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Published byErnest Lawrence Modified over 9 years ago
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Inductive Proofs Kangwon National University 임현승 Programming Languages These slides were originally created by Prof. Sungwoo Park at POSTECH.
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2 Natural numbers Regular binary trees Inductive Def. of Syntactic Categories
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3 Judgment Inference rules Inductive Definitions of Judgments
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4 Even and Odd Numbers Judgments Inference rules
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5 Derivable Rule and Admissible Rule Derivable rule Admissible rule
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6 But... What is the point of specifying a system and doing nothing else? –E.g., why do we define the two judgments n even and n odd at all? What if the definition is wrong? –E.g., what if we mistakenly introduced the rule: So we need "inductive proofs."
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7 Outline Inductive Proofs –Structural Induction –Rule Induction
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8 Proves a property of a syntactic category by analyzing the structure of its definition. I want to prove P(n) for every natural number n. –Examples of P(n) n has a successor. n is 0 or has a predecessor n'. n is a product of prime numbers. n is even (which cannot be proven). Structural Induction
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9 Structural Ind. ¼ Mathematical Ind.
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12 Structural Induction on Trees
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13 Example
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14 Structural Induction on mparen
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Here is the first theorem we prove in this course!
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17 Outline Inductive Proofs –Structural Induction V –Rule Induction similar to structural induction, but applied to derivation trees
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18 Rule Induction A judgment J with inference rules:
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19 Example
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20 How Rule Induction Works A judgment J with two inference rules:
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23 mparen and lparen From We obtain
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Sometimes we need a lemma if a direct proof attempt fails.
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26 But it is not of the form "If J holds, then P(J) holds." Trick: prove instead
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