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Static typing and proof in ATS language

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Static typing and proof in ATS language

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Static typing and proof in ATS language

  1. 1. Static typing and proof in ATS language Static typing and proof in ATS language Static typing and proof in ATS language Static typing and proof in ATS language Static typing and proof in ATS language Kiwamu OkabeKiwamu OkabeKiwamu OkabeKiwamu OkabeKiwamu Okabe
  2. 2. Let's make Rock-paper-scissors libraryLet's make Rock-paper-scissors libraryLet's make Rock-paper-scissors libraryLet's make Rock-paper-scissors libraryLet's make Rock-paper-scissors library ☆ �じゃんけん� in Japanese.☆ �じゃんけん� in Japanese.☆ �じゃんけん� in Japanese.☆ �じゃんけん� in Japanese.☆ �じゃんけん� in Japanese.
  3. 3. R-P-S in C language #1R-P-S in C language #1R-P-S in C language #1R-P-S in C language #1R-P-S in C language #1 #include�<stdio.h> #include�<stdlib.h> typedef�enum�{ R_RPS_T�=�0,�/*�Rock�*/ P_RPS_T,�����/*�Paper�*/ S_RPS_T������/*�Scissors�*/ }�rps_t; void�print_rps(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: printf("Rock");�break; case�P_RPS_T: printf("Paper");�break; case�S_RPS_T: printf("Scissors");�break; } } #include�<stdio.h> #include�<stdlib.h> typedef�enum�{ R_RPS_T�=�0,�/*�Rock�*/ P_RPS_T,�����/*�Paper�*/ S_RPS_T������/*�Scissors�*/ }�rps_t; void�print_rps(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: printf("Rock");�break; case�P_RPS_T: printf("Paper");�break; case�S_RPS_T: printf("Scissors");�break; } } #include�<stdio.h> #include�<stdlib.h> typedef�enum�{ R_RPS_T�=�0,�/*�Rock�*/ P_RPS_T,�����/*�Paper�*/ S_RPS_T������/*�Scissors�*/ }�rps_t; void�print_rps(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: printf("Rock");�break; case�P_RPS_T: printf("Paper");�break; case�S_RPS_T: printf("Scissors");�break; } } #include�<stdio.h> #include�<stdlib.h> typedef�enum�{ R_RPS_T�=�0,�/*�Rock�*/ P_RPS_T,�����/*�Paper�*/ S_RPS_T������/*�Scissors�*/ }�rps_t; void�print_rps(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: printf("Rock");�break; case�P_RPS_T: printf("Paper");�break; case�S_RPS_T: printf("Scissors");�break; } } #include�<stdio.h> #include�<stdlib.h> typedef enum�{ R_RPS_T�=�0,�/*�Rock�*/ P_RPS_T,�����/*�Paper�*/ S_RPS_T������/*�Scissors�*/ }�rps_t; void�print_rps(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: printf("Rock");�break; case�P_RPS_T: printf("Paper");�break; case�S_RPS_T: printf("Scissors");�break; } }
  4. 4. R-P-S in C language #2R-P-S in C language #2R-P-S in C language #2R-P-S in C language #2R-P-S in C language #2 int�main()�{ rps_t�paper_d�=�P_RPS_T; rps_t�win_d�=�rps_win_d�(paper_d); print_rps(win_d);�printf("�win�");�print_rps(paper_d);�printf(".n"); /*�=>�Scissors�win�Paper.�*/ rps_t�even_d�=�rps_even_d(win_d,�paper_d); print_rps(even_d);�printf("�is�even�between�");�print_rps(win_d); printf("�and�");�print_rps(paper_d);�printf(".n"); /*�=>�Rock�is�even�between�Scissors�and�Paper.�*/ } int�main()�{ rps_t�paper_d�=�P_RPS_T; rps_t�win_d�=�rps_win_d�(paper_d); print_rps(win_d);�printf("�win�");�print_rps(paper_d);�printf(".n"); /*�=>�Scissors�win�Paper.�*/ rps_t�even_d�=�rps_even_d(win_d,�paper_d); print_rps(even_d);�printf("�is�even�between�");�print_rps(win_d); printf("�and�");�print_rps(paper_d);�printf(".n"); /*�=>�Rock�is�even�between�Scissors�and�Paper.�*/ } int�main()�{ rps_t�paper_d�=�P_RPS_T; rps_t�win_d�=�rps_win_d�(paper_d); print_rps(win_d);�printf("�win�");�print_rps(paper_d);�printf(".n"); /*�=>�Scissors�win�Paper.�*/ rps_t�even_d�=�rps_even_d(win_d,�paper_d); print_rps(even_d);�printf("�is�even�between�");�print_rps(win_d); printf("�and�");�print_rps(paper_d);�printf(".n"); /*�=>�Rock�is�even�between�Scissors�and�Paper.�*/ } int�main()�{ rps_t�paper_d�=�P_RPS_T; rps_t�win_d�=�rps_win_d�(paper_d); print_rps(win_d);�printf("�win�");�print_rps(paper_d);�printf(".n"); /*�=>�Scissors�win�Paper.�*/ rps_t�even_d�=�rps_even_d(win_d,�paper_d); print_rps(even_d);�printf("�is�even�between�");�print_rps(win_d); printf("�and�");�print_rps(paper_d);�printf(".n"); /*�=>�Rock�is�even�between�Scissors�and�Paper.�*/ } int�main()�{ rps_t�paper_d�=�P_RPS_T; rps_t�win_d�=�rps_win_d�(paper_d); print_rps(win_d);�printf("�win�");�print_rps(paper_d);�printf(".n"); /*�=>�Scissors�win�Paper.�*/ rps_t�even_d�=�rps_even_d(win_d,�paper_d); print_rps(even_d);�printf("�is�even�between�");�print_rps(win_d); printf("�and�");�print_rps(paper_d);�printf(".n"); /*�=>�Rock�is�even�between�Scissors�and�Paper.�*/ }
  5. 5. R-P-S in C language #3R-P-S in C language #3R-P-S in C language #3R-P-S in C language #3R-P-S in C language #3 rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�P_RPS_T; case�P_RPS_T: return�S_RPS_T; case�S_RPS_T: return�R_RPS_T; default: abort(); } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(x); default: abort(); } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�P_RPS_T; case�P_RPS_T: return�S_RPS_T; case�S_RPS_T: return�R_RPS_T; default: abort(); } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(x); default: abort(); } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�P_RPS_T; case�P_RPS_T: return�S_RPS_T; case�S_RPS_T: return�R_RPS_T; default: abort(); } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(x); default: abort(); } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�P_RPS_T; case�P_RPS_T: return�S_RPS_T; case�S_RPS_T: return�R_RPS_T; default: abort(); } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(x); default: abort(); } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�P_RPS_T; case�P_RPS_T: return�S_RPS_T; case�S_RPS_T: return�R_RPS_T; default: abort(); } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(x); default: abort(); } }
  6. 6. However you can make run-time errorHowever you can make run-time errorHowever you can make run-time errorHowever you can make run-time errorHowever you can make run-time error rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�R_RPS_T;�/*�Run-time�error!�*/ case�P_RPS_T: return�S_RPS_T; /* case�S_RPS_T: return�R_RPS_T;�*/�/*�Not�exhaustive!�*/ default: abort();�/*�Abort�occurs!�*/ } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(y);�/*�Run-time�error!�*/ default: abort();�/*�Abort�occurs!�*/ } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�R_RPS_T;�/*�Run-time�error!�*/ case�P_RPS_T: return�S_RPS_T; /* case�S_RPS_T: return�R_RPS_T;�*/�/*�Not�exhaustive!�*/ default: abort();�/*�Abort�occurs!�*/ } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(y);�/*�Run-time�error!�*/ default: abort();�/*�Abort�occurs!�*/ } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�R_RPS_T;�/*�Run-time�error!�*/ case�P_RPS_T: return�S_RPS_T; /* case�S_RPS_T: return�R_RPS_T;�*/�/*�Not�exhaustive!�*/ default: abort();�/*�Abort�occurs!�*/ } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(y);�/*�Run-time�error!�*/ default: abort();�/*�Abort�occurs!�*/ } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�R_RPS_T;�/*�Run-time�error!�*/ case�P_RPS_T: return�S_RPS_T; /* case�S_RPS_T: return�R_RPS_T;�*/�/*�Not�exhaustive!�*/ default: abort();�/*�Abort�occurs!�*/ } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(y);�/*�Run-time�error!�*/ default: abort();�/*�Abort�occurs!�*/ } } rps_t�rps_win_d(rps_t�x)�{ switch�(x)�{ case�R_RPS_T: return�R_RPS_T;�/*�Run-time�error!�*/ case�P_RPS_T: return�S_RPS_T; /* case�S_RPS_T: return�R_RPS_T;�*/ /*�Not�exhaustive!�*/ default: abort();�/*�Abort�occurs!�*/ } } #define�PACK(X,Y)�((X)<<4�│�(Y)) rps_t�rps_even_d(rps_t�x,�rps_t�y)�{ switch�(PACK(x,�y))�{ case�PACK(P_RPS_T,�R_RPS_T): case�PACK(S_RPS_T,�P_RPS_T): case�PACK(R_RPS_T,�S_RPS_T): return�rps_win_d(y);�/*�Run-time�error!�*/ default: abort();�/*�Abort�occurs!�*/ } }
  7. 7. Please imagine specification on codePlease imagine specification on codePlease imagine specification on codePlease imagine specification on codePlease imagine specification on code ☆ Introduce specification using static typing.☆ Introduce specification using static typing.☆ Introduce specification using static typing.☆ Introduce specification using static typing.☆ Introduce specification using static typing. ☆ You will get more compile-time errors.☆ You will get more compile-time errors.☆ You will get more compile-time errors.☆ You will get more compile-time errors.☆ You will get more compile-time errors. ☆ You will get less run-time errors.☆ You will get less run-time errors.☆ You will get less run-time errors.☆ You will get less run-time errors.☆ You will get less run-time errors. ☆ You can do that in ATS language.☆ You can do that in ATS language.☆ You can do that in ATS language.☆ You can do that in ATS language.☆ You can do that in ATS language.
  8. 8. ATS languageATS languageATS languageATS languageATS language ☆ http://www.ats-lang.org/☆ http://www.ats-lang.org/☆ http://www.ats-lang.org/☆ http://www.ats-lang.org/☆ http://www.ats-lang.org/ ☆ DML-style dependent types☆ DML-style dependent types☆ DML-style dependent types☆ DML-style dependent types☆ DML-style dependent types ☆ Linear types☆ Linear types☆ Linear types☆ Linear types☆ Linear types ☆ Optional GC☆ Optional GC☆ Optional GC☆ Optional GC☆ Optional GC ☆ Optional malloc/free☆ Optional malloc/free☆ Optional malloc/free☆ Optional malloc/free☆ Optional malloc/free ☆ Optional run-time☆ Optional run-time☆ Optional run-time☆ Optional run-time☆ Optional run-time More strong type can be used on the ATS.More strong type can be used on the ATS.More strong type can be used on the ATS.More strong type can be used on the ATS.More strong type can be used on the ATS.
  9. 9. DML-style dependent types in ATSDML-style dependent types in ATSDML-style dependent types in ATSDML-style dependent types in ATSDML-style dependent types in ATS
  10. 10. Type of R-P-S in ATSType of R-P-S in ATSType of R-P-S in ATSType of R-P-S in ATSType of R-P-S in ATS datasort�rps_s�= ��│�r_rps_s�(*�Rock�*) ��│�p_rps_s�(*�Paper�*) ��│�s_rps_s�(*�Scissors�*) datatype�rps_t�(rps_s)�= ��│�r_rps_t�(r_rps_s)�(*�Rock�*) ��│�p_rps_t�(p_rps_s)�(*�Paper�*) ��│�s_rps_t�(s_rps_s)�(*�Scissors�*) typedef�rps_t_any�=�[x:rps_s]�rps_t�(x) datasort�rps_s�= ��│�r_rps_s�(*�Rock�*) ��│�p_rps_s�(*�Paper�*) ��│�s_rps_s�(*�Scissors�*) datatype�rps_t�(rps_s)�= ��│�r_rps_t�(r_rps_s)�(*�Rock�*) ��│�p_rps_t�(p_rps_s)�(*�Paper�*) ��│�s_rps_t�(s_rps_s)�(*�Scissors�*) typedef�rps_t_any�=�[x:rps_s]�rps_t�(x) datasort�rps_s�= ��│�r_rps_s�(*�Rock�*) ��│�p_rps_s�(*�Paper�*) ��│�s_rps_s�(*�Scissors�*) datatype�rps_t�(rps_s)�= ��│�r_rps_t�(r_rps_s)�(*�Rock�*) ��│�p_rps_t�(p_rps_s)�(*�Paper�*) ��│�s_rps_t�(s_rps_s)�(*�Scissors�*) typedef�rps_t_any�=�[x:rps_s]�rps_t�(x) datasort�rps_s�= ��│�r_rps_s�(*�Rock�*) ��│�p_rps_s�(*�Paper�*) ��│�s_rps_s�(*�Scissors�*) datatype�rps_t�(rps_s)�= ��│�r_rps_t�(r_rps_s)�(*�Rock�*) ��│�p_rps_t�(p_rps_s)�(*�Paper�*) ��│�s_rps_t�(s_rps_s)�(*�Scissors�*) typedef�rps_t_any�=�[x:rps_s]�rps_t�(x) datasort�rps_s�= ��│�r_rps_s�(*�Rock�*) ��│�p_rps_s�(*�Paper�*) ��│�s_rps_s�(*�Scissors�*) datatype�rps_t�(rps_s)�= ��│�r_rps_t�(r_rps_s)�(*�Rock�*) ��│�p_rps_t�(p_rps_s)�(*�Paper�*) ��│�s_rps_t�(s_rps_s)�(*�Scissors�*) typedef�rps_t_any�=�[x:rps_s]�rps_t�(x)
  11. 11. Only use Dynamics of ATSOnly use Dynamics of ATSOnly use Dynamics of ATSOnly use Dynamics of ATSOnly use Dynamics of ATS fun{}�rps_win_d�(x:�rps_t_any):�rps_t_any fun{}�rps_even_d�(x:�rps_t_any,�y:�rps_t_any):�rps_t_any�(*�The�x�should�win� the�y.�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(x) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.") fun{}�rps_win_d�(x:�rps_t_any):�rps_t_any fun{}�rps_even_d�(x:�rps_t_any,�y:�rps_t_any):�rps_t_any�(*�The�x�should�win� the�y.�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(x) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.") fun{}�rps_win_d�(x:�rps_t_any):�rps_t_any fun{}�rps_even_d�(x:�rps_t_any,�y:�rps_t_any):�rps_t_any�(*�The�x�should�win� the�y.�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(x) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.") fun{}�rps_win_d�(x:�rps_t_any):�rps_t_any fun{}�rps_even_d�(x:�rps_t_any,�y:�rps_t_any):�rps_t_any�(*�The�x�should�win� the�y.�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(x) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.") fun{}�rps_win_d�(x:�rps_t_any):�rps_t_any fun{}�rps_even_d�(x:�rps_t_any,�y:�rps_t_any):�rps_t_any�(*�The�x�should�win� the�y.�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(x) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")
  12. 12. Also you can make run-time errorAlso you can make run-time errorAlso you can make run-time errorAlso you can make run-time errorAlso you can make run-time error implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Run-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() (*│�s_rps_t()�=>�r_rps_t()�*)�(*�Compile-time�error!�*) implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(y)�(*�Run-time�error!�*) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")�(*�Exception� occurs!�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Run-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() (*│�s_rps_t()�=>�r_rps_t()�*)�(*�Compile-time�error!�*) implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(y)�(*�Run-time�error!�*) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")�(*�Exception� occurs!�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Run-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() (*│�s_rps_t()�=>�r_rps_t()�*)�(*�Compile-time�error!�*) implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(y)�(*�Run-time�error!�*) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")�(*�Exception� occurs!�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Run-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() (*│�s_rps_t()�=>�r_rps_t()�*)�(*�Compile-time�error!�*) implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(y)�(*�Run-time�error!�*) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")�(*�Exception� occurs!�*) implement{}�rps_win_d�(x)�= ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Run-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() (*│�s_rps_t()�=>�r_rps_t()�*) (*�Compile-time�error!�*) implement{}�rps_even_d�(x,�y)�= ��case+�(x,�y)�of ��│�(p_rps_t(),�r_rps_t())�=>�rps_win_d�(x) ��│�(s_rps_t(),�p_rps_t())�=>�rps_win_d�(y)�(*�Run-time�error!�*) ��│�(r_rps_t(),�s_rps_t())�=>�rps_win_d�(x) ��│�(_,�_)�=>�$raise�GenerallyExn("The�x�should�win�the�y.")�(*�Exception� occurs!�*) It's caused by having no specification in the implementation. ATS language can have specification as Statics or Proofs. It's caused by having no specification in the implementation. ATS language can have specification as Statics or Proofs. It's caused by having no specification in the implementation. ATS language can have specification as Statics or Proofs. It's caused by having no specification in the implementation. ATS language can have specification as Statics or Proofs. It's caused by having no specification in the implementation. ATS language can have specification as Statics or Proofs.
  13. 13. Specification using Statics of ATSSpecification using Statics of ATSSpecification using Statics of ATSSpecification using Statics of ATSSpecification using Statics of ATS stacst�rps_win_sta:�(rps_s,�rps_s)�->�bool fun{}�rps_win_s ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s�│�rps_win_sta�(r2,�r1)]�(rps_t�(r2)) stacst�rps_win_sta:�(rps_s,�rps_s)�->�bool fun{}�rps_win_s ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s�│�rps_win_sta�(r2,�r1)]�(rps_t�(r2)) stacst�rps_win_sta:�(rps_s,�rps_s)�->�bool fun{}�rps_win_s ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s�│�rps_win_sta�(r2,�r1)]�(rps_t�(r2)) stacst�rps_win_sta:�(rps_s,�rps_s)�->�bool fun{}�rps_win_s ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s�│�rps_win_sta�(r2,�r1)]�(rps_t�(r2)) stacst�rps_win_sta:�(rps_s,�rps_s)�->�bool fun{}�rps_win_s ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s�│�rps_win_sta�(r2,�r1)]�(rps_t�(r2))
  14. 14. Implementation using Statics of ATSImplementation using Statics of ATSImplementation using Statics of ATSImplementation using Statics of ATSImplementation using Statics of ATS extern�praxi�rps_win_r_s_pf�():�[rps_win_sta(r_rps_s,�s_rps_s)]�void�(*� Spec:�Paper�wins�Rock�*) extern�praxi�rps_win_p_r_pf�():�[rps_win_sta(p_rps_s,�r_rps_s)]�void�(*� Spec:�Scissors�win�Paper�*) extern�praxi�rps_win_s_p_pf�():�[rps_win_sta(s_rps_s,�p_rps_s)]�void�(*� Spec:�Rock�wins�Scissors�*) implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end extern�praxi�rps_win_r_s_pf�():�[rps_win_sta(r_rps_s,�s_rps_s)]�void�(*� Spec:�Paper�wins�Rock�*) extern�praxi�rps_win_p_r_pf�():�[rps_win_sta(p_rps_s,�r_rps_s)]�void�(*� Spec:�Scissors�win�Paper�*) extern�praxi�rps_win_s_p_pf�():�[rps_win_sta(s_rps_s,�p_rps_s)]�void�(*� Spec:�Rock�wins�Scissors�*) implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end extern�praxi�rps_win_r_s_pf�():�[rps_win_sta(r_rps_s,�s_rps_s)]�void�(*� Spec:�Paper�wins�Rock�*) extern�praxi�rps_win_p_r_pf�():�[rps_win_sta(p_rps_s,�r_rps_s)]�void�(*� Spec:�Scissors�win�Paper�*) extern�praxi�rps_win_s_p_pf�():�[rps_win_sta(s_rps_s,�p_rps_s)]�void�(*� Spec:�Rock�wins�Scissors�*) implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end extern�praxi�rps_win_r_s_pf�():�[rps_win_sta(r_rps_s,�s_rps_s)]�void�(*� Spec:�Paper�wins�Rock�*) extern�praxi�rps_win_p_r_pf�():�[rps_win_sta(p_rps_s,�r_rps_s)]�void�(*� Spec:�Scissors�win�Paper�*) extern�praxi�rps_win_s_p_pf�():�[rps_win_sta(s_rps_s,�p_rps_s)]�void�(*� Spec:�Rock�wins�Scissors�*) implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end extern praxi�rps_win_r_s_pf�():�[rps_win_sta(r_rps_s,�s_rps_s)]�void�(*� Spec:�Paper�wins�Rock�*) extern praxi�rps_win_p_r_pf�():�[rps_win_sta(p_rps_s,�r_rps_s)]�void�(*� Spec:�Scissors�win�Paper�*) extern praxi�rps_win_s_p_pf�():�[rps_win_sta(s_rps_s,�p_rps_s)]�void�(*� Spec:�Rock�wins�Scissors�*) implement{}�rps_win_s�(x)�=�let prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) prval�()�=�rps_win_p_r_pf() prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�p_rps_t() ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end
  15. 15. Statics can catch error at compile-timeStatics can catch error at compile-timeStatics can catch error at compile-timeStatics can catch error at compile-timeStatics can catch error at compile-time implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end implement{}�rps_win_s�(x)�=�let ��prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) ��prval�()�=�rps_win_p_r_pf() ��prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end implement{}�rps_win_s�(x)�=�let prval�()�=�rps_win_r_s_pf()�(*�Introduce�specification�*) prval�()�=�rps_win_p_r_pf() prval�()�=�rps_win_s_p_pf() in ��case+�x�of ��│�r_rps_t()�=>�r_rps_t()�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�s_rps_t() ��│�s_rps_t()�=>�r_rps_t() end $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�933(line=36,�offs=18)�--� 942(line=36,�offs=27):�error(3):�unsolved�constraint:�C3NSTRprop(C3TKmain();� S2Eapp(S2Ecst(rps_win_sta);�S2EVar(4434->S2Eapp(S2Ecst(r_rps_s);�)),�S2Evar (r1$2870(7929)))) $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�933(line=36,�offs=18)�--� 942(line=36,�offs=27):�error(3):�unsolved�constraint:�C3NSTRprop(C3TKmain();� S2Eapp(S2Ecst(rps_win_sta);�S2EVar(4434->S2Eapp(S2Ecst(r_rps_s);�)),�S2Evar (r1$2870(7929)))) $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�933(line=36,�offs=18)�--� 942(line=36,�offs=27):�error(3):�unsolved�constraint:�C3NSTRprop(C3TKmain();� S2Eapp(S2Ecst(rps_win_sta);�S2EVar(4434->S2Eapp(S2Ecst(r_rps_s);�)),�S2Evar (r1$2870(7929)))) $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�933(line=36,�offs=18)�--� 942(line=36,�offs=27):�error(3):�unsolved�constraint:�C3NSTRprop(C3TKmain();� S2Eapp(S2Ecst(rps_win_sta);�S2EVar(4434->S2Eapp(S2Ecst(r_rps_s);�)),�S2Evar (r1$2870(7929)))) $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�933(line=36,�offs=18)�--� 942(line=36,�offs=27):�error(3):�unsolved�constraint:�C3NSTRprop(C3TKmain();� S2Eapp(S2Ecst(rps_win_sta);�S2EVar(4434->S2Eapp(S2Ecst(r_rps_s);�)),�S2Evar (r1$2870(7929))))
  16. 16. Specification using Proofs of ATSSpecification using Proofs of ATSSpecification using Proofs of ATSSpecification using Proofs of ATSSpecification using Proofs of ATS dataprop�RPS_WIN�(rps_s,�rps_s)�= ��│�RPS_P_WIN_R�(p_rps_s,�r_rps_s)�(*�Spec:�Paper�wins�Rock�*) ��│�RPS_S_WIN_P�(s_rps_s,�p_rps_s)�(*�Spec:�Scissors�win�Paper�*) ��│�RPS_R_WIN_S�(r_rps_s,�s_rps_s)�(*�Spec:�Rock�wins�Scissors�*) fun{}�rps_win_p ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s]�(RPS_WIN�(r2,�r1)�│�rps_t�(r2)) dataprop�RPS_WIN�(rps_s,�rps_s)�= ��│�RPS_P_WIN_R�(p_rps_s,�r_rps_s)�(*�Spec:�Paper�wins�Rock�*) ��│�RPS_S_WIN_P�(s_rps_s,�p_rps_s)�(*�Spec:�Scissors�win�Paper�*) ��│�RPS_R_WIN_S�(r_rps_s,�s_rps_s)�(*�Spec:�Rock�wins�Scissors�*) fun{}�rps_win_p ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s]�(RPS_WIN�(r2,�r1)�│�rps_t�(r2)) dataprop�RPS_WIN�(rps_s,�rps_s)�= ��│�RPS_P_WIN_R�(p_rps_s,�r_rps_s)�(*�Spec:�Paper�wins�Rock�*) ��│�RPS_S_WIN_P�(s_rps_s,�p_rps_s)�(*�Spec:�Scissors�win�Paper�*) ��│�RPS_R_WIN_S�(r_rps_s,�s_rps_s)�(*�Spec:�Rock�wins�Scissors�*) fun{}�rps_win_p ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s]�(RPS_WIN�(r2,�r1)�│�rps_t�(r2)) dataprop�RPS_WIN�(rps_s,�rps_s)�= ��│�RPS_P_WIN_R�(p_rps_s,�r_rps_s)�(*�Spec:�Paper�wins�Rock�*) ��│�RPS_S_WIN_P�(s_rps_s,�p_rps_s)�(*�Spec:�Scissors�win�Paper�*) ��│�RPS_R_WIN_S�(r_rps_s,�s_rps_s)�(*�Spec:�Rock�wins�Scissors�*) fun{}�rps_win_p ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s]�(RPS_WIN�(r2,�r1)�│�rps_t�(r2)) dataprop�RPS_WIN�(rps_s,�rps_s)�= ��│�RPS_P_WIN_R�(p_rps_s,�r_rps_s)�(*�Spec:�Paper�wins�Rock�*) ��│�RPS_S_WIN_P�(s_rps_s,�p_rps_s)�(*�Spec:�Scissors�win�Paper�*) ��│�RPS_R_WIN_S�(r_rps_s,�s_rps_s)�(*�Spec:�Rock�wins�Scissors�*) fun{}�rps_win_p ��{r1:rps_s}�(x:�rps_t�(r1)): ��[r2:rps_s]�(RPS_WIN�(r2,�r1)�│�rps_t�(r2))
  17. 17. Implementation using Proofs of ATSImplementation using Proofs of ATSImplementation using Proofs of ATSImplementation using Proofs of ATSImplementation using Proofs of ATS implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_P_WIN_R()�│�p_rps_t()) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_P_WIN_R()�│�p_rps_t()) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_P_WIN_R()�│�p_rps_t()) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_P_WIN_R()�│�p_rps_t()) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_P_WIN_R()�│�p_rps_t()) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())
  18. 18. Proofs can catch error at compile-timeProofs can catch error at compile-timeProofs can catch error at compile-timeProofs can catch error at compile-timeProofs can catch error at compile-time implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) implement{}�rps_win_p�(x)�=�case+�x�of ��│�r_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t())�(*�Compile-time�error!�*) ��│�p_rps_t()�=>�(RPS_S_WIN_P()�│�s_rps_t()) ��│�s_rps_t()�=>�(RPS_R_WIN_S()�│�r_rps_t()) $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�1199(line=45,�offs=18)�--� 1226(line=45,�offs=45):�error(3):�the�dynamic�expression�cannot�be�assigned� the�type�[S2Eexi(r2$2876(7935);�;�S2Etyrec(flt0;�npf=1;�0=S2Eapp(S2Ecst (RPS_WIN);�S2Evar(r2$2876(7935)),�S2Evar(r1$2875(7934))),�1=S2Eapp(S2Ecst (rps_t);�S2Evar(r2$2876(7935)))))]. $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�1199(line=45,�offs=18)�--� 1226(line=45,�offs=45):�error(3):�the�dynamic�expression�cannot�be�assigned� the�type�[S2Eexi(r2$2876(7935);�;�S2Etyrec(flt0;�npf=1;�0=S2Eapp(S2Ecst (RPS_WIN);�S2Evar(r2$2876(7935)),�S2Evar(r1$2875(7934))),�1=S2Eapp(S2Ecst (rps_t);�S2Evar(r2$2876(7935)))))]. $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�1199(line=45,�offs=18)�--� 1226(line=45,�offs=45):�error(3):�the�dynamic�expression�cannot�be�assigned� the�type�[S2Eexi(r2$2876(7935);�;�S2Etyrec(flt0;�npf=1;�0=S2Eapp(S2Ecst (RPS_WIN);�S2Evar(r2$2876(7935)),�S2Evar(r1$2875(7934))),�1=S2Eapp(S2Ecst (rps_t);�S2Evar(r2$2876(7935)))))]. $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�1199(line=45,�offs=18)�--� 1226(line=45,�offs=45):�error(3):�the�dynamic�expression�cannot�be�assigned� the�type�[S2Eexi(r2$2876(7935);�;�S2Etyrec(flt0;�npf=1;�0=S2Eapp(S2Ecst (RPS_WIN);�S2Evar(r2$2876(7935)),�S2Evar(r1$2875(7934))),�1=S2Eapp(S2Ecst (rps_t);�S2Evar(r2$2876(7935)))))]. $�patscc�main.dats�-DATS_MEMALLOC_LIBC /home/kiwamu/src/practice-ats/static_rps/rps.dats:�1199(line=45,�offs=18)�--� 1226(line=45,�offs=45):�error(3):�the�dynamic�expression�cannot�be�assigned� the�type�[S2Eexi(r2$2876(7935);�;�S2Etyrec(flt0;�npf=1;�0=S2Eapp(S2Ecst (RPS_WIN);�S2Evar(r2$2876(7935)),�S2Evar(r1$2875(7934))),�1=S2Eapp(S2Ecst (rps_t);�S2Evar(r2$2876(7935)))))].
  19. 19. The code is found at GitHubThe code is found at GitHubThe code is found at GitHubThe code is found at GitHubThe code is found at GitHub https://github.com/jats-ug/practice-ats/tree/master/static_rpshttps://github.com/jats-ug/practice-ats/tree/master/static_rpshttps://github.com/jats-ug/practice-ats/tree/master/static_rpshttps://github.com/jats-ug/practice-ats/tree/master/static_rpshttps://github.com/jats-ug/practice-ats/tree/master/static_rps
  20. 20. 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