# Theory Multiset_Order

theory Multiset_Order
imports Multiset
```(*  Title:      HOL/Library/Multiset_Order.thy
Author:     Dmitriy Traytel, TU Muenchen
Author:     Jasmin Blanchette, Inria, LORIA, MPII
*)

section ‹More Theorems about the Multiset Order›

theory Multiset_Order
imports Multiset
begin

subsection ‹Alternative Characterizations›

context preorder
begin

lemma order_mult: "class.order
(λM N. (M, N) ∈ mult {(x, y). x < y} ∨ M = N)
(λM N. (M, N) ∈ mult {(x, y). x < y})"
(is "class.order ?le ?less")
proof -
have irrefl: "⋀M :: 'a multiset. ¬ ?less M M"
proof
fix M :: "'a multiset"
have "trans {(x'::'a, x). x' < x}"
by (rule transI) (blast intro: less_trans)
moreover
assume "(M, M) ∈ mult {(x, y). x < y}"
ultimately have "∃I J K. M = I + J ∧ M = I + K
∧ J ≠ {#} ∧ (∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(x, y). x < y})"
by (rule mult_implies_one_step)
then obtain I J K where "M = I + J" and "M = I + K"
and "J ≠ {#}" and "(∀k∈set_mset K. ∃j∈set_mset J. (k, j) ∈ {(x, y). x < y})" by blast
then have aux1: "K ≠ {#}" and aux2: "∀k∈set_mset K. ∃j∈set_mset K. k < j" by auto
have "finite (set_mset K)" by simp
moreover note aux2
ultimately have "set_mset K = {}"
by (induct rule: finite_induct)
(simp, metis (mono_tags) insert_absorb insert_iff insert_not_empty less_irrefl less_trans)
with aux1 show False by simp
qed
have trans: "⋀K M N :: 'a multiset. ?less K M ⟹ ?less M N ⟹ ?less K N"
unfolding mult_def by (blast intro: trancl_trans)
show "class.order ?le ?less"
by standard (auto simp add: less_eq_multiset_def irrefl dest: trans)
qed

text ‹The Dershowitz--Manna ordering:›

definition less_multiset⇩D⇩M where
"less_multiset⇩D⇩M M N ⟷
(∃X Y. X ≠ {#} ∧ X ⊆# N ∧ M = (N - X) + Y ∧ (∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ k < a)))"

text ‹The Huet--Oppen ordering:›

definition less_multiset⇩H⇩O where
"less_multiset⇩H⇩O M N ⟷ M ≠ N ∧ (∀y. count N y < count M y ⟶ (∃x. y < x ∧ count M x < count N x))"

lemma mult_imp_less_multiset⇩H⇩O:
"(M, N) ∈ mult {(x, y). x < y} ⟹ less_multiset⇩H⇩O M N"
proof (unfold mult_def, induct rule: trancl_induct)
case (base P)
then show ?case
by (auto elim!: mult1_lessE simp add: count_eq_zero_iff less_multiset⇩H⇩O_def split: if_splits dest!: Suc_lessD)
next
case (step N P)
from step(3) have "M ≠ N" and
**: "⋀y. count N y < count M y ⟹ (∃x>y. count M x < count N x)"
from step(2) obtain M0 a K where
*: "P = add_mset a M0" "N = M0 + K" "a ∉# K" "⋀b. b ∈# K ⟹ b < a"
by (blast elim: mult1_lessE)
from ‹M ≠ N› ** *(1,2,3) have "M ≠ P" by (force dest: *(4) elim!: less_asym split: if_splits )
moreover
{ assume "count P a ≤ count M a"
with ‹a ∉# K› have "count N a < count M a" unfolding *(1,2)
with ** obtain z where z: "z > a" "count M z < count N z"
by blast
with * have "count N z ≤ count P z"
by (auto elim: less_asym intro: count_inI)
with z have "∃z > a. count M z < count P z" by auto
} note count_a = this
{ fix y
assume count_y: "count P y < count M y"
have "∃x>y. count M x < count P x"
proof (cases "y = a")
case True
with count_y count_a show ?thesis by auto
next
case False
show ?thesis
proof (cases "y ∈# K")
case True
with *(4) have "y < a" by simp
then show ?thesis by (cases "count P a ≤ count M a") (auto dest: count_a intro: less_trans)
next
case False
with ‹y ≠ a› have "count P y = count N y" unfolding *(1,2)
with count_y ** obtain z where z: "z > y" "count M z < count N z" by auto
show ?thesis
proof (cases "z ∈# K")
case True
with *(4) have "z < a" by simp
with z(1) show ?thesis
by (cases "count P a ≤ count M a") (auto dest!: count_a intro: less_trans)
next
case False
with ‹a ∉# K› have "count N z ≤ count P z" unfolding *
with z show ?thesis by auto
qed
qed
qed
}
ultimately show ?case unfolding less_multiset⇩H⇩O_def by blast
qed

lemma less_multiset⇩D⇩M_imp_mult:
"less_multiset⇩D⇩M M N ⟹ (M, N) ∈ mult {(x, y). x < y}"
proof -
assume "less_multiset⇩D⇩M M N"
then obtain X Y where
"X ≠ {#}" and "X ⊆# N" and "M = N - X + Y" and "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ k < a)"
unfolding less_multiset⇩D⇩M_def by blast
then have "(N - X + Y, N - X + X) ∈ mult {(x, y). x < y}"
by (intro one_step_implies_mult) (auto simp: Bex_def trans_def)
with ‹M = N - X + Y› ‹X ⊆# N› show "(M, N) ∈ mult {(x, y). x < y}"
qed

lemma less_multiset⇩H⇩O_imp_less_multiset⇩D⇩M: "less_multiset⇩H⇩O M N ⟹ less_multiset⇩D⇩M M N"
unfolding less_multiset⇩D⇩M_def
proof (intro iffI exI conjI)
assume "less_multiset⇩H⇩O M N"
then obtain z where z: "count M z < count N z"
unfolding less_multiset⇩H⇩O_def by (auto simp: multiset_eq_iff nat_neq_iff)
define X where "X = N - M"
define Y where "Y = M - N"
from z show "X ≠ {#}" unfolding X_def by (auto simp: multiset_eq_iff not_less_eq_eq Suc_le_eq)
from z show "X ⊆# N" unfolding X_def by auto
show "M = (N - X) + Y" unfolding X_def Y_def multiset_eq_iff count_union count_diff by force
show "∀k. k ∈# Y ⟶ (∃a. a ∈# X ∧ k < a)"
proof (intro allI impI)
fix k
assume "k ∈# Y"
then have "count N k < count M k" unfolding Y_def
with ‹less_multiset⇩H⇩O M N› obtain a where "k < a" and "count M a < count N a"
unfolding less_multiset⇩H⇩O_def by blast
then show "∃a. a ∈# X ∧ k < a" unfolding X_def
qed
qed

lemma mult_less_multiset⇩D⇩M: "(M, N) ∈ mult {(x, y). x < y} ⟷ less_multiset⇩D⇩M M N"
by (metis less_multiset⇩D⇩M_imp_mult less_multiset⇩H⇩O_imp_less_multiset⇩D⇩M mult_imp_less_multiset⇩H⇩O)

lemma mult_less_multiset⇩H⇩O: "(M, N) ∈ mult {(x, y). x < y} ⟷ less_multiset⇩H⇩O M N"
by (metis less_multiset⇩D⇩M_imp_mult less_multiset⇩H⇩O_imp_less_multiset⇩D⇩M mult_imp_less_multiset⇩H⇩O)

lemmas mult⇩D⇩M = mult_less_multiset⇩D⇩M[unfolded less_multiset⇩D⇩M_def]
lemmas mult⇩H⇩O = mult_less_multiset⇩H⇩O[unfolded less_multiset⇩H⇩O_def]

end

lemma less_multiset_less_multiset⇩H⇩O: "M < N ⟷ less_multiset⇩H⇩O M N"
unfolding less_multiset_def mult⇩H⇩O less_multiset⇩H⇩O_def ..

lemmas less_multiset⇩D⇩M = mult⇩D⇩M[folded less_multiset_def]
lemmas less_multiset⇩H⇩O = mult⇩H⇩O[folded less_multiset_def]

lemma subset_eq_imp_le_multiset:
shows "M ⊆# N ⟹ M ≤ N"
unfolding less_eq_multiset_def less_multiset⇩H⇩O

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_right_total: "M < add_mset x M"
unfolding less_eq_multiset_def less_multiset⇩H⇩O by simp

lemma less_eq_multiset_empty_left[simp]:
shows "{#} ≤ M"

lemma ex_gt_imp_less_multiset: "(∃y. y ∈# N ∧ (∀x. x ∈# M ⟶ x < y)) ⟹ M < N"
unfolding less_multiset⇩H⇩O
by (metis count_eq_zero_iff count_greater_zero_iff less_le_not_le)

lemma less_eq_multiset_empty_right[simp]: "M ≠ {#} ⟹ ¬ M ≤ {#}"
by (metis less_eq_multiset_empty_left antisym)

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_left[simp]: "M ≠ {#} ⟹ {#} < M"

(* FIXME: "le" should be "less" in this and other names *)
lemma le_multiset_empty_right[simp]: "¬ M < {#}"
using subset_mset.le_zero_eq less_multiset⇩D⇩M by blast

(* FIXME: "le" should be "less" in this and other names *)
lemma union_le_diff_plus: "P ⊆# M ⟹ N < P ⟹ M - P + N < M"

begin

lemma less_eq_multiset⇩H⇩O:
"M ≤ N ⟷ (∀y. count N y < count M y ⟶ (∃x. y < x ∧ count M x < count N x))"
by (auto simp: less_eq_multiset_def less_multiset⇩H⇩O)

instance by standard (auto simp: less_eq_multiset⇩H⇩O)

lemma
fixes M N :: "'a multiset"
shows
less_eq_multiset_plus_left: "N ≤ (M + N)" and
less_eq_multiset_plus_right: "M ≤ (M + N)"
by simp_all

lemma
fixes M N :: "'a multiset"
shows
le_multiset_plus_left_nonempty: "M ≠ {#} ⟹ N < M + N" and
le_multiset_plus_right_nonempty: "N ≠ {#} ⟹ M < M + N"
by simp_all

end

lemma all_lt_Max_imp_lt_mset: "N ≠ {#} ⟹ (∀a ∈# M. a < Max (set_mset N)) ⟹ M < N"
by (meson Max_in[OF finite_set_mset] ex_gt_imp_less_multiset set_mset_eq_empty_iff)

lemma lt_imp_ex_count_lt: "M < N ⟹ ∃y. count M y < count N y"
by (meson less_eq_multiset⇩H⇩O less_le_not_le)

lemma subset_imp_less_mset: "A ⊂# B ⟹ A < B"

lemma image_mset_strict_mono:
assumes
mono_f: "∀x ∈ set_mset M. ∀y ∈ set_mset N. x < y ⟶ f x < f y" and
less: "M < N"
shows "image_mset f M < image_mset f N"
proof -
obtain Y X where
y_nemp: "Y ≠ {#}" and y_sub_N: "Y ⊆# N" and M_eq: "M = N - Y + X" and
ex_y: "∀x. x ∈# X ⟶ (∃y. y ∈# Y ∧ x < y)"
using less[unfolded less_multiset⇩D⇩M] by blast

have x_sub_M: "X ⊆# M"
using M_eq by simp

let ?fY = "image_mset f Y"
let ?fX = "image_mset f X"

show ?thesis
unfolding less_multiset⇩D⇩M
proof (intro exI conjI)
show "image_mset f M = image_mset f N - ?fY + ?fX"
using M_eq[THEN arg_cong, of "image_mset f"] y_sub_N
by (metis image_mset_Diff image_mset_union)
next
obtain y where y: "∀x. x ∈# X ⟶ y x ∈# Y ∧ x < y x"
using ex_y by moura

show "∀fx. fx ∈# ?fX ⟶ (∃fy. fy ∈# ?fY ∧ fx < fy)"
proof (intro allI impI)
fix fx
assume "fx ∈# ?fX"
then obtain x where fx: "fx = f x" and x_in: "x ∈# X"
by auto
hence y_in: "y x ∈# Y" and y_gt: "x < y x"
using y[rule_format, OF x_in] by blast+
hence "f (y x) ∈# ?fY ∧ f x < f (y x)"
using mono_f y_sub_N x_sub_M x_in
by (metis image_eqI in_image_mset mset_subset_eqD)
thus "∃fy. fy ∈# ?fY ∧ fx < fy"
unfolding fx by auto
qed
qed (auto simp: y_nemp y_sub_N image_mset_subseteq_mono)
qed

lemma image_mset_mono:
assumes
mono_f: "∀x ∈ set_mset M. ∀y ∈ set_mset N. x < y ⟶ f x < f y" and
less: "M ≤ N"
shows "image_mset f M ≤ image_mset f N"
by (metis eq_iff image_mset_strict_mono less less_imp_le mono_f order.not_eq_order_implies_strict)

lemma mset_lt_single_right_iff[simp]: "M < {#y#} ⟷ (∀x ∈# M. x < y)" for y :: "'a::linorder"
proof (rule iffI)
assume M_lt_y: "M < {#y#}"
show "∀x ∈# M. x < y"
proof
fix x
assume x_in: "x ∈# M"
hence M: "M - {#x#} + {#x#} = M"
by (meson insert_DiffM2)
hence "¬ {#x#} < {#y#} ⟹ x < y"
using x_in M_lt_y
by (metis diff_single_eq_union le_multiset_empty_left less_add_same_cancel2 mset_le_trans)
also have "¬ {#y#} < M"
using M_lt_y mset_le_not_sym by blast
ultimately show "x < y"
by (metis (no_types) Max_ge all_lt_Max_imp_lt_mset empty_iff finite_set_mset insertE
set_mset_eq_empty_iff x_in)
qed
next
assume y_max: "∀x ∈# M. x < y"
show "M < {#y#}"
by (rule all_lt_Max_imp_lt_mset) (auto intro!: y_max)
qed

lemma mset_le_single_right_iff[simp]:
"M ≤ {#y#} ⟷ M = {#y#} ∨ (∀x ∈# M. x < y)" for y :: "'a::linorder"
by (meson less_eq_multiset_def mset_lt_single_right_iff)

subsection ‹Simprocs›

"j ≤ (i::nat) ⟹ (repeat_mset i u + m ≤ repeat_mset j u + n) = (repeat_mset (i-j) u + m ≤ n)"
proof -
assume "j ≤ i"
then have "j + (i - j) = i"
then show ?thesis
qed

"i ≤ (j::nat) ⟹ (repeat_mset i u + m ≤ repeat_mset j u + n) = (m ≤ repeat_mset (j-i) u + n)"
proof -
assume "i ≤ j"
then have "i + (j - i) = j"
then show ?thesis
qed

simproc_setup msetless_cancel
("(l::'a::preorder multiset) + m < n" | "(l::'a multiset) < m + n" |
"add_mset a m < n" | "m < add_mset a n" |
"replicate_mset p a < n" | "m < replicate_mset p a" |
"repeat_mset p m < n" | "m < repeat_mset p n") =
‹fn phi => Cancel_Simprocs.less_cancel›

simproc_setup msetle_cancel
("(l::'a::preorder multiset) + m ≤ n" | "(l::'a multiset) ≤ m + n" |
"add_mset a m ≤ n" | "m ≤ add_mset a n" |
"replicate_mset p a ≤ n" | "m ≤ replicate_mset p a" |
"repeat_mset p m ≤ n" | "m ≤ repeat_mset p n") =
‹fn phi => Cancel_Simprocs.less_eq_cancel›

lemma ex_gt_count_imp_le_multiset:
"(∀y :: 'a :: order. y ∈# M + N ⟶ y ≤ x) ⟹ count M x < count N x ⟹ M < N"
unfolding less_multiset⇩H⇩O
by (metis count_greater_zero_iff le_imp_less_or_eq less_imp_not_less not_gr_zero union_iff)

lemma mset_lt_single_iff[iff]: "{#x#} < {#y#} ⟷ x < y"
unfolding less_multiset⇩H⇩O by simp

lemma mset_le_single_iff[iff]: "{#x#} ≤ {#y#} ⟷ x ≤ y" for x y :: "'a::order"
unfolding less_eq_multiset⇩H⇩O by force

by standard (metis less_eq_multiset⇩H⇩O not_less_iff_gr_or_eq)

lemma less_eq_multiset_total:
fixes M N :: "'a :: linorder multiset"
shows "¬ M ≤ N ⟹ N ≤ M"
by simp

instantiation multiset :: (wellorder) wellorder
begin

lemma wf_less_multiset: "wf {(M :: 'a multiset, N). M < N}"
unfolding less_multiset_def by (auto intro: wf_mult wf)

instance by standard (metis less_multiset_def wf wf_def wf_mult)

end

instantiation multiset :: (preorder) order_bot
begin

definition bot_multiset :: "'a multiset" where "bot_multiset = {#}"

instance by standard (simp add: bot_multiset_def)

end

instance multiset :: (preorder) no_top
proof standard
fix x :: "'a multiset"
obtain a :: 'a where True by simp
have "x < x + (x + {#a#})"
by simp
then show "∃y. x < y"
by blast
qed

by standard

instantiation multiset :: (linorder) distrib_lattice
begin

definition inf_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" where
"inf_multiset A B = (if A < B then A else B)"

definition sup_multiset :: "'a multiset ⇒ 'a multiset ⇒ 'a multiset" where
"sup_multiset A B = (if B > A then B else A)"

instance
by intro_classes (auto simp: inf_multiset_def sup_multiset_def)

end

end
```