Lvc.paco.paco3

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Section Arg3_1.

Definition monotone3 T0 T1 T2 (gf: rel3 T0 T1 T2 → rel3 T0 T1 T2) :=
  ∀ x0 x1 x2 r r´ (IN: gf r x0 x1 x2) (LE: r <3= r´), gf r´ x0 x1 x2.

Variable T0 : Type.
Variable T1 : ∀ (x0: @T0), Type.
Variable T2 : ∀ (x0: @T0) (x1: @T1 x0), Type.
Variable gf : rel3 T0 T1 T2 → rel3 T0 T1 T2.
Implicit Arguments gf [].

Theorem paco3_acc: ∀
  l r (OBG: ∀ rr (INC: r <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3 gf rr),
  l <3= paco3 gf r.

Theorem paco3_mon: monotone3 (paco3 gf).

Theorem paco3_mult_strong: ∀ r,
  paco3 gf (paco3 gf r \3/ r) <3= paco3 gf r.

Corollary paco3_mult: ∀ r,
  paco3 gf (paco3 gf r) <3= paco3 gf r.

Theorem paco3_fold: ∀ r,
  gf (paco3 gf r \3/ r) <3= paco3 gf r.

Theorem paco3_unfold: ∀ (MON: monotone3 gf) r,
  paco3 gf r <3= gf (paco3 gf r \3/ r).

End Arg3_1.

Hint Unfold monotone3.
Hint Resolve paco3_fold.

Implicit Arguments paco3_acc [ T0 T1 T2 ].
Implicit Arguments paco3_mon [ T0 T1 T2 ].
Implicit Arguments paco3_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_mult [ T0 T1 T2 ].
Implicit Arguments paco3_fold [ T0 T1 T2 ].
Implicit Arguments paco3_unfold [ T0 T1 T2 ].

Instance paco3_inst T0 T1 T2 (gf : rel3 T0 T1 T2→_) r x0 x1 x2 : paco_class (paco3 gf r x0 x1 x2) :=
{ pacoacc := paco3_acc gf;
  pacomult := paco3_mult gf;
  pacofold := paco3_fold gf;
  pacounfold := paco3_unfold gf }.

Section Arg3_2.

Definition monotone3_2 T0 T1 T2 (gf: rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2) :=
  ∀ x0 x1 x2 r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1 x0 x1 x2) (LE_0: r_0 <3= r´_0)(LE_1: r_1 <3= r´_1), gf r´_0 r´_1 x0 x1 x2.

Variable T0 : Type.
Variable T1 : ∀ (x0: @T0), Type.
Variable T2 : ∀ (x0: @T0) (x1: @T1 x0), Type.
Variable gf_0 gf_1 : rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco3_2_0_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_0 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_2_0 gf_0 gf_1 rr r_1),
  l <3= paco3_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_1_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_1 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_2_1 gf_0 gf_1 r_0 rr),
  l <3= paco3_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_0_mon: monotone3_2 (paco3_2_0 gf_0 gf_1).

Theorem paco3_2_1_mon: monotone3_2 (paco3_2_1 gf_0 gf_1).

Theorem paco3_2_0_mult_strong: ∀ r_0 r_1,
  paco3_2_0 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_1_mult_strong: ∀ r_0 r_1,
  paco3_2_1 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.

Corollary paco3_2_0_mult: ∀ r_0 r_1,
  paco3_2_0 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1) (paco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.

Corollary paco3_2_1_mult: ∀ r_0 r_1,
  paco3_2_1 gf_0 gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1) (paco3_2_1 gf_0 gf_1 r_0 r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_0_fold: ∀ r_0 r_1,
  gf_0 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1) <3= paco3_2_0 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_1_fold: ∀ r_0 r_1,
  gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1) <3= paco3_2_1 gf_0 gf_1 r_0 r_1.

Theorem paco3_2_0_unfold: ∀ (MON: monotone3_2 gf_0) (MON: monotone3_2 gf_1) r_0 r_1,
  paco3_2_0 gf_0 gf_1 r_0 r_1 <3= gf_0 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1).

Theorem paco3_2_1_unfold: ∀ (MON: monotone3_2 gf_0) (MON: monotone3_2 gf_1) r_0 r_1,
  paco3_2_1 gf_0 gf_1 r_0 r_1 <3= gf_1 (paco3_2_0 gf_0 gf_1 r_0 r_1 \3/ r_0) (paco3_2_1 gf_0 gf_1 r_0 r_1 \3/ r_1).

End Arg3_2.

Hint Unfold monotone3_2.
Hint Resolve paco3_2_0_fold.
Hint Resolve paco3_2_1_fold.

Implicit Arguments paco3_2_0_acc [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_acc [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mon [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mon [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_mult [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_mult [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_fold [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_fold [ T0 T1 T2 ].
Implicit Arguments paco3_2_0_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_2_1_unfold [ T0 T1 T2 ].

Instance paco3_2_0_inst T0 T1 T2 (gf_0 gf_1 : rel3 T0 T1 T2→_) r_0 r_1 x0 x1 x2 : paco_class (paco3_2_0 gf_0 gf_1 r_0 r_1 x0 x1 x2) :=
{ pacoacc := paco3_2_0_acc gf_0 gf_1;
  pacomult := paco3_2_0_mult gf_0 gf_1;
  pacofold := paco3_2_0_fold gf_0 gf_1;
  pacounfold := paco3_2_0_unfold gf_0 gf_1 }.

Instance paco3_2_1_inst T0 T1 T2 (gf_0 gf_1 : rel3 T0 T1 T2→_) r_0 r_1 x0 x1 x2 : paco_class (paco3_2_1 gf_0 gf_1 r_0 r_1 x0 x1 x2) :=
{ pacoacc := paco3_2_1_acc gf_0 gf_1;
  pacomult := paco3_2_1_mult gf_0 gf_1;
  pacofold := paco3_2_1_fold gf_0 gf_1;
  pacounfold := paco3_2_1_unfold gf_0 gf_1 }.

Section Arg3_3.

Definition monotone3_3 T0 T1 T2 (gf: rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2) :=
  ∀ x0 x1 x2 r_0 r_1 r_2 r´_0 r´_1 r´_2 (IN: gf r_0 r_1 r_2 x0 x1 x2) (LE_0: r_0 <3= r´_0)(LE_1: r_1 <3= r´_1)(LE_2: r_2 <3= r´_2), gf r´_0 r´_1 r´_2 x0 x1 x2.

Variable T0 : Type.
Variable T1 : ∀ (x0: @T0), Type.
Variable T2 : ∀ (x0: @T0) (x1: @T1 x0), Type.
Variable gf_0 gf_1 gf_2 : rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2 → rel3 T0 T1 T2.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco3_3_0_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_0 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_1_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_1 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_2_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_2 <3= rr) (CIH: l <_paco_3= rr), l <_paco_3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_0_mon: monotone3_3 (paco3_3_0 gf_0 gf_1 gf_2).

Theorem paco3_3_1_mon: monotone3_3 (paco3_3_1 gf_0 gf_1 gf_2).

Theorem paco3_3_2_mon: monotone3_3 (paco3_3_2 gf_0 gf_1 gf_2).

Theorem paco3_3_0_mult_strong: ∀ r_0 r_1 r_2,
  paco3_3_0 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_1_mult_strong: ∀ r_0 r_1 r_2,
  paco3_3_1 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_2_mult_strong: ∀ r_0 r_1 r_2,
  paco3_3_2 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco3_3_0_mult: ∀ r_0 r_1 r_2,
  paco3_3_0 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco3_3_1_mult: ∀ r_0 r_1 r_2,
  paco3_3_1 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Corollary paco3_3_2_mult: ∀ r_0 r_1 r_2,
  paco3_3_2 gf_0 gf_1 gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_0_fold: ∀ r_0 r_1 r_2,
  gf_0 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_1_fold: ∀ r_0 r_1 r_2,
  gf_1 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_2_fold: ∀ r_0 r_1 r_2,
  gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2) <3= paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.

Theorem paco3_3_0_unfold: ∀ (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
  paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_0 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2).

Theorem paco3_3_1_unfold: ∀ (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
  paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_1 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2).

Theorem paco3_3_2_unfold: ∀ (MON: monotone3_3 gf_0) (MON: monotone3_3 gf_1) (MON: monotone3_3 gf_2) r_0 r_1 r_2,
  paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <3= gf_2 (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_0) (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_1) (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \3/ r_2).

End Arg3_3.

Hint Unfold monotone3_3.
Hint Resolve paco3_3_0_fold.
Hint Resolve paco3_3_1_fold.
Hint Resolve paco3_3_2_fold.

Implicit Arguments paco3_3_0_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_acc [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mon [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mult_strong [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_mult [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_fold [ T0 T1 T2 ].
Implicit Arguments paco3_3_0_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_3_1_unfold [ T0 T1 T2 ].
Implicit Arguments paco3_3_2_unfold [ T0 T1 T2 ].

Instance paco3_3_0_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2→_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco3_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco3_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco3_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco3_3_1_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2→_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco3_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco3_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco3_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco3_3_2_inst T0 T1 T2 (gf_0 gf_1 gf_2 : rel3 T0 T1 T2→_) r_0 r_1 r_2 x0 x1 x2 : paco_class (paco3_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 x0 x1 x2) :=
{ pacoacc := paco3_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco3_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco3_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco3_3_2_unfold gf_0 gf_1 gf_2 }.