# Theory Misc

(*  Title:       Miscellaneous Definitions and Lemmas
Author:      Peter Lammich <peter.lammich@uni-muenster.de>
Maintainer:  Peter Lammich <peter.lammich@uni-muenster.de>
Thomas Tuerk <tuerk@in.tum.de>
*)

(*
CHANGELOG:
2010-05-09: Removed AC, AI locales, they are superseeded by concepts
from OrderedGroups
2010-09-22: Merges with ext/Aux
2017-06-12: Added a bunch of lemmas from various other projects

*)

section ‹Miscellaneous Definitions and Lemmas›

theory Misc
imports Main
"HOL-Library.Multiset"
"HOL-ex.Quicksort"
"HOL-Library.Option_ord"
"HOL-Library.Infinite_Set"
"HOL-Eisbach.Eisbach"
begin
text_raw ‹\label{thy:Misc}›

subsection ‹Isabelle Distribution Move Proposals›

subsubsection ‹Pure›
lemma TERMI: "TERM x" unfolding Pure.term_def .

subsubsection ‹HOL›
(* Stronger disjunction elimination rules. *)
lemma disjE1: "⟦ P ∨ Q; P ⟹ R; ⟦¬P;Q⟧ ⟹ R ⟧ ⟹ R"
by metis
lemma disjE2: "⟦ P ∨ Q; ⟦P; ¬Q⟧ ⟹ R; Q ⟹ R ⟧ ⟹ R"
by blast

lemma imp_mp_iff[simp]:
"a ∧ (a ⟶ b) ⟷ a ∧ b"
"(a ⟶ b) ∧ a ⟷ a ∧ b" (* is Inductive.imp_conj_iff, but not in simpset by default *)
by blast+

lemma atomize_Trueprop_eq[atomize]: "(Trueprop x ≡ Trueprop y) ≡ Trueprop (x=y)"
apply rule
apply (rule)
apply (erule equal_elim_rule1)
apply assumption
apply (erule equal_elim_rule2)
apply assumption
apply simp
done

subsubsection ‹Set›
lemma inter_compl_diff_conv[simp]: "A ∩ -B = A - B" by auto
lemma pair_set_inverse[simp]: "{(a,b). P a b}¯ = {(b,a). P a b}"
by auto

lemma card_doubleton_eq_2_iff[simp]: "card {a,b} = 2 ⟷ a≠b" by auto

subsubsection ‹List›
(* TODO: Move, analogous to List.length_greater_0_conv *)
thm List.length_greater_0_conv
lemma length_ge_1_conv[iff]: "Suc 0 ≤ length l ⟷ l≠[]"
by (cases l) auto

― ‹Obtains a list from the pointwise characterization of its elements›
lemma obtain_list_from_elements:
assumes A: "∀i<n. (∃li. P li i)"
obtains l where
"length l = n"
"∀i<n. P (l!i) i"
proof -
from A have "∃l. length l=n ∧ (∀i<n. P (l!i) i)"
proof (induct n)
case 0 thus ?case by simp
next
case (Suc n)
then obtain l where IH: "length l = n" "(∀i<n. P(l!i) i)" by auto
moreover from Suc.prems obtain ln where "P ln n" by auto
ultimately have "length (l@[ln]) = Suc n" "(∀i<Suc n. P((l@[ln])!i) i)"
by (auto simp add: nth_append dest: less_antisym)
thus ?case by blast
qed
thus ?thesis using that by (blast)
qed

lemma distinct_sorted_mono:
assumes S: "sorted l"
assumes D: "distinct l"
assumes B: "i<j" "j<length l"
shows "l!i < l!j"
proof -
from S B have "l!i ≤ l!j"
also from nth_eq_iff_index_eq[OF D] B have "l!i ≠ l!j"
by auto
finally show ?thesis .
qed

lemma distinct_sorted_strict_mono_iff:
assumes "distinct l" "sorted l"
assumes "i<length l" "j<length l"
shows "l!i<l!j ⟷ i<j"
using assms
by (metis distinct_sorted_mono leI less_le_not_le
order.strict_iff_order)

lemma distinct_sorted_mono_iff:
assumes "distinct l" "sorted l"
assumes "i<length l" "j<length l"
shows "l!i≤l!j ⟷ i≤j"
by (metis assms distinct_sorted_strict_mono_iff leD le_less_linear)

(* List.thy has:
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!]

We could, analogously, declare rules for "map _ _ = _@_" as dest!,
or use elim!, or declare the _conv-rule as simp
*)

lemma map_eq_appendE:
assumes "map f ls = fl@fl'"
obtains l l' where "ls=l@l'" and "map f l=fl" and  "map f l' = fl'"
using assms
proof (induction fl arbitrary: ls thesis)
case (Cons x xs)
then obtain l ls' where [simp]: "ls = l#ls'" "f l = x" by force
with Cons.prems(2) have "map f ls' = xs @ fl'" by simp
from Cons.IH[OF _ this] obtain ll ll' where "ls' = ll @ ll'" "map f ll = xs" "map f ll' = fl'" .
with Cons.prems(1)[of "l#ll" ll'] show thesis by simp
qed simp

lemma map_eq_append_conv: "map f ls = fl@fl' ⟷ (∃l l'. ls = l@l' ∧ map f l = fl ∧ map f l' = fl')"
by (auto elim!: map_eq_appendE)

lemmas append_eq_mapE = map_eq_appendE[OF sym]

lemma append_eq_map_conv: "fl@fl' = map f ls ⟷ (∃l l'. ls = l@l' ∧ map f l = fl ∧ map f l' = fl')"
by (auto elim!: append_eq_mapE)

lemma distinct_mapI: "distinct (map f l) ⟹ distinct l"
by (induct l) auto

lemma map_distinct_upd_conv:
"⟦i<length l; distinct l⟧ ⟹ (map f l)[i := x] = map (f(l!i := x)) l"
― ‹Updating a mapped distinct list is equal to updating the
mapping function›
by (auto simp: nth_eq_iff_index_eq intro: nth_equalityI)

lemma zip_inj: "⟦length a = length b; length a' = length b'; zip a b = zip a' b'⟧ ⟹ a=a' ∧ b=b'"
proof (induct a b arbitrary: a' b' rule: list_induct2)
case Nil
then show ?case by (cases a'; cases b'; auto)
next
case (Cons x xs y ys)
then show ?case by (cases a'; cases b'; auto)
qed

lemma zip_eq_zip_same_len[simp]:
"⟦ length a = length b; length a' = length b' ⟧ ⟹
zip a b = zip a' b' ⟷ a=a' ∧ b=b'"
by (auto dest: zip_inj)

lemma upt_merge[simp]: "i≤j ∧ j≤k ⟹ [i..<j]@[j..<k] = [i..<k]"

(* Maybe this should go into List.thy, next to snoc_eq_iff_butlast *)
lemma snoc_eq_iff_butlast':
"(ys = xs @ [x]) ⟷ (ys ≠ [] ∧ butlast ys = xs ∧ last ys = x)"
by auto

(* Case distinction how two elements of a list can be related to each other *)
lemma list_match_lel_lel:
assumes "c1 @ qs # c2 = c1' @ qs' # c2'"
obtains
(left) c21' where "c1 = c1' @ qs' # c21'" "c2' = c21' @ qs # c2"
| (center) "c1' = c1" "qs' = qs" "c2' = c2"
| (right) c21 where "c1' = c1 @ qs # c21" "c2 = c21 @ qs' # c2'"
using assms
apply (clarsimp simp: append_eq_append_conv2)
subgoal for us by (cases us) auto
done

lemma xy_in_set_cases[consumes 2, case_names EQ XY YX]:
assumes A: "x∈set l" "y∈set l"
and C:
"!!l1 l2. ⟦ x=y; l=l1@y#l2 ⟧ ⟹ P"
"!!l1 l2 l3. ⟦ x≠y; l=l1@x#l2@y#l3 ⟧ ⟹ P"
"!!l1 l2 l3. ⟦ x≠y; l=l1@y#l2@x#l3 ⟧ ⟹ P"
shows P
proof (cases "x=y")
case True with A(1) obtain l1 l2 where "l=l1@y#l2" by (blast dest: split_list)
with C(1) True show ?thesis by blast
next
case False
from A(1) obtain l1 l2 where S1: "l=l1@x#l2" by (blast dest: split_list)
from A(2) obtain l1' l2' where S2: "l=l1'@y#l2'" by (blast dest: split_list)
from S1 S2 have M: "l1@x#l2 = l1'@y#l2'" by simp
thus P proof (cases rule: list_match_lel_lel[consumes 1, case_names 1 2 3])
case (1 c) with S1 have "l=l1'@y#c@x#l2" by simp
with C(3) False show ?thesis by blast
next
case 2 with False have False by blast
thus ?thesis ..
next
case (3 c) with S1 have "l=l1@x#c@y#l2'" by simp
with C(2) False show ?thesis by blast
qed
qed

lemma list_e_eq_lel[simp]:
"[e] = l1@e'#l2 ⟷ l1=[] ∧ e'=e ∧ l2=[]"
"l1@e'#l2 = [e] ⟷ l1=[] ∧ e'=e ∧ l2=[]"
apply (cases l1, auto) []
apply (cases l1, auto) []
done

lemma list_ee_eq_leel[simp]:
"([e1,e2] = l1@e1'#e2'#l2) ⟷ (l1=[] ∧ e1=e1' ∧ e2=e2' ∧ l2=[])"
"(l1@e1'#e2'#l2 = [e1,e2]) ⟷ (l1=[] ∧ e1=e1' ∧ e2=e2' ∧ l2=[])"
apply (cases l1, auto) []
apply (cases l1, auto) []
done

subsubsection ‹Transitive Closure›

text ‹A point-free induction rule for elements reachable from an initial set›
lemma rtrancl_reachable_induct[consumes 0, case_names base step]:
assumes I0: "I ⊆ INV"
assumes IS: "EINV ⊆ INV"
shows "E⇧*I ⊆ INV"
by (metis I0 IS Image_closed_trancl Image_mono subset_refl)

lemma acyclic_empty[simp, intro!]: "acyclic {}" by (unfold acyclic_def) auto

lemma acyclic_union:
"acyclic (A∪B) ⟹ acyclic A"
"acyclic (A∪B) ⟹ acyclic B"
by (metis Un_upper1 Un_upper2 acyclic_subset)+

text ‹Here we provide a collection of miscellaneous definitions and helper lemmas›

subsection "Miscellaneous (1)"

text ‹This stuff is used in this theory itself, and thus occurs in first place or is simply not sorted into any other section of this theory.›

lemma IdD: "(a,b)∈Id ⟹ a=b" by simp

text ‹Conversion Tag›
definition [simp]: "CNV x y ≡ x=y"
lemma CNV_I: "CNV x x" by simp
lemma CNV_eqD: "CNV x y ⟹ x=y" by simp
lemma CNV_meqD: "CNV x y ⟹ x≡y" by simp

(* TODO: Move. Shouldn't this be detected by simproc? *)
lemma ex_b_b_and_simp[simp]: "(∃b. b ∧ Q b) ⟷ Q True" by auto
lemma ex_b_not_b_and_simp[simp]: "(∃b. ¬b ∧ Q b) ⟷ Q False" by auto

method repeat_all_new methods m = m;(repeat_all_new ‹m›)?

subsubsection "AC-operators"

text ‹Locale to declare AC-laws as simplification rules›
locale Assoc =
fixes f
assumes assoc[simp]: "f (f x y) z = f x (f y z)"

locale AC = Assoc +
assumes commute[simp]: "f x y = f y x"

lemma (in AC) left_commute[simp]: "f x (f y z) = f y (f x z)"
by (simp only: assoc[symmetric]) simp

lemmas (in AC) AC_simps = commute assoc left_commute

text ‹Locale to define functions from surjective, unique relations›
locale su_rel_fun =
fixes F and f
assumes unique: "⟦(A,B)∈F; (A,B')∈F⟧ ⟹ B=B'"
assumes surjective: "⟦!!B. (A,B)∈F ⟹ P⟧ ⟹ P"
assumes f_def: "f A == THE B. (A,B)∈F"

lemma (in su_rel_fun) repr1: "(A,f A)∈F" proof (unfold f_def)
obtain B where "(A,B)∈F" by (rule surjective)
with theI[where P="λB. (A,B)∈F", OF this] show "(A, THE x. (A, x) ∈ F) ∈ F" by (blast intro: unique)
qed

lemma (in su_rel_fun) repr2: "(A,B)∈F ⟹ B=f A" using repr1
by (blast intro: unique)

lemma (in su_rel_fun) repr: "(f A = B) = ((A,B)∈F)" using repr1 repr2
by (blast)

― ‹Contract quantification over two variables to pair›
lemma Ex_prod_contract: "(∃a b. P a b) ⟷ (∃z. P (fst z) (snd z))"
by auto

lemma All_prod_contract: "(∀a b. P a b) ⟷ (∀z. P (fst z) (snd z))"
by auto

lemma nat_geq_1_eq_neqz: "x≥1 ⟷ x≠(0::nat)"
by auto

lemma nat_in_between_eq:
"(a<b ∧ b≤Suc a) ⟷ b = Suc a"
"(a≤b ∧ b<Suc a) ⟷ b = a"
by auto

lemma Suc_n_minus_m_eq: "⟦ n≥m; m>1 ⟧ ⟹ Suc (n - m) = n - (m - 1)"
by simp

lemma Suc_to_right: "Suc n = m ⟹ n = m - Suc 0" by simp
lemma Suc_diff[simp]: "⋀n m. n≥m ⟹ m≥1 ⟹ Suc (n - m) = n - (m - 1)"
by simp

lemma if_not_swap[simp]: "(if ¬c then a else b) = (if c then b else a)" by auto
lemma all_to_meta: "Trueprop (∀a. P a) ≡ (⋀a. P a)"
apply rule
by auto

lemma imp_to_meta: "Trueprop (P⟶Q) ≡ (P⟹Q)"
apply rule
by auto

(* for some reason, there is no such rule in HOL *)
lemma iffI2: "⟦P ⟹ Q; ¬ P ⟹ ¬ Q⟧ ⟹ P ⟷ Q"
by metis

lemma iffExI:
"⟦ ⋀x. P x ⟹ Q x; ⋀x. Q x ⟹ P x ⟧ ⟹ (∃x. P x) ⟷ (∃x. Q x)"
by metis

lemma bex2I[intro?]: "⟦ (a,b)∈S; (a,b)∈S ⟹ P a b ⟧ ⟹ ∃a b. (a,b)∈S ∧ P a b"
by blast

(* TODO: Move lemma to HOL! *)
lemma cnv_conj_to_meta: "(P ∧ Q ⟹ PROP X) ≡ (⟦P;Q⟧ ⟹ PROP X)"
by (rule BNF_Fixpoint_Base.conj_imp_eq_imp_imp)

subsection ‹Sets›
lemma remove_subset: "x∈S ⟹ S-{x} ⊂ S" by auto

lemma subset_minus_empty: "A⊆B ⟹ A-B = {}" by auto

lemma insert_minus_eq: "x≠y ⟹ insert x A - {y} = insert x (A - {y})" by auto

lemma set_notEmptyE: "⟦S≠{}; !!x. x∈S ⟹ P⟧ ⟹ P"
by (metis equals0I)

lemma subset_Collect_conv: "S ⊆ Collect P ⟷ (∀x∈S. P x)"
by auto

lemma memb_imp_not_empty: "x∈S ⟹ S≠{}"
by auto

lemma disjoint_mono: "⟦ a⊆a'; b⊆b'; a'∩b'={} ⟧ ⟹ a∩b={}" by auto

lemma disjoint_alt_simp1: "A-B = A ⟷ A∩B = {}" by auto
lemma disjoint_alt_simp2: "A-B ≠ A ⟷ A∩B ≠ {}" by auto
lemma disjoint_alt_simp3: "A-B ⊂ A ⟷ A∩B ≠ {}" by auto

lemma disjointI[intro?]: "⟦ ⋀x. ⟦x∈a; x∈b⟧ ⟹ False ⟧ ⟹ a∩b={}"
by auto

lemmas set_simps = subset_minus_empty disjoint_alt_simp1 disjoint_alt_simp2 disjoint_alt_simp3 Un_absorb1 Un_absorb2

lemma set_minus_singleton_eq: "x∉X ⟹ X-{x} = X"
by auto

lemma set_diff_diff_left: "A-B-C = A-(B∪C)"
by auto

lemma image_update[simp]: "x∉A ⟹ f(x:=n)A = fA"
by auto

lemma eq_or_mem_image_simp[simp]: "{f l |l. l = a ∨ l∈B} = insert (f a) {f l|l. l∈B}" by blast

lemma set_union_code [code_unfold]:
"set xs ∪ set ys = set (xs @ ys)"
by auto

lemma in_fst_imageE:
assumes "x ∈ fstS"
obtains y where "(x,y)∈S"
using assms by auto

lemma in_snd_imageE:
assumes "y ∈ sndS"
obtains x where "(x,y)∈S"
using assms by auto

lemma fst_image_mp: "⟦fstA ⊆ B; (x,y)∈A ⟧ ⟹ x∈B"
by (metis Domain.DomainI fst_eq_Domain in_mono)

lemma snd_image_mp: "⟦sndA ⊆ B; (x,y)∈A ⟧ ⟹ y∈B"
by (metis Range.intros rev_subsetD snd_eq_Range)

lemma inter_eq_subsetI: "⟦ S⊆S'; A∩S' = B∩S' ⟧ ⟹ A∩S = B∩S"
by auto

text ‹
Decompose general union over sum types.
›
lemma Union_plus:
"(⋃ x ∈ A <+> B. f x) = (⋃ a ∈ A. f (Inl a)) ∪ (⋃b ∈ B. f (Inr b))"
by auto

lemma Union_sum:
"(⋃x. f (x::'a+'b)) = (⋃l. f (Inl l)) ∪ (⋃r. f (Inr r))"
(is "?lhs = ?rhs")
proof -
have "?lhs = (⋃x ∈ UNIV <+> UNIV. f x)"
by simp
thus ?thesis
by (simp only: Union_plus)
qed

subsubsection ‹Finite Sets›

lemma card_1_singletonI: "⟦finite S; card S = 1; x∈S⟧ ⟹ S={x}"
proof (safe, rule ccontr, goal_cases)
case prems: (1 x')
hence "finite (S-{x})" "S-{x} ≠ {}" by auto
hence "card (S-{x}) ≠ 0" by auto
moreover from prems(1-3) have "card (S-{x}) = 0" by auto
ultimately have False by simp
thus ?case ..
qed

lemma card_insert_disjoint': "⟦finite A; x ∉ A⟧ ⟹ card (insert x A) - Suc 0 = card A"
by (drule (1) card_insert_disjoint) auto

lemma card_eq_UNIV[simp]: "card (S::'a::finite set) = card (UNIV::'a set) ⟷ S=UNIV"
proof (auto)
fix x
assume A: "card S = card (UNIV::'a set)"
show "x∈S" proof (rule ccontr)
assume "x∉S" hence "S⊂UNIV" by auto
with psubset_card_mono[of UNIV S] have "card S < card (UNIV::'a set)" by auto
with A show False by simp
qed
qed

lemma card_eq_UNIV2[simp]: "card (UNIV::'a set) = card (S::'a::finite set) ⟷ S=UNIV"
using card_eq_UNIV[of S] by metis

lemma card_ge_UNIV[simp]: "card (UNIV::'a::finite set) ≤ card (S::'a set) ⟷ S=UNIV"
using card_mono[of "UNIV::'a::finite set" S, simplified]
by auto

lemmas length_remdups_card = length_remdups_concat[of "[l]", simplified] for l

lemma fs_contract: "fst  { p | p. f (fst p) (snd p) ∈ S } = { a . ∃b. f a b ∈ S }"

lemma finite_Collect: "finite S ⟹ inj f ⟹ finite {a. f a : S}"

― ‹Finite sets have an injective mapping to an initial segments of the
natural numbers›
(* This lemma is also in the standard library (from Isabelle2009-1 on)
as @{thm [source] Finite_Set.finite_imp_inj_to_nat_seg}. However, it is formulated with HOL's
∃ there rather then with the meta-logic obtain *)
lemma finite_imp_inj_to_nat_seg':
fixes A :: "'a set"
assumes A: "finite A"
obtains f::"'a ⇒ nat" and n::"nat" where
"fA = {i. i<n}"
"inj_on f A"
by (metis A finite_imp_inj_to_nat_seg)

lemma lists_of_len_fin1: "finite P ⟹ finite (lists P ∩ { l. length l = n })"
proof (induct n)
case 0 thus ?case by auto
next
case (Suc n)
have "lists P ∩ { l. length l = Suc n }
= (λ(a,l). a#l)  (P × (lists P ∩ {l. length l = n}))"
apply auto
apply (case_tac x)
apply auto
done
moreover from Suc have "finite …" by auto
ultimately show ?case by simp
qed

lemma lists_of_len_fin2: "finite P ⟹ finite (lists P ∩ { l. n = length l })"
proof -
assume A: "finite P"
have S: "{ l. n = length l } = { l. length l = n }" by auto
have "finite (lists P ∩ { l. n = length l })
⟷ finite (lists P ∩ { l. length l = n })"
by (subst S) simp

thus ?thesis using lists_of_len_fin1[OF A] by auto
qed

lemmas lists_of_len_fin = lists_of_len_fin1 lists_of_len_fin2

(* Try (simp only: cset_fin_simps, fastforce intro: cset_fin_intros) when reasoning about finiteness of collected sets *)
lemmas cset_fin_simps = Ex_prod_contract fs_contract[symmetric] image_Collect[symmetric]
lemmas cset_fin_intros = finite_imageI finite_Collect inj_onI

lemma Un_interval:
fixes b1 :: "'a::linorder"
assumes "b1≤b2" and "b2≤b3"
shows "{ f i | i. b1≤i ∧ i<b2 } ∪ { f i | i. b2≤i ∧ i<b3 }
= {f i | i. b1≤i ∧ i<b3}"
using assms
apply -
apply rule
apply safe []
apply (rule_tac x=i in exI, auto) []
apply (rule_tac x=i in exI, auto) []
apply rule
apply simp
apply (elim exE, simp)
apply (case_tac "i<b2")
apply (rule disjI1)
apply (rule_tac x=i in exI, auto) []
apply (rule disjI2)
apply (rule_tac x=i in exI, auto) []
done

text ‹
The standard library proves that a generalized union is finite
if the index set is finite and if for every index the component
set is itself finite. Conversely, we show that every component
set must be finite when the union is finite.
›
lemma finite_UNION_then_finite:
"finite (⋃(B  A)) ⟹ a ∈ A ⟹ finite (B a)"
by (metis Set.set_insert UN_insert Un_infinite)

lemma finite_if_eq_beyond_finite: "finite S ⟹ finite {s. s - S = s' - S}"
proof (rule finite_subset[where B="(λs. s ∪ (s' - S))  Pow S"], clarsimp)
fix s
have "s = (s ∩ S) ∪ (s - S)"
by auto
also assume "s - S = s' - S"
finally show "s ∈ (λs. s ∪ (s' - S))  Pow S" by blast
qed blast

lemma distinct_finite_subset:
assumes "finite x"
shows "finite {ys. set ys ⊆ x ∧ distinct ys}" (is "finite ?S")
proof (rule finite_subset)
from assms show "?S ⊆ {ys. set ys ⊆ x ∧ length ys ≤ card x}"
by clarsimp (metis distinct_card card_mono)
from assms show "finite ..." by (rule finite_lists_length_le)
qed

lemma distinct_finite_set:
shows "finite {ys. set ys = x ∧ distinct ys}" (is "finite ?S")
proof (cases "finite x")
case False hence "{ys. set ys = x} = {}" by auto
thus ?thesis by simp
next
case True show ?thesis
proof (rule finite_subset)
show "?S ⊆ {ys. set ys ⊆ x ∧ length ys ≤ card x}"
using distinct_card by force
from True show "finite ..." by (rule finite_lists_length_le)
qed
qed

lemma finite_set_image:
assumes f: "finite (set  A)"
and dist: "⋀xs. xs ∈ A ⟹ distinct xs"
shows "finite A"
proof (rule finite_subset)
from f show "finite (set - (set  A) ∩ {xs. distinct xs})"
proof (induct rule: finite_induct)
case (insert x F)
have "finite (set - {x} ∩ {xs. distinct xs})"
by (metis Collect_conj_eq distinct_finite_set)
with insert show ?case
apply (subst vimage_insert)
apply (subst Int_Un_distrib2)
apply (rule finite_UnI)
apply simp_all
done
qed simp
moreover from dist show "A ⊆ ..."
qed

subsubsection ‹Infinite Set›
lemma INFM_nat_inductI:
assumes P0: "P (0::nat)"
assumes PS: "⋀i. P i ⟹ ∃j>i. P j ∧ Q j"
shows "∃⇩∞i. Q i"
proof -
have "∀i. ∃j>i. P j ∧ Q j" proof
fix i
show "∃j>i. P j ∧ Q j"
apply (induction i)
using PS[OF P0] apply auto []
by (metis PS Suc_lessI)
qed
thus ?thesis unfolding INFM_nat by blast
qed

subsection ‹Functions›

lemma fun_neq_ext_iff: "m≠m' ⟷ (∃x. m x ≠ m' x)" by auto

definition "inv_on f A x == SOME y. y∈A ∧ f y = x"

lemma inv_on_f_f[simp]: "⟦inj_on f A; x∈A⟧ ⟹ inv_on f A (f x) = x"
by (auto simp add: inv_on_def inj_on_def)

lemma f_inv_on_f: "⟦ y∈fA ⟧ ⟹ f (inv_on f A y) = y"
by (auto simp add: inv_on_def intro: someI2)

lemma inv_on_f_range: "⟦ y ∈ fA ⟧ ⟹ inv_on f A y ∈ A"
by (auto simp add: inv_on_def intro: someI2)

lemma inj_on_map_inv_f [simp]: "⟦set l ⊆ A; inj_on f A⟧ ⟹ map (inv_on f A) (map f l) = l"
apply (simp)
apply (induct l)
apply auto
done

lemma comp_cong_right: "x = y ⟹ f o x = f o y" by (simp)
lemma comp_cong_left: "x = y ⟹ x o f = y o f" by (simp)

lemma fun_comp_eq_conv: "f o g = fg ⟷ (∀x. f (g x) = fg x)"
by auto

abbreviation comp2 (infixl "oo" 55) where "f oo g ≡ λx. f o (g x)"
abbreviation comp3 (infixl "ooo" 55) where "f ooo g ≡ λx. f oo (g x)"

notation
comp2  (infixl "∘∘" 55) and
comp3  (infixl "∘∘∘" 55)

definition [code_unfold, simp]: "swap_args2 f x y ≡ f y x"

subsection ‹Multisets›

(*
The following is a syntax extension for multisets. Unfortunately, it depends on a change in the Library/Multiset.thy, so it is commented out here, until it will be incorporated
into Library/Multiset.thy by its maintainers.

The required change in Library/Multiset.thy is removing the syntax for single:
- single :: "'a => 'a multiset"    ("{#_#}")
+ single :: "'a => 'a multiset"

+ syntax
+ "_multiset" :: "args ⇒ 'a multiset" ("{#(_)#}")

+ translations
+   "{#x, xs#}" == "{#x#} + {#xs#}"
+   "{# x #}" == "single x"

This translates "{# … #}" into a sum of singletons, that is parenthesized to the right. ?? Can we also achieve left-parenthesizing ??

*)

(* Let's try what happens if declaring AC-rules for multiset union as simp-rules *)
(*declare union_ac[simp] -- don't do it !*)

lemma count_mset_set_finite_iff:
"finite S ⟹ count (mset_set S) a = (if a ∈ S then 1 else 0)"
by simp

lemma ex_Melem_conv: "(∃x. x ∈# A) = (A ≠ {#})"

subsubsection ‹Count›
lemma count_ne_remove: "⟦ x ~= t⟧ ⟹ count S x = count (S-{#t#}) x"
by (auto)
lemma mset_empty_count[simp]: "(∀p. count M p = 0) = (M={#})"

subsubsection ‹Union, difference and intersection›

lemma size_diff_se: "t ∈# S ⟹ size S = size (S - {#t#}) + 1" proof (unfold size_multiset_overloaded_eq)
let ?SIZE = "sum (count S) (set_mset S)"
assume A: "t ∈# S"
from A have SPLITPRE: "finite (set_mset S) & {t}⊆(set_mset S)" by auto
hence "?SIZE = sum (count S) (set_mset S - {t}) + sum (count S) {t}" by (blast dest: sum.subset_diff)
hence "?SIZE = sum (count S) (set_mset S - {t}) + count (S) t" by auto
moreover with A have "count S t = count (S-{#t#}) t + 1" by auto
ultimately have D: "?SIZE = sum (count S) (set_mset S - {t}) + count (S-{#t#}) t + 1" by (arith)
moreover have "sum (count S) (set_mset S - {t}) = sum (count (S-{#t#})) (set_mset S - {t})" proof -
have "∀x∈(set_mset S - {t}). count S x = count (S-{#t#}) x" by (auto iff add: count_ne_remove)
thus ?thesis by simp
qed
ultimately have D: "?SIZE = sum (count (S-{#t#})) (set_mset S - {t}) + count (S-{#t#}) t + 1" by (simp)
moreover
{ assume CASE: "t ∉# S - {#t#}"
from CASE have "set_mset S - {t} = set_mset (S - {#t#})"
by (auto simp add: in_diff_count split: if_splits)
with CASE D have "?SIZE = sum (count (S-{#t#})) (set_mset (S - {#t#})) + 1"
}
moreover
{ assume CASE: "t ∈# S - {#t#}"
from CASE have "t ∈# S" by (rule in_diffD)
with CASE have 1: "set_mset S = set_mset (S-{#t#})"
by (auto simp add: in_diff_count split: if_splits)
moreover from D have "?SIZE = sum (count (S-{#t#})) (set_mset S - {t}) + sum (count (S-{#t#})) {t} + 1" by simp
moreover from SPLITPRE sum.subset_diff have "sum (count (S-{#t#})) (set_mset S) = sum (count (S-{#t#})) (set_mset S - {t}) + sum (count (S-{#t#})) {t}" by (blast)
ultimately have "?SIZE = sum (count (S-{#t#})) (set_mset (S-{#t#})) + 1" by simp
}
ultimately show "?SIZE = sum (count (S-{#t#})) (set_mset (S - {#t#})) + 1" by blast
qed

(* TODO: Check whether this proof can be done simpler *)
lemma mset_union_diff_comm: "t ∈# S ⟹ T + (S - {#t#}) = (T + S) - {#t#}" proof -
assume "t ∈# S"
then obtain S' where S: "S = add_mset t S'"
by (metis insert_DiffM)
then show ?thesis
by auto
qed

(*  lemma mset_diff_diff_left: "A-B-C = A-((B::'a multiset)+C)" proof -
have "∀e . count (A-B-C) e = count (A-(B+C)) e" by auto
thus ?thesis by (simp add: multiset_eq_conv_count_eq)
qed

lemma mset_diff_commute: "A-B-C = A-C-(B::'a multiset)" proof -
have "A-B-C = A-(B+C)" by (simp add: mset_diff_diff_left)
also have "… = A-(C+B)" by (simp add: union_commute)
thus ?thesis by (simp add: mset_diff_diff_left)
qed

lemma mset_diff_same_empty[simp]: "(S::'a multiset) - S = {#}"
proof -
have "∀e . count (S-S) e = 0" by auto
hence "∀e . ~ (e : set_mset (S-S))" by auto
hence "set_mset (S-S) = {}" by blast
thus ?thesis by (auto)
qed
*)

lemma mset_right_cancel_union: "⟦a ∈# A+B; ~(a ∈# B)⟧ ⟹ a∈#A"
by (simp)
lemma mset_left_cancel_union: "⟦a ∈# A+B; ~(a ∈# A)⟧ ⟹ a∈#B"
by (simp)

lemmas mset_cancel_union = mset_right_cancel_union mset_left_cancel_union

lemma mset_right_cancel_elem: "⟦a ∈# A+{#b#}; a~=b⟧ ⟹ a∈#A"
by simp

lemma mset_left_cancel_elem: "⟦a ∈# {#b#}+A; a~=b⟧ ⟹ a∈#A"
by simp

lemmas mset_cancel_elem = mset_right_cancel_elem mset_left_cancel_elem

lemma mset_diff_cancel1elem[simp]: "~(a ∈# B) ⟹ {#a#}-B = {#a#}"
by (auto simp add: not_in_iff intro!: multiset_eqI)

(*  lemma diff_union_inverse[simp]: "A + B - B = (A::'a multiset)"

lemma diff_union_inverse2[simp]: "B + A - B = (A::'a multiset)"
*)
(*lemma union_diff_assoc_se2: "t ∈# A ⟹ (A+B)-{#t#} = (A-{#t#}) + B"
lemmas union_diff_assoc_se = union_diff_assoc_se1 union_diff_assoc_se2*)

lemma union_diff_assoc: "C-B={#} ⟹ (A+B)-C = A + (B-C)"

lemmas mset_neutral_cancel1 = union_left_cancel[where N="{#}", simplified] union_right_cancel[where N="{#}", simplified]
declare mset_neutral_cancel1[simp]

lemma mset_union_2_elem: "{#a, b#} = add_mset c M ⟹ {#a#}=M & b=c | a=c & {#b#}=M"

lemma mset_un_cases[cases set, case_names left right]:
"⟦a ∈# A + B; a ∈# A ⟹ P; a ∈# B ⟹ P⟧ ⟹ P"
by (auto)

lemma mset_unplusm_dist_cases[cases set, case_names left right]:
assumes A: "{#s#}+A = B+C"
assumes L: "⟦B={#s#}+(B-{#s#}); A=(B-{#s#})+C⟧ ⟹ P"
assumes R: "⟦C={#s#}+(C-{#s#}); A=B+(C-{#s#})⟧ ⟹ P"
shows P
proof -
from A[symmetric] have "s ∈# B+C" by simp
thus ?thesis proof (cases rule: mset_un_cases)
case left hence 1: "B={#s#}+(B-{#s#})" by simp
with A have "{#s#}+A = {#s#}+((B-{#s#})+C)" by (simp add: union_ac)
hence 2: "A = (B-{#s#})+C" by (simp)
from L[OF 1 2] show ?thesis .
next
case right hence 1: "C={#s#}+(C-{#s#})" by simp
with A have "{#s#}+A = {#s#}+(B+(C-{#s#}))" by (simp add: union_ac)
hence 2: "A = B+(C-{#s#})" by (simp)
from R[OF 1 2] show ?thesis .
qed
qed

lemma mset_unplusm_dist_cases2[cases set, case_names left right]:
assumes A: "B+C = {#s#}+A"
assumes L: "⟦B={#s#}+(B-{#s#}); A=(B-{#s#})+C⟧ ⟹ P"
assumes R: "⟦C={#s#}+(C-{#s#}); A=B+(C-{#s#})⟧ ⟹ P"
shows P
using mset_unplusm_dist_cases[OF A[symmetric]] L R by blast

lemma mset_single_cases[cases set, case_names loc env]:
assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "⟦c'={#s#}+(c'-{#s#}); c={#r'#}+(c-{#r'#}); c-{#r'#} = c'-{#s#} ⟧ ⟹ P"
shows "P"
proof -
{ assume CASE: "s=r'"
with A have "c=c'" by simp
with CASE CASES have ?thesis by auto
} moreover {
assume CASE: "s≠r'"
have "s ∈# {#s#}+c" by simp
with A have "s ∈# {#r'#}+c'" by simp
with CASE have "s ∈# c'" by simp
hence 1: "c' = {#s#} + (c' - {#s#})" by simp
with A have "{#s#}+c = {#s#}+({#r'#}+(c' - {#s#}))" by (auto simp add: union_ac)
hence 2: "c={#r'#}+(c' - {#s#})" by (auto)
hence 3: "c-{#r'#} = (c' - {#s#})" by auto
from 1 2 3 CASES have ?thesis by auto
} ultimately show ?thesis by blast
qed

lemma mset_single_cases'[cases set, case_names loc env]:
assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "!!cc. ⟦c'={#s#}+cc; c={#r'#}+cc; c'-{#s#}=cc; c-{#r'#}=cc⟧ ⟹ P"
shows "P"
using A  CASES by (auto elim!: mset_single_cases)

lemma mset_single_cases2[cases set, case_names loc env]:
assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "⟦c'=(c'-{#s#})+{#s#}; c=(c-{#r'#})+{#r'#}; c-{#r'#} = c'-{#s#} ⟧ ⟹ P"
shows "P"
proof -
thus ?thesis using CASES by (cases rule: mset_single_cases) simp_all
qed

lemma mset_single_cases2'[cases set, case_names loc env]:
assumes CASES: "⟦s=r'; c=c'⟧ ⟹ P" "!!cc. ⟦c'=cc+{#s#}; c=cc+{#r'#}; c'-{#s#}=cc; c-{#r'#}=cc⟧ ⟹ P"
shows "P"
using A  CASES by (auto elim!: mset_single_cases2)

lemma mset_un_single_un_cases [consumes 1, case_names left right]:
assumes A: "add_mset a A = B + C"
and CASES: "a ∈# B ⟹ A = (B - {#a#}) + C ⟹ P"
"a ∈# C ⟹ A = B + (C - {#a#}) ⟹ P" shows "P"
proof -
have "a ∈# A+{#a#}" by simp
with A have "a ∈# B+C" by auto
thus ?thesis proof (cases rule: mset_un_cases)
case left hence "B=B-{#a#}+{#a#}" by auto
with A have "A+{#a#} = (B-{#a#})+C+{#a#}" by (auto simp add: union_ac)
hence "A=(B-{#a#})+C" by simp
with CASES(1)[OF left] show ?thesis by blast
next
case right hence "C=C-{#a#}+{#a#}" by auto
with A have "A+{#a#} = B+(C-{#a#})+{#a#}" by (auto simp add: union_ac)
hence "A=B+(C-{#a#})" by simp
with CASES(2)[OF right] show ?thesis by blast
qed
qed

(* TODO: Can this proof be done more automatically ? *)
lemma mset_distrib[consumes 1, case_names dist]: assumes A: "(A::'a multiset)+B = M+N" "!!Am An Bm Bn. ⟦A=Am+An; B=Bm+Bn; M=Am+Bm; N=An+Bn⟧ ⟹ P" shows "P"
proof -
have BN_MA: "B - N = M - A"
by (metis (no_types) add_diff_cancel_right assms(1) union_commute)
have H: "A = A∩# C + (A - C) ∩# D" if "A + B = C + D" for A B C D :: "'a multiset"
have A': "A = A∩# M + (A - M) ∩# N"
using A(1) H by blast
moreover have B': "B = (B - N) ∩# M + B∩# N"
using A(1) H[of B A N M] by (auto simp: ac_simps)
moreover have "M = A ∩# M + (B - N) ∩# M"
using H[of M N A B] BN_MA[symmetric] A(1) by (metis (no_types) diff_intersect_left_idem
union_commute)
moreover have "N = (A - M) ∩# N + B ∩# N"
by (metis A' assms(1) diff_union_cancelL inter_union_distrib_left inter_union_distrib_right
mset_subset_eq_multiset_union_diff_commute subset_mset.inf.cobounded1 subset_mset.inf.commute)
ultimately show P
using A(2) by blast
qed

subsubsection ‹Singleton multisets›

lemma mset_size_le1_cases[case_names empty singleton,consumes 1]: "⟦ size M ≤ Suc 0; M={#} ⟹ P; !!m. M={#m#} ⟹ P ⟧ ⟹ P"
by (cases M) auto

lemma diff_union_single_conv2: "a ∈# J ⟹ J + I - {#a#} = (J - {#a#}) + I"
by simp

lemmas diff_union_single_convs = diff_union_single_conv diff_union_single_conv2

lemma mset_contains_eq: "(m ∈# M) = ({#m#}+(M-{#m#})=M)"
using diff_single_trivial by fastforce

subsubsection ‹Pointwise ordering›

(*declare mset_le_trans[trans] Seems to be in there now. Why is this not done in Multiset.thy or order-class ? *)

lemma mset_le_incr_right1: "a⊆#(b::'a multiset) ⟹ a⊆#b+c" using mset_subset_eq_mono_add[of a b "{#}" c, simplified] .
lemma mset_le_incr_right2: "a⊆#(b::'a multiset) ⟹ a⊆#c+b" using mset_le_incr_right1
lemmas mset_le_incr_right = mset_le_incr_right1 mset_le_incr_right2

lemma mset_le_decr_left1: "a+c⊆#(b::'a multiset) ⟹ a⊆#b" using mset_le_incr_right1 mset_subset_eq_mono_add_right_cancel
by blast
lemma mset_le_decr_left2: "c+a⊆#(b::'a multiset) ⟹ a⊆#b" using mset_le_decr_left1

lemmas mset_le_decr_left = mset_le_decr_left1 mset_le_decr_left2 mset_le_add_mset_decr_left1

lemma mset_le_subtract: "A⊆#B ⟹ A-C ⊆# B-(C::'a multiset)"
apply (unfold subseteq_mset_def count_diff)
apply clarify
apply (subgoal_tac "count A a ≤ count B a")
apply arith
apply simp
done

lemma mset_union_subset: "A+B ⊆# C ⟹ A⊆#C ∧ B⊆#(C::'a multiset)"
by (auto dest: mset_le_decr_left)

lemma mset_le_add_mset: "add_mset x B ⊆# C ⟹ {#x#}⊆#C ∧ B⊆#(C::'a multiset)"
by (auto dest: mset_le_decr_left)

lemma mset_le_subtract_left: "A+B ⊆# (X::'a multiset) ⟹ B ⊆# X-A ∧ A⊆#X"
by (auto dest: mset_le_subtract[of "A+B" "X" "A"] mset_union_subset)
lemma mset_le_subtract_right: "A+B ⊆# (X::'a multiset) ⟹ A ⊆# X-B ∧ B⊆#X"
by (auto dest: mset_le_subtract[of "A+B" "X" "B"] mset_union_subset)

lemma mset_le_subtract_add_mset_left: "add_mset x B ⊆# (X::'a multiset) ⟹ B ⊆# X-{#x#} ∧ {#x#}⊆#X"

lemma mset_le_subtract_add_mset_right: "add_mset x B ⊆# (X::'a multiset) ⟹ {#x#} ⊆# X-B ∧ B⊆#X"

lemma mset_le_addE: "⟦ xs ⊆# (ys::'a multiset); !!zs. ys=xs+zs ⟹ P ⟧ ⟹ P" using mset_subset_eq_exists_conv
by blast

lemma mset_2dist2_cases:
assumes A: "{#a#}+{#b#} ⊆# A+B"
assumes CASES: "{#a#}+{#b#} ⊆# A ⟹ P" "{#a#}+{#b#} ⊆# B ⟹ P" "⟦a ∈# A; b ∈# B⟧ ⟹ P" "⟦a ∈# B; b ∈# A⟧ ⟹ P"
shows "P"
proof -
{ assume C: "a ∈# A" "b ∈# A-{#a#}"
with mset_subset_eq_mono_add[of "{#a#}" "{#a#}" "{#b#}" "A-{#a#}"] have "{#a#}+{#b#} ⊆# A" by auto
} moreover {
assume C: "a ∈# A" "¬ (b ∈# A-{#a#})"
with A have "b ∈# B"
by (metis diff_union_single_conv2 mset_le_subtract_left mset_subset_eq_insertD mset_un_cases)
} moreover {
assume C: "¬ (a ∈# A)" "b ∈# B-{#a#}"
with A have "a ∈# B"
by (auto dest: mset_subset_eqD)
with C mset_subset_eq_mono_add[of "{#a#}" "{#a#}" "{#b#}" "B-{#a#}"] have "{#a#}+{#b#} ⊆# B" by auto
} moreover {
assume C: "¬ (a ∈# A)" "¬ (b ∈# B-{#a#})"
with A have "a ∈# B ∧ b ∈# A"
apply (intro conjI)
apply (auto dest!: mset_subset_eq_insertD simp: insert_union_subset_iff; fail)[]
by (metis mset_diff_cancel1elem mset_le_subtract_left multiset_diff_union_assoc
single_subset_iff subset_eq_diff_conv)
} ultimately show P using CASES by blast
qed

lemma mset_union_subset_s: "{#a#}+B ⊆# C ⟹ a ∈# C ∧ B ⊆# C"
by (drule mset_union_subset) simp

(* TODO: Check which of these lemmas are already introduced by order-classes ! *)

lemma mset_le_single_cases[consumes 1, case_names empty singleton]: "⟦M⊆#{#a#}; M={#} ⟹ P; M={#a#} ⟹ P⟧ ⟹ P"
by (induct M) auto

lemma mset_le_distrib[consumes 1, case_names dist]: "⟦(X::'a multiset)⊆#A+B; !!Xa Xb. ⟦X=Xa+Xb; Xa⊆#A; Xb⊆#B⟧ ⟹ P ⟧ ⟹ P"

lemma mset_le_mono_add_single: "⟦a ∈# ys; b ∈# ws⟧ ⟹ {#a#} + {#b#} ⊆# ys + ws"

lemma mset_size1elem: "⟦size P ≤ 1; q ∈# P⟧ ⟹ P={#q#}"
by (auto elim: mset_size_le1_cases)
lemma mset_size2elem: "⟦size P ≤ 2; {#q#}+{#q'#} ⊆# P⟧ ⟹ P={#q#}+{#q'#}"

subsubsection ‹Image under function›

notation image_mset (infixr "#" 90)

lemma mset_map_split_orig: "!!M1 M2. ⟦f # P = M1+M2; !!P1 P2. ⟦P=P1+P2; f # P1 = M1; f # P2 = M2⟧ ⟹ Q ⟧ ⟹ Q"
by (induct P) (force elim!: mset_un_single_un_cases)+

lemma mset_map_id: "⟦!!x. f (g x) = x⟧ ⟹ f # g # X = X"
by (induct X) auto

text ‹The following is a very specialized lemma. Intuitively, it splits the original multiset
by a splitting of some pointwise supermultiset of its image.

Application:
This lemma came in handy when proving the correctness of a constraint system that collects at most k sized submultisets of the sets of spawned threads.
›
lemma mset_map_split_orig_le: assumes A: "f # P ⊆# M1+M2" and EX: "!!P1 P2. ⟦P=P1+P2; f # P1 ⊆# M1; f # P2 ⊆# M2⟧ ⟹ Q" shows "Q"
using A EX by (auto elim: mset_le_distrib mset_map_split_orig)

subsection ‹Lists›

lemma len_greater_imp_nonempty[simp]: "length l > x ⟹ l≠[]"
by auto

lemma list_take_induct_tl2:
"⟦length xs = length ys; ∀n<length xs. P (ys ! n) (xs ! n)⟧
⟹ ∀n < length (tl xs). P ((tl ys) ! n) ((tl xs) ! n)"
by (induct xs ys rule: list_induct2) auto

lemma not_distinct_split_distinct:
assumes "¬ distinct xs"
obtains y ys zs where "distinct ys" "y ∈ set ys" "xs = ys@[y]@zs"
using assms
proof (induct xs rule: rev_induct)
case Nil thus ?case by simp
next
case (snoc x xs) thus ?case by (cases "distinct xs") auto
qed

lemma distinct_length_le:
assumes d: "distinct ys"
and eq: "set ys = set xs"
shows "length ys ≤ length xs"
proof -
from d have "length ys = card (set ys)" by (simp add: distinct_card)
also from eq List.card_set have "card (set ys) = length (remdups xs)" by simp
also have "... ≤ length xs" by simp
finally show ?thesis .
qed

lemma find_SomeD:
"List.find P xs = Some x ⟹ P x"
"List.find P xs = Some x ⟹ x∈set xs"

lemma in_hd_or_tl_conv[simp]: "l≠[] ⟹ x=hd l ∨ x∈set (tl l) ⟷ x∈set l"
by (cases l) auto

lemma length_dropWhile_takeWhile:
assumes "x < length (dropWhile P xs)"
shows "x + length (takeWhile P xs) < length xs"
using assms
by (induct xs) auto

text ‹Elim-version of @{thm neq_Nil_conv}.›
lemma neq_NilE:
assumes "l≠[]"
obtains x xs where "l=x#xs"
using assms by (metis list.exhaust)

lemma length_Suc_rev_conv: "length xs = Suc n ⟷ (∃ys y. xs=ys@[y] ∧ length ys = n)"
by (cases xs rule: rev_cases) auto

subsubsection ‹List Destructors›
lemma not_hd_in_tl:
"x ≠ hd xs ⟹ x ∈ set xs ⟹ x ∈ set (tl xs)"
by (induct xs) simp_all

lemma distinct_hd_tl:
"distinct xs ⟹ x = hd xs ⟹ x ∉ set (tl (xs))"
by (induct xs) simp_all

lemma in_set_tlD: "x ∈ set (tl xs) ⟹ x ∈ set xs"
by (induct xs) simp_all

lemma nth_tl: "xs ≠ [] ⟹ tl xs ! n = xs ! Suc n"
by (induct xs) simp_all

lemma tl_subset:
"xs ≠ [] ⟹ set xs ⊆ A ⟹ set (tl xs) ⊆ A"
by (metis in_set_tlD rev_subsetD subsetI)

lemma tl_last:
"tl xs ≠ [] ⟹ last xs = last (tl xs)"
by (induct xs) simp_all

lemma tl_obtain_elem:
assumes "xs ≠ []" "tl xs = []"
obtains e where "xs = [e]"
using assms
by (induct xs rule: list_nonempty_induct) simp_all

lemma butlast_subset:
"xs ≠ [] ⟹ set xs ⊆ A ⟹ set (butlast xs) ⊆ A"
by (metis in_set_butlastD rev_subsetD subsetI)

lemma butlast_rev_tl:
"xs ≠ [] ⟹ butlast (rev xs) = rev (tl xs)"
by (induct xs rule: rev_induct) simp_all

lemma hd_butlast:
"length xs > 1 ⟹ hd (butlast xs) = hd xs"
by (induct xs) simp_all

lemma butlast_upd_last_eq[simp]: "length l ≥ 2 ⟹
(butlast l) [ length l - 2 := x ] = take (length l - 2) l @ [x]"
apply (case_tac l rule: rev_cases)
apply simp
apply simp
apply (case_tac ys rule: rev_cases)
apply simp
apply simp
done

lemma distinct_butlast_swap[simp]:
"distinct pq ⟹ distinct (butlast (pq[i := last pq]))"
apply (cases pq rule: rev_cases)
apply (auto simp: list_update_append distinct_list_update split: nat.split)
done

subsubsection ‹Splitting list according to structure of other list›
context begin
private definition "SPLIT_ACCORDING m l ≡ length l = length m"

private lemma SPLIT_ACCORDINGE:
assumes "length m = length l"
obtains "SPLIT_ACCORDING m l"
unfolding SPLIT_ACCORDING_def using assms by auto

private lemma SPLIT_ACCORDING_simp:
"SPLIT_ACCORDING m (l1@l2) ⟷ (∃m1 m2. m=m1@m2 ∧ SPLIT_ACCORDING m1 l1 ∧ SPLIT_ACCORDING m2 l2)"
"SPLIT_ACCORDING m (x#l') ⟷ (∃y m'. m=y#m' ∧ SPLIT_ACCORDING m' l')"
apply (fastforce simp: SPLIT_ACCORDING_def intro: exI[where x = "take (length l1) m"] exI[where x = "drop (length l1) m"])
apply (cases m;auto simp: SPLIT_ACCORDING_def)
done

text ‹Split structure of list @{term m} according to structure of list @{term l}.›
method split_list_according for m :: "'a list" and l :: "'b list" =
(rule SPLIT_ACCORDINGE[of m l],
(simp; fail),
( simp only: SPLIT_ACCORDING_simp,
elim exE conjE,
simp only: SPLIT_ACCORDING_def
)
)
end

subsubsection ‹‹list_all2››
lemma list_all2_induct[consumes 1, case_names Nil Cons]:
assumes "list_all2 P l l'"
assumes "Q [] []"
assumes "⋀x x' ls ls'. ⟦ P x x'; list_all2 P ls ls'; Q ls ls' ⟧
⟹ Q (x#ls) (x'#ls')"
shows "Q l l'"
using list_all2_lengthD[OF assms(1)] assms
apply (induct rule: list_induct2)
apply auto
done

subsubsection ‹Indexing›

lemma ran_nth_set_encoding_conv[simp]:
"ran (λi. if i<length l then Some (l!i) else None) = set l"
apply safe
apply (auto simp: ran_def split: if_split_asm) []
apply (auto simp: in_set_conv_nth intro: ranI) []
done

lemma nth_image_indices[simp]: "(!) l  {0..<length l} = set l"
by (auto simp: in_set_conv_nth)

lemma nth_update_invalid[simp]:"¬i<length l ⟹ l[j:=x]!i = l!i"
apply (induction l arbitrary: i j)
apply (auto split: nat.splits)
done

lemma nth_list_update': "l[i:=x]!j = (if i=j ∧ i<length l then x else l!j)"
by auto

lemma last_take_nth_conv: "n ≤ length l ⟹ n≠0 ⟹ last (take n l) = l!(n - 1)"
apply (induction l arbitrary: n)
apply (auto simp: take_Cons split: nat.split)
done

lemma nth_append_first[simp]: "i<length l ⟹ (l@l')!i = l!i"

lemma in_set_image_conv_nth: "f x ∈ fset l ⟷ (∃i<length l. f (l!i) = f x)"
by (auto simp: in_set_conv_nth) (metis image_eqI nth_mem)

lemma set_image_eq_pointwiseI:
assumes "length l = length l'"
assumes "⋀i. i<length l ⟹ f (l!i) = f (l'!i)"
shows "fset l = fset l'"
using assms
by (fastforce simp: in_set_conv_nth in_set_image_conv_nth)

lemma insert_swap_set_eq: "i<length l ⟹ insert (l!i) (set (l[i:=x])) = insert x (set l)"
by (auto simp: in_set_conv_nth nth_list_update split: if_split_asm)

subsubsection ‹Reverse lists›
lemma neq_Nil_revE:
assumes "l≠[]"
obtains ll e  where "l = ll@[e]"
using assms by (cases l rule: rev_cases) auto

lemma neq_Nil_rev_conv: "l≠[] ⟷ (∃xs x. l = xs@[x])"
by (cases l rule: rev_cases) auto

text ‹Caution: Same order of case variables in snoc-case as @{thm [source] rev_exhaust}, the other way round than @{thm [source] rev_induct} !›
lemma length_compl_rev_induct[case_names Nil snoc]: "⟦P []; !! l e . ⟦!! ll . length ll <= length l ⟹ P ll⟧ ⟹ P (l@[e])⟧ ⟹ P l"
apply(induct_tac l rule: length_induct)
apply(case_tac "xs" rule: rev_cases)
apply(auto)
done

lemma list_append_eq_Cons_cases[consumes 1]: "⟦ys@zs = x#xs; ⟦ys=[]; zs=x#xs⟧ ⟹ P; !!ys'. ⟦ ys=x#ys'; ys'@zs=xs ⟧ ⟹ P ⟧ ⟹ P"
lemma list_Cons_eq_append_cases[consumes 1]: "⟦x#xs = ys@zs; ⟦ys=[]; zs=x#xs⟧ ⟹ P; !!ys'. ⟦ ys=x#ys'; ys'@zs=xs ⟧ ⟹ P ⟧ ⟹ P"

lemma map_of_rev_distinct[simp]:
"distinct (map fst m) ⟹ map_of (rev m) = map_of m"
apply (induct m)
apply simp

apply simp
apply force
apply simp
done

― ‹Tail-recursive, generalized @{const rev}. May also be used for
tail-recursively getting a list with all elements of the two
operands, if the order does not matter, e.g. when implementing
sets by lists.›
fun revg where
"revg [] b = b" |
"revg (a#as) b = revg as (a#b)"

lemma revg_fun[simp]: "revg a b = rev a @ b"
by (induct a arbitrary: b)
auto

lemma rev_split_conv[simp]:
"l ≠ [] ⟹ rev (tl l) @ [hd l] = rev l"
by (induct l) simp_all

lemma rev_butlast_is_tl_rev: "rev (butlast l) = tl (rev l)"
by (induct l) auto

lemma hd_last_singletonI:
"⟦xs ≠ []; hd xs = last xs; distinct xs⟧ ⟹ xs = [hd xs]"
by (induct xs rule: list_nonempty_induct) auto

lemma last_filter:
"⟦xs ≠ []; P (last xs)⟧ ⟹ last (filter P xs) = last xs"
by (induct xs rule: rev_nonempty_induct) simp_all

(* As the following two rules are similar in nature to list_induct2',
they are named accordingly. *)
lemma rev_induct2' [case_names empty snocl snocr snoclr]:
assumes empty: "P [] []"
and snocl: "⋀x xs. P (xs@[x]) []"
and snocr: "⋀y ys. P [] (ys@[y])"
and snoclr: "⋀x xs y ys.  P xs ys  ⟹ P (xs@[x]) (ys@[y])"
shows "P xs ys"
proof (induct xs arbitrary: ys rule: rev_induct)
case Nil thus ?case using empty snocr
by (cases ys rule: rev_exhaust) simp_all
next
case snoc thus ?case using snocl snoclr
by (cases ys rule: rev_exhaust) simp_all
qed

lemma rev_nonempty_induct2' [case_names single snocl snocr snoclr, consumes 2]:
assumes "xs ≠ []" "ys ≠ []"
assumes single': "⋀x y. P [x] [y]"
and snocl: "⋀x xs y. xs ≠ [] ⟹ P (xs@[x]) [y]"
and snocr: "⋀x y ys. ys ≠ [] ⟹ P [x] (ys@[y])"
and snoclr: "⋀x xs y ys. ⟦P xs ys; xs ≠ []; ys≠[]⟧  ⟹ P (xs@[x]) (ys@[y])"
shows "P xs ys"
using assms(1,2)
proof (induct xs arbitrary: ys rule: rev_nonempty_induct)
case single then obtain z zs where "ys = zs@[z]" by (metis rev_exhaust)
thus ?case using single' snocr
by (cases "zs = []") simp_all
next
case (snoc x xs) then obtain z zs where zs: "ys = zs@[z]" by (metis rev_exhaust)
thus ?case using snocl snoclr snoc
by (cases "zs = []") simp_all
qed

subsubsection "Folding"

text "Ugly lemma about foldl over associative operator with left and right neutral element"
lemma foldl_A1_eq: "!!i. ⟦ !! e. f n e = e; !! e. f e n = e; !!a b c. f a (f b c) = f (f a b) c ⟧ ⟹ foldl f i ww = f i (foldl f n ww)"
proof (induct ww)
case Nil thus ?case by simp
next
case (Cons a ww i) note IHP[simplified]=this
have "foldl f i (a # ww) = foldl f (f i a) ww" by simp
also from IHP have "… = f (f i a) (foldl f n ww)" by blast
also from IHP(4) have "… = f i (f a (foldl f n ww))" by simp
also from IHP(1)[OF IHP(2,3,4), where i=a] have "… = f i (foldl f a ww)" by simp
also from IHP(2)[of a] have "… = f i (foldl f (f n a) ww)" by simp
also have "… = f i (foldl f n (a#ww))" by simp
finally show ?case .
qed

lemmas foldl_conc_empty_eq = foldl_A1_eq[of "(@)" "[]", simplified]
lemmas foldl_un_empty_eq = foldl_A1_eq[of "(∪)" "{}", simplified, OF Un_assoc[symmetric]]

lemma foldl_set: "foldl (∪) {} l = ⋃{x. x∈set l}"
apply (induct l)
apply simp_all
apply (subst foldl_un_empty_eq)
apply auto
done

lemma (in monoid_mult) foldl_absorb1: "x*foldl (*) 1 zs = foldl (*) x zs"
apply (rule sym)
apply (rule foldl_A1_eq)
done

text ‹Towards an invariant rule for foldl›
lemma foldl_rule_aux:
fixes I :: "'σ ⇒ 'a list ⇒ bool"
assumes initial: "I σ0 l0"
assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ (x#l2) ⟧ ⟹ I (f σ x) l2"
shows "I (foldl f σ0 l0) []"
using initial step
apply (induct l0 arbitrary: σ0)
apply auto
done

lemma foldl_rule_aux_P:
fixes I :: "'σ ⇒ 'a list ⇒ bool"
assumes initial: "I σ0 l0"
assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ (x#l2) ⟧ ⟹ I (f σ x) l2"
assumes final: "!!σ. I σ [] ⟹ P σ"
shows "P (foldl f σ0 l0)"
using foldl_rule_aux[of I σ0 l0, OF initial, OF step] final
by simp

lemma foldl_rule:
fixes I :: "'σ ⇒ 'a list ⇒ 'a list ⇒ bool"
assumes initial: "I σ0 [] l0"
assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ l1 (x#l2) ⟧ ⟹ I (f σ x) (l1@[x]) l2"
shows "I (foldl f σ0 l0) l0 []"
using initial step
apply (rule_tac I="λσ lr. ∃ll. l0=ll@lr ∧ I σ ll lr" in foldl_rule_aux_P)
apply auto
done

text ‹
Invariant rule for foldl. The invariant is parameterized with
the state, the list of items that have already been processed and
the list of items that still have to be processed.
›
lemma foldl_rule_P:
fixes I :: "'σ ⇒ 'a list ⇒ 'a list ⇒ bool"
― ‹The invariant holds for the initial state, no items processed yet and all items to be processed:›
assumes initial: "I σ0 [] l0"
― ‹The invariant remains valid if one item from the list is processed›
assumes step: "!!l1 l2 x σ. ⟦ l0=l1@x#l2; I σ l1 (x#l2) ⟧ ⟹ I (f σ x) (l1@[x]) l2"
― ‹The proposition follows from the invariant in the final state, i.e. all items processed and nothing to be processed›
assumes final: "!!σ. I σ l0 [] ⟹ P σ"
shows "P (foldl f σ0 l0)"
using foldl_rule[of I, OF initial step] by (simp add: final)

text ‹Invariant reasoning over @{const foldl} for distinct lists. Invariant rule makes no
lemma distinct_foldl_invar:
"⟦ distinct S; I (set S) σ0;
⋀x it σ. ⟦x ∈ it; it ⊆ set S; I it σ⟧ ⟹ I (it - {x}) (f σ x)
⟧ ⟹ I {} (foldl f σ0 S)"
proof (induct S arbitrary: σ0)
case Nil thus ?case by auto
next
case (Cons x S)

note [simp] = Cons.prems(1)[simplified]

show ?case
apply simp
apply (rule Cons.hyps)
proof -
from Cons.prems(1) show "distinct S" by simp
from Cons.prems(3)[of x "set (x#S)", simplified,
OF Cons.prems(2)[simplified]]
show "I (set S) (f σ0 x)" .
fix xx it σ
assume A: "xx∈it" "it ⊆ set S" "I it σ"
show "I (it - {xx}) (f σ xx)" using A(2)
apply (rule_tac Cons.prems(3))
apply blast
done
qed
qed

lemma foldl_length_aux: "foldl (λi x. Suc i) a l = a + length l"
by (induct l arbitrary: a) auto

lemmas foldl_length[simp] = foldl_length_aux[where a=0, simplified]

lemma foldr_length_aux: "foldr (λx i. Suc i) l a = a + length l"
by (induct l arbitrary: a rule: rev_induct) auto

lemmas foldr_length[simp] = foldr_length_aux[where a=0, simplified]

context comp_fun_commute begin

lemma foldl_f_commute: "f a (foldl (λa b. f b a) b xs) = foldl (λa b. f b a) (f a b) xs"
by(induct xs arbitrary: b)(simp_all add: fun_left_comm)

lemma foldr_conv_foldl: "foldr f xs a = foldl (λa b. f b a) a xs"
by(induct xs arbitrary: a)(simp_all add: foldl_f_commute)

end

lemma filter_conv_foldr:
"filter P xs = foldr (λx xs. if P x then x # xs else xs) xs []"
by(induct xs) simp_all

lemma foldr_Cons: "foldr Cons xs [] = xs"
by(induct xs) simp_all

lemma foldr_snd_zip:
"length xs ≥ length ys ⟹ foldr (λ(x, y). f y) (zip xs ys) b = foldr f ys b"
proof(induct ys arbitrary: xs)
case (Cons y ys) thus ?case by(cases xs) simp_all
qed simp

lemma foldl_snd_zip:
"length xs ≥ length ys ⟹ foldl (λb (x, y). f b y) b (zip xs ys) = foldl f b ys"
proof(induct ys arbitrary: xs b)
case (Cons y ys) thus ?case by(cases xs) simp_all
qed simp

lemma fst_foldl: "fst (foldl (λ(a, b) x. (f a x, g a b x)) (a, b) xs) = foldl f a xs"
by(induct xs arbitrary: a b) simp_all

lemma foldl_foldl_conv_concat: "foldl (foldl f) a xs = foldl f a (concat xs)"
by(induct xs arbitrary: a) simp_all

lemma foldl_list_update:
"n < length xs ⟹ foldl f a (xs[n := x]) = foldl f (f (foldl f a (take n xs)) x) (drop (Suc n) xs)"

lemma map_by_foldl:
fixes l :: "'a list" and f :: "'a ⇒ 'b"
shows "foldl (λl x. l@[f x]) [] l = map f l"
proof -
{
fix l'
have "foldl (λl x. l@[f x]) l' l = l'@map f l"
by (induct l arbitrary: l') auto
} thus ?thesis by simp
qed

subsubsection ‹Sorting›

lemma sorted_in_between:
assumes A: "0≤i" "i<j" "j<length l"
assumes S: "sorted l"
assumes E: "l!i ≤ x" "x<l!j"
obtains k where "i≤k" and "k<j" and "l!k≤x" and "x<l!(k+1)"
proof -
from A E have "∃k. i≤k ∧ k<j ∧ l!k≤x ∧ x<l!(k+1)"
proof (induct "j-i" arbitrary: i j)
case (Suc d)
show ?case proof (cases "l!(i+1) ≤ x")
case True
from True Suc.hyps have "d = j - (i + 1)" by simp
moreover from True have "i+1 < j"
by (metis Suc.prems Suc_eq_plus1 Suc_lessI not_less)
moreover from True have "0≤i+1" by simp
ultimately obtain k where
"i+1≤k" "k<j" "l!k ≤ x" "x<l!(k+1)"
using Suc.hyps(1)[of j "i+1"] Suc.prems True
by auto
thus ?thesis by (auto dest: Suc_leD)
next
case False
show ?thesis proof (cases "x<(l!(j - 1))")
case True
from True Suc.hyps have "d = j - (i + 1)" by simp
moreover from True Suc.prems have "i < j - 1"
by (metis Suc_eq_plus1 Suc_lessI diff_Suc_1 less_diff_conv not_le)
moreover from True Suc.prems have "j - 1 < length l" by simp
ultimately obtain k where
"i≤k" "k<j - 1" "l!k ≤ x" "x<l!(k+1)"
using Suc.hyps(1)[of "j - 1" i] Suc.prems True
by auto
thus ?thesis by (auto dest: Suc_leD)
next
case False thus ?thesis using Suc
apply clarsimp
not0_implies_Suc not_less)
qed
qed
qed simp
thus ?thesis by (blast intro: that)
qed

lemma sorted_hd_last:
"⟦sorted l; l≠[]⟧ ⟹ hd l ≤ last l"
by (metis eq_iff hd_Cons_tl last_in_set not_hd_in_tl sorted_wrt.simps(2))

lemma (in linorder) sorted_hd_min:
"⟦xs ≠ []; sorted xs⟧ ⟹ ∀x ∈ set xs. hd xs ≤ x"
by (induct xs, auto)

lemma sorted_append_bigger:
"⟦sorted xs; ∀x ∈ set xs. x ≤ y⟧ ⟹ sorted (xs @ [y])"
proof (induct xs)
case Nil
then show ?case by simp
next
case (Cons x xs)
then have s: "sorted xs" by (cases xs) simp_all
from Cons have a: "∀x∈set xs. x ≤ y" by simp
from Cons(1)[OF s a] Cons(2-) show ?case by (cases xs) simp_all
qed

lemma sorted_filter':
"sorted l ⟹ sorted (filter P l)"
using sorted_filter[where f=id, simplified] .

subsubsection ‹Map›
(* List.thy has:
declare map_eq_Cons_D [dest!]  Cons_eq_map_D [dest!] *)
lemma map_eq_consE: "⟦map f ls = fa#fl; !!a l. ⟦ ls=a#l; f a=fa; map f l = fl ⟧ ⟹ P⟧ ⟹ P"
by auto

lemma map_fst_mk_snd[simp]: "map fst (map (λx. (x,k)) l) = l" by (induct l) auto
lemma map_snd_mk_fst[simp]: "map snd (map (λx. (k,x)) l) = l" by (induct l) auto
lemma map_fst_mk_fst[simp]: "map fst (map (λx. (k,x)) l) = replicate (length l) k" by (induct l) auto
lemma map_snd_mk_snd[simp]: "map snd (map (λx. (x,k)) l) = replicate (length l) k" by (induct l) auto

lemma map_zip1: "map (λx. (x,k)) l = zip l (replicate (length l) k)" by (induct l) `