(* Title: HOL/Library/BigO.thy ID: $Id: BigO.thy,v 1.11 2007/06/23 17:33:23 nipkow Exp $ Authors: Jeremy Avigad and Kevin Donnelly *) header {* Big O notation *} theory BigO imports SetsAndFunctions begin text {* This library is designed to support asymptotic ``big O'' calculations, i.e.~reasoning with expressions of the form $f = O(g)$ and $f = g + O(h)$. An earlier version of this library is described in detail in \cite{Avigad-Donnelly}. The main changes in this version are as follows: \begin{itemize} \item We have eliminated the @{text O} operator on sets. (Most uses of this seem to be inessential.) \item We no longer use @{text "+"} as output syntax for @{text "+o"} \item Lemmas involving @{text "sumr"} have been replaced by more general lemmas involving `@{text "setsum"}. \item The library has been expanded, with e.g.~support for expressions of the form @{text "f < g + O(h)"}. \end{itemize} See \verb,Complex/ex/BigO_Complex.thy, for additional lemmas that require the \verb,HOL-Complex, logic image. Note also since the Big O library includes rules that demonstrate set inclusion, to use the automated reasoners effectively with the library one should redeclare the theorem @{text "subsetI"} as an intro rule, rather than as an @{text "intro!"} rule, for example, using \isa{\isakeyword{declare}}~@{text "subsetI [del, intro]"}. *} subsection {* Definitions *} definition bigo :: "('a => 'b::ordered_idom) => ('a => 'b) set" ("(1O'(_'))") where "O(f::('a => 'b)) = {h. EX c. ALL x. abs (h x) <= c * abs (f x)}" lemma bigo_pos_const: "(EX (c::'a::ordered_idom). ALL x. (abs (h x)) <= (c * (abs (f x)))) = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))" apply auto apply (case_tac "c = 0") apply simp apply (rule_tac x = "1" in exI) apply simp apply (rule_tac x = "abs c" in exI) apply auto apply (subgoal_tac "c * abs(f x) <= abs c * abs (f x)") apply (erule_tac x = x in allE) apply force apply (rule mult_right_mono) apply (rule abs_ge_self) apply (rule abs_ge_zero) done lemma bigo_alt_def: "O(f) = {h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}" by (auto simp add: bigo_def bigo_pos_const) lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)" apply (auto simp add: bigo_alt_def) apply (rule_tac x = "ca * c" in exI) apply (rule conjI) apply (rule mult_pos_pos) apply (assumption)+ apply (rule allI) apply (drule_tac x = "xa" in spec)+ apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))") apply (erule order_trans) apply (simp add: mult_ac) apply (rule mult_left_mono, assumption) apply (rule order_less_imp_le, assumption) done lemma bigo_refl [intro]: "f : O(f)" apply(auto simp add: bigo_def) apply(rule_tac x = 1 in exI) apply simp done lemma bigo_zero: "0 : O(g)" apply (auto simp add: bigo_def func_zero) apply (rule_tac x = 0 in exI) apply auto done lemma bigo_zero2: "O(%x.0) = {%x.0}" apply (auto simp add: bigo_def) apply (rule ext) apply auto done lemma bigo_plus_self_subset [intro]: "O(f) + O(f) <= O(f)" apply (auto simp add: bigo_alt_def set_plus) apply (rule_tac x = "c + ca" in exI) apply auto apply (simp add: ring_distribs func_plus) apply (rule order_trans) apply (rule abs_triangle_ineq) apply (rule add_mono) apply force apply force done lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)" apply (rule equalityI) apply (rule bigo_plus_self_subset) apply (rule set_zero_plus2) apply (rule bigo_zero) done lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)" apply (rule subsetI) apply (auto simp add: bigo_def bigo_pos_const func_plus set_plus) apply (subst bigo_pos_const [symmetric])+ apply (rule_tac x = "%n. if abs (g n) <= (abs (f n)) then x n else 0" in exI) apply (rule conjI) apply (rule_tac x = "c + c" in exI) apply (clarsimp) apply (auto) apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (f xa)") apply (erule_tac x = xa in allE) apply (erule order_trans) apply (simp) apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))") apply (erule order_trans) apply (simp add: ring_distribs) apply (rule mult_left_mono) apply assumption apply (simp add: order_less_le) apply (rule mult_left_mono) apply (simp add: abs_triangle_ineq) apply (simp add: order_less_le) apply (rule mult_nonneg_nonneg) apply (rule add_nonneg_nonneg) apply auto apply (rule_tac x = "%n. if (abs (f n)) < abs (g n) then x n else 0" in exI) apply (rule conjI) apply (rule_tac x = "c + c" in exI) apply auto apply (subgoal_tac "c * abs (f xa + g xa) <= (c + c) * abs (g xa)") apply (erule_tac x = xa in allE) apply (erule order_trans) apply (simp) apply (subgoal_tac "c * abs (f xa + g xa) <= c * (abs (f xa) + abs (g xa))") apply (erule order_trans) apply (simp add: ring_distribs) apply (rule mult_left_mono) apply (simp add: order_less_le) apply (simp add: order_less_le) apply (rule mult_left_mono) apply (rule abs_triangle_ineq) apply (simp add: order_less_le) apply (rule mult_nonneg_nonneg) apply (rule add_nonneg_nonneg) apply (erule order_less_imp_le)+ apply simp apply (rule ext) apply (auto simp add: if_splits linorder_not_le) done lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)" apply (subgoal_tac "A + B <= O(f) + O(f)") apply (erule order_trans) apply simp apply (auto del: subsetI simp del: bigo_plus_idemp) done lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==> O(f + g) = O(f) + O(g)" apply (rule equalityI) apply (rule bigo_plus_subset) apply (simp add: bigo_alt_def set_plus func_plus) apply clarify apply (rule_tac x = "max c ca" in exI) apply (rule conjI) apply (subgoal_tac "c <= max c ca") apply (erule order_less_le_trans) apply assumption apply (rule le_maxI1) apply clarify apply (drule_tac x = "xa" in spec)+ apply (subgoal_tac "0 <= f xa + g xa") apply (simp add: ring_distribs) apply (subgoal_tac "abs(a xa + b xa) <= abs(a xa) + abs(b xa)") apply (subgoal_tac "abs(a xa) + abs(b xa) <= max c ca * f xa + max c ca * g xa") apply (force) apply (rule add_mono) apply (subgoal_tac "c * f xa <= max c ca * f xa") apply (force) apply (rule mult_right_mono) apply (rule le_maxI1) apply assumption apply (subgoal_tac "ca * g xa <= max c ca * g xa") apply (force) apply (rule mult_right_mono) apply (rule le_maxI2) apply assumption apply (rule abs_triangle_ineq) apply (rule add_nonneg_nonneg) apply assumption+ done lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" apply (auto simp add: bigo_def) apply (rule_tac x = "abs c" in exI) apply auto apply (drule_tac x = x in spec)+ apply (simp add: abs_mult [symmetric]) done lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> f : O(g)" apply (erule bigo_bounded_alt [of f 1 g]) apply simp done lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==> f : lb +o O(g)" apply (rule set_minus_imp_plus) apply (rule bigo_bounded) apply (auto simp add: diff_minus func_minus func_plus) apply (drule_tac x = x in spec)+ apply force apply (drule_tac x = x in spec)+ apply force done lemma bigo_abs: "(%x. abs(f x)) =o O(f)" apply (unfold bigo_def) apply auto apply (rule_tac x = 1 in exI) apply auto done lemma bigo_abs2: "f =o O(%x. abs(f x))" apply (unfold bigo_def) apply auto apply (rule_tac x = 1 in exI) apply auto done lemma bigo_abs3: "O(f) = O(%x. abs(f x))" apply (rule equalityI) apply (rule bigo_elt_subset) apply (rule bigo_abs2) apply (rule bigo_elt_subset) apply (rule bigo_abs) done lemma bigo_abs4: "f =o g +o O(h) ==> (%x. abs (f x)) =o (%x. abs (g x)) +o O(h)" apply (drule set_plus_imp_minus) apply (rule set_minus_imp_plus) apply (subst func_diff) proof - assume a: "f - g : O(h)" have "(%x. abs (f x) - abs (g x)) =o O(%x. abs(abs (f x) - abs (g x)))" by (rule bigo_abs2) also have "... <= O(%x. abs (f x - g x))" apply (rule bigo_elt_subset) apply (rule bigo_bounded) apply force apply (rule allI) apply (rule abs_triangle_ineq3) done also have "... <= O(f - g)" apply (rule bigo_elt_subset) apply (subst func_diff) apply (rule bigo_abs) done also from a have "... <= O(h)" by (rule bigo_elt_subset) finally show "(%x. abs (f x) - abs (g x)) : O(h)". qed lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)" by (unfold bigo_def, auto) lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)" proof - assume "f : g +o O(h)" also have "... <= O(g) + O(h)" by (auto del: subsetI) also have "... = O(%x. abs(g x)) + O(%x. abs(h x))" apply (subst bigo_abs3 [symmetric])+ apply (rule refl) done also have "... = O((%x. abs(g x)) + (%x. abs(h x)))" by (rule bigo_plus_eq [symmetric], auto) finally have "f : ...". then have "O(f) <= ..." by (elim bigo_elt_subset) also have "... = O(%x. abs(g x)) + O(%x. abs(h x))" by (rule bigo_plus_eq, auto) finally show ?thesis by (simp add: bigo_abs3 [symmetric]) qed lemma bigo_mult [intro]: "O(f)*O(g) <= O(f * g)" apply (rule subsetI) apply (subst bigo_def) apply (auto simp add: bigo_alt_def set_times func_times) apply (rule_tac x = "c * ca" in exI) apply(rule allI) apply(erule_tac x = x in allE)+ apply(subgoal_tac "c * ca * abs(f x * g x) = (c * abs(f x)) * (ca * abs(g x))") apply(erule ssubst) apply (subst abs_mult) apply (rule mult_mono) apply assumption+ apply (rule mult_nonneg_nonneg) apply auto apply (simp add: mult_ac abs_mult) done lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)" apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult) apply (rule_tac x = c in exI) apply auto apply (drule_tac x = x in spec) apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))") apply (force simp add: mult_ac) apply (rule mult_left_mono, assumption) apply (rule abs_ge_zero) done lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)" apply (rule subsetD) apply (rule bigo_mult) apply (erule set_times_intro, assumption) done lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)" apply (drule set_plus_imp_minus) apply (rule set_minus_imp_plus) apply (drule bigo_mult3 [where g = g and j = g]) apply (auto simp add: ring_simps) done lemma bigo_mult5: "ALL x. f x ~= 0 ==> O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)" proof - assume "ALL x. f x ~= 0" show "O(f * g) <= f *o O(g)" proof fix h assume "h : O(f * g)" then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o O(f * g)" by auto also have "... <= O((%x. 1 / f x) * (f * g))" by (rule bigo_mult2) also have "(%x. 1 / f x) * (f * g) = g" apply (simp add: func_times) apply (rule ext) apply (simp add: prems nonzero_divide_eq_eq mult_ac) done finally have "(%x. (1::'b) / f x) * h : O(g)". then have "f * ((%x. (1::'b) / f x) * h) : f *o O(g)" by auto also have "f * ((%x. (1::'b) / f x) * h) = h" apply (simp add: func_times) apply (rule ext) apply (simp add: prems nonzero_divide_eq_eq mult_ac) done finally show "h : f *o O(g)". qed qed lemma bigo_mult6: "ALL x. f x ~= 0 ==> O(f * g) = (f::'a => ('b::ordered_field)) *o O(g)" apply (rule equalityI) apply (erule bigo_mult5) apply (rule bigo_mult2) done lemma bigo_mult7: "ALL x. f x ~= 0 ==> O(f * g) <= O(f::'a => ('b::ordered_field)) * O(g)" apply (subst bigo_mult6) apply assumption apply (rule set_times_mono3) apply (rule bigo_refl) done lemma bigo_mult8: "ALL x. f x ~= 0 ==> O(f * g) = O(f::'a => ('b::ordered_field)) * O(g)" apply (rule equalityI) apply (erule bigo_mult7) apply (rule bigo_mult) done lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)" by (auto simp add: bigo_def func_minus) lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)" apply (rule set_minus_imp_plus) apply (drule set_plus_imp_minus) apply (drule bigo_minus) apply (simp add: diff_minus) done lemma bigo_minus3: "O(-f) = O(f)" by (auto simp add: bigo_def func_minus abs_minus_cancel) lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)" proof - assume a: "f : O(g)" show "f +o O(g) <= O(g)" proof - have "f : O(f)" by auto then have "f +o O(g) <= O(f) + O(g)" by (auto del: subsetI) also have "... <= O(g) + O(g)" proof - from a have "O(f) <= O(g)" by (auto del: subsetI) thus ?thesis by (auto del: subsetI) qed also have "... <= O(g)" by (simp add: bigo_plus_idemp) finally show ?thesis . qed qed lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)" proof - assume a: "f : O(g)" show "O(g) <= f +o O(g)" proof - from a have "-f : O(g)" by auto then have "-f +o O(g) <= O(g)" by (elim bigo_plus_absorb_lemma1) then have "f +o (-f +o O(g)) <= f +o O(g)" by auto also have "f +o (-f +o O(g)) = O(g)" by (simp add: set_plus_rearranges) finally show ?thesis . qed qed lemma bigo_plus_absorb [simp]: "f : O(g) ==> f +o O(g) = O(g)" apply (rule equalityI) apply (erule bigo_plus_absorb_lemma1) apply (erule bigo_plus_absorb_lemma2) done lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)" apply (subgoal_tac "f +o A <= f +o O(g)") apply force+ done lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)" apply (subst set_minus_plus [symmetric]) apply (subgoal_tac "g - f = - (f - g)") apply (erule ssubst) apply (rule bigo_minus) apply (subst set_minus_plus) apply assumption apply (simp add: diff_minus add_ac) done lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" apply (rule iffI) apply (erule bigo_add_commute_imp)+ done lemma bigo_const1: "(%x. c) : O(%x. 1)" by (auto simp add: bigo_def mult_ac) lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)" apply (rule bigo_elt_subset) apply (rule bigo_const1) done lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)" apply (simp add: bigo_def) apply (rule_tac x = "abs(inverse c)" in exI) apply (simp add: abs_mult [symmetric]) done lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)" by (rule bigo_elt_subset, rule bigo_const3, assumption) lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==> O(%x. c) = O(%x. 1)" by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption) lemma bigo_const_mult1: "(%x. c * f x) : O(f)" apply (simp add: bigo_def) apply (rule_tac x = "abs(c)" in exI) apply (auto simp add: abs_mult [symmetric]) done lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)" by (rule bigo_elt_subset, rule bigo_const_mult1) lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)" apply (simp add: bigo_def) apply (rule_tac x = "abs(inverse c)" in exI) apply (simp add: abs_mult [symmetric] mult_assoc [symmetric]) done lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==> O(f) <= O(%x. c * f x)" by (rule bigo_elt_subset, rule bigo_const_mult3, assumption) lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==> O(%x. c * f x) = O(f)" by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4) lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==> (%x. c) *o O(f) = O(f)" apply (auto del: subsetI) apply (rule order_trans) apply (rule bigo_mult2) apply (simp add: func_times) apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times) apply (rule_tac x = "%y. inverse c * x y" in exI) apply (simp add: mult_assoc [symmetric] abs_mult) apply (rule_tac x = "abs (inverse c) * ca" in exI) apply (rule allI) apply (subst mult_assoc) apply (rule mult_left_mono) apply (erule spec) apply force done lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)" apply (auto intro!: subsetI simp add: bigo_def elt_set_times_def func_times) apply (rule_tac x = "ca * (abs c)" in exI) apply (rule allI) apply (subgoal_tac "ca * abs(c) * abs(f x) = abs(c) * (ca * abs(f x))") apply (erule ssubst) apply (subst abs_mult) apply (rule mult_left_mono) apply (erule spec) apply simp apply(simp add: mult_ac) done lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)" proof - assume "f =o O(g)" then have "(%x. c) * f =o (%x. c) *o O(g)" by auto also have "(%x. c) * f = (%x. c * f x)" by (simp add: func_times) also have "(%x. c) *o O(g) <= O(g)" by (auto del: subsetI) finally show ?thesis . qed lemma bigo_compose1: "f =o O(g) ==> (%x. f(k x)) =o O(%x. g(k x))" by (unfold bigo_def, auto) lemma bigo_compose2: "f =o g +o O(h) ==> (%x. f(k x)) =o (%x. g(k x)) +o O(%x. h(k x))" apply (simp only: set_minus_plus [symmetric] diff_minus func_minus func_plus) apply (erule bigo_compose1) done subsection {* Setsum *} lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==> EX c. ALL x. ALL y : A x. abs(f x y) <= c * (h x y) ==> (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)" apply (auto simp add: bigo_def) apply (rule_tac x = "abs c" in exI) apply (subst abs_of_nonneg) back back apply (rule setsum_nonneg) apply force apply (subst setsum_right_distrib) apply (rule allI) apply (rule order_trans) apply (rule setsum_abs) apply (rule setsum_mono) apply (rule order_trans) apply (drule spec)+ apply (drule bspec)+ apply assumption+ apply (drule bspec) apply assumption+ apply (rule mult_right_mono) apply (rule abs_ge_self) apply force done lemma bigo_setsum1: "ALL x y. 0 <= h x y ==> EX c. ALL x y. abs(f x y) <= c * (h x y) ==> (%x. SUM y : A x. f x y) =o O(%x. SUM y : A x. h x y)" apply (rule bigo_setsum_main) apply force apply clarsimp apply (rule_tac x = c in exI) apply force done lemma bigo_setsum2: "ALL y. 0 <= h y ==> EX c. ALL y. abs(f y) <= c * (h y) ==> (%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)" by (rule bigo_setsum1, auto) lemma bigo_setsum3: "f =o O(h) ==> (%x. SUM y : A x. (l x y) * f(k x y)) =o O(%x. SUM y : A x. abs(l x y * h(k x y)))" apply (rule bigo_setsum1) apply (rule allI)+ apply (rule abs_ge_zero) apply (unfold bigo_def) apply auto apply (rule_tac x = c in exI) apply (rule allI)+ apply (subst abs_mult)+ apply (subst mult_left_commute) apply (rule mult_left_mono) apply (erule spec) apply (rule abs_ge_zero) done lemma bigo_setsum4: "f =o g +o O(h) ==> (%x. SUM y : A x. l x y * f(k x y)) =o (%x. SUM y : A x. l x y * g(k x y)) +o O(%x. SUM y : A x. abs(l x y * h(k x y)))" apply (rule set_minus_imp_plus) apply (subst func_diff) apply (subst setsum_subtractf [symmetric]) apply (subst right_diff_distrib [symmetric]) apply (rule bigo_setsum3) apply (subst func_diff [symmetric]) apply (erule set_plus_imp_minus) done lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==> ALL x. 0 <= h x ==> (%x. SUM y : A x. (l x y) * f(k x y)) =o O(%x. SUM y : A x. (l x y) * h(k x y))" apply (subgoal_tac "(%x. SUM y : A x. (l x y) * h(k x y)) = (%x. SUM y : A x. abs((l x y) * h(k x y)))") apply (erule ssubst) apply (erule bigo_setsum3) apply (rule ext) apply (rule setsum_cong2) apply (subst abs_of_nonneg) apply (rule mult_nonneg_nonneg) apply auto done lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==> ALL x. 0 <= h x ==> (%x. SUM y : A x. (l x y) * f(k x y)) =o (%x. SUM y : A x. (l x y) * g(k x y)) +o O(%x. SUM y : A x. (l x y) * h(k x y))" apply (rule set_minus_imp_plus) apply (subst func_diff) apply (subst setsum_subtractf [symmetric]) apply (subst right_diff_distrib [symmetric]) apply (rule bigo_setsum5) apply (subst func_diff [symmetric]) apply (drule set_plus_imp_minus) apply auto done subsection {* Misc useful stuff *} lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)" apply (subst bigo_plus_idemp [symmetric]) apply (rule set_plus_mono2) apply assumption+ done lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)" apply (subst bigo_plus_idemp [symmetric]) apply (rule set_plus_intro) apply assumption+ done lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==> (%x. c) * f =o O(h) ==> f =o O(h)" apply (rule subsetD) apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)") apply assumption apply (rule bigo_const_mult6) apply (subgoal_tac "f = (%x. 1 / c) * ((%x. c) * f)") apply (erule ssubst) apply (erule set_times_intro2) apply (simp add: func_times) done lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==> f =o O(h)" apply (simp add: bigo_alt_def) apply auto apply (rule_tac x = c in exI) apply auto apply (case_tac "x = 0") apply simp apply (rule mult_nonneg_nonneg) apply force apply force apply (subgoal_tac "x = Suc (x - 1)") apply (erule ssubst) back apply (erule spec) apply simp done lemma bigo_fix2: "(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==> f 0 = g 0 ==> f =o g +o O(h)" apply (rule set_minus_imp_plus) apply (rule bigo_fix) apply (subst func_diff) apply (subst func_diff [symmetric]) apply (rule set_plus_imp_minus) apply simp apply (simp add: func_diff) done subsection {* Less than or equal to *} definition lesso :: "('a => 'b::ordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where "f <o g = (%x. max (f x - g x) 0)" lemma bigo_lesseq1: "f =o O(h) ==> ALL x. abs (g x) <= abs (f x) ==> g =o O(h)" apply (unfold bigo_def) apply clarsimp apply (rule_tac x = c in exI) apply (rule allI) apply (rule order_trans) apply (erule spec)+ done lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==> g =o O(h)" apply (erule bigo_lesseq1) apply (rule allI) apply (drule_tac x = x in spec) apply (rule order_trans) apply assumption apply (rule abs_ge_self) done lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==> g =o O(h)" apply (erule bigo_lesseq2) apply (rule allI) apply (subst abs_of_nonneg) apply (erule spec)+ done lemma bigo_lesseq4: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==> g =o O(h)" apply (erule bigo_lesseq1) apply (rule allI) apply (subst abs_of_nonneg) apply (erule spec)+ done lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)" apply (unfold lesso_def) apply (subgoal_tac "(%x. max (f x - g x) 0) = 0") apply (erule ssubst) apply (rule bigo_zero) apply (unfold func_zero) apply (rule ext) apply (simp split: split_max) done lemma bigo_lesso2: "f =o g +o O(h) ==> ALL x. 0 <= k x ==> ALL x. k x <= f x ==> k <o g =o O(h)" apply (unfold lesso_def) apply (rule bigo_lesseq4) apply (erule set_plus_imp_minus) apply (rule allI) apply (rule le_maxI2) apply (rule allI) apply (subst func_diff) apply (case_tac "0 <= k x - g x") apply simp apply (subst abs_of_nonneg) apply (drule_tac x = x in spec) back apply (simp add: compare_rls) apply (subst diff_minus)+ apply (rule add_right_mono) apply (erule spec) apply (rule order_trans) prefer 2 apply (rule abs_ge_zero) apply (simp add: compare_rls) done lemma bigo_lesso3: "f =o g +o O(h) ==> ALL x. 0 <= k x ==> ALL x. g x <= k x ==> f <o k =o O(h)" apply (unfold lesso_def) apply (rule bigo_lesseq4) apply (erule set_plus_imp_minus) apply (rule allI) apply (rule le_maxI2) apply (rule allI) apply (subst func_diff) apply (case_tac "0 <= f x - k x") apply simp apply (subst abs_of_nonneg) apply (drule_tac x = x in spec) back apply (simp add: compare_rls) apply (subst diff_minus)+ apply (rule add_left_mono) apply (rule le_imp_neg_le) apply (erule spec) apply (rule order_trans) prefer 2 apply (rule abs_ge_zero) apply (simp add: compare_rls) done lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==> g =o h +o O(k) ==> f <o h =o O(k)" apply (unfold lesso_def) apply (drule set_plus_imp_minus) apply (drule bigo_abs5) back apply (simp add: func_diff) apply (drule bigo_useful_add) apply assumption apply (erule bigo_lesseq2) back apply (rule allI) apply (auto simp add: func_plus func_diff compare_rls split: split_max abs_split) done lemma bigo_lesso5: "f <o g =o O(h) ==> EX C. ALL x. f x <= g x + C * abs(h x)" apply (simp only: lesso_def bigo_alt_def) apply clarsimp apply (rule_tac x = c in exI) apply (rule allI) apply (drule_tac x = x in spec) apply (subgoal_tac "abs(max (f x - g x) 0) = max (f x - g x) 0") apply (clarsimp simp add: compare_rls add_ac) apply (rule abs_of_nonneg) apply (rule le_maxI2) done lemma lesso_add: "f <o g =o O(h) ==> k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)" apply (unfold lesso_def) apply (rule bigo_lesseq3) apply (erule bigo_useful_add) apply assumption apply (force split: split_max) apply (auto split: split_max simp add: func_plus) done end
lemma bigo_pos_const:
(∃c. ∀x. ¦h x¦ ≤ c * ¦f x¦) = (∃c>0::'a. ∀x. ¦h x¦ ≤ c * ¦f x¦)
lemma bigo_alt_def:
O(f) = {h. ∃c>0::'b. ∀x. ¦h x¦ ≤ c * ¦f x¦}
lemma bigo_elt_subset:
f ∈ O(g) ==> O(f) ⊆ O(g)
lemma bigo_refl:
f ∈ O(f)
lemma bigo_zero:
0 ∈ O(g)
lemma bigo_zero2:
O(λx. 0::'b) = {λx. 0::'b}
lemma bigo_plus_self_subset:
O(f) + O(f) ⊆ O(f)
lemma bigo_plus_idemp:
O(f) + O(f) = O(f)
lemma bigo_plus_subset:
O(f + g) ⊆ O(f) + O(g)
lemma bigo_plus_subset2:
[| A ⊆ O(f); B ⊆ O(f) |] ==> A + B ⊆ O(f)
lemma bigo_plus_eq:
[| ∀x. (0::'b) ≤ f x; ∀x. (0::'b) ≤ g x |] ==> O(f + g) = O(f) + O(g)
lemma bigo_bounded_alt:
[| ∀x. (0::'b) ≤ f x; ∀x. f x ≤ c * g x |] ==> f ∈ O(g)
lemma bigo_bounded:
[| ∀x. (0::'b) ≤ f x; ∀x. f x ≤ g x |] ==> f ∈ O(g)
lemma bigo_bounded2:
[| ∀x. lb x ≤ f x; ∀x. f x ≤ lb x + g x |] ==> f ∈ lb +o O(g)
lemma bigo_abs:
(λx. ¦f x¦) ∈ O(f)
lemma bigo_abs2:
f ∈ O(λx. ¦f x¦)
lemma bigo_abs3:
O(f) = O(λx. ¦f x¦)
lemma bigo_abs4:
f ∈ g +o O(h) ==> (λx. ¦f x¦) ∈ (λx. ¦g x¦) +o O(h)
lemma bigo_abs5:
f ∈ O(g) ==> (λx. ¦f x¦) ∈ O(g)
lemma bigo_elt_subset2:
f ∈ g +o O(h) ==> O(f) ⊆ O(g) + O(h)
lemma bigo_mult:
O(f) * O(g) ⊆ O(f * g)
lemma bigo_mult2:
f *o O(g) ⊆ O(f * g)
lemma bigo_mult3:
[| f ∈ O(h); g ∈ O(j) |] ==> f * g ∈ O(h * j)
lemma bigo_mult4:
f ∈ k +o O(h) ==> g * f ∈ g * k +o O(g * h)
lemma bigo_mult5:
∀x. f x ≠ (0::'b) ==> O(f * g) ⊆ f *o O(g)
lemma bigo_mult6:
∀x. f x ≠ (0::'b) ==> O(f * g) = f *o O(g)
lemma bigo_mult7:
∀x. f x ≠ (0::'b) ==> O(f * g) ⊆ O(f) * O(g)
lemma bigo_mult8:
∀x. f x ≠ (0::'b) ==> O(f * g) = O(f) * O(g)
lemma bigo_minus:
f ∈ O(g) ==> - f ∈ O(g)
lemma bigo_minus2:
f ∈ g +o O(h) ==> - f ∈ - g +o O(h)
lemma bigo_minus3:
O(- f) = O(f)
lemma bigo_plus_absorb_lemma1:
f ∈ O(g) ==> f +o O(g) ⊆ O(g)
lemma bigo_plus_absorb_lemma2:
f ∈ O(g) ==> O(g) ⊆ f +o O(g)
lemma bigo_plus_absorb:
f ∈ O(g) ==> f +o O(g) = O(g)
lemma bigo_plus_absorb2:
[| f ∈ O(g); A ⊆ O(g) |] ==> f +o A ⊆ O(g)
lemma bigo_add_commute_imp:
f ∈ g +o O(h) ==> g ∈ f +o O(h)
lemma bigo_add_commute:
(f ∈ g +o O(h)) = (g ∈ f +o O(h))
lemma bigo_const1:
(λx. c) ∈ O(λx. 1::'b)
lemma bigo_const2:
O(λx. c) ⊆ O(λx. 1::'b)
lemma bigo_const3:
c ≠ (0::'a) ==> (λx. 1::'a) ∈ O(λx. c)
lemma bigo_const4:
c ≠ (0::'a) ==> O(λx. 1::'a) ⊆ O(λx. c)
lemma bigo_const:
c ≠ (0::'a) ==> O(λx. c) = O(λx. 1::'a)
lemma bigo_const_mult1:
(λx. c * f x) ∈ O(f)
lemma bigo_const_mult2:
O(λx. c * f x) ⊆ O(f)
lemma bigo_const_mult3:
c ≠ (0::'a) ==> f ∈ O(λx. c * f x)
lemma bigo_const_mult4:
c ≠ (0::'a) ==> O(f) ⊆ O(λx. c * f x)
lemma bigo_const_mult:
c ≠ (0::'a) ==> O(λx. c * f x) = O(f)
lemma bigo_const_mult5:
c ≠ (0::'a) ==> (λx. c) *o O(f) = O(f)
lemma bigo_const_mult6:
(λx. c) *o O(f) ⊆ O(f)
lemma bigo_const_mult7:
f ∈ O(g) ==> (λx. c * f x) ∈ O(g)
lemma bigo_compose1:
f ∈ O(g) ==> (λx. f (k x)) ∈ O(λx. g (k x))
lemma bigo_compose2:
f ∈ g +o O(h) ==> (λx. f (k x)) ∈ (λx. g (k x)) +o O(λx. h (k x))
lemma bigo_setsum_main:
[| ∀x. ∀y∈A x. (0::'c) ≤ h x y; ∃c. ∀x. ∀y∈A x. ¦f x y¦ ≤ c * h x y |]
==> (λx. setsum (f x) (A x)) ∈ O(λx. setsum (h x) (A x))
lemma bigo_setsum1:
[| ∀x y. (0::'c) ≤ h x y; ∃c. ∀x y. ¦f x y¦ ≤ c * h x y |]
==> (λx. setsum (f x) (A x)) ∈ O(λx. setsum (h x) (A x))
lemma bigo_setsum2:
[| ∀y. (0::'b) ≤ h y; ∃c. ∀y. ¦f y¦ ≤ c * h y |]
==> (λx. setsum f (A x)) ∈ O(λx. setsum h (A x))
lemma bigo_setsum3:
f ∈ O(h)
==> (λx. ∑y∈A x. l x y * f (k x y)) ∈ O(λx. ∑y∈A x. ¦l x y * h (k x y)¦)
lemma bigo_setsum4:
f ∈ g +o O(h)
==> (λx. ∑y∈A x. l x y * f (k x y))
∈ (λx. ∑y∈A x. l x y * g (k x y)) +o O(λx. ∑y∈A x. ¦l x y * h (k x y)¦)
lemma bigo_setsum5:
[| f ∈ O(h); ∀x y. (0::'b) ≤ l x y; ∀x. (0::'b) ≤ h x |]
==> (λx. ∑y∈A x. l x y * f (k x y)) ∈ O(λx. ∑y∈A x. l x y * h (k x y))
lemma bigo_setsum6:
[| f ∈ g +o O(h); ∀x y. (0::'b) ≤ l x y; ∀x. (0::'b) ≤ h x |]
==> (λx. ∑y∈A x. l x y * f (k x y))
∈ (λx. ∑y∈A x. l x y * g (k x y)) +o O(λx. ∑y∈A x. l x y * h (k x y))
lemma bigo_useful_intro:
[| A ⊆ O(f); B ⊆ O(f) |] ==> A + B ⊆ O(f)
lemma bigo_useful_add:
[| f ∈ O(h); g ∈ O(h) |] ==> f + g ∈ O(h)
lemma bigo_useful_const_mult:
[| c ≠ (0::'a); (λx. c) * f ∈ O(h) |] ==> f ∈ O(h)
lemma bigo_fix:
[| (λx. f (x + 1)) ∈ O(λx. h (x + 1)); f 0 = (0::'a) |] ==> f ∈ O(h)
lemma bigo_fix2:
[| (λx. f (x + 1)) ∈ (λx. g (x + 1)) +o O(λx. h (x + 1)); f 0 = g 0 |]
==> f ∈ g +o O(h)
lemma bigo_lesseq1:
[| f ∈ O(h); ∀x. ¦g x¦ ≤ ¦f x¦ |] ==> g ∈ O(h)
lemma bigo_lesseq2:
[| f ∈ O(h); ∀x. ¦g x¦ ≤ f x |] ==> g ∈ O(h)
lemma bigo_lesseq3:
[| f ∈ O(h); ∀x. (0::'b) ≤ g x; ∀x. g x ≤ f x |] ==> g ∈ O(h)
lemma bigo_lesseq4:
[| f ∈ O(h); ∀x. (0::'b) ≤ g x; ∀x. g x ≤ ¦f x¦ |] ==> g ∈ O(h)
lemma bigo_lesso1:
∀x. f x ≤ g x ==> f <o g ∈ O(h)
lemma bigo_lesso2:
[| f ∈ g +o O(h); ∀x. (0::'b) ≤ k x; ∀x. k x ≤ f x |] ==> k <o g ∈ O(h)
lemma bigo_lesso3:
[| f ∈ g +o O(h); ∀x. (0::'b) ≤ k x; ∀x. g x ≤ k x |] ==> f <o k ∈ O(h)
lemma bigo_lesso4:
[| f <o g ∈ O(k); g ∈ h +o O(k) |] ==> f <o h ∈ O(k)
lemma bigo_lesso5:
f <o g ∈ O(h) ==> ∃C. ∀x. f x ≤ g x + C * ¦h x¦
lemma lesso_add:
[| f <o g ∈ O(h); k <o l ∈ O(h) |] ==> (f + k) <o (g + l) ∈ O(h)