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1. TaPL #2
2012 2 18
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2. .
. . Mathematical Preliminaries
1
Sets, Relations, and Functions
Orderd Sets
Sequences
Induction
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7. 2.1.1 Notation of Sets
.
{. . . } .
..
. .
.
.
{ x ∈ S | ... } .
..
. .
.
.
φ .
..
. .
.
.
ST { x | x∈S∧x ∈T }
.
..
. .
.
.
|S| S .
..
. .
.
.
P(S) S powerset S .
..
. .
.
ex. S = {1, 2}, P(S) = {φ, {1}, {2}, {1, 2}}
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8. 2.1.2 Natural Number
.
.
.natural number
.
N : {0, 1, 2, 3, . . . }
..
. .
.
.
.
.countable
.
N 1 1 countable
..
. .
.
N
ex. etc.
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9. 2.1.3 n-Place Relation
.
n-place relation .
..
S1 , . . . , Sn R
S1 × · · · × Sn n-place relation
..
. .
.
ex. S1 = {1, 3}, S2 = {2, 4}, R = {(1, 2), (1, 4), (3, 4)}
R <
=
(1, 2) R
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10. 2.1.4 Predicate
.
.
.Predicate
.
.. S one-place relation P S predicate
. .
.
s∈S s∈P P s
λs.P (s) S
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11. 2.1.5 Binary Relation
.
Binary relation .
..
binary relation
.. two-place relation
. .
.
(s, t) ∈ R sRt
U U binary relation U binary
relation R
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12. 2.1.6 more notation
9 3
ex. Γ s:T Γ, s, T typing relation
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13. 2.1.7 Domain, Range
S T R
.
.
.domain
.
dom(R) = { s ∈ S | (s, t) ∈ R }
..
. .
.
.
range (codomain) .
..
range(R) = { t ∈ T | (s, t) ∈ R }
..
. .
.
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14. 2.1.8 Partial Function, Total
Function
.
partial function .
..
s ∈ S, t1 ∈ T, t2 ∈ T, (s, t1 ) ∈ R, (s, t2 ) ∈ R
t1 = t2 R S T partial
function
..
. .
.
.
.
.total function
.
partial function dom(R) = S R S
.. T total function function
. .
.
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15. 2.1.9 Defined, Undefined
.
.
.defined
.
S T partial function R s∈S
s∈R R s defined
defined undefined
f (χ) ↑ f (χ) =↑ f χ undefined
.. (χ) ↓
f
.
defined
.
.
( )
exception from S to T ∪ {f ail}
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16. 2.1.10 Preserved
.
preserved .
..
binary relation R S predicate P
sRs P (s) P (s ) P R
.. preserved
. .
.
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17. 2.2.1 Property of Binary Relation
S binary relation R
.
.
.reflexive
.
∀s∈S . sRs
..
. .
.
.
symmetric .
..
∀ s, t ∈ S . s R t → t R s
..
. .
.
.
.
.transitive
.
∀
.. s, t, u ∈ S . s R t ∧ t R u → s R u
. .
.
.
antisymmetric .
..
∀ s, t ∈ S . s R t ∧ t R s → s = t
..
. .
.
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18. 2.2.2 Preorder, Partial Order, Total
Order
.
preorder .
..
preorder R reflexive transitive preorder R
≤
preorderd set S S preorder R
..
. .
.
< s≤t∧s=t
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19. 2.2.2 Preorder, Partial Order, Total
Order
.
partial order .
..
partial order
.. preorder antisymmetric
. .
.
.
.
.total order
.
total order
.. partial order ∀ s, t ∈ S . s ≤ t ∨ t ≤ s
. .
.
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20. 2.2.3 Join, Meet
≤ S partial order s∈S t∈S
.
join least upper bound .
..
j∈S s t join
...
1 s≤j∧t≤j
.. ..
2. ∀ k ∈S . s≤k∧t≤k∧j ≤k
. .
.
.
meet greatest lower bound .
..
m∈S s t meet
...
1 m≤s∧m≤t
.. ...
.
2 ∀ n∈S . n≤s∧n≤t∧n≤m
.
.
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21. 2.2.4 Equivalence
.
equivalence .
..
S R equivalence R reflexive
transitive symmetric
..
. .
.
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22. 2.2.5
.
.
.reflexive closure
.
R
.. reflexive R
. .
.
.
.
.transitive closure
.
R
.. transitive R R+
. .
.
.
.
.reflexive and transitive closure
.
R
.. reflexive transitive R∗
. .
.
R∈R R ∀
Ri ∈ R . R ⊆ Ri
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23. 2.2.6 Exercise 2.2.6
S R R
R = R ∪ { (s, s) | s ∈ S }
R R reflexive closure
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24. 2.2.7 Exercise 2.2.7 -more
constructive definition of transitive
closure-
Ri
R0 = R
Ri+1 = Ri ∪ { (s, u) | ∃t ∈ R.(s, t) ∈ Ri ∧ (t, u) ∈ Ri }
R+ = Ri
i
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25. 2.2.8 Exercise 2.2.8
S binary relation R R preserved S
predicate P P R∗
preserved
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26. 2.2.9 Decreasing Chain
.
S preorder ≤ .
si ∈ S ∀ i ∈ N . si+1 < si
s1 , s2 , s3 , . . .
.. ≤ decreasing chain
. .
.
ex. ”5, 4, 3, 2, 1”
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27. 2.2.10 Well Founded
.
S preorder ≤ .
leq decresing chain ≤ well
founded
..
. .
.
ex.N < well founded
(0 < 1 < 2 < . . . )
ex.R not well founded
(· · · < −1 < 0 < 1 < . . . )
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28. 2.3.1 Sequences
.
sequence “,” .
“,” Cons Append
..
. .
.
.
1..n 1 n sequence .
..
. .
.
.
|a| sequence a .
..
. .
.
.
• sequence .
..
. .
.
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29. 2.4.1 AXIOM: Ordinary Induction
on N
P (0) ∀i ∈ N. P (i) → P (i + 1)
∀n ∈ N. P (n)
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30. 2.4.2 AXIOM: Complete Induction
on N
(∀i ∈ N, i < n. P (i)) → P (n)
∀n ∈ N. P (n)
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31. 2.4.3 Lexicographic Order
(Dictionary Order)
(m, n) ≤ (m , n )
⇔ m < m or (m = m and n ≤ n )
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32. 2.4.4 AXIOM: Lexicographic
Induction
(∀m , n ∈ N, (m , n ) < (m, n). P (m , n )) → P (m, n)
∀m, n ∈ N. P (m, n)
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33. 2.4.4 Lexicographic Induction
Lexicographic Induction nested induction
”by an inner
induction” 3 4
3
4
Chapter3 Theorem 3.3.4 structural
induction
term
Chapter 21
1
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