# Theory Multihole_Context

```(*
Author:  Bertram Felgenhauer <bertram.felgenhauer@uibk.ac.at> (2015)
Author:  Christian Sternagel <c.sternagel@gmail.com> (2013-2016)
Author:  Martin Avanzini <martin.avanzini@uibk.ac.at> (2014)
Author:  René Thiemann <rene.thiemann@uibk.ac.at> (2013-2015)
Author:  Julian Nagele <julian.nagele@uibk.ac.at> (2016)
License: LGPL (see file COPYING.LESSER)
*)

section ‹Preliminaries›
subsection ‹Multihole Contexts›

theory Multihole_Context
imports
Utils
begin

unbundle lattice_syntax

subsubsection ‹Partitioning lists into chunks of given length›

lemma concat_nth:
assumes "m < length xs" and "n < length (xs ! m)"
and "i = sum_list (map length (take m xs)) + n"
shows "concat xs ! i = xs ! m ! n"
using assms
proof (induct xs arbitrary: m n i)
case (Cons x xs)
show ?case
proof (cases m)
case 0
then show ?thesis using Cons by (simp add: nth_append)
next
case (Suc k)
with Cons(1) [of k n "i - length x"] and Cons(2-)
show ?thesis by (simp_all add: nth_append)
qed
qed simp

lemma sum_list_take_eq:
fixes xs :: "nat list"
shows "k < i ⟹ i < length xs ⟹ sum_list (take i xs) =
sum_list (take k xs) + xs ! k + sum_list (take (i - Suc k) (drop (Suc k) xs))"
by (subst id_take_nth_drop [of k]) (auto simp: min_def drop_take)

fun partition_by where
"partition_by xs [] = []" |
"partition_by xs (y#ys) = take y xs # partition_by (drop y xs) ys"

lemma partition_by_map0_append [simp]:
"partition_by xs (map (λx. 0) ys @ zs) = replicate (length ys) [] @ partition_by xs zs"
by (induct ys) simp_all

lemma concat_partition_by [simp]:
"sum_list ys = length xs ⟹ concat (partition_by xs ys) = xs"
by (induct ys arbitrary: xs) simp_all

definition partition_by_idx where
"partition_by_idx l ys i j = partition_by [0..<l] ys ! i ! j"

lemma partition_by_nth_nth_old:
assumes "i < length (partition_by xs ys)"
and "j < length (partition_by xs ys ! i)"
and "sum_list ys = length xs"
shows "partition_by xs ys ! i ! j = xs ! (sum_list (map length (take i (partition_by xs ys))) + j)"
using concat_nth [OF assms(1, 2) refl]
unfolding concat_partition_by [OF assms(3)] by simp

lemma map_map_partition_by:
"map (map f) (partition_by xs ys) = partition_by (map f xs) ys"
by (induct ys arbitrary: xs) (auto simp: take_map drop_map)

lemma length_partition_by [simp]:
"length (partition_by xs ys) = length ys"
by (induct ys arbitrary: xs) simp_all

lemma partition_by_Nil [simp]:
"partition_by [] ys = replicate (length ys) []"
by (induct ys) simp_all

lemma partition_by_concat_id [simp]:
assumes "length xss = length ys"
and "⋀i. i < length ys ⟹ length (xss ! i) = ys ! i"
shows "partition_by (concat xss) ys = xss"
using assms by (induct ys arbitrary: xss) (simp, case_tac xss, simp, fastforce)

lemma partition_by_nth:
"i < length ys ⟹ partition_by xs ys ! i = take (ys ! i) (drop (sum_list (take i ys)) xs)"
by (induct ys arbitrary: xs i) (simp, case_tac i, simp_all add: ac_simps)

lemma partition_by_nth_less:
assumes "k < i" and "i < length zs"
and "length xs = sum_list (take i zs) + j"
shows "partition_by (xs @ y # ys) zs ! k = take (zs ! k) (drop (sum_list (take k zs)) xs)"
proof -
have "partition_by (xs @ y # ys) zs ! k =
take (zs ! k) (drop (sum_list (take k zs)) (xs @ y # ys))"
using assms by (auto simp: partition_by_nth)
moreover have "zs ! k + sum_list (take k zs) ≤ length xs"
using assms by (simp add: sum_list_take_eq)
ultimately show ?thesis by simp
qed

lemma partition_by_nth_greater:
assumes "i < k" and "k < length zs" and "j < zs ! i"
and "length xs = sum_list (take i zs) + j"
shows "partition_by (xs @ y # ys) zs ! k =
take (zs ! k) (drop (sum_list (take k zs) - 1) (xs @ ys))"
proof -
have "partition_by (xs @ y # ys) zs ! k =
take (zs ! k) (drop (sum_list (take k zs)) (xs @ y # ys))"
using assms by (auto simp: partition_by_nth)
moreover have "sum_list (take k zs) > length xs"
using assms by (auto simp: sum_list_take_eq)
ultimately show ?thesis by (auto) (metis Suc_diff_Suc drop_Suc_Cons)
qed

lemma length_partition_by_nth:
"sum_list ys = length xs ⟹ i < length ys ⟹ length (partition_by xs ys ! i) = ys ! i"
by (induct ys arbitrary: xs i; case_tac i) auto

lemma partition_by_nth_nth_elem:
assumes "sum_list ys = length xs" "i < length ys" "j < ys ! i"
shows "partition_by xs ys ! i ! j ∈ set xs"
proof -
from assms have "j < length (partition_by xs ys ! i)" by (simp only: length_partition_by_nth)
then have "partition_by xs ys ! i ! j ∈ set (partition_by xs ys ! i)" by auto
with assms(2) have "partition_by xs ys ! i ! j ∈ set (concat (partition_by xs ys))" by auto
then show ?thesis using assms by simp
qed

lemma partition_by_nth_nth:
assumes "sum_list ys = length xs" "i < length ys" "j < ys ! i"
shows "partition_by xs ys ! i ! j = xs ! partition_by_idx (length xs) ys i j"
"partition_by_idx (length xs) ys i j < length xs"
unfolding partition_by_idx_def
proof -
let ?n = "partition_by [0..<length xs] ys ! i ! j"
show "?n < length xs"
using partition_by_nth_nth_elem[OF _ assms(2,3), of "[0..<length xs]"] assms(1) by simp
have li: "i < length (partition_by [0..<length xs] ys)" using assms(2) by simp
have lj: "j < length (partition_by [0..<length xs] ys ! i)"
using assms by (simp add: length_partition_by_nth)
have "partition_by (map ((!) xs) [0..<length xs]) ys ! i ! j = xs ! ?n"
by (simp only: map_map_partition_by[symmetric] nth_map[OF li] nth_map[OF lj])
then show "partition_by xs ys ! i ! j = xs ! ?n" by (simp add: map_nth)
qed

lemma map_length_partition_by [simp]:
"sum_list ys = length xs ⟹ map length (partition_by xs ys) = ys"
by (intro nth_equalityI, auto simp: length_partition_by_nth)

lemma map_partition_by_nth [simp]:
"i < length ys ⟹ map f (partition_by xs ys ! i) = partition_by (map f xs) ys ! i"
by (induct ys arbitrary: i xs) (simp, case_tac i, simp_all add: take_map drop_map)

lemma sum_list_partition_by [simp]:
"sum_list ys = length xs ⟹
sum_list (map (λx. sum_list (map f x)) (partition_by xs ys)) = sum_list (map f xs)"
by (induct ys arbitrary: xs) (simp_all, metis append_take_drop_id sum_list_append map_append)

lemma partition_by_map_conv:
"partition_by xs ys = map (λi. take (ys ! i) (drop (sum_list (take i ys)) xs)) [0 ..< length ys]"
by (rule nth_equalityI) (simp_all add: partition_by_nth)

lemma UN_set_partition_by_map:
"sum_list ys = length xs ⟹ (⋃x∈set (partition_by (map f xs) ys). ⋃ (set x)) = ⋃(set (map f xs))"
by (induct ys arbitrary: xs)
(simp_all add: drop_map take_map, metis UN_Un append_take_drop_id set_append)

lemma UN_set_partition_by:
"sum_list ys = length xs ⟹ (⋃zs ∈ set (partition_by xs ys). ⋃x ∈ set zs. f x) = (⋃x ∈ set xs. f x)"
by (induct ys arbitrary: xs) (simp_all, metis UN_Un append_take_drop_id set_append)

lemma Ball_atLeast0LessThan_partition_by_conv:
"(∀i∈{0..<length ys}. ∀x∈set (partition_by xs ys ! i). P x) =
(∀x ∈ ⋃(set (map set (partition_by xs ys))). P x)"
by auto (metis atLeast0LessThan in_set_conv_nth length_partition_by lessThan_iff)

lemma Ball_set_partition_by:
"sum_list ys = length xs ⟹
(∀x ∈ set (partition_by xs ys). ∀y ∈ set x. P y) = (∀x ∈ set xs. P x)"
proof (induct ys arbitrary: xs)
case (Cons y ys)
then show ?case
apply (subst (2) append_take_drop_id [of y xs, symmetric])
apply (simp only: set_append)
apply auto
done
qed simp

lemma partition_by_append2:
"partition_by xs (ys @ zs) = partition_by (take (sum_list ys) xs) ys @ partition_by (drop (sum_list ys) xs) zs"
by (induct ys arbitrary: xs) (auto simp: drop_take ac_simps split: split_min)

lemma partition_by_concat2:
"partition_by xs (concat ys) =
concat (map (λi . partition_by (partition_by xs (map sum_list ys) ! i) (ys ! i)) [0..<length ys])"
proof -
have *: "map (λi . partition_by (partition_by xs (map sum_list ys) ! i) (ys ! i)) [0..<length ys] =
map (λ(x,y). partition_by x y) (zip (partition_by xs (map sum_list ys)) ys)"
using zip_nth_conv[of "partition_by xs (map sum_list ys)" ys] by auto
show ?thesis unfolding * by (induct ys arbitrary: xs) (auto simp: partition_by_append2)
qed

lemma partition_by_partition_by:
"length xs = sum_list (map sum_list ys) ⟹
partition_by (partition_by xs (concat ys)) (map length ys) =
map (λi. partition_by (partition_by xs (map sum_list ys) ! i) (ys ! i)) [0..<length ys]"
by (auto simp: partition_by_concat2 intro: partition_by_concat_id)

subsubsection ‹Multihole contexts definition and functionalities›
datatype ('f, vars_mctxt : 'v) mctxt = MVar 'v | MHole | MFun 'f "('f, 'v) mctxt list"

subsubsection ‹Conversions from and to multihole contexts›

primrec mctxt_of_term :: "('f, 'v) term ⇒ ('f, 'v) mctxt" where
"mctxt_of_term (Var x) = MVar x" |
"mctxt_of_term (Fun f ts) = MFun f (map mctxt_of_term ts)"

primrec term_of_mctxt :: "('f, 'v) mctxt ⇒ ('f, 'v) term" where
"term_of_mctxt (MVar x) = Var x" |
"term_of_mctxt (MFun f Cs) = Fun f (map term_of_mctxt Cs)"

fun num_holes :: "('f, 'v) mctxt ⇒ nat" where
"num_holes (MVar _) = 0" |
"num_holes MHole = 1" |
"num_holes (MFun _ ctxts) = sum_list (map num_holes ctxts)"

fun ground_mctxt :: "('f, 'v) mctxt ⇒ bool" where
"ground_mctxt (MVar _) = False" |
"ground_mctxt MHole = True" |
"ground_mctxt (MFun f Cs) = Ball (set Cs) ground_mctxt"

fun map_mctxt :: "('f ⇒ 'g) ⇒ ('f, 'v) mctxt ⇒ ('g, 'v) mctxt"
where
"map_mctxt _ (MVar x) = (MVar x)" |
"map_mctxt _ (MHole) = MHole" |
"map_mctxt fg (MFun f Cs) = MFun (fg f) (map (map_mctxt fg) Cs)"

abbreviation "partition_holes xs Cs ≡ partition_by xs (map num_holes Cs)"
abbreviation "partition_holes_idx l Cs ≡ partition_by_idx l (map num_holes Cs)"

fun fill_holes :: "('f, 'v) mctxt ⇒ ('f, 'v) term list ⇒ ('f, 'v) term" where
"fill_holes (MVar x) _ = Var x" |
"fill_holes MHole [t] = t" |
"fill_holes (MFun f cs) ts = Fun f (map (λ i. fill_holes (cs ! i)
(partition_holes ts cs ! i)) [0 ..< length cs])"

fun fill_holes_mctxt :: "('f, 'v) mctxt ⇒ ('f, 'v) mctxt list ⇒ ('f, 'v) mctxt" where
"fill_holes_mctxt (MVar x) _ = MVar x" |
"fill_holes_mctxt MHole [] = MHole" |
"fill_holes_mctxt MHole [t] = t" |
"fill_holes_mctxt (MFun f cs) ts = (MFun f (map (λ i. fill_holes_mctxt (cs ! i)
(partition_holes ts cs ! i)) [0 ..< length cs]))"

fun unfill_holes :: "('f, 'v) mctxt ⇒ ('f, 'v) term ⇒ ('f, 'v) term list" where
"unfill_holes MHole t = [t]"
| "unfill_holes (MVar w) (Var v) = (if v = w then [] else undefined)"
| "unfill_holes (MFun g Cs) (Fun f ts) = (if f = g ∧ length ts = length Cs then
concat (map (λi. unfill_holes (Cs ! i) (ts ! i)) [0..<length ts]) else undefined)"

fun funas_mctxt where
"funas_mctxt (MFun f Cs) = {(f, length Cs)} ∪ ⋃(funas_mctxt ` set Cs)" |
"funas_mctxt _ = {}"

fun split_vars :: "('f, 'v) term ⇒ (('f, 'v) mctxt × 'v list)" where
"split_vars (Var x) = (MHole, [x])" |
"split_vars (Fun f ts) = (MFun f (map (fst ∘ split_vars) ts), concat (map (snd ∘ split_vars) ts))"

fun hole_poss_list :: "('f, 'v) mctxt ⇒ pos list" where
"hole_poss_list (MVar x) = []" |
"hole_poss_list MHole = [[]]" |
"hole_poss_list (MFun f cs) = concat (poss_args hole_poss_list cs)"

fun map_vars_mctxt :: "('v ⇒ 'w) ⇒ ('f, 'v) mctxt ⇒ ('f, 'w) mctxt"
where
"map_vars_mctxt vw MHole = MHole" |
"map_vars_mctxt vw (MVar v) = (MVar (vw v))" |
"map_vars_mctxt vw (MFun f Cs) = MFun f (map (map_vars_mctxt vw) Cs)"

inductive eq_fill :: "('f, 'v) term ⇒ ('f, 'v) mctxt × ('f, 'v) term list ⇒ bool" ("(_/ =⇩f _)" [51, 51] 50)
where
eqfI [intro]: "t = fill_holes D ss ⟹ num_holes D = length ss ⟹ t =⇩f (D, ss)"

subsubsection ‹Semilattice Structures›

instantiation mctxt :: (type, type) inf

begin

fun inf_mctxt :: "('a, 'b) mctxt ⇒ ('a, 'b) mctxt ⇒ ('a, 'b) mctxt"
where
"MHole ⊓ D = MHole" |
"C ⊓ MHole = MHole" |
"MVar x ⊓ MVar y = (if x = y then MVar x else MHole)" |
"MFun f Cs ⊓ MFun g Ds =
(if f = g ∧ length Cs = length Ds then MFun f (map (case_prod (⊓)) (zip Cs Ds))
else MHole)" |
"C ⊓ D = MHole"

instance ..

end

lemma inf_mctxt_idem [simp]:
fixes C :: "('f, 'v) mctxt"
shows "C ⊓ C = C"
by (induct C) (auto simp: zip_same_conv_map intro: map_idI)

lemma inf_mctxt_MHole2 [simp]:
"C ⊓ MHole = MHole"
by (induct C) simp_all

lemma inf_mctxt_comm [ac_simps]:
"(C :: ('f, 'v) mctxt) ⊓ D = D ⊓ C"
by (induct C D rule: inf_mctxt.induct) (fastforce simp: in_set_conv_nth intro!: nth_equalityI)+

lemma inf_mctxt_assoc [ac_simps]:
fixes C :: "('f, 'v) mctxt"
shows "C ⊓ D ⊓ E = C ⊓ (D ⊓ E)"
apply (induct C D arbitrary: E rule: inf_mctxt.induct)
apply auto
apply (case_tac E, auto)+
apply (fastforce simp: in_set_conv_nth intro!: nth_equalityI)
apply (case_tac E, auto)+
done

instantiation mctxt :: (type, type) order
begin

definition "(C :: ('a, 'b) mctxt) ≤ D ⟷ C ⊓ D = C"
definition "(C :: ('a, 'b) mctxt) < D ⟷ C ≤ D ∧ ¬ D ≤ C"

instance
by (standard, simp_all add: less_eq_mctxt_def less_mctxt_def ac_simps, metis inf_mctxt_assoc)

end

inductive less_eq_mctxt' :: "('f, 'v) mctxt ⇒ ('f,'v) mctxt ⇒ bool" where
"less_eq_mctxt' MHole u"
| "less_eq_mctxt' (MVar v) (MVar v)"
| "length cs = length ds ⟹ (⋀i. i < length cs ⟹ less_eq_mctxt' (cs ! i) (ds ! i)) ⟹ less_eq_mctxt' (MFun f cs) (MFun f ds)"

subsubsection ‹Lemmata›

lemma partition_holes_fill_holes_conv:
"fill_holes (MFun f cs) ts =
Fun f [fill_holes (cs ! i) (partition_holes ts cs ! i). i ← [0 ..< length cs]]"
by (simp add: partition_by_nth take_map)

lemma partition_holes_fill_holes_mctxt_conv:
"fill_holes_mctxt (MFun f Cs) ts =
MFun f [fill_holes_mctxt (Cs ! i) (partition_holes ts Cs ! i). i ← [0 ..< length Cs]]"
by (simp add: partition_by_nth take_map)

text ‹The following induction scheme provides the @{term MFun} case with the list argument split
according to the argument contexts. This feature is quite delicate: its benefit can be
destroyed by premature simplification using the @{thm concat_partition_by} simplification rule.›

lemma fill_holes_induct2[consumes 2, case_names MHole MVar MFun]:
fixes P :: "('f,'v) mctxt ⇒ 'a list ⇒ 'b list ⇒ bool"
assumes len1: "num_holes C = length xs" and len2: "num_holes C = length ys"
and Hole: "⋀x y. P MHole [x] [y]"
and Var: "⋀v. P (MVar v) [] []"
and Fun: "⋀f Cs xs ys.  sum_list (map num_holes Cs) = length xs ⟹
sum_list (map num_holes Cs) = length ys ⟹
(⋀i. i < length Cs ⟹ P (Cs ! i) (partition_holes xs Cs ! i) (partition_holes ys Cs ! i)) ⟹
P (MFun f Cs) (concat (partition_holes xs Cs)) (concat (partition_holes ys Cs))"
shows "P C xs ys"
proof (insert len1 len2, induct C arbitrary: xs ys)
case MHole then show ?case using Hole by (cases xs; cases ys) auto
next
case (MVar v) then show ?case using Var by auto
next
case (MFun f Cs) then show ?case using Fun[of Cs xs ys f] by (auto simp: length_partition_by_nth)
qed

lemma fill_holes_induct[consumes 1, case_names MHole MVar MFun]:
fixes P :: "('f,'v) mctxt ⇒ 'a list ⇒ bool"
assumes len: "num_holes C = length xs"
and Hole: "⋀x. P MHole [x]"
and Var: "⋀v. P (MVar v) []"
and Fun: "⋀f Cs xs. sum_list (map num_holes Cs) = length xs ⟹
(⋀i. i < length Cs ⟹ P (Cs ! i) (partition_holes xs Cs ! i)) ⟹
P (MFun f Cs) (concat (partition_holes xs Cs))"
shows "P C xs"
using fill_holes_induct2[of C xs xs "λ C xs _. P C xs"] assms by simp

lemma length_partition_holes_nth [simp]:
assumes "sum_list (map num_holes cs) = length ts"
and "i < length cs"
shows "length (partition_holes ts cs ! i) = num_holes (cs ! i)"
using assms by (simp add: length_partition_by_nth)

(*some compatibility lemmas (which should be dropped eventually)*)
lemmas
map_partition_holes_nth [simp] =
map_partition_by_nth [of _ "map num_holes Cs" for Cs, unfolded length_map] and
length_partition_holes [simp] =
length_partition_by [of _ "map num_holes Cs" for Cs, unfolded length_map]

lemma fill_holes_term_of_mctxt:
"num_holes C = 0 ⟹ fill_holes C [] = term_of_mctxt C"
by (induct C) (auto simp add: map_eq_nth_conv)

lemma fill_holes_MHole:
"length ts = Suc 0 ⟹ ts ! 0 = u ⟹ fill_holes MHole ts = u"
by (cases ts) simp_all

lemma fill_holes_arbitrary:
assumes lCs: "length Cs = length ts"
and lss: "length ss = length ts"
and rec: "⋀ i. i < length ts ⟹ num_holes (Cs ! i) = length (ss ! i) ∧ f (Cs ! i) (ss ! i) = ts ! i"
shows "map (λi. f (Cs ! i) (partition_holes (concat ss) Cs ! i)) [0 ..< length Cs] = ts"
proof -
have "sum_list (map num_holes Cs) = length (concat ss)" using assms
by (auto simp: length_concat map_nth_eq_conv intro: arg_cong[of _ _ "sum_list"])
moreover have "partition_holes (concat ss) Cs = ss"
using assms by (auto intro: partition_by_concat_id)
ultimately show ?thesis using assms by (auto intro: nth_equalityI)
qed

lemma fill_holes_MFun:
assumes lCs: "length Cs = length ts"
and lss: "length ss = length ts"
and rec: "⋀ i. i < length ts ⟹ num_holes (Cs ! i) = length (ss ! i) ∧ fill_holes (Cs ! i) (ss ! i) = ts ! i"
shows "fill_holes (MFun f Cs) (concat ss) = Fun f ts"
unfolding fill_holes.simps term.simps
by (rule conjI[OF refl], rule fill_holes_arbitrary[OF lCs lss rec])

lemma eqfE:
assumes "t =⇩f (D, ss)" shows "t = fill_holes D ss" "num_holes D = length ss"
using assms[unfolded eq_fill.simps] by auto

lemma eqf_MFunE:
assumes "s =⇩f (MFun f Cs,ss)"
obtains ts sss where "s = Fun f ts" "length ts = length Cs" "length sss = length Cs"
"⋀ i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, sss ! i)"
"ss = concat sss"
proof -
from eqfE[OF assms] have fh: "s = fill_holes (MFun f Cs) ss"
and nh: "sum_list (map num_holes Cs) = length ss" by auto
from fh obtain ts where s: "s = Fun f ts" by (cases s, auto)
from fh[unfolded s]
have ts: "ts = map (λi. fill_holes (Cs ! i) (partition_holes ss Cs ! i)) [0..<length Cs]"
(is "_ = map (?f Cs ss) _")
by auto
let ?sss = "partition_holes ss Cs"
from nh
have *: "length ?sss = length Cs" "⋀i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, ?sss ! i)" "ss = concat ?sss"
by (auto simp: ts)
have len: "length ts = length Cs" unfolding ts by auto
assume ass: "⋀ts sss. s = Fun f ts ⟹
length ts = length Cs ⟹
length sss = length Cs ⟹ (⋀i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, sss ! i)) ⟹ ss = concat sss ⟹ thesis"
show thesis
by (rule ass[OF s len *])
qed

lemma eqf_MFunI:
assumes "length sss = length Cs"
and "length ts = length Cs"
and"⋀ i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, sss ! i)"
shows "Fun f ts =⇩f (MFun f Cs, concat sss)"
proof
have "num_holes (MFun f Cs) = sum_list (map num_holes Cs)" by simp
also have "map num_holes Cs = map length sss"
by (rule nth_equalityI, insert assms eqfE[OF assms(3)], auto)
also have "sum_list (…) = length (concat sss)" unfolding length_concat ..
finally show "num_holes (MFun f Cs) = length (concat sss)" .
show "Fun f ts = fill_holes (MFun f Cs) (concat sss)"
by (rule fill_holes_MFun[symmetric], insert assms(1,2) eqfE[OF assms(3)], auto)
qed

lemma split_vars_ground_vars:
assumes "ground_mctxt C" and "num_holes C = length xs"
shows "split_vars (fill_holes C (map Var xs)) = (C, xs)" using assms
proof (induct C arbitrary: xs)
case (MHole xs)
then show ?case by (cases xs, auto)
next
case (MFun f Cs xs)
have "fill_holes (MFun f Cs) (map Var xs) =⇩f (MFun f Cs, map Var xs)"
by (rule eqfI, insert MFun(3), auto)
from eqf_MFunE[OF this]
obtain ts xss where fh: "fill_holes (MFun f Cs) (map Var xs) = Fun f ts"
and lent: "length ts = length Cs"
and lenx: "length xss = length Cs"
and args: "⋀i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, xss ! i)"
and id: "map Var xs = concat xss" by auto
from arg_cong[OF id, of "map the_Var"] have id2: "xs = concat (map (map the_Var) xss)"
by (metis map_concat length_map map_nth_eq_conv term.sel(1))
{
fix i
assume i: "i < length Cs"
then have mem: "Cs ! i ∈ set Cs" by auto
with MFun(2) have ground: "ground_mctxt (Cs ! i)" by auto
have "map Var (map the_Var (xss ! i)) = map id (xss ! i)" unfolding map_map o_def map_eq_conv
proof
fix x
assume "x ∈ set (xss ! i)"
with lenx i have "x ∈ set (concat xss)" by auto
from this[unfolded id[symmetric]] show "Var (the_Var x) = id x" by auto
qed
then have idxss: "map Var (map the_Var (xss ! i)) = xss ! i" by auto
note rec = eqfE[OF args[OF i]]
note IH = MFun(1)[OF mem ground, of "map the_Var (xss ! i)", unfolded rec(2) idxss rec(1)[symmetric]]
from IH have "split_vars (ts ! i) = (Cs ! i, map the_Var (xss ! i))" by auto
note this idxss
}
note IH = this
have "?case = (map fst (map split_vars ts) = Cs ∧ concat (map snd (map split_vars ts)) = concat (map (map the_Var) xss))"
unfolding fh unfolding id2 by auto
also have "…"
proof (rule conjI[OF nth_equalityI arg_cong[of _ _ concat, OF nth_equalityI, rule_format]], unfold length_map lent lenx)
fix i
assume i: "i < length Cs"
with arg_cong[OF IH(2)[OF this], of "map the_Var"]
IH[OF this] show "map snd (map split_vars ts) ! i = map (map the_Var) xss ! i" using lent lenx by auto
qed (insert IH lent, auto)
finally show ?case .
qed auto

lemma split_vars_vars_term_list: "snd (split_vars t) = vars_term_list t"
proof (induct t)
case (Fun f ts)
then show ?case by (auto simp: vars_term_list.simps o_def, induct ts, auto)
qed (auto simp: vars_term_list.simps)

lemma split_vars_num_holes: "num_holes (fst (split_vars t)) = length (snd (split_vars t))"
proof (induct t)
case (Fun f ts)
then show ?case by (induct ts, auto)
qed simp

lemma ground_eq_fill: "t =⇩f (C,ss) ⟹ ground t = (ground_mctxt C ∧ (∀ s ∈ set ss. ground s))"
proof (induct C arbitrary: t ss)
case (MVar x)
from eqfE[OF this] show ?case by simp
next
case (MHole t ss)
from eqfE[OF this] show ?case by (cases ss, auto)
next
case (MFun f Cs s ss)
from eqf_MFunE[OF MFun(2)] obtain ts sss where s: "s = Fun f ts" and len: "length ts = length Cs" "length sss = length Cs"
and IH: "⋀ i. i < length Cs ⟹ ts ! i =⇩f (Cs ! i, sss ! i)" and ss: "ss = concat sss" by metis
{
fix i
assume i: "i < length Cs"
then have "Cs ! i ∈ set Cs" by simp
from MFun(1)[OF this IH[OF i]]
have "ground (ts ! i) = (ground_mctxt (Cs ! i) ∧ (∀a∈set (sss ! i). ground a))" .
} note IH = this
note conv = set_conv_nth
have "?case = ((∀x∈set ts. ground x) = ((∀x∈set Cs. ground_mctxt x) ∧ (∀a∈set sss. ∀x∈set a. ground x)))"
unfolding s ss by simp
also have "..." unfolding conv[of ts] conv[of Cs] conv[of sss] len using IH by auto
finally show ?case by simp
qed

lemma ground_fill_holes:
assumes nh: "num_holes C = length ss"
shows "ground (fill_holes C ss) = (ground_mctxt C ∧ (∀ s ∈ set ss. ground s))"
by (rule ground_eq_fill[OF eqfI[OF refl nh]])

lemma split_vars_ground' [simp]:
"ground_mctxt (fst (split_vars t))"
by (induct t) auto

lemma split_vars_funas_mctxt [simp]:
"funas_mctxt (fst (split_vars t)) = funas_term t"
by (induct t) auto

lemma less_eq_mctxt_prime: "C ≤ D ⟷ less_eq_mctxt' C D"
proof
assume "less_eq_mctxt' C D" then show "C ≤ D"
by (induct C D rule: less_eq_mctxt'.induct) (auto simp: less_eq_mctxt_def intro: nth_equalityI)
next
assume "C ≤ D" then show "less_eq_mctxt' C D" unfolding less_eq_mctxt_def
by (induct C D rule: inf_mctxt.induct)
(auto split: if_splits simp: set_zip intro!: less_eq_mctxt'.intros nth_equalityI elim!: nth_equalityE, metis)
qed

lemmas less_eq_mctxt_induct = less_eq_mctxt'.induct[folded less_eq_mctxt_prime, consumes 1]
lemmas less_eq_mctxt_intros = less_eq_mctxt'.intros[folded less_eq_mctxt_prime]

lemma less_eq_mctxt_MHoleE2:
assumes "C ≤ MHole"
obtains (MHole) "C = MHole"
using assms unfolding less_eq_mctxt_prime by (cases C, auto)

lemma less_eq_mctxt_MVarE2:
assumes "C ≤ MVar v"
obtains (MHole) "C = MHole" | (MVar) "C = MVar v"
using assms unfolding less_eq_mctxt_prime by (cases C) auto

lemma less_eq_mctxt_MFunE2:
assumes "C ≤ MFun f ds"
obtains (MHole) "C = MHole"
| (MFun) cs where "C = MFun f cs" "length cs = length ds" "⋀i. i < length cs ⟹ cs ! i ≤ ds ! i"
using assms unfolding less_eq_mctxt_prime by (cases C) auto

lemmas less_eq_mctxtE2 = less_eq_mctxt_MHoleE2 less_eq_mctxt_MVarE2 less_eq_mctxt_MFunE2

lemma less_eq_mctxt_MVarE1:
assumes "MVar v ≤ D"
obtains (MVar) "D = MVar v"
using assms by (cases D) (auto elim: less_eq_mctxtE2)

lemma MHole_Bot [simp]: "MHole ≤ D"
by (simp add: less_eq_mctxt_intros(1))

lemma less_eq_mctxt_MFunE1:
assumes "MFun f cs ≤ D"
obtains (MFun) ds where "D = MFun f ds" "length cs = length ds" "⋀i. i < length cs ⟹ cs ! i ≤ ds ! i"
using assms by (cases D) (auto elim: less_eq_mctxtE2)

lemma length_unfill_holes [simp]:
assumes "C ≤ mctxt_of_term t"
shows "length (unfill_holes C t) = num_holes C"
using assms
proof (induct C t rule: unfill_holes.induct)
case (3 f Cs g ts) with 3(1)[OF _ nth_mem] 3(2) show ?case
by (auto simp: less_eq_mctxt_def length_concat
intro!: cong[of sum_list, OF refl] nth_equalityI elim!: nth_equalityE)
qed (auto simp: less_eq_mctxt_def)

lemma map_vars_mctxt_id [simp]:
"map_vars_mctxt (λ x. x) C = C"
by (induct C, auto intro: nth_equalityI)

lemma split_vars_eqf_subst_map_vars_term:
"t ⋅ σ =⇩f (map_vars_mctxt vw (fst (split_vars t)), map σ (snd (split_vars t)))"
proof (induct t)
case (Fun f ts)
have "?case = (Fun f (map (λt. t ⋅ σ) ts)
=⇩f (MFun f (map (map_vars_mctxt vw ∘ (fst ∘ split_vars)) ts), concat (map (map σ ∘ (snd ∘ split_vars)) ts)))"
by (simp add: map_concat)
also have "..."
proof (rule eqf_MFunI, simp, simp, unfold length_map)
fix i
assume i: "i < length ts"
then have mem: "ts ! i ∈ set ts" by auto
show "map (λt. t ⋅ σ) ts ! i =⇩f (map (map_vars_mctxt vw ∘ (fst ∘ split_vars)) ts ! i, map (map σ ∘ (snd ∘ split_vars)) ts ! i)"
using Fun[OF mem] i by auto
qed
finally show ?case by simp
qed auto

lemma split_vars_eqf_subst: "t ⋅ σ =⇩f (fst (split_vars t), (map σ (snd (split_vars t))))"
using split_vars_eqf_subst_map_vars_term[of t σ "λ x. x"] by simp

lemma split_vars_fill_holes:
assumes "C = fst (split_vars s)" and "ss = map Var (snd (split_vars s))"
shows "fill_holes C ss = s" using assms
by (metis eqfE(1) split_vars_eqf_subst subst_apply_term_empty)

lemma fill_unfill_holes:
assumes "C ≤ mctxt_of_term t"
shows "fill_holes C (unfill_holes C t) = t"
using assms
proof (induct C t rule: unfill_holes.induct)
case (3 f Cs g ts) with 3(1)[OF _ nth_mem] 3(2) show ?case
by (auto simp: less_eq_mctxt_def intro!: fill_holes_arbitrary elim!: nth_equalityE)
qed (auto simp: less_eq_mctxt_def split: if_splits)

lemma hole_poss_list_length:
"length (hole_poss_list D) = num_holes D"
by (induct D) (auto simp: length_concat intro!: nth_sum_listI)

lemma unfill_holles_hole_poss_list_length:
assumes "C ≤ mctxt_of_term t"
shows "length (unfill_holes C t) = length (hole_poss_list C)" using assms
proof (induct C arbitrary: t)
case (MVar x)
then have [simp]: "t = Var x" by (cases t) (auto dest: less_eq_mctxt_MVarE1)
show ?case by simp
next
case (MFun f ts) then show ?case
by (cases t) (auto simp: length_concat comp_def
elim!: less_eq_mctxt_MFunE1 less_eq_mctxt_MVarE1 intro!: nth_sum_listI)
qed auto

lemma unfill_holes_to_subst_at_hole_poss:
assumes "C ≤ mctxt_of_term t"
shows "unfill_holes C t = map ((|_) t) (hole_poss_list C)" using assms
proof (induct C arbitrary: t)
case (MVar x)
then show ?case by (cases t) (auto elim: less_eq_mctxt_MVarE1)
next
case (MFun f ts)
from MFun(2) obtain ss where [simp]: "t = Fun f ss" and l: "length ts = length ss"
by (cases t) (auto elim: less_eq_mctxt_MFunE1)
let ?ts = "map (λi. unfill_holes (ts ! i) (ss ! i)) [0..<length ts]"
let ?ss = "map (λ x. map ((|_) (Fun f ss)) (case x of (x, y) ⇒ map ((#) x) (hole_poss_list y))) (zip [0..<length ts] ts)"
have eq_l [simp]: "length (concat ?ts) = length (concat ?ss)" using MFun
by (auto simp: length_concat comp_def elim!: less_eq_mctxt_MFunE1 split!: prod.splits intro!: nth_sum_listI)
{fix i assume ass: "i < length (concat ?ts)"
then have lss: "i < length (concat ?ss)" by auto
obtain m n where [simp]: "concat_index_split (0, i) ?ts = (m, n)" by fastforce
then have [simp]: "concat_index_split (0, i) ?ss = (m, n)" using concat_index_split_unique[OF ass, of ?ss 0] MFun(2)
by (auto simp: unfill_holles_hole_poss_list_length[of "ts ! i" "ss ! i" for i]
simp del: length_unfill_holes elim!: less_eq_mctxt_MFunE1)
from concat_index_split_less_length_concat(2-)[OF ass ] concat_index_split_less_length_concat(2-)[OF lss]
have "concat ?ts ! i = concat ?ss! i" using MFun(1)[OF nth_mem, of m "ss ! m"] MFun(2)
by (auto elim!: less_eq_mctxt_MFunE1)} note nth = this
show ?case using MFun
by (auto simp: comp_def map_concat length_concat
elim!: less_eq_mctxt_MFunE1 split!: prod.splits
intro!: nth_equalityI nth_sum_listI nth)
qed auto

lemma hole_poss_split_varposs_list_length [simp]:
"length (hole_poss_list (fst (split_vars t))) = length (varposs_list t)"
by (induct t)(auto simp: length_concat comp_def intro!: nth_sum_listI)

lemma hole_poss_split_vars_varposs_list:
"hole_poss_list (fst (split_vars t)) = varposs_list t"
proof (induct t)
case (Fun f ts)
let ?ts = "poss_args hole_poss_list (map (fst ∘ split_vars) ts)"
let ?ss = "poss_args varposs_list ts"
have len: "length (concat ?ts) = length (concat ?ss)" "length ?ts = length ?ss"
"∀ i < length ?ts. length (?ts ! i) = length (?ss ! i)" by (auto intro: eq_length_concat_nth)
{fix i assume ass: "i < length (concat ?ts)"
then have lss: "i < length (concat ?ss)" using len by auto
obtain m n where int: "concat_index_split (0, i) ?ts = (m, n)" by fastforce
then have [simp]: "concat_index_split (0, i) ?ss = (m, n)" using concat_index_split_unique[OF ass len(2-)] by auto
from concat_index_split_less_length_concat(2-)[OF ass int] concat_index_split_less_length_concat(2-)[OF lss]
have "concat ?ts ! i = concat ?ss! i" using Fun[OF nth_mem, of m] by auto}
then show ?case using len by (auto intro: nth_equalityI)
qed auto

lemma funas_term_fill_holes_iff: "num_holes C = length ts ⟹
g ∈ funas_term (fill_holes C ts) ⟷ g ∈ funas_mctxt C ∨ (∃t ∈ set ts. g ∈ funas_term t)"
proof (induct C ts rule: fill_holes_induct)
case (MFun f Cs ts)
have "(∃i < length Cs. g ∈ funas_term (fill_holes (Cs ! i) (partition_holes (concat (partition_holes ts Cs)) Cs ! i)))
⟷ (∃C ∈ set Cs. g ∈ funas_mctxt C) ∨ (∃us ∈ set (partition_holes ts Cs). ∃t ∈ set us. g ∈ funas_term t)"
using MFun by (auto simp: ex_set_conv_ex_nth) blast
then show ?case by auto
qed auto

lemma vars_term_fill_holes [simp]:
"num_holes C = length ts ⟹ ground_mctxt C ⟹
vars_term (fill_holes C ts) = ⋃(vars_term ` set ts)"
proof (induct C arbitrary: ts)
case MHole
then show ?case by (cases ts) simp_all
next
case (MFun f Cs)
then have *: "length (partition_holes ts Cs) = length Cs" by simp
let ?f = "λx. ⋃y ∈ set x. vars_term y"
show ?case
using MFun
unfolding partition_holes_fill_holes_conv
by (simp add: UN_upt_len_conv [OF *, of ?f] UN_set_partition_by)
qed simp

lemma funas_mctxt_fill_holes [simp]:
assumes "num_holes C = length ts"
shows "funas_term (fill_holes C ts) = funas_mctxt C ∪ ⋃(set (map funas_term ts))"
using funas_term_fill_holes_iff[OF assms] by auto

lemma funas_mctxt_fill_holes_mctxt [simp]:
assumes "num_holes C = length Ds"
shows "funas_mctxt (fill_holes_mctxt C Ds) = funas_mctxt C ∪ ⋃(set (map funas_mctxt Ds))"
(is "?f C Ds = ?g C Ds")
using assms
proof (induct C arbitrary: Ds)
case MHole
then show ?case by (cases Ds) simp_all
next
case (MFun f Cs)
then have num_holes: "sum_list (map num_holes Cs) = length Ds" by simp
let ?ys = "partition_holes Ds Cs"
have "⋀i. i < length Cs ⟹ ?f (Cs ! i) (?ys ! i) = ?g (Cs ! i) (?ys ! i)"
using MFun by (metis nth_mem num_holes.simps(3) length_partition_holes_nth)
then have "(⋃i ∈ {0 ..< length Cs}. ?f (Cs ! i) (?ys ! i)) =
(⋃i ∈ {0 ..< length Cs}. ?g (Cs ! i) (?ys ! i))" by simp
then show ?case
using num_holes
unfolding partition_holes_fill_holes_mctxt_conv
by (simp add: UN_Un_distrib UN_upt_len_conv [of _ _ "λx. ⋃(set x)"] UN_set_partition_by_map)
qed simp

end
```