 lemma ex_has_greatest_nat:
  "P (k::nat) ==> ∀y. P y ⟶ y ≤ b ==> ∃x. P x ∧ (∀y. P y ⟶ y ≤ x)"
 proof (induction "b-k" arbitrary: b k rule: less_induct)
  case less
  show ?case
  proof cases
  assume "∃n>k. P n"
  then obtain n where "n>k" "P n" by blast
  have "n ≤ b" using ‹P n› less.prems(2) by auto
  hence "b-n 🚫"
  by(rule diff_less_mono2[OF ‹k🚫› less_le_trans[OF ‹k🚫›]])
  from less.hyps[OF this ‹P n› less.prems(2)]
  show ?thesis .
  next
  assume "¬ (∃n>k. P n)"
  hence "∀y. P y ⟶ y ≤ k" by (auto simp: not_less)
  thus ?thesis using less.prems(1) by auto
  qed
 qed
 
 lemma
  fixes k::nat
  assumes "P k" and minor: "∧y. P y ==> y ≤ b"
  shows GreatestI_nat: "P (Greatest P)"
  and Greatest_le_nat: "k ≤ Greatest P"
 proof -
  obtain x where "P x" "∧y. P y ==> y ≤ x"
  using assms ex_has_greatest_nat by blast
  with ‹P k› show "P (Greatest P)" "k ≤ Greatest P"
  using GreatestI2_order by blast+
 qed
 
 lemma GreatestI_ex_nat:
  "[ ∃k::nat. P k; ∧y. P y ==> y ≤ b ] ==> P (Greatest P)"
  by (blast intro: GreatestI_nat)
 
 
 subsection ‹Monotonicity of ‹funpow›\›
 
 lemma funpow_increasing: "m ≤ n ==> mono f ==> (f ^^ n) ⊤ ≤ (f ^^ m) ⊤"
  for f :: "'a::order_top ==> '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 ≤ n ==> mono f ==> (f ^^ m) ⊥ ≤ (f ^^ n) ⊥"
  for f :: "'a::order_bot ==> '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 ==> mono (λi. (Q ^^ i) ⊥)"
  for Q :: "'a::order_bot ==> 'a"
  by (auto intro!: funpow_decreasing simp: mono_def)
 
 lemma antimono_funpow: "mono Q ==> antimono (λi. (Q ^^ i) ⊤)"
  for Q :: "'a::order_top ==> 'a"
  by (auto intro!: funpow_increasing simp: antimono_def)
 
 
 subsection ‹Kleene's fixed point theorem for continuous functions›
 
 text ‹Kleene's fixed point theorem shows that the ‹lfp› of a omega-continuous function
 can be obtained as the supremum of an omega chain. It only requires an omega-complete partial order.
 We prove it here for complete lattices because the latter structures are not defined in Main
 but the theorem is also useful for complete lattices.›
 
 definition omega_chain :: "(nat ==> ('a::complete_lattice)) ==> bool" where
 "omega_chain C = (∀i. C i ≤ C(Suc i))"
 
 definition omega_cont :: "(('a::complete_lattice) ==> ('b::complete_lattice)) ==> bool" where
 "omega_cont f = (∀C. omega_chain C ⟶ f(SUP n. C n) = (SUP n. f(C n)))"
 
 lemma omega_chain_mono: "omega_chain C ==> i ≤ j ==> C i ≤ C j"
 unfolding omega_chain_def using lift_Suc_mono_le[of C]
 by(induction "j-i" arbitrary: i j)auto
 
 lemma mono_if_omega_cont: fixes f :: "('a::complete_lattice) ==> ('b::complete_lattice)"
  assumes "omega_cont f" shows "mono f"
 proof
  fix a b :: "'a" assume "a ≤ b"
  let ?C = "λn::nat. if n=0 then a else b"
  have *: "omega_chain ?C" using ‹a ≤ b› by(auto simp: omega_chain_def)
  have "f a ≤ sup (f a) (SUP n. f(?C n))" by(rule sup.cobounded1)
  also have "… = sup (f(?C 0)) (SUP n. f(?C n))" by (simp)
  also have "… = (SUP n. f (?C n))" using SUP_absorb[OF UNIV_I] .
  also have "… = f (SUP n. ?C n)"
  using assms * by (simp add: omega_cont_def del: if_image_distrib)
  also have "f (SUP n. ?C n) = f b"
  using ‹a ≤ b› by (auto simp add: gt_ex sup.absorb2 split: if_splits)
  finally show "f a ≤ f b" .
 qed
 
 lemma omega_chain_iterates: fixes f :: "('a::complete_lattice) ==> 'a"
  assumes "mono f" shows "omega_chain(λn. (f^^n) bot)"
 proof-
  have "(f ^^ n) bot ≤ (f ^^ Suc n) bot" for n
  proof (induction n)
  case 0 show ?case by simp
  next
  case (Suc n) thus ?case using assms by (auto simp: mono_def)
  qed
  thus ?thesis by(auto simp: omega_chain_def assms)
 qed
 
 theorem Kleene_lfp:
  assumes "omega_cont f" shows "lfp f = (SUP n. (f^^n) bot)" (is "_ = ?U")
 proof(rule Orderings.antisym)
  from assms mono_if_omega_cont
  have mono: "(f ^^ n) bot ≤ (f ^^ Suc n) bot" for n
  using funpow_decreasing [of n "Suc n"] by auto
  show "lfp f ≤ ?U"
  proof (rule lfp_lowerbound)
  have "f ?U = (SUP n. (f^^Suc n) bot)"
  using omega_chain_iterates[OF mono_if_omega_cont[OF assms]] assms
  by(simp add: omega_cont_def)
  also have "… = ?U" using mono by(blast intro: SUP_eq)
  finally show "f ?U ≤ ?U" by simp
  qed
 next
  have "(f^^n) bot ≤ p" if "f p ≤ p" for n p
  proof -
  show ?thesis
  proof(induction n)
  case 0 show ?case by simp
  next
  case Suc
  from monoD[OF mono_if_omega_cont[OF assms] Suc] ‹f p ≤ p›
  show ?case by simp
  qed
  qed
  thus "?U ≤ lfp f"
  using lfp_unfold[OF mono_if_omega_cont[OF assms]]
  by (simp add: SUP_le_iff)
 qed
 
 
 subsection ‹The divides relation on 🍋‹nat›\›
 
 lemma dvd_1_left [iff]: "Suc 0 dvd k"
  by (simp add: dvd_def)
 
 lemma dvd_1_iff_1 [simp]: "m dvd Suc 0 ⟷ m = Suc 0"
  by (simp add: dvd_def)
 
 lemma nat_dvd_1_iff_1 [simp]: "m dvd 1 ⟷ m = 1"
  for m :: nat
  by (simp add: dvd_def)
 
 lemma dvd_antisym: "m dvd n ==> n dvd m ==> 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 ==> k dvd n ==> k dvd (m - n)"
  for k m n :: nat
  unfolding dvd_def by (blast intro: right_diff_distrib' [symmetric])
 
 lemma dvd_diffD:
  fixes k m n :: nat
  assumes "k dvd m - n" "k dvd n" "n ≤ m"
  shows "k dvd m"
 proof -
  have "k dvd n + (m - n)"
  using assms by (blast intro: dvd_add)
  with assms show ?thesis
  by simp
 qed
 
 lemma dvd_diffD1: "k dvd m - n ==> k dvd m ==> n ≤ m ==> 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 🚫"
  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 ‹0 🚫› have "n = m * q" by (auto simp add: mult_left_cancel)
  then show ?thesis ..
 qed
 
 lemma dvd_mult_cancel1:
  fixes m n :: nat
  assumes "0 🚫"
  shows "m * n dvd m ⟷ n = 1"
 proof
  assume "m * n dvd m"
  then have "m * n dvd m * 1"
  by simp
  then have "n dvd 1"
  by (iprover intro: assms dvd_mult_cancel)
  then show "n = 1"
  by auto
 qed auto
 
 lemma dvd_mult_cancel2: "0 🚫 ==> n * m dvd m ⟷ 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 ==> 0 🚫 ==> k ≤ n"
  for k n :: nat
  by (auto elim!: dvdE) (auto simp add: gr0_conv_Suc)
 
 lemma nat_dvd_not_less: "0 🚫 ==> m 🚫 ==> ¬ 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 ≤ n"
  shows "m dvd n ⟷ 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 ⟷ n 🚫 ∨ m dvd n"
  for m n :: nat
  by (cases "n 🚫") (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 ≤ n" "q ≤ r * m"
  shows "m dvd n - q ⟷ m dvd n + (r * m - q)"
 proof -
  have "m dvd n - q ⟷ m dvd r * m + (n - q)"
  using dvd_add_times_triv_left_iff [of m r] by simp
  also from assms have "… ⟷ m dvd r * m + n - q" by simp
  also from assms have "… ⟷ m dvd (r * m - q) + n" by simp
  also have "… ⟷ m dvd n + (r * m - q)" by (simp add: add.commute)
  finally show ?thesis .
 qed
 
 
 subsection ‹Aliasses›
 
 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 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')
 
(*Used in AUTO2 and Groups.le_diff_conv2 (with variables renamed) doesn't work for some reason*)
lemma le_diff_conv2: "k ≤ j ==> (i ≤ j - k) = (i + k ≤ j)"
  for i j k :: nat
  by (fact le_diff_conv2) 

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 ‹Size of a datatype value›

class size =
  fixes size :: "'a ==> nat" 🍋 ‹see further theory ‹Wellfounded›\›

instantiation nat :: size
begin

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

instance ..

end

lemmas size_nat = size_nat_def

lemma size_neq_size_imp_neq: "size x ≠ size y ==> x ≠ y"
  by (erule contrapos_nn) (rule arg_cong)


subsection ‹Code module namespace›

code_identifier
  code_module Nat ⇀ (SML) Arith and (OCaml) Arith and (Haskell) Arith

hide_const (open) of_nat_aux

end