SLC2018: Programación funcional en Coq

En la sesión de hoy del Seminario de Lógica Computacional Ángel Ruiz Campos ha explicado la programación funcional en Coq.

La teoría, junto con los ejercicios, utilizados en la exposición son

(* T1: Programación funcional en Coq *)

Definition admit {T: Type} : T.  Admitted.

(* =====================================================================
   § Datos y funciones 
   ================================================================== *)

(* =====================================================================
   §§ Tipos enumerados  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir el tipo day cuyos constructores sean los días de
   la semana.
   ------------------------------------------------------------------ *)

Inductive day : Type :=
  | monday    : day
  | tuesday   : day
  | wednesday : day
  | thursday  : day
  | friday    : day
  | saturday  : day
  | sunday    : day.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función 
      next_weekday :: day -> day 
   tal que (next_weekday d) es el día laboral siguiente a d.
   ------------------------------------------------------------------ *)

Definition next_weekday (d:day) : day :=
  match d with
  | monday    => tuesday
  | tuesday   => wednesday
  | wednesday => thursday
  | thursday  => friday
  | friday    => monday
  | saturday  => monday
  | sunday    => monday
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Calcular el valor de las siguientes expresiones 
      + (next_weekday friday)
      + (next_weekday (next_weekday saturday))
   ------------------------------------------------------------------ *)

Compute (next_weekday friday).
(* ==> monday : day *)

Compute (next_weekday (next_weekday saturday)).
(* ==> tuesday : day *)

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que 
      (next_weekday (next_weekday saturday)) = tuesday
   ------------------------------------------------------------------ *)

Example test_next_weekday:
  (next_weekday (next_weekday saturday)) = tuesday.
Proof. simpl. reflexivity.  Qed.

(* =====================================================================
   §§ Booleanos  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir el tipo bool cuyos constructores son true y false.
   ------------------------------------------------------------------ *)

Inductive bool : Type :=
  | true  : bool
  | false : bool.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      negb :: bool -> bool
   tal que (negb b) es la negación de b.
   ------------------------------------------------------------------ *)

Definition negb (b:bool) : bool :=
  match b with
  | true  => false
  | false => true
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      andb :: bool -> bool -> bool
   tal que (andb b1 b2) es la conjunción de b1 y b2.
   ------------------------------------------------------------------ *)

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

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      orb :: bool -> bool -> bool
   tal que (orb b1 b2) es la disyunción de b1 y b2.
   ------------------------------------------------------------------ *)

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

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar las siguientes propiedades
      (orb true  false) = true.
      (orb false false) = false.
      (orb false true)  = true.
      (orb true  true)  = true.
   ------------------------------------------------------------------ *)

Example test_orb1: (orb true  false) = true.
Proof. simpl. reflexivity.  Qed.
Example test_orb2: (orb false false) = false.
Proof. simpl. reflexivity.  Qed.
Example test_orb3: (orb false true)  = true.
Proof. simpl. reflexivity.  Qed.
Example test_orb4: (orb true  true)  = true.
Proof. simpl. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Definir los operadores (&&) y (||) como abreviaturas de las
   funciones andb y orb.
   ------------------------------------------------------------------ *)

Notation "x && y" := (andb x y).
Notation "x || y" := (orb x y).

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que
      false || false || true = true.
   ------------------------------------------------------------------ *)

Example test_orb5: false || false || true = true.
Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------
   Ejercicio 1. Definir la función 
      nandb :: bool -> bool -> bool 
   tal que (nanb x y) se verifica si x e y no son verdaderos.

   Demostrar las siguientes propiedades de nand
      (nandb true  false) = true.
      (nandb false false) = true.
      (nandb false true)  = true.
      (nandb true  true)  = false.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Definition nandb (b1:bool) (b2:bool) : bool :=
  match b1 with 
  | true  => negb b2
  | false => true
  end.

Example prop_nandb1: (nandb true false) = true.
Proof. simpl. reflexivity. Qed. 
Example prop_nandb2: (nandb false false) = true.
Proof. simpl. reflexivity. Qed. 
Example prop_nandb3: (nandb false true) = true.
Proof. simpl. reflexivity. Qed. 
Example prop_nandb4: (nandb true true) = false.
Proof. simpl. reflexivity. Qed. 

(* angruicam1 *)
Definition nandb1 (b1:bool) (b2:bool) : bool :=
  match b1 with
  | true => negb b2
  | _    => true
  end.

Example test_nandb11: (nandb1 true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb12: (nandb1 false false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb13: (nandb1 false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb14: (nandb1 true true) = false.
Proof. simpl. reflexivity. Qed.

(* angruicam1 *)
Definition nandb2 (b1:bool) (b2:bool) : bool :=
  match b1, b2 with
  | true, true => false
  | _, _       => true
  end.

Example test_nandb21: (nandb2 true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb22: (nandb2 false false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb23: (nandb2 false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb24: (nandb2 true true) = false.
Proof. simpl. reflexivity. Qed.

(* angruicam1 *)
Definition nandb3 (b1:bool) (b2:bool) : bool :=
  if b1 then negb b2 else true.

Example test_nandb31: (nandb3 true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb32: (nandb3 false false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb33: (nandb3 false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb34: (nandb3 true true) = false.
Proof. simpl. reflexivity. Qed.

(* angruicam1 *)
Definition nandb4 (b1:bool) (b2:bool) : bool :=
  negb (b1 && b2).

Example test_nandb41: (nandb4 true false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb42: (nandb4 false false) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb43: (nandb4 false true) = true.
Proof. simpl. reflexivity. Qed.
Example test_nandb44: (nandb4 true true) = false.
Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------
   Ejercicio 2. Definir la función
      andb3 :: bool -> bool -> bool -> bool
   tal que (andb3 x y z) se verifica si x, y y z son verdaderos.

   Demostrar las siguientes propiedades de andb3
      (andb3 true  true  true)  = true.
      (andb3 false true  true)  = false.
      (andb3 true  false true)  = false.
      (andb3 true  true  false) = false.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool :=
  match b1 with
   | true  => andb b2 b3
   | false => false
  end.

Example prop_andb31: (andb3 true true true) = true.
Proof. simpl. reflexivity. Qed. 
Example prop_andb32: (andb3 false true true) = false.
Proof. simpl. reflexivity. Qed. 
Example prop_andb33: (andb3 true false true) = false.
Proof. simpl. reflexivity. Qed. 
Example prop_andb34: (andb3 true true false) = false.
Proof. simpl. reflexivity. Qed. 

(* angruicam1 *)
Notation "x && y" := (andb x y).

Definition andb32 (b1:bool) (b2:bool) (b3:bool) : bool :=
  b1 && b2 && b3.

Example test_andb321: (andb32 true true true) = true.
Proof. simpl. reflexivity. Qed.
Example test_andb322: (andb32 false true true) = false.
Proof. simpl. reflexivity. Qed.
Example test_andb323: (andb32 true false true) = false.
Proof. simpl. reflexivity. Qed.
Example test_andb324: (andb32 true true false) = false.
Proof. simpl. reflexivity. Qed.

(* =====================================================================
   §§ Tipos de las funciones  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Calcular el tipo de las siguientes expresiones
      + true
      + (negb true)
      + negb
   ------------------------------------------------------------------ *)

Check true.
(* ===> true : bool *)

Check (negb true).
(* ===> negb true : bool *)

Check negb.
(* ===> negb : bool -> bool *)

(* =====================================================================
   § Tipos compuestos  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir el tipo rgb cuyos constructores son red, green y
   blue. 
   ------------------------------------------------------------------ *)

Inductive rgb : Type :=
  | red   : rgb
  | green : rgb
  | blue  : rgb.

(* ---------------------------------------------------------------------
   Ejemplo. Definir el tipo color cuyos constructores son black, white y
   primary, donde primary es una función de rgb en color.
   ------------------------------------------------------------------ *)

Inductive color : Type :=
  | black   : color
  | white   : color
  | primary : rgb -> color.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      monochrome :: color -> bool
   tal que (monochrome c) se verifica si c es monocromático.
   ------------------------------------------------------------------ *)

Definition monochrome (c : color) : bool :=
  match c with
  | black     => true
  | white     => true
  | primary p => false
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      isred :: color -> bool
   tal que (isred c) se verifica si c es rojo.
   ------------------------------------------------------------------ *)

Definition isred (c : color) : bool :=
  match c with
  | black       => false
  | white       => false
  | primary red => true
  | primary _   => false
  end.

(* =====================================================================
   §§ Módulos  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Iniciar el módulo NatPlayground.
   ------------------------------------------------------------------ *)

Module NatPlayground.

(* =====================================================================
   §§ Números naturales  
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir el tipo nat de los números naturales con los
   constructores 0 (para el 0) y S (para el siguiente).
   ------------------------------------------------------------------ *)
  
Inductive nat : Type :=
  | O : nat
  | S : nat -> nat.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      pred :: nat -> nat
   tal que (pred n) es el predecesor de n.
   ------------------------------------------------------------------ *)


Definition pred (n : nat) : nat :=
  match n with
    | O    => O
    | S n' => n'
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Finaliz el módulo NatPlayground.
   ------------------------------------------------------------------ *)

End NatPlayground.

(* ---------------------------------------------------------------------
   Ejemplo. Calcular el tipo y valor de la expresión (S (S (S (S O)))).
   ------------------------------------------------------------------ *)

Check (S (S (S (S O)))).
  (* ===> 4 : nat *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      minustwo :: nat -> nat
   tal que (minustwo n) es n-2. 
   ------------------------------------------------------------------ *)

Definition minustwo (n : nat) : nat :=
  match n with
    | O        => O
    | S O      => O
    | S (S n') => n'
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Evaluar la expresión (minustwo 4).
   ------------------------------------------------------------------ *)

Compute (minustwo 4).
  (* ===> 2 : nat *)

(* ---------------------------------------------------------------------
   Ejemplo. Calcular et tipo de las funcionse S, pred y minustwo.
   ------------------------------------------------------------------ *)

Check S.
Check pred.
Check minustwo.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      evenb :: nat -> bool
   tal que (evenb n) se verifica si n es par.
   ------------------------------------------------------------------ *)

Fixpoint evenb (n:nat) : bool :=
  match n with
  | O        => true
  | S O      => false
  | S (S n') => evenb n'
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      oddb :: nat -> bool
   tal que (oddb n) se verifica si n es impar.
   ------------------------------------------------------------------ *)

Definition oddb (n:nat) : bool   :=   negb (evenb n).

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que
      + oddb 1 = true.
      + oddb 4 = false.
   ------------------------------------------------------------------ *)

Example test_oddb1: oddb 1 = true.
Proof. simpl. reflexivity.  Qed.
Example test_oddb2: oddb 4 = false.
Proof. simpl. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Iniciar el módulo NatPlayground2.
   ------------------------------------------------------------------ *)

Module NatPlayground2.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      plus :: nat -> nat -> nat 
   tal que (plus n m) es la suma de n y m. Por ejemplo,
      plus 3 2 = 5
   ------------------------------------------------------------------ *)
  
Fixpoint plus (n : nat) (m : nat) : nat :=
  match n with
    | O    => m
    | S n' => S (plus n' m)
  end.

Compute (plus 3 2).
(* ===> 5: nat *)

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      mult :: nat -> nat -> nat 
   tal que (mult n m) es el producto de n y m. Por ejemplo,
      mult 3 2 = 6
   ------------------------------------------------------------------ *)
  
Fixpoint mult (n m : nat) : nat :=
  match n with
    | O    => O
    | S n' => plus m (mult n' m)
  end.

Example test_mult1: (mult 2 3) = 6.
Proof. simpl. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      minus :: nat -> nat -> nat 
   tal que (minus n m) es la diferencia de n y m. Por ejemplo,
      mult 3 2 = 1
   ------------------------------------------------------------------ *)
  
Fixpoint minus (n m:nat) : nat :=
  match (n, m) with
  | (O   , _)    => O
  | (S _ , O)    => n
  | (S n', S m') => minus n' m'
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Cerrar el módulo NatPlayground2.
   ------------------------------------------------------------------ *)

End NatPlayground2.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      exp : nat ->  nat -> nat
   tal que (exp x n) es la potencia n-ésima de x. Por ejemplo,
      exp 2 3 = 8
   ------------------------------------------------------------------ *)

Fixpoint exp (base power : nat) : nat :=
  match power with
    | O   => S O
    | S p => mult base (exp base p)
  end.

Compute (exp 2 3).
(* ===> 8 : nat *)

(* ---------------------------------------------------------------------
   Ejercicio 3. Definir la función
      factorial :: nat -> nat1
   tal que (factorial n) es el factorial de n. 

      (factorial 3) = 6.
      (factorial 5) = (mult 10 12).
   ------------------------------------------------------------------ *)

(* alerodrod5, angruicam1 *)
Fixpoint factorial (n:nat) : nat := 
  match n with
  | O    => 1
  | S n' =>  S n' * factorial n'
  end.

Example prop_factorial1: (factorial 3) = 6.
Proof. simpl. reflexivity.  Qed.
Example prop_factorial2: (factorial 5) = (mult 10 12).
Proof. simpl. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Definir los operadores +, - y * como abreviaturas de  las
   funciones plus, minus y mult.  
   ------------------------------------------------------------------ *)

Notation "x + y" := (plus x y)
                       (at level 50, left associativity)
                       : nat_scope.
Notation "x - y" := (minus x y)
                       (at level 50, left associativity)
                       : nat_scope.
Notation "x * y" := (mult x y)
                       (at level 40, left associativity)
                       : nat_scope.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      beq_nat : nat -> nat -> bool
   tal que (beq_nat n m) se verifica si n y me son iguales.
   ------------------------------------------------------------------ *)

Fixpoint beq_nat (n m : nat) : bool :=
  match n with
  | O => match m with
         | O    => true
         | S m' => false
         end
  | S n' => match m with
            | O    => false
            | S m' => beq_nat n' m'
            end
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Definir la función
      leb : nat -> nat -> bool
   tal que (leb n m) se verifica si n es menor o igual que m.
   ------------------------------------------------------------------ *)

Fixpoint leb (n m : nat) : bool :=
  match n with
  | O    => true
  | S n' => match m with
            | O    => false
            | S m' => leb n' m'
            end
  end.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar las siguientes propiedades
      + (leb 2 2) = true.
      + (leb 2 4) = true.
      + (leb 4 2) = false.
   ------------------------------------------------------------------ *)

Example test_leb1: (leb 2 2) = true.
Proof. simpl. reflexivity.  Qed.
Example test_leb2: (leb 2 4) = true.
Proof. simpl. reflexivity.  Qed.
Example test_leb3: (leb 4 2) = false.
Proof. simpl. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejercicio 4. Definir la función
      blt_nat :: nat -> nat -> bool
   tal que (blt n m) se verifica si n es menor que m.

   Demostrar las siguientes propiedades
      (blt_nat 2 2) = false.
      (blt_nat 2 4) = true.
      (blt_nat 4 2) = false.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Definition blt_nat (n m : nat) : bool :=
  match n with
  | O => true
  | S n' =>
      match m with
      | O    => false
      | S m' => leb (S n')  m'
      end
  end.                                   

Example prop_blt_nat1: (blt_nat 2 2) = false.
Proof. simpl. reflexivity.  Qed.
Example prop_blt_nat2: (blt_nat 2 4) = true.
Proof. simpl. reflexivity.  Qed.
Example prop_blt_nat3: (blt_nat 4 2) = false.
Proof. simpl. reflexivity.  Qed.

(* angruicam1 *)
Definition blt_nat2 (n m : nat) : bool :=
  negb (beq_nat (m-n) 0).

Example test_blt_nat21: (blt_nat2 2 2) = false.
Proof. simpl. reflexivity. Qed.
Example test_blt_nat22: (blt_nat2 2 4) = true.
Proof. simpl. reflexivity. Qed.
Example test_blt_nat23: (blt_nat2 4 2) = false.
Proof. simpl. reflexivity. Qed.

(* =====================================================================
   § Demostraciones por simplificación 
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que el 0 es el elemento neutro por la izquierda de
   la suma de los números naturales.
   ------------------------------------------------------------------ *)

(* 1ª demostración *)
Theorem plus_O_n : forall n : nat, 0 + n = n.
Proof.
  intros n. simpl. reflexivity.  Qed.

(* 2ª demostración *)
Theorem plus_O_n' : forall n : nat, 0 + n = n.
Proof.
  intros n. reflexivity. Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que la suma de 1 y n es el siguiente de n.
   ------------------------------------------------------------------ *)

Theorem plus_1_l : forall n:nat, 1 + n = S n.
Proof.
  intros n. reflexivity.  Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que el producto de 0 por n es 0.
   ------------------------------------------------------------------ *)

Theorem mult_0_l : forall n:nat, 0 * n = 0.
Proof.
  intros n. reflexivity.  Qed.

(* =====================================================================
   § Demostraciones por reescritura 
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que si n = m, entonces n + n = m + m.
   ------------------------------------------------------------------ *)


Theorem plus_id_example : forall n m:nat,
  n = m ->
  n + n = m + m.

Proof.
  (* move both quantifiers into the context: *)
  intros n m.
  (* move the hypothesis into the context: *)
  intros H.
  (* rewrite the goal using the hypothesis: *)
  rewrite -> H.
  reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 5. Demostrar que si n = m y m = o, entonces n + m = m + o.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Theorem plus_id_exercise: forall n m o : nat,
  n = m -> m = o -> n + m = m + o.
Proof.
  intros n m.
  intros o Ho.
  rewrite -> Ho.
  intros H.
  rewrite -> H.
  reflexivity.
Qed.

(* angruicam1 *)
Theorem plus_id_exercise2 : forall n m o : nat,
  n = m -> m = o -> n + m = m + o.
Proof.
  intros n m o H1 H2.
  rewrite -> H1.
  rewrite -> H2.
  reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que (0 + n) * m = n * m.
   ------------------------------------------------------------------ *)

Theorem mult_0_plus : forall n m : nat,
  (0 + n) * m = n * m.
Proof.
  intros n m.
  rewrite -> plus_O_n.
  reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 6. Demostrar que si m = S n, entonces m * (1 + n) = m * m.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Theorem mult_S_1 : forall n m : nat,
  m = S n ->
  m * (1 + n) = m * m.
Proof.
  intros n m.
  intros H.
  rewrite -> H.
  reflexivity.
Qed. 

(* angruicam1 *)
Theorem mult_S_12 : forall n m : nat,
  m = S n ->
  m * (1 + n) = m * m.
Proof.
  intros n m H.
  rewrite H.
  reflexivity.
Qed.

(* =====================================================================
   § Demostraciones por análisis de casos 
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que n + 1 es distinto de 0.
   ------------------------------------------------------------------ *)

(* 1º intento *)
Theorem plus_1_neq_0_firsttry : forall n : nat,
  beq_nat (n + 1) 0 = false.
Proof.
  intros n.
  simpl.  (* does nothing! *)
Abort.

(* 2º intento *)
Theorem plus_1_neq_0 : forall n : nat,
  beq_nat (n + 1) 0 = false.
Proof.
  intros n. destruct n as [| n'].
  - reflexivity.
  - reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que la negación es involutiva; es decir, la
   negación de la negación de b es b.
   ------------------------------------------------------------------ *)

Theorem negb_involutive : forall b : bool,
  negb (negb b) = b.
Proof.
  intros b. destruct b.
  - reflexivity.
  - reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que la conjunción es conmutativa.
   ------------------------------------------------------------------ *)

(* 1ª demostración *)
Theorem andb_commutative : forall b c, andb b c = andb c b.
Proof.
  intros b c. destruct b.
  - destruct c.
    + reflexivity.
    + reflexivity.
  - destruct c.
    + reflexivity.
    + reflexivity.
Qed.

(* 2ª demostración *)
Theorem andb_commutative' : forall b c, andb b c = andb c b.
Proof.
  intros b c. destruct b.
  { destruct c.
    { reflexivity. }
    { reflexivity. } }
  { destruct c.
    { reflexivity. }
    { reflexivity. } }
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que la conjunción es asociativa.
   ------------------------------------------------------------------ *)

Theorem andb3_exchange :
  forall b c d, andb (andb b c) d = andb (andb b d) c.
Proof.
  intros b c d. destruct b.
  - destruct c.
    { destruct d.
      - reflexivity.
      - reflexivity. }
    { destruct d.
      - reflexivity.
      - reflexivity. }
  - destruct c.
    { destruct d.
      - reflexivity.
      - reflexivity. }
    { destruct d.
      - reflexivity.
      - reflexivity. }
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que n + 1 es distinto de 0.
   ------------------------------------------------------------------ *)

Theorem plus_1_neq_0' : forall n : nat,
  beq_nat (n + 1) 0 = false.
Proof.
  intros [|n].
  - reflexivity.
  - reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejemplo. Demostrar que la conjunción es conmutativa.
   ------------------------------------------------------------------ *)

Theorem andb_commutative'' :
  forall b c, andb b c = andb c b.
Proof.
  intros [] [].
  - reflexivity.
  - reflexivity.
  - reflexivity.
  - reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 7. Demostrar que si andb b c = true, entonces c = true.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Theorem andb_true_elim2 : forall b c : bool,
  andb b c = true -> c = true.
Proof.
  intros b c. destruct b.
  - intros H. rewrite <-H. reflexivity.
  - intros H. rewrite <-H. destruct false.
    + reflexivity.
    + destruct c. rewrite <-H. reflexivity. reflexivity. 
Qed.

(* angruicam1 *)
Theorem andb_true_elim22 : forall b c : bool,
  andb b c = true -> c = true.
Proof.
  intros [] [] [].
  - reflexivity.
  - reflexivity.
  - reflexivity.
  - reflexivity.
Qed.

(* jorcatote *)
Theorem andb_true_elim23 : forall b c : bool,
  andb b c = true -> c = true.
Proof.
  intros b c. destruct c.
    - reflexivity.   
    - destruct b.
      + simpl. intros h. rewrite h. reflexivity.
      + simpl. intros h. rewrite h. reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 8. Dmostrar que 0 es distinto de n + 1.
   ------------------------------------------------------------------ *)

(* alerodrod5 *)
Theorem zero_nbeq_plus_1: forall n : nat,
  beq_nat 0 (n + 1) = false.
Proof.
  intros n. destruct n.
  - reflexivity.
  - reflexivity.
Qed.

(* angruicam1 *)
Theorem zero_nbeq_plus_12 : forall n : nat,
  beq_nat 0 (n + 1) = false.
Proof.
  intros [| n'].
  - reflexivity.
  - reflexivity.
Qed.

(* =====================================================================
   § Ejercicios complementarios 
   ================================================================== *)

(* ---------------------------------------------------------------------
   Ejercicio 9. Demostrar que
      forall (f : bool -> bool),
        (forall (x : bool), f x = x) -> 
        forall (b : bool), f (f b) = b.
   ------------------------------------------------------------------ *)

(* angruicam1, alerodrod5 *)
Theorem identity_fn_applied_twice :
  forall (f : bool -> bool),
  (forall (x : bool), f x = x) ->
  forall (b : bool), f (f b) = b.
Proof.
  intros f x b.
  rewrite x.
  rewrite x.
  reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 10. Demostrar que
      forall (b c : bool),
        (andb b c = orb b c) -> b = c.
   ------------------------------------------------------------------ *)

(* angruicam1 alerodrod5 *)
Theorem andb_eq_orb :
  forall (b c : bool),
  (andb b c = orb b c) ->
  b = c.
Proof.
  intros [] c.
  - simpl. intros H. rewrite H. reflexivity.
  - simpl. intros H. rewrite H. reflexivity.
Qed.

(* ---------------------------------------------------------------------
   Ejercicio 11. En este ejercicio se considera la siguiente
   representación de los números naturales
      Inductive nat2 : Type :=
        | C  : nat2
        | D  : nat2 -> nat2
        | SD : nat2 -> nat2.
   donde C representa el cero, D el doble y SD el siguiente del doble.

   Definir la función
      nat2Anat :: nat2 -> nat
   tal que (nat2Anat x) es el número natural representado por x. 

   Demostrar que 
      nat2Anat (SD (SD C))     = 3
      nat2Anat (D (SD (SD C))) = 6.
   ------------------------------------------------------------------ *)

(* angruicam1, alerodrod5 *)
Inductive nat2 : Type :=
  | C  : nat2
  | D  : nat2 -> nat2
  | SD : nat2 -> nat2.
 
Fixpoint nat2Anat (x:nat2) : nat :=
  match x with
  | C => O
  | D n => 2*nat2Anat n
  | SD n => (2*nat2Anat n)+1
  end.
 
Example prop_nat2Anat1: (nat2Anat (SD (SD C))) = 3.
Proof. reflexivity. Qed.
Example prop_nat2Anat2: (nat2Anat (D (SD (SD C)))) = 6.
Proof. reflexivity. Qed.

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(* T1: Programación funcional en Coq *)

Definition admit {T: Type} : T. Admitted.

(* =====================================================================

§ Datos y funciones

================================================================== *)

(* =====================================================================

§§ Tipos enumerados

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Definir el tipo day cuyos constructores sean los días de

la semana.

------------------------------------------------------------------ *)

Inductive day : Type :=

| monday : day

| tuesday : day

| wednesday : day

| thursday : day

| friday : day

| saturday : day

| sunday : day.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

next_weekday :: day -> day

tal que (next_weekday d) es el día laboral siguiente a d.

------------------------------------------------------------------ *)

Definition next_weekday (d:day) : day :=

match d with

| monday => tuesday

| tuesday => wednesday

| wednesday => thursday

| thursday => friday

| friday => monday

| saturday => monday

| sunday => monday

end.

(* ---------------------------------------------------------------------

Ejemplo. Calcular el valor de las siguientes expresiones

+ (next_weekday friday)

+ (next_weekday (next_weekday saturday))

------------------------------------------------------------------ *)

Compute (next_weekday friday).

(* ==> monday : day *)

Compute (next_weekday (next_weekday saturday)).

(* ==> tuesday : day *)

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que

(next_weekday (next_weekday saturday)) = tuesday

------------------------------------------------------------------ *)

Example test_next_weekday:

(next_weekday (next_weekday saturday)) = tuesday.

Proof. simpl. reflexivity. Qed.

(* =====================================================================

§§ Booleanos

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Definir el tipo bool cuyos constructores son true y false.

------------------------------------------------------------------ *)

Inductive bool : Type :=

| true : bool

| false : bool.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

negb :: bool -> bool

tal que (negb b) es la negación de b.

------------------------------------------------------------------ *)

Definition negb (b:bool) : bool :=

match b with

| true => false

| false => true

end.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

andb :: bool -> bool -> bool

tal que (andb b1 b2) es la conjunción de b1 y b2.

------------------------------------------------------------------ *)

Definition andb (b1:bool) (b2:bool) : bool :=

match b1 with

| true => b2

| false => false

end.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

orb :: bool -> bool -> bool

tal que (orb b1 b2) es la disyunción de b1 y b2.

------------------------------------------------------------------ *)

Definition orb (b1:bool) (b2:bool) : bool :=

match b1 with

| true => true

| false => b2

end.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar las siguientes propiedades

(orb true false) = true.

(orb false false) = false.

(orb false true) = true.

(orb true true) = true.

------------------------------------------------------------------ *)

Example test_orb1: (orb true false) = true.

Proof. simpl. reflexivity. Qed.

Example test_orb2: (orb false false) = false.

Proof. simpl. reflexivity. Qed.

Example test_orb3: (orb false true) = true.

Proof. simpl. reflexivity. Qed.

Example test_orb4: (orb true true) = true.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Definir los operadores (&&) y (||) como abreviaturas de las

funciones andb y orb.

------------------------------------------------------------------ *)

Notation "x && y" := (andb x y).

Notation "x || y" := (orb x y).

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que

false || false || true = true.

------------------------------------------------------------------ *)

Example test_orb5: false || false || true = true.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejercicio 1. Definir la función

nandb :: bool -> bool -> bool

tal que (nanb x y) se verifica si x e y no son verdaderos.

Demostrar las siguientes propiedades de nand

(nandb true false) = true.

(nandb false false) = true.

(nandb false true) = true.

(nandb true true) = false.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Definition nandb (b1:bool) (b2:bool) : bool :=

match b1 with

| true => negb b2

| false => true

end.

Example prop_nandb1: (nandb true false) = true.

Proof. simpl. reflexivity. Qed.

Example prop_nandb2: (nandb false false) = true.

Proof. simpl. reflexivity. Qed.

Example prop_nandb3: (nandb false true) = true.

Proof. simpl. reflexivity. Qed.

Example prop_nandb4: (nandb true true) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Definition nandb1 (b1:bool) (b2:bool) : bool :=

match b1 with

| true => negb b2

| _ => true

end.

Example test_nandb11: (nandb1 true false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb12: (nandb1 false false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb13: (nandb1 false true) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb14: (nandb1 true true) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Definition nandb2 (b1:bool) (b2:bool) : bool :=

match b1, b2 with

| true, true => false

| _, _ => true

end.

Example test_nandb21: (nandb2 true false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb22: (nandb2 false false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb23: (nandb2 false true) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb24: (nandb2 true true) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Definition nandb3 (b1:bool) (b2:bool) : bool :=

if b1 then negb b2 else true.

Example test_nandb31: (nandb3 true false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb32: (nandb3 false false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb33: (nandb3 false true) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb34: (nandb3 true true) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Definition nandb4 (b1:bool) (b2:bool) : bool :=

negb (b1 && b2).

Example test_nandb41: (nandb4 true false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb42: (nandb4 false false) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb43: (nandb4 false true) = true.

Proof. simpl. reflexivity. Qed.

Example test_nandb44: (nandb4 true true) = false.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejercicio 2. Definir la función

andb3 :: bool -> bool -> bool -> bool

tal que (andb3 x y z) se verifica si x, y y z son verdaderos.

Demostrar las siguientes propiedades de andb3

(andb3 true true true) = true.

(andb3 false true true) = false.

(andb3 true false true) = false.

(andb3 true true false) = false.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Definition andb3 (b1:bool) (b2:bool) (b3:bool) : bool :=

match b1 with

| true => andb b2 b3

| false => false

end.

Example prop_andb31: (andb3 true true true) = true.

Proof. simpl. reflexivity. Qed.

Example prop_andb32: (andb3 false true true) = false.

Proof. simpl. reflexivity. Qed.

Example prop_andb33: (andb3 true false true) = false.

Proof. simpl. reflexivity. Qed.

Example prop_andb34: (andb3 true true false) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Notation "x && y" := (andb x y).

Definition andb32 (b1:bool) (b2:bool) (b3:bool) : bool :=

b1 && b2 && b3.

Example test_andb321: (andb32 true true true) = true.

Proof. simpl. reflexivity. Qed.

Example test_andb322: (andb32 false true true) = false.

Proof. simpl. reflexivity. Qed.

Example test_andb323: (andb32 true false true) = false.

Proof. simpl. reflexivity. Qed.

Example test_andb324: (andb32 true true false) = false.

Proof. simpl. reflexivity. Qed.

(* =====================================================================

§§ Tipos de las funciones

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Calcular el tipo de las siguientes expresiones

+ true

+ (negb true)

+ negb

------------------------------------------------------------------ *)

Check true.

(* ===> true : bool *)

Check (negb true).

(* ===> negb true : bool *)

Check negb.

(* ===> negb : bool -> bool *)

(* =====================================================================

§ Tipos compuestos

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Definir el tipo rgb cuyos constructores son red, green y

blue.

------------------------------------------------------------------ *)

Inductive rgb : Type :=

| red : rgb

| green : rgb

| blue : rgb.

(* ---------------------------------------------------------------------

Ejemplo. Definir el tipo color cuyos constructores son black, white y

primary, donde primary es una función de rgb en color.

------------------------------------------------------------------ *)

Inductive color : Type :=

| black : color

| white : color

| primary : rgb -> color.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

monochrome :: color -> bool

tal que (monochrome c) se verifica si c es monocromático.

------------------------------------------------------------------ *)

Definition monochrome (c : color) : bool :=

match c with

| black => true

| white => true

| primary p => false

end.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

isred :: color -> bool

tal que (isred c) se verifica si c es rojo.

------------------------------------------------------------------ *)

Definition isred (c : color) : bool :=

match c with

| black => false

| white => false

| primary red => true

| primary _ => false

end.

(* =====================================================================

§§ Módulos

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Iniciar el módulo NatPlayground.

------------------------------------------------------------------ *)

Module NatPlayground.

(* =====================================================================

§§ Números naturales

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Definir el tipo nat de los números naturales con los

constructores 0 (para el 0) y S (para el siguiente).

------------------------------------------------------------------ *)

Inductive nat : Type :=

| O : nat

| S : nat -> nat.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

pred :: nat -> nat

tal que (pred n) es el predecesor de n.

------------------------------------------------------------------ *)

Definition pred (n : nat) : nat :=

match n with

| O => O

| S n' => n'

end.

(* ---------------------------------------------------------------------

Ejemplo. Finaliz el módulo NatPlayground.

------------------------------------------------------------------ *)

End NatPlayground.

(* ---------------------------------------------------------------------

Ejemplo. Calcular el tipo y valor de la expresión (S (S (S (S O)))).

------------------------------------------------------------------ *)

Check (S (S (S (S O)))).

(* ===> 4 : nat *)

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

minustwo :: nat -> nat

tal que (minustwo n) es n-2.

------------------------------------------------------------------ *)

Definition minustwo (n : nat) : nat :=

match n with

| O => O

| S O => O

| S (S n') => n'

end.

(* ---------------------------------------------------------------------

Ejemplo. Evaluar la expresión (minustwo 4).

------------------------------------------------------------------ *)

Compute (minustwo 4).

(* ===> 2 : nat *)

(* ---------------------------------------------------------------------

Ejemplo. Calcular et tipo de las funcionse S, pred y minustwo.

------------------------------------------------------------------ *)

Check S.

Check pred.

Check minustwo.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

evenb :: nat -> bool

tal que (evenb n) se verifica si n es par.

------------------------------------------------------------------ *)

Fixpoint evenb (n:nat) : bool :=

match n with

| O => true

| S O => false

| S (S n') => evenb n'

end.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

oddb :: nat -> bool

tal que (oddb n) se verifica si n es impar.

------------------------------------------------------------------ *)

Definition oddb (n:nat) : bool := negb (evenb n).

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que

+ oddb 1 = true.

+ oddb 4 = false.

------------------------------------------------------------------ *)

Example test_oddb1: oddb 1 = true.

Proof. simpl. reflexivity. Qed.

Example test_oddb2: oddb 4 = false.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Iniciar el módulo NatPlayground2.

------------------------------------------------------------------ *)

Module NatPlayground2.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

plus :: nat -> nat -> nat

tal que (plus n m) es la suma de n y m. Por ejemplo,

plus 3 2 = 5

------------------------------------------------------------------ *)

Fixpoint plus (n : nat) (m : nat) : nat :=

match n with

| O => m

| S n' => S (plus n' m)

end.

Compute (plus 3 2).

(* ===> 5: nat *)

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

mult :: nat -> nat -> nat

tal que (mult n m) es el producto de n y m. Por ejemplo,

mult 3 2 = 6

------------------------------------------------------------------ *)

Fixpoint mult (n m : nat) : nat :=

match n with

| O => O

| S n' => plus m (mult n' m)

end.

Example test_mult1: (mult 2 3) = 6.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

minus :: nat -> nat -> nat

tal que (minus n m) es la diferencia de n y m. Por ejemplo,

mult 3 2 = 1

------------------------------------------------------------------ *)

Fixpoint minus (n m:nat) : nat :=

match (n, m) with

| (O , _) => O

| (S _ , O) => n

| (S n', S m') => minus n' m'

end.

(* ---------------------------------------------------------------------

Ejemplo. Cerrar el módulo NatPlayground2.

------------------------------------------------------------------ *)

End NatPlayground2.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

exp : nat -> nat -> nat

tal que (exp x n) es la potencia n-ésima de x. Por ejemplo,

exp 2 3 = 8

------------------------------------------------------------------ *)

Fixpoint exp (base power : nat) : nat :=

match power with

| O => S O

| S p => mult base (exp base p)

end.

Compute (exp 2 3).

(* ===> 8 : nat *)

(* ---------------------------------------------------------------------

Ejercicio 3. Definir la función

factorial :: nat -> nat1

tal que (factorial n) es el factorial de n.

(factorial 3) = 6.

(factorial 5) = (mult 10 12).

------------------------------------------------------------------ *)

(* alerodrod5, angruicam1 *)

Fixpoint factorial (n:nat) : nat :=

match n with

| O => 1

| S n' => S n' * factorial n'

end.

Example prop_factorial1: (factorial 3) = 6.

Proof. simpl. reflexivity. Qed.

Example prop_factorial2: (factorial 5) = (mult 10 12).

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Definir los operadores +, - y * como abreviaturas de las

funciones plus, minus y mult.

------------------------------------------------------------------ *)

Notation "x + y" := (plus x y)

(at level 50, left associativity)

: nat_scope.

Notation "x - y" := (minus x y)

(at level 50, left associativity)

: nat_scope.

Notation "x * y" := (mult x y)

(at level 40, left associativity)

: nat_scope.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

beq_nat : nat -> nat -> bool

tal que (beq_nat n m) se verifica si n y me son iguales.

------------------------------------------------------------------ *)

Fixpoint beq_nat (n m : nat) : bool :=

match n with

| O => match m with

| O => true

| S m' => false

end

| S n' => match m with

| O => false

| S m' => beq_nat n' m'

end

end.

(* ---------------------------------------------------------------------

Ejemplo. Definir la función

leb : nat -> nat -> bool

tal que (leb n m) se verifica si n es menor o igual que m.

------------------------------------------------------------------ *)

Fixpoint leb (n m : nat) : bool :=

match n with

| O => true

| S n' => match m with

| O => false

| S m' => leb n' m'

end

end.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar las siguientes propiedades

+ (leb 2 2) = true.

+ (leb 2 4) = true.

+ (leb 4 2) = false.

------------------------------------------------------------------ *)

Example test_leb1: (leb 2 2) = true.

Proof. simpl. reflexivity. Qed.

Example test_leb2: (leb 2 4) = true.

Proof. simpl. reflexivity. Qed.

Example test_leb3: (leb 4 2) = false.

Proof. simpl. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejercicio 4. Definir la función

blt_nat :: nat -> nat -> bool

tal que (blt n m) se verifica si n es menor que m.

Demostrar las siguientes propiedades

(blt_nat 2 2) = false.

(blt_nat 2 4) = true.

(blt_nat 4 2) = false.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Definition blt_nat (n m : nat) : bool :=

match n with

| O => true

| S n' =>

match m with

| O => false

| S m' => leb (S n') m'

end

end.

Example prop_blt_nat1: (blt_nat 2 2) = false.

Proof. simpl. reflexivity. Qed.

Example prop_blt_nat2: (blt_nat 2 4) = true.

Proof. simpl. reflexivity. Qed.

Example prop_blt_nat3: (blt_nat 4 2) = false.

Proof. simpl. reflexivity. Qed.

(* angruicam1 *)

Definition blt_nat2 (n m : nat) : bool :=

negb (beq_nat (m-n) 0).

Example test_blt_nat21: (blt_nat2 2 2) = false.

Proof. simpl. reflexivity. Qed.

Example test_blt_nat22: (blt_nat2 2 4) = true.

Proof. simpl. reflexivity. Qed.

Example test_blt_nat23: (blt_nat2 4 2) = false.

Proof. simpl. reflexivity. Qed.

(* =====================================================================

§ Demostraciones por simplificación

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que el 0 es el elemento neutro por la izquierda de

la suma de los números naturales.

------------------------------------------------------------------ *)

(* 1ª demostración *)

Theorem plus_O_n : forall n : nat, 0 + n = n.

Proof.

intros n. simpl. reflexivity. Qed.

(* 2ª demostración *)

Theorem plus_O_n' : forall n : nat, 0 + n = n.

Proof.

intros n. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que la suma de 1 y n es el siguiente de n.

------------------------------------------------------------------ *)

Theorem plus_1_l : forall n:nat, 1 + n = S n.

Proof.

intros n. reflexivity. Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que el producto de 0 por n es 0.

------------------------------------------------------------------ *)

Theorem mult_0_l : forall n:nat, 0 * n = 0.

Proof.

intros n. reflexivity. Qed.

(* =====================================================================

§ Demostraciones por reescritura

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que si n = m, entonces n + n = m + m.

------------------------------------------------------------------ *)

Theorem plus_id_example : forall n m:nat,

n = m ->

n + n = m + m.

Proof.

(* move both quantifiers into the context: *)

intros n m.

(* move the hypothesis into the context: *)

intros H.

(* rewrite the goal using the hypothesis: *)

rewrite -> H.

reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 5. Demostrar que si n = m y m = o, entonces n + m = m + o.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Theorem plus_id_exercise: forall n m o : nat,

n = m -> m = o -> n + m = m + o.

Proof.

intros n m.

intros o Ho.

rewrite -> Ho.

intros H.

rewrite -> H.

reflexivity.

Qed.

(* angruicam1 *)

Theorem plus_id_exercise2 : forall n m o : nat,

n = m -> m = o -> n + m = m + o.

Proof.

intros n m o H1 H2.

rewrite -> H1.

rewrite -> H2.

reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que (0 + n) * m = n * m.

------------------------------------------------------------------ *)

Theorem mult_0_plus : forall n m : nat,

(0 + n) * m = n * m.

Proof.

intros n m.

rewrite -> plus_O_n.

reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 6. Demostrar que si m = S n, entonces m * (1 + n) = m * m.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Theorem mult_S_1 : forall n m : nat,

m = S n ->

m * (1 + n) = m * m.

Proof.

intros n m.

intros H.

rewrite -> H.

reflexivity.

Qed.

(* angruicam1 *)

Theorem mult_S_12 : forall n m : nat,

m = S n ->

m * (1 + n) = m * m.

Proof.

intros n m H.

rewrite H.

reflexivity.

Qed.

(* =====================================================================

§ Demostraciones por análisis de casos

================================================================== *)

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que n + 1 es distinto de 0.

------------------------------------------------------------------ *)

(* 1º intento *)

Theorem plus_1_neq_0_firsttry : forall n : nat,

beq_nat (n + 1) 0 = false.

Proof.

intros n.

simpl. (* does nothing! *)

Abort.

(* 2º intento *)

Theorem plus_1_neq_0 : forall n : nat,

beq_nat (n + 1) 0 = false.

Proof.

intros n. destruct n as [| n'].

- reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que la negación es involutiva; es decir, la

negación de la negación de b es b.

------------------------------------------------------------------ *)

Theorem negb_involutive : forall b : bool,

negb (negb b) = b.

Proof.

intros b. destruct b.

- reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que la conjunción es conmutativa.

------------------------------------------------------------------ *)

(* 1ª demostración *)

Theorem andb_commutative : forall b c, andb b c = andb c b.

Proof.

intros b c. destruct b.

- destruct c.

+ reflexivity.

- destruct c.

+ reflexivity.

Qed.

(* 2ª demostración *)

Theorem andb_commutative' : forall b c, andb b c = andb c b.

Proof.

intros b c. destruct b.

{ destruct c.

{ reflexivity. }

{ reflexivity. } }

{ destruct c.

{ reflexivity. }

{ reflexivity. } }

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que la conjunción es asociativa.

------------------------------------------------------------------ *)

Theorem andb3_exchange :

forall b c d, andb (andb b c) d = andb (andb b d) c.

Proof.

intros b c d. destruct b.

- destruct c.

{ destruct d.

- reflexivity.

- reflexivity. }

{ destruct d.

- reflexivity.

- reflexivity. }

- destruct c.

{ destruct d.

- reflexivity.

- reflexivity. }

{ destruct d.

- reflexivity.

- reflexivity. }

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que n + 1 es distinto de 0.

------------------------------------------------------------------ *)

Theorem plus_1_neq_0' : forall n : nat,

beq_nat (n + 1) 0 = false.

Proof.

intros [|n].

- reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejemplo. Demostrar que la conjunción es conmutativa.

------------------------------------------------------------------ *)

Theorem andb_commutative'' :

forall b c, andb b c = andb c b.

Proof.

intros [] [].

- reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 7. Demostrar que si andb b c = true, entonces c = true.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Theorem andb_true_elim2 : forall b c : bool,

andb b c = true -> c = true.

Proof.

intros b c. destruct b.

- intros H. rewrite <-H. reflexivity.

- intros H. rewrite <-H. destruct false.

+ reflexivity.

+ destruct c. rewrite <-H. reflexivity. reflexivity.

Qed.

(* angruicam1 *)

Theorem andb_true_elim22 : forall b c : bool,

andb b c = true -> c = true.

Proof.

intros [] [] [].

- reflexivity.

Qed.

(* jorcatote *)

Theorem andb_true_elim23 : forall b c : bool,

andb b c = true -> c = true.

Proof.

intros b c. destruct c.

- reflexivity.

- destruct b.

+ simpl. intros h. rewrite h. reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 8. Dmostrar que 0 es distinto de n + 1.

------------------------------------------------------------------ *)

(* alerodrod5 *)

Theorem zero_nbeq_plus_1: forall n : nat,

beq_nat 0 (n + 1) = false.

Proof.

intros n. destruct n.

- reflexivity.

Qed.

(* angruicam1 *)

Theorem zero_nbeq_plus_12 : forall n : nat,

beq_nat 0 (n + 1) = false.

Proof.

intros [| n'].

- reflexivity.

Qed.

(* =====================================================================

§ Ejercicios complementarios

================================================================== *)

(* ---------------------------------------------------------------------

Ejercicio 9. Demostrar que

forall (f : bool -> bool),

(forall (x : bool), f x = x) ->

forall (b : bool), f (f b) = b.

------------------------------------------------------------------ *)

(* angruicam1, alerodrod5 *)

Theorem identity_fn_applied_twice :

forall (f : bool -> bool),

(forall (x : bool), f x = x) ->

forall (b : bool), f (f b) = b.

Proof.

intros f x b.

rewrite x.

reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 10. Demostrar que

forall (b c : bool),

(andb b c = orb b c) -> b = c.

------------------------------------------------------------------ *)

(* angruicam1 alerodrod5 *)

Theorem andb_eq_orb :

forall (b c : bool),

(andb b c = orb b c) ->

b = c.

Proof.

intros [] c.

- simpl. intros H. rewrite H. reflexivity.

Qed.

(* ---------------------------------------------------------------------

Ejercicio 11. En este ejercicio se considera la siguiente

representación de los números naturales

Inductive nat2 : Type :=

| C : nat2

| D : nat2 -> nat2

| SD : nat2 -> nat2.

donde C representa el cero, D el doble y SD el siguiente del doble.

Definir la función

nat2Anat :: nat2 -> nat

tal que (nat2Anat x) es el número natural representado por x.

Demostrar que

nat2Anat (SD (SD C)) = 3

nat2Anat (D (SD (SD C))) = 6.

------------------------------------------------------------------ *)

(* angruicam1, alerodrod5 *)

Inductive nat2 : Type :=

| C : nat2

| D : nat2 -> nat2

| SD : nat2 -> nat2.

Fixpoint nat2Anat (x:nat2) : nat :=

match x with

| C => O

| D n => 2*nat2Anat n

| SD n => (2*nat2Anat n)+1

end.

Example prop_nat2Anat1: (nat2Anat (SD (SD C))) = 3.

Proof. reflexivity. Qed.

Example prop_nat2Anat2: (nat2Anat (D (SD (SD C)))) = 6.

Proof. reflexivity. Qed.

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