Programming with Invariants through Targeted Synthesis. John Sarracino

Programming with Invariants through Targeted Synthesis John Sarracino 1

Programming is Hard

Programming is Hard Therac 25 1982

Programming is Hard Ariane 5 1996

Programming is Hard Maersk 2017

Programming is Hard Windows 2018

Programming is Hard Windows always

Computers are a Sadness and I am the cure

Programs are a Sadness and I am the cure

Software Invariants ubi_assert(e->num!= e1->num); if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); ensure cat_set : has_category(cat) end

Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); ensure cat_set : has_category(cat) end

Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Java ensure cat_set : has_category(cat) end

Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Java ensure cat_set : has_category(cat) end Eiffel

Software Invariants ubi_assert(e->num!= e1->num); C if (keyproto.getkeyvalue().size()!= Ed25519Verify.PUBLIC_KEY_LEN) { throw new GeneralSecurityException( invalid Ed25519 public key: incorrect key length ); Language agnostic, Manually maintained. Java ensure cat_set : has_category(cat) end Eiffel

Manual Invariant Programming Invariants Algorithm

Manual Invariant Programming Invariants Algorithm Code

Why are Manual Invariants difficult? Code changes => Programmer checks, maintains invariant Invariant changes => Programmer checks, maintains code

The code change problem: Maintaining Invariants Invariant: All urls are in the urlset

The code change problem: Maintaining Invariants Tedious and error-prone, particularly with a large codebase Invariant: All urls are in the urlset

The invariant change problem: Maintaining Code Invariant: All urls are in the urlset

The invariant change problem: Maintaining Code Invariant: All urls are in the urlset The allocset contains the urlset

The invariant change problem: Maintaining Code Even more tedious and error-prone Invariant: All urls are in the urlset The allocset contains the urlset

My thesis statement Compiling and generating invariants is useful, novel, and broadly applicable.

My thesis proposal Compiling and generating invariants is useful, novel, and broadly applicable. I propose a general framework for maintaining invariants

Proposed Work Data invariant calculus Extensions Invariants for Security GUI invariants

Outline Data invariant calculus Extensions Invariants for Security GUI invariants

Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i]

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc

Budgeting Code for (let index = 0; index!= week.length; ++index) { week[index].oninput = () => { let w = parsefloat(week[index].textcontent); month[index].textcontent = w * 4; year[index].textcontent = w * 12 * 4; refreshcolors(index); month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); year[index].oninput = () => { let y = parsefloat(year[index].textcontent); week[index].textcontent = y / (4 * 12); month[index].textcontent = y / 12; refreshcolors(index); document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

Budgeting Code document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].oninput = () => { week[index].style.backgroundcolor = evencolor; let w = parsefloat(week[index].textcontent); month[index].style.backgroundcolor = evencolor; month[index].textcontent = w * 4; year[index].style.backgroundcolor = evencolor; year[index].textcontent = w * 12 * 4; else { refreshcolors(index); week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; Tedious even for this year[index].style.backgroundcolor simple example = oddcolor; month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { year[index].oninput = () => { if (index % 2 == 0) { let y = parsefloat(year[index].textcontent); week[index].style.backgroundcolor = evencolor; week[index].textcontent = y / (4 * 12); month[index].style.backgroundcolor = evencolor; month[index].textcontent = y / 12; year[index].style.backgroundcolor = evencolor; refreshcolors(index); else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

Budgeting Code Invariants Algorithm for (let index = 0; index!= week.length; ++index) { week[index].oninput = () => { let w = parsefloat(week[index].textcontent); month[index].textcontent = w * 4; year[index].textcontent = w * 12 * 4; refreshcolors(index); month[index].oninput = () => { let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); year[index].oninput = () => { let y = parsefloat(year[index].textcontent); week[index].textcontent = y / (4 * 12); month[index].textcontent = y / 12; refreshcolors(index); document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].style.backgroundcolor = evencolor; month[index].style.backgroundcolor = evencolor; year[index].style.backgroundcolor = evencolor; else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

Budgeting Code Invariants Algorithm document.getelementbyid('evenpicker').oninput = () => { evencolor = document.getelementbyid('evenpicker').value; for (let index = 0; index!= week.length; ++index) { for (let index = 0; index!= week.length; ++index) { if (index % 2 == 0) { week[index].oninput = () => { week[index].style.backgroundcolor = evencolor; let w = parsefloat(week[index].textcontent); month[index].style.backgroundcolor = evencolor; month[index].textcontent = w * 4; year[index].style.backgroundcolor = evencolor; year[index].textcontent = w * 12 * 4; refreshcolors(index); Vast majority else of { code week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor; month[index].oninput manually = () => { maintains invariant let m = parsefloat(month[index].textcontent); week[index].textcontent = m / 4; year[index].textcontent = m * 12; refreshcolors(index); document.getelementbyid('oddpicker').oninput = () => { oddcolor = document.getelementbyid('oddpicker').value; for (let index = 0; index!= week.length; ++index) { year[index].oninput = () => { if (index % 2 == 0) { let y = parsefloat(year[index].textcontent); week[index].style.backgroundcolor = evencolor; week[index].textcontent = y / (4 * 12); month[index].style.backgroundcolor = evencolor; month[index].textcontent = y / 12; year[index].style.backgroundcolor = evencolor; refreshcolors(index); else { week[index].style.backgroundcolor = oddcolor; month[index].style.backgroundcolor = oddcolor; year[index].style.backgroundcolor = oddcolor;

Manual Invariant Programming Invariants Algorithm Code

Invariant Synthesis Invariants Algorithm Code

Language Design Goals Programmer thinks about invariants once Compiler automatically handles invariants

Compiling Invariants Program Synthesis compiles invariants* Inputs: 1) program with holes e.g. x :=??(y, x) 2) precondition e.g. y = 15 3) requirement e.g. x = y

Compiling Invariants Program Synthesis compiles invariants* Inputs: expression 1) program using with holes x and e.g. y x :=??(y, x) 2) precondition e.g. y = 15 3) requirement e.g. x = y

Naive Algorithm Synthesize at the end of every function. Use invariants as preconditions, requirements.

Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] =??(value, i, weeks[i]); years[i] =??(value, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] = 4*weeks[i]; years[i] = 4*12*weeks[i]; 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Naive Algorithm function updateweeks(value, i) { weeks[i] = value; Does not scale to realistic programs Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = value; months[i] =??(value, i, weeks[i]); years[i] =??(value, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Problem: Loops don t scale function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Problem: Loops don t scale function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Timeout even for this simple example Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Key insight: move hole inside loop function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; for (let i = 0; i < weeks.length; ++i) { months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Key insight: move hole inside loop function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Targeted Synthesis! Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] =??(scale, i, weeks[i]); years[i] =??(scale, i, weeks[i]); 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Simplification: No quantification!! function adjustforinflation(scale) { for (let i = 0; i < weeks.length; ++i) { weeks[i] = weeks[i] * scale; Pre Req 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] weeks[i] = weeks[i] * scale; months[i] = months[i] * scale; years[i] = years[i] * scale; 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i]

Spyder Separate invariant, imperative programs Compiler combines with targeted synthesis

Spyder Automatic Invariant maintenance Separate invariant, imperative programs Compiler combines with targeted synthesis

Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val;

Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val; Loops and Arrays

Imperative Language for (w: int) with idx in (weeks) { if (idx == i) { w = val; In-out iterators

Imperative AST

Invariant Language foreach (week, month) in (weeks, months) { week = 4 * month

Invariant Language foreach (week, month) in (weeks, months) { week = 4 * month Elementwise Quantification

Invariant AST

Compilation Algorithm

Compilation Algorithm 1) bring necessary variables into scope 2) specialize invariants to loops 3)??? (find template) 4) profit (synthesize)

Format Invariants Code Explanatory Caption Problem/Result

Simple Example x > 2 * y y = y + 1;

Simple Example x > 2 * y y = y + 1; X ~ Y

Simple Example x > 2 * y y = y + 1; X ~ Y Y edited

Simple Example x > 2 * y y = y + 1; X ~ Y Y edited x =??(x, y);

Simple Example x > 2 * y y = y + 1; X, Y in scope x =??(x, y);

Simple Example x > 2 * y y = y + 1; x =??(x, y);

Simple Example x > 2 * y y = y + 1; x = x + 2;

Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05;

Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05; weeks ~ months

Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int) in (weeks) { w = w * 1.05; weeks ~ months need months

Extending Loops foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; weeks ~ months need months

Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05;

Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant

Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant m =??(m, w);

Simplifying Invariants foreach (w, m) in (weeks, months){ w = 4 * m for (w: int, m: int) in (weeks, months){ w = w * 1.05; Key insight: foreach body <=> for invariant m = m * 1.05;

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] Need to synthesize loops, conditionals 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue;

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; bkg ~ bkgcolors bkg edited

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; bkg ~ bkgcolors bkg edited for (color: int) in (bkgcolors){ color =??(bkg, blue);

Loops foreach (color) in (bkgcolors) { color = bkg That s the skeleton bkg = blue; What about the invariant? for (color: int) in (bkgcolors){ color =??(bkg, blue);

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color =??(bkg, blue);

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; Use foreach spec for body for (color: int) in (bkgcolors){ color =??(bkg, blue);

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color =??(bkg, blue);

Synthesizing Loops foreach (color) in (bkgcolors) { color = bkg bkg = blue; for (color: int) in (bkgcolors){ color = blue;

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Example Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] =years[i] Need to synthesize loops, conditionals 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc

Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red;

Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; Implications as conditionals

Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; Implications as conditionals if (idx % 2 == 0) { color =??(even, red, idx);

Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; if (idx % 2 == 0) { color =??(even, red, idx);

Synthesizing Conditionals (idx % 2 = 0) ==> color = even even = red; if (idx % 2 == 0) { color = even;

Case studies* Game of Life (1d, 2d) Budgeting calculator Split expenses spreadsheet *

<latexit sha1_base64="tr8s+q96aeq/uprhjs8lwyt4d/o=">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</latexit> <latexit sha1_base64="tr8s+q96aeq/uprhjs8lwyt4d/o=">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</latexit> <latexit sha1_base64="tr8s+q96aeq/uprhjs8lwyt4d/o=">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</latexit> <latexit sha1_base64="tr8s+q96aeq/uprhjs8lwyt4d/o=">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</latexit> Game of Life Invariants 8i.model[i] =) view[i] =black 8i.!model[i] =) view[i] =white running () start enabled <latexit sha1_base64="xvg5tdsq10kirbnjgu7jdffp6pi=">aaacbhicbvc7tsmwfhxkq5rxglglryxevcuiccykfsyi0yfurjhj3lrwhseyhaqq6sdcr7awgbarh8hg3+c2gadlsjaozrnh9j1hxpnsjvntvdbwnza3qtu1nd29/qp78kir0lxs6ncup7ifegwccehopjn0mwkkctn0wvhnzo89gfqsffd6kogfkkfgmanegymw6zixgokh9lgcy6wj1f4agph8fngnp+nmgvejw5igkteo7c8vsmmegncue6ugrpnpvzb3msphwvnybrmhyzkegagcjkd8yr7efj8ajcjxks0rgs/v34mcjepnktbmjksp1li3e//zbrmor/yciszxiojiotjnwkd41giomasq+cqqqiuzf8v0rcsh2vrwmyw4yyuvku5503wa7t1fo3vd1lffdxsczpclllel3ai26ickhtezekvv1pp1yr1bh4vrilvmjtefwj8/6iyypw==</latexit> <latexit sha1_base64="xvg5tdsq10kirbnjgu7jdffp6pi=">aaacbhicbvc7tsmwfhxkq5rxglglryxevcuiccykfsyi0yfurjhj3lrwhseyhaqq6sdcr7awgbarh8hg3+c2gadlsjaozrnh9j1hxpnsjvntvdbwnza3qtu1nd29/qp78kir0lxs6ncup7ifegwccehopjn0mwkkctn0wvhnzo89gfqsffd6kogfkkfgmanegymw6zixgokh9lgcy6wj1f4agph8fngnp+nmgvejw5igkteo7c8vsmmegncue6ugrpnpvzb3msphwvnybrmhyzkegagcjkd8yr7efj8ajcjxks0rgs/v34mcjepnktbmjksp1li3e//zbrmor/yciszxiojiotjnwkd41giomasq+cqqqiuzf8v0rcsh2vrwmyw4yyuvku5503wa7t1fo3vd1lffdxsczpclllel3ai26ickhtezekvv1pp1yr1bh4vrilvmjtefwj8/6iyypw==</latexit> <latexit sha1_base64="xvg5tdsq10kirbnjgu7jdffp6pi=">aaacbhicbvc7tsmwfhxkq5rxglglryxevcuiccykfsyi0yfurjhj3lrwhseyhaqq6sdcr7awgbarh8hg3+c2gadlsjaozrnh9j1hxpnsjvntvdbwnza3qtu1nd29/qp78kir0lxs6ncup7ifegwccehopjn0mwkkctn0wvhnzo89gfqsffd6kogfkkfgmanegymw6zixgokh9lgcy6wj1f4agph8fngnp+nmgvejw5igkteo7c8vsmmegncue6ugrpnpvzb3msphwvnybrmhyzkegagcjkd8yr7efj8ajcjxks0rgs/v34mcjepnktbmjksp1li3e//zbrmor/yciszxiojiotjnwkd41giomasq+cqqqiuzf8v0rcsh2vrwmyw4yyuvku5503wa7t1fo3vd1lffdxsczpclllel3ai26ickhtezekvv1pp1yr1bh4vrilvmjtefwj8/6iyypw==</latexit> <latexit sha1_base64="xvg5tdsq10kirbnjgu7jdffp6pi=">aaacbhicbvc7tsmwfhxkq5rxglglryxevcuiccykfsyi0yfurjhj3lrwhseyhaqq6sdcr7awgbarh8hg3+c2gadlsjaozrnh9j1hxpnsjvntvdbwnza3qtu1nd29/qp78kir0lxs6ncup7ifegwccehopjn0mwkkctn0wvhnzo89gfqsffd6kogfkkfgmanegymw6zixgokh9lgcy6wj1f4agph8fngnp+nmgvejw5igkteo7c8vsmmegncue6ugrpnpvzb3msphwvnybrmhyzkegagcjkd8yr7efj8ajcjxks0rgs/v34mcjepnktbmjksp1li3e//zbrmor/yciszxiojiotjnwkd41giomasq+cqqqiuzf8v0rcsh2vrwmyw4yyuvku5503wa7t1fo3vd1lffdxsczpclllel3ai26ickhtezekvv1pp1yr1bh4vrilvmjtefwj8/6iyypw==</latexit> Code State transition Interactive display Start/Stop toggle * By Kieff - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=101736

<latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="4zi68pjubg9qod0oec37qfpgx9m=">aaacg3icbvc7tsmwfhxkq5rxgzhfokjiqpikczaichbgitgh1esv4960vu0ksh1qffu/wpgvfgyqykji4g9w0g7qcqajc+61zz1+zjnstv1tfvzw19y3ipulre2d3b3y/kfbrymk0kirj2txjwo4c6glmebqjsuq4xpo+oprzo/cg1qscu/0jazpkghiakajnlk/xhodsblomcnv/aawvj3m4cs6trhlhekacgdrqdo1jn1o6lhfrthvowdejs6cvnaczx750x1enbeqasqjuj3hjrwxeqkz5tatuymc2dxmhtazncqcljfmt03xivegjolmy+bc/b2reqhurphmuha9uotejv7n9ridxhgpc+neq0hnhwujxzrcwvf4wcrqzsegecqzyyrpiehctamzzepwfk9eju1a1bgrzu1zpxe1r6oijtaxokuookcndioaqiuoektp6bw9wu/wi/vufcxgc9z85xd9gfx1a/tqn34=</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="l1v+nvwnndd8x8c8v2qztoniuwe=">aaachhicbvdlssnafj3uv62vqes3g0vwfziq6eyounfzwt4gcwuymbrdzzjhziku0g9x46+4cagigxecf+m0zujbd1w4nhpvzl0nyhhv2nw/rcrk6tr6rnwztrw9s7tn7x90lmgljm0smjc9ccncaerammpgepkkieemdkprzczvphcpqejv9tgjiuedlcyui22kvn0wjeiixicfjqmaixg24bx0yec5wyaogi0gpg2njuk4elbv113hlqcxivesoijr6tufqsxwzkmqmunk+z6b6xccpkaykwktybxjeb6hafentrenkpwux03hivfialy0lwpyql8njogrnear6erid9winxp/8/xcj5fhhkzzrkmk5x8loynawflsmkasym3ghiasqdkv4igscgutz82e4c2evew6dcdzhe/uvn68luooginwde6bby5ae9ycfmgddb7bm3gfb9at9wk9wx/z1opvzhycp7c+fgcxfp84</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="c5tq+pzsrkxuddn5byednlyolcm=">aaachxicbvbnswmxem3wr1q/vj16crbbu9ktbb0irs8ek9hw6c4lm07b0cs7jnlckf0jxvwrxjwo4sgl+g9mt3vq1gcdj/dmkpkxjzxp43nftmftfwnzq7hd2tnd2z9wd49aok4vhsaneaweiqkbmwlnwwyhh0qberghdjs6mfvtmsjnynlvjgmeggwk6znkjjw6bi3ox4pwjhmu2ape3mnvfiu9hdbhfwcno9ggw0krwrhk9mlxlxsvlwnejx5oyihho+t+br2ypgkkozxo3fg9xirtogyjhgalinwqedoia+hykokahu6z62b4zco9bne0jq3o1n8tuyk0nojidgpihnrzm4v/ez3u9c/dkznjakdsxuf9lgmt43luumcuummnlhcqmn0v0yfrhbobammg4c+fvepa1yrvvfy7wrl+ncdrrcfofj0jh12gorpfddrefd2iz/sk3pwn58v5dz4wrqunnzlgf+b8/qcamp+4</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> <latexit sha1_base64="p4jtto0v5ou49ox5jwon3hnwr44=">aaacghicbvdlssnafj3uv62vqks3g0vwfrmrdkfqdooygn1agspkctmonuzczeqpoz/hxl9x40irt935n07tlrr6vodz7uwee4kum6ud58sqls2vrk6v1ysbm1vbo9xdvzzkmkmhsroeye5afhamokmz5tbjjza44naohjdtv/0aurfe3otrcn5m+ojfjbjtpf71pbslkncogbbxi8bqeczhl9jbxrabakbwlag9fa+whlbxrtm2uwd/je6c1nacjv510g0tmsugnovekc91uu3nrgpgoywr3uxbsuiq9mezvjaylj8xj43xkvfce0awixch/tzisazuka7mzez0qc16u/e/z8t0dohntkszbkfnh6kmy53gaus4zbko5indcjxmzmv0qcsh2nrzmsw4iy//ja1t23vs9+6svr+e11fgb+gqhsmxnam6ukun1equpaex9iberwfr1fqwpmejjwu+s49+wzp8a9ehnl8=</latexit> Budgeting Invariants 8i.4 weeks[i] =months[i] 8i.12 months[i] = years[i] 8i.weeks[i] < 0 =) font[i] =red 8i.weeks[i] >= 0 =) font[i] =black 8i.i 8i.i mod 2 = 0 =) bkg[i] =evenc mod 2 = 1 =) bkg[i] =oddc Code Interactive cells Inflation Backgrounds *

<latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> <latexit sha1_base64="ytonubwk7z5fmdpm5iu7okouxp8=">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</latexit> Split expenses Invariants 8i.2 mid[i] =me[i]+you[i] 8i.debt[i] =mid[i] you[i] 8i.debt[i] < 0 =) color[i] =black 8i.debt[i] >= 0 =) color[i] =red Code Interactive cells Backgrounds *

Formalisms

Formalisms Interesting Translation Rule

Formalisms What we need Formal problem definition Full translation semantics Proof of correctness

Formal Correctness Problem: Correctness:

Formal Correctness Problem: Given a program with arrays and invariants Correctness:

Formal Correctness Problem: Given a program with arrays and invariants Correctness: We find a program with augmented semantics

Formal Correctness Problem: Given a program with arrays and invariants Thank you, Reviewer A!!! Correctness: We find a program with augmented semantics

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Consider Sortedness

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j]

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Limited to maplike properties

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Cannot express :(

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j] Key insight: change of perspective!! Cannot express :(

<latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> <latexit sha1_base64="w/rt8x1midbnkxmdveacwkmailc=">aaachnicbvbns8naen3ur1q/oh69lbbbg5refd14khrxwmf+qbrkzjntt91swu5gkkx9i178k148kcj40n/jts1bwx8mpn6bywzekhcmton8w7ml5zxvtfx6ywnza3vh3t2rqtivfko05rfsbeqbzwkqmmkojuqciqio9ab/m/hrdyavi8w9hitgr6qjwjtroo3uss+b7vgszje7wb0sho8xw1e4h5ssmutbyrvldahhfcnnvoe37kjtcqbai8tnsbflqltsz2yy0zqcosknsnmuk2h/skrmlmoo0ewvjit2sqc8qwwjqpnd6xsjfgsuejs7tqmnp+rvisgjlbpegemmio6qew8i/ud5qw5f+kmmklsdolnf7zrjhenjvjhkeqjma0milczcimmxsek1sbrgqndnx14ktdos65tcu7ni+tqli48o0ce6ri66qgv0iyqoiih6rm/ofb1zt9al9w59zfpzvjazj/7a+vobsv+haq==</latexit> Stronger Invariants 8i, j. i < j =) sorted[i] <sorted[j]

<latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> 8i, j. i < j =) sorted[i] <sorted[i + 1] 8i. sorted[i] <sorted[i + 1]

<latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qq0fpwg5lceds6ddut6clsgrw9k=">aaacdnicbzbps8mwgmbt+w/of1wpxojjiailfuephozepe5wm9cvkabpfpymjumfubyv4mwv4swdil49e/pbmg1fdpobwi/nfv/yvk+ymqq0635zpaxlldw18nply3nre8fe3wspkulmmlgwie9cpaijndq11yzcpzkgjgskhq6ujvx2pzgkcn6rhykjettjnkyyawn17vonfhixbqkdx2oohnqk8mkal3742au6dtv13kngingfveghrtf+7eqczwnhgjoklo+5qq5yjdxfjiwqnuyrfoeb6hhfiecjuue+pwcea8ajonnlpk7h1p09kaneqwesms4e6b6ar03m/2p+pupzikc8ztthepzrndgobzxkaymqcdzsaabhsc2ueperrfibbcsmbg/+5evontie63g3p9x6zrfhgryaq3aephag6uaaneatypaansaleluerwfrzxqftzasymyf/jh18q2v5zso</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> <latexit sha1_base64="qzfx2szoil8eexwtgl/a7rtix+o=">aaacinicbvdjsgnbeo2jw4zbqecvjueqlgfgbbu8bl14jgawmblct6esdnkz0n0jhch5fs/+ihcpinos/bg7i6cjdwoe71vrvs9iojpktj+n3mli0vjkfrwwtr6xuwvu71rlnaokfrrzwnqdiogzccqkkq71raajaw61ohc98mv3icslozvvt8apsttiluaj0lldvpbassccy3bctfbwibm+xf3ssvbvb4k9d8tykgi6znfodz9y/izztc17ddxpnckpoinkdfpda8y0dsfslbmpxcdolj8rorjlmch4qyse0b5pg6tprekqfjz+cyaptnle+lzdkcjj9fderkip+2ggo0oionlwg4n/ew6qwud+xqikvrdryajwyrgk8sgv3gqcqoj9tqgvtn+kayciqpvotabdcgzfnifve8uxlef2tfi6msarr3tohx0ib52herpbzvrbfd2gj/scxo1h49l4mz4mrtljorol/sd4+gy4nqi8</latexit> 8i, j. i < j =) Hard sorted[i] <sorted[i + 1] Easy 8i. sorted[i] <sorted[i + 1]

<latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> Hard 8i, j. P (i, j) Easy 8i.P (i, i + 1)

<latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="9j1rtaqqu6p8b0ow3gywf5fy8kq=">aaacahicbvbns8naej34wetx1imhl4tfqfbcioiei148vraf0iay2w7abtebslsrsmkp/huvhhtx6s/w5r9x2+agrq8ghu/nmdmvsdht2nw/rzxvtfwnzdxwfntnd2/fpjisqtivhfzjzgpzclcinala1uxz2kgkxvhaat0y3e79+iovisxiqq8t6ke4k1jicnzgatvhrtcwmhpesn0htsaoumql1d9v2wxxcwday8tlsaeyvnr2v6stkzsiqhoolwp6bql9ezaaeu7h+vaqailjahdp01cbi6r80eybmtozsgezs0wjjwbq74krjpqarohpjlduquvvkv7nnvmdxvsjjpjuu0hmi8kuix2jarqowyqlmg8nwuqycysipswx0sazvanbw3x5mdquhm91vpvlqvkmiymhj3akrfdgcspwbxwoaoexpmmrvflp1ov1bn3mw1esboyi/sd6/aecs5s2</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> <latexit sha1_base64="/u9z2c+nkvumm7lkdb869+4hdso=">aaab/xicbvdlssnafl2pr1pf8bfzm1ieihisexrzdooygn1ag8pkommhtizhziluupwvny4ucet/upnvnlzzaoubgcm593lpncdhtgnx/byks8srq2vf9dlg5tb2jr2711bxkgmtk5jhshvgrtkttk6z5rsvsiqjgnnmmlyz+m0hkhwlxb0ejdspcf+wkbgsjds1dzphldhnidmovmfnij16j1277drufgiredkpq45a1/7q9gksrlrowrfsbc9ntj9hqrnhdfzqpiommaxxn7ynftiiys+m6cfo2cg9zgkyjzsaqr83mhwpnyocmxlhpvdz3kt8z2unorzymyasvfnbzofclcmdo0kvqmckjzqpdmfempmvkqgwmghtwmmu4m1/eze0zh3pdby7i3l1oq+jcidwbbxw4bkqcas1qaobr3igv3iznqwx6936mi0wrhxnh/7a+vwbc4stqq==</latexit> Hard 8i, j. P (i, j) Thanks Nadia!! Easy 8i.P (i, i + 1)

Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi

Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem

Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem sorted = foldl adder ([], false) sorted adder (es, d) next =...

Synthesizing Insert adjacent (lo, hi) in (sorted) { lo < hi sorted = sorted + newelem But in imperative language sorted = foldl adder ([], false) sorted adder (es, d) next =...

Implications Not clear need to adjust at least 1) loop translation 2) synthesis Formal correctness would help!!

Security is really hard

Hot off the presses!! September 28

Manual Security Invariants Invariant: All urls are in the urlset

Manual Security Invariants Invariant: All urls are in the urlset All connections use the urlset