# HG changeset patch # User blanchet # Date 1321613232 -3600 # Node ID 3a865fc42bbff8424cb037ff7611099cbeaebe70 # Parent 7a39df11bcf6f28b0a0b28bed27a6fe99da53f35 more "metis" calls in example diff -r 7a39df11bcf6 -r 3a865fc42bbf src/HOL/Metis_Examples/Big_O.thy --- a/src/HOL/Metis_Examples/Big_O.thy Fri Nov 18 11:47:12 2011 +0100 +++ b/src/HOL/Metis_Examples/Big_O.thy Fri Nov 18 11:47:12 2011 +0100 @@ -10,36 +10,31 @@ theory Big_O imports "~~/src/HOL/Decision_Procs/Dense_Linear_Order" - Main "~~/src/HOL/Library/Function_Algebras" "~~/src/HOL/Library/Set_Algebras" begin -declare [[metis_new_skolemizer]] - subsection {* Definitions *} -definition bigo :: "('a => 'b::{linordered_idom,number_ring}) => ('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,number_ring}) => ('a => 'b) set" ("(1O'(_'))") where + "O(f\('a => 'b)) == {h. \c. \x. abs (h x) <= c * abs (f x)}" -declare [[ sledgehammer_problem_prefix = "BigO__bigo_pos_const" ]] -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)))))" - apply auto - apply (case_tac "c = 0", simp) - apply (rule_tac x = "1" in exI, simp) - apply (rule_tac x = "abs c" in exI, auto) - apply (metis abs_ge_zero abs_of_nonneg Orderings.xt1(6) abs_mult) - done +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)))))" +by (metis (hide_lams, no_types) abs_ge_zero + comm_semiring_1_class.normalizing_semiring_rules(7) mult.comm_neutral + mult_nonpos_nonneg not_leE order_trans zero_less_one) (*** Now various verions with an increasing shrink factor ***) sledgehammer_params [isar_proof, isar_shrink_factor = 1] -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 + "(\(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", simp) apply (rule_tac x = "1" in exI, simp) @@ -67,9 +62,10 @@ sledgehammer_params [isar_proof, isar_shrink_factor = 2] -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 + "(\(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", simp) apply (rule_tac x = "1" in exI, simp) @@ -89,9 +85,10 @@ sledgehammer_params [isar_proof, isar_shrink_factor = 3] -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 + "(\(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", simp) apply (rule_tac x = "1" in exI, simp) @@ -108,9 +105,10 @@ sledgehammer_params [isar_proof, isar_shrink_factor = 4] -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 + "(\(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", simp) apply (rule_tac x = "1" in exI, simp) @@ -127,142 +125,109 @@ sledgehammer_params [isar_proof, isar_shrink_factor = 1] -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) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_elt_subset" ]] -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) +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)+ -(*sledgehammer*) - 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 +(* sledgehammer *) +apply (rule allI) +apply (drule_tac x = "xa" in spec)+ +apply (subgoal_tac "ca * abs (f xa) <= ca * (c * abs (g xa))") + apply (metis comm_semiring_1_class.normalizing_semiring_rules(19) + comm_semiring_1_class.normalizing_semiring_rules(7) order_trans) +by (metis mult_le_cancel_left_pos) - -declare [[ sledgehammer_problem_prefix = "BigO__bigo_refl" ]] lemma bigo_refl [intro]: "f : O(f)" apply (auto simp add: bigo_def) by (metis mult_1 order_refl) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_zero" ]] lemma bigo_zero: "0 : O(g)" apply (auto simp add: bigo_def func_zero) by (metis mult_zero_left order_refl) -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)" - apply (auto simp add: bigo_alt_def set_plus_def) - apply (rule_tac x = "c + ca" in exI) - apply auto - apply (simp add: ring_distribs func_plus) - apply (blast intro:order_trans abs_triangle_ineq add_mono elim:) -done +apply (auto simp add: bigo_alt_def set_plus_def) +apply (rule_tac x = "c + ca" in exI) +apply auto +apply (simp add: ring_distribs func_plus) +by (metis order_trans abs_triangle_ineq add_mono) 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 +by (metis bigo_plus_self_subset bigo_zero set_eq_subset set_zero_plus2) 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 conjI) - apply (rule_tac x = "c + c" in exI) - apply (clarsimp) - apply (auto) +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 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 (metis mult_2 order_trans) 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: abs_triangle_ineq) + 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 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 (metis order_trans semiring_mult_2) + 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: abs_triangle_ineq) - 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 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 (rule abs_triangle_ineq) - apply (simp add: order_less_le) -apply (metis abs_not_less_zero even_less_0_iff less_not_permute linorder_not_less mult_less_0_iff) -done + apply (metis abs_triangle_ineq mult_le_cancel_left_pos) +by (metis abs_ge_zero abs_of_pos zero_le_mult_iff) -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_subset2 [intro]: "A <= O(f) \ B <= O(f) \ A \ B <= O(f)" +by (metis bigo_plus_idemp set_plus_mono2) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq" ]] -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_def func_plus) - apply clarify -(*sledgehammer*) - apply (rule_tac x = "max c ca" in exI) - apply (rule conjI) - apply (metis Orderings.less_max_iff_disj) - 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 (blast intro: order_trans) +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 +(* sledgehammer *) +apply (rule_tac x = "max c ca" in exI) +apply (rule conjI) + apply (metis less_max_iff_disj) +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 (metis order_trans) defer 1 - apply (rule abs_triangle_ineq) - apply (metis add_nonneg_nonneg) - apply (rule add_mono) -using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]] - apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6)) - apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans) -done + apply (metis abs_triangle_ineq) + apply (metis add_nonneg_nonneg) +apply (rule add_mono) + apply (metis le_maxI2 linorder_linear min_max.sup_absorb1 mult_right_mono xt1(6)) +by (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt" ]] -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) +lemma bigo_bounded_alt: "\x. 0 <= f x \ \x. f x <= c * g x \ f : O(g)" +apply (auto simp add: bigo_def) (* Version 1: one-line proof *) - apply (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult) - done +by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult) -lemma (*bigo_bounded_alt:*) "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> - f : O(g)" +lemma "\x. 0 <= f x \ \x. f x <= c * g x \ f : O(g)" apply (auto simp add: bigo_def) (* Version 2: structured proof *) proof - @@ -270,32 +235,11 @@ thus "\c. \x. f x \ c * \g x\" by (metis abs_mult abs_ge_self order_trans) qed -text{*So here is the easier (and more natural) problem using transitivity*} -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]] -lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" -apply (auto simp add: bigo_def) -(* Version 1: one-line proof *) -by (metis abs_ge_self abs_mult order_trans) +lemma bigo_bounded: "\x. 0 <= f x \ \x. f x <= g x \ f : O(g)" +apply (erule bigo_bounded_alt [of f 1 g]) +by (metis mult_1) -text{*So here is the easier (and more natural) problem using transitivity*} -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded_alt_trans" ]] -lemma "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==> f : O(g)" - apply (auto simp add: bigo_def) -(* Version 2: structured proof *) -proof - - assume "\x. f x \ c * g x" - thus "\c. \x. f x \ c * \g x\" by (metis abs_mult abs_ge_self order_trans) -qed - -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 - -declare [[ sledgehammer_problem_prefix = "BigO__bigo_bounded2" ]] -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: diff_minus fun_Compl_def func_plus) @@ -308,19 +252,17 @@ thus "(0\'b) \ f x + - lb x" by (metis not_leE diff_minus less_iff_diff_less_0 less_le_not_le) qed -declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]] -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 by (metis mult_1 order_refl) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]] -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 by (metis mult_1 order_refl) -lemma bigo_abs3: "O(f) = O(%x. abs(f x))" +lemma bigo_abs3: "O(f) = O(\x. abs(f x))" proof - have F1: "\v u. u \ O(v) \ O(u) \ O(v)" by (metis bigo_elt_subset) have F2: "\u. (\R. \u R\) \ O(u)" by (metis bigo_abs) @@ -328,16 +270,15 @@ thus "O(f) = O(\x. \f x\)" using F1 F2 by auto qed -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)))" + 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 @@ -351,45 +292,43 @@ done also have "... <= O(h)" using a 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)" 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)))" + 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 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]] lemma bigo_mult [intro]: "O(f)\O(g) <= O(f * g)" apply (rule subsetI) apply (subst bigo_def) apply (auto simp del: abs_mult mult_ac simp add: bigo_alt_def set_times_def func_times) -(*sledgehammer*) +(* sledgehammer *) 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))") -using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]] prefer 2 apply (metis mult_assoc mult_left_commute abs_of_pos mult_left_commute @@ -400,14 +339,12 @@ abs_mult has just been done *) by (metis abs_ge_zero mult_mono') -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult2" ]] 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) -(*sledgehammer*) +(* sledgehammer *) apply (rule_tac x = c in exI) apply clarify apply (drule_tac x = x in spec) -using [[ sledgehammer_problem_prefix = "BigO__bigo_mult2_simpler" ]] (*sledgehammer [no luck]*) apply (subgoal_tac "abs(f x) * abs(b x) <= abs(f x) * (c * abs(g x))") apply (simp add: mult_ac) @@ -415,36 +352,33 @@ apply (rule abs_ge_zero) done -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]] -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)" by (metis bigo_mult set_rev_mp set_times_intro) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]] -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)" by (metis bigo_mult2 set_plus_mono_b set_times_intro2 set_times_plus_distrib) - -lemma bigo_mult5: "ALL x. f x ~= 0 ==> - O(f * g) <= (f::'a => ('b::{linordered_field,number_ring})) *o O(g)" +lemma bigo_mult5: "\x. f x ~= 0 \ + O(f * g) <= (f\'a => ('b\{linordered_field,number_ring})) *o O(g)" proof - - assume a: "ALL x. f x ~= 0" + assume a: "\x. f x ~= 0" show "O(f * g) <= f *o O(g)" proof fix h assume h: "h : O(f * g)" - then have "(%x. 1 / (f x)) * h : (%x. 1 / f x) *o 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" + also have "(\x. 1 / f x) * (f * g) = g" apply (simp add: func_times) apply (rule ext) apply (simp add: a h 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" + also have "f * ((\x. (1\'b) / f x) * h) = h" apply (simp add: func_times) apply (rule ext) apply (simp add: a h nonzero_divide_eq_eq mult_ac) @@ -453,34 +387,32 @@ qed qed -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]] -lemma bigo_mult6: "ALL x. f x ~= 0 ==> - O(f * g) = (f::'a => ('b::{linordered_field,number_ring})) *o O(g)" +lemma bigo_mult6: "\x. f x ~= 0 \ + O(f * g) = (f\'a => ('b\{linordered_field,number_ring})) *o O(g)" by (metis bigo_mult2 bigo_mult5 order_antisym) (*proof requires relaxing relevance: 2007-01-25*) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult7" ]] declare bigo_mult6 [simp] -lemma bigo_mult7: "ALL x. f x ~= 0 ==> - O(f * g) <= O(f::'a => ('b::{linordered_field,number_ring})) \ O(g)" -(*sledgehammer*) +lemma bigo_mult7: "\x. f x ~= 0 \ + O(f * g) <= O(f\'a => ('b\{linordered_field,number_ring})) \ O(g)" +(* sledgehammer *) apply (subst bigo_mult6) apply assumption apply (rule set_times_mono3) apply (rule bigo_refl) done - declare bigo_mult6 [simp del] -declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult8" ]] - declare bigo_mult7[intro!] -lemma bigo_mult8: "ALL x. f x ~= 0 ==> - O(f * g) = O(f::'a => ('b::{linordered_field,number_ring})) \ O(g)" +declare bigo_mult6 [simp del] +declare bigo_mult7 [intro!] + +lemma bigo_mult8: "\x. f x ~= 0 \ + O(f * g) = O(f\'a => ('b\{linordered_field,number_ring})) \ O(g)" by (metis bigo_mult bigo_mult7 order_antisym_conv) -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) @@ -490,7 +422,7 @@ lemma bigo_minus3: "O(-f) = O(f)" by (auto simp add: bigo_def fun_Compl_def abs_minus_cancel) -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)" @@ -508,7 +440,7 @@ 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)" @@ -522,23 +454,22 @@ qed qed -declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]] -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)" by (metis bigo_plus_absorb_lemma1 bigo_plus_absorb_lemma2 order_eq_iff) -lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= 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) apply (rule bigo_minus) apply (subst set_minus_plus) apply assumption - apply (simp add: diff_minus add_ac) + apply (simp add: diff_minus add_ac) done lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))" @@ -546,67 +477,60 @@ 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) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]] -lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)" +lemma (*bigo_const2 [intro]:*) "O(\x. c) <= O(\x. 1)" by (metis bigo_const1 bigo_elt_subset) -lemma bigo_const2 [intro]: "O(%x. c::'b::{linordered_idom,number_ring}) <= O(%x. 1)" -(* "thus" had to be replaced by "show" with an explicit reference to "F1" *) +lemma bigo_const2 [intro]: "O(\x. c\'b\{linordered_idom,number_ring}) <= O(\x. 1)" proof - - have F1: "\u. (\Q. u) \ O(\Q. 1)" by (metis bigo_const1) - show "O(\x. c) \ O(\x. 1)" by (metis F1 bigo_elt_subset) + have "\u. (\Q. u) \ O(\Q. 1)" by (metis bigo_const1) + thus "O(\x. c) \ O(\x. 1)" by (metis bigo_elt_subset) qed -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]] -lemma bigo_const3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> (%x. 1) : O(%x. c)" +lemma bigo_const3: "(c\'a\{linordered_field,number_ring}) ~= 0 \ (\x. 1) : O(\x. c)" apply (simp add: bigo_def) by (metis abs_eq_0 left_inverse order_refl) -lemma bigo_const4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> O(%x. 1) <= O(%x. c)" +lemma bigo_const4: "(c\'a\{linordered_field,number_ring}) ~= 0 \ O(\x. 1) <= O(\x. c)" by (rule bigo_elt_subset, rule bigo_const3, assumption) -lemma bigo_const [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> - O(%x. c) = O(%x. 1)" +lemma bigo_const [simp]: "(c\'a\{linordered_field,number_ring}) ~= 0 \ + O(\x. c) = O(\x. 1)" by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]] -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 abs_mult) by (metis le_less) -lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)" +lemma bigo_const_mult2: "O(\x. c * f x) <= O(f)" by (rule bigo_elt_subset, rule bigo_const_mult1) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]] -lemma bigo_const_mult3: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> f : O(%x. c * f x)" - apply (simp add: bigo_def) -(*sledgehammer [no luck]*) - apply (rule_tac x = "abs(inverse c)" in exI) - apply (simp only: abs_mult [symmetric] mult_assoc [symmetric]) +lemma bigo_const_mult3: "(c\'a\{linordered_field,number_ring}) ~= 0 \ f : O(\x. c * f x)" +apply (simp add: bigo_def) +(* sledgehammer *) +apply (rule_tac x = "abs(inverse c)" in exI) +apply (simp only: abs_mult [symmetric] mult_assoc [symmetric]) apply (subst left_inverse) -apply (auto ) -done +by auto -lemma bigo_const_mult4: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> - O(f) <= O(%x. c * f x)" +lemma bigo_const_mult4: "(c\'a\{linordered_field,number_ring}) ~= 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::{linordered_field,number_ring}) ~= 0 ==> - O(%x. c * f x) = O(f)" +lemma bigo_const_mult [simp]: "(c\'a\{linordered_field,number_ring}) ~= 0 \ + O(\x. c * f x) = O(f)" by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]] -lemma bigo_const_mult5 [simp]: "(c::'a::{linordered_field,number_ring}) ~= 0 ==> - (%x. c) *o O(f) = O(f)" +lemma bigo_const_mult5 [simp]: "(c\'a\{linordered_field,number_ring}) ~= 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 (rule_tac x = "\y. inverse c * x y" in exI) apply (rename_tac g d) apply safe apply (rule_tac [2] ext) @@ -633,13 +557,11 @@ using A2 F4 by (metis ab_semigroup_mult_class.mult_ac(1) comm_mult_left_mono) qed - -declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]] -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!: subsetI simp add: bigo_def elt_set_times_def func_times simp del: abs_mult mult_ac) -(*sledgehammer*) +(* sledgehammer *) 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))") @@ -651,23 +573,23 @@ 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))" +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))" +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 fun_Compl_def func_plus) apply (erule bigo_compose1) @@ -675,9 +597,9 @@ 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. 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 @@ -691,61 +613,50 @@ apply (blast intro: order_trans mult_right_mono abs_ge_self) done -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum1" ]] -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) -(*sledgehammer*) - apply force - apply clarsimp - apply (rule_tac x = c in exI) - apply force -done +lemma bigo_setsum1: "\x y. 0 <= h x y \ + \c. \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)" +by (metis (no_types) bigo_setsum_main) -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)" +lemma bigo_setsum2: "\y. 0 <= h y \ + \c. \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) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum3" ]] -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 simp add: abs_mult) -(*sledgehammer*) - apply (rule_tac x = c in exI) - apply (rule allI)+ - apply (subst mult_left_commute) - apply (rule mult_left_mono) - apply (erule spec) - apply (rule abs_ge_zero) -done +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 simp add: abs_mult) +(* sledgehammer *) +apply (rule_tac x = c in exI) +apply (rule allI)+ +apply (subst mult_left_commute) +apply (rule mult_left_mono) + apply (erule spec) +by (rule abs_ge_zero) -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 fun_diff_def) - apply (subst setsum_subtractf [symmetric]) - apply (subst right_diff_distrib [symmetric]) - apply (rule bigo_setsum3) - apply (subst fun_diff_def [symmetric]) - apply (erule set_plus_imp_minus) -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 fun_diff_def) +apply (subst setsum_subtractf [symmetric]) +apply (subst right_diff_distrib [symmetric]) +apply (rule bigo_setsum3) +apply (subst fun_diff_def [symmetric]) +by (erule set_plus_imp_minus) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_setsum5" ]] -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. 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) @@ -754,11 +665,11 @@ apply (metis abs_of_nonneg zero_le_mult_iff) 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. 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 fun_diff_def) apply (subst setsum_subtractf [symmetric]) @@ -771,50 +682,39 @@ subsection {* Misc useful stuff *} -lemma bigo_useful_intro: "A <= O(f) ==> 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,number_ring}) ~= 0 ==> - (%x. c) * f =o O(h) ==> f =o O(h)" +lemma bigo_useful_const_mult: "(c\'a\{linordered_field,number_ring}) ~= 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 -declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]] -lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==> +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) -(*sledgehammer*) - apply clarify - apply (rule_tac x = c in exI) - apply safe - apply (case_tac "x = 0") -apply (metis abs_ge_zero abs_zero order_less_le split_mult_pos_le) - apply (subgoal_tac "x = Suc (x - 1)") - apply metis - apply simp - done - +apply (simp add: bigo_alt_def) +by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc) 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)" + "(\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) @@ -826,23 +726,23 @@ 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) ==> +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 (blast intro: order_trans) done -lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==> +lemma bigo_lesseq2: "f =o O(h) \ \x. abs (g x) <= f x \ g =o O(h)" apply (erule bigo_lesseq1) apply (blast intro: abs_ge_self order_trans) done -lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==> +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) @@ -850,8 +750,8 @@ apply (erule spec)+ done -lemma bigo_lesseq4: "f =o O(h) ==> - ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==> +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) @@ -859,23 +759,15 @@ apply (erule spec)+ done -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]] -lemma bigo_lesso1: "ALL x. f x <= g x ==> f x. f x <= g x \ f x. max (f x - g x) 0) = 0" - thus "(\x. max (f x - g x) 0) \ O(h)" by (metis bigo_zero) -next - show "\x\'a. f x \ g x \ (\x\'a. max (f x - g x) (0\'b)) = (0\'a \ 'b)" - apply (unfold func_zero) - apply (rule ext) - by (simp split: split_max) -qed +apply (subgoal_tac "(\x. max (f x - g x) 0) = 0") + apply (metis bigo_zero) +by (metis (lam_lifting, no_types) func_zero le_fun_def le_iff_diff_le_0 + min_max.sup_absorb2 order_eq_iff) -declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso2" ]] -lemma bigo_lesso2: "f =o g +o O(h) ==> - ALL x. 0 <= k x ==> ALL x. k x <= f x ==> +lemma bigo_lesso2: "f =o g +o O(h) \ + \x. 0 <= k x \ \x. k x <= f x \ k x\'a. k x \ f x" - hence F1: "\x\<^isub>1\'a. max (k x\<^isub>1) (f x\<^isub>1) = f x\<^isub>1" by (metis min_max.sup_absorb2) - assume "(0\'b) \ k x - g x" - hence F2: "max (0\'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2) - have F3: "\x\<^isub>1\'b. x\<^isub>1 \ \x\<^isub>1\" by (metis abs_le_iff le_less) - have "\(x\<^isub>2\'b) x\<^isub>1\'b. max x\<^isub>1 x\<^isub>2 \ x\<^isub>2 \ max x\<^isub>1 x\<^isub>2 \ x\<^isub>1" by (metis le_less le_max_iff_disj) - hence "\(x\<^isub>3\'b) (x\<^isub>2\'b) x\<^isub>1\'b. x\<^isub>1 - x\<^isub>2 \ x\<^isub>3 - x\<^isub>2 \ x\<^isub>3 \ x\<^isub>1" by (metis add_le_imp_le_right diff_minus min_max.le_supE) - hence "k x - g x \ f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute) - hence "k x - g x \ \f x - g x\" by (metis F3 le_max_iff_disj min_max.sup_absorb2) - thus "max (k x - g x) (0\'b) \ \f x - g x\" by (metis F2 min_max.sup_commute) -next - show "\x\'a. - \\x\'a. (0\'b) \ k x; \x\'a. k x \ f x; \ (0\'b) \ k x - g x\ - \ max (k x - g x) (0\'b) \