# HG changeset patch # User wenzelm # Date 1393675546 -3600 # Node ID 44055f07cbd8831d4b4d9704209834dacaeac2c5 # Parent 61869776ce1fc5a29c1bce5bea7459e9c6d119b9 more symbols, less parentheses; diff -r 61869776ce1f -r 44055f07cbd8 src/HOL/Library/BigO.thy --- a/src/HOL/Library/BigO.thy Sat Mar 01 12:07:26 2014 +0100 +++ b/src/HOL/Library/BigO.thy Sat Mar 01 13:05:46 2014 +0100 @@ -19,7 +19,7 @@ \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 \item We no longer use @{text "+"} as output syntax for @{text "+o"} -\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas +\item Lemmas involving @{text "sumr"} have been replaced by more general lemmas involving `@{text "setsum"}.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)"}. @@ -34,14 +34,12 @@ subsection {* Definitions *} -definition - bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where - "O(f::('a => 'b)) = - {h. EX c. ALL x. abs (h x) <= c * abs (f x)}" +definition bigo :: "('a \ 'b::linordered_idom) \ ('a \ 'b) set" ("(1O'(_'))") + where "O(f:: 'a \ 'b) = {h. \c. \x. abs (h x) \ c * abs (f x)}" -lemma bigo_pos_const: "(EX (c::'a::linordered_idom). - ALL x. (abs (h x)) <= (c * (abs (f x)))) - = (EX c. 0 < c & (ALL x. (abs(h x)) <= (c * (abs (f x)))))" +lemma bigo_pos_const: + "(\c::'a::linordered_idom. \x. abs (h x) \ c * abs (f x)) \ + (\c. 0 < c \ (\x. abs (h x) \ c * abs (f x)))" apply auto apply (case_tac "c = 0") apply simp @@ -49,7 +47,7 @@ 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 (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) @@ -57,42 +55,40 @@ 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)))}" +lemma bigo_alt_def: "O(f) = {h. \c. 0 < c \ (\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)" +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 assumption+ apply (rule allI) apply (drule_tac x = "xa" in spec)+ - apply (subgoal_tac "ca * abs(f xa) <= ca * (c * abs(g xa))") + 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)" +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)" +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}" - by (auto simp add: bigo_def) +lemma bigo_zero2: "O(\x. 0) = {\x. 0}" + by (auto simp add: bigo_def) -lemma bigo_plus_self_subset [intro]: - "O(f) + O(f) <= O(f)" +lemma bigo_plus_self_subset [intro]: "O(f) + O(f) \ O(f)" apply (auto simp add: bigo_alt_def set_plus_def) apply (rule_tac x = "c + ca" in exI) apply auto @@ -102,30 +98,29 @@ apply (rule add_mono) apply force apply force -done + 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 set_zero_plus2) apply (rule bigo_zero) done -lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)" +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_def) 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_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 (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 (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) @@ -133,16 +128,15 @@ apply (simp add: order_less_le) apply (rule mult_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_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 (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 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) @@ -153,41 +147,39 @@ apply simp 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)") +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)" +lemma bigo_plus_eq: "\x. 0 \ f x \ \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_def func_plus) apply clarify apply (rule_tac x = "max c ca" in exI) apply (rule conjI) - apply (subgoal_tac "c <= max c ca") + apply (subgoal_tac "c \ max c ca") apply (erule order_less_le_trans) apply assumption apply (rule max.cobounded1) apply clarify apply (drule_tac x = "xa" in spec)+ - apply (subgoal_tac "0 <= f xa + g xa") + 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 (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 (subgoal_tac "c * f xa \ max c ca * f xa") + apply force apply (rule mult_right_mono) apply (rule max.cobounded1) apply assumption - apply (subgoal_tac "ca * g xa <= max c ca * g xa") - apply (force) + apply (subgoal_tac "ca * g xa \ max c ca * g xa") + apply force apply (rule mult_right_mono) apply (rule max.cobounded2) apply assumption @@ -196,8 +188,7 @@ apply assumption+ done -lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> - f : O(g)" +lemma bigo_bounded_alt: "\x. 0 \ f x \ \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 @@ -205,14 +196,12 @@ apply (simp add: abs_mult [symmetric]) done -lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==> - f : O(g)" +lemma bigo_bounded: "\x. 0 \ f x \ \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)" +lemma bigo_bounded2: "\x. lb x \ f x \ \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: fun_Compl_def func_plus) @@ -222,21 +211,21 @@ apply force done -lemma bigo_abs: "(%x. abs(f x)) =o O(f)" +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))" +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))" +lemma bigo_abs3: "O(f) = O(\x. abs (f x))" apply (rule equalityI) apply (rule bigo_elt_subset) apply (rule bigo_abs2) @@ -244,65 +233,63 @@ 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)" +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 fun_diff_def) 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)))" + 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))" + 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)" + also have "\ \ O(f - g)" apply (rule bigo_elt_subset) apply (subst fun_diff_def) apply (rule bigo_abs) done - also from a have "... <= O(h)" + also from a have "\ \ O(h)" by (rule bigo_elt_subset) - finally show "(%x. abs (f x) - abs (g x)) : O(h)". + 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)" +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)" +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)" + 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))" + 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) <= ..." + 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))" + 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)" +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_def 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 (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+ @@ -311,24 +298,24 @@ apply (simp add: mult_ac abs_mult) done -lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)" +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 (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)" +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)" +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]) @@ -336,110 +323,117 @@ done lemma bigo_mult5: - assumes "ALL x. f x ~= 0" - shows "O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)" + fixes f :: "'a \ 'b::linordered_field" + assumes "\x. f x \ 0" + shows "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)" + 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))" + 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) + also have "(\x. 1 / f x) * (f * g) = g" + apply (simp add: func_times) apply (rule ext) apply (simp add: assms 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)" + 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) + also have "f * ((\x. (1::'b) / f x) * h) = h" + apply (simp add: func_times) apply (rule ext) apply (simp add: assms nonzero_divide_eq_eq mult_ac) done - finally show "h : f *o O(g)" . + finally show "h \ f *o O(g)" . qed -lemma bigo_mult6: "ALL x. f x ~= 0 ==> - O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)" +lemma bigo_mult6: + fixes f :: "'a \ 'b::linordered_field" + shows "\x. f x \ 0 \ O(f * g) = f *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::linordered_field)) * O(g)" +lemma bigo_mult7: + fixes f :: "'a \ 'b::linordered_field" + shows "\x. f x \ 0 \ O(f * g) \ O(f) * 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::linordered_field)) * O(g)" +lemma bigo_mult8: + fixes f :: "'a \ 'b::linordered_field" + shows "\x. f x \ 0 \ O(f * g) = O(f) * O(g)" apply (rule equalityI) apply (erule bigo_mult7) apply (rule bigo_mult) done -lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)" +lemma bigo_minus [intro]: "f \ O(g) \ - f \ O(g)" by (auto simp add: bigo_def fun_Compl_def) -lemma bigo_minus2: "f : g +o O(h) ==> -f : -g +o O(h)" +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 done -lemma bigo_minus3: "O(-f) = O(f)" +lemma bigo_minus3: "O(- f) = O(f)" by (auto simp add: bigo_def fun_Compl_def) -lemma bigo_plus_absorb_lemma1: "f : O(g) ==> f +o O(g) <= O(g)" +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)" + 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)" + 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)" + also have "\ \ O(g) + O(g)" proof - - from a have "O(f) <= O(g)" by (auto del: subsetI) + from a have "O(f) \ O(g)" by (auto del: subsetI) thus ?thesis by (auto del: subsetI) qed - also have "... <= O(g)" by simp + also have "\ \ O(g)" by simp finally show ?thesis . qed qed -lemma bigo_plus_absorb_lemma2: "f : O(g) ==> O(g) <= f +o O(g)" +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)" + 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)" + 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)" +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)") +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)" +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) @@ -449,63 +443,88 @@ apply (simp add: add_ac) done -lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" +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)" +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)" +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::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)" +lemma bigo_const3: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ (\x. 1) \ O(\x. c)" apply (simp add: bigo_def) - apply (rule_tac x = "abs(inverse c)" in exI) + apply (rule_tac x = "abs (inverse c)" in exI) apply (simp add: abs_mult [symmetric]) done -lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)" - by (rule bigo_elt_subset, rule bigo_const3, assumption) +lemma bigo_const4: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ O(\x. 1) \ O(\x. c)" + apply (rule bigo_elt_subset) + apply (rule bigo_const3) + apply assumption + done -lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ==> - O(%x. c) = O(%x. 1)" - by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption) +lemma bigo_const [simp]: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ O(\x. c) = O(\x. 1)" + apply (rule equalityI) + apply (rule bigo_const2) + apply (rule bigo_const4) + apply assumption + done -lemma bigo_const_mult1: "(%x. c * f x) : O(f)" +lemma bigo_const_mult1: "(\x. c * f x) \ O(f)" apply (simp add: bigo_def) - apply (rule_tac x = "abs(c)" in exI) + 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_mult2: "O(\x. c * f x) \ O(f)" + apply (rule bigo_elt_subset) + apply (rule bigo_const_mult1) + done -lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)" +lemma bigo_const_mult3: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ f \ O(\x. c * f x)" apply (simp add: bigo_def) - apply (rule_tac x = "abs(inverse c)" in exI) + 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::linordered_field) ~= 0 ==> - O(f) <= O(%x. c * f x)" - by (rule bigo_elt_subset, rule bigo_const_mult3, assumption) +lemma bigo_const_mult4: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ O(f) \ O(\x. c * f x)" + apply (rule bigo_elt_subset) + apply (rule bigo_const_mult3) + apply assumption + done -lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ==> - O(%x. c * f x) = O(f)" - by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4) +lemma bigo_const_mult [simp]: + fixes c :: "'a::linordered_field" + shows "c \ 0 \ O(\x. c * f x) = O(f)" + apply (rule equalityI) + apply (rule bigo_const_mult2) + apply (erule bigo_const_mult4) + done -lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==> - (%x. c) *o O(f) = O(f)" +lemma bigo_const_mult5 [simp]: + fixes c :: "'a::linordered_field" + shows "c \ 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!: simp add: bigo_def elt_set_times_def func_times) - apply (rule_tac x = "%y. inverse c * x y" in exI) + 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) @@ -515,11 +534,11 @@ apply force done -lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)" +lemma bigo_const_mult6 [intro]: "(\x. c) *o O(f) \ O(f)" apply (auto intro!: simp add: bigo_def elt_set_times_def func_times) - apply (rule_tac x = "ca * (abs c)" in exI) + 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 (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) @@ -528,33 +547,34 @@ apply(simp add: mult_ac) done -lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)" +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)" + then have "(\x. c) * f =o (\x. c) *o O(g)" by auto - also have "(%x. c) * f = (%x. c * f x)" + also have "(\x. c) * f = (\x. c * f x)" by (simp add: func_times) - also have "(%x. c) *o O(g) <= O(g)" + 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_compose1: "f =o O(g) \ (\x. f (k x)) =o O(\x. g (k x))" + unfolding bigo_def by 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))" +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] fun_Compl_def func_plus) - apply (drule bigo_compose1) apply (simp add: fun_diff_def) + apply (drule bigo_compose1) + apply (simp add: fun_diff_def) 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)" +lemma bigo_setsum_main: "\x. \y \ A x. 0 \ h x y \ + \c. \x. \y \ A x. abs (f x y) \ c * (h x y) \ + (\x. \y \ A x. f x y) =o O(\x. \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 @@ -571,14 +591,14 @@ apply assumption+ apply (drule bspec) apply assumption+ - apply (rule mult_right_mono) + 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)" +lemma bigo_setsum1: "\x y. 0 \ h x y \ + \c. \x y. abs (f x y) \ c * h x y \ + (\x. \y \ A x. f x y) =o O(\x. \y \ A x. h x y)" apply (rule bigo_setsum_main) apply force apply clarsimp @@ -586,14 +606,13 @@ 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_setsum2: "\y. 0 \ h y \ + \c. \y. abs (f y) \ c * (h y) \ + (\x. \y \ A x. f y) =o O(\x. \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)))" +lemma bigo_setsum3: "f =o O(h) \ + (\x. \y \ A x. l x y * f (k x y)) =o O(\x. \y \ A x. abs (l x y * h (k x y)))" apply (rule bigo_setsum1) apply (rule allI)+ apply (rule abs_ge_zero) @@ -608,10 +627,10 @@ 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)))" +lemma bigo_setsum4: "f =o g +o O(h) \ + (\x. \y \ A x. l x y * f (k x y)) =o + (\x. \y \ A x. l x y * g (k x y)) +o + O(\x. \y \ A x. abs (l x y * h (k x y)))" apply (rule set_minus_imp_plus) apply (subst fun_diff_def) apply (subst setsum_subtractf [symmetric]) @@ -621,12 +640,12 @@ 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)))") +lemma bigo_setsum5: "f =o O(h) \ \x y. 0 \ l x y \ + \x. 0 \ h x \ + (\x. \y \ A x. l x y * f (k x y)) =o + O(\x. \y \ A x. l x y * h (k x y))" + apply (subgoal_tac "(\x. \y \ A x. l x y * h (k x y)) = + (\x. \y \ A x. abs (l x y * h (k x y)))") apply (erule ssubst) apply (erule bigo_setsum3) apply (rule ext) @@ -636,11 +655,11 @@ 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))" +lemma bigo_setsum6: "f =o g +o O(h) \ \x y. 0 \ l x y \ + \x. 0 \ h x \ + (\x. \y \ A x. l x y * f (k x y)) =o + (\x. \y \ A x. l x y * g (k x y)) +o + O(\x. \y \ A x. l x y * h (k x y))" apply (rule set_minus_imp_plus) apply (subst fun_diff_def) apply (subst setsum_subtractf [symmetric]) @@ -654,33 +673,32 @@ subsection {* Misc useful stuff *} -lemma bigo_useful_intro: "A <= O(f) ==> B <= O(f) ==> - A + B <= O(f)" +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)" +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::linordered_field) ~= 0 ==> - (%x. c) * f =o O(h) ==> f =o O(h)" + +lemma bigo_useful_const_mult: + fixes c :: "'a::linordered_field" + shows "c \ 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 (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 (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)" +lemma bigo_fix: "(\x::nat. f (x + 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) @@ -696,9 +714,9 @@ 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)" +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 fun_diff_def) @@ -711,13 +729,10 @@ subsection {* Less than or equal to *} -definition - lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" - (infixl " 'b::linordered_idom) \ ('a \ 'b) \ 'a \ 'b" (infixl "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)" +lemma bigo_lesseq1: "f =o O(h) \ \x. abs (g x) \ abs (f x) \ g =o O(h)" apply (unfold bigo_def) apply clarsimp apply (rule_tac x = c in exI) @@ -726,8 +741,7 @@ apply (erule spec)+ done -lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==> - g =o O(h)" +lemma bigo_lesseq2: "f =o O(h) \ \x. abs (g x) \ f x \ g =o O(h)" apply (erule bigo_lesseq1) apply (rule allI) apply (drule_tac x = x in spec) @@ -736,26 +750,24 @@ 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)" +lemma bigo_lesseq3: "f =o O(h) \ \x. 0 \ g x \ \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)" +lemma bigo_lesseq4: "f =o O(h) \ + \x. 0 \ g x \ \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 x. f x \ g x \ f x. max (f x - g x) 0) = 0") apply (erule ssubst) apply (rule bigo_zero) apply (unfold func_zero) @@ -763,9 +775,8 @@ 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 + \x. 0 \ k x \ \x. k x \ f x \ k k x - g x") apply simp apply (subst abs_of_nonneg) apply (drule_tac x = x in spec) back @@ -781,15 +792,14 @@ apply (subst diff_conv_add_uminus)+ apply (rule add_right_mono) apply (erule spec) - apply (rule order_trans) + apply (rule order_trans) prefer 2 apply (rule abs_ge_zero) apply (simp add: algebra_simps) done -lemma bigo_lesso3: "f =o g +o O(h) ==> - ALL x. 0 <= k x ==> ALL x. g x <= k x ==> - f + \x. 0 \ k x \ \x. g x \ k x \ f f x - k x") apply simp apply (subst abs_of_nonneg) apply (drule_tac x = x in spec) back @@ -806,14 +816,15 @@ apply (rule add_left_mono) apply (rule le_imp_neg_le) apply (erule spec) - apply (rule order_trans) + apply (rule order_trans) prefer 2 apply (rule abs_ge_zero) apply (simp add: algebra_simps) done -lemma bigo_lesso4: "f 'b::linordered_field) ==> - g =o h +o O(k) ==> f 'b::linordered_field" + shows "f g =o h +o O(k) \ f - EX C. ALL x. f x <= g x + C * abs(h x)" +lemma bigo_lesso5: "f \C. \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: algebra_simps) + apply (subgoal_tac "abs (max (f x - g x) 0) = max (f x - g x) 0") + apply (clarsimp simp add: algebra_simps) apply (rule abs_of_nonneg) apply (rule max.cobounded2) done -lemma lesso_add: "f - k (f + k) k (f + k) g ----> 0 ==> f ----> (0::real)" +lemma bigo_LIMSEQ1: "f =o O(g) \ g ----> 0 \ f ----> (0::real)" apply (simp add: LIMSEQ_iff bigo_alt_def) apply clarify apply (drule_tac x = "r / c" in spec) @@ -866,21 +874,20 @@ apply (rule order_le_less_trans) apply assumption apply (rule order_less_le_trans) - apply (subgoal_tac "c * abs(g n) < c * (r / c)") + apply (subgoal_tac "c * abs (g n) < c * (r / c)") apply assumption apply (erule mult_strict_left_mono) apply assumption apply simp -done + done -lemma bigo_LIMSEQ2: "f =o g +o O(h) ==> h ----> 0 ==> f ----> a - ==> g ----> (a::real)" +lemma bigo_LIMSEQ2: "f =o g +o O(h) \ h ----> 0 \ f ----> a \ g ----> (a::real)" apply (drule set_plus_imp_minus) apply (drule bigo_LIMSEQ1) apply assumption apply (simp only: fun_diff_def) apply (erule LIMSEQ_diff_approach_zero2) apply assumption -done + done end