federicobruzzone / software-foundations

Solutions (in Coq) of the exercises in the software foundation books.

Software Foundations

Solutions (in Coq) of the exercises in the software foundation books.

Useful links

• Coq Library 8.9

• Coq Library 8.15

• Software foundations

  • Volume 1

Contributing

Contributions to this project are welcome! If you have any suggestions, improvements, or bug fixes, feel free to submit a pull request.

License

This repository is licensed under the GNU General Public License (GPL). Please review the license file provided in the repository for more information regarding the terms and conditions of the GPL license.

Contact

If you have any questions, suggestions, or feedback, do not hesitate to contact me.

software-foundations's Issues

Issue with double_injective_take2 theorem

There is the Theorem on the Vol. 1 - Tactics page

Theorem double_injective_take2 : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n m.
  (* [n] and [m] are both in the context *)
  generalize dependent n.
  (* Now [n] is back in the goal and we can do induction on
     [m] and get a sufficiently general IH. *)
  induction m as [| m' IHm'].
  - (* m = O *) simpl. intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) reflexivity.
    + (* n = S n' *) discriminate eq.
  - (* m = S m' *) intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) discriminate eq.
    + (* n = S n' *) f_equal.
      apply IHm'. injection eq as goal. apply goal. Qed.

Coq 8.17.1 fails on apply goal. command with the following error message:

In environment
m' : nat
IHm' : forall n : nat, double n = double m' -> n = m'
n, n' : nat
E : n = S n'
goal : n' + S n' = m' + S m'
Unable to unify "n' + S n' = m' + S m'" with
 "double n' = double m'".

It seems that there is typo in the theorem. I have fixed it cumbersome way:

Theorem double_injective_take2 : forall n m,
  double n = double m ->
  n = m.
Proof.
  intros n m.
  (* [n] and [m] are both in the context *)
  generalize dependent n.
  (* Now [n] is back in the goal and we can do induction on
     [m] and get a sufficiently general IH. *)
  induction m as [| m' IHm'].
  - (* m = O *) simpl. intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) reflexivity.
    + (* n = S n' *) discriminate eq.
  - (* m = S m' *) intros n eq. destruct n as [| n'] eqn:E.
    + (* n = O *) discriminate eq.
    + (* n = S n' *) apply f_equal.
      apply IHm'.
      injection eq as goal. 
      rewrite add_comm in goal. simpl in goal.
      symmetry in goal.
      rewrite add_comm in goal. simpl in goal. 
      symmetry in goal.
      apply S_injective in  goal.
      apply goal.      
Qed.