(* week-07_soundness-and-completeness-of-equality-predicates-revisited.v *)
(* LPP 2024 - S3C234 2023-2024, Sem2 *)
(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
(* Version of 08 Mar 2024 *)

(* ********** *)

Ltac fold_unfold_tactic name := intros; unfold name; fold name; reflexivity.

Require Import Arith Bool List.

(* ********** *)

Definition is_a_sound_and_complete_equality_predicate (V : Type) (eqb_V : V -> V -> bool) :=
  forall v1 v2 : V,
    eqb_V v1 v2 = true <-> v1 = v2.

(* ********** *)

Check Bool.eqb.
(* eqb : bool -> bool -> bool *)

Definition eqb_bool (b1 b2 : bool) : bool :=
  match b1 with
    true =>
    match b2 with
      true =>
      true
    | false =>
      false
    end
  | false =>
    match b2 with
      true =>
      false
    | false =>
      true
    end
  end.

Lemma eqb_bool_is_reflexive :
  forall b : bool,
    eqb_bool b b = true.
Proof.
Abort.

Search (eqb _ _ = _ -> _ = _).
(* eqb_prop: forall a b : bool, eqb a b = true -> a = b *)

Proposition soundness_and_completeness_of_eqb_bool :
  is_a_sound_and_complete_equality_predicate bool eqb_bool.
Proof.
  unfold is_a_sound_and_complete_equality_predicate.
Proof.
Abort.

(* ***** *)

Proposition soundness_and_completeness_of_eqb_bool :
  is_a_sound_and_complete_equality_predicate bool eqb.
Proof.
Abort.

(* ********** *)

Check Nat.eqb.
(* Nat.eqb : nat -> nat -> bool *)

Fixpoint eqb_nat (n1 n2 : nat) : bool :=
  match n1 with
    O =>
    match n2 with
      O =>
      true
    | S n2' =>
      false
    end
  | S n1' =>
    match n2 with
      O =>
      false
    | S n2' =>
      eqb_nat n1' n2'
    end
  end.

Lemma fold_unfold_eqb_nat_O :
  forall n2 : nat,
    eqb_nat 0 n2 =
    match n2 with
      O =>
      true
    | S _ =>
      false
    end.
Proof.
  fold_unfold_tactic eqb_nat.
Qed.

Lemma fold_unfold_eqb_nat_S :
  forall n1' n2 : nat,
    eqb_nat (S n1') n2 =
    match n2 with
      O =>
      false
    | S n2' =>
      eqb_nat n1' n2'
    end.
Proof.
  fold_unfold_tactic eqb_nat.
Qed.

Search (Nat.eqb _ _ = true -> _ = _).
(* beq_nat_true: forall n m : nat, (n =? m) = true -> n = m *)

Proposition soundness_and_completeness_of_eqb_nat :
  is_a_sound_and_complete_equality_predicate nat eqb_nat.
Proof.
Abort.

(* ***** *)

Lemma fold_unfold_eqb_Nat_O :
  forall n2 : nat,
    0 =? n2 =
    match n2 with
      O =>
      true
    | S _ =>
      false
    end.
Proof.
  fold_unfold_tactic Nat.eqb.
Qed.

Lemma fold_unfold_eqb_Nat_S :
  forall n1' n2 : nat,
    S n1' =? n2 =
    match n2 with
      O =>
      false
    | S n2' =>
      n1' =? n2'
    end.
Proof.
  fold_unfold_tactic Nat.eqb.
Qed.

Proposition soundness_and_completeness_of_eqb_nat :
  is_a_sound_and_complete_equality_predicate nat Nat.eqb.
Proof.
Abort.

(* ********** *)

Definition eqb_option (V : Type) (eqb_V : V -> V -> bool) (ov1 ov2 : option V) : bool :=
  match ov1 with
    Some v1 =>
    match ov2 with
      Some v2 =>
      eqb_V v1 v2
    | None =>
      false
    end
  | None =>
    match ov2 with
      Some v2 =>
      false
    | None =>
      true
    end
  end.

Proposition soundness_and_completeness_of_eqb_option :
  forall (V : Type)
         (eqb_V : V -> V -> bool),
    is_a_sound_and_complete_equality_predicate V eqb_V ->
    is_a_sound_and_complete_equality_predicate (option V) (eqb_option V eqb_V).
Proof.
Abort.

(* ***** *)

(*
Definition eqb_option_option (V : Type) (eqb_V : V -> V -> bool) (oov1 oov2 : option (option V)) : bool :=
*)

(*
Proposition soundness_and_completeness_of_eqb_option_option :
  forall (V : Type)
         (eqb_V : V -> V -> bool),
    is_a_sound_and_complete_equality_predicate V eqb_V ->
    is_a_sound_and_complete_equality_predicate (option (option V)) (eqb_option_option V eqb_V).
Proof.
Abort.
*)

(* ********** *)

Definition eqb_pair (V : Type) (eqb_V : V -> V -> bool) (W : Type) (eqb_W : W -> W -> bool) (p1 p2 : V * W) : bool :=
  match p1 with
    (v1, w1) =>
    match p2 with
      (v2, w2) =>
      eqb_V v1 v2 && eqb_W w1 w2
    end
  end.

Proposition soundness_and_completeness_of_eqb_pair :
  forall (V : Type)
         (eqb_V : V -> V -> bool)
         (W : Type)
         (eqb_W : W -> W -> bool),
    is_a_sound_and_complete_equality_predicate V eqb_V ->
    is_a_sound_and_complete_equality_predicate W eqb_W ->
    is_a_sound_and_complete_equality_predicate (V * W) (eqb_pair V eqb_V W eqb_W).
Proof.
Abort.

(* ********** *)

Fixpoint eqb_list (V : Type) (eqb_V : V -> V -> bool) (v1s v2s : list V) : bool :=
  match v1s with
    nil =>
    match v2s with
      nil =>
      true
    | v2 :: v2s' =>
      false
    end
  | v1 :: v1s' =>
    match v2s with
      nil =>
      false
    | v2 :: v2s' =>
      eqb_V v1 v2 && eqb_list V eqb_V v1s' v2s'
    end
  end.

Lemma fold_unfold_eqb_list_nil :
  forall (V : Type)
         (eqb_V : V -> V -> bool)
         (v2s : list V),
    eqb_list V eqb_V nil v2s =
    match v2s with
      nil =>
      true
    | v2 :: v2s' =>
      false
    end.
Proof.
  fold_unfold_tactic eqb_list.
Qed.

Lemma fold_unfold_eqb_list_cons :
  forall (V : Type)
         (eqb_V : V -> V -> bool)
         (v1 : V)
         (v1s' v2s : list V),
    eqb_list V eqb_V (v1 :: v1s') v2s =
    match v2s with
      nil =>
      false
    | v2 :: v2s' =>
      eqb_V v1 v2 && eqb_list V eqb_V v1s' v2s'
    end.
Proof.
  fold_unfold_tactic eqb_list.
Qed.

Proposition soundness_and_completeness_of_eqb_list :
  forall (V : Type)
         (eqb_V : V -> V -> bool),
    is_a_sound_and_complete_equality_predicate V eqb_V ->
    is_a_sound_and_complete_equality_predicate (list V) (eqb_list V eqb_V).
Proof.
Abort.

(* ********** *)

(* end of week-07_soundness-and-completeness-of-equality-predicates-revisited.v *)