# Theory Big_O

theory Big_O
imports Dense_Linear_Order Function_Algebras Set_Algebras
```(*  Title:      HOL/Metis_Examples/Big_O.thy
Author:     Lawrence C. Paulson, Cambridge University Computer Laboratory
Author:     Jasmin Blanchette, TU Muenchen

Metis example featuring the Big O notation.
*)

section ‹Metis Example Featuring the Big O Notation›

theory Big_O
imports
"HOL-Decision_Procs.Dense_Linear_Order"
"HOL-Library.Function_Algebras"
"HOL-Library.Set_Algebras"
begin

subsection ‹Definitions›

definition bigo :: "('a => 'b::linordered_idom) => ('a => 'b) set" ("(1O'(_'))") where
"O(f::('a => 'b)) == {h. ∃c. ∀x. ¦h x¦ <= c * ¦f x¦}"

lemma bigo_pos_const:
"(∃c::'a::linordered_idom.
∀x. ¦h x¦ ≤ c * ¦f x¦)
⟷ (∃c. 0 < c ∧ (∀x. ¦h x¦ ≤ c * ¦f x¦))"
by (metis (no_types) abs_ge_zero
algebra_simps mult.comm_neutral
mult_nonpos_nonneg not_le_imp_less order_trans zero_less_one)

(*** Now various verions with an increasing shrink factor ***)

sledgehammer_params [isar_proofs, compress = 1]

lemma
"(∃c::'a::linordered_idom.
∀x. ¦h x¦ ≤ c * ¦f x¦)
⟷ (∃c. 0 < c ∧ (∀x. ¦h x¦ ≤ c * ¦f x¦))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "¦c¦" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "∀x. ¦h x¦ ≤ c * ¦f x¦"
have F1: "∀x⇩1::'a::linordered_idom. 0 ≤ ¦x⇩1¦" by (metis abs_ge_zero)
have F2: "∀x⇩1::'a::linordered_idom. 1 * x⇩1 = x⇩1" by (metis mult_1)
have F3: "∀x⇩1 x⇩3. x⇩3 ≤ ¦h x⇩1¦ ⟶ x⇩3 ≤ c * ¦f x⇩1¦" by (metis A1 order_trans)
have F4: "∀x⇩2 x⇩3::'a::linordered_idom. ¦x⇩3¦ * ¦x⇩2¦ = ¦x⇩3 * x⇩2¦"
by (metis abs_mult)
have F5: "∀x⇩3 x⇩1::'a::linordered_idom. 0 ≤ x⇩1 ⟶ ¦x⇩3 * x⇩1¦ = ¦x⇩3¦ * x⇩1"
by (metis abs_mult_pos)
hence "∀x⇩1≥0. ¦x⇩1::'a::linordered_idom¦ = ¦1¦ * x⇩1" by (metis F2)
hence "∀x⇩1≥0. ¦x⇩1::'a::linordered_idom¦ = x⇩1" by (metis F2 abs_one)
hence "∀x⇩3. 0 ≤ ¦h x⇩3¦ ⟶ ¦c * ¦f x⇩3¦¦ = c * ¦f x⇩3¦" by (metis F3)
hence "∀x⇩3. ¦c * ¦f x⇩3¦¦ = c * ¦f x⇩3¦" by (metis F1)
hence "∀x⇩3. (0::'a) ≤ ¦f x⇩3¦ ⟶ c * ¦f x⇩3¦ = ¦c¦ * ¦f x⇩3¦" by (metis F5)
hence "∀x⇩3. (0::'a) ≤ ¦f x⇩3¦ ⟶ c * ¦f x⇩3¦ = ¦c * f x⇩3¦" by (metis F4)
hence "∀x⇩3. c * ¦f x⇩3¦ = ¦c * f x⇩3¦" by (metis F1)
hence "¦h x¦ ≤ ¦c * f x¦" by (metis A1)
thus "¦h x¦ ≤ ¦c¦ * ¦f x¦" by (metis F4)
qed

sledgehammer_params [isar_proofs, compress = 2]

lemma
"(∃c::'a::linordered_idom.
∀x. ¦h x¦ ≤ c * ¦f x¦)
⟷ (∃c. 0 < c ∧ (∀x. ¦h x¦ ≤ c * ¦f x¦))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "¦c¦" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "∀x. ¦h x¦ ≤ c * ¦f x¦"
have F1: "∀x⇩1::'a::linordered_idom. 1 * x⇩1 = x⇩1" by (metis mult_1)
have F2: "∀x⇩2 x⇩3::'a::linordered_idom. ¦x⇩3¦ * ¦x⇩2¦ = ¦x⇩3 * x⇩2¦"
by (metis abs_mult)
have "∀x⇩1≥0. ¦x⇩1::'a::linordered_idom¦ = x⇩1" by (metis F1 abs_mult_pos abs_one)
hence "∀x⇩3. ¦c * ¦f x⇩3¦¦ = c * ¦f x⇩3¦" by (metis A1 abs_ge_zero order_trans)
hence "∀x⇩3. 0 ≤ ¦f x⇩3¦ ⟶ c * ¦f x⇩3¦ = ¦c * f x⇩3¦" by (metis F2 abs_mult_pos)
hence "¦h x¦ ≤ ¦c * f x¦" by (metis A1 abs_ge_zero)
thus "¦h x¦ ≤ ¦c¦ * ¦f x¦" by (metis F2)
qed

sledgehammer_params [isar_proofs, compress = 3]

lemma
"(∃c::'a::linordered_idom.
∀x. ¦h x¦ ≤ c * ¦f x¦)
⟷ (∃c. 0 < c ∧ (∀x. ¦h x¦ ≤ c * ¦f x¦))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "¦c¦" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "∀x. ¦h x¦ ≤ c * ¦f x¦"
have F1: "∀x⇩1::'a::linordered_idom. 1 * x⇩1 = x⇩1" by (metis mult_1)
have F2: "∀x⇩3 x⇩1::'a::linordered_idom. 0 ≤ x⇩1 ⟶ ¦x⇩3 * x⇩1¦ = ¦x⇩3¦ * x⇩1" by (metis abs_mult_pos)
hence "∀x⇩1≥0. ¦x⇩1::'a::linordered_idom¦ = x⇩1" by (metis F1 abs_one)
hence "∀x⇩3. 0 ≤ ¦f x⇩3¦ ⟶ c * ¦f x⇩3¦ = ¦c¦ * ¦f x⇩3¦" by (metis F2 A1 abs_ge_zero order_trans)
thus "¦h x¦ ≤ ¦c¦ * ¦f x¦" by (metis A1 abs_ge_zero)
qed

sledgehammer_params [isar_proofs, compress = 4]

lemma
"(∃c::'a::linordered_idom.
∀x. ¦h x¦ ≤ c * ¦f x¦)
⟷ (∃c. 0 < c ∧ (∀x. ¦h x¦ ≤ c * ¦f x¦))"
apply auto
apply (case_tac "c = 0", simp)
apply (rule_tac x = "1" in exI, simp)
apply (rule_tac x = "¦c¦" in exI, auto)
proof -
fix c :: 'a and x :: 'b
assume A1: "∀x. ¦h x¦ ≤ c * ¦f x¦"
have "∀x⇩1::'a::linordered_idom. 1 * x⇩1 = x⇩1" by (metis mult_1)
hence "∀x⇩3. ¦c * ¦f x⇩3¦¦ = c * ¦f x⇩3¦"
by (metis A1 abs_ge_zero order_trans abs_mult_pos abs_one)
hence "¦h x¦ ≤ ¦c * f x¦" by (metis A1 abs_ge_zero abs_mult_pos abs_mult)
thus "¦h x¦ ≤ ¦c¦ * ¦f x¦" by (metis abs_mult)
qed

sledgehammer_params [isar_proofs, compress = 1]

lemma bigo_alt_def: "O(f) = {h. ∃c. (0 < c ∧ (∀x. ¦h x¦ <= c * ¦f x¦))}"
by (auto simp add: bigo_def bigo_pos_const)

lemma bigo_elt_subset [intro]: "f : O(g) ⟹ O(f) ≤ O(g)"
apply (rule_tac x = "ca * c" in exI)
apply (metis algebra_simps mult_le_cancel_left_pos order_trans mult_pos_pos)
done

lemma bigo_refl [intro]: "f : O(f)"
unfolding bigo_def mem_Collect_eq
by (metis mult_1 order_refl)

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(λx. 0) = {λx. 0}"

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

lemma bigo_plus_idemp [simp]: "O(f) + O(f) = O(f)"
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 = "λn. if ¦g n¦ <= ¦f n¦ then x n else 0" in exI)
apply (rule conjI)
apply (rule_tac x = "c + c" in exI)
apply clarsimp
apply (subgoal_tac "c * ¦f xa + g xa¦ <= (c + c) * ¦f xa¦")
apply (metis mult_2 order_trans)
apply (subgoal_tac "c * ¦f xa + g xa¦ <= c * (¦f xa¦ + ¦g xa¦)")
apply (erule order_trans)
apply (rule mult_left_mono)
apply (rule_tac x = "λn. if ¦f n¦ < ¦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 * ¦f xa + g xa¦ <= (c + c) * ¦g xa¦")
apply (metis order_trans mult_2)
apply (subgoal_tac "c * ¦f xa + g xa¦ <= c * (¦f xa¦ + ¦g xa¦)")
apply (erule order_trans)
by (metis abs_triangle_ineq mult_le_cancel_left_pos)

lemma bigo_plus_subset2 [intro]: "A <= O(f) ⟹ B <= O(f) ⟹ A + B <= O(f)"
by (metis bigo_plus_idemp set_plus_mono2)

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 (subgoal_tac "¦a xa + b xa¦ <= ¦a xa¦ + ¦b xa¦")
apply (subgoal_tac "¦a xa¦ + ¦b xa¦ <= max c ca * f xa + max c ca * g xa")
apply (metis order_trans)
defer 1
apply (metis abs_triangle_ineq)
apply (metis max.cobounded2 linorder_linear max.absorb1 mult_right_mono xt1(6))
by (metis max.cobounded2 linorder_not_le mult_le_cancel_right order_trans)

lemma bigo_bounded_alt: "∀x. 0 <= f x ⟹ ∀x. f x <= c * g x ⟹ f : O(g)"
(* Version 1: one-line proof *)
by (metis abs_le_D1 linorder_class.not_less order_less_le Orderings.xt1(12) abs_mult)

lemma "∀x. 0 <= f x ⟹ ∀x. f x <= c * g x ⟹ f : O(g)"
(* 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: "∀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)

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)
algebra_simps)
algebra_simps)

lemma bigo_abs: "(λx. ¦f x¦) =o O(f)"
apply (unfold bigo_def)
apply auto
by (metis mult_1 order_refl)

lemma bigo_abs2: "f =o O(λx. ¦f x¦)"
apply (unfold bigo_def)
apply auto
by (metis mult_1 order_refl)

lemma bigo_abs3: "O(f) = O(λx. ¦f x¦)"
proof -
have F1: "∀v u. u ∈ O(v) ⟶ O(u) ⊆ O(v)" by (metis bigo_elt_subset)
have F2: "∀u. (λR. ¦u R¦) ∈ O(u)" by (metis bigo_abs)
have "∀u. u ∈ O(λR. ¦u R¦)" by (metis bigo_abs2)
thus "O(f) = O(λx. ¦f x¦)" using F1 F2 by auto
qed

lemma bigo_abs4: "f =o g +o O(h) ⟹ (λx. ¦f x¦) =o (λx. ¦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 "(λx. ¦f x¦ - ¦g x¦) =o O(λx. ¦¦f x¦ - ¦g x¦¦)"
by (rule bigo_abs2)
also have "... <= O(λx. ¦f x - g x¦)"
apply (rule bigo_elt_subset)
apply (rule bigo_bounded)
apply (metis abs_ge_zero)
by (metis abs_triangle_ineq3)
also have "... <= O(f - g)"
apply (rule bigo_elt_subset)
apply (subst fun_diff_def)
apply (rule bigo_abs)
done
also have "... <= O(h)"
using a by (rule bigo_elt_subset)
finally show "(λx. ¦f x¦ - ¦g x¦) : O(h)" .
qed

lemma bigo_abs5: "f =o O(g) ⟹ (λx. ¦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)"
proof -
assume "f : g +o O(h)"
also have "... <= O(g) + O(h)"
by (auto del: subsetI)
also have "... = O(λx. ¦g x¦) + O(λx. ¦h x¦)"
by (metis bigo_abs3)
also have "... = O((λx. ¦g x¦) + (λx. ¦h x¦))"
by (rule bigo_plus_eq [symmetric], auto)
finally have "f : ...".
then have "O(f) <= ..."
by (elim bigo_elt_subset)
also have "... = O(λx. ¦g x¦) + O(λx. ¦h x¦)"
by (rule bigo_plus_eq, auto)
finally show ?thesis
qed

lemma bigo_mult [intro]: "O(f) * O(g) <= O(f * g)"
apply (rule subsetI)
apply (subst bigo_def)
apply (auto simp del: abs_mult ac_simps
(* sledgehammer *)
apply (rule_tac x = "c * ca" in exI)
apply (rule allI)
apply (erule_tac x = x in allE)+
apply (subgoal_tac "c * ca * ¦f x * g x¦ = (c * ¦f x¦) * (ca * ¦g x¦)")
apply (metis (no_types) abs_ge_zero abs_mult mult_mono')
by (metis mult.assoc mult.left_commute abs_of_pos mult.left_commute abs_mult)

lemma bigo_mult2 [intro]: "f *o O(g) <= O(f * g)"
by (metis bigo_mult bigo_refl set_times_mono3 subset_trans)

lemma bigo_mult3: "f : O(h) ⟹ g : O(j) ⟹ f * g : O(h * j)"
by (metis bigo_mult set_rev_mp set_times_intro)

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: "∀x. f x ~= 0 ⟹
O(f * g) <= (f::'a => ('b::linordered_field)) *o O(g)"
proof -
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 "(λx. 1 / (f x)) * h : (λx. 1 / f x) *o O(f * g)"
by auto
also have "... <= O((λx. 1 / f x) * (f * g))"
by (rule bigo_mult2)
also have "(λx. 1 / f x) * (f * g) = g"
finally have "(λx. (1::'b) / f x) * h : O(g)".
then have "f * ((λx. (1::'b) / f x) * h) : f *o O(g)"
by auto
also have "f * ((λx. (1::'b) / f x) * h) = h"
by (simp add: func_times fun_eq_iff a)
finally show "h : f *o O(g)".
qed
qed

lemma bigo_mult6:
"∀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 bigo_mult6 [simp]

lemma bigo_mult7:
"∀x. f x ≠ 0 ⟹ O(f * g) ≤ O(f::'a ⇒ ('b::linordered_field)) * O(g)"
by (metis bigo_refl bigo_mult6 set_times_mono3)

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)) * O(g)"
by (metis bigo_mult bigo_mult7 order_antisym_conv)

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)"
minus_minus set_minus_imp_plus set_plus_imp_minus)

lemma bigo_minus3: "O(-f) = O(f)"
by (metis bigo_elt_subset bigo_minus bigo_refl equalityI minus_minus)

lemma bigo_plus_absorb_lemma1: "f : O(g) ⟹ f +o O(g) ≤ O(g)"
by (metis bigo_plus_idemp set_plus_mono3)

lemma bigo_plus_absorb_lemma2: "f : O(g) ⟹ O(g) ≤ f +o O(g)"
by (metis (no_types) bigo_minus bigo_plus_absorb_lemma1 right_minus
set_plus_mono set_plus_rearrange2 set_zero_plus subsetD subset_refl
subset_trans)

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)"
by (metis bigo_plus_absorb set_plus_mono)

lemma bigo_add_commute_imp: "f : g +o O(h) ⟹ g : f +o O(h)"
by (metis bigo_minus minus_diff_eq set_plus_imp_minus set_minus_plus)

lemma bigo_add_commute: "(f : g +o O(h)) = (g : f +o O(h))"

lemma bigo_const1: "(λx. c) : O(λx. 1)"
by (auto simp add: bigo_def ac_simps)

lemma bigo_const2 [intro]: "O(λx. c) ≤ O(λx. 1)"
by (metis bigo_const1 bigo_elt_subset)

lemma bigo_const3: "(c::'a::linordered_field) ~= 0 ⟹ (λx. 1) : O(λx. c)"
by (metis abs_eq_0 left_inverse order_refl)

lemma bigo_const4: "(c::'a::linordered_field) ~= 0 ⟹ O(λx. 1) <= O(λx. c)"
by (metis bigo_elt_subset bigo_const3)

lemma bigo_const [simp]: "(c::'a::linordered_field) ~= 0 ⟹
O(λx. c) = O(λx. 1)"
by (metis bigo_const2 bigo_const4 equalityI)

lemma bigo_const_mult1: "(λx. c * f x) : O(f)"
by (metis le_less)

lemma bigo_const_mult2: "O(λx. c * f x) ≤ O(f)"
by (rule bigo_elt_subset, rule bigo_const_mult1)

lemma bigo_const_mult3: "(c::'a::linordered_field) ~= 0 ⟹ f : O(λx. c * f x)"
by (metis (no_types) abs_mult mult.assoc mult_1 order_refl left_inverse)

lemma bigo_const_mult4:
"(c::'a::linordered_field) ≠ 0 ⟹ O(f) ≤ O(λx. c * f x)"
by (metis bigo_elt_subset bigo_const_mult3)

lemma bigo_const_mult [simp]: "(c::'a::linordered_field) ~= 0 ⟹
O(λx. c * f x) = O(f)"
by (metis equalityI bigo_const_mult2 bigo_const_mult4)

lemma bigo_const_mult5 [simp]: "(c::'a::linordered_field) ~= 0 ⟹
(λx. 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 = "λy. inverse c * x y" in exI)
apply (rename_tac g d)
apply safe
apply (rule_tac [2] ext)
prefer 2
apply simp
apply (simp add: mult.assoc [symmetric] abs_mult)
(* couldn't get this proof without the step above *)
proof -
fix g :: "'b ⇒ 'a" and d :: 'a
assume A1: "c ≠ (0::'a)"
assume A2: "∀x::'b. ¦g x¦ ≤ d * ¦f x¦"
have F1: "inverse ¦c¦ = ¦inverse c¦" using A1 by (metis nonzero_abs_inverse)
have F2: "(0::'a) < ¦c¦" using A1 by (metis zero_less_abs_iff)
have "(0::'a) < ¦c¦ ⟶ (0::'a) < ¦inverse c¦" using F1 by (metis positive_imp_inverse_positive)
hence "(0::'a) < ¦inverse c¦" using F2 by metis
hence F3: "(0::'a) ≤ ¦inverse c¦" by (metis order_le_less)
have "∃(u::'a) SKF⇩7::'a ⇒ 'b. ¦g (SKF⇩7 (¦inverse c¦ * u))¦ ≤ u * ¦f (SKF⇩7 (¦inverse c¦ * u))¦"
using A2 by metis
hence F4: "∃(u::'a) SKF⇩7::'a ⇒ 'b. ¦g (SKF⇩7 (¦inverse c¦ * u))¦ ≤ u * ¦f (SKF⇩7 (¦inverse c¦ * u))¦ ∧ (0::'a) ≤ ¦inverse c¦"
using F3 by metis
hence "∃(v::'a) (u::'a) SKF⇩7::'a ⇒ 'b. ¦inverse c¦ * ¦g (SKF⇩7 (u * v))¦ ≤ u * (v * ¦f (SKF⇩7 (u * v))¦)"
by (metis mult_left_mono)
then show "∃ca::'a. ∀x::'b. inverse ¦c¦ * ¦g x¦ ≤ ca * ¦f x¦"
using A2 F4 by (metis F1 ‹0 < ¦inverse c¦› linordered_field_class.sign_simps(23) mult_le_cancel_left_pos)
qed

lemma bigo_const_mult6 [intro]: "(λx. c) *o O(f) <= O(f)"
apply (auto intro!: subsetI
simp del: abs_mult ac_simps)
(* sledgehammer *)
apply (rule_tac x = "ca * ¦c¦" in exI)
apply (rule allI)
apply (subgoal_tac "ca * ¦c¦ * ¦f x¦ = ¦c¦ * (ca * ¦f x¦)")
apply (erule ssubst)
apply (subst abs_mult)
apply (rule mult_left_mono)
apply (erule spec)
apply simp
done

lemma bigo_const_mult7 [intro]: "f =o O(g) ⟹ (λx. c * f x) =o O(g)"
by (metis bigo_const_mult1 bigo_elt_subset order_less_le psubsetD)

lemma bigo_compose1: "f =o O(g) ⟹ (λx. f(k x)) =o O(λx. g(k x))"
by (unfold bigo_def, auto)

lemma bigo_compose2:
"f =o g +o O(h) ⟹ (λx. f(k x)) =o (λx. g(k x)) +o O(λx. h(k x))"
apply (simp only: set_minus_plus [symmetric] fun_Compl_def func_plus)
apply (drule bigo_compose1 [of "f - g" h k])
done

subsection ‹Sum›

lemma bigo_sum_main: "∀x. ∀y ∈ A x. 0 <= h x y ⟹
∃c. ∀x. ∀y ∈ A x. ¦f x y¦ <= c * (h x y) ⟹
(λx. ∑y ∈ A x. f x y) =o O(λx. ∑y ∈ A x. h x y)"
apply (rule_tac x = "¦c¦" in exI)
apply (subst abs_of_nonneg) back back
apply (rule sum_nonneg)
apply force
apply (subst sum_distrib_left)
apply (rule allI)
apply (rule order_trans)
apply (rule sum_abs)
apply (rule sum_mono)
by (metis abs_ge_self abs_mult_pos order_trans)

lemma bigo_sum1: "∀x y. 0 <= h x y ⟹
∃c. ∀x y. ¦f x y¦ <= c * (h x y) ⟹
(λx. ∑y ∈ A x. f x y) =o O(λx. ∑y ∈ A x. h x y)"
by (metis (no_types) bigo_sum_main)

lemma bigo_sum2: "∀y. 0 <= h y ⟹
∃c. ∀y. ¦f y¦ <= c * (h y) ⟹
(λx. ∑y ∈ A x. f y) =o O(λx. ∑y ∈ A x. h y)"
apply (rule bigo_sum1)
by metis+

lemma bigo_sum3: "f =o O(h) ⟹
(λx. ∑y ∈ A x. (l x y) * f(k x y)) =o
O(λx. ∑y ∈ A x. ¦l x y * h(k x y)¦)"
apply (rule bigo_sum1)
apply (rule allI)+
apply (rule abs_ge_zero)
apply (unfold bigo_def)
by (metis abs_ge_zero mult.left_commute mult_left_mono)

lemma bigo_sum4: "f =o g +o O(h) ⟹
(λx. ∑y ∈ A x. l x y * f(k x y)) =o
(λx. ∑y ∈ A x. l x y * g(k x y)) +o
O(λx. ∑y ∈ A x. ¦l x y * h(k x y)¦)"
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
apply (subst sum_subtractf [symmetric])
apply (subst right_diff_distrib [symmetric])
apply (rule bigo_sum3)
by (metis (lifting, no_types) fun_diff_def set_plus_imp_minus ext)

lemma bigo_sum5: "f =o O(h) ⟹ ∀x y. 0 <= l x y ⟹
∀x. 0 <= h x ⟹
(λx. ∑y ∈ A x. (l x y) * f(k x y)) =o
O(λx. ∑y ∈ A x. (l x y) * h(k x y))"
apply (subgoal_tac "(λx. ∑y ∈ A x. (l x y) * h(k x y)) =
(λx. ∑y ∈ A x. ¦(l x y) * h(k x y)¦)")
apply (erule ssubst)
apply (erule bigo_sum3)
apply (rule ext)
apply (rule sum.cong)
apply (rule refl)
by (metis abs_of_nonneg zero_le_mult_iff)

lemma bigo_sum6: "f =o g +o O(h) ⟹ ∀x y. 0 <= l x y ⟹
∀x. 0 <= h x ⟹
(λx. ∑y ∈ A x. (l x y) * f(k x y)) =o
(λx. ∑y ∈ A x. (l x y) * g(k x y)) +o
O(λx. ∑y ∈ A x. (l x y) * h(k x y))"
apply (rule set_minus_imp_plus)
apply (subst fun_diff_def)
apply (subst sum_subtractf [symmetric])
apply (subst right_diff_distrib [symmetric])
apply (rule bigo_sum5)
apply (subst fun_diff_def [symmetric])
apply (drule set_plus_imp_minus)
apply auto
done

subsection ‹Misc useful stuff›

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)"
apply (subst bigo_plus_idemp [symmetric])
apply (rule set_plus_intro)
apply assumption+
done

lemma bigo_useful_const_mult: "(c::'a::linordered_field) ~= 0 ⟹
(λx. c) * f =o O(h) ⟹ f =o O(h)"
apply (rule subsetD)
apply (subgoal_tac "(λx. 1 / c) *o O(h) <= O(h)")
apply assumption
apply (rule bigo_const_mult6)
apply (subgoal_tac "f = (λx. 1 / c) * ((λx. c) * f)")
apply (erule ssubst)
apply (erule set_times_intro2)
done

lemma bigo_fix: "(λx. f ((x::nat) + 1)) =o O(λx. h(x + 1)) ⟹ f 0 = 0 ⟹
f =o O(h)"
by (metis abs_ge_zero abs_mult abs_of_pos abs_zero not0_implies_Suc)

lemma bigo_fix2:
"(λx. f ((x::nat) + 1)) =o (λx. g(x + 1)) +o O(λx. 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)
apply (subst fun_diff_def [symmetric])
apply (rule set_plus_imp_minus)
apply simp
done

subsection ‹Less than or equal to›

definition lesso :: "('a => 'b::linordered_idom) => ('a => 'b) => ('a => 'b)" (infixl "<o" 70) where
"f <o g == (λx. max (f x - g x) 0)"

lemma bigo_lesseq1: "f =o O(h) ⟹ ∀x. ¦g x¦ <= ¦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) ⟹ ∀x. ¦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) ⟹ ∀x. 0 <= g x ⟹ ∀x. g x <= f x ⟹
g =o O(h)"
apply (erule bigo_lesseq2)
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
done

lemma bigo_lesseq4: "f =o O(h) ⟹
∀x. 0 <= g x ⟹ ∀x. g x <= ¦f x¦ ⟹
g =o O(h)"
apply (erule bigo_lesseq1)
apply (rule allI)
apply (subst abs_of_nonneg)
apply (erule spec)+
done

lemma bigo_lesso1: "∀x. f x <= g x ⟹ f <o g =o O(h)"
apply (unfold lesso_def)
apply (subgoal_tac "(λx. max (f x - g x) 0) = 0")
apply (metis bigo_zero)
by (metis (lifting, no_types) func_zero le_fun_def le_iff_diff_le_0
max.absorb2 order_eq_iff)

lemma bigo_lesso2: "f =o g +o O(h) ⟹
∀x. 0 <= k x ⟹ ∀x. k x <= f x ⟹
k <o g =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
apply (erule set_plus_imp_minus)
apply (rule allI)
apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (erule thin_rl)
(* sledgehammer *)
apply (case_tac "0 <= k x - g x")
apply (metis (lifting) abs_le_D1 linorder_linear min_diff_distrib_left
min.absorb1 min.absorb2 max.absorb1)
by (metis abs_ge_zero le_cases max.absorb2)

lemma bigo_lesso3: "f =o g +o O(h) ⟹
∀x. 0 <= k x ⟹ ∀x. g x <= k x ⟹
f <o k =o O(h)"
apply (unfold lesso_def)
apply (rule bigo_lesseq4)
apply (erule set_plus_imp_minus)
apply (rule allI)
apply (rule max.cobounded2)
apply (rule allI)
apply (subst fun_diff_def)
apply (erule thin_rl)
(* sledgehammer *)
apply (case_tac "0 <= f x - k x")
apply simp
apply (subst abs_of_nonneg)
apply (drule_tac x = x in spec) back
apply (metis diff_less_0_iff_less linorder_not_le not_le_imp_less xt1(12) xt1(6))
by (metis abs_ge_zero linorder_linear max.absorb1 max.commute)

lemma bigo_lesso4:
"f <o g =o O(k::'a=>'b::{linordered_field}) ⟹
g =o h +o O(k) ⟹ f <o h =o O(k)"
apply (unfold lesso_def)
apply (drule set_plus_imp_minus)
apply (drule bigo_abs5) back