Lvc.paco.paco4

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Section Arg4_1.

Definition monotone4 T0 T1 T2 T3 (gf: rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3) :=
  ∀ x0 x1 x2 x3 r r´ (IN: gf r x0 x1 x2 x3) (LE: r <4= r´), gf r´ x0 x1 x2 x3.

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 gf : rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3.
Implicit Arguments gf [].

Theorem paco4_acc: ∀
  l r (OBG: ∀ rr (INC: r <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4 gf rr),
  l <4= paco4 gf r.

Theorem paco4_mon: monotone4 (paco4 gf).

Theorem paco4_mult_strong: ∀ r,
  paco4 gf (paco4 gf r \4/ r) <4= paco4 gf r.

Corollary paco4_mult: ∀ r,
  paco4 gf (paco4 gf r) <4= paco4 gf r.

Theorem paco4_fold: ∀ r,
  gf (paco4 gf r \4/ r) <4= paco4 gf r.

Theorem paco4_unfold: ∀ (MON: monotone4 gf) r,
  paco4 gf r <4= gf (paco4 gf r \4/ r).

End Arg4_1.

Hint Unfold monotone4.
Hint Resolve paco4_fold.

Implicit Arguments paco4_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_unfold [ T0 T1 T2 T3 ].

Instance paco4_inst T0 T1 T2 T3 (gf : rel4 T0 T1 T2 T3→_) r x0 x1 x2 x3 : paco_class (paco4 gf r x0 x1 x2 x3) :=
{ pacoacc := paco4_acc gf;
  pacomult := paco4_mult gf;
  pacofold := paco4_fold gf;
  pacounfold := paco4_unfold gf }.

Section Arg4_2.

Definition monotone4_2 T0 T1 T2 T3 (gf: rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3) :=
  ∀ x0 x1 x2 x3 r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1 x0 x1 x2 x3) (LE_0: r_0 <4= r´_0)(LE_1: r_1 <4= r´_1), gf r´_0 r´_1 x0 x1 x2 x3.

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 gf_0 gf_1 : rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco4_2_0_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_0 <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4_2_0 gf_0 gf_1 rr r_1),
  l <4= paco4_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_1_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_1 <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4_2_1 gf_0 gf_1 r_0 rr),
  l <4= paco4_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_0_mon: monotone4_2 (paco4_2_0 gf_0 gf_1).

Theorem paco4_2_1_mon: monotone4_2 (paco4_2_1 gf_0 gf_1).

Theorem paco4_2_0_mult_strong: ∀ r_0 r_1,
  paco4_2_0 gf_0 gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1) <4= paco4_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_1_mult_strong: ∀ r_0 r_1,
  paco4_2_1 gf_0 gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1) <4= paco4_2_1 gf_0 gf_1 r_0 r_1.

Corollary paco4_2_0_mult: ∀ r_0 r_1,
  paco4_2_0 gf_0 gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1) (paco4_2_1 gf_0 gf_1 r_0 r_1) <4= paco4_2_0 gf_0 gf_1 r_0 r_1.

Corollary paco4_2_1_mult: ∀ r_0 r_1,
  paco4_2_1 gf_0 gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1) (paco4_2_1 gf_0 gf_1 r_0 r_1) <4= paco4_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_0_fold: ∀ r_0 r_1,
  gf_0 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1) <4= paco4_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_1_fold: ∀ r_0 r_1,
  gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1) <4= paco4_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco4_2_0_unfold: ∀ (MON: monotone4_2 gf_0) (MON: monotone4_2 gf_1) r_0 r_1,
  paco4_2_0 gf_0 gf_1 r_0 r_1 <4= gf_0 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1).

Theorem paco4_2_1_unfold: ∀ (MON: monotone4_2 gf_0) (MON: monotone4_2 gf_1) r_0 r_1,
  paco4_2_1 gf_0 gf_1 r_0 r_1 <4= gf_1 (paco4_2_0 gf_0 gf_1 r_0 r_1 \4/ r_0) (paco4_2_1 gf_0 gf_1 r_0 r_1 \4/ r_1).

End Arg4_2.

Hint Unfold monotone4_2.
Hint Resolve paco4_2_0_fold.
Hint Resolve paco4_2_1_fold.

Implicit Arguments paco4_2_0_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_0_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_0_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_0_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_0_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_0_unfold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_2_1_unfold [ T0 T1 T2 T3 ].

Instance paco4_2_0_inst T0 T1 T2 T3 (gf_0 gf_1 : rel4 T0 T1 T2 T3→_) r_0 r_1 x0 x1 x2 x3 : paco_class (paco4_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3) :=
{ pacoacc := paco4_2_0_acc gf_0 gf_1;
  pacomult := paco4_2_0_mult gf_0 gf_1;
  pacofold := paco4_2_0_fold gf_0 gf_1;
  pacounfold := paco4_2_0_unfold gf_0 gf_1 }.

Instance paco4_2_1_inst T0 T1 T2 T3 (gf_0 gf_1 : rel4 T0 T1 T2 T3→_) r_0 r_1 x0 x1 x2 x3 : paco_class (paco4_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2 x3) :=
{ pacoacc := paco4_2_1_acc gf_0 gf_1;
  pacomult := paco4_2_1_mult gf_0 gf_1;
  pacofold := paco4_2_1_fold gf_0 gf_1;
  pacounfold := paco4_2_1_unfold gf_0 gf_1 }.

Section Arg4_3.

Definition monotone4_3 T0 T1 T2 T3 (gf: rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3) :=
  ∀ x0 x1 x2 x3 r_0 r_1 r_2 r´_0 r´_1 r´_2 (IN: gf r_0 r_1 r_2 x0 x1 x2 x3) (LE_0: r_0 <4= r´_0)(LE_1: r_1 <4= r´_1)(LE_2: r_2 <4= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2 x3.

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 gf_0 gf_1 gf_2 : rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3 → rel4 T0 T1 T2 T3.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco4_3_0_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_0 <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <4= paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_1_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_1 <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <4= paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_2_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_2 <4= rr) (CIH: l <_paco_4= rr), l <_paco_4= paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <4= paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_0_mon: monotone4_3 (paco4_3_0 gf_0 gf_1 gf_2).

Theorem paco4_3_1_mon: monotone4_3 (paco4_3_1 gf_0 gf_1 gf_2).

Theorem paco4_3_2_mon: monotone4_3 (paco4_3_2 gf_0 gf_1 gf_2).

Theorem paco4_3_0_mult_strong: ∀ r_0 r_1 r_2,
  paco4_3_0 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_1_mult_strong: ∀ r_0 r_1 r_2,
  paco4_3_1 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_2_mult_strong: ∀ r_0 r_1 r_2,
  paco4_3_2 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco4_3_0_mult: ∀ r_0 r_1 r_2,
  paco4_3_0 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <4= paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco4_3_1_mult: ∀ r_0 r_1 r_2,
  paco4_3_1 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <4= paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco4_3_2_mult: ∀ r_0 r_1 r_2,
  paco4_3_2 gf_0 gf_1 gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <4= paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_0_fold: ∀ r_0 r_1 r_2,
  gf_0 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_1_fold: ∀ r_0 r_1 r_2,
  gf_1 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_2_fold: ∀ r_0 r_1 r_2,
  gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2) <4= paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco4_3_0_unfold: ∀ (MON: monotone4_3 gf_0) (MON: monotone4_3 gf_1) (MON: monotone4_3 gf_2) r_0 r_1 r_2,
  paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <4= gf_0 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2).

Theorem paco4_3_1_unfold: ∀ (MON: monotone4_3 gf_0) (MON: monotone4_3 gf_1) (MON: monotone4_3 gf_2) r_0 r_1 r_2,
  paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <4= gf_1 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2).

Theorem paco4_3_2_unfold: ∀ (MON: monotone4_3 gf_0) (MON: monotone4_3 gf_1) (MON: monotone4_3 gf_2) r_0 r_1 r_2,
  paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <4= gf_2 (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_0) (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_1) (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \4/ r_2).

End Arg4_3.

Hint Unfold monotone4_3.
Hint Resolve paco4_3_0_fold.
Hint Resolve paco4_3_1_fold.
Hint Resolve paco4_3_2_fold.

Implicit Arguments paco4_3_0_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_acc [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_0_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_mon [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_0_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_mult_strong [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_0_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_mult [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_0_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_fold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_0_unfold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_1_unfold [ T0 T1 T2 T3 ].
Implicit Arguments paco4_3_2_unfold [ T0 T1 T2 T3 ].

Instance paco4_3_0_inst T0 T1 T2 T3 (gf_0 gf_1 gf_2 : rel4 T0 T1 T2 T3→_) r_0 r_1 r_2 x0 x1 x2 x3 : paco_class (paco4_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3) :=
{ pacoacc := paco4_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco4_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco4_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco4_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco4_3_1_inst T0 T1 T2 T3 (gf_0 gf_1 gf_2 : rel4 T0 T1 T2 T3→_) r_0 r_1 r_2 x0 x1 x2 x3 : paco_class (paco4_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3) :=
{ pacoacc := paco4_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco4_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco4_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco4_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco4_3_2_inst T0 T1 T2 T3 (gf_0 gf_1 gf_2 : rel4 T0 T1 T2 T3→_) r_0 r_1 r_2 x0 x1 x2 x3 : paco_class (paco4_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2 x3) :=
{ pacoacc := paco4_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco4_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco4_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco4_3_2_unfold gf_0 gf_1 gf_2 }.