--- a/src/HOL/Library/BigO.thy Fri Apr 13 21:26:34 2007 +0200
+++ b/src/HOL/Library/BigO.thy Fri Apr 13 21:26:35 2007 +0200
@@ -59,11 +59,11 @@
apply (rule mult_right_mono)
apply (rule abs_ge_self)
apply (rule abs_ge_zero)
-done
+ done
lemma bigo_alt_def: "O(f) =
{h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
-by (auto simp add: bigo_def bigo_pos_const)
+ by (auto simp add: bigo_def bigo_pos_const)
lemma bigo_elt_subset [intro]: "f : O(g) ==> O(f) <= O(g)"
apply (auto simp add: bigo_alt_def)
@@ -78,25 +78,25 @@
apply (simp add: mult_ac)
apply (rule mult_left_mono, assumption)
apply (rule order_less_imp_le, assumption)
-done
+ done
lemma bigo_refl [intro]: "f : O(f)"
apply(auto simp add: bigo_def)
apply(rule_tac x = 1 in exI)
apply simp
-done
+ done
lemma bigo_zero: "0 : O(g)"
apply (auto simp add: bigo_def func_zero)
apply (rule_tac x = 0 in exI)
apply auto
-done
+ done
lemma bigo_zero2: "O(%x.0) = {%x.0}"
apply (auto simp add: bigo_def)
apply (rule ext)
apply auto
-done
+ done
lemma bigo_plus_self_subset [intro]:
"O(f) + O(f) <= O(f)"
@@ -116,7 +116,7 @@
apply (rule bigo_plus_self_subset)
apply (rule set_zero_plus2)
apply (rule bigo_zero)
-done
+ done
lemma bigo_plus_subset [intro]: "O(f + g) <= O(f) + O(g)"
apply (rule subsetI)
@@ -168,17 +168,17 @@
apply simp
apply (rule ext)
apply (auto simp add: if_splits linorder_not_le)
-done
+ done
lemma bigo_plus_subset2 [intro]: "A <= O(f) ==> B <= O(f) ==> A + B <= O(f)"
apply (subgoal_tac "A + B <= O(f) + O(f)")
apply (erule order_trans)
apply simp
apply (auto del: subsetI simp del: bigo_plus_idemp)
-done
+ done
lemma bigo_plus_eq: "ALL x. 0 <= f x ==> ALL x. 0 <= g x ==>
- O(f + g) = O(f) + O(g)"
+ O(f + g) = O(f) + O(g)"
apply (rule equalityI)
apply (rule bigo_plus_subset)
apply (simp add: bigo_alt_def set_plus func_plus)
@@ -211,7 +211,7 @@
apply (rule abs_triangle_ineq)
apply (rule add_nonneg_nonneg)
apply assumption+
-done
+ done
lemma bigo_bounded_alt: "ALL x. 0 <= f x ==> ALL x. f x <= c * g x ==>
f : O(g)"
@@ -220,13 +220,13 @@
apply auto
apply (drule_tac x = x in spec)+
apply (simp add: abs_mult [symmetric])
-done
+ done
lemma bigo_bounded: "ALL x. 0 <= f x ==> ALL x. f x <= g x ==>
f : O(g)"
apply (erule bigo_bounded_alt [of f 1 g])
apply simp
-done
+ done
lemma bigo_bounded2: "ALL x. lb x <= f x ==> ALL x. f x <= lb x + g x ==>
f : lb +o O(g)"
@@ -237,21 +237,21 @@
apply force
apply (drule_tac x = x in spec)+
apply force
-done
+ done
lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
apply (unfold bigo_def)
apply auto
apply (rule_tac x = 1 in exI)
apply auto
-done
+ done
lemma bigo_abs2: "f =o O(%x. abs(f x))"
apply (unfold bigo_def)
apply auto
apply (rule_tac x = 1 in exI)
apply auto
-done
+ done
lemma bigo_abs3: "O(f) = O(%x. abs(f x))"
apply (rule equalityI)
@@ -259,7 +259,7 @@
apply (rule bigo_abs2)
apply (rule bigo_elt_subset)
apply (rule bigo_abs)
-done
+ done
lemma bigo_abs4: "f =o g +o O(h) ==>
(%x. abs (f x)) =o (%x. abs (g x)) +o O(h)"
@@ -288,7 +288,7 @@
qed
lemma bigo_abs5: "f =o O(g) ==> (%x. abs(f x)) =o O(g)"
-by (unfold bigo_def, auto)
+ by (unfold bigo_def, auto)
lemma bigo_elt_subset2 [intro]: "f : g +o O(h) ==> O(f) <= O(g) + O(h)"
proof -
@@ -326,7 +326,7 @@
apply (rule mult_nonneg_nonneg)
apply auto
apply (simp add: mult_ac abs_mult)
-done
+ done
lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
apply (auto simp add: bigo_def elt_set_times_def func_times abs_mult)
@@ -337,20 +337,20 @@
apply (force simp add: mult_ac)
apply (rule mult_left_mono, assumption)
apply (rule abs_ge_zero)
-done
+ done
lemma bigo_mult3: "f : O(h) ==> g : O(j) ==> f * g : O(h * j)"
apply (rule subsetD)
apply (rule bigo_mult)
apply (erule set_times_intro, assumption)
-done
+ done
lemma bigo_mult4 [intro]:"f : k +o O(h) ==> g * f : (g * k) +o O(g * h)"
apply (drule set_plus_imp_minus)
apply (rule set_minus_imp_plus)
apply (drule bigo_mult3 [where g = g and j = g])
apply (auto simp add: ring_eq_simps mult_ac)
-done
+ done
lemma bigo_mult5: "ALL x. f x ~= 0 ==>
O(f * g) <= (f::'a => ('b::ordered_field)) *o O(g)"
@@ -386,7 +386,7 @@
apply (rule equalityI)
apply (erule bigo_mult5)
apply (rule bigo_mult2)
-done
+ done
lemma bigo_mult7: "ALL x. f x ~= 0 ==>
O(f * g) <= O(f::'a => ('b::ordered_field)) * O(g)"
@@ -394,14 +394,14 @@
apply assumption
apply (rule set_times_mono3)
apply (rule bigo_refl)
-done
+ done
lemma bigo_mult8: "ALL x. f x ~= 0 ==>
O(f * g) = O(f::'a => ('b::ordered_field)) * O(g)"
apply (rule equalityI)
apply (erule bigo_mult7)
apply (rule bigo_mult)
-done
+ done
lemma bigo_minus [intro]: "f : O(g) ==> - f : O(g)"
by (auto simp add: bigo_def func_minus)
@@ -411,7 +411,7 @@
apply (drule set_plus_imp_minus)
apply (drule bigo_minus)
apply (simp add: diff_minus)
-done
+ done
lemma bigo_minus3: "O(-f) = O(f)"
by (auto simp add: bigo_def func_minus abs_minus_cancel)
@@ -452,12 +452,12 @@
apply (rule equalityI)
apply (erule bigo_plus_absorb_lemma1)
apply (erule bigo_plus_absorb_lemma2)
-done
+ done
lemma bigo_plus_absorb2 [intro]: "f : O(g) ==> A <= O(g) ==> f +o A <= O(g)"
apply (subgoal_tac "f +o A <= f +o O(g)")
apply force+
-done
+ done
lemma bigo_add_commute_imp: "f : g +o O(h) ==> g : f +o O(h)"
apply (subst set_minus_plus [symmetric])
@@ -467,56 +467,56 @@
apply (subst set_minus_plus)
apply assumption
apply (simp add: diff_minus add_ac)
-done
+ done
lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"
apply (rule iffI)
apply (erule bigo_add_commute_imp)+
-done
+ done
lemma bigo_const1: "(%x. c) : O(%x. 1)"
-by (auto simp add: bigo_def mult_ac)
+ by (auto simp add: bigo_def mult_ac)
lemma bigo_const2 [intro]: "O(%x. c) <= O(%x. 1)"
apply (rule bigo_elt_subset)
apply (rule bigo_const1)
-done
+ done
lemma bigo_const3: "(c::'a::ordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
apply (simp add: bigo_def)
apply (rule_tac x = "abs(inverse c)" in exI)
apply (simp add: abs_mult [symmetric])
-done
+ done
lemma bigo_const4: "(c::'a::ordered_field) ~= 0 ==> O(%x. 1) <= O(%x. c)"
-by (rule bigo_elt_subset, rule bigo_const3, assumption)
+ by (rule bigo_elt_subset, rule bigo_const3, assumption)
lemma bigo_const [simp]: "(c::'a::ordered_field) ~= 0 ==>
O(%x. c) = O(%x. 1)"
-by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
+ by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
apply (simp add: bigo_def)
apply (rule_tac x = "abs(c)" in exI)
apply (auto simp add: abs_mult [symmetric])
-done
+ done
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
-by (rule bigo_elt_subset, rule bigo_const_mult1)
+ by (rule bigo_elt_subset, rule bigo_const_mult1)
lemma bigo_const_mult3: "(c::'a::ordered_field) ~= 0 ==> f : O(%x. c * f x)"
apply (simp add: bigo_def)
apply (rule_tac x = "abs(inverse c)" in exI)
apply (simp add: abs_mult [symmetric] mult_assoc [symmetric])
-done
+ done
lemma bigo_const_mult4: "(c::'a::ordered_field) ~= 0 ==>
O(f) <= O(%x. c * f x)"
-by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
+ by (rule bigo_elt_subset, rule bigo_const_mult3, assumption)
lemma bigo_const_mult [simp]: "(c::'a::ordered_field) ~= 0 ==>
O(%x. c * f x) = O(f)"
-by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
+ by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
lemma bigo_const_mult5 [simp]: "(c::'a::ordered_field) ~= 0 ==>
(%x. c) *o O(f) = O(f)"
@@ -533,7 +533,7 @@
apply (rule mult_left_mono)
apply (erule spec)
apply force
-done
+ done
lemma bigo_const_mult6 [intro]: "(%x. c) *o O(f) <= O(f)"
apply (auto intro!: subsetI
@@ -547,7 +547,7 @@
apply (erule spec)
apply simp
apply(simp add: mult_ac)
-done
+ done
lemma bigo_const_mult7 [intro]: "f =o O(g) ==> (%x. c * f x) =o O(g)"
proof -
@@ -571,6 +571,7 @@
apply (erule bigo_compose1)
done
+
subsection {* Setsum *}
lemma bigo_setsum_main: "ALL x. ALL y : A x. 0 <= h x y ==>
@@ -595,7 +596,7 @@
apply (rule mult_right_mono)
apply (rule abs_ge_self)
apply force
-done
+ done
lemma bigo_setsum1: "ALL x y. 0 <= h x y ==>
EX c. ALL x y. abs(f x y) <= c * (h x y) ==>
@@ -605,12 +606,12 @@
apply clarsimp
apply (rule_tac x = c in exI)
apply force
-done
+ 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)
+ 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
@@ -627,7 +628,7 @@
apply (rule mult_left_mono)
apply (erule spec)
apply (rule abs_ge_zero)
-done
+ done
lemma bigo_setsum4: "f =o g +o O(h) ==>
(%x. SUM y : A x. l x y * f(k x y)) =o
@@ -640,7 +641,7 @@
apply (rule bigo_setsum3)
apply (subst func_diff [symmetric])
apply (erule set_plus_imp_minus)
-done
+ done
lemma bigo_setsum5: "f =o O(h) ==> ALL x y. 0 <= l x y ==>
ALL x. 0 <= h x ==>
@@ -655,7 +656,7 @@
apply (subst abs_of_nonneg)
apply (rule mult_nonneg_nonneg)
apply auto
-done
+ done
lemma bigo_setsum6: "f =o g +o O(h) ==> ALL x y. 0 <= l x y ==>
ALL x. 0 <= h x ==>
@@ -670,7 +671,8 @@
apply (subst func_diff [symmetric])
apply (drule set_plus_imp_minus)
apply auto
-done
+ done
+
subsection {* Misc useful stuff *}
@@ -679,13 +681,13 @@
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_mono2)
apply assumption+
-done
+ done
lemma bigo_useful_add: "f =o O(h) ==> g =o O(h) ==> f + g =o O(h)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_intro)
apply assumption+
-done
+ done
lemma bigo_useful_const_mult: "(c::'a::ordered_field) ~= 0 ==>
(%x. c) * f =o O(h) ==> f =o O(h)"
@@ -701,7 +703,7 @@
apply (subst times_divide_eq_left [symmetric])
apply (subst divide_self)
apply (assumption, simp)
-done
+ done
lemma bigo_fix: "(%x. f ((x::nat) + 1)) =o O(%x. h(x + 1)) ==> f 0 = 0 ==>
f =o O(h)"
@@ -718,7 +720,7 @@
apply (erule ssubst) back
apply (erule spec)
apply simp
-done
+ done
lemma bigo_fix2:
"(%x. f ((x::nat) + 1)) =o (%x. g(x + 1)) +o O(%x. h(x + 1)) ==>
@@ -730,7 +732,8 @@
apply (rule set_plus_imp_minus)
apply simp
apply (simp add: func_diff)
-done
+ done
+
subsection {* Less than or equal to *}
@@ -747,7 +750,7 @@
apply (rule allI)
apply (rule order_trans)
apply (erule spec)+
-done
+ done
lemma bigo_lesseq2: "f =o O(h) ==> ALL x. abs (g x) <= f x ==>
g =o O(h)"
@@ -757,15 +760,15 @@
apply (rule order_trans)
apply assumption
apply (rule abs_ge_self)
-done
+ done
lemma bigo_lesseq3: "f =o O(h) ==> ALL x. 0 <= g x ==> ALL x. g x <= f x ==>
- g =o O(h)"
+ g =o O(h)"
apply (erule bigo_lesseq2)
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
-done
+ done
lemma bigo_lesseq4: "f =o O(h) ==>
ALL x. 0 <= g x ==> ALL x. g x <= abs (f x) ==>
@@ -774,7 +777,7 @@
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
-done
+ done
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
apply (unfold lesso_def)
@@ -784,7 +787,7 @@
apply (unfold func_zero)
apply (rule ext)
apply (simp split: split_max)
-done
+ done
lemma bigo_lesso2: "f =o g +o O(h) ==>
ALL x. 0 <= k x ==> ALL x. k x <= f x ==>
@@ -808,7 +811,7 @@
prefer 2
apply (rule abs_ge_zero)
apply (simp add: compare_rls)
-done
+ done
lemma bigo_lesso3: "f =o g +o O(h) ==>
ALL x. 0 <= k x ==> ALL x. g x <= k x ==>
@@ -833,7 +836,7 @@
prefer 2
apply (rule abs_ge_zero)
apply (simp add: compare_rls)
-done
+ done
lemma bigo_lesso4: "f <o g =o O(k::'a=>'b::ordered_field) ==>
g =o h +o O(k) ==> f <o h =o O(k)"
@@ -847,7 +850,7 @@
apply (rule allI)
apply (auto simp add: func_plus func_diff compare_rls
split: split_max abs_split)
-done
+ done
lemma bigo_lesso5: "f <o g =o O(h) ==>
EX C. ALL x. f x <= g x + C * abs(h x)"
@@ -860,7 +863,7 @@
apply (clarsimp simp add: compare_rls add_ac)
apply (rule abs_of_nonneg)
apply (rule le_maxI2)
-done
+ done
lemma lesso_add: "f <o g =o O(h) ==>
k <o l =o O(h) ==> (f + k) <o (g + l) =o O(h)"
@@ -870,6 +873,6 @@
apply assumption
apply (force split: split_max)
apply (auto split: split_max simp add: func_plus)
-done
+ done
end