# [Solved] CIS301 Homework #4-Propositional Proofs with negation

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Purpose of Assignment:

1. natural deduction rules for propositional logic
2. how to apply the rules for proving validity of sequents
3. some properties of propositional logic connectives and how they can be proven using natural deduction

We have created some automated grading tools to speed the grading of homeworks. To apply those tools, we need to make sure that each student uses a consistent naming for all their solutions file. Therefore, we have created an IntelliJ project with template files for your solution. DON’T CHANGE THE NAME OF ANY OF THE FILES that we give you.

# Hints:

If you get stuck in a proof, take a look at the tactics given in the lecture slides associated with the operators involved in the formulae.

Typically, you’ll introduce from the inside-out and eliminate from the outside-in

If you can’t build something directly, use `pbc`

# Getting started

You can find examples of completed Logika propositional proofs in the Logika example repository (included in the class examples that you downloaded (as illustrated in the “Step #2” videos). Here is a direct link to the propositional proofs portion of those examples:https://github.com/sireum/v3-logika-examples/tree/release/src/propositional

# Considerations

1. All files must run in the Sireum IVE
2. “Proof” means “a Logika 2 column proof”. And “Prove” means “provide a proof”
3. To receive any points a problem must:
• be a Logika Propositional Proof
• contain the correct logical sequent
1. Partial credit may be received if the proof is not finished.
2. Correctly proven claims that do not progress the proof will not impact your grade
3. Each provable sequent can be proven in at most 25 steps.
4. Point values are proportional to difficulty.

# Problems

2. (5 points) As it got later in the night, Stan and Lacy got some coffee and continued their discussion, Lacy found it easier to think about requirement (1) as follows:
• If the system is not in maintenance mode, then we really don’t care about the queue. Just the fact that the system is not in maintenance mode will make the requirement true.
• However, if the the system is in maintenance mode (i.e., if it is not the case that it is not in maintenance mode), then we need to have the queue to be empty.Again, using the our proposition mapping above, we would havep = System is in Maintenance mode q = Operational Tasking queue is emptyand Lacy’s idea would be formalized as¬p v qHelp Stan and Lacy confirm that Lacy’s alternative statement of the requirement is appropriate by giving a proof demonstrating that the following sequent holds¬p ∨ q ⊢ p → qi.e., the original requirement is entailed by (follows from) Lacy concept. (Note that if we wanted to show that the two versions of the requirement were equivalent, we would would need to show p → q ⊢ ¬p ∨ q, but that is not part of this question).
1. (5 points) Now Stan and Lacy were trying to come up with some test scenarios for their systemThey want to explore what the system should do if it was not the case that… …The system is in maintenance mode OR … The Operational Tasking queue is empty OR … The MAIN task is in IDLE modeFred, who is always trying to simplify things stuck his head and the door and said, “Well, that scenario would be more directly captured as
• It is not the case that the system is in maintenance mode and
• it is not the case that the operational tasking queue is empty and
• it is not the case that the MAIN task is in IDLE mode (which could also be stated as “an operational task is executing”)”

Show that Fred is on to something by proving the following sequent¬(p ∨ q ∨ r) ⊢ ¬p ∧ ¬q ∧ ¬r(Note that we should also prove the reverse direction, but you don’t have to consider that for this assignment).

2. (5 points) Fred had a report due the following morning, but he was sick of working on it, so he decided to sit in with Stan and Lacy for a while.Stan, Lacy, and Fred started talking about another test scenario…It was not the case
• The system is in maintenance mode AND
• The Operational Tasking queue is empty AND
• The MAIN task is in IDLE mode

They decided that if that scenario held, it would be the case that…

• The system is NOT in maintenance mode OR
• The Operational Tasking queue is NOT empty OR
• It is not the case that the MAIN tasks is in IDLE mode

Show that they are correct by proving the following sequent…¬(p ∧ q ∧ r) ⊢ ¬p ∨ ¬q ∨ ¬r

3. (6 points) Ritesh was getting a Red Bull from the frig in company meeting room and overheard the conversation. Ritesh had found an old logic puzzle book at a library book sale, and he said “Since you guys are so into logic, you should try to prove that the following sequents. These are formulas that are tautologies: you don’t have any premises, the formulas should always hold true.”⊢ (p ∨ ¬p) ∧ ¬(p ∧ ¬p)Something to think about: can you think of a short intuitive explanation for why this formula is a tautology based on the semantics of “->”?
4. (6 points) Ritesh’s second brain teaser was…⊢ (p → q) ∨ (q → r)Something to think about: can you think of a short intuitive explanation for why this formula is a tautology based on the semantics of “->”?

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[Solved] CIS301 Homework #4-Propositional Proofs with negation
10 USD \$