(* Title: HOL/Archimedean_Field.thy Author: Brian Huffman *) section ‹Archimedean Fields, Floor and Ceiling Functions› theory Archimedean_Field imports Main begin lemma cInf_abs_ge: fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" assumes "S ≠ {}" and bdd: "⋀x. x∈S ⟹ ¦x¦ ≤ a" shows "¦Inf S¦ ≤ a" proof - have "Sup (uminus ` S) = - (Inf S)" proof (rule antisym) show "- (Inf S) ≤ Sup (uminus ` S)" apply (subst minus_le_iff) apply (rule cInf_greatest [OF ‹S ≠ {}›]) apply (subst minus_le_iff) apply (rule cSup_upper) apply force using bdd apply (force simp: abs_le_iff bdd_above_def) done next have *: "⋀x. x ∈ S ⟹ Inf S ≤ x" by (meson abs_le_iff bdd bdd_below_def cInf_lower minus_le_iff) show "Sup (uminus ` S) ≤ - Inf S" using ‹S ≠ {}› by (force intro: * cSup_least) qed with cSup_abs_le [of "uminus ` S"] assms show ?thesis by fastforce qed lemma cSup_asclose: fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" assumes S: "S ≠ {}" and b: "∀x∈S. ¦x - l¦ ≤ e" shows "¦Sup S - l¦ ≤ e" proof - have *: "¦x - l¦ ≤ e ⟷ l - e ≤ x ∧ x ≤ l + e" for x l e :: 'a by arith have "bdd_above S" using b by (auto intro!: bdd_aboveI[of _ "l + e"]) with S b show ?thesis unfolding * by (auto intro!: cSup_upper2 cSup_least) qed lemma cInf_asclose: fixes S :: "'a::{linordered_idom,conditionally_complete_linorder} set" assumes S: "S ≠ {}" and b: "∀x∈S. ¦x - l¦ ≤ e" shows "¦Inf S - l¦ ≤ e" proof - have *: "¦x - l¦ ≤ e ⟷ l - e ≤ x ∧ x ≤ l + e" for x l e :: 'a by arith have "bdd_below S" using b by (auto intro!: bdd_belowI[of _ "l - e"]) with S b show ?thesis unfolding * by (auto intro!: cInf_lower2 cInf_greatest) qed subsection ‹Class of Archimedean fields› text ‹Archimedean fields have no infinite elements.› class archimedean_field = linordered_field + assumes ex_le_of_int: "∃z. x ≤ of_int z" lemma ex_less_of_int: "∃z. x < of_int z" for x :: "'a::archimedean_field" proof - from ex_le_of_int obtain z where "x ≤ of_int z" .. then have "x < of_int (z + 1)" by simp then show ?thesis .. qed lemma ex_of_int_less: "∃z. of_int z < x" for x :: "'a::archimedean_field" proof - from ex_less_of_int obtain z where "- x < of_int z" .. then have "of_int (- z) < x" by simp then show ?thesis .. qed lemma reals_Archimedean2: "∃n. x < of_nat n" for x :: "'a::archimedean_field" proof - obtain z where "x < of_int z" using ex_less_of_int .. also have "… ≤ of_int (int (nat z))" by simp also have "… = of_nat (nat z)" by (simp only: of_int_of_nat_eq) finally show ?thesis .. qed lemma real_arch_simple: "∃n. x ≤ of_nat n" for x :: "'a::archimedean_field" proof - obtain n where "x < of_nat n" using reals_Archimedean2 .. then have "x ≤ of_nat n" by simp then show ?thesis .. qed text ‹Archimedean fields have no infinitesimal elements.› lemma reals_Archimedean: fixes x :: "'a::archimedean_field" assumes "0 < x" shows "∃n. inverse (of_nat (Suc n)) < x" proof - from ‹0 < x› have "0 < inverse x" by (rule positive_imp_inverse_positive) obtain n where "inverse x < of_nat n" using reals_Archimedean2 .. then obtain m where "inverse x < of_nat (Suc m)" using ‹0 < inverse x› by (cases n) (simp_all del: of_nat_Suc) then have "inverse (of_nat (Suc m)) < inverse (inverse x)" using ‹0 < inverse x› by (rule less_imp_inverse_less) then have "inverse (of_nat (Suc m)) < x" using ‹0 < x› by (simp add: nonzero_inverse_inverse_eq) then show ?thesis .. qed lemma ex_inverse_of_nat_less: fixes x :: "'a::archimedean_field" assumes "0 < x" shows "∃n>0. inverse (of_nat n) < x" using reals_Archimedean [OF ‹0 < x›] by auto lemma ex_less_of_nat_mult: fixes x :: "'a::archimedean_field" assumes "0 < x" shows "∃n. y < of_nat n * x" proof - obtain n where "y / x < of_nat n" using reals_Archimedean2 .. with ‹0 < x› have "y < of_nat n * x" by (simp add: pos_divide_less_eq) then show ?thesis .. qed subsection ‹Existence and uniqueness of floor function› lemma exists_least_lemma: assumes "¬ P 0" and "∃n. P n" shows "∃n. ¬ P n ∧ P (Suc n)" proof - from ‹∃n. P n› have "P (Least P)" by (rule LeastI_ex) with ‹¬ P 0› obtain n where "Least P = Suc n" by (cases "Least P") auto then have "n < Least P" by simp then have "¬ P n" by (rule not_less_Least) then have "¬ P n ∧ P (Suc n)" using ‹P (Least P)› ‹Least P = Suc n› by simp then show ?thesis .. qed lemma floor_exists: fixes x :: "'a::archimedean_field" shows "∃z. of_int z ≤ x ∧ x < of_int (z + 1)" proof (cases "0 ≤ x") case True then have "¬ x < of_nat 0" by simp then have "∃n. ¬ x < of_nat n ∧ x < of_nat (Suc n)" using reals_Archimedean2 by (rule exists_least_lemma) then obtain n where "¬ x < of_nat n ∧ x < of_nat (Suc n)" .. then have "of_int (int n) ≤ x ∧ x < of_int (int n + 1)" by simp then show ?thesis .. next case False then have "¬ - x ≤ of_nat 0" by simp then have "∃n. ¬ - x ≤ of_nat n ∧ - x ≤ of_nat (Suc n)" using real_arch_simple by (rule exists_least_lemma) then obtain n where "¬ - x ≤ of_nat n ∧ - x ≤ of_nat (Suc n)" .. then have "of_int (- int n - 1) ≤ x ∧ x < of_int (- int n - 1 + 1)" by simp then show ?thesis .. qed lemma floor_exists1: "∃!z. of_int z ≤ x ∧ x < of_int (z + 1)" for x :: "'a::archimedean_field" proof (rule ex_ex1I) show "∃z. of_int z ≤ x ∧ x < of_int (z + 1)" by (rule floor_exists) next fix y z assume "of_int y ≤ x ∧ x < of_int (y + 1)" and "of_int z ≤ x ∧ x < of_int (z + 1)" with le_less_trans [of "of_int y" "x" "of_int (z + 1)"] le_less_trans [of "of_int z" "x" "of_int (y + 1)"] show "y = z" by (simp del: of_int_add) qed subsection ‹Floor function› class floor_ceiling = archimedean_field + fixes floor :: "'a ⇒ int" ("⌊_⌋") assumes floor_correct: "of_int ⌊x⌋ ≤ x ∧ x < of_int (⌊x⌋ + 1)" lemma floor_unique: "of_int z ≤ x ⟹ x < of_int z + 1 ⟹ ⌊x⌋ = z" using floor_correct [of x] floor_exists1 [of x] by auto lemma floor_eq_iff: "⌊x⌋ = a ⟷ of_int a ≤ x ∧ x < of_int a + 1" using floor_correct floor_unique by auto lemma of_int_floor_le [simp]: "of_int ⌊x⌋ ≤ x" using floor_correct .. lemma le_floor_iff: "z ≤ ⌊x⌋ ⟷ of_int z ≤ x" proof assume "z ≤ ⌊x⌋" then have "(of_int z :: 'a) ≤ of_int ⌊x⌋" by simp also have "of_int ⌊x⌋ ≤ x" by (rule of_int_floor_le) finally show "of_int z ≤ x" . next assume "of_int z ≤ x" also have "x < of_int (⌊x⌋ + 1)" using floor_correct .. finally show "z ≤ ⌊x⌋" by (simp del: of_int_add) qed lemma floor_less_iff: "⌊x⌋ < z ⟷ x < of_int z" by (simp add: not_le [symmetric] le_floor_iff) lemma less_floor_iff: "z < ⌊x⌋ ⟷ of_int z + 1 ≤ x" using le_floor_iff [of "z + 1" x] by auto lemma floor_le_iff: "⌊x⌋ ≤ z ⟷ x < of_int z + 1" by (simp add: not_less [symmetric] less_floor_iff) lemma floor_split[linarith_split]: "P ⌊t⌋ ⟷ (∀i. of_int i ≤ t ∧ t < of_int i + 1 ⟶ P i)" by (metis floor_correct floor_unique less_floor_iff not_le order_refl) lemma floor_mono: assumes "x ≤ y" shows "⌊x⌋ ≤ ⌊y⌋" proof - have "of_int ⌊x⌋ ≤ x" by (rule of_int_floor_le) also note ‹x ≤ y› finally show ?thesis by (simp add: le_floor_iff) qed lemma floor_less_cancel: "⌊x⌋ < ⌊y⌋ ⟹ x < y" by (auto simp add: not_le [symmetric] floor_mono) lemma floor_of_int [simp]: "⌊of_int z⌋ = z" by (rule floor_unique) simp_all lemma floor_of_nat [simp]: "⌊of_nat n⌋ = int n" using floor_of_int [of "of_nat n"] by simp lemma le_floor_add: "⌊x⌋ + ⌊y⌋ ≤ ⌊x + y⌋" by (simp only: le_floor_iff of_int_add add_mono of_int_floor_le) text ‹Floor with numerals.› lemma floor_zero [simp]: "⌊0⌋ = 0" using floor_of_int [of 0] by simp lemma floor_one [simp]: "⌊1⌋ = 1" using floor_of_int [of 1] by simp lemma floor_numeral [simp]: "⌊numeral v⌋ = numeral v" using floor_of_int [of "numeral v"] by simp lemma floor_neg_numeral [simp]: "⌊- numeral v⌋ = - numeral v" using floor_of_int [of "- numeral v"] by simp lemma zero_le_floor [simp]: "0 ≤ ⌊x⌋ ⟷ 0 ≤ x" by (simp add: le_floor_iff) lemma one_le_floor [simp]: "1 ≤ ⌊x⌋ ⟷ 1 ≤ x" by (simp add: le_floor_iff) lemma numeral_le_floor [simp]: "numeral v ≤ ⌊x⌋ ⟷ numeral v ≤ x" by (simp add: le_floor_iff) lemma neg_numeral_le_floor [simp]: "- numeral v ≤ ⌊x⌋ ⟷ - numeral v ≤ x" by (simp add: le_floor_iff) lemma zero_less_floor [simp]: "0 < ⌊x⌋ ⟷ 1 ≤ x" by (simp add: less_floor_iff) lemma one_less_floor [simp]: "1 < ⌊x⌋ ⟷ 2 ≤ x" by (simp add: less_floor_iff) lemma numeral_less_floor [simp]: "numeral v < ⌊x⌋ ⟷ numeral v + 1 ≤ x" by (simp add: less_floor_iff) lemma neg_numeral_less_floor [simp]: "- numeral v < ⌊x⌋ ⟷ - numeral v + 1 ≤ x" by (simp add: less_floor_iff) lemma floor_le_zero [simp]: "⌊x⌋ ≤ 0 ⟷ x < 1" by (simp add: floor_le_iff) lemma floor_le_one [simp]: "⌊x⌋ ≤ 1 ⟷ x < 2" by (simp add: floor_le_iff) lemma floor_le_numeral [simp]: "⌊x⌋ ≤ numeral v ⟷ x < numeral v + 1" by (simp add: floor_le_iff) lemma floor_le_neg_numeral [simp]: "⌊x⌋ ≤ - numeral v ⟷ x < - numeral v + 1" by (simp add: floor_le_iff) lemma floor_less_zero [simp]: "⌊x⌋ < 0 ⟷ x < 0" by (simp add: floor_less_iff) lemma floor_less_one [simp]: "⌊x⌋ < 1 ⟷ x < 1" by (simp add: floor_less_iff) lemma floor_less_numeral [simp]: "⌊x⌋ < numeral v ⟷ x < numeral v" by (simp add: floor_less_iff) lemma floor_less_neg_numeral [simp]: "⌊x⌋ < - numeral v ⟷ x < - numeral v" by (simp add: floor_less_iff) lemma le_mult_floor_Ints: assumes "0 ≤ a" "a ∈ Ints" shows "of_int (⌊a⌋ * ⌊b⌋) ≤ (of_int⌊a * b⌋ :: 'a :: linordered_idom)" by (metis Ints_cases assms floor_less_iff floor_of_int linorder_not_less mult_left_mono of_int_floor_le of_int_less_iff of_int_mult) text ‹Addition and subtraction of integers.› lemma floor_add_int: "⌊x⌋ + z = ⌊x + of_int z⌋" using floor_correct [of x] by (simp add: floor_unique[symmetric]) lemma int_add_floor: "z + ⌊x⌋ = ⌊of_int z + x⌋" using floor_correct [of x] by (simp add: floor_unique[symmetric]) lemma one_add_floor: "⌊x⌋ + 1 = ⌊x + 1⌋" using floor_add_int [of x 1] by simp lemma floor_diff_of_int [simp]: "⌊x - of_int z⌋ = ⌊x⌋ - z" using floor_add_int [of x "- z"] by (simp add: algebra_simps) lemma floor_uminus_of_int [simp]: "⌊- (of_int z)⌋ = - z" by (metis floor_diff_of_int [of 0] diff_0 floor_zero) lemma floor_diff_numeral [simp]: "⌊x - numeral v⌋ = ⌊x⌋ - numeral v" using floor_diff_of_int [of x "numeral v"] by simp lemma floor_diff_one [simp]: "⌊x - 1⌋ = ⌊x⌋ - 1" using floor_diff_of_int [of x 1] by simp lemma le_mult_floor: assumes "0 ≤ a" and "0 ≤ b" shows "⌊a⌋ * ⌊b⌋ ≤ ⌊a * b⌋" proof - have "of_int ⌊a⌋ ≤ a" and "of_int ⌊b⌋ ≤ b" by (auto intro: of_int_floor_le) then have "of_int (⌊a⌋ * ⌊b⌋) ≤ a * b" using assms by (auto intro!: mult_mono) also have "a * b < of_int (⌊a * b⌋ + 1)" using floor_correct[of "a * b"] by auto finally show ?thesis unfolding of_int_less_iff by simp qed lemma floor_divide_of_int_eq: "⌊of_int k / of_int l⌋ = k div l" for k l :: int proof (cases "l = 0") case True then show ?thesis by simp next case False have *: "⌊of_int (k mod l) / of_int l :: 'a⌋ = 0" proof (cases "l > 0") case True then show ?thesis by (auto intro: floor_unique) next case False obtain r where "r = - l" by blast then have l: "l = - r" by simp with ‹l ≠ 0› False have "r > 0" by simp with l show ?thesis using pos_mod_bound [of r] by (auto simp add: zmod_zminus2_eq_if less_le field_simps intro: floor_unique) qed have "(of_int k :: 'a) = of_int (k div l * l + k mod l)" by simp also have "… = (of_int (k div l) + of_int (k mod l) / of_int l) * of_int l" using False by (simp only: of_int_add) (simp add: field_simps) finally have "(of_int k / of_int l :: 'a) = … / of_int l" by simp then have "(of_int k / of_int l :: 'a) = of_int (k div l) + of_int (k mod l) / of_int l" using False by (simp only:) (simp add: field_simps) then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k div l) + of_int (k mod l) / of_int l :: 'a⌋" by simp then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k mod l) / of_int l + of_int (k div l) :: 'a⌋" by (simp add: ac_simps) then have "⌊of_int k / of_int l :: 'a⌋ = ⌊of_int (k mod l) / of_int l :: 'a⌋ + k div l" by (simp add: floor_add_int) with * show ?thesis by simp qed lemma floor_divide_of_nat_eq: "⌊of_nat m / of_nat n⌋ = of_nat (m div n)" for m n :: nat proof (cases "n = 0") case True then show ?thesis by simp next case False then have *: "⌊of_nat (m mod n) / of_nat n :: 'a⌋ = 0" by (auto intro: floor_unique) have "(of_nat m :: 'a) = of_nat (m div n * n + m mod n)" by simp also have "… = (of_nat (m div n) + of_nat (m mod n) / of_nat n) * of_nat n" using False by (simp only: of_nat_add) (simp add: field_simps) finally have "(of_nat m / of_nat n :: 'a) = … / of_nat n" by simp then have "(of_nat m / of_nat n :: 'a) = of_nat (m div n) + of_nat (m mod n) / of_nat n" using False by (simp only:) simp then have "⌊of_nat m / of_nat n :: 'a⌋ = ⌊of_nat (m div n) + of_nat (m mod n) / of_nat n :: 'a⌋" by simp then have "⌊of_nat m / of_nat n :: 'a⌋ = ⌊of_nat (m mod n) / of_nat n + of_nat (m div n) :: 'a⌋" by (simp add: ac_simps) moreover have "(of_nat (m div n) :: 'a) = of_int (of_nat (m div n))" by simp ultimately have "⌊of_nat m / of_nat n :: 'a⌋ = ⌊of_nat (m mod n) / of_nat n :: 'a⌋ + of_nat (m div n)" by (simp only: floor_add_int) with * show ?thesis by simp qed lemma floor_divide_lower: fixes q :: "'a::floor_ceiling" shows "q > 0 ⟹ of_int ⌊p / q⌋ * q ≤ p" using of_int_floor_le pos_le_divide_eq by blast lemma floor_divide_upper: fixes q :: "'a::floor_ceiling" shows "q > 0 ⟹ p < (of_int ⌊p / q⌋ + 1) * q" by (meson floor_eq_iff pos_divide_less_eq) subsection ‹Ceiling function› definition ceiling :: "'a::floor_ceiling ⇒ int" ("⌈_⌉") where "⌈x⌉ = - ⌊- x⌋" lemma ceiling_correct: "of_int ⌈x⌉ - 1 < x ∧ x ≤ of_int ⌈x⌉" unfolding ceiling_def using floor_correct [of "- x"] by (simp add: le_minus_iff) lemma ceiling_unique: "of_int z - 1 < x ⟹ x ≤ of_int z ⟹ ⌈x⌉ = z" unfolding ceiling_def using floor_unique [of "- z" "- x"] by simp lemma ceiling_eq_iff: "⌈x⌉ = a ⟷ of_int a - 1 < x ∧ x ≤ of_int a" using ceiling_correct ceiling_unique by auto lemma le_of_int_ceiling [simp]: "x ≤ of_int ⌈x⌉" using ceiling_correct .. lemma ceiling_le_iff: "⌈x⌉ ≤ z ⟷ x ≤ of_int z" unfolding ceiling_def using le_floor_iff [of "- z" "- x"] by auto lemma less_ceiling_iff: "z < ⌈x⌉ ⟷ of_int z < x" by (simp add: not_le [symmetric] ceiling_le_iff) lemma ceiling_less_iff: "⌈x⌉ < z ⟷ x ≤ of_int z - 1" using ceiling_le_iff [of x "z - 1"] by simp lemma le_ceiling_iff: "z ≤ ⌈x⌉ ⟷ of_int z - 1 < x" by (simp add: not_less [symmetric] ceiling_less_iff) lemma ceiling_mono: "x ≥ y ⟹ ⌈x⌉ ≥ ⌈y⌉" unfolding ceiling_def by (simp add: floor_mono) lemma ceiling_less_cancel: "⌈x⌉ < ⌈y⌉ ⟹ x < y" by (auto simp add: not_le [symmetric] ceiling_mono) lemma ceiling_of_int [simp]: "⌈of_int z⌉ = z" by (rule ceiling_unique) simp_all lemma ceiling_of_nat [simp]: "⌈of_nat n⌉ = int n" using ceiling_of_int [of "of_nat n"] by simp lemma ceiling_add_le: "⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉" by (simp only: ceiling_le_iff of_int_add add_mono le_of_int_ceiling) lemma mult_ceiling_le: assumes "0 ≤ a" and "0 ≤ b" shows "⌈a * b⌉ ≤ ⌈a⌉ * ⌈b⌉" by (metis assms ceiling_le_iff ceiling_mono le_of_int_ceiling mult_mono of_int_mult) lemma mult_ceiling_le_Ints: assumes "0 ≤ a" "a ∈ Ints" shows "(of_int ⌈a * b⌉ :: 'a :: linordered_idom) ≤ of_int(⌈a⌉ * ⌈b⌉)" by (metis Ints_cases assms ceiling_le_iff ceiling_of_int le_of_int_ceiling mult_left_mono of_int_le_iff of_int_mult) lemma finite_int_segment: fixes a :: "'a::floor_ceiling" shows "finite {x ∈ ℤ. a ≤ x ∧ x ≤ b}" proof - have "finite {ceiling a..floor b}" by simp moreover have "{x ∈ ℤ. a ≤ x ∧ x ≤ b} = of_int ` {ceiling a..floor b}" by (auto simp: le_floor_iff ceiling_le_iff elim!: Ints_cases) ultimately show ?thesis by simp qed corollary finite_abs_int_segment: fixes a :: "'a::floor_ceiling" shows "finite {k ∈ ℤ. ¦k¦ ≤ a}" using finite_int_segment [of "-a" a] by (auto simp add: abs_le_iff conj_commute minus_le_iff) subsubsection ‹Ceiling with numerals.› lemma ceiling_zero [simp]: "⌈0⌉ = 0" using ceiling_of_int [of 0] by simp lemma ceiling_one [simp]: "⌈1⌉ = 1" using ceiling_of_int [of 1] by simp lemma ceiling_numeral [simp]: "⌈numeral v⌉ = numeral v" using ceiling_of_int [of "numeral v"] by simp lemma ceiling_neg_numeral [simp]: "⌈- numeral v⌉ = - numeral v" using ceiling_of_int [of "- numeral v"] by simp lemma ceiling_le_zero [simp]: "⌈x⌉ ≤ 0 ⟷ x ≤ 0" by (simp add: ceiling_le_iff) lemma ceiling_le_one [simp]: "⌈x⌉ ≤ 1 ⟷ x ≤ 1" by (simp add: ceiling_le_iff) lemma ceiling_le_numeral [simp]: "⌈x⌉ ≤ numeral v ⟷ x ≤ numeral v" by (simp add: ceiling_le_iff) lemma ceiling_le_neg_numeral [simp]: "⌈x⌉ ≤ - numeral v ⟷ x ≤ - numeral v" by (simp add: ceiling_le_iff) lemma ceiling_less_zero [simp]: "⌈x⌉ < 0 ⟷ x ≤ -1" by (simp add: ceiling_less_iff) lemma ceiling_less_one [simp]: "⌈x⌉ < 1 ⟷ x ≤ 0" by (simp add: ceiling_less_iff) lemma ceiling_less_numeral [simp]: "⌈x⌉ < numeral v ⟷ x ≤ numeral v - 1" by (simp add: ceiling_less_iff) lemma ceiling_less_neg_numeral [simp]: "⌈x⌉ < - numeral v ⟷ x ≤ - numeral v - 1" by (simp add: ceiling_less_iff) lemma zero_le_ceiling [simp]: "0 ≤ ⌈x⌉ ⟷ -1 < x" by (simp add: le_ceiling_iff) lemma one_le_ceiling [simp]: "1 ≤ ⌈x⌉ ⟷ 0 < x" by (simp add: le_ceiling_iff) lemma numeral_le_ceiling [simp]: "numeral v ≤ ⌈x⌉ ⟷ numeral v - 1 < x" by (simp add: le_ceiling_iff) lemma neg_numeral_le_ceiling [simp]: "- numeral v ≤ ⌈x⌉ ⟷ - numeral v - 1 < x" by (simp add: le_ceiling_iff) lemma zero_less_ceiling [simp]: "0 < ⌈x⌉ ⟷ 0 < x" by (simp add: less_ceiling_iff) lemma one_less_ceiling [simp]: "1 < ⌈x⌉ ⟷ 1 < x" by (simp add: less_ceiling_iff) lemma numeral_less_ceiling [simp]: "numeral v < ⌈x⌉ ⟷ numeral v < x" by (simp add: less_ceiling_iff) lemma neg_numeral_less_ceiling [simp]: "- numeral v < ⌈x⌉ ⟷ - numeral v < x" by (simp add: less_ceiling_iff) lemma ceiling_altdef: "⌈x⌉ = (if x = of_int ⌊x⌋ then ⌊x⌋ else ⌊x⌋ + 1)" by (intro ceiling_unique; simp, linarith?) lemma floor_le_ceiling [simp]: "⌊x⌋ ≤ ⌈x⌉" by (simp add: ceiling_altdef) subsubsection ‹Addition and subtraction of integers.› lemma ceiling_add_of_int [simp]: "⌈x + of_int z⌉ = ⌈x⌉ + z" using ceiling_correct [of x] by (simp add: ceiling_def) lemma ceiling_add_numeral [simp]: "⌈x + numeral v⌉ = ⌈x⌉ + numeral v" using ceiling_add_of_int [of x "numeral v"] by simp lemma ceiling_add_one [simp]: "⌈x + 1⌉ = ⌈x⌉ + 1" using ceiling_add_of_int [of x 1] by simp lemma ceiling_diff_of_int [simp]: "⌈x - of_int z⌉ = ⌈x⌉ - z" using ceiling_add_of_int [of x "- z"] by (simp add: algebra_simps) lemma ceiling_diff_numeral [simp]: "⌈x - numeral v⌉ = ⌈x⌉ - numeral v" using ceiling_diff_of_int [of x "numeral v"] by simp lemma ceiling_diff_one [simp]: "⌈x - 1⌉ = ⌈x⌉ - 1" using ceiling_diff_of_int [of x 1] by simp lemma ceiling_split[linarith_split]: "P ⌈t⌉ ⟷ (∀i. of_int i - 1 < t ∧ t ≤ of_int i ⟶ P i)" by (auto simp add: ceiling_unique ceiling_correct) lemma ceiling_diff_floor_le_1: "⌈x⌉ - ⌊x⌋ ≤ 1" proof - have "of_int ⌈x⌉ - 1 < x" using ceiling_correct[of x] by simp also have "x < of_int ⌊x⌋ + 1" using floor_correct[of x] by simp_all finally have "of_int (⌈x⌉ - ⌊x⌋) < (of_int 2::'a)" by simp then show ?thesis unfolding of_int_less_iff by simp qed lemma nat_approx_posE: fixes e:: "'a::{archimedean_field,floor_ceiling}" assumes "0 < e" obtains n :: nat where "1 / of_nat(Suc n) < e" proof have "(1::'a) / of_nat (Suc (nat ⌈1/e⌉)) < 1 / of_int (⌈1/e⌉)" proof (rule divide_strict_left_mono) show "(of_int ⌈1 / e⌉::'a) < of_nat (Suc (nat ⌈1 / e⌉))" using assms by (simp add: field_simps) show "(0::'a) < of_nat (Suc (nat ⌈1 / e⌉)) * of_int ⌈1 / e⌉" using assms by (auto simp: zero_less_mult_iff pos_add_strict) qed auto also have "1 / of_int (⌈1/e⌉) ≤ 1 / (1/e)" by (rule divide_left_mono) (auto simp: ‹0 < e› ceiling_correct) also have "… = e" by simp finally show "1 / of_nat (Suc (nat ⌈1 / e⌉)) < e" by metis qed lemma ceiling_divide_upper: fixes q :: "'a::floor_ceiling" shows "q > 0 ⟹ p ≤ of_int (ceiling (p / q)) * q" by (meson divide_le_eq le_of_int_ceiling) lemma ceiling_divide_lower: fixes q :: "'a::floor_ceiling" shows "q > 0 ⟹ (of_int ⌈p / q⌉ - 1) * q < p" by (meson ceiling_eq_iff pos_less_divide_eq) subsection ‹Negation› lemma floor_minus: "⌊- x⌋ = - ⌈x⌉" unfolding ceiling_def by simp lemma ceiling_minus: "⌈- x⌉ = - ⌊x⌋" unfolding ceiling_def by simp subsection ‹Natural numbers› lemma of_nat_floor: "r≥0 ⟹ of_nat (nat ⌊r⌋) ≤ r" by simp lemma of_nat_ceiling: "of_nat (nat ⌈r⌉) ≥ r" by (cases "r≥0") auto subsection ‹Frac Function› definition frac :: "'a ⇒ 'a::floor_ceiling" where "frac x ≡ x - of_int ⌊x⌋" lemma frac_lt_1: "frac x < 1" by (simp add: frac_def) linarith lemma frac_eq_0_iff [simp]: "frac x = 0 ⟷ x ∈ ℤ" by (simp add: frac_def) (metis Ints_cases Ints_of_int floor_of_int ) lemma frac_ge_0 [simp]: "frac x ≥ 0" unfolding frac_def by linarith lemma frac_gt_0_iff [simp]: "frac x > 0 ⟷ x ∉ ℤ" by (metis frac_eq_0_iff frac_ge_0 le_less less_irrefl) lemma frac_of_int [simp]: "frac (of_int z) = 0" by (simp add: frac_def) lemma frac_frac [simp]: "frac (frac x) = frac x" by (simp add: frac_def) lemma floor_add: "⌊x + y⌋ = (if frac x + frac y < 1 then ⌊x⌋ + ⌊y⌋ else (⌊x⌋ + ⌊y⌋) + 1)" proof - have "x + y < 1 + (of_int ⌊x⌋ + of_int ⌊y⌋) ⟹ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋" by (metis add.commute floor_unique le_floor_add le_floor_iff of_int_add) moreover have "¬ x + y < 1 + (of_int ⌊x⌋ + of_int ⌊y⌋) ⟹ ⌊x + y⌋ = 1 + (⌊x⌋ + ⌊y⌋)" apply (simp add: floor_eq_iff) apply (auto simp add: algebra_simps) apply linarith done ultimately show ?thesis by (auto simp add: frac_def algebra_simps) qed lemma floor_add2[simp]: "x ∈ ℤ ∨ y ∈ ℤ ⟹ ⌊x + y⌋ = ⌊x⌋ + ⌊y⌋" by (metis add.commute add.left_neutral frac_lt_1 floor_add frac_eq_0_iff) lemma frac_add: "frac (x + y) = (if frac x + frac y < 1 then frac x + frac y else (frac x + frac y) - 1)" by (simp add: frac_def floor_add) lemma frac_unique_iff: "frac x = a ⟷ x - a ∈ ℤ ∧ 0 ≤ a ∧ a < 1" for x :: "'a::floor_ceiling" apply (auto simp: Ints_def frac_def algebra_simps floor_unique) apply linarith+ done lemma frac_eq: "frac x = x ⟷ 0 ≤ x ∧ x < 1" by (simp add: frac_unique_iff) lemma frac_neg: "frac (- x) = (if x ∈ ℤ then 0 else 1 - frac x)" for x :: "'a::floor_ceiling" apply (auto simp add: frac_unique_iff) apply (simp add: frac_def) apply (meson frac_lt_1 less_iff_diff_less_0 not_le not_less_iff_gr_or_eq) done lemma frac_in_Ints_iff [simp]: "frac x ∈ ℤ ⟷ x ∈ ℤ" proof safe assume "frac x ∈ ℤ" hence "of_int ⌊x⌋ + frac x ∈ ℤ" by auto also have "of_int ⌊x⌋ + frac x = x" by (simp add: frac_def) finally show "x ∈ ℤ" . qed (auto simp: frac_def) lemma frac_1_eq: "frac (x+1) = frac x" by (simp add: frac_def) subsection ‹Rounding to the nearest integer› definition round :: "'a::floor_ceiling ⇒ int" where "round x = ⌊x + 1/2⌋" lemma of_int_round_ge: "of_int (round x) ≥ x - 1/2" and of_int_round_le: "of_int (round x) ≤ x + 1/2" and of_int_round_abs_le: "¦of_int (round x) - x¦ ≤ 1/2" and of_int_round_gt: "of_int (round x) > x - 1/2" proof - from floor_correct[of "x + 1/2"] have "x + 1/2 < of_int (round x) + 1" by (simp add: round_def) from add_strict_right_mono[OF this, of "-1"] show A: "of_int (round x) > x - 1/2" by simp then show "of_int (round x) ≥ x - 1/2" by simp from floor_correct[of "x + 1/2"] show "of_int (round x) ≤ x + 1/2" by (simp add: round_def) with A show "¦of_int (round x) - x¦ ≤ 1/2" by linarith qed lemma round_of_int [simp]: "round (of_int n) = n" unfolding round_def by (subst floor_eq_iff) force lemma round_0 [simp]: "round 0 = 0" using round_of_int[of 0] by simp lemma round_1 [simp]: "round 1 = 1" using round_of_int[of 1] by simp lemma round_numeral [simp]: "round (numeral n) = numeral n" using round_of_int[of "numeral n"] by simp lemma round_neg_numeral [simp]: "round (-numeral n) = -numeral n" using round_of_int[of "-numeral n"] by simp lemma round_of_nat [simp]: "round (of_nat n) = of_nat n" using round_of_int[of "int n"] by simp lemma round_mono: "x ≤ y ⟹ round x ≤ round y" unfolding round_def by (intro floor_mono) simp lemma round_unique: "of_int y > x - 1/2 ⟹ of_int y ≤ x + 1/2 ⟹ round x = y" unfolding round_def proof (rule floor_unique) assume "x - 1 / 2 < of_int y" from add_strict_left_mono[OF this, of 1] show "x + 1 / 2 < of_int y + 1" by simp qed lemma round_unique': "¦x - of_int n¦ < 1/2 ⟹ round x = n" by (subst (asm) abs_less_iff, rule round_unique) (simp_all add: field_simps) lemma round_altdef: "round x = (if frac x ≥ 1/2 then ⌈x⌉ else ⌊x⌋)" by (cases "frac x ≥ 1/2") (rule round_unique, ((simp add: frac_def field_simps ceiling_altdef; linarith)+)[2])+ lemma floor_le_round: "⌊x⌋ ≤ round x" unfolding round_def by (intro floor_mono) simp lemma ceiling_ge_round: "⌈x⌉ ≥ round x" unfolding round_altdef by simp lemma round_diff_minimal: "¦z - of_int (round z)¦ ≤ ¦z - of_int m¦" for z :: "'a::floor_ceiling" proof (cases "of_int m ≥ z") case True then have "¦z - of_int (round z)¦ ≤ ¦of_int ⌈z⌉ - z¦" unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith also have "of_int ⌈z⌉ - z ≥ 0" by linarith with True have "¦of_int ⌈z⌉ - z¦ ≤ ¦z - of_int m¦" by (simp add: ceiling_le_iff) finally show ?thesis . next case False then have "¦z - of_int (round z)¦ ≤ ¦of_int ⌊z⌋ - z¦" unfolding round_altdef by (simp add: field_simps ceiling_altdef frac_def) linarith also have "z - of_int ⌊z⌋ ≥ 0" by linarith with False have "¦of_int ⌊z⌋ - z¦ ≤ ¦z - of_int m¦" by (simp add: le_floor_iff) finally show ?thesis . qed end