Library HW1

Homework Assignment 1
Interactive Computer Theorem Proving
CS294-9, Fall 2006
UC Berkeley

Submit your solution file for this assignment by e-mail with the subject "ICTP HW1" by the start of class on September 7. You should write your solutions entirely on your own, since that's the only way to develop the intuitions needed to do this kind of stuff!

OK, to start, grab the template source file.

Meta-note: Since this is the first time we're offering a class like this, any feedback you can provide on this and future homework assignments would be very helpful. If something is too hard or too easy, we'd love to hear about it in a comment in your solution file. (Comments are the things bracketed between parentheses and stars.)

Some useful tactics

Here is some support code to help you build proofs using the same natural deduction terminology that we used in class. It's safe to skip over these.

One potential gotcha that's important to remember: For tactics that take arguments, when one of the arguments contains spaces, enclose it in parentheses to avoid scary error messages.

Ltac elimer e := try exact e; let H := fresh "H" in (generalize e; intro H).

Ltac true_i := elimer I.

Ltac false_e Hfalse := destruct Hfalse.

Ltac imp_i := intro.
Ltac imp_e Himp Hant := elimer (Himp Hant).

Ltac and_i := split.
Ltac and_e1 Hand := elimer (proj1 Hand).
Ltac and_e2 Hand := elimer (proj2 Hand).

Ltac or_i1 := left.
Ltac or_i2 := right.
Ltac or_e Hor := destruct Hor.

Ltac not_i := intro.
Ltac not_e Hno Hyes := destruct (Hno Hyes).

Ltac forall_i := intro.
Ltac forall_e Hall x := elimer (Hall x).

Ltac exists_i x := exists x.
Ltac exists_e Hex := destruct Hex.

Plus the predefined tactics assumption, assert, reflexivity, and rewrite

Propositional Logic

We open a Section, which is a mechanism for controlling the scoping of variables and assumptions.
Section propositional.
We want to prove some theorems about arbitary propositions, so we declare three propositional variables, scoped within this Section only.
   Variables P Q R : Prop.

OK, here's where the interactive part starts. Prove these simple theorems in propositional logic. Use a natural deduction style, by way of the tactics defined in the beginning of this source file. They're the same that we used in class. Coq has lots of other tactics available, but we'll consider them off-limits just for this assignment.

Each problem is given in the form: Theorem name : formula. Admitted.

This essentially states the theorem as an axiom. Your job is to make it a bonafide theorem by replacing Admitted with a proof script and Qed, as in this example:

  Theorem grand_challenge_problem : P -> P.

Truth is true.
  Theorem truth : True.

False implies anything
  Theorem contradiction_P : False -> P.

/\ is associative.
  Theorem and_associates : (P /\ Q) /\ R -> P /\ (Q /\ R).

Dropping conjuncts preserves truth.
  Theorem cut_out_the_middle_man : P /\ Q /\ R -> P /\ R.

One or the other is true! Can you figure out which?
  Theorem make_up_your_mind : True \/ False.

\/ is commutative.
  Theorem or_commutes : P \/ Q -> Q \/ P.

\/ is associative.
  Theorem or_associates : (P \/ Q) \/ R -> P \/ (Q \/ R).

Something is fishy about the hypotheses, so we can deduce anything.
  Theorem something_fishy : (P -> Q) -> P -> ~Q -> R.

The law of contraposition holds only in this direction in constructive logic.
  Theorem contrapositive : (P -> Q) -> ~Q -> ~P.
End propositional.

First-order logic

Section firstorder.
We'll prove some theorems that draw their individuals an unspecified set A.
   Variable A : Set.

Here's an arbitrary individual in A.
   Variable e : A.

Here's a unary function symbol.
   Variable f : A -> A.

Here are two unary predicate symbols.
   Variables P P' : A -> Prop.

Here's a binary predicate symbol.
   Variable Q : A -> A -> Prop.

Prove the following theorems using only the natural deduction tactics introduced in class:

If P holds for any value, then it must hold for f e.
  Theorem indeed : (forall x : A, P x) -> P (f e).

If P holds for anything f returns, then P holds of something.
  Theorem ofcourse : (forall x : A, P (f x)) -> (exists y : A, P y).

Swapping exists and forall in this way is always legal.
  Theorem swap_quantifiers :
    (exists x : A, forall y : A, Q x y)
    -> (forall y : A, exists x : A, Q x y).

If P maps everything to false, then we can't find anything it maps to true.
  Theorem nobody :
    (forall x : A, ~(P x))
    -> ~(exists x : A, P x).

We can push a conjunction inside universal quantifications.
  Theorem coalesce1 : (forall x : A, P x) /\ (forall x : A, P' x)
    -> (forall x : A, P x /\ P' x).

We can push a universal quantification inside a conjunction.
  Theorem coalesce2 : (forall x : A, P x /\ P' x)
    -> (forall x : A, P x) /\ (forall x : A, P' x).

This one doesn't have any deeper meaning. :-)
  Theorem one_or_the_other :
    (forall x : A, P x -> P' e)
    -> ((forall y : A, P' y) \/ (exists z : A, P z))
    -> P' e.

A straightforward fact about equality and functions
  Theorem reverse_congruence : forall x : A, forall y : A,
    x = y -> f y = f x.

If the set A has only one member, then we can derive a particular equality.
  Theorem small_world : (exists x : A, forall y : A, x = y)
    -> e = f e.

A complicated theorem about equality and functions
  Theorem nasty : forall x : A, forall y : A, forall z : A,
    f (f x) = x
    -> f z = f x
    -> f y = f z
    -> f (f (f (f (f x)))) = f y.
End firstorder.

When you're done, you'll know that your solutions are all correct if there are no more occurrences of Admitted and you haven't erased any of the Theorem lines that were in the template file.

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