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

```
```Abella < Define ctx : olist -> prop by
ctx nil;
ctx (hyp B :: L) := {form B} /\ ctx L.

```
```Abella < Theorem ctx_member :
forall E L, ctx L -> member E L -> (exists B, E = hyp B /\ {form B}).

```
```
============================
forall E L, ctx L -> member E L -> (exists B, E = hyp B /\ {form B})

ctx_member < induction on 1.
```
```
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
============================
forall E L, ctx L @ -> member E L -> (exists B, E = hyp B /\ {form B})

ctx_member < intros.
```
```
Variables: E L
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H1 : ctx L @
H2 : member E L
============================
exists B, E = hyp B /\ {form B}

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

Variables: E
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H2 : member E nil
============================
exists B, E = hyp B /\ {form B}

Subgoal 2 is:
exists B, E = hyp B /\ {form B}

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

Variables: E L1 B
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H2 : member E (hyp B :: L1)
H3 : {form B}
H4 : ctx L1 *
============================
exists B, E = hyp B /\ {form B}

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

Variables: L1 B
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H3 : {form B}
H4 : ctx L1 *
============================
exists B1, hyp B = hyp B1 /\ {form B1}

Subgoal 2.2 is:
exists B, E = hyp B /\ {form B}

ctx_member < search.
```
```Subgoal 2.2:

Variables: E L1 B
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H3 : {form B}
H4 : ctx L1 *
H5 : member E L1
============================
exists B, E = hyp B /\ {form B}

ctx_member < apply IH to H4 H5.
```
```Subgoal 2.2:

Variables: L1 B B1
IH : forall E L, ctx L * -> member E L -> (exists B, E = hyp B /\ {form B})
H3 : {form B}
H4 : ctx L1 *
H5 : member (hyp B1) L1
H6 : {form B1}
============================
exists B, hyp B1 = hyp B /\ {form B}

ctx_member < search.
Proof completed.
```
```Abella < Theorem hyp_form :
forall L B, ctx L -> {L |- hyp B} -> {form B}.

```
```
============================
forall L B, ctx L -> {L |- hyp B} -> {form B}

hyp_form < intros.
```
```
Variables: L B
H1 : ctx L
H2 : {L |- hyp B}
============================
{form B}

hyp_form < case H2.
```
```
Variables: L B F
H1 : ctx L
H3 : {L, [F] |- hyp B}
H4 : member F L
============================
{form B}

hyp_form < apply ctx_member to H1 H4.
```
```
Variables: L B B1
H1 : ctx L
H3 : {L, [hyp B1] |- hyp B}
H4 : member (hyp B1) L
H5 : {form B1}
============================
{form B}

hyp_form < case H3.
```
```
Variables: L B
H1 : ctx L
H4 : member (hyp B) L
H5 : {form B}
============================
{form B}

hyp_form < search.
Proof completed.
```
```Abella < Theorem hyp_imp_form :
forall L B1 B2, ctx L -> {L |- hyp (imp B1 B2)} -> {form B1} /\ {form B2}.

```
```
============================
forall L B1 B2, ctx L -> {L |- hyp (imp B1 B2)} -> {form B1} /\ {form B2}

hyp_imp_form < intros.
```
```
Variables: L B1 B2
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
============================
{form B1} /\ {form B2}

hyp_imp_form < apply hyp_form to H1 H2.
```
```
Variables: L B1 B2
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (imp B1 B2)}
============================
{form B1} /\ {form B2}

hyp_imp_form < case H3.
```
```
Variables: L B1 B2
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {form B1}
H5 : {form B2}
============================
{form B1} /\ {form B2}

hyp_imp_form < search.
Proof completed.
```
```Abella < Theorem sound :
(forall L A B, ctx L -> {form B} -> {L |- focus B A} ->
{L, hyp B |- conc (atom A)}) /\
(forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B}).

```
```
============================
(forall L A B, ctx L -> {form B} -> {L |- focus B A} ->
{L, hyp B |- conc (atom A)}) /\
(forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B})

sound < induction on 3 3.
```
```
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
============================
(forall L A B, ctx L -> {form B} -> {L |- focus B A}@ ->
{L, hyp B |- conc (atom A)}) /\
(forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B})

sound < split.
```
```Subgoal 1:

IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
============================
forall L A B, ctx L -> {form B} -> {L |- focus B A}@ ->
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < intros.
```
```Subgoal 1:

Variables: L A B
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H3 : {L |- focus B A}@
============================
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < case H3.
```
```Subgoal 1.1:

Variables: L A
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (atom A)}
============================
{L, hyp (atom A) |- conc (atom A)}

Subgoal 1.2 is:
{L, hyp (imp B1 C) |- conc (atom A)}

Subgoal 1.3 is:
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < search.
```
```Subgoal 1.2:

Variables: L A C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (imp B1 C)}
H4 : {L |- unfocus B1}*
H5 : {L, hyp C |- focus C A}*
============================
{L, hyp (imp B1 C) |- conc (atom A)}

Subgoal 1.3 is:
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < case H2.
```
```Subgoal 1.2:

Variables: L A C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H4 : {L |- unfocus B1}*
H5 : {L, hyp C |- focus C A}*
H6 : {form B1}
H7 : {form C}
============================
{L, hyp (imp B1 C) |- conc (atom A)}

Subgoal 1.3 is:
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < apply IH1 to H1 H6 H4.
```
```Subgoal 1.2:

Variables: L A C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H4 : {L |- unfocus B1}*
H5 : {L, hyp C |- focus C A}*
H6 : {form B1}
H7 : {form C}
H8 : {L |- conc B1}
============================
{L, hyp (imp B1 C) |- conc (atom A)}

Subgoal 1.3 is:
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < apply IH to _ H7 H5.
```
```Subgoal 1.2:

Variables: L A C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H4 : {L |- unfocus B1}*
H5 : {L, hyp C |- focus C A}*
H6 : {form B1}
H7 : {form C}
H8 : {L |- conc B1}
H9 : {L, hyp C |- conc (atom A)}
============================
{L, hyp (imp B1 C) |- conc (atom A)}

Subgoal 1.3 is:
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < search.
```
```Subgoal 1.3:

Variables: L A B F
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H4 : {L, [F] |- focus B A}*
H5 : member F L
============================
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < apply ctx_member to H1 H5.
```
```Subgoal 1.3:

Variables: L A B B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H4 : {L, [hyp B1] |- focus B A}*
H5 : member (hyp B1) L
H6 : {form B1}
============================
{L, hyp B |- conc (atom A)}

Subgoal 2 is:
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < case H4.
```
```Subgoal 2:

IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
============================
forall L B, ctx L -> {form B} -> {L |- unfocus B}@ -> {L |- conc B}

sound < intros.
```
```Subgoal 2:

Variables: L B
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H3 : {L |- unfocus B}@
============================
{L |- conc B}

sound < case H3.
```
```Subgoal 2.1:

Variables: L C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (imp B1 C)}
H4 : {L, hyp B1 |- unfocus C}*
============================
{L |- conc (imp B1 C)}

Subgoal 2.2 is:
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

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

Variables: L C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H4 : {L, hyp B1 |- unfocus C}*
H5 : {form B1}
H6 : {form C}
============================
{L |- conc (imp B1 C)}

Subgoal 2.2 is:
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < apply IH1 to _ H6 H4.
```
```Subgoal 2.1:

Variables: L C B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H4 : {L, hyp B1 |- unfocus C}*
H5 : {form B1}
H6 : {form C}
H7 : {L, hyp B1 |- conc C}
============================
{L |- conc (imp B1 C)}

Subgoal 2.2 is:
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < search.
```
```Subgoal 2.2:

Variables: L A B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (atom A)}
H4 : {L |- hyp B1}*
H5 : {L |- focus B1 A}*
============================
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < apply hyp_form to H1 H4.
```
```Subgoal 2.2:

Variables: L A B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (atom A)}
H4 : {L |- hyp B1}*
H5 : {L |- focus B1 A}*
H6 : {form B1}
============================
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < apply IH to H1 H6 H5.
```
```Subgoal 2.2:

Variables: L A B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (atom A)}
H4 : {L |- hyp B1}*
H5 : {L |- focus B1 A}*
H6 : {form B1}
H7 : {L, hyp B1 |- conc (atom A)}
============================
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < cut H7 with H4.
```
```Subgoal 2.2:

Variables: L A B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form (atom A)}
H4 : {L |- hyp B1}*
H5 : {L |- focus B1 A}*
H6 : {form B1}
H7 : {L, hyp B1 |- conc (atom A)}
H8 : {L |- conc (atom A)}
============================
{L |- conc (atom A)}

Subgoal 2.3 is:
{L |- conc B}

sound < search.
```
```Subgoal 2.3:

Variables: L B F
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H4 : {L, [F] |- unfocus B}*
H5 : member F L
============================
{L |- conc B}

sound < apply ctx_member to H1 H5.
```
```Subgoal 2.3:

Variables: L B B1
IH : forall L A B, ctx L -> {form B} -> {L |- focus B A}* ->
{L, hyp B |- conc (atom A)}
IH1 : forall L B, ctx L -> {form B} -> {L |- unfocus B}* -> {L |- conc B}
H1 : ctx L
H2 : {form B}
H4 : {L, [hyp B1] |- unfocus B}*
H5 : member (hyp B1) L
H6 : {form B1}
============================
{L |- conc B}

sound < case H4.
Proof completed.
```
```Abella < Theorem soundness :
forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B}.

```
```
============================
forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B}

soundness < apply sound.
```
```
H1 : forall L A B, ctx L -> {form B} -> {L |- focus B A} ->
{L, hyp B |- conc (atom A)}
H2 : forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B}
============================
forall L B, ctx L -> {form B} -> {L |- unfocus B} -> {L |- conc B}

soundness < search.
Proof completed.
```
```Abella < Theorem init_form :
forall L B, {form B} -> member (hyp B) L -> {L |- conc-i B}.

```
```
============================
forall L B, {form B} -> member (hyp B) L -> {L |- conc-i B}

init_form < induction on 1.
```
```
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
============================
forall L B, {form B}@ -> member (hyp B) L -> {L |- conc-i B}

init_form < intros.
```
```
Variables: L B
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
H1 : {form B}@
H2 : member (hyp B) L
============================
{L |- conc-i B}

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

Variables: L A
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
H2 : member (hyp (atom A)) L
============================
{L |- conc-i (atom A)}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

init_form < search.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
H2 : member (hyp (imp B1 C)) L
H3 : {form B1}*
H4 : {form C}*
============================
{L |- conc-i (imp B1 C)}

init_form < apply IH to H3 _ with L = hyp B1 :: L.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
H2 : member (hyp (imp B1 C)) L
H3 : {form B1}*
H4 : {form C}*
H5 : {L, hyp B1 |- conc-i B1}
============================
{L |- conc-i (imp B1 C)}

init_form < apply IH to H4 _ with L = hyp C :: L.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, {form B}* -> member (hyp B) L -> {L |- conc-i B}
H2 : member (hyp (imp B1 C)) L
H3 : {form B1}*
H4 : {form C}*
H5 : {L, hyp B1 |- conc-i B1}
H6 : {L, hyp C |- conc-i C}
============================
{L |- conc-i (imp B1 C)}

init_form < search.
Proof completed.
```
```Abella < Theorem restrict_init :
forall L B, ctx L -> {form B} -> {L |- conc B} -> {L |- conc-i B}.

```
```
============================
forall L B, ctx L -> {form B} -> {L |- conc B} -> {L |- conc-i B}

restrict_init < induction on 3.
```
```
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
============================
forall L B, ctx L -> {form B} -> {L |- conc B}@ -> {L |- conc-i B}

restrict_init < intros.
```
```
Variables: L B
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H3 : {L |- conc B}@
============================
{L |- conc-i B}

restrict_init < case H3.
```
```Subgoal 1:

Variables: L B
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp B}*
============================
{L |- conc-i B}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < case H4.
```
```Subgoal 1:

Variables: L B F
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H5 : {L, [F] |- hyp B}*
H6 : member F L
============================
{L |- conc-i B}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply ctx_member to H1 H6.
```
```Subgoal 1:

Variables: L B B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H5 : {L, [hyp B1] |- hyp B}*
H6 : member (hyp B1) L
H7 : {form B1}
============================
{L |- conc-i B}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply init_form to H7 H6.
```
```Subgoal 1:

Variables: L B B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H5 : {L, [hyp B1] |- hyp B}*
H6 : member (hyp B1) L
H7 : {form B1}
H8 : {L |- conc-i B1}
============================
{L |- conc-i B}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < case H5.
```
```Subgoal 1:

Variables: L B
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H6 : member (hyp B) L
H7 : {form B}
H8 : {L |- conc-i B}
============================
{L |- conc-i B}

Subgoal 2 is:
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < search.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form (imp B1 C)}
H4 : {L, hyp B1 |- conc C}*
============================
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

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

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H4 : {L, hyp B1 |- conc C}*
H5 : {form B1}
H6 : {form C}
============================
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply IH to _ H6 H4.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H4 : {L, hyp B1 |- conc C}*
H5 : {form B1}
H6 : {form C}
H7 : {L, hyp B1 |- conc-i C}
============================
{L |- conc-i (imp B1 C)}

Subgoal 3 is:
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < search.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc B1}*
H6 : {L, hyp C |- conc B}*
============================
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply hyp_imp_form to H1 H4.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc B1}*
H6 : {L, hyp C |- conc B}*
H7 : {form B1}
H8 : {form C}
============================
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply IH to H1 H7 H5.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc B1}*
H6 : {L, hyp C |- conc B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- conc-i B1}
============================
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < apply IH to _ H2 H6.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc B1}*
H6 : {L, hyp C |- conc B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- conc-i B1}
H10 : {L, hyp C |- conc-i B}
============================
{L |- conc-i B}

Subgoal 4 is:
{L |- conc-i B}

restrict_init < search.
```
```Subgoal 4:

Variables: L B F
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L, [F] |- conc B}*
H5 : member F L
============================
{L |- conc-i B}

restrict_init < apply ctx_member to H1 H5.
```
```Subgoal 4:

Variables: L B B1
IH : forall L B, ctx L -> {form B} -> {L |- conc B}* -> {L |- conc-i B}
H1 : ctx L
H2 : {form B}
H4 : {L, [hyp B1] |- conc B}*
H5 : member (hyp B1) L
H6 : {form B1}
============================
{L |- conc-i B}

restrict_init < case H4.
Proof completed.
```
```Abella < Theorem lemma :
forall B2, (forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C} ->
{L |- unfocus C}) /\
(forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A} -> {L |- focus B A}).

```
```
============================
forall B2, (forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C} ->
{L |- unfocus C}) /\
(forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A} -> {L |- focus B A})

lemma < intros.
```
```
Variables: B2
============================
(forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C} -> {L |- unfocus C}) /\
(forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A} -> {L |- focus B A})

lemma < induction on 5 5.
```
```
Variables: B2
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
============================
(forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}@ -> {L |- unfocus C}) /\
(forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A})

lemma < split.
```
```Subgoal 1:

Variables: B2
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
============================
forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}@ -> {L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < intros.
```
```Subgoal 1:

Variables: B2 L B1 C
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form C}
H4 : {L |- unfocus B1}
H5 : {L, hyp B2 |- unfocus C}@
============================
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H5.
```
```Subgoal 1.1:

Variables: B2 L B1 C1 B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (imp B C1)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, hyp B |- unfocus C1}*
============================
{L |- unfocus (imp B C1)}

Subgoal 1.2 is:
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H3.
```
```Subgoal 1.1:

Variables: B2 L B1 C1 B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, hyp B |- unfocus C1}*
H7 : {form B}
H8 : {form C1}
============================
{L |- unfocus (imp B C1)}

Subgoal 1.2 is:
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < apply IH to _ _ H8 _ H6.
```
```Subgoal 1.1:

Variables: B2 L B1 C1 B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, hyp B |- unfocus C1}*
H7 : {form B}
H8 : {form C1}
H9 : {L, hyp B |- unfocus C1}
============================
{L |- unfocus (imp B C1)}

Subgoal 1.2 is:
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < search.
```
```Subgoal 1.2:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- hyp B}*
H7 : {L, hyp B2 |- focus B A}*
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H6.
```
```Subgoal 1.2:

Variables: B2 L B1 A B F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H8 : {L, hyp B2, [F] |- hyp B}*
H9 : member F (hyp B2 :: L)
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H9.
```
```Subgoal 1.2.1:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H8 : {L, hyp B2, [hyp B2] |- hyp B}*
============================
{L |- unfocus (atom A)}

Subgoal 1.2.2 is:
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H8.
```
```Subgoal 1.2.1:

Variables: L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B2, ctx L -> {L |- hyp (imp B1 B)} -> {form B2} ->
{L |- unfocus B1} -> {L, hyp B |- focus B2 A}* -> {L |- focus B2 A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B |- focus B A}*
============================
{L |- unfocus (atom A)}

Subgoal 1.2.2 is:
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < search.
```
```Subgoal 1.2.2:

Variables: B2 L B1 A B F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H8 : {L, hyp B2, [F] |- hyp B}*
H10 : member F L
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < apply ctx_member to H1 H10.
```
```Subgoal 1.2.2:

Variables: B2 L B1 A B B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H8 : {L, hyp B2, [hyp B3] |- hyp B}*
H10 : member (hyp B3) L
H11 : {form B3}
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H8.
```
```Subgoal 1.2.2:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H10 : member (hyp B) L
H11 : {form B}
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < apply IH1 to H1 H2 H11 H4 H7.
```
```Subgoal 1.2.2:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
H7 : {L, hyp B2 |- focus B A}*
H10 : member (hyp B) L
H11 : {form B}
H12 : {L |- focus B A}
============================
{L |- unfocus (atom A)}

Subgoal 1.3 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < search.
```
```Subgoal 1.3:

Variables: B2 L B1 C F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form C}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [F] |- unfocus C}*
H7 : member F (hyp B2 :: L)
============================
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H7.
```
```Subgoal 1.3.1:

Variables: B2 L B1 C
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form C}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [hyp B2] |- unfocus C}*
============================
{L |- unfocus C}

Subgoal 1.3.2 is:
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H6.
```
```Subgoal 1.3.2:

Variables: B2 L B1 C F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form C}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [F] |- unfocus C}*
H8 : member F L
============================
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < apply ctx_member to H1 H8.
```
```Subgoal 1.3.2:

Variables: B2 L B1 C B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form C}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [hyp B] |- unfocus C}*
H8 : member (hyp B) L
H9 : {form B}
============================
{L |- unfocus C}

Subgoal 2 is:
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < case H6.
```
```Subgoal 2:

Variables: B2
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
============================
forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}@ -> {L |- focus B A}

lemma < intros.
```
```Subgoal 2:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form B}
H4 : {L |- unfocus B1}
H5 : {L, hyp B2 |- focus B A}@
============================
{L |- focus B A}

lemma < case H5.
```
```Subgoal 2.1:

Variables: B2 L B1 A
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (atom A)}
H4 : {L |- unfocus B1}
============================
{L |- focus (atom A) A}

Subgoal 2.2 is:
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < search.
```
```Subgoal 2.2:

Variables: B2 L B1 A C B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form (imp B3 C)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- unfocus B3}*
H7 : {L, hyp B2, hyp C |- focus C A}*
============================
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < case H3.
```
```Subgoal 2.2:

Variables: B2 L B1 A C B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- unfocus B3}*
H7 : {L, hyp B2, hyp C |- focus C A}*
H8 : {form B3}
H9 : {form C}
============================
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < apply hyp_imp_form to H1 H2.
```
```Subgoal 2.2:

Variables: B2 L B1 A C B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- unfocus B3}*
H7 : {L, hyp B2, hyp C |- focus C A}*
H8 : {form B3}
H9 : {form C}
H10 : {form B1}
H11 : {form B2}
============================
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < apply IH to _ _ H8 _ H6.
```
```Subgoal 2.2:

Variables: B2 L B1 A C B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- unfocus B3}*
H7 : {L, hyp B2, hyp C |- focus C A}*
H8 : {form B3}
H9 : {form C}
H10 : {form B1}
H11 : {form B2}
H12 : {L |- unfocus B3}
============================
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < apply IH1 to _ _ H9 _ H7.
```
```Subgoal 2.2:

Variables: B2 L B1 A C B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2 |- unfocus B3}*
H7 : {L, hyp B2, hyp C |- focus C A}*
H8 : {form B3}
H9 : {form C}
H10 : {form B1}
H11 : {form B2}
H12 : {L |- unfocus B3}
H13 : {L, hyp C |- focus C A}
============================
{L |- focus (imp B3 C) A}

Subgoal 2.3 is:
{L |- focus B A}

lemma < search.
```
```Subgoal 2.3:

Variables: B2 L B1 A B F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form B}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [F] |- focus B A}*
H7 : member F (hyp B2 :: L)
============================
{L |- focus B A}

lemma < case H7.
```
```Subgoal 2.3.1:

Variables: B2 L B1 A B
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form B}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [hyp B2] |- focus B A}*
============================
{L |- focus B A}

Subgoal 2.3.2 is:
{L |- focus B A}

lemma < case H6.
```
```Subgoal 2.3.2:

Variables: B2 L B1 A B F
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form B}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [F] |- focus B A}*
H8 : member F L
============================
{L |- focus B A}

lemma < apply ctx_member to H1 H8.
```
```Subgoal 2.3.2:

Variables: B2 L B1 A B B3
IH : forall L B1 C, ctx L -> {L |- hyp (imp B1 B2)} -> {form C} ->
{L |- unfocus B1} -> {L, hyp B2 |- unfocus C}* -> {L |- unfocus C}
IH1 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 B2)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp B2 |- focus B A}* -> {L |- focus B A}
H1 : ctx L
H2 : {L |- hyp (imp B1 B2)}
H3 : {form B}
H4 : {L |- unfocus B1}
H6 : {L, hyp B2, [hyp B3] |- focus B A}*
H8 : member (hyp B3) L
H9 : {form B3}
============================
{L |- focus B A}

lemma < case H6.
Proof completed.
```
```Abella < Theorem conc-i_complete :
forall L B, ctx L -> {form B} -> {L |- conc-i B} -> {L |- unfocus B}.

```
```
============================
forall L B, ctx L -> {form B} -> {L |- conc-i B} -> {L |- unfocus B}

conc-i_complete < induction on 3.
```
```
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
============================
forall L B, ctx L -> {form B} -> {L |- conc-i B}@ -> {L |- unfocus B}

conc-i_complete < intros.
```
```
Variables: L B
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H3 : {L |- conc-i B}@
============================
{L |- unfocus B}

conc-i_complete < case H3.
```
```Subgoal 1:

Variables: L A
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form (atom A)}
H4 : {L |- hyp (atom A)}*
============================
{L |- unfocus (atom A)}

Subgoal 2 is:
{L |- unfocus (imp B1 C)}

Subgoal 3 is:
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < search.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form (imp B1 C)}
H4 : {L, hyp B1 |- conc-i C}*
============================
{L |- unfocus (imp B1 C)}

Subgoal 3 is:
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < case H2.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H4 : {L, hyp B1 |- conc-i C}*
H5 : {form B1}
H6 : {form C}
============================
{L |- unfocus (imp B1 C)}

Subgoal 3 is:
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply IH to _ H6 H4.
```
```Subgoal 2:

Variables: L C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H4 : {L, hyp B1 |- conc-i C}*
H5 : {form B1}
H6 : {form C}
H7 : {L, hyp B1 |- unfocus C}
============================
{L |- unfocus (imp B1 C)}

Subgoal 3 is:
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < search.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply hyp_imp_form to H1 H4.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
H7 : {form B1}
H8 : {form C}
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply IH to H1 H7 H5.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- unfocus B1}
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply IH to _ H2 H6.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- unfocus B1}
H10 : {L, hyp C |- unfocus B}
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply lemma with B2 = C.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- unfocus B1}
H10 : {L, hyp C |- unfocus B}
H11 : forall L B1 C1, ctx L -> {L |- hyp (imp B1 C)} -> {form C1} ->
{L |- unfocus B1} -> {L, hyp C |- unfocus C1} -> {L |- unfocus C1}
H12 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 C)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp C |- focus B A} -> {L |- focus B A}
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < apply H11 to H1 H4 H2 H9 H10.
```
```Subgoal 3:

Variables: L B C B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L |- hyp (imp B1 C)}*
H5 : {L |- conc-i B1}*
H6 : {L, hyp C |- conc-i B}*
H7 : {form B1}
H8 : {form C}
H9 : {L |- unfocus B1}
H10 : {L, hyp C |- unfocus B}
H11 : forall L B1 C1, ctx L -> {L |- hyp (imp B1 C)} -> {form C1} ->
{L |- unfocus B1} -> {L, hyp C |- unfocus C1} -> {L |- unfocus C1}
H12 : forall L B1 A B, ctx L -> {L |- hyp (imp B1 C)} -> {form B} ->
{L |- unfocus B1} -> {L, hyp C |- focus B A} -> {L |- focus B A}
H13 : {L |- unfocus B}
============================
{L |- unfocus B}

Subgoal 4 is:
{L |- unfocus B}

conc-i_complete < search.
```
```Subgoal 4:

Variables: L B F
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L, [F] |- conc-i B}*
H5 : member F L
============================
{L |- unfocus B}

conc-i_complete < apply ctx_member to H1 H5.
```
```Subgoal 4:

Variables: L B B1
IH : forall L B, ctx L -> {form B} -> {L |- conc-i B}* -> {L |- unfocus B}
H1 : ctx L
H2 : {form B}
H4 : {L, [hyp B1] |- conc-i B}*
H5 : member (hyp B1) L
H6 : {form B1}
============================
{L |- unfocus B}

conc-i_complete < case H4.
Proof completed.
```
```Abella < Theorem completeness :
forall L B, ctx L -> {form B} -> {L |- conc B} -> {L |- unfocus B}.

```
```
============================
forall L B, ctx L -> {form B} -> {L |- conc B} -> {L |- unfocus B}

completeness < intros.
```
```
Variables: L B
H1 : ctx L
H2 : {form B}
H3 : {L |- conc B}
============================
{L |- unfocus B}

completeness < apply restrict_init to H1 H2 H3.
```
```
Variables: L B
H1 : ctx L
H2 : {form B}
H3 : {L |- conc B}
H4 : {L |- conc-i B}
============================
{L |- unfocus B}

completeness < apply conc-i_complete to H1 H2 H4.
```
```
Variables: L B
H1 : ctx L
H2 : {form B}
H3 : {L |- conc B}
H4 : {L |- conc-i B}
H5 : {L |- unfocus B}
============================
{L |- unfocus B}

completeness < search.
Proof completed.
```
```Abella <
```