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Proof Tactics, Strategies and Derived Rules
CS 270 Math Foundations of CS Jeremy Johnson
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Outline Review Rules Positive subformulas and extraction Proof tactics
Extraction, Conversion, Inversion, Division, and Refutation Finding contradictions Proof strategy Search tree and an algorithm to find a proof Derived rules
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Conjunction Rules Introduction Rule Elimination Rule I
I ER EL
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Implication Rules Introduction Rule Assume and show
Elimination Rule (Modus Ponens) … I E
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Disjunction Rules Introduction Rule
Elimination Rule (proof by case analysis) IR IL … … E
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Negation Rules Introduce the symbol (⊥ =bottom) to encode a contradiction Bottom elimination ⊥ can prove anything Bottom introduction ⊥ ⊥E ⊥I ⊥
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Negation Rules Introduction and elimination rules Double negation …
⊥ I … ⊥ E E
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] … 1 p p goal
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] 1 (p p) assumption … 2 ⊥ goal 3 p p ¬E 2
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] 1 (p p) assumption … 2 p p Goal 3 ⊥ ⊥I 1,2 4 ¬E 3
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] 1 (p p) assumption … 2 p Goal 3 p p ∨IR 2 4 ⊥ ⊥I 1,3 5 ¬E 4
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] 1 (p p) assumption 2 𝑝 … 3 ⊥ Goal 4 p ¬E 3 5 p p ∨IR 4 6 ⊥I 1,5 7 ¬E 6
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Law of the Excluded Middle
𝑝∨¬𝑝 [derived rule LEM] 1 (p p) assumption 2 𝑝 3 (p p) ∨IL 2 4 ⊥ ⊥i 3,1 5 p ¬E 4 6 p p ∨IR 5 7 ⊥I 6,1 8 ¬E 7
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Search Tree ? P P IR IL E ? P ? P (P P) ? ⊥
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Search Tree P ? ⊥ E ? P ? P (P P) ? ⊥ IL IR E ? P P
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Search Tree P ? P ⊥I The only possible contradictory pair is P and P
IL ? P E P ? ⊥ ⊥I P ? P The only possible contradictory pair is P and P and P ? P can only be addressed by E and we are back where we started. backtrack
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Search Tree ? P P IR IL E ? P ? P (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 Extraction Conversion Inversion Division 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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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 M assumption 3 M O ∨IR 1 4 M O Df I 5 M E1,4 6 ⊥ ⊥I2,5 7 ¬E 6
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Derived Rules Commutative rules Associative rules Idempotence rules
Associative rules ( ) ( ) ( ) ( ) Idempotence rules and and
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Derived Rules Distributive rules Disjunctive syllogism
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Disjunctive syllogism ( ), Cut (resolution) ( ), ( ) ( )
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Derived Rules DeMorgan’s rules ( ) ( )
( ) ( )
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Derived Rules Modus Tollens Transposition Hypothetical Syllogism
( , ) Transposition ( ) ( ) Hypothetical Syllogism ( , ) ( ) Exportation and Importation (( ) ) ( ( )) ( ( )) (( ) )
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Derived Rules Definition of conditional Negated 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 2 P assumption 3 Q DSL 1,2 4 P Q I3 1
P P LEM 3 P assumption 4 Q E1,3 5 P Q IL4 6 P 7 IR6 8 E2,5,7
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