(* week-02_formalizing-a-proof-in-tcpa.v *)
(* LPP 2024 - CS3234 2023-2024, Sem2 *)
(* Olivier Danvy <danvy@yale-nus.edu.sg> *)
(* Version of 26 Jan 2024 with a correct definition of the specifications *)
(* was: *)
(* Version of 25 Jan 2024 *)

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

Require Import Arith Bool.

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

Definition lambda_dropped_specification_of_the_addition_function (add : nat -> nat -> nat) : Prop :=
  forall j : nat,
    (add O j = j)
    /\
    (forall i' ih : nat,
        add i' j = ih ->
        add (S i') j = S ih).

Definition lambda_lifted_specification_of_the_addition_function (add : nat -> nat -> nat) : Prop :=
  (forall j : nat,
      add O j = j)
  /\
  (forall j i' ih : nat,
      add i' j = ih ->
      add (S i') j = S ih).

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

Lemma soundness_of_evenp (evenp : nat -> bool) :
  forall n : nat,
    evenp n = true ->
    exists m : nat,
      n = 2 * m.
Admitted.

Lemma completeness_of_evenp (evenp : nat -> bool) :
  forall n : nat,
    (exists m : nat,
        n = 2 * m) ->
    evenp n = true.
Admitted.

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

(* Exercise 01 *)

Proposition soundness_of_the_lambda_dropped_specification_of_the_addition_function :
  forall add : nat -> nat -> nat,
    lambda_dropped_specification_of_the_addition_function add ->
    forall x y z : nat,
      add x y = z ->
      z = x + y.
Admitted.

Proposition completeness_of_the_lambda_lifted_specification_of_the_addition_function :
  forall add : nat -> nat -> nat,
    lambda_lifted_specification_of_the_addition_function add ->
    forall x y z : nat,
      x + y = z ->
      z = add x y.
Admitted.

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

(* Exercise 02 *)


Definition specification_of_negation (candidate : bool -> bool) : Prop :=
  (candidate true = false) /\
  (candidate false = true).

Theorem soundness_of_the_specification_of_negation_with_respect_to_negb :
  forall candidate: bool -> bool,
    specification_of_negation candidate ->
    forall b b': bool,
      candidate b = b' ->
      b' = negb b.
Admitted.

Theorem completeness_of_the_specification_of_negation_with_respect_to_negb :
  forall candidate: bool -> bool,
    specification_of_negation candidate ->
    forall b b': bool,
      negb b = b' ->
      candidate b = b'.
Admitted.


Definition negation (b : bool) : bool :=
  if b then false else true.

Theorem negation_satisfies_the_specification_of_negation :
  specification_of_negation negation.
  Admitted.

Theorem soundness_of_negation_with_respect_to_negb :
  forall b b' : bool,
    negation b = b' ->
    b' = negb b.
Admitted.

Theorem completeness_of_negation_with_respect_to_negb :
  forall b b' : bool,
    negb b = b' ->
    b' = negation b.
Admitted.

Theorem soundness_of_negb_with_respect_to_negation :
  forall b b' : bool,
    negb b = b' ->
    b' = negation b.
Admitted.

Theorem completeness_of_negb_with_respect_to_negation :
  forall b b' : bool,
    negation b = b' ->
    b' = negb b.
Admitted.
(* ********** *)

Search (_ + 0 = _).

Check Nat.add_0_r.

Proposition first_formal_proof :
  forall n : nat,
    n + 0 = 0 + n.
Proof.
  intro n.
  Check (Nat.add_0_r n).
  rewrite -> (Nat.add_0_r n).
  Check (Nat.add_0_l n).
  rewrite -> (Nat.add_0_l n).
  reflexivity.
Qed.

Check first_formal_proof.

Proposition first_formal_proof' :
  forall n : nat,
    n + 0 = 0 + n.
Proof.
  intro n.
  rewrite -> (Nat.add_0_l n).
  rewrite -> (Nat.add_0_r n).
  reflexivity.
Qed.

Proposition first_formal_proof'' :
  forall n : nat,
    n + 0 = 0 + n.
Proof.
  intro n.
  rewrite -> (Nat.add_0_r n).
  rewrite -> (Nat.add_0_l n).
  reflexivity.

  Restart.

  intro n.
  rewrite -> (Nat.add_0_l n).
  rewrite -> (Nat.add_0_r n).
  reflexivity.
Qed.

Proposition first_formal_proof''' :
  forall n : nat,
    n + 0 = 0 + n.
Proof.
  intro n.
  Check (Nat.add_comm n 0).
  exact (Nat.add_comm n 0).

  Restart.

  intro n.
  Check (Nat.add_comm 0 n).
  symmetry.
  exact (Nat.add_comm 0 n).
Qed.

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


Search (_ + _ = _ + _).
Proposition first_formal_proof'''' :
  forall n : nat,
    n + 0 = 0 + n.
Proof.
  intro n.
  exact (Nat.add_comm n 0).

  Restart.
  intro n.
  symmetry.
  exact (Nat.add_comm 0 n).
  Qed.
(*
Nat.add_comm: forall n m : nat, n + m = m + n
Nat.add_assoc: forall n m p : nat, n + (m + p) = n + m + p
*)

Check (Nat.add_comm 2 3).

Proposition second_formal_proof:
  forall x y z : nat,
    x + (y + z) = y + (x + z).
Proof.
  intro x.
  intro y.
  intro z.
  Search (_ + (_ + _)).
  Check (Nat.add_assoc x y z).

  rewrite -> (Nat.add_assoc x y z).
  Check Nat.add_comm.
  rewrite -> (Nat.add_comm x y).
  rewrite <- (Nat.add_assoc).
  reflexivity.
  Restart.
  intros x y z.
  rewrite -> (Nat.add_assoc x y z).
  rewrite -> (Nat.add_comm x y).
  rewrite <- (Nat.add_assoc y x z).
  reflexivity.
Qed.



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

Proposition third_formal_proof:
  forall n : nat,
    n + 0 + 0 = 0 + 0 + n.
Proof.
  intro n.
  rewrite -> (Nat.add_0_r n).
  rewrite -> (Nat.add_0_r n).

  Check (Nat.add_0_l 0).
  rewrite -> (Nat.add_0_l 0).
  rewrite -> (Nat.add_0_l n).
  reflexivity.
  Restart.
  intro n.
  rewrite -> (Nat.add_0_r n).
  rewrite -> (Nat.add_0_r n).
  rewrite -> (Nat.add_0_l 0).
  exact (Nat.add_0_l n).

  Restart.
  intro n.
  Check (first_formal_proof (n + 0)).
  rewrite -> (first_formal_proof (n + 0)).
  Check (first_formal_proof n).
  rewrite ->  (first_formal_proof n).
  exact (Nat.add_assoc 0 0 n).

  Restart.
  intro n.
  Check (Nat.add_0_l 0).
  rewrite -> (Nat.add_0_l 0).
  rewrite -> (Nat.add_0_r n).
  Check first_formal_proof.
  exact (first_formal_proof n).
  Qed.

(* ********** *)
(* Exercise 3 *)
Proposition foo :
  forall n1 n2 n3 n4 : nat,
    n1 + (n2 + (n3 + n4)) = ((n1 + n2) + n3) + n4.
Proof.
  intros a b c d.
  Search (_ + (_ + _) = _ + _ + _).
  rewrite -> (Nat.add_assoc a b (c + d)).
  Check Nat.add_assoc.
  rewrite -> (Nat.add_assoc (a + b) c d).
  reflexivity.
  Qed.
(* ********** *)
(* Exercise 4 *)
(* ********** *)
(* Exercise 5 *)
Proposition binomial_expansion_2_warmup :
  forall x y : nat,
    (x + y) * (x + y) = x * x + x * y + x * y + y * y.
Proof.
  intros x y.
  Search (_ * (_ + _) = _).
  Check Nat.mul_add_distr_l.
  Check Nat.mul_add_distr_r.
  rewrite -> (Nat.mul_add_distr_l (x + y) x y).
  rewrite -> (Nat.mul_add_distr_r x y x ).
  rewrite -> (Nat.mul_comm y x).
  rewrite -> (Nat.mul_add_distr_r x y y ).
  Check Nat.add_assoc.
  rewrite -> (Nat.add_assoc (x * x + x * y) (x * y) (y * y)).
  reflexivity.
  Qed.

Proposition binomial_expansion_2 :
  forall x y : nat,
    (x + y) * (x + y) = x * x + 2 * (x * y) + y * y.
Proof.
  intros x y.

  Check (Nat.mul_succ_l 1 (x * y)).
  rewrite -> (Nat.mul_succ_l 1 (x * y)).
  Search (1 * _).
  rewrite -> (Nat.mul_1_l (x * y)).
  rewrite -> (Nat.add_assoc (x * x) (x * y) (x * y)).

  exact (binomial_expansion_2_warmup x y).

  Qed.

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



(* end of week-02_formalizing-a-proof-in-tcpa.v *)