# Theory Int

```(*  Title:      HOL/Int.thy
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
Author:     Tobias Nipkow, Florian Haftmann, TU Muenchen
*)

section ‹The Integers as Equivalence Classes over Pairs of Natural Numbers›

theory Int
imports Quotient Groups_Big Fun_Def
begin

subsection ‹Definition of integers as a quotient type›

definition intrel :: "(nat × nat) ⇒ (nat × nat) ⇒ bool"
where "intrel = (λ(x, y) (u, v). x + v = u + y)"

lemma intrel_iff [simp]: "intrel (x, y) (u, v) ⟷ x + v = u + y"

quotient_type int = "nat × nat" / "intrel"
morphisms Rep_Integ Abs_Integ
proof (rule equivpI)
show "reflp intrel" by (auto simp: reflp_def)
show "symp intrel" by (auto simp: symp_def)
show "transp intrel" by (auto simp: transp_def)
qed

lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
"(⋀x y. z = Abs_Integ (x, y) ⟹ P) ⟹ P"
by (induct z) auto

subsection ‹Integers form a commutative ring›

instantiation int :: comm_ring_1
begin

lift_definition zero_int :: "int" is "(0, 0)" .

lift_definition one_int :: "int" is "(1, 0)" .

lift_definition plus_int :: "int ⇒ int ⇒ int"
is "λ(x, y) (u, v). (x + u, y + v)"
by clarsimp

lift_definition uminus_int :: "int ⇒ int"
is "λ(x, y). (y, x)"
by clarsimp

lift_definition minus_int :: "int ⇒ int ⇒ int"
is "λ(x, y) (u, v). (x + v, y + u)"
by clarsimp

lift_definition times_int :: "int ⇒ int ⇒ int"
is "λ(x, y) (u, v). (x*u + y*v, x*v + y*u)"
proof (unfold intrel_def, clarify)
fix s t u v w x y z :: nat
assume "s + v = u + t" and "w + z = y + x"
then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
(u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
by simp
then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
qed

instance
by standard (transfer; clarsimp simp: algebra_simps)+

end

abbreviation int :: "nat ⇒ int"
where "int ≡ of_nat"

lemma int_def: "int n = Abs_Integ (n, 0)"

lemma int_transfer [transfer_rule]:
includes lifting_syntax
shows "rel_fun (=) pcr_int (λn. (n, 0)) int"
by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)

lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
by transfer clarsimp

subsection ‹Integers are totally ordered›

instantiation int :: linorder
begin

lift_definition less_eq_int :: "int ⇒ int ⇒ bool"
is "λ(x, y) (u, v). x + v ≤ u + y"
by auto

lift_definition less_int :: "int ⇒ int ⇒ bool"
is "λ(x, y) (u, v). x + v < u + y"
by auto

instance
by standard (transfer, force)+

end

instantiation int :: distrib_lattice
begin

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

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

instance
by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)

end

subsection ‹Ordering properties of arithmetic operations›

proof
fix i j k :: int
show "i ≤ j ⟹ k + i ≤ k + j"
by transfer clarsimp
qed

text ‹Strict Monotonicity of Multiplication.›

text ‹Strict, in 1st argument; proof is by induction on ‹k > 0›.›
lemma zmult_zless_mono2_lemma: "i < j ⟹ 0 < k ⟹ int k * i < int k * j"
for i j :: int
proof (induct k)
case 0
then show ?case by simp
next
case (Suc k)
then show ?case
qed

lemma zero_le_imp_eq_int:
assumes "k ≥ (0::int)" shows "∃n. k = int n"
proof -
have "b ≤ a ⟹ ∃n::nat. a = n + b" for a b
using exI[of _ "a - b"] by simp
with assms show ?thesis
by transfer auto
qed

lemma zero_less_imp_eq_int:
assumes "k > (0::int)" shows "∃n>0. k = int n"
proof -
have "b < a ⟹ ∃n::nat. n>0 ∧ a = n + b" for a b
using exI[of _ "a - b"] by simp
with assms show ?thesis
by transfer auto
qed

lemma zmult_zless_mono2: "i < j ⟹ 0 < k ⟹ k * i < k * j"
for i j k :: int
by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)

text ‹The integers form an ordered integral domain.›

instantiation int :: linordered_idom
begin

definition zabs_def: "¦i::int¦ = (if i < 0 then - i else i)"

definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"

instance
proof
fix i j k :: int
show "i < j ⟹ 0 < k ⟹ k * i < k * j"
by (rule zmult_zless_mono2)
show "¦i¦ = (if i < 0 then -i else i)"
by (simp only: zabs_def)
show "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
by (simp only: zsgn_def)
qed

end

lemma zless_imp_add1_zle: "w < z ⟹ w + 1 ≤ z"
for w z :: int
by transfer clarsimp

lemma zless_iff_Suc_zadd: "w < z ⟷ (∃n. z = w + int (Suc n))"
for w z :: int
proof -
have "⋀a b c d. a + d < c + b ⟹ ∃n. c + b = Suc (a + n + d)"
proof -
fix a b c d :: nat
assume "a + d < c + b"
then have "c + b = Suc (a + (c + b - Suc (a + d)) + d) "
by arith
then show "∃n. c + b = Suc (a + n + d)"
by (rule exI)
qed
then show ?thesis
by transfer auto
qed

lemma zabs_less_one_iff [simp]: "¦z¦ < 1 ⟷ z = 0" (is "?lhs ⟷ ?rhs")
for z :: int
proof
assume ?rhs
then show ?lhs by simp
next
assume ?lhs
with zless_imp_add1_zle [of "¦z¦" 1] have "¦z¦ + 1 ≤ 1" by simp
then have "¦z¦ ≤ 0" by simp
then show ?rhs by simp
qed

subsection ‹Embedding of the Integers into any ‹ring_1›: ‹of_int››

context ring_1
begin

lift_definition of_int :: "int ⇒ 'a"
is "λ(i, j). of_nat i - of_nat j"

lemma of_int_0 [simp]: "of_int 0 = 0"
by transfer simp

lemma of_int_1 [simp]: "of_int 1 = 1"
by transfer simp

lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
by transfer (clarsimp simp add: algebra_simps)

lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
by (transfer fixing: uminus) clarsimp

lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
using of_int_add [of w "- z"] by simp

lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
by (transfer fixing: times) (clarsimp simp add: algebra_simps)

lemma mult_of_int_commute: "of_int x * y = y * of_int x"
by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)

text ‹Collapse nested embeddings.›
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
by (induct n) auto

lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])

lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
by simp

lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
by (induct n) simp_all

lemma of_int_of_bool [simp]:
"of_int (of_bool P) = of_bool P"
by auto

end

context ring_char_0
begin

lemma of_int_eq_iff [simp]: "of_int w = of_int z ⟷ w = z"

text ‹Special cases where either operand is zero.›
lemma of_int_eq_0_iff [simp]: "of_int z = 0 ⟷ z = 0"
using of_int_eq_iff [of z 0] by simp

lemma of_int_0_eq_iff [simp]: "0 = of_int z ⟷ z = 0"
using of_int_eq_iff [of 0 z] by simp

lemma of_int_eq_1_iff [iff]: "of_int z = 1 ⟷ z = 1"
using of_int_eq_iff [of z 1] by simp

lemma numeral_power_eq_of_int_cancel_iff [simp]:
"numeral x ^ n = of_int y ⟷ numeral x ^ n = y"
using of_int_eq_iff[of "numeral x ^ n" y, unfolded of_int_numeral of_int_power] .

lemma of_int_eq_numeral_power_cancel_iff [simp]:
"of_int y = numeral x ^ n ⟷ y = numeral x ^ n"
using numeral_power_eq_of_int_cancel_iff [of x n y] by (metis (mono_tags))

lemma neg_numeral_power_eq_of_int_cancel_iff [simp]:
"(- numeral x) ^ n = of_int y ⟷ (- numeral x) ^ n = y"
using of_int_eq_iff[of "(- numeral x) ^ n" y]
by simp

lemma of_int_eq_neg_numeral_power_cancel_iff [simp]:
"of_int y = (- numeral x) ^ n ⟷ y = (- numeral x) ^ n"
using neg_numeral_power_eq_of_int_cancel_iff[of x n y] by (metis (mono_tags))

lemma of_int_eq_of_int_power_cancel_iff[simp]: "(of_int b) ^ w = of_int x ⟷ b ^ w = x"
by (metis of_int_power of_int_eq_iff)

lemma of_int_power_eq_of_int_cancel_iff[simp]: "of_int x = (of_int b) ^ w ⟷ x = b ^ w"
by (metis of_int_eq_of_int_power_cancel_iff)

end

context linordered_idom
begin

text ‹Every ‹linordered_idom› has characteristic zero.›
subclass ring_char_0 ..

lemma of_int_le_iff [simp]: "of_int w ≤ of_int z ⟷ w ≤ z"
by (transfer fixing: less_eq)

lemma of_int_less_iff [simp]: "of_int w < of_int z ⟷ w < z"

lemma of_int_0_le_iff [simp]: "0 ≤ of_int z ⟷ 0 ≤ z"
using of_int_le_iff [of 0 z] by simp

lemma of_int_le_0_iff [simp]: "of_int z ≤ 0 ⟷ z ≤ 0"
using of_int_le_iff [of z 0] by simp

lemma of_int_0_less_iff [simp]: "0 < of_int z ⟷ 0 < z"
using of_int_less_iff [of 0 z] by simp

lemma of_int_less_0_iff [simp]: "of_int z < 0 ⟷ z < 0"
using of_int_less_iff [of z 0] by simp

lemma of_int_1_le_iff [simp]: "1 ≤ of_int z ⟷ 1 ≤ z"
using of_int_le_iff [of 1 z] by simp

lemma of_int_le_1_iff [simp]: "of_int z ≤ 1 ⟷ z ≤ 1"
using of_int_le_iff [of z 1] by simp

lemma of_int_1_less_iff [simp]: "1 < of_int z ⟷ 1 < z"
using of_int_less_iff [of 1 z] by simp

lemma of_int_less_1_iff [simp]: "of_int z < 1 ⟷ z < 1"
using of_int_less_iff [of z 1] by simp

lemma of_int_pos: "z > 0 ⟹ of_int z > 0"
by simp

lemma of_int_nonneg: "z ≥ 0 ⟹ of_int z ≥ 0"
by simp

lemma of_int_abs [simp]: "of_int ¦x¦ = ¦of_int x¦"

lemma of_int_lessD:
assumes "¦of_int n¦ < x"
shows "n = 0 ∨ x > 1"
proof (cases "n = 0")
case True
then show ?thesis by simp
next
case False
then have "¦n¦ ≠ 0" by simp
then have "¦n¦ > 0" by simp
then have "¦n¦ ≥ 1"
using zless_imp_add1_zle [of 0 "¦n¦"] by simp
then have "¦of_int n¦ ≥ 1"
unfolding of_int_1_le_iff [of "¦n¦", symmetric] by simp
then have "1 < x" using assms by (rule le_less_trans)
then show ?thesis ..
qed

lemma of_int_leD:
assumes "¦of_int n¦ ≤ x"
shows "n = 0 ∨ 1 ≤ x"
proof (cases "n = 0")
case True
then show ?thesis by simp
next
case False
then have "¦n¦ ≠ 0" by simp
then have "¦n¦ > 0" by simp
then have "¦n¦ ≥ 1"
using zless_imp_add1_zle [of 0 "¦n¦"] by simp
then have "¦of_int n¦ ≥ 1"
unfolding of_int_1_le_iff [of "¦n¦", symmetric] by simp
then have "1 ≤ x" using assms by (rule order_trans)
then show ?thesis ..
qed

lemma numeral_power_le_of_int_cancel_iff [simp]:
"numeral x ^ n ≤ of_int a ⟷ numeral x ^ n ≤ a"
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_le_iff)

lemma of_int_le_numeral_power_cancel_iff [simp]:
"of_int a ≤ numeral x ^ n ⟷ a ≤ numeral x ^ n"
by (metis (mono_tags) local.numeral_power_eq_of_int_cancel_iff of_int_le_iff)

lemma numeral_power_less_of_int_cancel_iff [simp]:
"numeral x ^ n < of_int a ⟷ numeral x ^ n < a"
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)

lemma of_int_less_numeral_power_cancel_iff [simp]:
"of_int a < numeral x ^ n ⟷ a < numeral x ^ n"
by (metis (mono_tags) local.of_int_eq_numeral_power_cancel_iff of_int_less_iff)

lemma neg_numeral_power_le_of_int_cancel_iff [simp]:
"(- numeral x) ^ n ≤ of_int a ⟷ (- numeral x) ^ n ≤ a"
by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)

lemma of_int_le_neg_numeral_power_cancel_iff [simp]:
"of_int a ≤ (- numeral x) ^ n ⟷ a ≤ (- numeral x) ^ n"
by (metis (mono_tags) of_int_le_iff of_int_neg_numeral of_int_power)

lemma neg_numeral_power_less_of_int_cancel_iff [simp]:
"(- numeral x) ^ n < of_int a ⟷ (- numeral x) ^ n < a"
using of_int_less_iff[of "(- numeral x) ^ n" a]
by simp

lemma of_int_less_neg_numeral_power_cancel_iff [simp]:
"of_int a < (- numeral x) ^ n ⟷ a < (- numeral x::int) ^ n"
using of_int_less_iff[of a "(- numeral x) ^ n"]
by simp

lemma of_int_le_of_int_power_cancel_iff[simp]: "(of_int b) ^ w ≤ of_int x ⟷ b ^ w ≤ x"
by (metis (mono_tags) of_int_le_iff of_int_power)

lemma of_int_power_le_of_int_cancel_iff[simp]: "of_int x ≤ (of_int b) ^ w⟷ x ≤ b ^ w"
by (metis (mono_tags) of_int_le_iff of_int_power)

lemma of_int_less_of_int_power_cancel_iff[simp]: "(of_int b) ^ w < of_int x ⟷ b ^ w < x"
by (metis (mono_tags) of_int_less_iff of_int_power)

lemma of_int_power_less_of_int_cancel_iff[simp]: "of_int x < (of_int b) ^ w⟷ x < b ^ w"
by (metis (mono_tags) of_int_less_iff of_int_power)

lemma of_int_max: "of_int (max x y) = max (of_int x) (of_int y)"
by (auto simp: max_def)

lemma of_int_min: "of_int (min x y) = min (of_int x) (of_int y)"
by (auto simp: min_def)

end

context division_ring
begin

lemmas mult_inverse_of_int_commute =
mult_commute_imp_mult_inverse_commute[OF mult_of_int_commute]

end

text ‹Comparisons involving \<^term>‹of_int›.›

lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) ⟷ z = numeral n"
using of_int_eq_iff by fastforce

lemma of_int_le_numeral_iff [simp]:
"of_int z ≤ (numeral n :: 'a::linordered_idom) ⟷ z ≤ numeral n"
using of_int_le_iff [of z "numeral n"] by simp

lemma of_int_numeral_le_iff [simp]:
"(numeral n :: 'a::linordered_idom) ≤ of_int z ⟷ numeral n ≤ z"
using of_int_le_iff [of "numeral n"] by simp

lemma of_int_less_numeral_iff [simp]:
"of_int z < (numeral n :: 'a::linordered_idom) ⟷ z < numeral n"
using of_int_less_iff [of z "numeral n"] by simp

lemma of_int_numeral_less_iff [simp]:
"(numeral n :: 'a::linordered_idom) < of_int z ⟷ numeral n < z"
using of_int_less_iff [of "numeral n" z] by simp

lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x ⟷ int n < x"
by (metis of_int_of_nat_eq of_int_less_iff)

lemma of_int_eq_id [simp]: "of_int = id"
proof
show "of_int z = id z" for z
by (cases z rule: int_diff_cases) simp
qed

instance int :: no_top
proof
fix x::int
have "x < x + 1"
by simp
then show "∃y. x < y"
by (rule exI)
qed

instance int :: no_bot
proof
fix x::int
have "x - 1< x"
by simp
then show "∃y. y < x"
by (rule exI)
qed

subsection ‹Magnitude of an Integer, as a Natural Number: ‹nat››

lift_definition nat :: "int ⇒ nat" is "λ(x, y). x - y"
by auto

lemma nat_int [simp]: "nat (int n) = n"
by transfer simp

lemma int_nat_eq [simp]: "int (nat z) = (if 0 ≤ z then z else 0)"
by transfer clarsimp

lemma nat_0_le: "0 ≤ z ⟹ int (nat z) = z"
by simp

lemma nat_le_0 [simp]: "z ≤ 0 ⟹ nat z = 0"
by transfer clarsimp

lemma nat_le_eq_zle: "0 < w ∨ 0 ≤ z ⟹ nat w ≤ nat z ⟷ w ≤ z"
by transfer (clarsimp, arith)

text ‹An alternative condition is \<^term>‹0 ≤ w›.›
lemma nat_mono_iff: "0 < z ⟹ nat w < nat z ⟷ w < z"
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

lemma nat_less_eq_zless: "0 ≤ w ⟹ nat w < nat z ⟷ w < z"
by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

lemma zless_nat_conj [simp]: "nat w < nat z ⟷ 0 < z ∧ w < z"
by transfer (clarsimp, arith)

lemma nonneg_int_cases:
assumes "0 ≤ k"
obtains n where "k = int n"
proof -
from assms have "k = int (nat k)"
by simp
then show thesis
by (rule that)
qed

lemma pos_int_cases:
assumes "0 < k"
obtains n where "k = int n" and "n > 0"
proof -
from assms have "0 ≤ k"
by simp
then obtain n where "k = int n"
by (rule nonneg_int_cases)
moreover have "n > 0"
using ‹k = int n› assms by simp
ultimately show thesis
by (rule that)
qed

lemma nonpos_int_cases:
assumes "k ≤ 0"
obtains n where "k = - int n"
proof -
from assms have "- k ≥ 0"
by simp
then obtain n where "- k = int n"
by (rule nonneg_int_cases)
then have "k = - int n"
by simp
then show thesis
by (rule that)
qed

lemma neg_int_cases:
assumes "k < 0"
obtains n where "k = - int n" and "n > 0"
proof -
from assms have "- k > 0"
by simp
then obtain n where "- k = int n" and "- k > 0"
by (blast elim: pos_int_cases)
then have "k = - int n" and "n > 0"
by simp_all
then show thesis
by (rule that)
qed

lemma nat_eq_iff: "nat w = m ⟷ (if 0 ≤ w then w = int m else m = 0)"

lemma nat_eq_iff2: "m = nat w ⟷ (if 0 ≤ w then w = int m else m = 0)"
using nat_eq_iff [of w m] by auto

lemma nat_0 [simp]: "nat 0 = 0"

lemma nat_1 [simp]: "nat 1 = Suc 0"

lemma nat_numeral [simp]: "nat (numeral k) = numeral k"

lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
by simp

lemma nat_2: "nat 2 = Suc (Suc 0)"
by simp

lemma nat_less_iff: "0 ≤ w ⟹ nat w < m ⟷ w < of_nat m"
by transfer (clarsimp, arith)

lemma nat_le_iff: "nat x ≤ n ⟷ x ≤ int n"
by transfer (clarsimp simp add: le_diff_conv)

lemma nat_mono: "x ≤ y ⟹ nat x ≤ nat y"
by transfer auto

lemma nat_0_iff[simp]: "nat i = 0 ⟷ i ≤ 0"
for i :: int
by transfer clarsimp

lemma int_eq_iff: "of_nat m = z ⟷ m = nat z ∧ 0 ≤ z"

lemma zero_less_nat_eq [simp]: "0 < nat z ⟷ 0 < z"
using zless_nat_conj [of 0] by auto

lemma nat_add_distrib: "0 ≤ z ⟹ 0 ≤ z' ⟹ nat (z + z') = nat z + nat z'"
by transfer clarsimp

lemma nat_diff_distrib': "0 ≤ x ⟹ 0 ≤ y ⟹ nat (x - y) = nat x - nat y"
by transfer clarsimp

lemma nat_diff_distrib: "0 ≤ z' ⟹ z' ≤ z ⟹ nat (z - z') = nat z - nat z'"
by (rule nat_diff_distrib') auto

lemma nat_zminus_int [simp]: "nat (- int n) = 0"
by transfer simp

lemma le_nat_iff: "k ≥ 0 ⟹ n ≤ nat k ⟷ int n ≤ k"
by transfer auto

lemma zless_nat_eq_int_zless: "m < nat z ⟷ int m < z"
by transfer (clarsimp simp add: less_diff_conv)

lemma (in ring_1) of_nat_nat [simp]: "0 ≤ z ⟹ of_nat (nat z) = of_int z"
by transfer (clarsimp simp add: of_nat_diff)

lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)

lemma nat_abs_triangle_ineq:
"nat ¦k + l¦ ≤ nat ¦k¦ + nat ¦l¦"

lemma nat_of_bool [simp]:
"nat (of_bool P) = of_bool P"
by auto

lemma split_nat [linarith_split]: "P (nat i) ⟷ ((∀n. i = int n ⟶ P n) ∧ (i < 0 ⟶ P 0))"
(is "?P = (?L ∧ ?R)")
for i :: int
proof (cases "i < 0")
case True
then show ?thesis
by auto
next
case False
have "?P = ?L"
proof
assume ?P
then show ?L using False by auto
next
assume ?L
moreover from False have "int (nat i) = i"
ultimately show ?P
by simp
qed
with False show ?thesis by simp
qed

lemma all_nat: "(∀x. P x) ⟷ (∀x≥0. P (nat x))"
by (auto split: split_nat)

lemma ex_nat: "(∃x. P x) ⟷ (∃x. 0 ≤ x ∧ P (nat x))"
proof
assume "∃x. P x"
then obtain x where "P x" ..
then have "int x ≥ 0 ∧ P (nat (int x))" by simp
then show "∃x≥0. P (nat x)" ..
next
assume "∃x≥0. P (nat x)"
then show "∃x. P x" by auto
qed

text ‹For termination proofs:›
lemma measure_function_int[measure_function]: "is_measure (nat ∘ abs)" ..

subsection ‹Lemmas about the Function \<^term>‹of_nat› and Orderings›

lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
by (simp add: order_less_le del: of_nat_Suc)

lemma negative_zless [iff]: "- (int (Suc n)) < int m"
by (rule negative_zless_0 [THEN order_less_le_trans], simp)

lemma negative_zle_0: "- int n ≤ 0"

lemma negative_zle [iff]: "- int n ≤ int m"
by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])

lemma not_zle_0_negative [simp]: "¬ 0 ≤ - int (Suc n)"
by (subst le_minus_iff) (simp del: of_nat_Suc)

lemma int_zle_neg: "int n ≤ - int m ⟷ n = 0 ∧ m = 0"
by transfer simp

lemma not_int_zless_negative [simp]: "¬ int n < - int m"

lemma negative_eq_positive [simp]: "- int n = of_nat m ⟷ n = 0 ∧ m = 0"
by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)

lemma zle_iff_zadd: "w ≤ z ⟷ (∃n. z = w + int n)"
(is "?lhs ⟷ ?rhs")
proof
assume ?rhs
then show ?lhs by auto
next
assume ?lhs
then have "0 ≤ z - w" by simp
then obtain n where "z - w = int n"
using zero_le_imp_eq_int [of "z - w"] by blast
then have "z = w + int n" by simp
then show ?rhs ..
qed

lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
by simp

lemma negD:
assumes "x < 0" shows "∃n. x = - (int (Suc n))"
proof -
have "⋀a b. a < b ⟹ ∃n. Suc (a + n) = b"
proof -
fix a b:: nat
assume "a < b"
then have "Suc (a + (b - Suc a)) = b"
by arith
then show "∃n. Suc (a + n) = b"
by (rule exI)
qed
with assms show ?thesis
by transfer auto
qed

subsection ‹Cases and induction›

text ‹
Now we replace the case analysis rule by a more conventional one:
whether an integer is negative or not.
›

text ‹This version is symmetric in the two subgoals.›
lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
"(⋀n. z = int n ⟹ P) ⟹ (⋀n. z = - (int n) ⟹ P) ⟹ P"
by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])

text ‹This is the default, with a negative case.›
lemma int_cases [case_names nonneg neg, cases type: int]:
assumes pos: "⋀n. z = int n ⟹ P" and neg: "⋀n. z = - (int (Suc n)) ⟹ P"
shows P
proof (cases "z < 0")
case True
with neg show ?thesis
by (blast dest!: negD)
next
case False
with pos show ?thesis
by (force simp add: linorder_not_less dest: nat_0_le [THEN sym])
qed

lemma int_cases3 [case_names zero pos neg]:
fixes k :: int
assumes "k = 0 ⟹ P" and "⋀n. k = int n ⟹ n > 0 ⟹ P"
and "⋀n. k = - int n ⟹ n > 0 ⟹ P"
shows "P"
proof (cases k "0::int" rule: linorder_cases)
case equal
with assms(1) show P by simp
next
case greater
then have *: "nat k > 0" by simp
moreover from * have "k = int (nat k)" by auto
ultimately show P using assms(2) by blast
next
case less
then have *: "nat (- k) > 0" by simp
moreover from * have "k = - int (nat (- k))" by auto
ultimately show P using assms(3) by blast
qed

lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
"(⋀n. P (int n)) ⟹ (⋀n. P (- (int (Suc n)))) ⟹ P z"
by (cases z) auto

lemma sgn_mult_dvd_iff [simp]:
"sgn r * l dvd k ⟷ l dvd k ∧ (r = 0 ⟶ k = 0)" for k l r :: int
by (cases r rule: int_cases3) auto

lemma mult_sgn_dvd_iff [simp]:
"l * sgn r dvd k ⟷ l dvd k ∧ (r = 0 ⟶ k = 0)" for k l r :: int
using sgn_mult_dvd_iff [of r l k] by (simp add: ac_simps)

lemma dvd_sgn_mult_iff [simp]:
"l dvd sgn r * k ⟷ l dvd k ∨ r = 0" for k l r :: int
by (cases r rule: int_cases3) simp_all

lemma dvd_mult_sgn_iff [simp]:
"l dvd k * sgn r ⟷ l dvd k ∨ r = 0" for k l r :: int
using dvd_sgn_mult_iff [of l r k] by (simp add: ac_simps)

lemma int_sgnE:
fixes k :: int
obtains n and l where "k = sgn l * int n"
proof -
have "k = sgn k * int (nat ¦k¦)"
then show ?thesis ..
qed

subsubsection ‹Binary comparisons›

text ‹Preliminaries›

lemma le_imp_0_less:
fixes z :: int
assumes le: "0 ≤ z"
shows "0 < 1 + z"
proof -
have "0 ≤ z" by fact
also have "… < z + 1" by (rule less_add_one)
also have "… = 1 + z" by (simp add: ac_simps)
finally show "0 < 1 + z" .
qed

lemma odd_less_0_iff: "1 + z + z < 0 ⟷ z < 0"
for z :: int
proof (cases z)
case (nonneg n)
then show ?thesis
next
case (neg n)
then show ?thesis
by (simp del: of_nat_Suc of_nat_add of_nat_1
qed

subsubsection ‹Comparisons, for Ordered Rings›

lemma odd_nonzero: "1 + z + z ≠ 0"
for z :: int
proof (cases z)
case (nonneg n)
have le: "0 ≤ z + z"
then show ?thesis
using le_imp_0_less [OF le] by (auto simp: ac_simps)
next
case (neg n)
show ?thesis
proof
assume eq: "1 + z + z = 0"
have "0 < 1 + (int n + int n)"
also have "… = - (1 + z + z)"
also have "… = 0" by (simp add: eq)
finally have "0<0" ..
then show False by blast
qed
qed

subsection ‹The Set of Integers›

context ring_1
begin

definition Ints :: "'a set"  ("ℤ")
where "ℤ = range of_int"

lemma Ints_of_int [simp]: "of_int z ∈ ℤ"

lemma Ints_of_nat [simp]: "of_nat n ∈ ℤ"
using Ints_of_int [of "of_nat n"] by simp

lemma Ints_0 [simp]: "0 ∈ ℤ"
using Ints_of_int [of "0"] by simp

lemma Ints_1 [simp]: "1 ∈ ℤ"
using Ints_of_int [of "1"] by simp

lemma Ints_numeral [simp]: "numeral n ∈ ℤ"
by (subst of_nat_numeral [symmetric], rule Ints_of_nat)

lemma Ints_add [simp]: "a ∈ ℤ ⟹ b ∈ ℤ ⟹ a + b ∈ ℤ"

lemma Ints_minus [simp]: "a ∈ ℤ ⟹ -a ∈ ℤ"
by (force simp add: Ints_def simp flip: of_int_minus intro: range_eqI)

lemma minus_in_Ints_iff: "-x ∈ ℤ ⟷ x ∈ ℤ"
using Ints_minus[of x] Ints_minus[of "-x"] by auto

lemma Ints_diff [simp]: "a ∈ ℤ ⟹ b ∈ ℤ ⟹ a - b ∈ ℤ"
by (force simp add: Ints_def simp flip: of_int_diff intro: range_eqI)

lemma Ints_mult [simp]: "a ∈ ℤ ⟹ b ∈ ℤ ⟹ a * b ∈ ℤ"
by (force simp add: Ints_def simp flip: of_int_mult intro: range_eqI)

lemma Ints_power [simp]: "a ∈ ℤ ⟹ a ^ n ∈ ℤ"
by (induct n) simp_all

lemma Ints_cases [cases set: Ints]:
assumes "q ∈ ℤ"
obtains (of_int) z where "q = of_int z"
unfolding Ints_def
proof -
from ‹q ∈ ℤ› have "q ∈ range of_int" unfolding Ints_def .
then obtain z where "q = of_int z" ..
then show thesis ..
qed

lemma Ints_induct [case_names of_int, induct set: Ints]:
"q ∈ ℤ ⟹ (⋀z. P (of_int z)) ⟹ P q"
by (rule Ints_cases) auto

lemma Nats_subset_Ints: "ℕ ⊆ ℤ"
unfolding Nats_def Ints_def
by (rule subsetI, elim imageE, hypsubst, subst of_int_of_nat_eq[symmetric], rule imageI) simp_all

lemma Nats_altdef1: "ℕ = {of_int n |n. n ≥ 0}"
proof (intro subsetI equalityI)
fix x :: 'a
assume "x ∈ {of_int n |n. n ≥ 0}"
then obtain n where "x = of_int n" "n ≥ 0"
by (auto elim!: Ints_cases)
then have "x = of_nat (nat n)"
by (subst of_nat_nat) simp_all
then show "x ∈ ℕ"
by simp
next
fix x :: 'a
assume "x ∈ ℕ"
then obtain n where "x = of_nat n"
by (auto elim!: Nats_cases)
then have "x = of_int (int n)" by simp
also have "int n ≥ 0" by simp
then have "of_int (int n) ∈ {of_int n |n. n ≥ 0}" by blast
finally show "x ∈ {of_int n |n. n ≥ 0}" .
qed

end

lemma Ints_sum [intro]: "(⋀x. x ∈ A ⟹ f x ∈ ℤ) ⟹ sum f A ∈ ℤ"
by (induction A rule: infinite_finite_induct) auto

lemma Ints_prod [intro]: "(⋀x. x ∈ A ⟹ f x ∈ ℤ) ⟹ prod f A ∈ ℤ"
by (induction A rule: infinite_finite_induct) auto

lemma (in linordered_idom) Ints_abs [simp]:
shows "a ∈ ℤ ⟹ abs a ∈ ℤ"
by (auto simp: abs_if)

lemma (in linordered_idom) Nats_altdef2: "ℕ = {n ∈ ℤ. n ≥ 0}"
proof (intro subsetI equalityI)
fix x :: 'a
assume "x ∈ {n ∈ ℤ. n ≥ 0}"
then obtain n where "x = of_int n" "n ≥ 0"
by (auto elim!: Ints_cases)
then have "x = of_nat (nat n)"
by (subst of_nat_nat) simp_all
then show "x ∈ ℕ"
by simp
qed (auto elim!: Nats_cases)

lemma (in idom_divide) of_int_divide_in_Ints:
"of_int a div of_int b ∈ ℤ" if "b dvd a"
proof -
from that obtain c where "a = b * c" ..
then show ?thesis
by (cases "of_int b = 0") simp_all
qed

text ‹The premise involving \<^term>‹Ints› prevents \<^term>‹a = 1/2›.›

lemma Ints_double_eq_0_iff:
fixes a :: "'a::ring_char_0"
assumes in_Ints: "a ∈ ℤ"
shows "a + a = 0 ⟷ a = 0"
(is "?lhs ⟷ ?rhs")
proof -
from in_Ints have "a ∈ range of_int"
unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
assume ?rhs
then show ?lhs by simp
next
assume ?lhs
with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
then have "z + z = 0" by (simp only: of_int_eq_iff)
then have "z = 0" by (simp only: double_zero)
with a show ?rhs by simp
qed
qed

lemma Ints_odd_nonzero:
fixes a :: "'a::ring_char_0"
assumes in_Ints: "a ∈ ℤ"
shows "1 + a + a ≠ 0"
proof -
from in_Ints have "a ∈ range of_int"
unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
assume "1 + a + a = 0"
with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
with odd_nonzero show False by blast
qed
qed

lemma Nats_numeral [simp]: "numeral w ∈ ℕ"
using of_nat_in_Nats [of "numeral w"] by simp

lemma Ints_odd_less_0:
fixes a :: "'a::linordered_idom"
assumes in_Ints: "a ∈ ℤ"
shows "1 + a + a < 0 ⟷ a < 0"
proof -
from in_Ints have "a ∈ range of_int"
unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
with a have "1 + a + a < 0 ⟷ of_int (1 + z + z) < (of_int 0 :: 'a)"
by simp
also have "… ⟷ z < 0"
by (simp only: of_int_less_iff odd_less_0_iff)
also have "… ⟷ a < 0"
finally show ?thesis .
qed

subsection ‹\<^term>‹sum› and \<^term>‹prod››

context semiring_1
begin

lemma of_nat_sum [simp]:
"of_nat (sum f A) = (∑x∈A. of_nat (f x))"
by (induction A rule: infinite_finite_induct) auto

end

context ring_1
begin

lemma of_int_sum [simp]:
"of_int (sum f A) = (∑x∈A. of_int (f x))"
by (induction A rule: infinite_finite_induct) auto

end

context comm_semiring_1
begin

lemma of_nat_prod [simp]:
"of_nat (prod f A) = (∏x∈A. of_nat (f x))"
by (induction A rule: infinite_finite_induct) auto

end

context comm_ring_1
begin

lemma of_int_prod [simp]:
"of_int (prod f A) = (∏x∈A. of_int (f x))"
by (induction A rule: infinite_finite_induct) auto

end

subsection ‹Setting up simplification procedures›

ML_file ‹Tools/int_arith.ML›

declaration ‹K (
#> Lin_Arith.add_inj_thms @{thms of_nat_le_iff [THEN iffD2] of_nat_eq_iff [THEN iffD2]}
#> Lin_Arith.add_inj_const (\<^const_name>‹of_nat›, \<^typ>‹nat ⇒ int›)
@{thms of_int_0 of_int_1 of_int_add of_int_mult of_int_numeral of_int_neg_numeral nat_0 nat_1 diff_nat_numeral nat_numeral
neg_less_iff_less
True_implies_equals
distrib_left [where a = "numeral v" for v]
distrib_left [where a = "- numeral v" for v]
div_by_1 div_0
times_divide_eq_right times_divide_eq_left
minus_divide_left [THEN sym] minus_divide_right [THEN sym]
of_int_minus of_int_diff
of_int_of_nat_eq}
)›

simproc_setup fast_arith
("(m::'a::linordered_idom) < n" |
"(m::'a::linordered_idom) ≤ n" |
"(m::'a::linordered_idom) = n") =
‹K Lin_Arith.simproc›

subsection‹More Inequality Reasoning›

lemma zless_add1_eq: "w < z + 1 ⟷ w < z ∨ w = z"
for w z :: int
by arith

lemma add1_zle_eq: "w + 1 ≤ z ⟷ w < z"
for w z :: int
by arith

lemma zle_diff1_eq [simp]: "w ≤ z - 1 ⟷ w < z"
for w z :: int
by arith

lemma zle_add1_eq_le [simp]: "w < z + 1 ⟷ w ≤ z"
for w z :: int
by arith

lemma int_one_le_iff_zero_less: "1 ≤ z ⟷ 0 < z"
for z :: int
by arith

lemma Ints_nonzero_abs_ge1:
fixes x:: "'a :: linordered_idom"
assumes "x ∈ Ints" "x ≠ 0"
shows "1 ≤ abs x"
proof (rule Ints_cases [OF ‹x ∈ Ints›])
fix z::int
assume "x = of_int z"
with ‹x ≠ 0›
show "1 ≤ ¦x¦"
apply (auto simp: abs_if)
by (metis diff_0 of_int_1 of_int_le_iff of_int_minus zle_diff1_eq)
qed

lemma Ints_nonzero_abs_less1:
fixes x:: "'a :: linordered_idom"
shows "⟦x ∈ Ints; abs x < 1⟧ ⟹ x = 0"
using Ints_nonzero_abs_ge1 [of x] by auto

lemma Ints_eq_abs_less1:
fixes x:: "'a :: linordered_idom"
shows "⟦x ∈ Ints; y ∈ Ints⟧ ⟹ x = y ⟷ abs (x-y) < 1"
using eq_iff_diff_eq_0 by (fastforce intro: Ints_nonzero_abs_less1)

subsection ‹The functions \<^term>‹nat› and \<^term>‹int››

text ‹Simplify the term \<^term>‹w + - z›.›

lemma one_less_nat_eq [simp]: "Suc 0 < nat z ⟷ 1 < z"
using zless_nat_conj [of 1 z] by auto

lemma int_eq_iff_numeral [simp]:
"int m = numeral v ⟷ m = numeral v"

lemma nat_abs_int_diff:
"nat ¦int a - int b¦ = (if a ≤ b then b - a else a - b)"
by auto

lemma nat_int_add: "nat (int a + int b) = a + b"
by auto

context ring_1
begin

lemma of_int_of_nat [nitpick_simp]:
"of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
proof (cases "k < 0")
case True
then have "0 ≤ - k" by simp
then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
with True show ?thesis by simp
next
case False
then show ?thesis by (simp add: not_less)
qed

end

lemma transfer_rule_of_int:
includes lifting_syntax
fixes R :: "'a::ring_1 ⇒ 'b::ring_1 ⇒ bool"
assumes [transfer_rule]: "R 0 0" "R 1 1"
"(R ===> R ===> R) (+) (+)"
"(R ===> R) uminus uminus"
shows "((=) ===> R) of_int of_int"
proof -
note assms
note transfer_rule_of_nat [transfer_rule]
have [transfer_rule]: "((=) ===> R) of_nat of_nat"
by transfer_prover
show ?thesis
by (unfold of_int_of_nat [abs_def]) transfer_prover
qed

lemma nat_mult_distrib:
fixes z z' :: int
assumes "0 ≤ z"
shows "nat (z * z') = nat z * nat z'"
proof (cases "0 ≤ z'")
case False
with assms have "z * z' ≤ 0"
then have "nat (z * z') = 0" by simp
moreover from False have "nat z' = 0" by simp
ultimately show ?thesis by simp
next
case True
with assms have ge_0: "z * z' ≥ 0" by (simp add: zero_le_mult_iff)
show ?thesis
by (rule injD [of "of_nat :: nat ⇒ int", OF inj_of_nat])
(simp only: of_nat_mult of_nat_nat [OF True]
of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
qed

lemma nat_mult_distrib_neg:
assumes "z ≤ (0::int)" shows "nat (z * z') = nat (- z) * nat (- z')" (is "?L = ?R")
proof -
have "?L = nat (- z * - z')"
using assms by auto
also have "... = ?R"
by (rule nat_mult_distrib) (use assms in auto)
finally show ?thesis .
qed

lemma nat_abs_mult_distrib: "nat ¦w * z¦ = nat ¦w¦ * nat ¦z¦"
by (cases "z = 0 ∨ w = 0")
(auto simp add: abs_if nat_mult_distrib [symmetric]
nat_mult_distrib_neg [symmetric] mult_less_0_iff)

lemma int_in_range_abs [simp]: "int n ∈ range abs"
proof (rule range_eqI)
show "int n = ¦int n¦" by simp
qed

lemma range_abs_Nats [simp]: "range abs = (ℕ :: int set)"
proof -
have "¦k¦ ∈ ℕ" for k :: int
by (cases k) simp_all
moreover have "k ∈ range abs" if "k ∈ ℕ" for k :: int
using that by induct simp
ultimately show ?thesis by blast
qed

lemma Suc_nat_eq_nat_zadd1: "0 ≤ z ⟹ Suc (nat z) = nat (1 + z)"
for z :: int
by (rule sym) (simp add: nat_eq_iff)

lemma diff_nat_eq_if:
"nat z - nat z' =
(if z' < 0 then nat z
else
let d = z - z'
in if d < 0 then 0 else nat d)"
by (simp add: Let_def nat_diff_distrib [symmetric])

lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
using diff_nat_numeral [of v Num.One] by simp

subsection ‹Induction principles for int›

text ‹Well-founded segments of the integers.›

definition int_ge_less_than :: "int ⇒ (int × int) set"
where "int_ge_less_than d = {(z', z). d ≤ z' ∧ z' < z}"

lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
proof -
have "int_ge_less_than d ⊆ measure (λz. nat (z - d))"
then show ?thesis
by (rule wf_subset [OF wf_measure])
qed

text ‹
This variant looks odd, but is typical of the relations suggested
by RankFinder.›

definition int_ge_less_than2 :: "int ⇒ (int × int) set"
where "int_ge_less_than2 d = {(z',z). d ≤ z ∧ z' < z}"

lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
proof -
have "int_ge_less_than2 d ⊆ measure (λz. nat (1 + z - d))"
then show ?thesis
by (rule wf_subset [OF wf_measure])
qed

(* `set:int': dummy construction *)
theorem int_ge_induct [case_names base step, induct set: int]:
fixes i :: int
assumes ge: "k ≤ i"
and base: "P k"
and step: "⋀i. k ≤ i ⟹ P i ⟹ P (i + 1)"
shows "P i"
proof -
have "⋀i::int. n = nat (i - k) ⟹ k ≤ i ⟹ P i" for n
proof (induct n)
case 0
then have "i = k" by arith
with base show "P i" by simp
next
case (Suc n)
then have "n = nat ((i - 1) - k)" by arith
moreover have k: "k ≤ i - 1" using Suc.prems by arith
ultimately have "P (i - 1)" by (rule Suc.hyps)
from step [OF k this] show ?case by simp
qed
with ge show ?thesis by fast
qed

(* `set:int': dummy construction *)
theorem int_gr_induct [case_names base step, induct set: int]:
fixes i k :: int
assumes "k < i" "P (k + 1)" "⋀i. k < i ⟹ P i ⟹ P (i + 1)"
shows "P i"
proof -
have "k+1 ≤ i"
using assms by auto
then show ?thesis
by (induction i rule: int_ge_induct) (auto simp: assms)
qed

theorem int_le_induct [consumes 1, case_names base step]:
fixes i k :: int
assumes le: "i ≤ k"
and base: "P k"
and step: "⋀i. i ≤ k ⟹ P i ⟹ P (i - 1)"
shows "P i"
proof -
have "⋀i::int. n = nat(k-i) ⟹ i ≤ k ⟹ P i" for n
proof (induct n)
case 0
then have "i = k" by arith
with base show "P i" by simp
next
case (Suc n)
then have "n = nat (k - (i + 1))" by arith
moreover have k: "i + 1 ≤ k" using Suc.prems by arith
ultimately have "P (i + 1)" by (rule Suc.hyps)
from step[OF k this] show ?case by simp
qed
with le show ?thesis by fast
qed

theorem int_less_induct [consumes 1, case_names base step]:
fixes i k :: int
assumes "i < k" "P (k - 1)" "⋀i. i < k ⟹ P i ⟹ P (i - 1)"
shows "P i"
proof -
have "i ≤ k-1"
using assms by auto
then show ?thesis
by (induction i rule: int_le_induct) (auto simp: assms)
qed

theorem int_induct [case_names base step1 step2]:
fixes k :: int
assumes base: "P k"
and step1: "⋀i. k ≤ i ⟹ P i ⟹ P (i + 1)"
and step2: "⋀i. k ≥ i ⟹ P i ⟹ P (i - 1)"
shows "P i"
proof -
have "i ≤ k ∨ i ≥ k" by arith
then show ?thesis
proof
assume "i ≥ k"
then show ?thesis
using base by (rule int_ge_induct) (fact step1)
next
assume "i ≤ k"
then show ?thesis
using base by (rule int_le_induct) (fact step2)
qed
qed

subsection ‹Intermediate value theorems›

lemma nat_ivt_aux:
"⟦∀i<n. ¦f (Suc i) - f i¦ ≤ 1; f 0 ≤ k; k ≤ f n⟧ ⟹ ∃i ≤ n. f i = k"
for m n :: nat and k :: int
proof (induct n)
case (Suc n)
show ?case
proof (cases "k = f (Suc n)")
case False
with Suc have "k ≤ f n"
by auto
with Suc show ?thesis
by (auto simp add: abs_if split: if_split_asm intro: le_SucI)
qed (use Suc in auto)
qed auto

lemma nat_intermed_int_val:
fixes m n :: nat and k :: int
assumes "∀i. m ≤ i ∧ i < n ⟶ ¦f (Suc i) - f i¦ ≤ 1" "m ≤ n" "f m ≤ k" "k ≤ f n"
shows "∃i. m ≤ i ∧ i ≤ n ∧ f i = k"
proof -
obtain i where "i ≤ n - m" "k = f (m + i)"
using nat_ivt_aux [of "n - m" "f ∘ plus m" k] assms by auto
with assms show ?thesis
using exI[of _ "m + i"] by auto
qed

lemma nat0_intermed_int_val:
"∃i≤n. f i = k"
if "∀i<n. ¦f (i + 1) - f i¦ ≤ 1" "f 0 ≤ k" "k ≤ f n"
for n :: nat and k :: int
using nat_intermed_int_val [of 0 n f k] that by auto

subsection ‹Products and 1, by T. M. Rasmussen›

lemma abs_zmult_eq_1:
fixes m n :: int
assumes mn: "¦m * n¦ = 1"
shows "¦m¦ = 1"
proof -
from mn have 0: "m ≠ 0" "n ≠ 0" by auto
have "¬ 2 ≤ ¦m¦"
proof
assume "2 ≤ ¦m¦"
then have "2 * ¦n¦ ≤ ¦m¦ * ¦n¦" by (simp add: mult_mono 0)
also have "… = ¦m * n¦" by (simp add: abs_mult)
also from mn have "… = 1" by simp
finally have "2 * ¦n¦ ≤ 1" .
with 0 show "False" by arith
qed
with 0 show ?thesis by auto
qed

lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 ⟹ m = 1 ∨ m = - 1"
for m n :: int
using abs_zmult_eq_1 [of m n] by arith

lemma pos_zmult_eq_1_iff:
fixes m n :: int
assumes "0 < m"
shows "m * n = 1 ⟷ m = 1 ∧ n = 1"
proof -
from assms have "m * n = 1 ⟹ m = 1"
by (auto dest: pos_zmult_eq_1_iff_lemma)
then show ?thesis
by (auto dest: pos_zmult_eq_1_iff_lemma)
qed

lemma zmult_eq_1_iff: "m * n = 1 ⟷ (m = 1 ∧ n = 1) ∨ (m = - 1 ∧ n = - 1)" (is "?L = ?R")
for m n :: int
proof
assume L: ?L show ?R
using pos_zmult_eq_1_iff_lemma [OF L] L by force
qed auto

lemma infinite_UNIV_int [simp]: "¬ finite (UNIV::int set)"
proof
assume "finite (UNIV::int set)"
moreover have "inj (λi::int. 2 * i)"
by (rule injI) simp
ultimately have "surj (λi::int. 2 * i)"
by (rule finite_UNIV_inj_surj)
then obtain i :: int where "1 = 2 * i" by (rule surjE)
then show False by (simp add: pos_zmult_eq_1_iff)
qed

subsection ‹The divides relation›

lemma zdvd_antisym_nonneg: "0 ≤ m ⟹ 0 ≤ n ⟹ m dvd n ⟹ n dvd m ⟹ m = n"
for m n :: int
by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)

lemma zdvd_antisym_abs:
fixes a b :: int
assumes "a dvd b" and "b dvd a"
shows "¦a¦ = ¦b¦"
proof (cases "a = 0")
case True
with assms show ?thesis by simp
next
case False
from ‹a dvd b› obtain k where k: "b = a * k"
unfolding dvd_def by blast
from ‹b dvd a› obtain k' where k': "a = b * k'"
unfolding dvd_def by blast
from k k' have "a = a * k * k'" by simp
with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
using ‹a ≠ 0› by (simp add: mult.assoc)
then have "k = 1 ∧ k' = 1 ∨ k = -1 ∧ k' = -1"
with k k' show ?thesis by auto
qed

lemma zdvd_zdiffD: "k dvd m - n ⟹ k dvd n ⟹ k dvd m"
for k m n :: int
using dvd_add_right_iff [of k "- n" m] by simp

lemma zdvd_reduce: "k dvd n + k * m ⟷ k dvd n"
for k m n :: int

lemma dvd_imp_le_int:
fixes d i :: int
assumes "i ≠ 0" and "d dvd i"
shows "¦d¦ ≤ ¦i¦"
proof -
from ‹d dvd i› obtain k where "i = d * k" ..
with ‹i ≠ 0› have "k ≠ 0" by auto
then have "1 ≤ ¦k¦" and "0 ≤ ¦d¦" by auto
then have "¦d¦ * 1 ≤ ¦d¦ * ¦k¦" by (rule mult_left_mono)
with ‹i = d * k› show ?thesis by (simp add: abs_mult)
qed

lemma zdvd_not_zless:
fixes m n :: int
assumes "0 < m" and "m < n"
shows "¬ n dvd m"
proof
from assms have "0 < n" by auto
assume "n dvd m" then obtain k where k: "m = n * k" ..
with ‹0 < m› have "0 < n * k" by auto
with ‹0 < n› have "0 < k" by (simp add: zero_less_mult_iff)
with k ‹0 < n› ‹m < n› have "n * k < n * 1" by simp
with ‹0 < n› ‹0 < k› show False unfolding mult_less_cancel_left by auto
qed

lemma zdvd_mult_cancel:
fixes k m n :: int
assumes d: "k * m dvd k * n"
and "k ≠ 0"
shows "m dvd n"
proof -
from d obtain h where h: "k * n = k * m * h"
unfolding dvd_def by blast
have "n = m * h"
proof (rule ccontr)
assume "¬ ?thesis"
with ‹k ≠ 0› have "k * n ≠ k * (m * h)" by simp
with h show False
qed
then show ?thesis by simp
qed

lemma int_dvd_int_iff [simp]:
"int m dvd int n ⟷ m dvd n"
proof -
have "m dvd n" if "int n = int m * k" for k
proof (cases k)
case (nonneg q)
with that have "n = m * q"
by (simp del: of_nat_mult add: of_nat_mult [symmetric])
then show ?thesis ..
next
case (neg q)
with that have "int n = int m * (- int (Suc q))"
by simp
also have "… = - (int m * int (Suc q))"
by (simp only: mult_minus_right)
also have "… = - int (m * Suc q)"
by (simp only: of_nat_mult [symmetric])
finally have "- int (m * Suc q) = int n" ..
then show ?thesis
by (simp only: negative_eq_positive) auto
qed
then show ?thesis by (auto simp add: dvd_def)
qed

lemma dvd_nat_abs_iff [simp]:
"n dvd nat ¦k¦ ⟷ int n dvd k"
proof -
have "n dvd nat ¦k¦ ⟷ int n dvd int (nat ¦k¦)"
by (simp only: int_dvd_int_iff)
then show ?thesis
by simp
qed

lemma nat_abs_dvd_iff [simp]:
"nat ¦k¦ dvd n ⟷ k dvd int n"
proof -
have "nat ¦k¦ dvd n ⟷ int (nat ¦k¦) dvd int n"
by (simp only: int_dvd_int_iff)
then show ?thesis
by simp
qed

lemma zdvd1_eq [simp]: "x dvd 1 ⟷ ¦x¦ = 1" (is "?lhs ⟷ ?rhs")
for x :: int
proof
assume ?lhs
then have "nat ¦x¦ dvd nat ¦1¦"
by (simp only: nat_abs_dvd_iff) simp
then have "nat ¦x¦ = 1"
by simp
then show ?rhs
by (cases "x < 0") simp_all
next
assume ?rhs
then have "x = 1 ∨ x = - 1"
by auto
then show ?lhs
by (auto intro: dvdI)
qed

lemma zdvd_mult_cancel1:
fixes m :: int
assumes mp: "m ≠ 0"
shows "m * n dvd m ⟷ ¦n¦ = 1"
(is "?lhs ⟷ ?rhs")
proof
assume ?rhs
then show ?lhs
by (cases "n > 0") (auto simp add: minus_equation_iff)
next
assume ?lhs
then have "m * n dvd m * 1" by simp
from zdvd_mult_cancel[OF this mp] show ?rhs
by (simp only: zdvd1_eq)
qed

lemma nat_dvd_iff: "nat z dvd m ⟷ (if 0 ≤ z then z dvd int m else m = 0)"
using nat_abs_dvd_iff [of z m] by (cases "z ≥ 0") auto

lemma eq_nat_nat_iff: "0 ≤ z ⟹ 0 ≤ z' ⟹ nat z = nat z' ⟷ z = z'"
by (auto elim: nonneg_int_cases)

lemma nat_power_eq: "0 ≤ z ⟹ nat (z ^ n) = nat z ^ n"
by (induct n) (simp_all add: nat_mult_distrib)

lemma numeral_power_eq_nat_cancel_iff [simp]:
"numeral x ^ n = nat y ⟷ numeral x ^ n = y"
using nat_eq_iff2 by auto

lemma nat_eq_numeral_power_cancel_iff [simp]:
"nat y = numeral x ^ n ⟷ y = numeral x ^ n"
using numeral_power_eq_nat_cancel_iff[of x n y]
by (metis (mono_tags))

lemma numeral_power_le_nat_cancel_iff [simp]:
"numeral x ^ n ≤ nat a ⟷ numeral x ^ n ≤ a"
using nat_le_eq_zle[of "numeral x ^ n" a]
by (auto simp: nat_power_eq)

lemma nat_le_numeral_power_cancel_iff [simp]:
"nat a ≤ numeral x ^ n ⟷ a ≤ numeral x ^ n"

lemma numeral_power_less_nat_cancel_iff [simp]:
"numeral x ^ n < nat a ⟷ numeral x ^ n < a"
using nat_less_eq_zless[of "numeral x ^ n" a]
by (auto simp: nat_power_eq)

lemma nat_less_numeral_power_cancel_iff [simp]:
"nat a < numeral x ^ n ⟷ a < numeral x ^ n"
using nat_less_eq_zless[of a "numeral x ^ n"]
by (cases "a < 0") (auto simp: nat_power_eq less_le_trans[where y=0])

lemma zdvd_imp_le: "z ≤ n" if "z dvd n" "0 < n" for n z :: int
proof (cases n)
case (nonneg n)
show ?thesis
by (cases z) (use nonneg dvd_imp_le that in auto)
qed (use that in auto)

lemma zdvd_period:
fixes a d :: int
assumes "a dvd d"
shows "a dvd (x + t) ⟷ a dvd ((x + c * d) + t)"
(is "?lhs ⟷ ?rhs")
proof -
from assms have "a dvd (x + t) ⟷ a dvd ((x + t) + c * d)"
then show ?thesis
qed

subsection ‹Powers with integer exponents›

text ‹
The following allows writing powers with an integer exponent. While the type signature
is very generic, most theorems will assume that the underlying type is a division ring or
a field.

The notation `powi' is inspired by the `powr' notation for real/complex exponentiation.
›
definition power_int :: "'a :: {inverse, power} ⇒ int ⇒ 'a" (infixr "powi" 80) where
"power_int x n = (if n ≥ 0 then x ^ nat n else inverse x ^ (nat (-n)))"

lemma power_int_0_right [simp]: "power_int x 0 = 1"
and power_int_1_right [simp]:
"power_int (y :: 'a :: {power, inverse, monoid_mult}) 1 = y"
and power_int_minus1_right [simp]:
"power_int (y :: 'a :: {power, inverse, monoid_mult}) (-1) = inverse y"

lemma power_int_of_nat [simp]: "power_int x (int n) = x ^ n"

lemma power_int_numeral [simp]: "power_int x (numeral n) = x ^ numeral n"

lemma int_cases4 [case_names nonneg neg]:
fixes m :: int
obtains n where "m = int n" | n where "n > 0" "m = -int n"
proof (cases "m ≥ 0")
case True
thus ?thesis using that(1)[of "nat m"] by auto
next
case False
thus ?thesis using that(2)[of "nat (-m)"] by auto
qed

context
assumes "SORT_CONSTRAINT('a::division_ring)"
begin

lemma power_int_minus: "power_int (x::'a) (-n) = inverse (power_int x n)"
by (auto simp: power_int_def power_inverse)

lemma power_int_eq_0_iff [simp]: "power_int (x::'a) n = 0 ⟷ x = 0 ∧ n ≠ 0"
by (auto simp: power_int_def)

lemma power_int_0_left_If: "power_int (0 :: 'a) m = (if m = 0 then 1 else 0)"
by (auto simp: power_int_def)

lemma power_int_0_left [simp]: "m ≠ 0 ⟹ power_int (0 :: 'a) m = 0"

lemma power_int_1_left [simp]: "power_int 1 n = (1 :: 'a :: division_ring)"
by (auto simp: power_int_def)

lemma power_diff_conv_inverse: "x ≠ 0 ⟹ m ≤ n ⟹ (x :: 'a) ^ (n - m) = x ^ n * inverse x ^ m"

lemma power_mult_inverse_distrib: "x ^ m * inverse (x :: 'a) = inverse x * x ^ m"
proof (cases "x = 0")
case [simp]: False
show ?thesis
proof (cases m)
case (Suc m')
have "x ^ Suc m' * inverse x = x ^ m'"
by (subst power_Suc2) (auto simp: mult.assoc)
also have "… = inverse x * x ^ Suc m'"
by (subst power_Suc) (auto simp: mult.assoc [symmetric])
finally show ?thesis using Suc by simp
qed auto
qed auto

lemma power_mult_power_inverse_commute:
"x ^ m * inverse (x :: 'a) ^ n = inverse x ^ n * x ^ m"
proof (induction n)
case (Suc n)
have "x ^ m * inverse x ^ Suc n = (x ^ m * inverse x ^ n) * inverse x"
by (simp only: power_Suc2 mult.assoc)
also have "x ^ m * inverse x ^ n = inverse x ^ n * x ^ m"
by (rule Suc)
also have "… ```