Lvc.paco.paco13

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Predicates of Arity 13

1 Mutual Coinduction

Section Arg13_1.

Definition monotone13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 r (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) (LE: r <13= ), gf x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12.

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 T8 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
Variable T9 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Variable T12 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
Variable gf : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12.
Implicit Arguments gf [].

Theorem paco13_acc:
  l r (OBG: rr (INC: r <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13 gf rr),
  l <13= paco13 gf r.
Proof.
  intros; assert (SIM: paco13 gf (r \13/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_mon: monotone13 (paco13 gf).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_mult_strong: r,
  paco13 gf (paco13 gf r \13/ r) <13= paco13 gf r.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Corollary paco13_mult: r,
  paco13 gf (paco13 gf r) <13= paco13 gf r.
Proof. intros; eapply paco13_mult_strong, paco13_mon; eauto. Qed.

Theorem paco13_fold: r,
  gf (paco13 gf r \13/ r) <13= paco13 gf r.
Proof. intros; econstructor; [ |eauto]; eauto. Qed.

Theorem paco13_unfold: (MON: monotone13 gf) r,
  paco13 gf r <13= gf (paco13 gf r \13/ r).
Proof. unfold monotone13; intros; destruct PR; eauto. Qed.

End Arg13_1.

Hint Unfold monotone13.
Hint Resolve paco13_fold.

Implicit Arguments paco13_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].

Instance paco13_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_acc gf;
  pacomult := paco13_mult gf;
  pacofold := paco13_fold gf;
  pacounfold := paco13_unfold gf }.

2 Mutual Coinduction

Section Arg13_2.

Definition monotone13_2 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) (LE_0: r_0 <13= r´_0)(LE_1: r_1 <13= r´_1), gf r´_0 r´_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12.

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 T8 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
Variable T9 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Variable T12 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
Variable gf_0 gf_1 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco13_2_0_acc:
  l r_0 r_1 (OBG: rr (INC: r_0 <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13_2_0 gf_0 gf_1 rr r_1),
  l <13= paco13_2_0 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco13_2_0 gf_0 gf_1 (r_0 \13/ l) r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_2_1_acc:
  l r_0 r_1 (OBG: rr (INC: r_1 <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13_2_1 gf_0 gf_1 r_0 rr),
  l <13= paco13_2_1 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco13_2_1 gf_0 gf_1 r_0 (r_1 \13/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_2_0_mon: monotone13_2 (paco13_2_0 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_2_1_mon: monotone13_2 (paco13_2_1 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_2_0_mult_strong: r_0 r_1,
  paco13_2_0 gf_0 gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1) <13= paco13_2_0 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_2_1_mult_strong: r_0 r_1,
  paco13_2_1 gf_0 gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1) <13= paco13_2_1 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Corollary paco13_2_0_mult: r_0 r_1,
  paco13_2_0 gf_0 gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1) (paco13_2_1 gf_0 gf_1 r_0 r_1) <13= paco13_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco13_2_0_mult_strong, paco13_2_0_mon; eauto. Qed.

Corollary paco13_2_1_mult: r_0 r_1,
  paco13_2_1 gf_0 gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1) (paco13_2_1 gf_0 gf_1 r_0 r_1) <13= paco13_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco13_2_1_mult_strong, paco13_2_1_mon; eauto. Qed.

Theorem paco13_2_0_fold: r_0 r_1,
  gf_0 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1) <13= paco13_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco13_2_1_fold: r_0 r_1,
  gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1) <13= paco13_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco13_2_0_unfold: (MON: monotone13_2 gf_0) (MON: monotone13_2 gf_1) r_0 r_1,
  paco13_2_0 gf_0 gf_1 r_0 r_1 <13= gf_0 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1).
Proof. unfold monotone13_2; intros; destruct PR; eauto. Qed.

Theorem paco13_2_1_unfold: (MON: monotone13_2 gf_0) (MON: monotone13_2 gf_1) r_0 r_1,
  paco13_2_1 gf_0 gf_1 r_0 r_1 <13= gf_1 (paco13_2_0 gf_0 gf_1 r_0 r_1 \13/ r_0) (paco13_2_1 gf_0 gf_1 r_0 r_1 \13/ r_1).
Proof. unfold monotone13_2; intros; destruct PR; eauto. Qed.

End Arg13_2.

Hint Unfold monotone13_2.
Hint Resolve paco13_2_0_fold.
Hint Resolve paco13_2_1_fold.

Implicit Arguments paco13_2_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_2_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].

Instance paco13_2_0_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf_0 gf_1 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_2_0_acc gf_0 gf_1;
  pacomult := paco13_2_0_mult gf_0 gf_1;
  pacofold := paco13_2_0_fold gf_0 gf_1;
  pacounfold := paco13_2_0_unfold gf_0 gf_1 }.

Instance paco13_2_1_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf_0 gf_1 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_2_1_acc gf_0 gf_1;
  pacomult := paco13_2_1_mult gf_0 gf_1;
  pacofold := paco13_2_1_fold gf_0 gf_1;
  pacounfold := paco13_2_1_unfold gf_0 gf_1 }.

3 Mutual Coinduction

Section Arg13_3.

Definition monotone13_3 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf: rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 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 x8 x9 x10 x11 x12) (LE_0: r_0 <13= r´_0)(LE_1: r_1 <13= r´_1)(LE_2: r_2 <13= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12.

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 T8 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6), Type.
Variable T9 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7), Type.
Variable T10 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8), Type.
Variable T11 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9), Type.
Variable T12 : (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) (x7: @T7 x0 x1 x2 x3 x4 x5 x6) (x8: @T8 x0 x1 x2 x3 x4 x5 x6 x7) (x9: @T9 x0 x1 x2 x3 x4 x5 x6 x7 x8) (x10: @T10 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9) (x11: @T11 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10), Type.
Variable gf_0 gf_1 gf_2 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco13_3_0_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_0 <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <13= paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco13_3_0 gf_0 gf_1 gf_2 (r_0 \13/ l) r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_3_1_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_1 <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <13= paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco13_3_1 gf_0 gf_1 gf_2 r_0 (r_1 \13/ l) r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_3_2_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_2 <13= rr) (CIH: l <_paco_13= rr), l <_paco_13= paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <13= paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 (r_2 \13/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) by eauto.
  clear PR; repeat (try left; do 14 paco_revert; paco_cofix_auto).
Qed.

Theorem paco13_3_0_mon: monotone13_3 (paco13_3_0 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_3_1_mon: monotone13_3 (paco13_3_1 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_3_2_mon: monotone13_3 (paco13_3_2 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_3_0_mult_strong: r_0 r_1 r_2,
  paco13_3_0 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_3_1_mult_strong: r_0 r_1 r_2,
  paco13_3_1 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Theorem paco13_3_2_mult_strong: r_0 r_1 r_2,
  paco13_3_2 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 14 paco_revert; paco_cofix_auto). Qed.

Corollary paco13_3_0_mult: r_0 r_1 r_2,
  paco13_3_0 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <13= paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco13_3_0_mult_strong, paco13_3_0_mon; eauto. Qed.

Corollary paco13_3_1_mult: r_0 r_1 r_2,
  paco13_3_1 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <13= paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco13_3_1_mult_strong, paco13_3_1_mon; eauto. Qed.

Corollary paco13_3_2_mult: r_0 r_1 r_2,
  paco13_3_2 gf_0 gf_1 gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <13= paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco13_3_2_mult_strong, paco13_3_2_mon; eauto. Qed.

Theorem paco13_3_0_fold: r_0 r_1 r_2,
  gf_0 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco13_3_1_fold: r_0 r_1 r_2,
  gf_1 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco13_3_2_fold: r_0 r_1 r_2,
  gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2) <13= paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco13_3_0_unfold: (MON: monotone13_3 gf_0) (MON: monotone13_3 gf_1) (MON: monotone13_3 gf_2) r_0 r_1 r_2,
  paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <13= gf_0 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2).
Proof. unfold monotone13_3; intros; destruct PR; eauto. Qed.

Theorem paco13_3_1_unfold: (MON: monotone13_3 gf_0) (MON: monotone13_3 gf_1) (MON: monotone13_3 gf_2) r_0 r_1 r_2,
  paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <13= gf_1 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2).
Proof. unfold monotone13_3; intros; destruct PR; eauto. Qed.

Theorem paco13_3_2_unfold: (MON: monotone13_3 gf_0) (MON: monotone13_3 gf_1) (MON: monotone13_3 gf_2) r_0 r_1 r_2,
  paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <13= gf_2 (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_0) (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_1) (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \13/ r_2).
Proof. unfold monotone13_3; intros; destruct PR; eauto. Qed.

End Arg13_3.

Hint Unfold monotone13_3.
Hint Resolve paco13_3_0_fold.
Hint Resolve paco13_3_1_fold.
Hint Resolve paco13_3_2_fold.

Implicit Arguments paco13_3_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].
Implicit Arguments paco13_3_2_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 ].

Instance paco13_3_0_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf_0 gf_1 gf_2 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco13_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco13_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco13_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco13_3_1_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf_0 gf_1 gf_2 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco13_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco13_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco13_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco13_3_2_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 (gf_0 gf_1 gf_2 : rel13 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 : paco_class (paco13_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12) :=
{ pacoacc := paco13_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco13_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco13_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco13_3_2_unfold gf_0 gf_1 gf_2 }.