Lvc.paco.paco5

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Section Arg5_1.

Definition monotone5 T0 T1 T2 T3 T4 (gf: rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4) :=
  ∀ x0 x1 x2 x3 x4 r r´ (IN: gf r x0 x1 x2 x3 x4) (LE: r <5= r´), gf r´ x0 x1 x2 x3 x4.

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 gf : rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4.
Implicit Arguments gf [].

Theorem paco5_acc: ∀
  l r (OBG: ∀ rr (INC: r <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5 gf rr),
  l <5= paco5 gf r.

Theorem paco5_mon: monotone5 (paco5 gf).

Theorem paco5_mult_strong: ∀ r,
  paco5 gf (paco5 gf r \5/ r) <5= paco5 gf r.

Corollary paco5_mult: ∀ r,
  paco5 gf (paco5 gf r) <5= paco5 gf r.

Theorem paco5_fold: ∀ r,
  gf (paco5 gf r \5/ r) <5= paco5 gf r.

Theorem paco5_unfold: ∀ (MON: monotone5 gf) r,
  paco5 gf r <5= gf (paco5 gf r \5/ r).

End Arg5_1.

Hint Unfold monotone5.
Hint Resolve paco5_fold.

Implicit Arguments paco5_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_unfold [ T0 T1 T2 T3 T4 ].

Instance paco5_inst T0 T1 T2 T3 T4 (gf : rel5 T0 T1 T2 T3 T4→_) r x0 x1 x2 x3 x4 : paco_class (paco5 gf r x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_acc gf;
  pacomult := paco5_mult gf;
  pacofold := paco5_fold gf;
  pacounfold := paco5_unfold gf }.

Section Arg5_2.

Definition monotone5_2 T0 T1 T2 T3 T4 (gf: rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4) :=
  ∀ x0 x1 x2 x3 x4 r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1 x0 x1 x2 x3 x4) (LE_0: r_0 <5= r´_0)(LE_1: r_1 <5= r´_1), gf r´_0 r´_1 x0 x1 x2 x3 x4.

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 gf_0 gf_1 : rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco5_2_0_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_0 <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5_2_0 gf_0 gf_1 rr r_1),
  l <5= paco5_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_1_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_1 <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5_2_1 gf_0 gf_1 r_0 rr),
  l <5= paco5_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_0_mon: monotone5_2 (paco5_2_0 gf_0 gf_1).

Theorem paco5_2_1_mon: monotone5_2 (paco5_2_1 gf_0 gf_1).

Theorem paco5_2_0_mult_strong: ∀ r_0 r_1,
  paco5_2_0 gf_0 gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1) <5= paco5_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_1_mult_strong: ∀ r_0 r_1,
  paco5_2_1 gf_0 gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1) <5= paco5_2_1 gf_0 gf_1 r_0 r_1.

Corollary paco5_2_0_mult: ∀ r_0 r_1,
  paco5_2_0 gf_0 gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1) (paco5_2_1 gf_0 gf_1 r_0 r_1) <5= paco5_2_0 gf_0 gf_1 r_0 r_1.

Corollary paco5_2_1_mult: ∀ r_0 r_1,
  paco5_2_1 gf_0 gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1) (paco5_2_1 gf_0 gf_1 r_0 r_1) <5= paco5_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_0_fold: ∀ r_0 r_1,
  gf_0 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1) <5= paco5_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_1_fold: ∀ r_0 r_1,
  gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1) <5= paco5_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco5_2_0_unfold: ∀ (MON: monotone5_2 gf_0) (MON: monotone5_2 gf_1) r_0 r_1,
  paco5_2_0 gf_0 gf_1 r_0 r_1 <5= gf_0 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1).

Theorem paco5_2_1_unfold: ∀ (MON: monotone5_2 gf_0) (MON: monotone5_2 gf_1) r_0 r_1,
  paco5_2_1 gf_0 gf_1 r_0 r_1 <5= gf_1 (paco5_2_0 gf_0 gf_1 r_0 r_1 \5/ r_0) (paco5_2_1 gf_0 gf_1 r_0 r_1 \5/ r_1).

End Arg5_2.

Hint Unfold monotone5_2.
Hint Resolve paco5_2_0_fold.
Hint Resolve paco5_2_1_fold.

Implicit Arguments paco5_2_0_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_0_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_0_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_0_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_0_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_0_unfold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_2_1_unfold [ T0 T1 T2 T3 T4 ].

Instance paco5_2_0_inst T0 T1 T2 T3 T4 (gf_0 gf_1 : rel5 T0 T1 T2 T3 T4→_) r_0 r_1 x0 x1 x2 x3 x4 : paco_class (paco5_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_2_0_acc gf_0 gf_1;
  pacomult := paco5_2_0_mult gf_0 gf_1;
  pacofold := paco5_2_0_fold gf_0 gf_1;
  pacounfold := paco5_2_0_unfold gf_0 gf_1 }.

Instance paco5_2_1_inst T0 T1 T2 T3 T4 (gf_0 gf_1 : rel5 T0 T1 T2 T3 T4→_) r_0 r_1 x0 x1 x2 x3 x4 : paco_class (paco5_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_2_1_acc gf_0 gf_1;
  pacomult := paco5_2_1_mult gf_0 gf_1;
  pacofold := paco5_2_1_fold gf_0 gf_1;
  pacounfold := paco5_2_1_unfold gf_0 gf_1 }.

Section Arg5_3.

Definition monotone5_3 T0 T1 T2 T3 T4 (gf: rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4) :=
  ∀ x0 x1 x2 x3 x4 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) (LE_0: r_0 <5= r´_0)(LE_1: r_1 <5= r´_1)(LE_2: r_2 <5= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2 x3 x4.

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 gf_0 gf_1 gf_2 : rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4 → rel5 T0 T1 T2 T3 T4.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco5_3_0_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_0 <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <5= paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_1_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_1 <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <5= paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_2_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_2 <5= rr) (CIH: l <_paco_5= rr), l <_paco_5= paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <5= paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_0_mon: monotone5_3 (paco5_3_0 gf_0 gf_1 gf_2).

Theorem paco5_3_1_mon: monotone5_3 (paco5_3_1 gf_0 gf_1 gf_2).

Theorem paco5_3_2_mon: monotone5_3 (paco5_3_2 gf_0 gf_1 gf_2).

Theorem paco5_3_0_mult_strong: ∀ r_0 r_1 r_2,
  paco5_3_0 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_1_mult_strong: ∀ r_0 r_1 r_2,
  paco5_3_1 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_2_mult_strong: ∀ r_0 r_1 r_2,
  paco5_3_2 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco5_3_0_mult: ∀ r_0 r_1 r_2,
  paco5_3_0 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <5= paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco5_3_1_mult: ∀ r_0 r_1 r_2,
  paco5_3_1 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <5= paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco5_3_2_mult: ∀ r_0 r_1 r_2,
  paco5_3_2 gf_0 gf_1 gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <5= paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_0_fold: ∀ r_0 r_1 r_2,
  gf_0 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_1_fold: ∀ r_0 r_1 r_2,
  gf_1 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_2_fold: ∀ r_0 r_1 r_2,
  gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2) <5= paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco5_3_0_unfold: ∀ (MON: monotone5_3 gf_0) (MON: monotone5_3 gf_1) (MON: monotone5_3 gf_2) r_0 r_1 r_2,
  paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <5= gf_0 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2).

Theorem paco5_3_1_unfold: ∀ (MON: monotone5_3 gf_0) (MON: monotone5_3 gf_1) (MON: monotone5_3 gf_2) r_0 r_1 r_2,
  paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <5= gf_1 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2).

Theorem paco5_3_2_unfold: ∀ (MON: monotone5_3 gf_0) (MON: monotone5_3 gf_1) (MON: monotone5_3 gf_2) r_0 r_1 r_2,
  paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <5= gf_2 (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_0) (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_1) (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \5/ r_2).

End Arg5_3.

Hint Unfold monotone5_3.
Hint Resolve paco5_3_0_fold.
Hint Resolve paco5_3_1_fold.
Hint Resolve paco5_3_2_fold.

Implicit Arguments paco5_3_0_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_acc [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_0_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_mon [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_0_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_mult_strong [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_0_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_mult [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_0_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_fold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_0_unfold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_1_unfold [ T0 T1 T2 T3 T4 ].
Implicit Arguments paco5_3_2_unfold [ T0 T1 T2 T3 T4 ].

Instance paco5_3_0_inst T0 T1 T2 T3 T4 (gf_0 gf_1 gf_2 : rel5 T0 T1 T2 T3 T4→_) r_0 r_1 r_2 x0 x1 x2 x3 x4 : paco_class (paco5_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco5_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco5_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco5_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco5_3_1_inst T0 T1 T2 T3 T4 (gf_0 gf_1 gf_2 : rel5 T0 T1 T2 T3 T4→_) r_0 r_1 r_2 x0 x1 x2 x3 x4 : paco_class (paco5_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco5_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco5_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco5_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco5_3_2_inst T0 T1 T2 T3 T4 (gf_0 gf_1 gf_2 : rel5 T0 T1 T2 T3 T4→_) r_0 r_1 r_2 x0 x1 x2 x3 x4 : paco_class (paco5_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3 x4) :=
{ pacoacc := paco5_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco5_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco5_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco5_3_2_unfold gf_0 gf_1 gf_2 }.