Lvc.paco.paco8

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Predicates of Arity 8

1 Mutual Coinduction

Section Arg8_1.

Definition monotone8 T0 T1 T2 T3 T4 T5 T6 T7 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
   x0 x1 x2 x3 x4 x5 x6 x7 r (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7) (LE: r <8= ), gf x0 x1 x2 x3 x4 x5 x6 x7.

Variable T0 : Type.
Variable T1 : (x0: @T0), Type.
Variable T2 : (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
Variable T4 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Variable T5 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
Variable gf : rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7.
Implicit Arguments gf [].

Theorem paco8_acc:
  l r (OBG: rr (INC: r <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8 gf rr),
  l <8= paco8 gf r.
Proof.
  intros; assert (SIM: paco8 gf (r \8/ l) x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_mon: monotone8 (paco8 gf).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_mult_strong: r,
  paco8 gf (paco8 gf r \8/ r) <8= paco8 gf r.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Corollary paco8_mult: r,
  paco8 gf (paco8 gf r) <8= paco8 gf r.
Proof. intros; eapply paco8_mult_strong, paco8_mon; eauto. Qed.

Theorem paco8_fold: r,
  gf (paco8 gf r \8/ r) <8= paco8 gf r.
Proof. intros; econstructor; [ |eauto]; eauto. Qed.

Theorem paco8_unfold: (MON: monotone8 gf) r,
  paco8 gf r <8= gf (paco8 gf r \8/ r).
Proof. unfold monotone8; intros; destruct PR; eauto. Qed.

End Arg8_1.

Hint Unfold monotone8.
Hint Resolve paco8_fold.

Implicit Arguments paco8_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].

Instance paco8_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8 gf r x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_acc gf;
  pacomult := paco8_mult gf;
  pacofold := paco8_fold gf;
  pacounfold := paco8_unfold gf }.

2 Mutual Coinduction

Section Arg8_2.

Definition monotone8_2 T0 T1 T2 T3 T4 T5 T6 T7 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
   x0 x1 x2 x3 x4 x5 x6 x7 r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7) (LE_0: r_0 <8= r´_0)(LE_1: r_1 <8= r´_1), gf r´_0 r´_1 x0 x1 x2 x3 x4 x5 x6 x7.

Variable T0 : Type.
Variable T1 : (x0: @T0), Type.
Variable T2 : (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
Variable T4 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Variable T5 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
Variable gf_0 gf_1 : rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco8_2_0_acc:
  l r_0 r_1 (OBG: rr (INC: r_0 <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8_2_0 gf_0 gf_1 rr r_1),
  l <8= paco8_2_0 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco8_2_0 gf_0 gf_1 (r_0 \8/ l) r_1 x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_2_1_acc:
  l r_0 r_1 (OBG: rr (INC: r_1 <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8_2_1 gf_0 gf_1 r_0 rr),
  l <8= paco8_2_1 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco8_2_1 gf_0 gf_1 r_0 (r_1 \8/ l) x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_2_0_mon: monotone8_2 (paco8_2_0 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_2_1_mon: monotone8_2 (paco8_2_1 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_2_0_mult_strong: r_0 r_1,
  paco8_2_0 gf_0 gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1) <8= paco8_2_0 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_2_1_mult_strong: r_0 r_1,
  paco8_2_1 gf_0 gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1) <8= paco8_2_1 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Corollary paco8_2_0_mult: r_0 r_1,
  paco8_2_0 gf_0 gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1) (paco8_2_1 gf_0 gf_1 r_0 r_1) <8= paco8_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco8_2_0_mult_strong, paco8_2_0_mon; eauto. Qed.

Corollary paco8_2_1_mult: r_0 r_1,
  paco8_2_1 gf_0 gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1) (paco8_2_1 gf_0 gf_1 r_0 r_1) <8= paco8_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco8_2_1_mult_strong, paco8_2_1_mon; eauto. Qed.

Theorem paco8_2_0_fold: r_0 r_1,
  gf_0 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1) <8= paco8_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco8_2_1_fold: r_0 r_1,
  gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1) <8= paco8_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco8_2_0_unfold: (MON: monotone8_2 gf_0) (MON: monotone8_2 gf_1) r_0 r_1,
  paco8_2_0 gf_0 gf_1 r_0 r_1 <8= gf_0 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1).
Proof. unfold monotone8_2; intros; destruct PR; eauto. Qed.

Theorem paco8_2_1_unfold: (MON: monotone8_2 gf_0) (MON: monotone8_2 gf_1) r_0 r_1,
  paco8_2_1 gf_0 gf_1 r_0 r_1 <8= gf_1 (paco8_2_0 gf_0 gf_1 r_0 r_1 \8/ r_0) (paco8_2_1 gf_0 gf_1 r_0 r_1 \8/ r_1).
Proof. unfold monotone8_2; intros; destruct PR; eauto. Qed.

End Arg8_2.

Hint Unfold monotone8_2.
Hint Resolve paco8_2_0_fold.
Hint Resolve paco8_2_1_fold.

Implicit Arguments paco8_2_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_2_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].

Instance paco8_2_0_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf_0 gf_1 : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_2_0_acc gf_0 gf_1;
  pacomult := paco8_2_0_mult gf_0 gf_1;
  pacofold := paco8_2_0_fold gf_0 gf_1;
  pacounfold := paco8_2_0_unfold gf_0 gf_1 }.

Instance paco8_2_1_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf_0 gf_1 : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_2_1_acc gf_0 gf_1;
  pacomult := paco8_2_1_mult gf_0 gf_1;
  pacofold := paco8_2_1_fold gf_0 gf_1;
  pacounfold := paco8_2_1_unfold gf_0 gf_1 }.

3 Mutual Coinduction

Section Arg8_3.

Definition monotone8_3 T0 T1 T2 T3 T4 T5 T6 T7 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7) :=
   x0 x1 x2 x3 x4 x5 x6 x7 r_0 r_1 r_2 r´_0 r´_1 r´_2 (IN: gf r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7) (LE_0: r_0 <8= r´_0)(LE_1: r_1 <8= r´_1)(LE_2: r_2 <8= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2 x3 x4 x5 x6 x7.

Variable T0 : Type.
Variable T1 : (x0: @T0), Type.
Variable T2 : (x0: @T0) (x1: @T1 x0), Type.
Variable T3 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1), Type.
Variable T4 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2), Type.
Variable T5 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3), Type.
Variable T6 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4), Type.
Variable T7 : (x0: @T0) (x1: @T1 x0) (x2: @T2 x0 x1) (x3: @T3 x0 x1 x2) (x4: @T4 x0 x1 x2 x3) (x5: @T5 x0 x1 x2 x3 x4) (x6: @T6 x0 x1 x2 x3 x4 x5), Type.
Variable gf_0 gf_1 gf_2 : rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7rel8 T0 T1 T2 T3 T4 T5 T6 T7.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco8_3_0_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_0 <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <8= paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco8_3_0 gf_0 gf_1 gf_2 (r_0 \8/ l) r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_3_1_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_1 <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <8= paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco8_3_1 gf_0 gf_1 gf_2 r_0 (r_1 \8/ l) r_2 x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_3_2_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_2 <8= rr) (CIH: l <_paco_8= rr), l <_paco_8= paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <8= paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 (r_2 \8/ l) x0 x1 x2 x3 x4 x5 x6 x7) by eauto.
  clear PR; repeat (try left; do 9 paco_revert; paco_cofix_auto).
Qed.

Theorem paco8_3_0_mon: monotone8_3 (paco8_3_0 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_3_1_mon: monotone8_3 (paco8_3_1 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_3_2_mon: monotone8_3 (paco8_3_2 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_3_0_mult_strong: r_0 r_1 r_2,
  paco8_3_0 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_3_1_mult_strong: r_0 r_1 r_2,
  paco8_3_1 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Theorem paco8_3_2_mult_strong: r_0 r_1 r_2,
  paco8_3_2 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 9 paco_revert; paco_cofix_auto). Qed.

Corollary paco8_3_0_mult: r_0 r_1 r_2,
  paco8_3_0 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <8= paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco8_3_0_mult_strong, paco8_3_0_mon; eauto. Qed.

Corollary paco8_3_1_mult: r_0 r_1 r_2,
  paco8_3_1 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <8= paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco8_3_1_mult_strong, paco8_3_1_mon; eauto. Qed.

Corollary paco8_3_2_mult: r_0 r_1 r_2,
  paco8_3_2 gf_0 gf_1 gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <8= paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco8_3_2_mult_strong, paco8_3_2_mon; eauto. Qed.

Theorem paco8_3_0_fold: r_0 r_1 r_2,
  gf_0 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco8_3_1_fold: r_0 r_1 r_2,
  gf_1 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco8_3_2_fold: r_0 r_1 r_2,
  gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2) <8= paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco8_3_0_unfold: (MON: monotone8_3 gf_0) (MON: monotone8_3 gf_1) (MON: monotone8_3 gf_2) r_0 r_1 r_2,
  paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <8= gf_0 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2).
Proof. unfold monotone8_3; intros; destruct PR; eauto. Qed.

Theorem paco8_3_1_unfold: (MON: monotone8_3 gf_0) (MON: monotone8_3 gf_1) (MON: monotone8_3 gf_2) r_0 r_1 r_2,
  paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <8= gf_1 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2).
Proof. unfold monotone8_3; intros; destruct PR; eauto. Qed.

Theorem paco8_3_2_unfold: (MON: monotone8_3 gf_0) (MON: monotone8_3 gf_1) (MON: monotone8_3 gf_2) r_0 r_1 r_2,
  paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <8= gf_2 (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_0) (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_1) (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \8/ r_2).
Proof. unfold monotone8_3; intros; destruct PR; eauto. Qed.

End Arg8_3.

Hint Unfold monotone8_3.
Hint Resolve paco8_3_0_fold.
Hint Resolve paco8_3_1_fold.
Hint Resolve paco8_3_2_fold.

Implicit Arguments paco8_3_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_acc [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_mon [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_mult [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_fold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].
Implicit Arguments paco8_3_2_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 ].

Instance paco8_3_0_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf_0 gf_1 gf_2 : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco8_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco8_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco8_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco8_3_1_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf_0 gf_1 gf_2 : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco8_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco8_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco8_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco8_3_2_inst T0 T1 T2 T3 T4 T5 T6 T7 (gf_0 gf_1 gf_2 : rel8 T0 T1 T2 T3 T4 T5 T6 T7_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 : paco_class (paco8_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7) :=
{ pacoacc := paco8_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco8_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco8_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco8_3_2_unfold gf_0 gf_1 gf_2 }.