Lvc.paco.paco12

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Predicates of Arity 12

1 Mutual Coinduction

Section Arg12_1.

Definition monotone12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf: rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 r (IN: gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) (LE: r <12= ), gf x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.

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 gf : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11.
Implicit Arguments gf [].

Theorem paco12_acc:
  l r (OBG: rr (INC: r <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12 gf rr),
  l <12= paco12 gf r.
Proof.
  intros; assert (SIM: paco12 gf (r \12/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_mon: monotone12 (paco12 gf).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_mult_strong: r,
  paco12 gf (paco12 gf r \12/ r) <12= paco12 gf r.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Corollary paco12_mult: r,
  paco12 gf (paco12 gf r) <12= paco12 gf r.
Proof. intros; eapply paco12_mult_strong, paco12_mon; eauto. Qed.

Theorem paco12_fold: r,
  gf (paco12 gf r \12/ r) <12= paco12 gf r.
Proof. intros; econstructor; [ |eauto]; eauto. Qed.

Theorem paco12_unfold: (MON: monotone12 gf) r,
  paco12 gf r <12= gf (paco12 gf r \12/ r).
Proof. unfold monotone12; intros; destruct PR; eauto. Qed.

End Arg12_1.

Hint Unfold monotone12.
Hint Resolve paco12_fold.

Implicit Arguments paco12_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].

Instance paco12_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12 gf r x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) :=
{ pacoacc := paco12_acc gf;
  pacomult := paco12_mult gf;
  pacofold := paco12_fold gf;
  pacounfold := paco12_unfold gf }.

2 Mutual Coinduction

Section Arg12_2.

Definition monotone12_2 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf: rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 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) (LE_0: r_0 <12= r´_0)(LE_1: r_1 <12= r´_1), gf r´_0 r´_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.

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 gf_0 gf_1 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco12_2_0_acc:
  l r_0 r_1 (OBG: rr (INC: r_0 <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12_2_0 gf_0 gf_1 rr r_1),
  l <12= paco12_2_0 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco12_2_0 gf_0 gf_1 (r_0 \12/ l) r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_2_1_acc:
  l r_0 r_1 (OBG: rr (INC: r_1 <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12_2_1 gf_0 gf_1 r_0 rr),
  l <12= paco12_2_1 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco12_2_1 gf_0 gf_1 r_0 (r_1 \12/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_2_0_mon: monotone12_2 (paco12_2_0 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_2_1_mon: monotone12_2 (paco12_2_1 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_2_0_mult_strong: r_0 r_1,
  paco12_2_0 gf_0 gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1) <12= paco12_2_0 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_2_1_mult_strong: r_0 r_1,
  paco12_2_1 gf_0 gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1) <12= paco12_2_1 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Corollary paco12_2_0_mult: r_0 r_1,
  paco12_2_0 gf_0 gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1) (paco12_2_1 gf_0 gf_1 r_0 r_1) <12= paco12_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco12_2_0_mult_strong, paco12_2_0_mon; eauto. Qed.

Corollary paco12_2_1_mult: r_0 r_1,
  paco12_2_1 gf_0 gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1) (paco12_2_1 gf_0 gf_1 r_0 r_1) <12= paco12_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco12_2_1_mult_strong, paco12_2_1_mon; eauto. Qed.

Theorem paco12_2_0_fold: r_0 r_1,
  gf_0 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1) <12= paco12_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco12_2_1_fold: r_0 r_1,
  gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1) <12= paco12_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco12_2_0_unfold: (MON: monotone12_2 gf_0) (MON: monotone12_2 gf_1) r_0 r_1,
  paco12_2_0 gf_0 gf_1 r_0 r_1 <12= gf_0 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1).
Proof. unfold monotone12_2; intros; destruct PR; eauto. Qed.

Theorem paco12_2_1_unfold: (MON: monotone12_2 gf_0) (MON: monotone12_2 gf_1) r_0 r_1,
  paco12_2_1 gf_0 gf_1 r_0 r_1 <12= gf_1 (paco12_2_0 gf_0 gf_1 r_0 r_1 \12/ r_0) (paco12_2_1 gf_0 gf_1 r_0 r_1 \12/ r_1).
Proof. unfold monotone12_2; intros; destruct PR; eauto. Qed.

End Arg12_2.

Hint Unfold monotone12_2.
Hint Resolve paco12_2_0_fold.
Hint Resolve paco12_2_1_fold.

Implicit Arguments paco12_2_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_2_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].

Instance paco12_2_0_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf_0 gf_1 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) :=
{ pacoacc := paco12_2_0_acc gf_0 gf_1;
  pacomult := paco12_2_0_mult gf_0 gf_1;
  pacofold := paco12_2_0_fold gf_0 gf_1;
  pacounfold := paco12_2_0_unfold gf_0 gf_1 }.

Instance paco12_2_1_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf_0 gf_1 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) :=
{ pacoacc := paco12_2_1_acc gf_0 gf_1;
  pacomult := paco12_2_1_mult gf_0 gf_1;
  pacofold := paco12_2_1_fold gf_0 gf_1;
  pacounfold := paco12_2_1_unfold gf_0 gf_1 }.

3 Mutual Coinduction

Section Arg12_3.

Definition monotone12_3 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf: rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11) :=
   x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 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) (LE_0: r_0 <12= r´_0)(LE_1: r_1 <12= r´_1)(LE_2: r_2 <12= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11.

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 gf_0 gf_1 gf_2 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco12_3_0_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_0 <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <12= paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco12_3_0 gf_0 gf_1 gf_2 (r_0 \12/ l) r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_3_1_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_1 <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <12= paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco12_3_1 gf_0 gf_1 gf_2 r_0 (r_1 \12/ l) r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_3_2_acc:
  l r_0 r_1 r_2 (OBG: rr (INC: r_2 <12= rr) (CIH: l <_paco_12= rr), l <_paco_12= paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <12= paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 (r_2 \12/ l) x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) by eauto.
  clear PR; repeat (try left; do 13 paco_revert; paco_cofix_auto).
Qed.

Theorem paco12_3_0_mon: monotone12_3 (paco12_3_0 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_3_1_mon: monotone12_3 (paco12_3_1 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_3_2_mon: monotone12_3 (paco12_3_2 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_3_0_mult_strong: r_0 r_1 r_2,
  paco12_3_0 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_3_1_mult_strong: r_0 r_1 r_2,
  paco12_3_1 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Theorem paco12_3_2_mult_strong: r_0 r_1 r_2,
  paco12_3_2 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 13 paco_revert; paco_cofix_auto). Qed.

Corollary paco12_3_0_mult: r_0 r_1 r_2,
  paco12_3_0 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <12= paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco12_3_0_mult_strong, paco12_3_0_mon; eauto. Qed.

Corollary paco12_3_1_mult: r_0 r_1 r_2,
  paco12_3_1 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <12= paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco12_3_1_mult_strong, paco12_3_1_mon; eauto. Qed.

Corollary paco12_3_2_mult: r_0 r_1 r_2,
  paco12_3_2 gf_0 gf_1 gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <12= paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco12_3_2_mult_strong, paco12_3_2_mon; eauto. Qed.

Theorem paco12_3_0_fold: r_0 r_1 r_2,
  gf_0 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco12_3_1_fold: r_0 r_1 r_2,
  gf_1 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco12_3_2_fold: r_0 r_1 r_2,
  gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2) <12= paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco12_3_0_unfold: (MON: monotone12_3 gf_0) (MON: monotone12_3 gf_1) (MON: monotone12_3 gf_2) r_0 r_1 r_2,
  paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <12= gf_0 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2).
Proof. unfold monotone12_3; intros; destruct PR; eauto. Qed.

Theorem paco12_3_1_unfold: (MON: monotone12_3 gf_0) (MON: monotone12_3 gf_1) (MON: monotone12_3 gf_2) r_0 r_1 r_2,
  paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <12= gf_1 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2).
Proof. unfold monotone12_3; intros; destruct PR; eauto. Qed.

Theorem paco12_3_2_unfold: (MON: monotone12_3 gf_0) (MON: monotone12_3 gf_1) (MON: monotone12_3 gf_2) r_0 r_1 r_2,
  paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <12= gf_2 (paco12_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_0) (paco12_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_1) (paco12_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \12/ r_2).
Proof. unfold monotone12_3; intros; destruct PR; eauto. Qed.

End Arg12_3.

Hint Unfold monotone12_3.
Hint Resolve paco12_3_0_fold.
Hint Resolve paco12_3_1_fold.
Hint Resolve paco12_3_2_fold.

Implicit Arguments paco12_3_0_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_acc [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_0_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_mon [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_0_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_mult_strong [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_0_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_mult [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_0_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_fold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_0_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_1_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].
Implicit Arguments paco12_3_2_unfold [ T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 ].

Instance paco12_3_0_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf_0 gf_1 gf_2 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12_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) :=
{ pacoacc := paco12_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco12_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco12_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco12_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco12_3_1_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf_0 gf_1 gf_2 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12_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) :=
{ pacoacc := paco12_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco12_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco12_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco12_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco12_3_2_inst T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 (gf_0 gf_1 gf_2 : rel12 T0 T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11_) r_0 r_1 r_2 x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 : paco_class (paco12_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) :=
{ pacoacc := paco12_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco12_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco12_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco12_3_2_unfold gf_0 gf_1 gf_2 }.