Proof Tactics, Strategies and Derived Rules

CS 270 Math Foundations of CSJeremy Johnson

Outline

1. Review Rules2. Positive subformulas and extraction3. Proof tactics• Extraction, Conversion, Inversion, Division,

and Refutation• Finding contradictions

4. Proof strategy• Search tree and an algorithm to find a proof

5. Derived rules

Conjunction Rules

• Introduction Rule

• Elimination Rule

I

ER

EL

Implication Rules

• Introduction Rule

Assume and show

• Elimination Rule (Modus Ponens) E

… I

Disjunction Rules

• Introduction Rule

• Elimination Rule (proof by case analysis)

IR

E

IL

…

…

Negation Rules

• Introduce the symbol ( to encode a contradiction

• Bottom elimination can prove anything

• Bottom introduction

I

Negation Rules

• Introduction and elimination rules

• Double negation E

… I

… E

Law of the Excluded Middle

• [derived rule LEM]

…

1 p p

Law of the Excluded Middle

• [derived rule LEM]

1 (p p) assumption

…

2

3 p p

Law of the Excluded Middle

• [derived rule LEM]

1 (p p) assumption

…

2 p p

3

4 p p

Law of the Excluded Middle

• [derived rule LEM]

1 (p p) assumption

…

2 p Goal

3 p p

4

5 p p

Law of the Excluded Middle

• [derived rule LEM]

1 (p p) assumption

2 assumption

…

3 Goal4 p

5 p p

6

7 p p

Law of the Excluded Middle

• [derived rule LEM]

1 (p p) assumption

2 assumption

3 (p p)

4 3,15 p 4

6 p p

7 6,1

8 p p 7

Search Tree

? P P? P P

IR IL E

? P ? P (P P) ?

Search Tree

? P P? P P

IR IL E

? P ? P (P P) ?

E

P ?

Search Tree

IL

? P

E

P ?

P ?

The only possible contradictory pair is P and Pand P ? can only be addressed by E and we are back where we started.

backtrack

Search Tree

? P P? P P

IR IL E

? P ? P (P P) ?

Proof Tactics

• Systematically search for a proof• Apply (,,) elimination rules forward• Apply introduction rules backwards• No extraneous steps• Backtrack when dead-end reached

1. Extraction2. Conversion3. Inversion4. Division5. Refutation

Positive Subformulas

• PS()• If is an atom return • If = return • If = then return PS() PS() • If = then return PS() PS() • If = then return PS()

Extraction

• Apply elimination rules forward in order to extract goals that occur as positive subformulae of the formulae on available lines.

Conversion

• Use disjunction elimination in order to obtain goal disjunctions.

Inversion

• Invert non-atomic goals by applying introduction rules backward to them.

Division

• Use disjunction elimination on any goals for which the previous three tactics have either not applied, or not been successful.

Refutation

• Apply negation elimination backward to goals that cannot be obtained by any other means.

Possible Contradictions

• Form a list of all negations that appear as a positive subformulas of all premises and available assumptions.

• Pair each negation with its immediate subformula .

• These pairs are the only possible contradictions that must be considered.

Exercise

• Prove the definition of conditional• ( ) • ( )

Deadend

Solution

Solution

Algorithm

Using Derived Rules

• Once you have proven a rule from the basic rules you may use it in your proofs• Derive M from (M O) M

1 (M O) M premise

2 M assumption

3 M O

4 M O Df I

5 M E1,46 I2,5

7 M 6

Derived Rules

• Commutative rules• •

• Associative rules• ( ) ( ) • ( ) ( )

• Idempotence rules• and • and

Derived Rules

• Distributive rules• ( ) ( ) ( )• ( ) ( ) ( ) • ( ) ( ) ( )• ( ) ( ) ( )

• Disjunctive syllogism• ( ),

• Cut (resolution)• ( ), ( ) ( )

Derived Rules

• DeMorgan’s rules• ( ) • ( ) • ( ) • ( )

Derived Rules

• Modus Tollens• ( , )

• Transposition• ( ) ( )

• Hypothetical Syllogism• ( , ) ( )

• Exportation and Importation• (( ) ) ( ( )) • ( ( )) (( ) )

Derived Rules

• Definition of conditional• ( ) • ( )

• Negated conditional• ( ) • ( )

Exercise

• Prove the definition of the conditional using Disjunctive Syllogism and LEM

Solution1 P Q premise

2 P assumption

3 Q DSL 1,2

4 P Q I3

1 P Q premise

2 P P LEM

3 P assumption

4 Q E1,3

5 P Q IL4

6 P assumption

7 P Q IR6

8 P Q E2,5,7