Lvc.paco.paco0

Require Export paconotation pacotac pacodef pacotacuser.
Set Implicit Arguments.

Section Arg0_1.

Definition monotone0 (gf: rel0 → rel0) :=
  ∀ r r´ (IN: gf r) (LE: r <0= r´), gf r´.

Variable gf : rel0 → rel0.
Implicit Arguments gf [].

Theorem paco0_acc: ∀
  l r (OBG: ∀ rr (INC: r <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0 gf rr),
  l <0= paco0 gf r.
Proof.
  intros; assert (SIM: paco0 gf (r \0/ l)) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_mon: monotone0 (paco0 gf).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_mult_strong: ∀ r,
  paco0 gf (paco0 gf r \0/ r) <0= paco0 gf r.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Corollary paco0_mult: ∀ r,
  paco0 gf (paco0 gf r) <0= paco0 gf r.
Proof. intros; eapply paco0_mult_strong, paco0_mon; eauto. Qed.

Theorem paco0_fold: ∀ r,
  gf (paco0 gf r \0/ r) <0= paco0 gf r.
Proof. intros; econstructor; [ |eauto]; eauto. Qed.

Theorem paco0_unfold: ∀ (MON: monotone0 gf) r,
  paco0 gf r <0= gf (paco0 gf r \0/ r).
Proof. unfold monotone0; intros; destruct PR; eauto. Qed.

End Arg0_1.

Hint Unfold monotone0.
Hint Resolve paco0_fold.

Implicit Arguments paco0_acc [ ].
Implicit Arguments paco0_mon [ ].
Implicit Arguments paco0_mult_strong [ ].
Implicit Arguments paco0_mult [ ].
Implicit Arguments paco0_fold [ ].
Implicit Arguments paco0_unfold [ ].

Instance paco0_inst (gf : rel0→_) r : paco_class (paco0 gf r) :=
{ pacoacc := paco0_acc gf;
  pacomult := paco0_mult gf;
  pacofold := paco0_fold gf;
  pacounfold := paco0_unfold gf }.

Section Arg0_2.

Definition monotone0_2 (gf: rel0 → rel0 → rel0) :=
  ∀ r_0 r_1 r´_0 r´_1 (IN: gf r_0 r_1) (LE_0: r_0 <0= r´_0)(LE_1: r_1 <0= r´_1), gf r´_0 r´_1.

Variable gf_0 gf_1 : rel0 → rel0 → rel0.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].

Theorem paco0_2_0_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_0 <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0_2_0 gf_0 gf_1 rr r_1),
  l <0= paco0_2_0 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco0_2_0 gf_0 gf_1 (r_0 \0/ l) r_1) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_2_1_acc: ∀
  l r_0 r_1 (OBG: ∀ rr (INC: r_1 <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0_2_1 gf_0 gf_1 r_0 rr),
  l <0= paco0_2_1 gf_0 gf_1 r_0 r_1.
Proof.
  intros; assert (SIM: paco0_2_1 gf_0 gf_1 r_0 (r_1 \0/ l)) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_2_0_mon: monotone0_2 (paco0_2_0 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_2_1_mon: monotone0_2 (paco0_2_1 gf_0 gf_1).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_2_0_mult_strong: ∀ r_0 r_1,
  paco0_2_0 gf_0 gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1) <0= paco0_2_0 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_2_1_mult_strong: ∀ r_0 r_1,
  paco0_2_1 gf_0 gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1) <0= paco0_2_1 gf_0 gf_1 r_0 r_1.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Corollary paco0_2_0_mult: ∀ r_0 r_1,
  paco0_2_0 gf_0 gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1) (paco0_2_1 gf_0 gf_1 r_0 r_1) <0= paco0_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco0_2_0_mult_strong, paco0_2_0_mon; eauto. Qed.

Corollary paco0_2_1_mult: ∀ r_0 r_1,
  paco0_2_1 gf_0 gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1) (paco0_2_1 gf_0 gf_1 r_0 r_1) <0= paco0_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; eapply paco0_2_1_mult_strong, paco0_2_1_mon; eauto. Qed.

Theorem paco0_2_0_fold: ∀ r_0 r_1,
  gf_0 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1) <0= paco0_2_0 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco0_2_1_fold: ∀ r_0 r_1,
  gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1) <0= paco0_2_1 gf_0 gf_1 r_0 r_1.
Proof. intros; econstructor; [ | |eauto]; eauto. Qed.

Theorem paco0_2_0_unfold: ∀ (MON: monotone0_2 gf_0) (MON: monotone0_2 gf_1) r_0 r_1,
  paco0_2_0 gf_0 gf_1 r_0 r_1 <0= gf_0 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1).
Proof. unfold monotone0_2; intros; destruct PR; eauto. Qed.

Theorem paco0_2_1_unfold: ∀ (MON: monotone0_2 gf_0) (MON: monotone0_2 gf_1) r_0 r_1,
  paco0_2_1 gf_0 gf_1 r_0 r_1 <0= gf_1 (paco0_2_0 gf_0 gf_1 r_0 r_1 \0/ r_0) (paco0_2_1 gf_0 gf_1 r_0 r_1 \0/ r_1).
Proof. unfold monotone0_2; intros; destruct PR; eauto. Qed.

End Arg0_2.

Hint Unfold monotone0_2.
Hint Resolve paco0_2_0_fold.
Hint Resolve paco0_2_1_fold.

Implicit Arguments paco0_2_0_acc [ ].
Implicit Arguments paco0_2_1_acc [ ].
Implicit Arguments paco0_2_0_mon [ ].
Implicit Arguments paco0_2_1_mon [ ].
Implicit Arguments paco0_2_0_mult_strong [ ].
Implicit Arguments paco0_2_1_mult_strong [ ].
Implicit Arguments paco0_2_0_mult [ ].
Implicit Arguments paco0_2_1_mult [ ].
Implicit Arguments paco0_2_0_fold [ ].
Implicit Arguments paco0_2_1_fold [ ].
Implicit Arguments paco0_2_0_unfold [ ].
Implicit Arguments paco0_2_1_unfold [ ].

Instance paco0_2_0_inst (gf_0 gf_1 : rel0→_) r_0 r_1 : paco_class (paco0_2_0 gf_0 gf_1 r_0 r_1) :=
{ pacoacc := paco0_2_0_acc gf_0 gf_1;
  pacomult := paco0_2_0_mult gf_0 gf_1;
  pacofold := paco0_2_0_fold gf_0 gf_1;
  pacounfold := paco0_2_0_unfold gf_0 gf_1 }.

Instance paco0_2_1_inst (gf_0 gf_1 : rel0→_) r_0 r_1 : paco_class (paco0_2_1 gf_0 gf_1 r_0 r_1) :=
{ pacoacc := paco0_2_1_acc gf_0 gf_1;
  pacomult := paco0_2_1_mult gf_0 gf_1;
  pacofold := paco0_2_1_fold gf_0 gf_1;
  pacounfold := paco0_2_1_unfold gf_0 gf_1 }.

Section Arg0_3.

Definition monotone0_3 (gf: rel0 → rel0 → rel0 → rel0) :=
  ∀ r_0 r_1 r_2 r´_0 r´_1 r´_2 (IN: gf r_0 r_1 r_2) (LE_0: r_0 <0= r´_0)(LE_1: r_1 <0= r´_1)(LE_2: r_2 <0= r´_2), gf r´_0 r´_1 r´_2.

Variable gf_0 gf_1 gf_2 : rel0 → rel0 → rel0 → rel0.
Implicit Arguments gf_0 [].
Implicit Arguments gf_1 [].
Implicit Arguments gf_2 [].

Theorem paco0_3_0_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_0 <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0_3_0 gf_0 gf_1 gf_2 rr r_1 r_2),
  l <0= paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco0_3_0 gf_0 gf_1 gf_2 (r_0 \0/ l) r_1 r_2) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_3_1_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_1 <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0_3_1 gf_0 gf_1 gf_2 r_0 rr r_2),
  l <0= paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco0_3_1 gf_0 gf_1 gf_2 r_0 (r_1 \0/ l) r_2) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_3_2_acc: ∀
  l r_0 r_1 r_2 (OBG: ∀ rr (INC: r_2 <0= rr) (CIH: l <_paco_0= rr), l <_paco_0= paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 rr),
  l <0= paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof.
  intros; assert (SIM: paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 (r_2 \0/ l)) by eauto.
  clear PR; repeat (try left; do 1 paco_revert; paco_cofix_auto).
Qed.

Theorem paco0_3_0_mon: monotone0_3 (paco0_3_0 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_3_1_mon: monotone0_3 (paco0_3_1 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_3_2_mon: monotone0_3 (paco0_3_2 gf_0 gf_1 gf_2).
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_3_0_mult_strong: ∀ r_0 r_1 r_2,
  paco0_3_0 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_3_1_mult_strong: ∀ r_0 r_1 r_2,
  paco0_3_1 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Theorem paco0_3_2_mult_strong: ∀ r_0 r_1 r_2,
  paco0_3_2 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. paco_cofix_auto; repeat (left; do 1 paco_revert; paco_cofix_auto). Qed.

Corollary paco0_3_0_mult: ∀ r_0 r_1 r_2,
  paco0_3_0 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <0= paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco0_3_0_mult_strong, paco0_3_0_mon; eauto. Qed.

Corollary paco0_3_1_mult: ∀ r_0 r_1 r_2,
  paco0_3_1 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <0= paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco0_3_1_mult_strong, paco0_3_1_mon; eauto. Qed.

Corollary paco0_3_2_mult: ∀ r_0 r_1 r_2,
  paco0_3_2 gf_0 gf_1 gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) <0= paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; eapply paco0_3_2_mult_strong, paco0_3_2_mon; eauto. Qed.

Theorem paco0_3_0_fold: ∀ r_0 r_1 r_2,
  gf_0 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco0_3_1_fold: ∀ r_0 r_1 r_2,
  gf_1 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco0_3_2_fold: ∀ r_0 r_1 r_2,
  gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2) <0= paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2.
Proof. intros; econstructor; [ | | |eauto]; eauto. Qed.

Theorem paco0_3_0_unfold: ∀ (MON: monotone0_3 gf_0) (MON: monotone0_3 gf_1) (MON: monotone0_3 gf_2) r_0 r_1 r_2,
  paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 <0= gf_0 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2).
Proof. unfold monotone0_3; intros; destruct PR; eauto. Qed.

Theorem paco0_3_1_unfold: ∀ (MON: monotone0_3 gf_0) (MON: monotone0_3 gf_1) (MON: monotone0_3 gf_2) r_0 r_1 r_2,
  paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 <0= gf_1 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2).
Proof. unfold monotone0_3; intros; destruct PR; eauto. Qed.

Theorem paco0_3_2_unfold: ∀ (MON: monotone0_3 gf_0) (MON: monotone0_3 gf_1) (MON: monotone0_3 gf_2) r_0 r_1 r_2,
  paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 <0= gf_2 (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_0) (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_1) (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2 \0/ r_2).
Proof. unfold monotone0_3; intros; destruct PR; eauto. Qed.

End Arg0_3.

Hint Unfold monotone0_3.
Hint Resolve paco0_3_0_fold.
Hint Resolve paco0_3_1_fold.
Hint Resolve paco0_3_2_fold.

Implicit Arguments paco0_3_0_acc [ ].
Implicit Arguments paco0_3_1_acc [ ].
Implicit Arguments paco0_3_2_acc [ ].
Implicit Arguments paco0_3_0_mon [ ].
Implicit Arguments paco0_3_1_mon [ ].
Implicit Arguments paco0_3_2_mon [ ].
Implicit Arguments paco0_3_0_mult_strong [ ].
Implicit Arguments paco0_3_1_mult_strong [ ].
Implicit Arguments paco0_3_2_mult_strong [ ].
Implicit Arguments paco0_3_0_mult [ ].
Implicit Arguments paco0_3_1_mult [ ].
Implicit Arguments paco0_3_2_mult [ ].
Implicit Arguments paco0_3_0_fold [ ].
Implicit Arguments paco0_3_1_fold [ ].
Implicit Arguments paco0_3_2_fold [ ].
Implicit Arguments paco0_3_0_unfold [ ].
Implicit Arguments paco0_3_1_unfold [ ].
Implicit Arguments paco0_3_2_unfold [ ].

Instance paco0_3_0_inst (gf_0 gf_1 gf_2 : rel0→_) r_0 r_1 r_2 : paco_class (paco0_3_0 gf_0 gf_1 gf_2 r_0 r_1 r_2) :=
{ pacoacc := paco0_3_0_acc gf_0 gf_1 gf_2;
  pacomult := paco0_3_0_mult gf_0 gf_1 gf_2;
  pacofold := paco0_3_0_fold gf_0 gf_1 gf_2;
  pacounfold := paco0_3_0_unfold gf_0 gf_1 gf_2 }.

Instance paco0_3_1_inst (gf_0 gf_1 gf_2 : rel0→_) r_0 r_1 r_2 : paco_class (paco0_3_1 gf_0 gf_1 gf_2 r_0 r_1 r_2) :=
{ pacoacc := paco0_3_1_acc gf_0 gf_1 gf_2;
  pacomult := paco0_3_1_mult gf_0 gf_1 gf_2;
  pacofold := paco0_3_1_fold gf_0 gf_1 gf_2;
  pacounfold := paco0_3_1_unfold gf_0 gf_1 gf_2 }.

Instance paco0_3_2_inst (gf_0 gf_1 gf_2 : rel0→_) r_0 r_1 r_2 : paco_class (paco0_3_2 gf_0 gf_1 gf_2 r_0 r_1 r_2) :=
{ pacoacc := paco0_3_2_acc gf_0 gf_1 gf_2;
  pacomult := paco0_3_2_mult gf_0 gf_1 gf_2;
  pacofold := paco0_3_2_fold gf_0 gf_1 gf_2;
  pacounfold := paco0_3_2_unfold gf_0 gf_1 gf_2 }.