src/HOL/Nat.thy
 author nipkow Thu, 14 May 2020 23:44:01 +0200 changeset 71841 f4626b1f1b96 parent 71836 c095d3143047 child 71851 34ecb540a079 permissions -rw-r--r--
added lemma
```
(*  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 \<Rightarrow> ind"
\<comment> \<open>The axiom of infinity in 2 parts:\<close>
where Suc_Rep_inject: "Suc_Rep x = Suc_Rep y \<Longrightarrow> x = y"
and Suc_Rep_not_Zero_Rep: "Suc_Rep x \<noteq> Zero_Rep"

subsection \<open>Type nat\<close>

text \<open>Type definition\<close>

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

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

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

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

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

instantiation nat :: zero
begin

definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"

instance ..

end

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

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

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

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

lemma nat_induct0:
assumes "P 0" and "\<And>n. P n \<Longrightarrow> P (Suc n)"
shows "P n"
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)"
apply atomize_elim
apply (rename_tac n, induct_tac n rule: nat_induct0, auto)
apply (simp add: Suc_def Nat_Abs_Nat_inject Nat_Rep_Nat Suc_RepI Suc_Rep_inject' Rep_Nat_inject)
apply (simp only: Suc_not_Zero)
done

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

old_rep_datatype "0 :: nat" Suc
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 \<Rightarrow> (nat \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
where "rec_nat \<equiv> old.rec_nat"

declare nat.sel[code del]

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

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

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

hide_fact
nat_exhaust
nat_induct0

ML \<open>
val nat_basic_lfp_sugar =
let
val ctr_sugar = the (Ctr_Sugar.ctr_sugar_of_global \<^theory> \<^type_name>\<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 \<longleftrightarrow> Suc ` M = N"
by (simp add: bij_betw_def)

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

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

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

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

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

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

subsection \<open>Arithmetic operators\<close>

instantiation nat :: comm_monoid_diff
begin

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

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

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

declare add_0 [code]

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

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

declare diff_Suc [simp del]

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

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

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

end

hide_fact (open) add_0 add_0_right diff_0

instantiation nat :: comm_semiring_1_cancel
begin

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

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

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

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

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

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

end

subsubsection \<open>Addition\<close>

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

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

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

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

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

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

subsubsection \<open>Multiplication\<close>

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

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

lemma one_eq_mult_iff [simp]: "Suc 0 = m * n \<longleftrightarrow> m = Suc 0 \<and> n = Suc 0"
by (simp add: eq_commute flip: mult_eq_1_iff)

lemma nat_mult_eq_1_iff [simp]: "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
and nat_1_eq_mult_iff [simp]: "1 = m * n \<longleftrightarrow> m = 1 \<and> n = 1" for m n :: nat
by auto

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

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

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

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

subsubsection \<open>Operation definition\<close>

instantiation nat :: linorder
begin

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

declare less_eq_nat.simps [simp del]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

end

instantiation nat :: order_bot
begin

definition bot_nat :: nat
where "bot_nat = 0"

instance
by standard (simp add: bot_nat_def)

end

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

subsubsection \<open>Introduction properties\<close>

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

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

subsubsection \<open>Elimination properties\<close>

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma less_SucE:
assumes major: "m < Suc n"
and less: "m < n \<Longrightarrow> P"
and eq: "m = n \<Longrightarrow> P"
shows P
proof (rule major [THEN lessE])
show "Suc n = Suc m \<Longrightarrow> P"
using eq by blast
show "\<And>j. \<lbrakk>m < j; Suc n = Suc j\<rbrakk> \<Longrightarrow> P"
by (blast intro: less)
qed

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

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

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

lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

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

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

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

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

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

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

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

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

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

lemma All_less_Suc: "(\<forall>i < Suc n. P i) = (P n \<and> (\<forall>i < n. P i))"
by (auto simp: less_Suc_eq)

lemma All_less_Suc2: "(\<forall>i < Suc n. P i) = (P 0 \<and> (\<forall>i < n. P(Suc i)))"
by (auto simp: less_Suc_eq_0_disj)

lemma Ex_less_Suc: "(\<exists>i < Suc n. P i) = (P n \<or> (\<exists>i < n. P i))"
by (auto simp: less_Suc_eq)

lemma Ex_less_Suc2: "(\<exists>i < Suc n. P i) = (P 0 \<or> (\<exists>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 \<ge> 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 \<Longrightarrow> Suc (n - Suc 0) = n"
by (simp add: diff_Suc split: nat.split)

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

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

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

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

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

text \<open>strict, in both arguments\<close>
lemma add_less_mono:
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 \<Longrightarrow> \<exists>k. n = Suc (m + k)"
proof (induct n)
case 0
then show ?case by simp
next
case Suc
then show ?case
by (simp add: order_le_less)
(blast elim!: less_SucE intro!: Nat.add_0_right [symmetric] add_Suc_right [symmetric])
qed

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

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: "\<not> m < n \<Longrightarrow> n + (m - n) = m"
for m n :: nat
by (induct m n rule: diff_induct) simp_all

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

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

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

instance nat :: dioid
by standard (rule nat_le_iff_add)

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

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

subsubsection \<open>\<^term>\<open>min\<close> and \<^term>\<open>max\<close>\<close>

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

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

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

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

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

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

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

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

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

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

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

lemma max_0_iff[simp]: "max m n = (0::nat) \<longleftrightarrow> m = 0 \<and> n = 0"
by(cases m, auto simp: max_Suc1 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: "(\<And>m. m < n \<Longrightarrow> P m) \<Longrightarrow> P n" for n :: nat
have "\<And>q. q \<le> n \<Longrightarrow> P q"
proof (induct n)
case (0 n)
have "P 0" by (rule step) auto
with 0 show ?case by auto
next
case (Suc m n)
then have "n \<le> m \<or> n = Suc m"
by (simp add: le_Suc_eq)
then show ?case
proof
assume "n \<le> m"
then show "P n" by (rule Suc(1))
next
assume n: "n = Suc m"
show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
qed
qed
then show "P n" by auto
qed

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

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

lemma ex_least_nat_le:
fixes P :: "nat \<Rightarrow> bool"
assumes "P n" "\<not> P 0"
shows "\<exists>k\<le>n. (\<forall>i<k. \<not> P i) \<and> 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 \<Rightarrow> bool"
assumes "P n" "\<not> P 0"
shows "\<exists>k<n. (\<forall>i\<le>k. \<not> P i) \<and> P (Suc k)"
proof (cases n)
case (Suc m)
then obtain k where k: "k \<le> n" "\<forall>i<k. \<not> P i" "P k"
using ex_least_nat_le [OF assms] 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 \<Rightarrow> bool"
assumes "\<And>n. \<forall>m. m < n \<longrightarrow> P m \<Longrightarrow> P n"
shows "P n"
using assms less_induct by blast

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

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

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

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

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

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

lemma infinite_descent0 [case_names 0 smaller]:
fixes P :: "nat \<Rightarrow> bool"
assumes "P 0"
and "\<And>n. n > 0 \<Longrightarrow> \<not> P n \<Longrightarrow> \<exists>m. m < n \<and> \<not> P m"
shows "P n"
proof (rule infinite_descent)
show "\<And>n. \<not> P n \<Longrightarrow> \<exists>m<n. \<not> P m"
using assms by (case_tac "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 \<Rightarrow> nat"
assumes 1: "\<And>x. V x = 0 \<Longrightarrow> P x"
and 2: "\<And>x. V x > 0 \<Longrightarrow> \<not> P x \<Longrightarrow> \<exists>y. V y < V x \<and> \<not> P y"
shows "P x"
proof -
obtain n where "n = V x" by auto
moreover have "\<And>x. V x = n \<Longrightarrow> P x"
proof (induct n rule: infinite_descent0)
case 0
with 1 show "P x" by auto
next
case (smaller n)
then obtain x where *: "V x = n " and "V x > 0 \<and> \<not> P x" by auto
with 2 obtain y where "V y < V x \<and> \<not> P y" by auto
with * obtain m where "m = V y \<and> m < n \<and> \<not> P y" by auto
then show ?case by auto
qed
ultimately show "P x" by auto
qed

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma not_add_less1 [iff]: "\<not> i + j < i"
for i j :: nat
by simp

lemma not_add_less2 [iff]: "\<not> j + i < i"
for i j :: nat
by simp

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

lemma add_leD2: "m + k \<le> n \<Longrightarrow> k \<le> n"
for k m n :: nat
by (force simp add: add.commute dest: add_leD1)

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

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

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

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

lemma diff_less_Suc: "m - n < Suc m"
by (induct m n rule: diff_induct) (auto simp: less_Suc_eq)

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

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

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

lemma diff_add_assoc: "k \<le> j \<Longrightarrow> (i + j) - k = i + (j - k)"
for i j k :: nat
by (fact ordered_cancel_comm_monoid_diff_class.diff_add_assoc)

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

lemma mult_less_cancel2 [simp]: "m * k < n * k \<longleftrightarrow> 0 < k \<and> m < n"
for k m n :: nat
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 \<and> m < n"
then show "m * k < n * k"
by (blast intro: mult_less_mono1)
qed

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

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

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

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

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

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

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

text \<open>Lemma for \<open>gcd\<close>\<close>
lemma mult_eq_self_implies_10:
fixes m n :: nat
assumes "m = m * n" shows "n = 1 \<or> m = 0"
proof (rule disjCI)
assume "m \<noteq> 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 \<le> q"
with assms show "n * m \<le> n * q" by simp
qed

text \<open>The lattice order on \<^typ>\<open>nat\<close>.\<close>

instantiation nat :: distrib_lattice
begin

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

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

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

end

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

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

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

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

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

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

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

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

end

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

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

lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

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

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

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

hide_const (open) funpow

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

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

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

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

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

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

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

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

lemma inj_fn[simp]:
fixes f::"'a \<Rightarrow> '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 \<Rightarrow> '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 \<Rightarrow> '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]]])

subsection \<open>Kleene iteration\<close>

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

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

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

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

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

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

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

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

context semiring_1
begin

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

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

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

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

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

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

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

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

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:
\<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

end

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

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

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

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

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

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

lemma of_nat_1_eq_iff [simp]: "1 = of_nat n \<longleftrightarrow> n=1"
using of_nat_eq_iff by fastforce

lemma of_nat_eq_1_iff [simp]: "of_nat n = 1 \<longleftrightarrow> n=1"
using of_nat_eq_iff by fastforce

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

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

end

class ring_char_0 = ring_1 + semiring_char_0

context linordered_nonzero_semiring
begin

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

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

lemma of_nat_mono[simp]: "i \<le> j \<Longrightarrow> of_nat i \<le> of_nat j"
by (auto simp: le_iff_add intro!: add_increasing2)

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

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

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

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

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

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

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

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

end

context linordered_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]:
"\<bar>of_nat n\<bar> = 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"  ("\<nat>")
where "\<nat> = range of_nat"

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

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

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

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

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

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

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

end

lemma Nats_diff [simp]:
fixes a:: "'a::linordered_idom"
assumes "a \<in> \<nat>" "b \<in> \<nat>" "b \<le> a" shows "a - b \<in> \<nat>"
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 \<le> 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) \<equiv> k + a \<Longrightarrow> A + b \<equiv> k + (a + b)"
by (simp only: ac_simps)

lemma add2: "(B::'a::comm_monoid_add) \<equiv> k + b \<Longrightarrow> a + B \<equiv> k + (a + b)"
by (simp only: ac_simps)

lemma suc1: "A == k + a \<Longrightarrow> Suc A \<equiv> k + Suc a"
by (simp only: add_Suc_right)

lemma rule0: "(a::'a::comm_monoid_add) \<equiv> 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>fn phi => try o Nat_Arith.cancel_eq_conv\<close>

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

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

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

context order
begin

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

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

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

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

end

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

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

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

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

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

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

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

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

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

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

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

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

hide_fact (open) diff_diff_eq

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

text \<open>Induction starting beyond zero\<close>

lemma nat_induct_at_least [consumes 1, case_names base Suc]:
"P n" if "n \<ge> m" "P m" "\<And>n. n \<ge> m \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
proof -
define q where "q = n - m"
with \<open>n \<ge> m\<close> 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" "\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
proof -
from \<open>n > 0\<close> have "n \<ge> 1"
by (cases n) simp_all
moreover note \<open>P 1\<close>
moreover have "\<And>n. n \<ge> 1 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)"
using \<open>\<And>n. n > 0 \<Longrightarrow> P n \<Longrightarrow> P (Suc n)\<close>
by (simp add: Suc_le_eq)
ultimately show "P n"
by (rule nat_induct_at_least)
qed

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

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

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

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

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

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

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

lemma transitive_stepwise_le:
assumes "m \<le> n" "\<And>x. R x x" "\<And>x y z. R x y \<Longrightarrow> R y z \<Longrightarrow> R x z" and "\<And>n. R n (Suc n)"
shows "R m n"
using \<open>m \<le> n\<close>
by (induction rule: dec_induct) (use assms in blast)+

subsubsection \<open>Greatest operator\<close>

lemma ex_has_greatest_nat:
"P (k::nat) \<Longrightarrow> \<forall>y. P y \<longrightarrow> y \<le> b \<Longrightarrow> \<exists>x. P x \<and> (\<forall>y. P y \<longrightarrow> y \<le> x)"
proof (induction "b-k" arbitrary: b k rule: less_induct)
case less
show ?case
proof cases
assume "\<exists>n>k. P n"
then obtain n where "n>k" "P n" by blast
have "n \<le> b" using \<open>P n\<close> less.prems(2) by auto
hence "b-n < b-k"
by(rule diff_less_mono2[OF \<open>k<n\<close> less_le_trans[OF \<open>k<n\<close>]])
from less.hyps[OF this \<open>P n\<close> less.prems(2)]
show ?thesis .
next
assume "\<not> (\<exists>n>k. P n)"
hence "\<forall>y. P y \<longrightarrow> y \<le> 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: "\<And>y. P y \<Longrightarrow> y \<le> b"
shows GreatestI_nat: "P (Greatest P)"
and Greatest_le_nat: "k \<le> Greatest P"
proof -
obtain x where "P x" "\<And>y. P y \<Longrightarrow> y \<le> x"
using assms ex_has_greatest_nat by blast
with \<open>P k\<close> show "P (Greatest P)" "k \<le> Greatest P"
using GreatestI2_order by blast+
qed

lemma GreatestI_ex_nat:
"\<lbrakk> \<exists>k::nat. P k;  \<And>y. P y \<Longrightarrow> y \<le> b \<rbrakk> \<Longrightarrow> P (Greatest P)"
by (blast intro: GreatestI_nat)

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

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

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

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

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

subsection \<open>The divides relation on \<^typ>\<open>nat\<close>\<close>

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

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

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

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

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

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

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

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

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

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

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

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

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

subsection \<open>Aliasses\<close>

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

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

lemma 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 \<le> j \<Longrightarrow> (i \<le> j - k) = (i + k \<le> 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 \<open>Size of a datatype value\<close>

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

instantiation nat :: size
begin

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

instance ..

end

lemmas size_nat = size_nat_def

lemma size_neq_size_imp_neq: "size x \<noteq> size y \<Longrightarrow> x \<noteq> y"
by (erule contrapos_nn) (rule arg_cong)

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

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

hide_const (open) of_nat_aux

end
```