(* 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 \<^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)" 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) ⟷ (∀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" 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" ("ℤ") where "ℤ = range of_int" lemma Ints_of_int [simp]: "of_int z ∈ ℤ" by (simp add: Ints_def) 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" by (force simp add: Ints_def simp flip: of_int_add intro: range_eqI) 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" by (simp add: a) 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_discrete_type \<^type_name>‹Int.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_name>‹of_nat›, \<^typ>‹nat ⇒ 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 \<^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" 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 = (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))" 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: "∃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 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