TaPL                      #2       2012          2    18()            TaPL       #2         2012   2   18   1 / 34
.. . Mathematical Preliminaries  1     Sets, Relations, and Functions     Orderd Sets     Sequences     Induction         ...
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()   TaPL   #2   2012   2   18   6 / 34
2.1.1 Notation of Sets.{. . . }                                                       ....                                ...
2.1.2 Natural Number .                                                                          ..natural number . N : {0,...
2.1.3 n-Place Relation . n-place relation                                                 ...      S1 , . . . , Sn        ...
2.1.4 Predicate .                                                                    ..Predicate . ..  S one-place relatio...
2.1.5 Binary Relation . Binary relation                                                 ... binary relation ..            ...
2.1.6 more notation               9             3 ex. Γ   s:T       Γ, s, T       typing relation         ()              ...
2.1.7 Domain, Range     S T         R .                                                          ..domain . dom(R) = { s ∈...
2.1.8 Partial Function, TotalFunction . partial function                                            ... s ∈ S, t1 ∈ T, t2 ...
2.1.9 Defined, Undefined .                                                                             ..defined .     S     ...
2.1.10 Preserved . preserved                                                          ... binary relation R         S     ...
2.2.1 Property of Binary Relation S         binary relation R .                                                           ...
2.2.2 Preorder, Partial Order, TotalOrder . preorder                                                                ... pr...
2.2.2 Preorder, Partial Order, TotalOrder . partial order                                            ... partial order .. ...
2.2.3 Join, Meet ≤        S        partial order          s∈S   t∈S . join least upper bound                              ...
2.2.4 Equivalence . equivalence                                                     ... S       R equivalence             ...
2.2.5 .                                                                  ..reflexive closure . R ..              reflexive  ...
2.2.6 Exercise 2.2.6    S          R                     R             R = R ∪ { (s, s) | s ∈ S }R   R   reflexive closure ...
2.2.7 Exercise 2.2.7 -moreconstructive definition of transitiveclosure-RiR0 = RRi+1 = Ri ∪ { (s, u) | ∃t ∈ R.(s, t) ∈ Ri ∧ ...
2.2.8 Exercise 2.2.8S       binary relation   R R          preserved          S      predicate P                     P R∗ ...
2.2.9 Decreasing Chain.S          preorder        ≤                                       .si ∈ S               ∀ i ∈ N . ...
2.2.10 Well Founded.S     preorder        ≤                                               .leq         decresing chain    ...
2.3.1 Sequences.sequence              “,”                                       .                        “,”     Cons   Ap...
2.4.1 AXIOM: Ordinary Inductionon N           P (0)   ∀i ∈ N. P (i) → P (i + 1)                    ∀n ∈ N. P (n)      ()  ...
2.4.2 AXIOM: Complete Inductionon N           (∀i ∈ N, i < n. P (i)) → P (n)                  ∀n ∈ N. P (n)      ()       ...
2.4.3 Lexicographic Order(Dictionary Order)(m, n) ≤ (m , n )  ⇔       m < m or (m = m and n ≤ n )        ()          TaPL ...
2.4.4 AXIOM: LexicographicInduction (∀m , n ∈ N, (m , n ) < (m, n). P (m , n )) → P (m, n)                 ∀m, n ∈ N. P (m...
2.4.4 Lexicographic Induction   Lexicographic Induction    nested induction                                 ”by an inner  ...
()   TaPL   #2   2012   2   18   34 / 34
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TaPL名古屋 Chap2

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TaPL名古屋 Chap2

  1. 1. TaPL #2 2012 2 18() TaPL #2 2012 2 18 1 / 34
  2. 2. .. . Mathematical Preliminaries 1 Sets, Relations, and Functions Orderd Sets Sequences Induction () TaPL #2 2012 2 18 2 / 34
  3. 3. () TaPL #2 2012 2 18 3 / 34
  4. 4. () TaPL #2 2012 2 18 4 / 34
  5. 5. () TaPL #2 2012 2 18 5 / 34
  6. 6. () TaPL #2 2012 2 18 6 / 34
  7. 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}} () TaPL #2 2012 2 18 7 / 34
  8. 8. 2.1.2 Natural Number . ..natural number . N : {0, 1, 2, 3, . . . } .. . . . . ..countable . N 1 1 countable .. . . . N ex. etc. () TaPL #2 2012 2 18 8 / 34
  9. 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 () TaPL #2 2012 2 18 9 / 34
  10. 10. 2.1.4 Predicate . ..Predicate . .. S one-place relation P S predicate . . . s∈S s∈P P s λs.P (s) S () TaPL #2 2012 2 18 10 / 34
  11. 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 () TaPL #2 2012 2 18 11 / 34
  12. 12. 2.1.6 more notation 9 3 ex. Γ s:T Γ, s, T typing relation () TaPL #2 2012 2 18 12 / 34
  13. 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 } .. . . . () TaPL #2 2012 2 18 13 / 34
  14. 14. 2.1.8 Partial Function, TotalFunction . 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 . . . () TaPL #2 2012 2 18 14 / 34
  15. 15. 2.1.9 Defined, Undefined . ..defined . S T partial function R s∈S s∈R R s defined defined undefinedf (χ) ↑ f (χ) =↑ f χ undefined.. (χ) ↓f. defined . . ( ) exception from S to T ∪ {f ail} () TaPL #2 2012 2 18 15 / 34
  16. 16. 2.1.10 Preserved . preserved ... binary relation R S predicate P sRs P (s) P (s ) P R .. preserved . . . () TaPL #2 2012 2 18 16 / 34
  17. 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 .. . . . () TaPL #2 2012 2 18 17 / 34
  18. 18. 2.2.2 Preorder, Partial Order, TotalOrder . preorder ... preorder R reflexive transitive preorder R ≤ preorderd set S S preorder R .. . . . < s≤t∧s=t () TaPL #2 2012 2 18 18 / 34
  19. 19. 2.2.2 Preorder, Partial Order, TotalOrder . partial order ... partial order .. preorder antisymmetric . . . . ..total order . total order .. partial order ∀ s, t ∈ S . s ≤ t ∨ t ≤ s . . . () TaPL #2 2012 2 18 19 / 34
  20. 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 . . () TaPL #2 2012 2 18 20 / 34
  21. 21. 2.2.4 Equivalence . equivalence ... S R equivalence R reflexive transitive symmetric .. . . . () TaPL #2 2012 2 18 21 / 34
  22. 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 () TaPL #2 2012 2 18 22 / 34
  23. 23. 2.2.6 Exercise 2.2.6 S R R R = R ∪ { (s, s) | s ∈ S }R R reflexive closure () TaPL #2 2012 2 18 23 / 34
  24. 24. 2.2.7 Exercise 2.2.7 -moreconstructive definition of transitiveclosure-RiR0 = RRi+1 = Ri ∪ { (s, u) | ∃t ∈ R.(s, t) ∈ Ri ∧ (t, u) ∈ Ri } R+ = Ri i () TaPL #2 2012 2 18 24 / 34
  25. 25. 2.2.8 Exercise 2.2.8S binary relation R R preserved S predicate P P R∗ preserved () TaPL #2 2012 2 18 25 / 34
  26. 26. 2.2.9 Decreasing Chain.S preorder ≤ .si ∈ S ∀ i ∈ N . si+1 < sis1 , s2 , s3 , . . ... ≤ decreasing chain. . . ex. ”5, 4, 3, 2, 1” () TaPL #2 2012 2 18 26 / 34
  27. 27. 2.2.10 Well Founded.S preorder ≤ .leq decresing chain ≤ wellfounded... . . ex.N < well founded (0 < 1 < 2 < . . . ) ex.R not well founded (· · · < −1 < 0 < 1 < . . . ) () TaPL #2 2012 2 18 27 / 34
  28. 28. 2.3.1 Sequences.sequence “,” . “,” Cons Append... . ..1..n 1 n sequence .... . ..|a| sequence a .... . ..• sequence .... . . () TaPL #2 2012 2 18 28 / 34
  29. 29. 2.4.1 AXIOM: Ordinary Inductionon N P (0) ∀i ∈ N. P (i) → P (i + 1) ∀n ∈ N. P (n) () TaPL #2 2012 2 18 29 / 34
  30. 30. 2.4.2 AXIOM: Complete Inductionon N (∀i ∈ N, i < n. P (i)) → P (n) ∀n ∈ N. P (n) () TaPL #2 2012 2 18 30 / 34
  31. 31. 2.4.3 Lexicographic Order(Dictionary Order)(m, n) ≤ (m , n ) ⇔ m < m or (m = m and n ≤ n ) () TaPL #2 2012 2 18 31 / 34
  32. 32. 2.4.4 AXIOM: LexicographicInduction (∀m , n ∈ N, (m , n ) < (m, n). P (m , n )) → P (m, n) ∀m, n ∈ N. P (m, n) () TaPL #2 2012 2 18 32 / 34
  33. 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 () TaPL #2 2012 2 18 33 / 34
  34. 34. () TaPL #2 2012 2 18 34 / 34
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