--- a/src/HOL/Metis_Examples/BigO.thy Wed Sep 01 00:03:15 2010 +0200
+++ b/src/HOL/Metis_Examples/BigO.thy Wed Sep 01 00:07:31 2010 +0200
@@ -15,7 +15,7 @@
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)}"
-declare [[ atp_problem_prefix = "BigO__bigo_pos_const" ]]
+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)))))"
@@ -124,7 +124,7 @@
{h. EX c. (0 < c & (ALL x. abs (h x) <= c * abs (f x)))}"
by (auto simp add: bigo_def bigo_pos_const)
-declare [[ atp_problem_prefix = "BigO__bigo_elt_subset" ]]
+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)
@@ -142,12 +142,12 @@
done
-declare [[ atp_problem_prefix = "BigO__bigo_refl" ]]
+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 [[ atp_problem_prefix = "BigO__bigo_zero" ]]
+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)
@@ -230,7 +230,7 @@
apply (auto del: subsetI simp del: bigo_plus_idemp)
done
-declare [[ atp_problem_prefix = "BigO__bigo_plus_eq" ]]
+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)
@@ -253,13 +253,13 @@
apply (rule abs_triangle_ineq)
apply (metis add_nonneg_nonneg)
apply (rule add_mono)
-using [[ atp_problem_prefix = "BigO__bigo_plus_eq_simpler" ]]
+using [[ sledgehammer_problem_prefix = "BigO__bigo_plus_eq_simpler" ]]
(*Found by SPASS; SLOW*)
apply (metis le_maxI2 linorder_linear linorder_not_le min_max.sup_absorb1 mult_le_cancel_right order_trans)
apply (metis le_maxI2 linorder_not_le mult_le_cancel_right order_trans)
done
-declare [[ atp_problem_prefix = "BigO__bigo_bounded_alt" ]]
+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)
@@ -277,14 +277,14 @@
qed
text{*So here is the easier (and more natural) problem using transitivity*}
-declare [[ atp_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
+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)
text{*So here is the easier (and more natural) problem using transitivity*}
-declare [[ atp_problem_prefix = "BigO__bigo_bounded_alt_trans" ]]
+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 *)
@@ -299,7 +299,7 @@
apply simp
done
-declare [[ atp_problem_prefix = "BigO__bigo_bounded2" ]]
+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)"
apply (rule set_minus_imp_plus)
@@ -314,13 +314,13 @@
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 [[ atp_problem_prefix = "BigO__bigo_abs" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs" ]]
lemma bigo_abs: "(%x. abs(f x)) =o O(f)"
apply (unfold bigo_def)
apply auto
by (metis mult_1 order_refl)
-declare [[ atp_problem_prefix = "BigO__bigo_abs2" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_abs2" ]]
lemma bigo_abs2: "f =o O(%x. abs(f x))"
apply (unfold bigo_def)
apply auto
@@ -383,7 +383,7 @@
by (simp add: bigo_abs3 [symmetric])
qed
-declare [[ atp_problem_prefix = "BigO__bigo_mult" ]]
+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)
@@ -395,7 +395,7 @@
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 [[ atp_problem_prefix = "BigO__bigo_mult_simpler" ]]
+using [[ sledgehammer_problem_prefix = "BigO__bigo_mult_simpler" ]]
prefer 2
apply (metis mult_assoc mult_left_commute
abs_of_pos mult_left_commute
@@ -406,14 +406,14 @@
abs_mult has just been done *)
by (metis abs_ge_zero mult_mono')
-declare [[ atp_problem_prefix = "BigO__bigo_mult2" ]]
+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*);
apply (rule_tac x = c in exI)
apply clarify
apply (drule_tac x = x in spec)
-using [[ atp_problem_prefix = "BigO__bigo_mult2_simpler" ]]
+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)
@@ -421,11 +421,11 @@
apply (rule abs_ge_zero)
done
-declare [[ atp_problem_prefix = "BigO__bigo_mult3" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult3" ]]
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 [[ atp_problem_prefix = "BigO__bigo_mult4" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult4" ]]
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)
@@ -459,13 +459,13 @@
qed
qed
-declare [[ atp_problem_prefix = "BigO__bigo_mult6" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_mult6" ]]
lemma bigo_mult6: "ALL x. f x ~= 0 ==>
O(f * g) = (f::'a => ('b::linordered_field)) *o O(g)"
by (metis bigo_mult2 bigo_mult5 order_antisym)
(*proof requires relaxing relevance: 2007-01-25*)
-declare [[ atp_problem_prefix = "BigO__bigo_mult7" ]]
+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)) \<otimes> O(g)"
@@ -477,7 +477,7 @@
done
declare bigo_mult6 [simp del]
-declare [[ atp_problem_prefix = "BigO__bigo_mult8" ]]
+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)) \<otimes> O(g)"
@@ -528,7 +528,7 @@
qed
qed
-declare [[ atp_problem_prefix = "BigO__bigo_plus_absorb" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_plus_absorb" ]]
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);
@@ -555,7 +555,7 @@
lemma bigo_const1: "(%x. c) : O(%x. 1)"
by (auto simp add: bigo_def mult_ac)
-declare [[ atp_problem_prefix = "BigO__bigo_const2" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const2" ]]
lemma (*bigo_const2 [intro]:*) "O(%x. c) <= O(%x. 1)"
by (metis bigo_const1 bigo_elt_subset);
@@ -566,7 +566,7 @@
show "O(\<lambda>x. c) \<subseteq> O(\<lambda>x. 1)" by (metis F1 bigo_elt_subset)
qed
-declare [[ atp_problem_prefix = "BigO__bigo_const3" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const3" ]]
lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ==> (%x. 1) : O(%x. c)"
apply (simp add: bigo_def)
by (metis abs_eq_0 left_inverse order_refl)
@@ -578,7 +578,7 @@
O(%x. c) = O(%x. 1)"
by (rule equalityI, rule bigo_const2, rule bigo_const4, assumption)
-declare [[ atp_problem_prefix = "BigO__bigo_const_mult1" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult1" ]]
lemma bigo_const_mult1: "(%x. c * f x) : O(f)"
apply (simp add: bigo_def abs_mult)
by (metis le_less)
@@ -586,7 +586,7 @@
lemma bigo_const_mult2: "O(%x. c * f x) <= O(f)"
by (rule bigo_elt_subset, rule bigo_const_mult1)
-declare [[ atp_problem_prefix = "BigO__bigo_const_mult3" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult3" ]]
lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ==> f : O(%x. c * f x)"
apply (simp add: bigo_def)
(*sledgehammer [no luck]*)
@@ -604,7 +604,7 @@
O(%x. c * f x) = O(f)"
by (rule equalityI, rule bigo_const_mult2, erule bigo_const_mult4)
-declare [[ atp_problem_prefix = "BigO__bigo_const_mult5" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult5" ]]
lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ==>
(%x. c) *o O(f) = O(f)"
apply (auto del: subsetI)
@@ -624,7 +624,7 @@
done
-declare [[ atp_problem_prefix = "BigO__bigo_const_mult6" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_const_mult6" ]]
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
@@ -681,7 +681,7 @@
apply (blast intro: order_trans mult_right_mono abs_ge_self)
done
-declare [[ atp_problem_prefix = "BigO__bigo_setsum1" ]]
+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)"
@@ -698,7 +698,7 @@
(%x. SUM y : A x. f y) =o O(%x. SUM y : A x. h y)"
by (rule bigo_setsum1, auto)
-declare [[ atp_problem_prefix = "BigO__bigo_setsum3" ]]
+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)))"
@@ -729,7 +729,7 @@
apply (erule set_plus_imp_minus)
done
-declare [[ atp_problem_prefix = "BigO__bigo_setsum5" ]]
+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
@@ -786,7 +786,7 @@
apply (simp add: func_times)
done
-declare [[ atp_problem_prefix = "BigO__bigo_fix" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_fix" ]]
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)
@@ -849,7 +849,7 @@
apply (erule spec)+
done
-declare [[ atp_problem_prefix = "BigO__bigo_lesso1" ]]
+declare [[ sledgehammer_problem_prefix = "BigO__bigo_lesso1" ]]
lemma bigo_lesso1: "ALL x. f x <= g x ==> f <o g =o O(h)"
apply (unfold lesso_def)
apply (subgoal_tac "(%x. max (f x - g x) 0) = 0")
@@ -863,7 +863,7 @@
by (simp split: split_max)
qed
-declare [[ atp_problem_prefix = "BigO__bigo_lesso2" ]]
+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 ==>
k <o g =o O(h)"
@@ -899,7 +899,7 @@
by (metis abs_ge_zero le_cases min_max.sup_absorb2)
qed
-declare [[ atp_problem_prefix = "BigO__bigo_lesso3" ]]
+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 ==>
f <o k =o O(h)"
@@ -916,7 +916,7 @@
apply (simp)
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
-using [[ atp_problem_prefix = "BigO__bigo_lesso3_simpler" ]]
+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 add_minus_cancel diff_le_eq le_diff_eq uminus_add_conv_diff)
apply (metis abs_ge_zero linorder_linear min_max.sup_absorb1 min_max.sup_commute)
@@ -936,7 +936,7 @@
split: split_max abs_split)
done
-declare [[ atp_problem_prefix = "BigO__bigo_lesso5" ]]
+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)"
apply (simp only: lesso_def bigo_alt_def)