Lvc.paco.paco8

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Section Arg8_1.

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

Section Arg8_2.

Definition monotone8_2 T0 T1 T2 T3 T4 T5 T6 T7 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 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 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 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 }.

Section Arg8_3.

Definition monotone8_3 T0 T1 T2 T3 T4 T5 T6 T7 (gf: rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 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 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 T0 T1 T2 T3 T4 T5 T6 T7 → rel8 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 }.