Formal Methods in Computer Science CS1502 Proofs with Boolean Connectives

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Formal Methods in Computer ScienceCS1502Proofs
with Boolean Connectives

• Patchrawat Uthaisombut
• University of Pittsburgh

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Goals

• To learn 4 simple proof rules
• /\ Elim, /\ Intro, \/ Intro, Reit
• To learn about subproof.
• To learn the method of proof by cases
• \/ Elim
• To gain skills in constructing proofs using proof
  by cases as well as simple proof rules.

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Rules learned so far
Line 1 says ab. In line 3, we replace the first
a in line 2 with a b.
If ab, anything that is true with a is also true
with b. We can replace a with b (but not vice
versa).
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Motivation

• Why do we need proofs? Truth table method
  already works fine.
• If there are many atomic sentences, a large truth
  table is needed.
• Truth table method does not work when we
  introduce quantifiers ?, ?. Methods of proof can
  be extended to accommodate this.

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Simple Rules (Cut-and-Paste Rules)

• /\ Elim /\ Intro \/ Intro
• P1 /\ P2 /\ P3 P1 Pi
• P2
• P3
• P1 P1 /\ P2 /\ P3 P1 \/ P2 \/ P3

(or P2 or P3)
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Reit(eration)

• Reit
• P
• P

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Simple Rules (Cut-and-Paste Rules)

• /\ Elim
• Select the conjunct that you want
• /\ Intro
• Combine multiple sentences into one
• \/ Intro
• Add extra stuff to a sentence

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Example Cut and Paste Rules

• A /\ B
• C
• A \/ (B /\ C)

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Exercise Cut and Paste Rules

• (A \/ B) /\ C (A \/ B) /\ C
• A /\ D A /\ D
• C /\ A B /\ D
• Is each of the arguments valid?
• Give a proof or a counterexample.

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Goals

• To learn 4 simple proof rules
• /\ Elim, /\ Intro, \/ Intro, Reit
• To learn about subproof.
• To learn the method of proof by cases
• \/ Elim
• To gain skills in constructing proofs using proof
  by cases as well as simple proof rules.

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Example Proof by cases

• (Cube(a) /\ Small(a)) \/ (Cube(a) /\ Large(a))
• Cube(a)
• It is given that (Cube(a) /\ Small(a)) \/
  (Cube(a) /\ Large(a)).
• Well break it into 2 cases.
• First, assume that (Cube(a) /\ Small(a)) holds.
• Then it implies that Cube(a) holds.
• In the second case, we assume that Cube(a) /\
  Large(a) holds.
• Thus, it follows that Cube(a).
• In either case, we have Cube(a), as desired.

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Subproofs

• A subproof is a proof that occurs within the
  context of a larger proof.
• The assumption of a subproof can be used only in
  the subproof. It cannot be used outside the
  scope of the subproof.
• This is similar to a local variable in a
  subroutine.

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\/ Elim (Proof by cases)
P1 or P2 is true

• P1 \/ P2
• P1
• S
• P2
• S
• S

S follows from P1
S follows from P2
Since P1 or P2 is true, then S is true.
PAUSE
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Example \/ Elim (Proof by cases)

• 1. (A /\ B) \/ (C /\ A)
• 2. A /\ B
• 3. A /\ Elim 2
• 4. C /\ A
• 5. A /\ Elim 4
• 6. A \/ Elim 1, 2-3, 4-5

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Example \/ Elim with \/ Intro

• 1. (A /\ B) \/ (C /\ D)
• 2. A /\ B
• 3. B /\ Elim 2
• 4. B \/ D \/ Intro 3
• 5. C /\ D
• 6. D /\ Elim 5
• 7. B \/ D \/ Intro 6
• 8. B \/ D \/ Elim 1, 2-4, 5-7
• Use \/ Intro to put conclusion in the right form

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Example \/ Elim with Reit

• 1. (A /\ B) \/ A
• 2. A /\ B
• 3. A /\ Elim 2
• 4. A
• 5. A Reit 4
• 6. A \/ Elim 1, 2-3, 4-5
• What if the conclusion is A \/ B?

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Exercises \/ Elim

• (A /\ B) \/ C
• A \/ B \/ C
• (A /\ B) \/ (B /\ C) \/ (C /\ A)
• B \/ C

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Exercise

• What can be concluded from this premise?
• What cannot?
• Prove or give counterexample.
• (A /\ (B \/ C)) \/ (C /\ (A \/ B))
• ???

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Exercise (more detail)

• Which of these are consequences of(A /\ (B \/
  C)) \/ (C /\ (A \/ B))?
• Prove or find a counterexample
• A
• B
• C
• A \/ B
• A \/ C
• B \/ C
• A \/ B \/ C

• At least 2 of the 3 variables are true.
• How do you express this in logic?