# Theory Nat

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

section ‹Natural numbers›

theory Nat
imports Inductive Typedef Fun Rings
begin

subsection ‹Type ‹ind››

typedecl ind

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

subsection ‹Type nat›

text ‹Type definition›

inductive Nat :: "ind ⇒ bool"
where
Zero_RepI: "Nat Zero_Rep"
| Suc_RepI: "Nat i ⟹ Nat (Suc_Rep i)"

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

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

lemma Nat_Abs_Nat_inverse: "Nat n ⟹ Rep_Nat (Abs_Nat n) = n"
using Abs_Nat_inverse by simp

lemma Nat_Abs_Nat_inject: "Nat n ⟹ Nat m ⟹ Abs_Nat n = Abs_Nat m ⟷ n = m"
using Abs_Nat_inject by simp

instantiation nat :: zero
begin

definition Zero_nat_def: "0 = Abs_Nat Zero_Rep"

instance ..

end

definition Suc :: "nat ⇒ nat"
where "Suc n = Abs_Nat (Suc_Rep (Rep_Nat n))"

lemma Suc_not_Zero: "Suc m ≠ 0"
by (simp add: Zero_nat_def Suc_def Suc_RepI Zero_RepI
Nat_Abs_Nat_inject Suc_Rep_not_Zero_Rep Nat_Rep_Nat)

lemma Zero_not_Suc: "0 ≠ Suc m"
by (rule not_sym) (rule Suc_not_Zero)

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

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

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

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

setup ‹Sign.parent_path›

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

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

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

setup ‹Sign.parent_path›

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

declare nat.sel[code del]

hide_const (open) Nat.pred ― ‹hide everything related to the selector›
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 ⟹ P) ⟹ (⋀nat. y = Suc nat ⟹ P) ⟹ P"
― ‹for backward compatibility -- names of variables differ›
by (rule old.nat.exhaust)

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

hide_fact
nat_exhaust
nat_induct0

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

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

text ‹Injectiveness and distinctness lemmas›

lemma inj_Suc [simp]:
"inj_on Suc N"

lemma bij_betw_Suc [simp]:
"bij_betw Suc M N ⟷ Suc ` M = N"

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

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

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

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

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

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

subsection ‹Arithmetic operators›

instantiation nat :: comm_monoid_diff
begin

primrec plus_nat
where
add_0: "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

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)"

lemma add_mult_distrib: "(m + n) * k = (m * k) + (n * k)"
for m n k :: nat

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)"
show "(n + m) * q = n * q + m * q"
show "k * (m - n) = (k * m) - (k * n)"
by (induct m n rule: diff_induct) simp_all
qed

end

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

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

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

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

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

lemma plus_1_eq_Suc:
"plus 1 = Suc"

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

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

subsubsection ‹Difference›

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

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

subsubsection ‹Multiplication›

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

lemma mult_eq_1_iff [simp]: "m * n = Suc 0 ⟷ m = Suc 0 ∧ n = Suc 0"
proof (induct m)
case 0
then show ?case by simp
next
case (Suc m)
then show ?case by (induct n) auto
qed

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

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

lemma mult_cancel1 [simp]: "k * m = k * n ⟷ m = n ∨ k = 0"
for k m n :: nat
proof -
have "k ≠ 0 ⟹ k * m = k * n ⟹ m = n"
proof (induct n arbitrary: m)
case 0
then show "m = 0" by simp
next
case (Suc n)
then show "m = Suc n"
by (cases m) (simp_all add: eq_commute [of 0])
qed
then show ?thesis by auto
qed

lemma mult_cancel2 [simp]: "m * k = n * k ⟷ m = n ∨ k = 0"
for k m n :: nat

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

subsection ‹Orders on \<^typ>‹nat››

subsubsection ‹Operation definition›

instantiation nat :: linorder
begin

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

declare less_eq_nat.simps [simp del]

lemma le0 [iff]: "0 ≤ n" for
n :: nat

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"

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

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"

lemma less_Suc_eq_le [code]: "m < Suc n ⟷ m ≤ n"

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)
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)
qed

end

instantiation nat :: order_bot
begin

definition bot_nat :: nat
where "bot_nat = 0"

instance

end

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

subsubsection ‹Introduction properties›

lemma lessI [iff]: "n < Suc n"

lemma zero_less_Suc [iff]: "0 < Suc n"

subsubsection ‹Elimination properties›

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

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

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

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

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

lemma less_Suc_eq: "m < Suc n ⟷ m < n ∨ m = n"
unfolding less_Suc_eq_le le_less ..

lemma less_Suc0 [iff]: "(n < Suc 0) = (n = 0)"

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

lemma Suc_mono: "m < n ⟹ Suc m < Suc n"
by simp

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

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

subsubsection ‹Inductive (?) properties›

lemma Suc_lessI: "m < n ⟹ Suc m ≠ n ⟹ Suc m < n"
unfolding less_eq_Suc_le [of m] le_less by simp

lemma lessE:
assumes major: "i < k"
and 1: "k = Suc i ⟹ P"
and 2: "⋀j. i < j ⟹ k = Suc j ⟹ P"
shows P
proof -
from major have "∃j. i ≤ j ∧ k = Suc j"
unfolding less_eq_Suc_le by (induct k) simp_all
then have "(∃j. i < j ∧ k = Suc j) ∨ k = Suc i"
with 1 2 show P by auto
qed

lemma less_SucE:
assumes major: "m < Suc n"
and less: "m < n ⟹ P"
and eq: "m = n ⟹ P"
shows P
proof (rule major [THEN lessE])
show "Suc n = Suc m ⟹ P"
using eq by blast
show "⋀j. ⟦m < j; Suc n = Suc j⟧ ⟹ P"
by (blast intro: less)
qed

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

lemma Suc_less_SucD: "Suc m < Suc n ⟹ m < n"
by simp

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

text ‹Can be used with ‹less_Suc_eq› to get \<^prop>‹n = m ∨ n < m›.›
lemma not_less_eq: "¬ m < n ⟷ n < Suc m"
by (simp only: not_less less_Suc_eq_le)

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

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

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

lemma Suc_n_not_le_n: "¬ Suc n ≤ n"

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

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

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

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

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

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

text ‹Equivalence of ‹m ≤ n› and ‹m < n ∨ m = n››

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

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

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

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

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

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

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

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

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

lemmas linorder_neqE_nat = linorder_neqE [where 'a = nat]

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

lemma not_less_less_Suc_eq: "¬ n < m ⟹ n < Suc m ⟷ n = m"
unfolding not_less by (rule le_less_Suc_eq)

lemmas not_less_simps = not_less_less_Suc_eq le_less_Suc_eq

lemma not0_implies_Suc: "n ≠ 0 ⟹ ∃m. n = Suc m"
by (cases n) simp_all

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

text ‹strict, in both arguments›
fixes i j k l :: nat
assumes "i < j" "k < l" shows "i + k < j + l"
proof -
have "i + k < j + k"
also have "...  < j + l"
using ‹i < j› 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
qed

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

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

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

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

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

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

instance nat :: linordered_semidom
proof
fix m n q :: nat
show "0 < (1::nat)"
by simp
show "m ≤ n ⟹ q + m ≤ q + n"
by simp
show "m < n ⟹ 0 < q ⟹ q * m < q * n"
show "m ≠ 0 ⟹ n ≠ 0 ⟹ m * n ≠ 0"
by simp
show "n ≤ m ⟹ (m - n) + n = m"
qed

instance nat :: dioid

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

instance nat :: ordered_cancel_comm_monoid_diff ..

subsubsection ‹\<^term>‹min› and \<^term>‹max››

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

global_interpretation max_nat: semilattice_neutr_order max ‹0::nat› ‹(≥)› ‹(>)›

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)"

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)"

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
(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
(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

lemma nat_add_max_right: "m + max n q = max (m + n) (m + q)"
for m n q :: nat

lemma nat_mult_max_left: "max m n * q = max (m * q) (n * q)"
for m n q :: nat
(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
(auto dest: mult_left_le_imp_le mult_left_less_imp_less le_less_trans)

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

instance nat :: wellorder
proof
fix P and n :: nat
assume step: "(⋀m. m < n ⟹ P m) ⟹ P n" for n :: nat
have "⋀q. q ≤ n ⟹ P q"
proof (induct n)
case (0 n)
have "P 0" by (rule step) auto
with 0 show ?case by auto
next
case (Suc m n)
then have "n ≤ m ∨ n = Suc m"
then show ?case
proof
assume "n ≤ m"
then show "P n" by (rule Suc(1))
next
assume n: "n = Suc m"
show "P n" by (rule step) (rule Suc(1), simp add: n le_simps)
qed
qed
then show "P n" by auto
qed

lemma Least_eq_0[simp]: "P 0 ⟹ Least P = 0"
for P :: "nat ⇒ bool"
by (rule Least_equality[OF _ le0])

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

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

lemma ex_least_nat_le:
fixes P :: "nat ⇒ bool"
assumes "P n" "¬ P 0"
shows "∃k≤n. (∀i<k. ¬ P i) ∧ P k"
proof (cases n)
case (Suc m)
with assms show ?thesis
by (blast intro: Least_le LeastI_ex dest: not_less_Least)
qed (use assms in auto)

lemma ex_least_nat_less:
fixes P :: "nat ⇒ bool"
assumes "P n" "¬ P 0"
shows "∃k<n. (∀i≤k. ¬ P i) ∧ P (Suc k)"
proof (cases n)
case (Suc m)
then obtain k where k: "k ≤ n" "∀i<k. ¬ 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 ⇒ bool"
assumes "⋀n. ∀m. m < n ⟶ P m ⟹ P n"
shows "P n"
using assms less_induct by blast

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

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

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

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

text ‹
The method of infinite descent, frequently used in number theory.
Provided by Roelof Oosterhuis.
‹P n› is true for all natural numbers if
▪ case ``0'': given ‹n = 0› prove ‹P n›
▪ case ``smaller'': given ‹n > 0› and ‹¬ P n› prove there exists
a smaller natural number ‹m› such that ‹¬ P m›.
›

lemma infinite_descent: "(⋀n. ¬ P n ⟹ ∃m<n. ¬ P m) ⟹ P n" for P :: "nat ⇒ bool"
― ‹compact version without explicit base case›
by (induct n rule: less_induct) auto

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

text ‹
Infinite descent using a mapping to ‹nat›:
‹P x› is true for all ‹x ∈ D› if there exists a ‹V ∈ D ⇒ nat› and
▪ case ``0'': given ‹V x = 0› prove ‹P x›
▪ ``smaller'': given ‹V x > 0› and ‹¬ P x› prove
there exists a ‹y ∈ D› such that ‹V y < V x› and ‹¬ P y›.
›
corollary infinite_descent0_measure [case_names 0 smaller]:
fixes V :: "'a ⇒ nat"
assumes 1: "⋀x. V x = 0 ⟹ P x"
and 2: "⋀x. V x > 0 ⟹ ¬ P x ⟹ ∃y. V y < V x ∧ ¬ P y"
shows "P x"
proof -
obtain n where "n = V x" by auto
moreover have "⋀x. V x = n ⟹ P x"
proof (induct n rule: infinite_descent0)
case 0
with 1 show "P x" by auto
next
case (smaller n)
then obtain x where *: "V x = n " and "V x > 0 ∧ ¬ P x" by auto
with 2 obtain y where "V y < V x ∧ ¬ P y" by auto
with * obtain m where "m = V y ∧ m < n ∧ ¬ P y" by auto
then show ?case by auto
qed
ultimately show "P x" by auto
qed

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

text ‹A (clumsy) way of lifting ‹<› monotonicity to ‹≤› monotonicity›
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 ‹non-strict, in 1st argument›
lemma add_le_mono1: "i ≤ j ⟹ i + k ≤ j + k"
for i j k :: nat

text ‹non-strict, in both arguments›
lemma add_le_mono: "i ≤ j ⟹ k ≤ l ⟹ i + k ≤ j + l"
for i j k l :: nat

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))"

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

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

lemma add_leD2: "m + k ≤ n ⟹ k ≤ n"
for k m n :: nat

lemma add_leE: "m + k ≤ n ⟹ (m ≤ n ⟹ k ≤ n ⟹ R) ⟹ R"
for k m n :: nat

text ‹needs ‹⋀k› for ‹ac_simps› to work›
lemma less_add_eq_less: "⋀k. k < l ⟹ m + l = k + n ⟹ m < n"
for l m n :: nat

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

lemma add_diff_assoc [simp]: "k ≤ j ⟹ i + (j - k) = i + j - k"
for i j k :: nat

lemma diff_add_assoc2: "k ≤ j ⟹ (j + i) - k = (j - k) + i"
for i j k :: nat

lemma add_diff_assoc2 [simp]: "k ≤ j ⟹ j - k + i = j + i - k"
for i j k :: nat

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 (rule iffD2, rule diff_is_0_eq)

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

assumes "i < j"
shows "∃k::nat. 0 < k ∧ i + k = j"
proof
from assms show "0 < j - i ∧ i + (j - i) = j"
qed

text ‹a nice rewrite for bounded subtraction›
lemma nat_minus_add_max: "n - m + m = max n m"
for m n :: nat
by (simp add: max_def not_le order_less_imp_le)

lemma nat_diff_split: "P (a - b) ⟷ (a < b ⟶ P 0) ∧ (∀d. a = b + d ⟶ P d)"
for a b :: nat
― ‹elimination of ‹-› on ‹nat››
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
― ‹elimination of ‹-› on ‹nat› in assumptions›
by (auto split: nat_diff_split)

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 ‹Monotonicity of multiplication›

lemma mult_le_mono1: "i ≤ j ⟹ i * k ≤ j * k"
for i j k :: nat

lemma mult_le_mono2: "i ≤ j ⟹ k * i ≤ k * j"
for i j k :: nat

text ‹‹≤› monotonicity, BOTH arguments›
lemma mult_le_mono: "i ≤ j ⟹ k ≤ l ⟹ i * k ≤ j * l"
for i j k l :: nat

lemma mult_less_mono1: "i < j ⟹ 0 < k ⟹ i * k < j * k"
for i j k :: nat

text ‹Differs from the standard ‹zero_less_mult_iff› in that there are no negative numbers.›
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 ‹Lemma for ‹gcd››
lemma mult_eq_self_implies_10:
fixes m n :: nat
assumes "m = m * n" shows "n = 1 ∨ m = 0"
proof (rule disjCI)
assume "m ≠ 0"
show "n = 1"
proof (cases n "1::nat" rule: linorder_cases)
case greater
show ?thesis
using assms mult_less_mono2 [OF greater, of m] ‹m ≠ 0› by auto
qed (use assms ‹m ≠ 0› in auto)
qed

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

text ‹The lattice order on \<^typ>‹nat›.›

instantiation nat :: distrib_lattice
begin

definition "(inf :: nat ⇒ nat ⇒ nat) = min"

definition "(sup :: nat ⇒ nat ⇒ nat) = max"

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

end

subsection ‹Natural operation of natural numbers on functions›

text ‹
We use the same logical constant for the power operations on
functions and relations, in order to share the same syntax.
›

consts compow :: "nat ⇒ 'a ⇒ 'a"

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

notation (latex output)
compower ("(_⇗_⇖)" [1000] 1000)

text ‹‹f ^^ n = f ∘ … ∘ f›, the ‹n›-fold composition of ‹f››

funpow ≡ "compow :: nat ⇒ ('a ⇒ 'a) ⇒ ('a ⇒ 'a)"
begin

primrec funpow :: "nat ⇒ ('a ⇒ 'a) ⇒ 'a ⇒ 'a"
where
"funpow 0 f = id"
| "funpow (Suc n) f = f ∘ funpow n f"

end

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

lemma funpow_Suc_right: "f ^^ Suc n = f ^^ n ∘ f"
proof (induct n)
case 0
then show ?case by simp
next
fix n
assume "f ^^ Suc n = f ^^ n ∘ f"
then show "f ^^ Suc (Suc n) = f ^^ Suc n ∘ f"
qed

lemmas funpow_simps_right = funpow.simps(1) funpow_Suc_right

text ‹For code generation.›

context
begin

qualified definition funpow :: "nat ⇒ ('a ⇒ 'a) ⇒ 'a ⇒ 'a"
where funpow_code_def [code_abbrev]: "funpow = compow"

lemma [code]:
"funpow (Suc n) f = f ∘ funpow n f"
"funpow 0 f = id"

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"

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 arbitrary: A B)
(auto simp del: funpow.simps(2) simp add: funpow_Suc_right 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)
qed
qed

lemma inj_fn[simp]:
fixes f::"'a ⇒ 'a"
assumes "inj f"
shows "inj (f^^n)"
proof (induction n)
case Suc thus ?case using inj_compose[OF assms Suc.IH] by (simp del: comp_apply)
qed simp

lemma surj_fn[simp]:
fixes f::"'a ⇒ 'a"
assumes "surj f"
shows "surj (f^^n)"
proof (induction n)
case Suc thus ?case by (simp add: comp_surj[OF Suc.IH assms] del: comp_apply)
qed simp

lemma bij_fn[simp]:
fixes f::"'a ⇒ 'a"
assumes "bij f"
shows "bij (f^^n)"
by (rule bijI[OF inj_fn[OF bij_is_inj[OF assms]] surj_fn[OF bij_is_surj[OF assms]]])

lemma bij_betw_funpow: ✐‹contributor ‹Lars Noschinski››
assumes "bij_betw f S S" shows "bij_betw (f ^^ n) S S"
proof (induct n)
case 0 then show ?case by (auto simp: id_def[symmetric])
next
case (Suc n)
then show ?case unfolding funpow.simps using assms by (rule bij_betw_trans)
qed

subsection ‹Kleene iteration›

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

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

lemma mono_pow: "mono f ⟹ mono (f ^^ n)"
for f :: "'a ⇒ 'a::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 ≤ lfp (f ^^ Suc n)"
proof (rule lfp_lowerbound)
have "f (lfp (f ^^ Suc n)) = lfp (λx. f ((f ^^ n) x))"
unfolding funpow_Suc_right by (simp add: lfp_rolling f mono_pow comp_def)
then show "f (lfp (f ^^ Suc n)) ≤ lfp (f ^^ Suc n)"
qed
have "(f ^^ n) (lfp f) = lfp f" for n
by (induct n) (auto intro: f lfp_fixpoint)
then show "lfp (f ^^ Suc n) ≤ lfp f"
by (intro lfp_lowerbound) (simp del: funpow.simps)
qed

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

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

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

subsection ‹Embedding of the naturals into any ‹semiring_1›: \<^term>‹of_nat››

context semiring_1
begin

definition of_nat :: "nat ⇒ 'a"
where "of_nat n = (plus 1 ^^ n) 0"

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

lemma of_nat_1 [simp]: "of_nat 1 = 1"

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

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

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

primrec of_nat_aux :: "('a ⇒ 'a) ⇒ nat ⇒ 'a ⇒ 'a"
where
"of_nat_aux inc 0 i = i"
| "of_nat_aux inc (Suc n) i = of_nat_aux inc n (inc i)" ― ‹tail recursive›

lemma of_nat_code: "of_nat n = of_nat_aux (λi. i + 1) n 0"
proof (induct n)
case 0
then show ?case by simp
next
case (Suc n)
have "⋀i. of_nat_aux (λi. i + 1) n (i + 1) = of_nat_aux (λi. i + 1) n i + 1"
by (induct n) simp_all
from this [of 0] have "of_nat_aux (λi. i + 1) n 1 = of_nat_aux (λi. i + 1) n 0 + 1"
by simp
with Suc show ?case
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:
‹of_nat (m - n) = of_nat m - of_nat n› if ‹n ≤ m›
proof -
from that obtain q where ‹m = n + q›
by (blast dest: le_Suc_ex)
then show ?thesis
by simp
qed

end

text ‹Class for unital semirings with characteristic zero.
Includes non-ordered rings like the complex numbers.›

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

lemma of_nat_eq_iff [simp]: "of_nat m = of_nat n ⟷ m = n"
by (auto intro: inj_of_nat injD)

text ‹Special cases where either operand is zero›

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

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

lemma of_nat_1_eq_iff [simp]: "1 = of_nat n ⟷ n=1"
using of_nat_eq_iff by fastforce

lemma of_nat_eq_1_iff [simp]: "of_nat n = 1 ⟷ n=1"
using of_nat_eq_iff by fastforce

lemma of_nat_neq_0 [simp]: "of_nat (Suc n) ≠ 0"
unfolding of_nat_eq_0_iff by simp

lemma of_nat_0_neq [simp]: "0 ≠ of_nat (Suc n)"
unfolding of_nat_0_eq_iff by simp

end

class ring_char_0 = ring_1 + semiring_char_0

context linordered_nonzero_semiring
begin

lemma of_nat_0_le_iff [simp]: "0 ≤ of_nat n"
by (induct n) simp_all

lemma of_nat_less_0_iff [simp]: "¬ of_nat m < 0"

lemma of_nat_mono[simp]: "i ≤ j ⟹ of_nat i ≤ of_nat j"

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
next
case (3 m n)
then show ?case
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 ‹Every ‹linordered_nonzero_semiring› has characteristic zero.›

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

text ‹Special cases where either operand is zero›

lemma of_nat_le_0_iff [simp]: "of_nat m ≤ 0 ⟷ m = 0"
by (rule of_nat_le_iff [of _ 0, simplified])

lemma of_nat_0_less_iff [simp]: "0 < 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"

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"

subsection ‹The set of natural numbers›

context semiring_1
begin

definition Nats :: "'a set"  ("ℕ")
where "ℕ = range of_nat"

lemma of_nat_in_Nats [simp]: "of_nat n ∈ ℕ"

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

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

lemma Nats_add [simp]: "a ∈ ℕ ⟹ b ∈ ℕ ⟹ a + b ∈ ℕ"
by (blast intro: range_eqI)

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

lemma Nats_cases [cases set: Nats]:
assumes "x ∈ ℕ"
obtains (of_nat) n where "x = of_nat n"
unfolding Nats_def
proof -
from ‹x ∈ ℕ› have "x ∈ range of_nat" unfolding Nats_def .
then obtain n where "x = of_nat n" ..
then show thesis ..
qed

lemma Nats_induct [case_names of_nat, induct set: Nats]: "x ∈ ℕ ⟹ (⋀n. P (of_nat n)) ⟹ P x"
by (rule Nats_cases) auto

end

lemma Nats_diff [simp]:
fixes a:: "'a::linordered_idom"
assumes "a ∈ ℕ" "b ∈ ℕ" "b ≤ a" shows "a - b ∈ ℕ"
proof -
obtain i where i: "a = of_nat i"
using Nats_cases assms by blast
obtain j where j: "b = of_nat j"
using Nats_cases assms by blast
have "j ≤ i"
using ‹b ≤ a› i j of_nat_le_iff by blast
then have *: "of_nat i - of_nat j = (of_nat (i-j) :: 'a)"
then show ?thesis
by (simp add: * i j)
qed

subsection ‹Further arithmetic facts concerning the natural numbers›

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

locale nat_arith
begin

lemma add1: "(A::'a::comm_monoid_add) ≡ k + a ⟹ A + b ≡ k + (a + b)"
by (simp only: ac_simps)

lemma add2: "(B::'a::comm_monoid_add) ≡ k + b ⟹ a + B ≡ k + (a + b)"
by (simp only: ac_simps)

lemma suc1: "A == k + a ⟹ Suc A ≡ k + Suc a"

lemma rule0: "(a::'a::comm_monoid_add) ≡ a + 0"

end

ML_file ‹Tools/nat_arith.ML›

simproc_setup nateq_cancel_sums
("(l::nat) + m = n" | "(l::nat) = m + n" | "Suc m = n" | "m = Suc n") =
‹fn phi => try o Nat_Arith.cancel_eq_conv›

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

simproc_setup natle_cancel_sums
("(l::nat) + m ≤ n" | "(l::nat) ≤ m + n" | "Suc m ≤ n" | "m ≤ Suc n") =
‹fn phi => try o Nat_Arith.cancel_le_conv›

simproc_setup natdiff_cancel_sums
("(l::nat) + m - n" | "(l::nat) - (m + n)" | "Suc m - n" | "m - Suc n") =
‹fn phi => try o Nat_Arith.cancel_diff_conv›

context order
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)
next
case False
with ‹n ≤ n'› show ?thesis by auto
qed

lemma lift_Suc_antimono_le:
assumes mono: "⋀n. f n ≥ f (Suc n)"
and "n ≤ n'"
shows "f n ≥ f n'"
proof (cases "n < n'")
case True
then show ?thesis
by</```