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cs3234/labs/week-07_soundness-and-completeness-of-equality-predicates-revisited.v

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+(* 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 *)