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products/sources/formale Sprachen/PVS/analysis/ (Beweissystem der NASA Version 6.0.9^©) Datei vom 28.9.2014 mit Größe 17 kB

Quelle Nat.thy Sprache: Isabelle

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

section \<open>Natural numbers\<close>

theory Nat
imports Inductive Typedef Fun Rings
begin

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

typedecl ind

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

subsection \<open>Type nat\<close>

text \<open>Type definition\<close>

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

\<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
  by (erule nat_induct0) auto

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 \ (nat \ 'a \ 'a) \ nat \ 'a"
  where "rec_nat \ old.rec_nat"

declare nat.sel[code del]

hide_const (open) Nat.pred \<comment> \<open>hide everything related to the selector\<close>

lemma nat_exhaust [case_names 0 Suc, cases type: nat]:
  "(y = 0 \ P) \ (\nat. y = Suc nat \ P) \ 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 "\n. P n \ 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>\<open>nat\<close>);
    val recx = Logic.varify_types_global \<^term>\<open>rec_nat\<close>;
    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>\<open>nat\<close>] _ _ 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 = [], 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
\<close>

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

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 \<open>A special form of induction for reasoning about \<^term>\<open>m < n\<close> and \<^term>\<open>m - n\<close>.\<close>
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 \<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 [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 \<open>Addition\<close>

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

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 \<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: "Suc n - 1 = n"
  by simp

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

subsubsection \<open>Multiplication\<close>

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 \<open>Orders on \<^typ>\<open>nat\<close>\<close>

subsubsection \<open>Operation definition\<close>

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 \<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: "\ 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 \<open>"Less than" is antisymmetric, sort of.\<close>
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 \<open>Inductive (?) properties\<close>

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 \<open>Can be used with \<open>less_Suc_eq\<close> to get \<^prop>\<open>n = m \<or> n < m\<close>.\<close>
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 \<open>Properties of "less than or equal".\<close>

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 \<open>Stronger version of \<open>Suc_leD\<close>.\<close>
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 \<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 \ 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 \<open>Useful with \<open>blast\<close>.\<close>
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 \<open>This theorem is useful with \<open>blast\<close>\<close>
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 \<open>Useful in certain inductive arguments\<close>
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 \<open>@{term mono} (non-strict) doesn't imply increasing, as the function could be constant\<close>
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 \<open>Monotonicity of Addition\<close>

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 \<open>strict, in 1st argument\<close>
lemma add_less_mono1: "i < j \ 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:
  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 \<open>i < j\<close> 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 \<open>m < n\<close>
  obtains q where \<open>n = Suc (m + q)\<close>
  using assms by (auto dest: less_imp_Suc_add intro: that)

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>\<open>n \<le> m\<close> then \<^term>\<open>n + (m - n) = m\<close>.\<close>
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 \<open>The naturals form an ordered \<open>semidom\<close> and a \<open>dioid\<close>.\<close>

instance nat :: discrete_linordered_semidom
proof
  fix m n q :: nat
  show \<open>0 < (1::nat)\<close>
    by simp
  show \<open>m \<le> n \<Longrightarrow> q + m \<le> q + n\<close>
    by simp
  show \<open>m < n \<Longrightarrow> 0 < q \<Longrightarrow> q * m < q * n\<close>
    by (simp add: mult_less_mono2)
  show \<open>m \<noteq> 0 \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> m * n \<noteq> 0\<close>
    by simp
  show \<open>n \<le> m \<Longrightarrow> (m - n) + n = m\<close>
    by (simp add: add_diff_inverse_nat add.commute linorder_not_less)
  show \<open>m < n \<longleftrightarrow> m + 1 \<le> n\<close>
    by (simp add: Suc_le_eq)
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>\<open>min\<close> and \<^term>\<open>max\<close>\<close>

global_interpretation bot_nat_0: ordering_top \<open>(\<ge>)\<close> \<open>(>)\<close> \<open>0::nat\<close>
  by standard simp

global_interpretation max_nat: semilattice_neutr_order max \<open>0::nat\<close> \<open>(\<ge>)\<close> \<open>(>)\<close>
  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 \<open>Additional theorems about \<^term>\<open>(\<le>)\<close>\<close>

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

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 \<section>: "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 \<section> by (simp add: assms)
    next
      case Suc
      with \<section> 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 "\ki\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 \<open>P n\<close> 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 \<equiv> "f a" arbitrary: a rule: less_induct) (auto intro: step)

text \<open>old style induction rules:\<close>
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\<open>An induction rule for establishing binary relations\<close>
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 \<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: "(\n. \ P n \ \m P m) \ P n" for P :: "nat \ 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 \ 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 \<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 \ 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 \<open>Again, without explicit base case:\<close>
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 \<open>A (clumsy) way of lifting \<open><\<close> monotonicity to \<open>\<le>\<close> monotonicity\<close>
lemma less_mono_imp_le_mono:
  fixes f :: "nat \ nat"
    and i j :: nat
  assumes "\i j::nat. i < j \ f i < f j"
    and "i \ j"
  shows "f i \ f j"
  using assms by (auto simp add: order_le_less)

text \<open>non-strict, in 1st argument\<close>
lemma add_le_mono1: "i \ j \ i + k \ 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 \ 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 < 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 \ (\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 < j \ i < j + m"
  for i j m :: nat
  by (rule less_le_trans, assumption, rule le_add1)

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

lemma add_lessD1: "i + j < k \ 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]: "\ i + j < i"
  for i j :: nat
  by simp

lemma not_add_less2 [iff]: "\ j + i < 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 \<open>needs \<open>\<And>k\<close> for \<open>ac_simps\<close> to work\<close>
lemma less_add_eq_less: "\k. k < l \ m + l = k + n \ 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 \ m \ Suc m - n = Suc (m - n)"
  by (induct m n rule: diff_induct) simp_all

lemma diff_less_Suc: "m - n < Suc 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 < k \ j - n < k"
  for j k n :: nat
  by (rule le_less_trans, rule diff_le_self)

lemma diff_Suc_less [simp]: "0 < n \ n - Suc i < n"
  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 < n - m \ m < n"
  for m n :: nat
  by (induct m n rule: diff_induct) simp_all

lemma less_imp_add_positive:
  assumes "i < j"
  shows "\k::nat. 0 < k \ i + k = j"
proof
  from assms show "0 < j - i \ 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) \ (a < b \ P 0) \ (\d. a = b + d \ 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) \ \ (a < b \ \ P 0 \ (\d. a = b + d \ \ 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)

lemmas nat_diff_splits = nat_diff_split nat_diff_split_asm

lemma Suc_pred': "0 < n \ 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 \ 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 \<open>Monotonicity of multiplication\<close>

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 \<open>\<open>\<le>\<close> monotonicity, BOTH arguments\<close>
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 < j \ 0 < k \ 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 \ 0 < m \ 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 \ 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 < n * k \ 0 < k \ m < n"
  for k m n :: nat
proof (intro iffI conjI)
  assume m: "m * k < n * k"
  then show "0 < k"
    by (cases k) auto
  show "m < n"
  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 < k \ m < n"
  then show "m * k < n * k"
    by (blast intro: mult_less_mono1)
qed

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

lemma mult_le_cancel1 [simp]: "k * m \ k * n \ (0 < k \ 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 < k \ m \ 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 \ m < n"
  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 \<open>Lemma for \<open>gcd\<close>\<close>
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] \<open>m \<noteq> 0\<close> by auto
  qed (use assms \<open>m \<noteq> 0\<close> 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 \<open>The lattice order on \<^typ>\<open>nat\<close>.\<close>

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 \<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 \ 'a \ 'a"

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

notation (latex output)
  compower (\<open>(_\<^bsup>_\<^esup>)\<close> [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 \ ('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 \<open>For code generation.\<close>

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: \<^marker>\<open>contributor \<open>Lars Noschinski\<close>\<close>
  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 \<open>Kleene iteration\<close>

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 (\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)) \ 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 (\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)) \ 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 \<open>Embedding of the naturals into any \<open>semiring_1\<close>: \<^term>\<open>of_nat\<close>\<close>

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)" \<comment> \<open>tail recursive\<close>

lemma of_nat_code: "of_nat n = of_nat_aux (\i. i + 1) n 0"
proof (induct n)
  case 0
  then show ?case by simp
next
  case (Suc n)
  have "\i. of_nat_aux (\i. i + 1) n (i + 1) = of_nat_aux (\i. i + 1) n i + 1"
    by (induct n) simp_all
  from this [of 0] have "of_nat_aux (\i. i + 1) n 1 = of_nat_aux (\i. 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]:
  \<open>of_nat (m - n) = of_nat m - of_nat n\<close> if \<open>n \<le> m\<close>
proof -
  from that obtain q where \<open>m = n + q\<close>
    by (blast dest: le_Suc_ex)
  then show ?thesis
    by simp
qed

lemma of_nat_diff_if: \<open>of_nat (m - n) = (if n\<le>m then of_nat m - of_nat n else 0)\<close>
  by (simp add: not_le less_imp_le)

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 \ 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 \ 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 < 0"
  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 < of_nat n \ m < n"
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 < n \ of_nat m < of_nat n"
  by simp

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

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

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

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

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 < of_nat n \ 0 < n"
  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 \<open>The set of natural numbers\<close>

context semiring_1
begin

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

lemma of_nat_in_Nats [simp]: "of_nat n \ \"
  by (simp add: Nats_def)

lemma Nats_0 [simp]: "0 \ \"
  using of_nat_0 [symmetric] unfolding Nats_def
  by (rule range_eqI)

lemma Nats_1 [simp]: "1 \ \"
  using of_nat_1 [symmetric] unfolding Nats_def
  by (rule range_eqI)

lemma Nats_add [simp]: "a \ \ \ b \ \ \ a + b \ \"
  unfolding Nats_def using of_nat_add [symmetric]
  by (blast intro: range_eqI)

lemma Nats_mult [simp]: "a \ \ \ b \ \ \ a * b \ \"
  unfolding Nats_def using of_nat_mult [symmetric]
  by (blast intro: range_eqI)

lemma Nats_cases [cases set: Nats]:
  assumes "x \ \"
  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 \ \ \ (\n. P (of_nat n)) \ P x"
  by (rule Nats_cases) auto

lemma Nats_nonempty [simp]: "\ \ {}"
  unfolding Nats_def by auto

end

lemma Nats_diff [simp]:
  fixes a:: "'a::linordered_idom"
  assumes "a \ \" "b \ \" "b \ a" shows "a - b \ \"
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 \<open>b \<le> a\<close> 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 \<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)

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 \<open>Tools/nat_arith.ML\<close>

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

simproc_setup natle_cancel_sums
  ("(l::nat) + m \ n" | "(l::nat) \ m + n" | "Suc m \ n" | "m \ Suc n") =
  \<open>K (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>K (try o Nat_Arith.cancel_diff_conv)\<close>

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 < n'")
  case True
  then show ?thesis
    by (induct n n' rule: less_Suc_induct) (auto intro: mono order.trans)
next
  case False
  with \<open>n \<le> n'\<close> 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 < n'")
  case True
  then show ?thesis
    by (induct n n' rule: less_Suc_induct) (auto intro: mono order.trans)
next
  case False
  with \<open>n \<le> n'\<close> show ?thesis by auto
qed

lemma lift_Suc_mono_less:
  assumes mono: "\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 order.strict_trans)

lemma lift_Suc_mono_less_iff: "(\n. f n < f (Suc n)) \ f n < f m \ 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 \ (\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 < f (Suc n))"
proof (intro iffI strict_monoI)
  assume *: "\n. f n < f (Suc n)"
  fix m n :: nat assume "m < n"
  thus "f m < f n"
    by (induction rule: less_Suc_induct) (use * in auto)
qed (auto simp: strict_mono_def)

lemma strict_mono_add: "strict_mono (\n::'a::linordered_semidom. n + k)"
  by (auto simp: strict_mono_def)

lemma mono_nat_linear_lb:
  fixes f :: "nat \ nat"
  assumes "\m n. m < n \ f m < f 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 < n \ \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) < 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 < n"
      thus ?thesis using Suc.IH less_SucI by blast
    next
      assume "f n = n"
      with \<open>f(Suc n) = n\<close> 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 \<open>Subtraction laws, mostly by Clemens Ballarin\<close>

lemma diff_less_mono:
  fixes a b c :: nat
  assumes "a < b" and "c \ 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 \ i + k < j"
  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 < i \ j < i + 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 < n \ 0 < m \ 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 \ 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)

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