Welcome to Abella 2.0.5-dev.
```Abella < Specification "copy".

```
```Abella < Theorem member_fresh :
forall L E, nabla x, member (E x) L -> (exists F, E = y\F).

```
```
============================
forall L E, nabla x, member (E x) L -> (exists F, E = y\F)

member_fresh < induction on 1.
```
```
IH : forall L E, nabla x, member (E x) L * -> (exists F, E = y\F)
============================
forall L E, nabla x, member (E x) L @ -> (exists F, E = y\F)

member_fresh < intros.
```
```
Variables: L E
IH : forall L E, nabla x, member (E x) L * -> (exists F, E = y\F)
H1 : member (E n1) L @
============================
exists F, E = y\F

member_fresh < case H1.
```
```Subgoal 1:

Variables: L3 L2
IH : forall L E, nabla x, member (E x) L * -> (exists F, E = y\F)
============================
exists F, z1\L2 = y\F

Subgoal 2 is:
exists F, E = y\F

member_fresh < search.
```
```Subgoal 2:

Variables: E L3 L2
IH : forall L E, nabla x, member (E x) L * -> (exists F, E = y\F)
H2 : member (E n1) L3 *
============================
exists F, E = y\F

member_fresh < apply IH to H2.
```
```Subgoal 2:

Variables: L3 L2 F
IH : forall L E, nabla x, member (E x) L * -> (exists F, E = y\F)
H2 : member F L3 *
============================
exists F1, z1\F = y\F1

member_fresh < search.
Proof completed.
```
```Abella < Define ctxs : olist -> olist -> prop by
ctxs nil nil;
ctxs (copy X Y :: L1) (copy2 X Y :: L2) := ctxs L1 L2.

```
```Abella < Define name : tm -> prop by
nabla x, name x.

```
```Abella < Theorem ctx_mem1 :
forall F L1 L2, ctxs L1 L2 -> member F L1 -> (exists X Y, F = copy X Y).

```
```
============================
forall F L1 L2, ctxs L1 L2 -> member F L1 -> (exists X Y, F = copy X Y)

ctx_mem1 < induction on 1.
```
```
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
============================
forall F L1 L2, ctxs L1 L2 @ -> member F L1 -> (exists X Y, F = copy X Y)

ctx_mem1 < intros.
```
```
Variables: F L1 L2
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H1 : ctxs L1 L2 @
H2 : member F L1
============================
exists X Y, F = copy X Y

ctx_mem1 < case H1.
```
```Subgoal 1:

Variables: F
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H2 : member F nil
============================
exists X Y, F = copy X Y

Subgoal 2 is:
exists X Y, F = copy X Y

ctx_mem1 < case H2.
```
```Subgoal 2:

Variables: F L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H2 : member F (copy X Y :: L3)
H3 : ctxs L3 L4 *
============================
exists X Y, F = copy X Y

ctx_mem1 < case H2.
```
```Subgoal 2.1:

Variables: L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H3 : ctxs L3 L4 *
============================
exists X1 Y1, copy X Y = copy X1 Y1

Subgoal 2.2 is:
exists X Y, F = copy X Y

ctx_mem1 < search.
```
```Subgoal 2.2:

Variables: F L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H3 : ctxs L3 L4 *
H4 : member F L3
============================
exists X Y, F = copy X Y

ctx_mem1 < apply IH to H3 H4.
```
```Subgoal 2.2:

Variables: L4 Y X L3 X1 Y1
IH : forall F L1 L2, ctxs L1 L2 * -> member F L1 ->
(exists X Y, F = copy X Y)
H3 : ctxs L3 L4 *
H4 : member (copy X1 Y1) L3
============================
exists X Y, copy X1 Y1 = copy X Y

ctx_mem1 < search.
Proof completed.
```
```Abella < Theorem ctx_mem2 :
forall F L1 L2, ctxs L1 L2 -> member F L2 -> (exists X Y, F = copy2 X Y).

```
```
============================
forall F L1 L2, ctxs L1 L2 -> member F L2 -> (exists X Y, F = copy2 X Y)

ctx_mem2 < induction on 1.
```
```
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
============================
forall F L1 L2, ctxs L1 L2 @ -> member F L2 -> (exists X Y, F = copy2 X Y)

ctx_mem2 < intros.
```
```
Variables: F L1 L2
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H1 : ctxs L1 L2 @
H2 : member F L2
============================
exists X Y, F = copy2 X Y

ctx_mem2 < case H1.
```
```Subgoal 1:

Variables: F
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H2 : member F nil
============================
exists X Y, F = copy2 X Y

Subgoal 2 is:
exists X Y, F = copy2 X Y

ctx_mem2 < case H2.
```
```Subgoal 2:

Variables: F L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H2 : member F (copy2 X Y :: L4)
H3 : ctxs L3 L4 *
============================
exists X Y, F = copy2 X Y

ctx_mem2 < case H2.
```
```Subgoal 2.1:

Variables: L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H3 : ctxs L3 L4 *
============================
exists X1 Y1, copy2 X Y = copy2 X1 Y1

Subgoal 2.2 is:
exists X Y, F = copy2 X Y

ctx_mem2 < search.
```
```Subgoal 2.2:

Variables: F L4 Y X L3
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H3 : ctxs L3 L4 *
H4 : member F L4
============================
exists X Y, F = copy2 X Y

ctx_mem2 < apply IH to H3 H4.
```
```Subgoal 2.2:

Variables: L4 Y X L3 X1 Y1
IH : forall F L1 L2, ctxs L1 L2 * -> member F L2 ->
(exists X Y, F = copy2 X Y)
H3 : ctxs L3 L4 *
H4 : member (copy2 X1 Y1) L4
============================
exists X Y, copy2 X1 Y1 = copy2 X Y

ctx_mem2 < search.
Proof completed.
```
```Abella < Theorem copy2_align :
forall M N K L, nabla z, ctxs K L -> {L, copy2 z z |- copy2 (M z) (N z)} ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)}).

```
```
============================
forall M N K L, nabla z, ctxs K L -> {L, copy2 z z |- copy2 (M z) (N z)} ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})

copy2_align < induction on 2.
```
```
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
============================
forall M N K L, nabla z, ctxs K L -> {L, copy2 z z |- copy2 (M z) (N z)}@ ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})

copy2_align < intros.
```
```
Variables: M N K L
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H2 : {L, copy2 n1 n1 |- copy2 (M n1) (N n1)}@
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < case H2.
```
```Subgoal 1:

Variables: K L Q M1 P N1
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1 |- copy2 (N1 n1) (P n1)}*
H4 : {L, copy2 n1 n1 |- copy2 (M1 n1) (Q n1)}*
============================
{L, copy2 n1 n2 |- copy2 (app (N1 n1) (M1 n1)) (app (P n2) (Q n2))}

Subgoal 2 is:
{L, copy2 n1 n2 |- copy2 (abs (R n1)) (abs (S n2))}

Subgoal 3 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < apply IH to H1 H3.
```
```Subgoal 1:

Variables: K L Q M1 P N1
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1 |- copy2 (N1 n1) (P n1)}*
H4 : {L, copy2 n1 n1 |- copy2 (M1 n1) (Q n1)}*
H5 : {L, copy2 n1 n2 |- copy2 (N1 n1) (P n2)}
============================
{L, copy2 n1 n2 |- copy2 (app (N1 n1) (M1 n1)) (app (P n2) (Q n2))}

Subgoal 2 is:
{L, copy2 n1 n2 |- copy2 (abs (R n1)) (abs (S n2))}

Subgoal 3 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < apply IH to H1 H4.
```
```Subgoal 1:

Variables: K L Q M1 P N1
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1 |- copy2 (N1 n1) (P n1)}*
H4 : {L, copy2 n1 n1 |- copy2 (M1 n1) (Q n1)}*
H5 : {L, copy2 n1 n2 |- copy2 (N1 n1) (P n2)}
H6 : {L, copy2 n1 n2 |- copy2 (M1 n1) (Q n2)}
============================
{L, copy2 n1 n2 |- copy2 (app (N1 n1) (M1 n1)) (app (P n2) (Q n2))}

Subgoal 2 is:
{L, copy2 n1 n2 |- copy2 (abs (R n1)) (abs (S n2))}

Subgoal 3 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < search.
```
```Subgoal 2:

Variables: K L S R
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, copy2 n2 n3 |- copy2 (R n1 n2) (S n1 n3)}*
============================
{L, copy2 n1 n2 |- copy2 (abs (R n1)) (abs (S n2))}

Subgoal 3 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < apply IH to _ H3.
```
```Subgoal 2:

Variables: K L S R
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, copy2 n2 n3 |- copy2 (R n1 n2) (S n1 n3)}*
H4 : {L, copy2 n2 n3, copy2 n1 n4 |- copy2 (R n1 n2) (S n4 n3)}
============================
{L, copy2 n1 n2 |- copy2 (abs (R n1)) (abs (S n2))}

Subgoal 3 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < search.
```
```Subgoal 3:

Variables: M N K L F
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, [F n1] |- copy2 (M n1) (N n1)}*
H4 : member (F n1) (copy2 n1 n1 :: L)
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < case H4.
```
```Subgoal 3.1:

Variables: M N K L
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, [copy2 n1 n1] |- copy2 (M n1) (N n1)}*
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

Subgoal 3.2 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < case H3.
```
```Subgoal 3.1:

Variables: K L
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
============================
{L, copy2 n1 n2 |- copy2 n1 n2}

Subgoal 3.2 is:
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < search.
```
```Subgoal 3.2:

Variables: M N K L F
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, [F n1] |- copy2 (M n1) (N n1)}*
H5 : member (F n1) L
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < apply member_fresh to H5.
```
```Subgoal 3.2:

Variables: M N K L F1
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, [F1] |- copy2 (M n1) (N n1)}*
H5 : member F1 L
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < apply ctx_mem2 to H1 H5.
```
```Subgoal 3.2:

Variables: M N K L X Y
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H3 : {L, copy2 n1 n1, [copy2 X Y] |- copy2 (M n1) (N n1)}*
H5 : member (copy2 X Y) L
============================
{L, copy2 n1 n2 |- copy2 (M n1) (N n2)}

copy2_align < case H3.
```
```Subgoal 3.2:

Variables: K L X Y
IH : forall M N K L, nabla z, ctxs K L ->
{L, copy2 z z |- copy2 (M z) (N z)}* ->
(nabla x y, {L, copy2 x y |- copy2 (M x) (N y)})
H1 : ctxs K L
H5 : member (copy2 X Y) L
============================
{L, copy2 n1 n2 |- copy2 X Y}

copy2_align < search.
Proof completed.
```
```Abella < Theorem ctxs_member1 :
forall X Y L K, ctxs L K -> member (copy X Y) L -> member (copy2 X Y) K.

```
```
============================
forall X Y L K, ctxs L K -> member (copy X Y) L -> member (copy2 X Y) K

ctxs_member1 < induction on 1.
```
```
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
============================
forall X Y L K, ctxs L K @ -> member (copy X Y) L -> member (copy2 X Y) K

ctxs_member1 < intros.
```
```
Variables: X Y L K
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H1 : ctxs L K @
H2 : member (copy X Y) L
============================
member (copy2 X Y) K

ctxs_member1 < case H1.
```
```Subgoal 1:

Variables: X Y
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H2 : member (copy X Y) nil
============================
member (copy2 X Y) nil

Subgoal 2 is:
member (copy2 X Y) (copy2 X1 Y1 :: L2)

ctxs_member1 < case H2.
```
```Subgoal 2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H2 : member (copy X Y) (copy X1 Y1 :: L1)
H3 : ctxs L1 L2 *
============================
member (copy2 X Y) (copy2 X1 Y1 :: L2)

ctxs_member1 < case H2.
```
```Subgoal 2.1:

Variables: L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H3 : ctxs L1 L2 *
============================
member (copy2 X1 Y1) (copy2 X1 Y1 :: L2)

Subgoal 2.2 is:
member (copy2 X Y) (copy2 X1 Y1 :: L2)

ctxs_member1 < search.
```
```Subgoal 2.2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H3 : ctxs L1 L2 *
H4 : member (copy X Y) L1
============================
member (copy2 X Y) (copy2 X1 Y1 :: L2)

ctxs_member1 < apply IH to H3 H4.
```
```Subgoal 2.2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy X Y) L ->
member (copy2 X Y) K
H3 : ctxs L1 L2 *
H4 : member (copy X Y) L1
H5 : member (copy2 X Y) L2
============================
member (copy2 X Y) (copy2 X1 Y1 :: L2)

ctxs_member1 < search.
Proof completed.
```
```Abella < Theorem copy_copy2 :
forall L K M N, ctxs L K -> {L |- copy M N} -> {K |- copy2 M N}.

```
```
============================
forall L K M N, ctxs L K -> {L |- copy M N} -> {K |- copy2 M N}

copy_copy2 < induction on 2.
```
```
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
============================
forall L K M N, ctxs L K -> {L |- copy M N}@ -> {K |- copy2 M N}

copy_copy2 < intros.
```
```
Variables: L K M N
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H2 : {L |- copy M N}@
============================
{K |- copy2 M N}

copy_copy2 < case H2.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L |- copy N1 P}*
H4 : {L |- copy M1 Q}*
============================
{K |- copy2 (app N1 M1) (app P Q)}

Subgoal 2 is:
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < apply IH to H1 H3.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L |- copy N1 P}*
H4 : {L |- copy M1 Q}*
H5 : {K |- copy2 N1 P}
============================
{K |- copy2 (app N1 M1) (app P Q)}

Subgoal 2 is:
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < apply IH to H1 H4.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L |- copy N1 P}*
H4 : {L |- copy M1 Q}*
H5 : {K |- copy2 N1 P}
H6 : {K |- copy2 M1 Q}
============================
{K |- copy2 (app N1 M1) (app P Q)}

Subgoal 2 is:
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < search.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L, copy n1 n1 |- copy (R n1) (S n1)}*
============================
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < apply IH to _ H3.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L, copy n1 n1 |- copy (R n1) (S n1)}*
H4 : {K, copy2 n1 n1 |- copy2 (R n1) (S n1)}
============================
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < apply copy2_align to H1 H4.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L, copy n1 n1 |- copy (R n1) (S n1)}*
H4 : {K, copy2 n1 n1 |- copy2 (R n1) (S n1)}
H5 : {K, copy2 n1 n2 |- copy2 (R n1) (S n2)}
============================
{K |- copy2 (abs R) (abs S)}

Subgoal 3 is:
{K |- copy2 M N}

copy_copy2 < search.
```
```Subgoal 3:

Variables: L K M N F
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L, [F] |- copy M N}*
H4 : member F L
============================
{K |- copy2 M N}

copy_copy2 < apply ctx_mem1 to H1 H4.
```
```Subgoal 3:

Variables: L K M N X Y
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H3 : {L, [copy X Y] |- copy M N}*
H4 : member (copy X Y) L
============================
{K |- copy2 M N}

copy_copy2 < case H3.
```
```Subgoal 3:

Variables: L K M N
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H4 : member (copy M N) L
============================
{K |- copy2 M N}

copy_copy2 < apply ctxs_member1 to H1 H4.
```
```Subgoal 3:

Variables: L K M N
IH : forall L K M N, ctxs L K -> {L |- copy M N}* -> {K |- copy2 M N}
H1 : ctxs L K
H4 : member (copy M N) L
H5 : member (copy2 M N) K
============================
{K |- copy2 M N}

copy_copy2 < search.
Proof completed.
```
```Abella < Theorem copy_align :
forall M N L, nabla x y, {L, copy x y |- copy (M x) (N y)} ->
(nabla z, {L, copy z z |- copy (M z) (N z)}).

```
```
============================
forall M N L, nabla x y, {L, copy x y |- copy (M x) (N y)} ->
(nabla z, {L, copy z z |- copy (M z) (N z)})

copy_align < intros.
```
```
Variables: M N L
H1 : {L, copy n1 n2 |- copy (M n1) (N n2)}
============================
{L, copy n1 n1 |- copy (M n1) (N n1)}

copy_align <  inst H1 with n2 = n1.
```
```
Variables: M N L
H1 : {L, copy n1 n2 |- copy (M n1) (N n2)}
H2 : {L, copy n1 n1 |- copy (M n1) (N n1)}
============================
{L, copy n1 n1 |- copy (M n1) (N n1)}

copy_align < search.
Proof completed.
```
```Abella < Theorem ctxs_member2 :
forall X Y L K, ctxs L K -> member (copy2 X Y) K -> member (copy X Y) L.

```
```
============================
forall X Y L K, ctxs L K -> member (copy2 X Y) K -> member (copy X Y) L

ctxs_member2 < induction on 1.
```
```
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
============================
forall X Y L K, ctxs L K @ -> member (copy2 X Y) K -> member (copy X Y) L

ctxs_member2 < intros.
```
```
Variables: X Y L K
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H1 : ctxs L K @
H2 : member (copy2 X Y) K
============================
member (copy X Y) L

ctxs_member2 < case H1.
```
```Subgoal 1:

Variables: X Y
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H2 : member (copy2 X Y) nil
============================
member (copy X Y) nil

Subgoal 2 is:
member (copy X Y) (copy X1 Y1 :: L1)

ctxs_member2 < case H2.
```
```Subgoal 2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H2 : member (copy2 X Y) (copy2 X1 Y1 :: L2)
H3 : ctxs L1 L2 *
============================
member (copy X Y) (copy X1 Y1 :: L1)

ctxs_member2 < case H2.
```
```Subgoal 2.1:

Variables: L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H3 : ctxs L1 L2 *
============================
member (copy X1 Y1) (copy X1 Y1 :: L1)

Subgoal 2.2 is:
member (copy X Y) (copy X1 Y1 :: L1)

ctxs_member2 < search.
```
```Subgoal 2.2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H3 : ctxs L1 L2 *
H4 : member (copy2 X Y) L2
============================
member (copy X Y) (copy X1 Y1 :: L1)

ctxs_member2 < apply IH to H3 H4.
```
```Subgoal 2.2:

Variables: X Y L2 Y1 X1 L1
IH : forall X Y L K, ctxs L K * -> member (copy2 X Y) K ->
member (copy X Y) L
H3 : ctxs L1 L2 *
H4 : member (copy2 X Y) L2
H5 : member (copy X Y) L1
============================
member (copy X Y) (copy X1 Y1 :: L1)

ctxs_member2 < search.
Proof completed.
```
```Abella < Theorem copy2_copy :
forall L K M N, ctxs L K -> {K |- copy2 M N} -> {L |- copy M N}.

```
```
============================
forall L K M N, ctxs L K -> {K |- copy2 M N} -> {L |- copy M N}

copy2_copy < induction on 2.
```
```
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
============================
forall L K M N, ctxs L K -> {K |- copy2 M N}@ -> {L |- copy M N}

copy2_copy < intros.
```
```
Variables: L K M N
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H2 : {K |- copy2 M N}@
============================
{L |- copy M N}

copy2_copy < case H2.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K |- copy2 N1 P}*
H4 : {K |- copy2 M1 Q}*
============================
{L |- copy (app N1 M1) (app P Q)}

Subgoal 2 is:
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < apply IH to H1 H3.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K |- copy2 N1 P}*
H4 : {K |- copy2 M1 Q}*
H5 : {L |- copy N1 P}
============================
{L |- copy (app N1 M1) (app P Q)}

Subgoal 2 is:
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < apply IH to H1 H4.
```
```Subgoal 1:

Variables: L K Q M1 P N1
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K |- copy2 N1 P}*
H4 : {K |- copy2 M1 Q}*
H5 : {L |- copy N1 P}
H6 : {L |- copy M1 Q}
============================
{L |- copy (app N1 M1) (app P Q)}

Subgoal 2 is:
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < search.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K, copy2 n1 n2 |- copy2 (R n1) (S n2)}*
============================
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < apply IH to _ H3.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K, copy2 n1 n2 |- copy2 (R n1) (S n2)}*
H4 : {L, copy n1 n2 |- copy (R n1) (S n2)}
============================
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < apply copy_align to H4.
```
```Subgoal 2:

Variables: L K S R
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K, copy2 n1 n2 |- copy2 (R n1) (S n2)}*
H4 : {L, copy n1 n2 |- copy (R n1) (S n2)}
H5 : {L, copy n1 n1 |- copy (R n1) (S n1)}
============================
{L |- copy (abs R) (abs S)}

Subgoal 3 is:
{L |- copy M N}

copy2_copy < search.
```
```Subgoal 3:

Variables: L K M N F
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K, [F] |- copy2 M N}*
H4 : member F K
============================
{L |- copy M N}

copy2_copy < apply ctx_mem2 to H1 H4.
```
```Subgoal 3:

Variables: L K M N X Y
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H3 : {K, [copy2 X Y] |- copy2 M N}*
H4 : member (copy2 X Y) K
============================
{L |- copy M N}

copy2_copy < case H3.
```
```Subgoal 3:

Variables: L K M N
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H4 : member (copy2 M N) K
============================
{L |- copy M N}

copy2_copy < apply ctxs_member2 to H1 H4.
```
```Subgoal 3:

Variables: L K M N
IH : forall L K M N, ctxs L K -> {K |- copy2 M N}* -> {L |- copy M N}
H1 : ctxs L K
H4 : member (copy2 M N) K
H5 : member (copy M N) L
============================
{L |- copy M N}

copy2_copy < search.
Proof completed.
```
```Abella <
```