Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

## Predicates of Arity 3

1 Mutual Coinduction

Section Arg3_1.

Definition monotone3 T0 T1 T2 (gf: rel3 T0 T1 T2 -> rel3 T0 T1 T2) :=
forall x0 x1 x2 r r' (IN: gf r x0 x1 x2) (LE: r <3= r'), gf r' x0 x1 x2.

Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable gf : rel3 T0 T1 T2 -> rel3 T0 T1 T2.
Implicit Arguments gf [].

Theorem paco3_acc: forall
l r (OBG: forall rr (INC: r <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3 gf rr),
l <3= paco3 gf r.
Proof.
intros; assert (SIM: paco3 gf (r \3/ l) x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_mon: monotone3 (paco3 gf).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_mult_strong: forall r,
paco3 gf (upaco3 gf r) <3= paco3 gf r.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Corollary paco3_mult: forall r,
paco3 gf (paco3 gf r) <3= paco3 gf r.
Proof. intros; eapply paco3_mult_strong, paco3_mon; eauto. Qed.

Theorem paco3_fold: forall r,
gf (upaco3 gf r) <3= paco3 gf r.
Proof. intros; econstructor; [ |eauto]; eauto. Qed.

Theorem paco3_unfold: forall (MON: monotone3 gf) r,
paco3 gf r <3= gf (upaco3 gf r).
Proof. unfold monotone3; intros; destruct PR; eauto. Qed.

End Arg3_1.

Hint Unfold monotone3.
Hint Resolve paco3_fold.

Implicit Arguments paco3_acc [ T0 T1 T2 ].
Implicit Arguments paco3_mon [ T0 T1 T2 ].
Implicit Arguments paco3_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_mult [ T0 T1 T2 ].
Implicit Arguments paco3_fold [ T0 T1 T2 ].
Implicit Arguments paco3_unfold [ T0 T1 T2 ].

Instance paco3_inst T0 T1 T2 (gf : rel3 T0 T1 T2->_) r x0 x1 x2 : paco_class (paco3 gf r x0 x1 x2) :=
{ pacoacc := paco3_acc gf;
pacomult := paco3_mult gf;
pacofold := paco3_fold gf;
pacounfold := paco3_unfold gf }.

2 Mutual Coinduction

Section Arg3_2.

Definition monotone3_2 T0 T1 T2 (gf: rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2) :=
forall x0 x1 x2 r_0 r_1 r'_0 r'_1 (IN: gf r_0 r_1 x0 x1 x2) (LE_0: r_0 <3= r'_0)(LE_1: r_1 <3= r'_1), gf r'_0 r'_1 x0 x1 x2.

Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable gf_0 gf_1 : rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco3_2_0_acc: forall
l r_0 r_1 (OBG: forall rr (INC: r_0 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_2_0 gf_0 gf_1 rr r_1),
l <3= paco3_2_0 gf_0 gf_1 r_0 r_1.
Proof.
intros; assert (SIM: paco3_2_0 gf_0 gf_1 (r_0 \3/ l) r_1 x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_2_1_acc: forall
l r_0 r_1 (OBG: forall rr (INC: r_1 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_2_1 gf_0 gf_1 r_0 rr),
l <3= paco3_2_1 gf_0 gf_1 r_0 r_1.
Proof.
intros; assert (SIM: paco3_2_1 gf_0 gf_1 r_0 (r_1 \3/ l) x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_2_0_mon: monotone3_2 (paco3_2_0 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_2_1_mon: monotone3_2 (paco3_2_1 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_2_0_mult_strong: forall r_0 r_1,
paco3_2_0 gf_0 gf_1 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_2_1_mult_strong: forall r_0 r_1,
paco3_2_1 gf_0 gf_1 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Corollary paco3_2_0_mult: forall r_0 r_1,
paco3_2_0 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1) (paco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco3_2_0_mult_strong, paco3_2_0_mon; eauto. Qed.

Corollary paco3_2_1_mult: forall r_0 r_1,
paco3_2_1 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1) (paco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco3_2_1_mult_strong, paco3_2_1_mon; eauto. Qed.

Theorem paco3_2_0_fold: forall r_0 r_1,
gf_0 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco3_2_1_fold: forall r_0 r_1,
gf_1 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco3_2_0_unfold: forall (MON: monotone3_2 gf_0) (MON: monotone3_2 gf_1) r_0 r_1,
paco3_2_0 gf_0 gf_1 r_0 r_1 <3= gf_0 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1).
Proof. unfold monotone3_2; intros; destruct PR; eauto. Qed.

Theorem paco3_2_1_unfold: forall (MON: monotone3_2 gf_0) (MON: monotone3_2 gf_1) r_0 r_1,
paco3_2_1 gf_0 gf_1 r_0 r_1 <3= gf_1 (upaco3_2_0 gf_0 gf_1 r_0 r_1) (upaco3_2_1 gf_0 gf_1 r_0 r_1).
Proof. unfold monotone3_2; intros; destruct PR; eauto. Qed.

End Arg3_2.

Hint Unfold monotone3_2.
Hint Resolve paco3_2_0_fold.
Hint Resolve paco3_2_1_fold.

Implicit Arguments paco3_2_0_acc [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_acc [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mon [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mon [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mult [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mult [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_fold [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_fold [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_unfold [ T0 T1 T2 ].

Instance paco3_2_0_inst T0 T1 T2 (gf_0 gf_1 : rel3 T0 T1 T2->_) r_0 r_1 x0 x1 x2 : paco_class (paco3_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2) :=
{ pacoacc := paco3_2_0_acc gf_0 gf_1;
pacomult := paco3_2_0_mult gf_0 gf_1;
pacofold := paco3_2_0_fold gf_0 gf_1;
pacounfold := paco3_2_0_unfold gf_0 gf_1 }.

Instance paco3_2_1_inst T0 T1 T2 (gf_0 gf_1 : rel3 T0 T1 T2->_) r_0 r_1 x0 x1 x2 : paco_class (paco3_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2) :=
{ pacoacc := paco3_2_1_acc gf_0 gf_1;
pacomult := paco3_2_1_mult gf_0 gf_1;
pacofold := paco3_2_1_fold gf_0 gf_1;
pacounfold := paco3_2_1_unfold gf_0 gf_1 }.

3 Mutual Coinduction

Section Arg3_3.

Definition monotone3_3 T0 T1 T2 (gf: rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2) :=
forall x0 x1 x2 r_0 r_1 r_2 r'_0 r'_1 r'_2 (IN: gf r_0 r_1 r_2 x0 x1 x2) (LE_0: r_0 <3= r'_0)(LE_1: r_1 <3= r'_1)(LE_2: r_2 <3= r'_2), gf r'_0 r'_1 r'_2 x0 x1 x2.

Variable T0 : Type.
Variable T1 : forall (x0: @T0), Type.
Variable T2 : forall (x0: @T0) (x1: @T1 x0), Type.
Variable gf_0 gf_1 gf_2 : rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2 -> rel3 T0 T1 T2.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco3_3_0_acc: forall
l r_0 r_1 r_2 (OBG: forall rr (INC: r_0 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
l <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
intros; assert (SIM: paco3_3_0 gf_0 gf_1 gf_2 (r_0 \3/ l) r_1 r_2 x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_3_1_acc: forall
l r_0 r_1 r_2 (OBG: forall rr (INC: r_1 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
l <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
intros; assert (SIM: paco3_3_1 gf_0 gf_1 gf_2 r_0 (r_1 \3/ l) r_2 x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_3_2_acc: forall
l r_0 r_1 r_2 (OBG: forall rr (INC: r_2 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
l <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
intros; assert (SIM: paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 (r_2 \3/ l) x0 x1 x2) by eauto.
clear PR; repeat (try left; do 4 paco_revert; paco_cofix_auto).
Qed.

Theorem paco3_3_0_mon: monotone3_3 (paco3_3_0 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_3_1_mon: monotone3_3 (paco3_3_1 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_3_2_mon: monotone3_3 (paco3_3_2 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_3_0_mult_strong: forall r_0 r_1 r_2,
paco3_3_0 gf_0 gf_1 gf_2 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_3_1_mult_strong: forall r_0 r_1 r_2,
paco3_3_1 gf_0 gf_1 gf_2 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Theorem paco3_3_2_mult_strong: forall r_0 r_1 r_2,
paco3_3_2 gf_0 gf_1 gf_2 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 4 paco_revert; paco_cofix_auto). Qed.

Corollary paco3_3_0_mult: forall r_0 r_1 r_2,
paco3_3_0 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco3_3_0_mult_strong, paco3_3_0_mon; eauto. Qed.

Corollary paco3_3_1_mult: forall r_0 r_1 r_2,
paco3_3_1 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco3_3_1_mult_strong, paco3_3_1_mon; eauto. Qed.

Corollary paco3_3_2_mult: forall r_0 r_1 r_2,
paco3_3_2 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco3_3_2_mult_strong, paco3_3_2_mon; eauto. Qed.

Theorem paco3_3_0_fold: forall r_0 r_1 r_2,
gf_0 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco3_3_1_fold: forall r_0 r_1 r_2,
gf_1 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco3_3_2_fold: forall r_0 r_1 r_2,
gf_2 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco3_3_0_unfold: forall (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_0 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2).
Proof. unfold monotone3_3; intros; destruct PR; eauto. Qed.

Theorem paco3_3_1_unfold: forall (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_1 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2).
Proof. unfold monotone3_3; intros; destruct PR; eauto. Qed.

Theorem paco3_3_2_unfold: forall (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_2 (upaco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (upaco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2).
Proof. unfold monotone3_3; intros; destruct PR; eauto. Qed.

End Arg3_3.

Hint Unfold monotone3_3.
Hint Resolve paco3_3_0_fold.
Hint Resolve paco3_3_1_fold.
Hint Resolve paco3_3_2_fold.

Implicit Arguments paco3_3_0_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_unfold [ T0 T1 T2 ].

Instance paco3_3_0_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2->_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_0_acc gf_0 gf_1 gf_2;
pacomult := paco3_3_0_mult gf_0 gf_1 gf_2;
pacofold := paco3_3_0_fold gf_0 gf_1 gf_2;
pacounfold := paco3_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco3_3_1_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2->_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_1_acc gf_0 gf_1 gf_2;
pacomult := paco3_3_1_mult gf_0 gf_1 gf_2;
pacofold := paco3_3_1_fold gf_0 gf_1 gf_2;
pacounfold := paco3_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco3_3_2_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2->_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_2_acc gf_0 gf_1 gf_2;
pacomult := paco3_3_2_mult gf_0 gf_1 gf_2;
pacofold := paco3_3_2_fold gf_0 gf_1 gf_2;
pacounfold := paco3_3_2_unfold gf_0 gf_1 gf_2 }.