対話的定理証明支援系 Agda の紹介です

1. Agda
@ikegami _ _
Proof Summit 2011
2011-09-25

2. Agda
•
• =
• =
•
• tactic

4. Agda
• tactic
• =
• =
• =
• Agda
• Agda

5. Universe
Unicode subscript
Set1 Set1

6. TAB

7. TAB

8. TAB

9. TAB

10. TAB

11. TAB

12. TAB

13. ℕ
ℕ
ℕ ℕ

17. ℕ ℕ
total

18. Hidden arguments

19. lambda

20. Unicode / ASCII
UTF8

21. mixfix
ℕ ℕ ℕ

22. mixfix

23. ℕ
∷ ℕ

24. ℕ
∷ ℕ

25. ℕ
∷ ℕ

26. ℕ

28. Agda
Agda
A B
A
A B
A B

29. Proposition as Set
•
• proof object
•

30. A B
•
•
•

31. :A A

38. Agda
Agda
A B
A
A B
A B

40. A
•
•
•

41. Agda
Agda
A B
A
A B
A B

42. disjoint union
⊎
⊎
⊎

43. Agda
Agda
A B
A
A B ⊎
A B

44. sigma

45. times

46. Agda
Agda
A B
A
A B ⊎
A B

47. • double negation elimination
•
• Law of the excluded middle
• ⊎
•
• etc.

48. ℕ ℕ

49. Agda

50. forall
ℕ
ℕ ℕ

51. forall
ℕ
ℕ
ℕ

52. Hidden argument
ℕ
m

53. Hidden argument
ℕ
m

55. let where

56. ()

57. wildcard

58. with

59. dot
ℕ
ℕ

60. dot
ℕ
ℕ
ℕ

61. dot
ℕ
ℕ
ℕ
m

62. dot
ℕ
ℕ
ℕ

63. dot
ℕ
ℕ
ℕ

64. dot
ℕ
ℕ
ℕ

65. dot
ℕ
ℕ
ℕ

66. mutual
Agda 2.2.12

67. mutual
Agda 2.2.12

68. mutual
Agda 2.2.12

69. postulate
Agda

70. postulate
Agda

71. postulate
Agda

72. parametrized module

73. Agda module
Ulf (2006) 30 …

74. module
lib/src/Data/Nat.agda
(add-to-list ʻagda2-include-dirs “.../lib/src”)
Agda .../lib/src

75. module
MyProject/Foo.agda
(add-to-list ʻagda2-include-dirs “.../MyProject”)

76. open import
• import
• Nat.zero
• open import
• zero

77. module using

78. module renaming

79. module hiding
ℕ

80. module as

81. Agda

82. agda2-goal-and-context
Ctrl-c Ctrl-,

83. agda2-goal-and-context-
and-inferred
Ctrl-c Ctrl-.

84. agda2-refine
Ctrl-c Ctrl-r

85. agda2-refine
Ctrl-c Ctrl-r

86. /
• Ctrl-C Ctrl-l (agda2-load)
• Ctrl-C Ctrl-x Ctrl-r (agda2-restart)

87. compile
⊤
C-c C-x C-c (agda2-compile)
MAlonzo

88. automation proving
Ctrl-c Ctrl-a
Agsy

89. automation proving
Ctrl-c Ctrl-a
Agsy

90. automation proving
Ctrl-c Ctrl-a
Agsy

91. advanced topic

92. equality
Leibnitz equality

93. ℕ

94. ℕ

95. ℕ

96. rewrite
ℕ

97. rewrite
ℕ

98. rewrite
ℕ

99. rewrite
ℕ

100. rewrite
ℕ

101. rewrite
ℕ

102. rewrite

103. equality reasoning
⟨ ⟩
∎

104. equality reasoning
⟨ ⟩
∎

105. equality reasoning
⟨ ⟩
∎

106. tactic

107. emptiness check
ℕ ℕ
Agda 2.2.12 Feature

109. --universe-polymorphism
Universe
Agda
Agda 2.2.12

110. agda
• --compile
• MAlonzo
• --html
• HTML
•
•
• etc.

111. Agda release notes
• Agda 2-2-0 (2009-03-18)
• Sized type

112. sized types
ℕ ℕ ℕ
ℕ ℕ ℕ

113. sized types
ℕ ℕ ℕ
ℕ ℕ ℕ

114. Sized Type

115. Agda release notes
• Agda 2.2.10 (2011-02-21)
• --without-K flag
• irrelevant declarations
• termination checker with projections
• compiler backend
• Epic
• MAlonzo

116. Epic
• Epic
• http://www.cs.st-andrews.ac.uk/~eb/
epic.php
• Agda Epic
• Epic C

117. MAlonzo
• Agda
• Haskell (!?)
Agda Haskell

118. (experimental)
• JavaScript Compiler Backend
• Ruby Compiler Backend
• https://github.com/larrytheliquid/agda-rb

119. Agda release notes
• Agda 2.2.12 (current development)
• coming soon (maybe within this week?)
• instance arguments {{ }}
• pattern matching lambda

120. Agda in browser
• Agda + JavaScript Compiler
• Functional Reactive Programming
• Example : Clock

122. Demo
KeyCastr.app
Emacs.app (Command + T )
[1B_Maze-Black 25pt font]

123. • Agda standard library • irrelevant declaration
• projection and
termination
• well-founded
induction • instance arguments
• coinduction • pattern matching lambda
• --without-K • reflection
• etc. (too much)