From QuickChick Require Import QuickChick Tactics TacticsUtil Instances Classes
     DependentClasses CheckerProofs EnumProofs.

Require Import String. Open Scope string.
Require Import List micromega.Lia.
Require Import enumProofs. 

Import ListNotations.
From mathcomp Require Import ssreflect ssrfun ssrbool ssrnat eqtype seq.
Set SsrOldRewriteGoalsOrder.  (* change Set to Unset when porting the file, then remove the line when requiring MathComp >= 2.6 *)

From Ltac2 Require Import Ltac2.

#[local] Open Scope set_scope.

Inductive square_of : nat -> nat -> Prop :=
  sq : forall n m, m = n * n -> square_of n m.

QCDerive EnumSizedSuchThat for (fun x => square_of x n).

Inductive tree1 :=
| Leaf1 : tree1
| Node1 : nat -> tree1 -> tree1 -> tree1.

Inductive perfect' : nat -> tree1 -> Prop :=
| PerfectLeaf : perfect' 0 Leaf1
| PerfectNode : forall x l r n, perfect' n l -> perfect' n r ->
                                perfect' (S n) (Node1 x l r).

QCDerive DecOpt for (perfect' n t).

QCDerive EnumSizedSuchThat for (fun n => perfect' n t).

Inductive perfect : tree1 -> Prop :=
| Perfect : forall n t, perfect' n t -> perfect t.

QCDerive DecOpt for (perfect t).

Lemma semProdSizeOpt_bicupNone A s (S : set A) :
  (\bigcup_(x in [:: returnEnum (@None A)]) semProdSizeOpt x s \subset S).
Proof.
  intros x Hin. inv Hin. inv H. inv H0.
  - inv H1. congruence.
    inv H.
  - inv H.
Qed.

Set Bullet Behavior "Strict Subproofs".

Inductive In' {A} : A -> list A -> Prop :=
| In_hd :
    forall x l, In' x (cons x l)
| In_tl :
    forall x y l, In' x l -> In' x (cons y l).


QCDerive DecOpt for (In' a l).
 
Instance DecOptIn'_listSizeMonotonic A {_ : Enum A} {_ : Dec_Eq A}
         (x : A) (l : list A) : DecOptSizeMonotonic (In' x l).
Proof. derive_mon (). Qed.

Instance DecOptIn'_list_sound A {_ : Enum A} {_ : Dec_Eq A} (x : A) (l : list A) :
  DecOptSoundPos (In' x l).
Proof. derive_sound (). Qed.

Instance DecOptIn'_list_complete A {_ : Enum A} {_ : Dec_Eq A} (x : A) (l : list A) :
  DecOptCompletePos (In' x l).
Proof. derive_complete (). Qed.

QCDerive ArbitrarySizedSuchThat for (fun x => In' x l).
QCDerive EnumSizedSuchThat for (fun x => In' x l).

Instance EnumSizedSuchThatIn'_SizedMonotonic A {_ : Enum A} {_ : Dec_Eq A} l :
  SizedMonotonicOpt (@enumSizeST A _ (EnumSizedSuchThatIn' l)).
Proof. derive_enumST_SizedMonotonic (). Qed.

Instance EnumSizedSuchThatIn'_SizeMonotonic  A {_ : Enum A} {_ : Dec_Eq A}
         (* `{EnumMonotonic A} *) l :
  forall s, SizeMonotonicOpt (@enumSizeST _ _ (EnumSizedSuchThatIn' l) s).
Proof. derive_enumST_SizeMonotonic (). Qed.


Instance EnumSizedSuchThatIn'_Correct A {_ : Enum A}  {_ : Dec_Eq A}
         (* `{EnumMonotonicCorrect A} *) l :
  CorrectSizedST (fun x => In' x l) (@enumSizeST _ _ (EnumSizedSuchThatIn' l)).
Proof. derive_enumST_Correct (). Admitted. (* TODO *)
 
QCDerive EnumSizedSuchThat for (fun l => In' x l).


Inductive bst : nat -> nat -> tree1 -> Prop :=
| BstLeaf : forall n1 n2, bst n1 n2 Leaf1
| BstNode : forall min max n t1 t2,
    le min max -> le min n -> le n max ->
    bst min n t1 -> bst n max t2 ->
    bst min max (Node1 n t1 t2).

QCDerive DecOpt for (le min max).
QCDerive EnumSizedSuchThat for (fun m => le n m).

QCDerive EnumSizedSuchThat for (fun t => bst min max t).

QCDerive ArbitrarySizedSuchThat for (fun m => le n m).
QCDerive ArbitrarySizedSuchThat for (fun t => bst min max t).

QCDerive DecOpt for (bst min max t).
  
Instance EnumSizedSuchThatle_SizedAMonotonic n :
  SizedMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatle n)).
Proof. derive_enumST_SizedMonotonicFP (). Qed. 

Instance EnumSizedSuchThatle_SizedMonotonic n :
  SizedMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatle n)).
Proof. derive_enumST_SizedMonotonic (). Qed.

Instance EnumSizedSuchThatle_SizeMonotonic n :
  forall s, SizeMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatle n) s).
Proof. derive_enumST_SizeMonotonic (). Qed.

Instance EnumSizedSuchThatle_SizeMonotonicFP n :
  forall s, SizeMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatle n) s).
Proof. derive_enumST_SizeMonotonicFP (). Qed. 

(* XXX predicate must be eta expanded, otherwise typeclass resolution fails *)
Instance EnumSizedSuchThatle_Correct n :
  CorrectSizedST [eta le n] (@enumSizeST _ _ (@EnumSizedSuchThatle n)).
Proof. derive_enumST_Correct (). Qed.

Instance EnumSizedSuchThatbst_SizedMonotonicFP min max :
  SizedMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatbst min max)).
Proof. derive_enumST_SizedMonotonicFP (). Qed.

Instance EnumSizedSuchThatbst_SizedMonotonic min max :
  SizedMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatbst min max)).
Proof. derive_enumST_SizedMonotonic (). Qed.  

Instance EnumSizedSuchThatbst_SizeMonotonic min max :
  forall s, SizeMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatbst min max) s).
Proof. derive_enumST_SizeMonotonic (). Qed.

Instance EnumSizedSuchThatbst_SizeMonotonicFP min max :
  forall s, SizeMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatbst min max) s).
Proof. derive_enumST_SizeMonotonicFP (). Qed.
  
Instance EnumSizedSuchThatbst_Correct n m :
  CorrectSizedST (bst n m) (@enumSizeST _ _ (@EnumSizedSuchThatbst n m)).
Proof. derive_enumST_Correct (). Qed.


(* XXX missing enum list instances. *) 
(* Instance EnumSizedSuchThatIn'0_SizedMonotonic A {_ : Enum A} x : *)
(*   SizedMonotonicOpt (@enumSizeST _ _ (EnumSizedSuchThatIn'0 x)). *)
(* Proof. derive_enumST_SizedMonotonic (). Qed. *)


Inductive ltest : list nat -> nat -> Prop :=
  | ltestnil :
      ltest [] 0
  | ltestcons :
      forall x m' m l,
        (m' + 1) = m ->
        (* In' m' l -> *)
        ltest l m' ->
        ltest (x :: l) m.


QCDerive EnumSizedSuchThat for (fun n => eq x n).
QCDerive EnumSizedSuchThat for (fun n => eq n x).

QCDerive DecOpt for (ltest l n).

Instance DecOptltest_listSizeMonotonic l x : DecOptSizeMonotonic (ltest l x).
Proof. derive_mon (). Qed.

Instance DecOptltest_listsound l x : DecOptSoundPos (ltest l x).
Proof. derive_sound (). Qed. 

Instance DecOptIn'ltest_complete A {_ : Enum A} {_ : Dec_Eq A} x l :
  DecOptCompletePos (ltest x l).
Proof. derive_complete (). Qed.

(* Set Typeclasses Debug. *)
(* QuickChickDebug Debug On. *)

(* XXX error *)
(* Derive EnumSizedSuchThat for (fun l => ltest l n). *)

Inductive goodTree : nat -> tree nat  -> Prop :=
| GL : forall a, goodTree 0 (Leaf nat a)
| GN :
    forall k t1 t2 n (* m : nat)*),
      (* le m n -> *)
      goodTree n t1 ->
      goodTree n t1 ->
      goodTree (S n) (Node nat k t1 t2).

QCDerive DecOpt for (goodTree n t).

(* XXX this fails if tree has type param A ... *) 

Instance DecOptgoodTree_listSizeMonotonic n t : DecOptSizeMonotonic (goodTree n t).
Proof. derive_mon (). Qed.

Instance DecOptgoodTree_list_sound n t : DecOptSoundPos (goodTree n t).
Proof. derive_sound (). Qed.

Instance DecOptgoodTree_list_complete n t : DecOptCompletePos (goodTree n t).
Proof. derive_complete (). Qed.

QCDerive EnumSizedSuchThat for (fun t => goodTree k t).



Instance EnumSizedSuchThatgoodTree_SizedMonotonic n :
  SizedMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatgoodTree n)).
Proof. derive_enumST_SizedMonotonic (). Qed.

Instance EnumSizedSuchThatgoodTree_SizeMonotonic n :
  forall s, SizeMonotonicOpt (@enumSizeST _ _ (@EnumSizedSuchThatgoodTree n) s).
Proof. derive_enumST_SizeMonotonic (). Qed.

Instance EnumSizedSuchThatgoodTree_SizedMonotonicFP n :
  SizedMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatgoodTree n)).
Proof. derive_enumST_SizedMonotonicFP (). Qed.
   
Instance EnumSizedSuchThatgoodTree_SizeMonotonicFP n :
  forall s, SizeMonotonicOptFP (@enumSizeST _ _ (@EnumSizedSuchThatgoodTree n) s).
Proof. derive_enumST_SizeMonotonicFP (). Qed.


Instance EnumSizedSuchThatgoodTree_Correct n :
  CorrectSizedST (goodTree n) (@enumSizeST _ _ (@EnumSizedSuchThatgoodTree n)).
Proof. derive_enumST_Correct (). Qed.
