author blanchet Fri, 18 Nov 2011 11:47:12 +0100 changeset 45575 3a865fc42bbf parent 45574 7a39df11bcf6 child 45576 6ea2bba2694a
more "metis" calls in example
```--- 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\<Colon>{linordered_idom,number_ring}) => ('a => 'b) set" ("(1O'(_'))") where
+  "O(f\<Colon>('a => 'b)) == {h. \<exists>c. \<forall>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:
+  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
+    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
+      = (\<exists>c. 0 < c & (\<forall>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
+  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
+    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
+      = (\<exists>c. 0 < c & (\<forall>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
+  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
+    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
+      = (\<exists>c. 0 < c & (\<forall>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
+  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
+    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
+      = (\<exists>c. 0 < c & (\<forall>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
+  "(\<exists>(c\<Colon>'a\<Colon>linordered_idom).
+    \<forall>x. (abs (h x)) <= (c * (abs (f x))))
+      = (\<exists>c. 0 < c & (\<forall>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. \<exists>c. (0 < c & (\<forall>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) \<Longrightarrow> O(f) <= O(g)"
+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 (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)"
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(\<lambda>x. 0) = {\<lambda>x. 0}"

lemma bigo_plus_self_subset [intro]:
"O(f) \<oplus> 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

lemma bigo_plus_idemp [simp]: "O(f) \<oplus> 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) \<oplus> 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 = "\<lambda>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 (erule order_trans)
apply (rule mult_left_mono)
-  apply (rule mult_nonneg_nonneg)
+ 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 (rule_tac x = "\<lambda>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 (rule mult_left_mono)
-  apply (rule abs_triangle_ineq)
-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 \<oplus> B <= O(f)"
-  apply (subgoal_tac "A \<oplus> B <= O(f) \<oplus> 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) \<Longrightarrow> B <= O(f) \<Longrightarrow> A \<oplus> 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) \<oplus> 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 (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: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> O(f + g) = O(f) \<oplus> 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 (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)
-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 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: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
(* 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 "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= c * g x \<Longrightarrow> f : O(g)"
(* Version 2: structured proof *)
proof -
@@ -270,32 +235,11 @@
thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" 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)"
-(* Version 1: one-line proof *)
-by (metis abs_ge_self abs_mult order_trans)
+lemma bigo_bounded: "\<forall>x. 0 <= f x \<Longrightarrow> \<forall>x. f x <= g x \<Longrightarrow> 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 "\<forall>x. f x \<le> c * g x"
-  thus "\<exists>c. \<forall>x. f x \<le> c * \<bar>g x\<bar>" 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: "\<forall>x. lb x <= f x \<Longrightarrow> \<forall>x. f x <= lb x + g x \<Longrightarrow> 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\<Colon>'b) \<le> 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: "(\<lambda>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(\<lambda>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(\<lambda>x. abs(f x))"
proof -
have F1: "\<forall>v u. u \<in> O(v) \<longrightarrow> O(u) \<subseteq> O(v)" by (metis bigo_elt_subset)
have F2: "\<forall>u. (\<lambda>R. \<bar>u R\<bar>) \<in> O(u)" by (metis bigo_abs)
@@ -328,16 +270,15 @@
thus "O(f) = O(\<lambda>x. \<bar>f x\<bar>)" 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) \<Longrightarrow> (\<lambda>x. abs (f x)) =o (\<lambda>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 "(\<lambda>x. abs (f x) - abs (g x)) =o O(\<lambda>x. abs(abs (f x) - abs (g x)))"
by (rule bigo_abs2)
-  also have "... <= O(%x. abs (f x - g x))"
+  also have "... <= O(\<lambda>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 "(\<lambda>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) \<Longrightarrow> (\<lambda>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) \<oplus> O(h)"
+lemma bigo_elt_subset2 [intro]: "f : g +o O(h) \<Longrightarrow> O(f) <= O(g) \<oplus> O(h)"
proof -
assume "f : g +o O(h)"
also have "... <= O(g) \<oplus> O(h)"
by (auto del: subsetI)
-  also have "... = O(%x. abs(g x)) \<oplus> O(%x. abs(h x))"
+  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>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((\<lambda>x. abs(g x)) + (\<lambda>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)) \<oplus> O(%x. abs(h x))"
+  also have "... = O(\<lambda>x. abs(g x)) \<oplus> O(\<lambda>x. abs(h x))"
by (rule bigo_plus_eq, auto)
finally show ?thesis
qed

-declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult" ]]
lemma bigo_mult [intro]: "O(f)\<otimes>O(g) <= O(f * g)"
apply (rule subsetI)
apply (subst bigo_def)
apply (auto simp del: abs_mult mult_ac
-(*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))")
@@ -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) \<Longrightarrow> g : O(j) \<Longrightarrow> 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) \<Longrightarrow> 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: "\<forall>x. f x ~= 0 \<Longrightarrow>
+    O(f * g) <= (f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) *o O(g)"
proof -
-  assume a: "ALL x. f x ~= 0"
+  assume a: "\<forall>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 "(\<lambda>x. 1 / (f x)) * h : (\<lambda>x. 1 / f x) *o O(f * g)"
by auto
-    also have "... <= O((%x. 1 / f x) * (f * g))"
+    also have "... <= O((\<lambda>x. 1 / f x) * (f * g))"
by (rule bigo_mult2)
-    also have "(%x. 1 / f x) * (f * g) = g"
+    also have "(\<lambda>x. 1 / f x) * (f * g) = g"
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 "(\<lambda>x. (1\<Colon>'b) / f x) * h : O(g)".
+    then have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) : f *o O(g)"
by auto
-    also have "f * ((%x. (1::'b) / f x) * h) = h"
+    also have "f * ((\<lambda>x. (1\<Colon>'b) / f x) * h) = h"
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: "\<forall>x. f x ~= 0 \<Longrightarrow>
+    O(f * g) = (f\<Colon>'a => ('b\<Colon>{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})) \<otimes> O(g)"
-(*sledgehammer*)
+lemma bigo_mult7: "\<forall>x. f x ~= 0 \<Longrightarrow>
+    O(f * g) <= O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> 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})) \<otimes> O(g)"
+declare bigo_mult6 [simp del]
+declare bigo_mult7 [intro!]
+
+lemma bigo_mult8: "\<forall>x. f x ~= 0 \<Longrightarrow>
+    O(f * g) = O(f\<Colon>'a => ('b\<Colon>{linordered_field,number_ring})) \<otimes> 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) \<Longrightarrow> - 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) \<Longrightarrow> -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) \<Longrightarrow> 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) \<Longrightarrow> 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) \<Longrightarrow> 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) \<Longrightarrow> A <= O(g) \<Longrightarrow> 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) \<Longrightarrow> 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
done

lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
@@ -546,67 +477,60 @@
done

-lemma bigo_const1: "(%x. c) : O(%x. 1)"
+lemma bigo_const1: "(\<lambda>x. c) : O(\<lambda>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(\<lambda>x. c) <= O(\<lambda>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(\<lambda>x. c\<Colon>'b\<Colon>{linordered_idom,number_ring}) <= O(\<lambda>x. 1)"
proof -
-  have F1: "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
-  show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset)
+  have "\<forall>u. (\<lambda>Q. u) \<in> O(\<lambda>Q. 1)" by (metis bigo_const1)
+  thus "O(\<lambda>x. c) \<subseteq> O(\<lambda>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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> (\<lambda>x. 1) : O(\<lambda>x. c)"
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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> O(\<lambda>x. 1) <= O(\<lambda>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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
+    O(\<lambda>x. c) = O(\<lambda>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: "(\<lambda>x. c * f x) : O(f)"
by (metis le_less)

-lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
+lemma bigo_const_mult2: "O(\<lambda>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)"
-(*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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow> f : O(\<lambda>x. c * f x)"
+(* 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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
+    O(f) <= O(\<lambda>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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
+    O(\<lambda>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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
+    (\<lambda>x. c) *o O(f) = O(f)"
apply (auto del: subsetI)
apply (rule order_trans)
apply (rule bigo_mult2)
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 = "\<lambda>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]: "(\<lambda>x. c) *o O(f) <= O(f)"
apply (auto intro!: subsetI
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 @@
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) \<Longrightarrow> (\<lambda>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 "(\<lambda>x. c) * f =o (\<lambda>x. c) *o O(g)"
by auto
-  also have "(%x. c) * f = (%x. c * f x)"
+  also have "(\<lambda>x. c) * f = (\<lambda>x. c * f x)"
-  also have "(%x. c) *o O(g) <= O(g)"
+  also have "(\<lambda>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) \<Longrightarrow> (\<lambda>x. f(k x)) =o O(\<lambda>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) \<Longrightarrow> (\<lambda>x. f(k x)) =o (\<lambda>x. g(k x)) +o
+    O(\<lambda>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: "\<forall>x. \<forall>y \<in> A x. 0 <= h x y \<Longrightarrow>
+    \<exists>c. \<forall>x. \<forall>y \<in> A x. abs (f x y) <= c * (h x y) \<Longrightarrow>
+      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>x. SUM y : A x. h x y)"
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: "\<forall>x y. 0 <= h x y \<Longrightarrow>
+    \<exists>c. \<forall>x y. abs (f x y) <= c * (h x y) \<Longrightarrow>
+      (\<lambda>x. SUM y : A x. f x y) =o O(\<lambda>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: "\<forall>y. 0 <= h y \<Longrightarrow>
+    \<exists>c. \<forall>y. abs(f y) <= c * (h y) \<Longrightarrow>
+      (\<lambda>x. SUM y : A x. f y) =o O(\<lambda>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) \<Longrightarrow>
+    (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
+      O(\<lambda>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)
+(* 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) \<Longrightarrow>
+    (\<lambda>x. SUM y : A x. l x y * f(k x y)) =o
+      (\<lambda>x. SUM y : A x. l x y * g(k x y)) +o
+        O(\<lambda>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) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
+    \<forall>x. 0 <= h x \<Longrightarrow>
+      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
+        O(\<lambda>x. SUM y : A x. (l x y) * h(k x y))"
+  apply (subgoal_tac "(\<lambda>x. SUM y : A x. (l x y) * h(k x y)) =
+      (\<lambda>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) \<Longrightarrow> \<forall>x y. 0 <= l x y \<Longrightarrow>
+    \<forall>x. 0 <= h x \<Longrightarrow>
+      (\<lambda>x. SUM y : A x. (l x y) * f(k x y)) =o
+        (\<lambda>x. SUM y : A x. (l x y) * g(k x y)) +o
+          O(\<lambda>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) \<Longrightarrow> B <= O(f) \<Longrightarrow>
A \<oplus> 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) \<Longrightarrow> g =o O(h) \<Longrightarrow> 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\<Colon>'a\<Colon>{linordered_field,number_ring}) ~= 0 \<Longrightarrow>
+    (\<lambda>x. c) * f =o O(h) \<Longrightarrow> f =o O(h)"
apply (rule subsetD)
-  apply (subgoal_tac "(%x. 1 / c) *o O(h) <= O(h)")
+  apply (subgoal_tac "(\<lambda>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 = (\<lambda>x. 1 / c) * ((\<lambda>x. c) * f)")
apply (erule ssubst)
apply (erule set_times_intro2)
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: "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o O(\<lambda>x. h(x + 1)) \<Longrightarrow> f 0 = 0 \<Longrightarrow>
f =o O(h)"
-(*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
-
+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)"
+    "(\<lambda>x. f ((x\<Colon>nat) + 1)) =o (\<lambda>x. g(x + 1)) +o O(\<lambda>x. h(x + 1)) \<Longrightarrow>
+       f 0 = g 0 \<Longrightarrow> 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 "<o" 70) where
-  "f <o g == (%x. max (f x - g x) 0)"
+definition lesso :: "('a => 'b\<Colon>linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
+  "f <o g == (\<lambda>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) \<Longrightarrow> \<forall>x. abs (g x) <= abs (f x) \<Longrightarrow>
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) \<Longrightarrow> \<forall>x. abs (g x) <= f x \<Longrightarrow>
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) \<Longrightarrow> \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= f x \<Longrightarrow>
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) \<Longrightarrow>
+    \<forall>x. 0 <= g x \<Longrightarrow> \<forall>x. g x <= abs (f x) \<Longrightarrow>
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 <o g =o O(h)"
+lemma bigo_lesso1: "\<forall>x. f x <= g x \<Longrightarrow> f <o g =o O(h)"
apply (unfold lesso_def)
-apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
-proof -
-  assume "(\<lambda>x. max (f x - g x) 0) = 0"
-  thus "(\<lambda>x. max (f x - g x) 0) \<in> O(h)" by (metis bigo_zero)
-next
-  show "\<forall>x\<Colon>'a. f x \<le> g x \<Longrightarrow> (\<lambda>x\<Colon>'a. max (f x - g x) (0\<Colon>'b)) = (0\<Colon>'a \<Rightarrow> 'b)"
-  apply (unfold func_zero)
-  apply (rule ext)
-  by (simp split: split_max)
-qed
+apply (subgoal_tac "(\<lambda>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) \<Longrightarrow>
+    \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. k x <= f x \<Longrightarrow>
k <o g =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
@@ -885,33 +777,15 @@
apply (rule allI)
apply (subst fun_diff_def)
apply (erule thin_rl)
-(*sledgehammer*)
-  apply (case_tac "0 <= k x - g x")
-(* apply (metis abs_le_iff add_le_imp_le_right diff_minus le_less
-                le_max_iff_disj min_max.le_supE min_max.sup_absorb2
-                min_max.sup_commute) *)
-proof -
-  fix x :: 'a
-  assume "\<forall>x\<Colon>'a. k x \<le> f x"
-  hence F1: "\<forall>x\<^isub>1\<Colon>'a. max (k x\<^isub>1) (f x\<^isub>1) = f x\<^isub>1" by (metis min_max.sup_absorb2)
-  assume "(0\<Colon>'b) \<le> k x - g x"
-  hence F2: "max (0\<Colon>'b) (k x - g x) = k x - g x" by (metis min_max.sup_absorb2)
-  have F3: "\<forall>x\<^isub>1\<Colon>'b. x\<^isub>1 \<le> \<bar>x\<^isub>1\<bar>" by (metis abs_le_iff le_less)
-  have "\<forall>(x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>2 \<or> max x\<^isub>1 x\<^isub>2 \<le> x\<^isub>1" by (metis le_less le_max_iff_disj)
-  hence "\<forall>(x\<^isub>3\<Colon>'b) (x\<^isub>2\<Colon>'b) x\<^isub>1\<Colon>'b. x\<^isub>1 - x\<^isub>2 \<le> x\<^isub>3 - x\<^isub>2 \<or> x\<^isub>3 \<le> x\<^isub>1" by (metis add_le_imp_le_right diff_minus min_max.le_supE)
-  hence "k x - g x \<le> f x - g x" by (metis F1 le_less min_max.sup_absorb2 min_max.sup_commute)
-  hence "k x - g x \<le> \<bar>f x - g x\<bar>" by (metis F3 le_max_iff_disj min_max.sup_absorb2)
-  thus "max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>" by (metis F2 min_max.sup_commute)
-next
-  show "\<And>x\<Colon>'a.
-       \<lbrakk>\<forall>x\<Colon>'a. (0\<Colon>'b) \<le> k x; \<forall>x\<Colon>'a. k x \<le> f x; \<not> (0\<Colon>'b) \<le> k x - g x\<rbrakk>
-       \<Longrightarrow> max (k x - g x) (0\<Colon>'b) \<le> \<bar>f x - g x\<bar>"
-    by (metis abs_ge_zero le_cases min_max.sup_absorb2)
-qed
+(* sledgehammer *)
+apply (case_tac "0 <= k x - g x")
+ apply (metis (hide_lams, no_types) abs_le_iff add_le_imp_le_right diff_minus le_less
+          le_max_iff_disj min_max.le_supE min_max.sup_absorb2
+          min_max.sup_commute)
+by (metis abs_ge_zero le_cases min_max.sup_absorb2)

-declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3" ]]
-lemma bigo_lesso3: "f =o g +o O(h) ==>
-    ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
+lemma bigo_lesso3: "f =o g +o O(h) \<Longrightarrow>
+    \<forall>x. 0 <= k x \<Longrightarrow> \<forall>x. g x <= k x \<Longrightarrow>
f <o k =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
@@ -920,20 +794,19 @@
apply (rule le_maxI2)
apply (rule allI)
apply (subst fun_diff_def)
-apply (erule thin_rl)
-(*sledgehammer*)
+  apply (erule thin_rl)
+  (* sledgehammer *)
apply (case_tac "0 <= f x - k x")
-  apply (simp)
+  apply simp
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
-using [[ sledgehammer_problem_prefix = "BigO__bigo_lesso3_simpler" ]]
-apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
+  apply (metis diff_less_0_iff_less linorder_not_le not_leE uminus_add_conv_diff xt1(12) xt1(6))
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
done

-lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::{linordered_field,number_ring}) ==>
-    g =o h +o O(k) ==> f <o h =o O(k)"
+lemma bigo_lesso4: "f <o g =o O(k\<Colon>'a=>'b\<Colon>{linordered_field,number_ring}) \<Longrightarrow>
+    g =o h +o O(k) \<Longrightarrow> f <o h =o O(k)"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
apply (drule bigo_abs5) back
@@ -946,9 +819,7 @@
split: split_max abs_split)
done

-declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso5" ]]
-lemma bigo_lesso5: "f <o g =o O(h) ==>
-    EX C. ALL x. f x <= g x + C * abs(h x)"
+lemma bigo_lesso5: "f <o g =o O(h) \<Longrightarrow> \<exists>C. \<forall>x. f x <= g x + C * abs(h x)"
apply (simp only: lesso_def bigo_alt_def)
apply clarsimp
apply (metis abs_if abs_mult add_commute diff_le_eq less_not_permute)```