# Theory Utils

```theory Utils
imports Regular_Tree_Relations.Term_Context
Regular_Tree_Relations.FSet_Utils
begin

subsection ‹Misc›

definition "funas_trs ℛ = ⋃ ((λ (s, t). funas_term s ∪ funas_term t) ` ℛ)"

fun linear_term :: "('f, 'v) term ⇒ bool" where
"linear_term (Var _) = True" |
"linear_term (Fun _ ts) = (is_partition (map vars_term ts) ∧ (∀t∈set ts. linear_term t))"

fun vars_term_list :: "('f, 'v) term ⇒ 'v list" where
"vars_term_list (Var x) = [x]" |
"vars_term_list (Fun _ ts) = concat (map vars_term_list ts)"

fun varposs :: "('f, 'v) term ⇒ pos set" where
"varposs (Var x) = {[]}" |
"varposs (Fun f ts) = (⋃i<length ts. {i # p | p. p ∈ varposs (ts ! i)})"

abbreviation "poss_args f ts ≡ map2 (λ i t. map ((#) i) (f t)) ([0 ..< length ts]) ts"

fun varposs_list :: "('f, 'v) term ⇒ pos list" where
"varposs_list (Var x) = [[]]" |
"varposs_list (Fun f ts) = concat (poss_args varposs_list ts)"

fun concat_index_split where
"concat_index_split (o_idx, i_idx) (x # xs) =
(if i_idx < length x
then (o_idx, i_idx)
else concat_index_split (Suc o_idx, i_idx - length x) xs)"

inductive_set trancl_list for ℛ where
base[intro, Pure.intro] : "length xs = length ys ⟹
(∀ i < length ys. (xs ! i, ys ! i) ∈ ℛ) ⟹ (xs, ys) ∈ trancl_list ℛ"
| list_trancl [Pure.intro]: "(xs, ys) ∈ trancl_list ℛ ⟹ i < length ys ⟹ (ys ! i, z) ∈ ℛ ⟹
(xs, ys[i := z]) ∈ trancl_list ℛ"

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

lemma sum_list_replicate_length' [simp]:
"sum_list (replicate n (Suc 0)) = n"
by (induct n) simp_all

lemma arg_subteq [simp]:
assumes "t ∈ set ts" shows "Fun f ts ⊵ t"
using assms by auto

lemma finite_funas_term: "finite (funas_term s)"
by (induct s) auto

lemma finite_funas_trs:
"finite ℛ ⟹ finite (funas_trs ℛ)"
by (induct rule: finite.induct) (auto simp: finite_funas_term funas_trs_def)

fun subterms where
"subterms (Var x) = {Var x}"|
"subterms (Fun f ts) = {Fun f ts} ∪ (⋃ (subterms ` set ts))"

lemma finite_subterms_fun: "finite (subterms s)"
by (induct s) auto

lemma subterms_supteq_conv: "t ∈ subterms s ⟷ s ⊵ t"
by (induct s) (auto elim: supteq.cases)

lemma set_all_subteq_subterms:
"subterms s = {t. s ⊵ t}"
using subterms_supteq_conv by auto

lemma finite_subterms: "finite {t. s ⊵ t}"
unfolding set_all_subteq_subterms[symmetric]

lemma finite_strict_subterms: "finite {t. s ⊳ t}"
by (intro finite_subset[OF _ finite_subterms]) auto

lemma finite_UN_I2:
"finite A ⟹ (∀ B ∈ A. finite B) ⟹ finite (⋃ A)"
by blast

lemma root_substerms_funas_term:
"the ` (root ` (subterms s) - {None}) = funas_term s" (is "?Ls = ?Rs")
proof -
thm subterms.induct
{fix x assume "x ∈ ?Ls" then have "x ∈ ?Rs"
proof (induct s arbitrary: x)
case (Fun f ts)
then show ?case
by auto (metis DiffI Fun.hyps imageI option.distinct(1) singletonD)
qed auto}
moreover
{fix g assume "g ∈ ?Rs" then have "g ∈ ?Ls"
proof (induct s arbitrary: g)
case (Fun f ts)
from Fun(2) consider "g = (f, length ts)" | "∃ t ∈ set ts. g ∈ funas_term t"
by (force simp: in_set_conv_nth)
then show ?case
proof cases
case 1 then show ?thesis
by (auto simp: image_iff intro: bexI[of _ "Some (f, length ts)"])
next
case 2
then obtain t where wit: "t ∈ set ts" "g ∈ funas_term t" by blast
have "g ∈ the ` (root ` subterms t - {None})" using Fun(1)[OF wit] .
then show ?thesis using wit(1)
by (auto simp: image_iff)
qed
qed auto}
ultimately show ?thesis by auto
qed

lemma root_substerms_funas_term_set:
"the ` (root ` ⋃ (subterms ` R) - {None}) = ⋃ (funas_term ` R)"
using root_substerms_funas_term
by auto (smt DiffE DiffI UN_I image_iff)

lemma subst_merge:
assumes part: "is_partition (map vars_term ts)"
shows "∃σ. ∀i<length ts. ∀x∈vars_term (ts ! i). σ x = τ i x"
proof -
let ?τ = "map τ [0 ..< length ts]"
let ?σ = "fun_merge ?τ (map vars_term ts)"
show ?thesis
by (rule exI[of _ ?σ], intro allI impI ballI,
insert fun_merge_part[OF part, of _ _ ?τ], auto)
qed

lemma rel_comp_empty_trancl_simp: "R O R = {} ⟹ R⇧+ = R"
by (metis O_assoc relcomp_empty2 sup_bot_right trancl_unfold trancl_unfold_right)

lemma choice_nat:
assumes "∀i<n. ∃x. P x i"
shows "∃f. ∀x<n. P (f x) x" using assms
proof -
from assms have "∀ i. ∃ x. i < n ⟶ P x i" by simp
from choice[OF this] show ?thesis by auto
qed

lemma subseteq_set_conv_nth:
"(∀ i < length ss. ss ! i ∈ T) ⟷ set ss ⊆ T"
by (metis in_set_conv_nth subset_code(1))

lemma singelton_trancl [simp]: "{a}⇧+ = {a}"
using tranclD tranclD2 by fastforce

context
includes fset.lifting
begin
lemmas frelcomp_empty_ftrancl_simp = rel_comp_empty_trancl_simp [Transfer.transferred]
lemmas in_fset_idx = in_set_idx [Transfer.transferred]
lemmas fsubseteq_fset_conv_nth = subseteq_set_conv_nth [Transfer.transferred]
lemmas singelton_ftrancl [simp] = singelton_trancl [Transfer.transferred]
end

lemma set_take_nth:
assumes "x ∈ set (take i xs)"
shows "∃ j < length xs. j < i ∧ xs ! j = x" using assms
by (metis in_set_conv_nth length_take min_less_iff_conj nth_take)

lemma nth_sum_listI:
assumes "length xs = length ys"
and "∀ i < length xs. xs ! i = ys ! i"
shows "sum_list xs = sum_list ys"
using assms nth_equalityI by blast

lemma concat_nth_length:
"i < length uss ⟹ j < length (uss ! i) ⟹
sum_list (map length (take i uss)) + j < length (concat uss)"
by (induct uss arbitrary: i j) (simp, case_tac i, auto)

lemma sum_list_1_E [elim]:
assumes "sum_list xs = Suc 0"
obtains i where "i < length xs" "xs ! i = Suc 0" "∀ j < length xs. j ≠ i ⟶ xs ! j = 0"
proof -
have "∃ i < length xs. xs ! i = Suc 0 ∧ (∀ j < length xs. j ≠ i ⟶ xs ! j = 0)" using assms
proof (induct xs)
case (Cons a xs) show ?case
proof (cases a)
case [simp]: 0
obtain i where "i < length xs" "xs ! i = Suc 0" "(∀ j < length xs. j ≠ i ⟶ xs ! j = 0)"
using Cons by auto
then show ?thesis using less_Suc_eq_0_disj
by (intro exI[of _ "Suc i"]) auto
next
case (Suc nat) then show ?thesis using Cons by auto
qed
qed auto
then show " (⋀i. i < length xs ⟹ xs ! i = Suc 0 ⟹ ∀j<length xs. j ≠ i ⟶ xs ! j = 0 ⟹ thesis) ⟹ thesis"
by blast
qed

lemma nth_equalityE:
"xs = ys ⟹ (length xs = length ys ⟹ (⋀i. i < length xs ⟹ xs ! i = ys ! i) ⟹ P) ⟹ P"
by simp

lemma map_cons_presv_distinct:
"distinct t ⟹ distinct (map ((#) i) t)"

lemma concat_nth_nthI:
assumes "length ss = length ts" "∀ i < length ts. length (ss ! i) = length (ts ! i)"
and "∀ i < length ts. ∀ j < length (ts ! i). P (ss ! i ! j) (ts ! i ! j)"
shows "∀ i < length (concat ts). P (concat ss ! i) (concat ts ! i)"
using assms by (metis nth_concat_two_lists)

lemma last_nthI:
assumes "i < length ts" "¬ i < length ts - Suc 0"
shows "ts ! i = last ts" using assms
by (induct ts arbitrary: i)
(auto, metis last_conv_nth length_0_conv less_antisym nth_Cons')

(* induction scheme for transitive closures of lists *)
lemma trancl_list_appendI [simp, intro]:
"(xs, ys) ∈ trancl_list ℛ ⟹ (x, y) ∈ ℛ ⟹ (x # xs, y # ys) ∈ trancl_list ℛ"
proof (induct rule: trancl_list.induct)
case (base xs ys)
then show ?case using less_Suc_eq_0_disj
by (intro trancl_list.base) auto
next
case (list_trancl xs ys i z)
from list_trancl(3) have *: "y # ys[i := z] = (y # ys)[Suc i := z]" by auto
show ?case using list_trancl unfolding *
by (intro trancl_list.list_trancl) auto
qed

lemma trancl_list_append_tranclI [intro]:
"(x, y) ∈ ℛ⇧+ ⟹ (xs, ys) ∈ trancl_list ℛ ⟹ (x # xs, y # ys) ∈ trancl_list ℛ"
proof (induct rule: trancl.induct)
case (trancl_into_trancl a b c)
then have "(a # xs, b # ys) ∈ trancl_list ℛ" by auto
from trancl_list.list_trancl[OF this, of 0 c]
show ?case using trancl_into_trancl(3)
by auto
qed auto

lemma trancl_list_conv:
"(xs, ys) ∈ trancl_list ℛ ⟷ length xs = length ys ∧ (∀ i < length ys. (xs ! i, ys ! i) ∈ ℛ⇧+)" (is "?Ls ⟷ ?Rs")
proof
assume "?Ls" then show ?Rs
proof (induct)
case (list_trancl xs ys i z)
then show ?case
by auto (metis nth_list_update trancl.trancl_into_trancl)
qed auto
next
assume ?Rs then show ?Ls
proof (induct ys arbitrary: xs)
case Nil
then show ?case by (cases xs) auto
next
case (Cons y ys)
from Cons(2) obtain x xs' where *: "xs = x # xs'" and
inv: "(x, y) ∈ ℛ⇧+"
by (cases xs) auto
show ?case using Cons(1)[of "tl xs"] Cons(2) unfolding *
by (intro trancl_list_append_tranclI[OF inv]) force
qed
qed

lemma trancl_list_induct [consumes 2, case_names base step]:
assumes "length ss = length ts" "∀ i < length ts. (ss ! i, ts ! i) ∈ ℛ⇧+"
and "⋀xs ys. length xs = length ys ⟹ ∀ i < length ys. (xs ! i, ys ! i) ∈ ℛ ⟹ P xs ys"
and "⋀xs ys i z. length xs = length ys ⟹ ∀ i < length ys. (xs ! i, ys ! i) ∈ ℛ⇧+ ⟹ P xs ys
⟹ i < length ys ⟹ (ys ! i, z) ∈ ℛ ⟹ P xs (ys[i := z])"
shows "P ss ts" using assms
by (intro trancl_list.induct[of ss ts ℛ P]) (auto simp: trancl_list_conv)

lemma swap_trancl:
"(prod.swap ` R)⇧+ = prod.swap ` (R⇧+)"
proof -
have [simp]: "prod.swap ` X = X¯" for X by auto
show ?thesis by (simp add: trancl_converse)
qed

lemma swap_rtrancl:
"(prod.swap ` R)⇧* = prod.swap ` (R⇧*)"
proof -
have [simp]: "prod.swap ` X = X¯" for X by auto
show ?thesis by (simp add: rtrancl_converse)
qed

lemma Restr_simps:
"R ⊆ X × X ⟹ Restr (R⇧+) X = R⇧+"
"R ⊆ X × X ⟹ Restr Id X O R = R"
"R ⊆ X × X ⟹ R O Restr Id X = R"
"R ⊆ X × X ⟹ S ⊆ X × X ⟹ Restr (R O S) X = R O S"
"R ⊆ X × X ⟹ R⇧+ ⊆ X × X"
subgoal using trancl_mono_set[of R "X × X"] by (auto simp: trancl_full_on)
subgoal by auto
subgoal by auto
subgoal by auto
subgoal using trancl_subset_Sigma .
done

lemma Restr_tracl_comp_simps:
"ℛ ⊆ X × X ⟹ ℒ ⊆ X × X ⟹ ℒ⇧+ O ℛ ⊆ X × X"
"ℛ ⊆ X × X ⟹ ℒ ⊆ X × X ⟹ ℒ O ℛ⇧+ ⊆ X × X"
"ℛ ⊆ X × X ⟹ ℒ ⊆ X × X ⟹ ℒ⇧+ O ℛ O ℒ⇧+ ⊆ X × X"
by (auto dest: subsetD[OF Restr_simps(5)[of ℒ]] subsetD[OF Restr_simps(5)[of ℛ]])

text ‹Conversions of the Nth function between lists and a spliting of the list into lists of lists›

lemma concat_index_split_mono_first_arg:
"i < length (concat xs) ⟹ o_idx ≤ fst (concat_index_split (o_idx, i) xs)"

lemma concat_index_split_sound_fst_arg_aux:
"i < length (concat xs) ⟹ fst (concat_index_split (o_idx, i) xs) < length xs + o_idx"

lemma concat_index_split_sound_fst_arg:
"i < length (concat xs) ⟹ fst (concat_index_split (0, i) xs) < length xs"
using concat_index_split_sound_fst_arg_aux[of i xs 0] by auto

lemma concat_index_split_sound_snd_arg_aux:
assumes "i < length (concat xs)"
shows "snd (concat_index_split (n, i) xs) < length (xs ! (fst (concat_index_split (n, i) xs) - n))" using assms
proof (induct xs arbitrary: i n)
case (Cons x xs)
show ?case proof (cases "i < length x")
case False then have size: "i - length x < length (concat xs)"
using Cons(2) False by auto
obtain k j where [simp]: "concat_index_split (Suc n, i - length x) xs = (k, j)"
using old.prod.exhaust by blast
show ?thesis using False Cons(1)[OF size, of "Suc n"] concat_index_split_mono_first_arg[OF size, of "Suc n"]
by (auto simp: nth_append)
qed auto

lemma concat_index_split_sound_snd_arg:
assumes "i < length (concat xs)"
shows "snd (concat_index_split (0, i) xs) < length (xs ! fst (concat_index_split (0, i) xs))"
using concat_index_split_sound_snd_arg_aux[OF assms, of 0] by auto

lemma reconstr_1d_concat_index_split:
assumes "i < length (concat xs)"
shows "i = (λ (m, j). sum_list (map length (take (m - n) xs)) + j) (concat_index_split (n, i) xs)" using assms
proof (induct xs arbitrary: i n)
case (Cons x xs) show ?case
proof (cases "i < length x")
case False
obtain m k where res: "concat_index_split (Suc n, i - length x) xs = (m, k)"
using prod_decode_aux.cases by blast
then have unf_ind: "concat_index_split (n, i) (x # xs) = concat_index_split (Suc n, i - length x) xs" and
size: "i - length x < length (concat xs)" using Cons(2) False by auto
have "Suc n ≤ m" using concat_index_split_mono_first_arg[OF size, of "Suc n"] by (auto simp: res)
then have [simp]: "sum_list (map length (take (m - n) (x # xs))) = sum_list (map length (take (m - Suc n) xs)) + length x"
show ?thesis using Cons(1)[OF size, of "Suc n"] False unfolding unf_ind res by auto
qed auto
qed auto

lemma concat_index_split_larger_lists [simp]:
assumes "i < length (concat xs)"
shows "concat_index_split (n, i) (xs @ ys) = concat_index_split (n, i) xs" using assms
by (induct xs arbitrary: ys n i) auto

lemma concat_index_split_split_sound_aux:
assumes "i < length (concat xs)"
shows "concat xs ! i = (λ (k, j). xs ! (k - n) ! j) (concat_index_split (n, i) xs)" using assms
proof (induct xs arbitrary: i n)
case (Cons x xs)
show ?case proof (cases "i < length x")
case False then have size: "i - length x < length (concat xs)"
using Cons(2) False by auto
obtain k j where [simp]: "concat_index_split (Suc n, i - length x) xs = (k, j)"
using prod_decode_aux.cases by blast
show ?thesis using False Cons(1)[OF size, of "Suc n"]
using concat_index_split_mono_first_arg[OF size, of "Suc n"]
by (auto simp: nth_append)
qed auto

lemma concat_index_split_sound:
assumes "i < length (concat xs)"
shows "concat xs ! i = (λ (k, j). xs ! k ! j) (concat_index_split (0, i) xs)"
using concat_index_split_split_sound_aux[OF assms, of 0] by auto

lemma concat_index_split_sound_bounds:
assumes "i < length (concat xs)" and "concat_index_split (0, i) xs = (m, n)"
shows "m < length xs" "n < length (xs ! m)"
using concat_index_split_sound_fst_arg[OF assms(1)] concat_index_split_sound_snd_arg[OF assms(1)]
by (auto simp: assms(2))

lemma concat_index_split_less_length_concat:
assumes "i < length (concat xs)" and "concat_index_split (0, i) xs = (m, n)"
shows "i = sum_list (map length (take m xs)) + n" "m < length xs" "n < length (xs ! m)"
"concat xs ! i = xs ! m ! n"
using concat_index_split_sound[OF assms(1)] reconstr_1d_concat_index_split[OF assms(1), of 0]
using concat_index_split_sound_fst_arg[OF assms(1)] concat_index_split_sound_snd_arg[OF assms(1)] assms(2)
by auto

lemma nth_concat_split':
assumes "i < length (concat xs)"
obtains j k where "j < length xs" "k < length (xs ! j)" "concat xs ! i = xs ! j ! k" "i = sum_list (map length (take j xs)) + k"
using concat_index_split_less_length_concat[OF assms]
by (meson old.prod.exhaust)

lemma sum_list_split [dest!, consumes 1]:
assumes "sum_list (map length (take i xs)) + j = sum_list (map length (take k xs)) + l"
and "i < length xs" "k < length xs"
and "j < length (xs ! i)" "l < length (xs ! k)"
shows "i = k ∧ j = l" using assms
proof (induct xs rule: rev_induct)
case (snoc x xs)
then show ?case
by (auto simp: nth_append split: if_splits)
qed auto

lemma concat_index_split_unique:
assumes "i < length (concat xs)" and "length xs = length ys"
and "∀ i < length xs. length (xs ! i) = length (ys ! i)"
shows "concat_index_split (n, i) xs = concat_index_split (n, i) ys" using assms
proof (induct xs arbitrary: ys n i)
case (Cons x xs) note IH = this show ?case
proof (cases ys)
case Nil then show ?thesis using Cons(3) by auto
next
case [simp]: (Cons y ys')
have [simp]: "length y = length x" using IH(4) by force
have [simp]: "¬ i < length x ⟹ i - length x < length (concat xs)" using IH(2) by auto
have [simp]: "i < length ys' ⟹ length (xs ! i) = length (ys' ! i)" for i using IH(3, 4)
by (auto simp: All_less_Suc) (metis IH(4) Suc_less_eq length_Cons Cons nth_Cons_Suc)
show ?thesis using IH(2-) IH(1)[of "i - length x" ys' "Suc n"] by auto
qed
qed auto

lemma set_vars_term_list [simp]:
"set (vars_term_list t) = vars_term t"
by (induct t) simp_all

lemma vars_term_list_empty_ground [simp]:
"vars_term_list t = [] ⟷ ground t"
by (induct t) auto

lemma varposs_imp_poss:
assumes "p ∈ varposs t"
shows "p ∈ poss t"
using assms by (induct t arbitrary: p) auto

lemma vaposs_list_fun:
assumes "p ∈ set (varposs_list (Fun f ts))"
obtains i ps where "i < length ts" "p = i # ps"
using assms set_zip_leftD by fastforce

lemma varposs_list_distinct:
"distinct (varposs_list t)"
proof (induct t)
case (Fun f ts)
then show ?case proof (induct ts rule: rev_induct)
case (snoc x xs)
then have "distinct (varposs_list (Fun f xs))" "distinct (varposs_list x)" by auto
then show ?case using snoc by (auto simp add: map_cons_presv_distinct dest: set_zip_leftD)
qed auto
qed auto

lemma varposs_append:
"varposs (Fun f (ts @ [t])) = varposs (Fun f ts) ∪ ((#) (length ts)) ` varposs t"
by (auto simp: nth_append split: if_splits)

lemma varposs_eq_varposs_list:
"set (varposs_list t) = varposs t"
proof (induct t)
case (Fun f ts)
then show ?case proof (induct ts rule: rev_induct)
case (snoc x xs)
then have "varposs (Fun f xs) = set (varposs_list (Fun f xs))"
"varposs x = set (varposs_list x)" by auto
then show ?case using snoc unfolding varposs_append
by auto
qed auto
qed auto

lemma varposs_list_var_terms_length:
"length (varposs_list t) = length (vars_term_list t)"
by (induct t) (auto simp: vars_term_list.simps intro: eq_length_concat_nth)

lemma vars_term_list_nth:
assumes "i < length (vars_term_list (Fun f ts))"
and "concat_index_split (0, i) (map vars_term_list ts) = (k, j)"
shows "k < length ts ∧ j < length (vars_term_list (ts ! k)) ∧
vars_term_list (Fun f ts) ! i = map vars_term_list ts ! k ! j ∧
i = sum_list (map length (map vars_term_list (take k ts))) + j"
using assms concat_index_split_less_length_concat[of i "map vars_term_list ts" k j]
by (auto simp: vars_term_list.simps comp_def take_map)

lemma varposs_list_nth:
assumes "i < length (varposs_list (Fun f ts))"
and "concat_index_split (0, i) (poss_args varposs_list ts) = (k, j)"
shows "k < length ts ∧ j < length (varposs_list (ts ! k)) ∧
varposs_list (Fun f ts) ! i = k # (map varposs_list ts) ! k ! j ∧
i = sum_list (map length (map varposs_list (take k ts))) + j"
using assms concat_index_split_less_length_concat[of i "poss_args varposs_list ts" k j]
by (auto simp: comp_def take_map intro: nth_sum_listI)

lemma varposs_list_to_var_term_list:
assumes "i < length (varposs_list t)"
shows "the_Var (t |_ (varposs_list t ! i)) = (vars_term_list t) ! i" using assms
proof (induct t arbitrary: i)
case (Fun f ts)
have "concat_index_split (0, i) (poss_args varposs_list ts) = concat_index_split (0, i) (map vars_term_list ts)"
using Fun(2) concat_index_split_unique[of i "poss_args varposs_list ts" "map vars_term_list ts" 0]
using varposs_list_var_terms_length[of "ts ! i" for i]
by (auto simp: vars_term_list.simps)
then obtain k j where "concat_index_split (0, i) (poss_args varposs_list ts) = (k, j)"
"concat_index_split (0, i) (map vars_term_list ts) = (k, j)" by fastforce
from varposs_list_nth[OF Fun(2) this(1)] vars_term_list_nth[OF _ this(2)]
show ?case using Fun(2) Fun(1)[OF nth_mem] varposs_list_var_terms_length[of "Fun f ts"] by auto
qed (auto simp: vars_term_list.simps)

end```