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CSE-321 Programming Languages
Inductive Proofs
박성우
POSTECH
March 19, 2007
Inductive Definitions of Syntactic Categories
• Natural numbers
• Regular binary trees
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Inductive Definitions of Judgments
• Judgment
• Inference rules
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Even and Odd Numbers
• Judgments
• Inference rules
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Derivable Rule and Admissible Rule
• Derivable rule
• Admissible rule
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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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Outline
• Inductive proofs
– Structural induction
– Rule induction
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Structural Induction
• 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).
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Structural Ind. ¼ Mathematical Ind.
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Structural Induction on Trees
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Example
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Structural Induction on mparen
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Here is the first theorem
we prove in this course!
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Outline
• Inductive proofs
– Structural induction V
– Rule induction
• similar to structural induction, but applied to
derivation trees
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Rule Induction
• A judgment J with inference rules:
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Example
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How Rule Induction Works
• A judgment J with two inference rules:
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mparen and lparen
• From
• We obtain
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Sometimes we need a lemma
if a direct proof attempt fails.
• But it is not of the form
"If J holds, then P(J) holds."
• Trick: prove instead
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Current Schedule
March 2007
Su Mo Tu We Th
1
4 5 6 7 8
11 12 13 14 15
18 19 20 21 22
A2
25 26 27 28 29
Q1
A3
Fr
2
9
16
23
Sa
3
10
17
24
30 31
April 2007
Su Mo Tu We Th Fr Sa
1 2 3 4 5 6 7
8 9 10 11 12 13 14
Q2
A4
15 16 17 18 19 20 21
22 23 24 25 26 27 28
<midterm>
29 30 1 2
A5
27
New Schedule?
March 2007
Su Mo Tu We Th
1
4 5 6 7 8
11 12 13 14 15
18 19 20 21 22
A2
25 26 27 28 29
Q1
Fr
2
9
16
23
Sa
3
10
17
24
30 31
April 2007
Su Mo Tu We Th
1 2 3 4 5
A3
8 9 10 11 12
Q2
15 16 17 18 19
A4
22 23 24 25 26
<midterm>
29 30 1 2
A5
Fr Sa
6 7
13 14
20 21
27 28
28