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"
  by (simp add: intrel_def)

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


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)"
    by (simp add: algebra_simps)
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)"
  by (induct n) (simp add: zero_int.abs_eq, simp add: one_int.abs_eq plus_int.abs_eq)

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›

instance int :: ordered_cancel_ab_semigroup_add
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
    by (cases "k = 0") (simp_all add: distrib_right add_strict_mono)
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

instance int :: discrete_linordered_semidom
proof
  fix k l :: int
  show k < l  k + 1  l (is ?P  ?Q)
  proof
    assume ?Q
    then show ?P
      by simp
  next
    assume ?P
    then have l - k > 0
      by simp
    with zero_less_imp_eq_int obtain n where l - k = int n
      by blast
    then have n > 0
      using l - k > 0 by simp
    then have n  1
      by simp
    then have int n  int 1
      by (rule of_nat_mono)
    with l - k = int n show ?Q
      by simp
  qed
qed

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"
  by (clarsimp simp add: diff_eq_eq eq_diff_eq diff_add_eq
      of_nat_add [symmetric] simp del: of_nat_add)

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"
  by transfer (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)

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)
    (clarsimp simp add: algebra_simps of_nat_add [symmetric] simp del: of_nat_add)

lemma of_int_less_iff [simp]: "of_int w < of_int z  w < z"
  by (simp add: less_le order_less_le)

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¦"
  by (auto simp add: abs_if)

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 termof_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 term0  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)"
  by transfer (clarsimp simp add: le_imp_diff_is_add)

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"
  by (simp add: nat_eq_iff)

lemma nat_1 [simp]: "nat 1 = Suc 0"
  by (simp add: nat_eq_iff)

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

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"
  by (auto simp add: nat_eq_iff2)

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¦"
  by (simp add: nat_add_distrib [symmetric] nat_le_eq_zle abs_triangle_ineq)

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"
      by (simp add: not_less)
    ultimately show ?P
      by simp
  qed
  with False show ?thesis by simp
qed

lemma all_nat: "(x. P x)  (x0. 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 "x0. P (nat x)" ..
next
  assume "x0. 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 termof_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"
  by (simp add: minus_le_iff)

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"
  by (simp add: linorder_not_less)

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¦)"
    by (simp add: sgn_mult_abs)
  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
    by (simp add: linorder_not_less add.assoc add_increasing le_imp_0_less [THEN order_less_imp_le])
next
  case (neg n)
  then show ?thesis
    by (simp del: of_nat_Suc of_nat_add of_nat_1
        add: algebra_simps of_nat_1 [where 'a=int, symmetric] of_nat_add [symmetric])
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"
    by (simp add: nonneg add_increasing)
  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)"
      by (simp add: le_imp_0_less add_increasing)
    also have " = - (1 + z + z)"
      by (simp add: neg add.assoc [symmetric])
    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"  ("")
  where " = range of_int"

lemma Ints_of_int [simp]: "of_int z  "
  by (simp add: Ints_def)

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

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

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

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

lemma Ints_add [simp]: "a    b    a + b  "
  by (force simp add: Ints_def simp flip: of_int_add intro: range_eqI)

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

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

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

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

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

lemma Ints_cases [cases set: Ints]:
  assumes "q  "
  obtains (of_int) z where "q = of_int z"
  unfolding Ints_def
proof -
  from q   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. P (of_int z))  P q"
  by (rule Ints_cases) auto

lemma Nats_subset_Ints: "  "
  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: " = {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  "
    by simp
next
  fix x :: 'a
  assume "x  "
  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  )  sum f A  "
  by (induction A rule: infinite_finite_induct) auto

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

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

lemma (in linordered_idom) Nats_altdef2: " = {n  . n  0}"
proof (intro subsetI equalityI)
  fix x :: 'a
  assume "x  {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  "
    by simp
qed (auto elim!: Nats_cases)

lemma (in idom_divide) of_int_divide_in_Ints: 
  "of_int a div of_int b  " 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 termInts prevents terma = 1/2.›

lemma Ints_double_eq_0_iff:
  fixes a :: "'a::ring_char_0"
  assumes in_Ints: "a  "
  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  "
  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  "
  using of_nat_in_Nats [of "numeral w"] by simp

lemma Ints_odd_less_0:
  fixes a :: "'a::linordered_idom"
  assumes in_Ints: "a  "
  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"
    by (simp add: a)
  finally show ?thesis .
qed


subsection termsum and termprod

context semiring_1
begin

lemma of_nat_sum [simp]:
  "of_nat (sum f A) = (xA. 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) = (xA. 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) = (xA. 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) = (xA. 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_discrete_type type_nameInt.int
  #> Lin_Arith.add_lessD @{thm zless_imp_add1_zle}
  #> Lin_Arith.add_inj_thms @{thms of_nat_le_iff [THEN iffD2] of_nat_eq_iff [THEN iffD2]}
  #> Lin_Arith.add_inj_const (const_nameof_nat, typnat  int)
  #> Lin_Arith.add_simps
      @{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]
      add_divide_distrib diff_divide_distrib
      of_int_minus of_int_diff
      of_int_of_nat_eq}
  #> Lin_Arith.add_simprocs [Int_Arith.zero_one_idom_simproc]
)

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 termnat and termint

text ‹Simplify the term termw + - 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"
  by (simp add: int_eq_iff)

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"
    by (simp add: not_le mult_le_0_iff)
  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 = ( :: int set)"
proof -
  have "¦k¦  " for k :: int
    by (cases k) simp_all
  moreover have "k  range abs" if "k  " 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))"
    by (auto simp add: int_ge_less_than_def)
  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))"
    by (auto simp add: int_ge_less_than2_def)
  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:
  "in. 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 zmult_eq_neg1_iff: "a * b = (-1 :: int)  a = 1  b = -1  a = -1  b = 1"
  using zmult_eq_1_iff[of a "-b"] by 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"
    by (simp add: zmult_eq_1_iff)
  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
  using dvd_add_times_triv_right_iff [of k n m] by (simp add: ac_simps)

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
      by (simp add: mult.assoc)
  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"
  by (simp add: nat_le_iff)

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)"
    by (simp add: dvd_add_left_iff)
  then show ?thesis
    by (simp add: ac_simps)
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"
  by (simp_all add: power_int_def)

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

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

lemma powi_numeral_reduce: "x powi numeral n = x * x powi int (pred_numeral n)"
  by (simp add: numeral_eq_Suc)

lemma powi_minus_numeral_reduce: "x powi - (numeral n) = inverse x * x powi - int(pred_numeral n)"
  by (simp add: numeral_eq_Suc power_int_def)

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_minus_divide: "power_int (x::'a) (-n) = 1 / (power_int x n)"
  by (simp add: divide_inverse power_int_minus)

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"
  by (simp add: power_int_0_left_If)

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"
  by (simp add: field_simps flip: power_add)

lemma