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Proof Tactics, Strategies and Derived Rules CS 270 Math Foundations of CS Jeremy Johnson
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Outline 1.Review Rules 2.Positive subformulas and extraction 3.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
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Conjunction Rules Introduction Rule Elimination Rule I ER EL
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Implication Rules Introduction Rule Assume and show Elimination Rule (Modus Ponens) E … I
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Disjunction Rules Introduction Rule Elimination Rule (proof by case analysis) IR E IL …… ……
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Negation Rules
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Introduction and elimination rules Double negation E
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Law of the Excluded Middle … 1 p p
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Law of the Excluded Middle 1 (p p) assumption … 2 3 p p
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Law of the Excluded Middle 1 (p p) assumption … 2 p p 3 4
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Law of the Excluded Middle 1 (p p) assumption … 2pGoal 3 p p 4 5
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Law of the Excluded Middle 1 (p p) assumption 2 … 3Goal 4p 5 p p 6 7
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Law of the Excluded Middle 1 (p p) assumption 2 3 (p p) 4 5p 6 p p 7 8
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Search Tree ? P P IR IL EE ? P ? P
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Search Tree ? P P IR IL EE ? P ? P EE
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Search Tree IL ? P EE
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Search Tree ? P P IR IL EE ? P ? P
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Proof Tactics Systematically search for a proof Apply ( , , ) elimination rules forward Apply introduction rules backwards No extraneous steps Backtrack when dead-end reached 1.Extraction 2.Conversion 3.Inversion 4.Division 5.Refutation
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Positive Subformulas PS( ) If is an atom return If = return If = then return PS( ) PS( ) If = then return PS( ) PS( ) If = then return PS( )
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Extraction Apply elimination rules forward in order to extract goals that occur as positive subformulae of the formulae on available lines.
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Conversion Use disjunction elimination in order to obtain goal disjunctions.
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Inversion Invert non-atomic goals by applying introduction rules backward to them.
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Division Use disjunction elimination on any goals for which the previous three tactics have either not applied, or not been successful.
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Refutation Apply negation elimination backward to goals that cannot be obtained by any other means.
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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.
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Exercise Prove the definition of conditional ( ) ( )
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Deadend
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Solution
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Algorithm
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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 MM assumption 3 M O 4 M ODf I 5M E1,4 6 7M
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Derived Rules Commutative rules Associative rules ( ) ( ) ( ) ( ) Idempotence rules and and
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Derived Rules Distributive rules ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Disjunctive syllogism ( ), Cut (resolution) ( ), ( ) ( )
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Derived Rules DeMorgan’s rules ( ) ( ) ( ) ( )
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Derived Rules Modus Tollens ( , ) Transposition ( ) ( ) Hypothetical Syllogism ( , ) ( ) Exportation and Importation (( ) ) ( ( )) ( ( )) (( ) )
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Derived Rules Definition of conditional ( ) ( ) Negated conditional ( ) ( )
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Exercise Prove the definition of the conditional using Disjunctive Syllogism and LEM
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Solution 1 P Q premise 2Passumption 3QDSL 1,2 4 P Q I3 1 P Q premise 2 P PP P LEM 3Passumption 4Q E1,3 5 P Q IL4 6 PP assumption 7 P Q IR6 8 P Q E2,5,7
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