2d:null 2e:["$","title",null,{"children":"Expression 2 | Vinícius Mello | Observable"}] 33:T3d37,Ex = { const Ex = {}; const Form = (name, attr = {}) => { const obj = { name: name, up: [], down: [], attr: attr } const fn = (...ex) => { if(obj.attr.atom) return [obj, ...ex]; ex = ex.map( (o) => { switch (typeof o) { case 'string': return Ex.Literal(o); case 'number': return Ex.Integer(o); case 'function': return Ex.Function(o); default: return o; } }); return [obj, ...ex]; } Ex[name] = fn; return fn; } const Integer = Form('Integer', {atom: true}); const Rational = Form('Rational', {atom: true}); const Literal = Form('Literal', {atom: true}); const Function = Form('Function', {atom: true}); const List = Form('List'); const Plus = Form('Plus', {flat: true, orderless: true}); const Times = Form('Times', {flat: true, orderless: true}); const Divide = Form('Divide'); const Power = Form('Power'); const Cat = Form('Cat'); const Apply = Form('Apply'); const Map = Form('Map'); const Sqrt = Form('Sqrt'); const Sequence = Form('Sequence'); const Blank = Form('Blank'); const BlankSequence = Form('BlankSequence'); const BlankNullSequence = Form('BlankNullSequence'); const LaTeX = Form('LaTeX'); const isAtom = (e) => e[0].attr.atom === true; const same = (a, b) => a[0] === b[0]; const apply = (h, e) => e[0]=h()[0]; const equal = (a, b) => { if(!same(a, b)) return false; if(a.length != b.length) return false; if(isAtom(a) && isAtom(b)) { for(let i=1; i { const matchR = (ex, pat, cap) => { if(isAtom(pat)) return equal(ex, pat); if(same(pat, Blank())) { const name = pat[1] ? pat[1][1] : undefined; const head = pat[2] ? pat[2][1] : undefined; if(head && ex[0].name !== head) return false; if(!name) return true; const en = cap[name]; if(en) return equal(ex, en); cap[name] = ex; return true; } if(!same(ex, pat)) return false; for(let i=1;i (same(e, Integer()) || same(e, Rational())); const numericValue = (e) => same(e, Rational()) ? e[1]/e[2] : e[1]; const less = (a, b) => { if(isNumeric(a) && isNumeric(b)) return numericValue(a) < numericValue(b); if(same(a, Literal()) && same(b, Literal())) return a[1] < b[1]; if((same(a, Plus()) && same(b, Plus())) || (same(a, Times()) && same(b, Times()))) { let m = a.length-1; let n = b.length-1; while(m>=1 && n>=1) { if(equal(a[m], b[n])) { m = m-1; n = n-1; } else { return less(a[m], b[n]); } } return m < n; } if((same(a, Power()) && same(b, Power()))) { if(equal(a[1], b[1])) return less(a[2], b[2]); return less(a[1], b[1]); } if(same(a, b)) { let m = a.length; let n = b.length; let i = 1; while(i { if(isAtom(e)) return e; let ex = []; if(e[0].attr.holdAll) { for(let i=1;i less(a,b) ? -1 : 1); ex.splice(0, 0, e[0]); let tex; for(let i=1; i { let tab; if(up) tab = up[0].up; else tab = rule[0].down; tab.push( (ex) => { let cap = {} if(match(ex, rule, cap)) return fn(cap); return null; }); } const toString = (e) => { if(same(e, Integer())) return e[1].toString(); if(same(e, Rational())) return `Rational(${e[1]}, ${e[2]})`; if(same(e, Literal())) return "'"+e[1]+"'"; if(same(e, Function())) return '#func'; let r = e[0].name+'('; for(let i=1; i { const s = b.split('_'); switch(s.length) { case 2: return Blank(s[0], s[1]); case 3: return BlankSequence(s[0], s[2]); case 4: return BlankNullSequence(s[0], s[3]); } } addRule(Map(_('f_Function'), _('a_')), ({f,a}) => { for(let i=1; i { a[0] = f[1]()[0]; return a; }); addRule(Cat(_('l___')), ({l}) => { let r = ''; for(let i=1; i Integer(0)); addRule(Plus(_('a_')), ({a}) => a); addRule(Plus(0,_('a__')), ({a}) => Plus(a)); addRule(Plus(_('a_Integer'),_('b_Integer'),_('c___')), ({a,b,c}) => Plus(a[1]+b[1], c)); const fracNorm = (num, den) => { const g = gcd(num, den); num = num/g; den = den/g; if(den<0) { num = -num; den = -den; } return [num, den]; } addRule(Divide(_('a_Integer'),_('b_Integer')), ({a,b}) => { const [num, den] = fracNorm(a[1], b[1]); if(den==1) return Integer(num); else return Rational(num, den); }); addRule(Divide(_('a_'),_('b_')), ({a,b}) => Times(a, Power(b, -1))); addRule(Plus(_('a_Integer'),_('b_Rational'),_('c___')), ({a,b,c}) => Plus(Divide(a[1]*b[2]+b[1], b[2]), c)); addRule(Plus(_('b_Rational'),_('a_Integer'),_('c___')), ({a,b,c}) => Plus(Divide(a[1]*b[2]+b[1], b[2]), c)); addRule(Plus(_('a_Rational'),_('b_Rational'),_('c___')), ({a,b,c}) => Plus(Divide(a[1]*b[2]+b[1]*a[2], a[2]*b[2]), c)); addRule(Times(), () => Integer(1)); addRule(Times(_('a_')), ({a}) => a); addRule(Times(1,_('a__')), ({a}) => Times(a)); addRule(Times(_('a_Integer'),_('b_Integer'),_('c___')), ({a,b,c}) => Times(a[1]*b[1], c)); addRule(Times(_('a_Integer'),_('b_Rational'),_('c___')), ({a,b,c}) => Times(Divide(a[1]*b[1], b[2]), c)); addRule(Times(_('b_Rational'),_('a_Integer'),_('c___')), ({a,b,c}) => Times(Divide(a[1]*b[1], b[2]), c)); addRule(Times(_('a_Rational'),_('b_Rational'),_('c___')), ({a,b,c}) => Times(Divide(a[1]*b[1], a[2]*b[2]), c)); addRule(Times(-1, Plus(_('a__'))), ({a}) => { const r = Plus(); for(let i = 1; i { const sa = toString(a); if(!t[sa]) { t[sa] = [a, Sequence(b)]; return false; } else { t[sa][1].push(b); return true; } } addRule(Plus(_('c__')), ({c}) => { const r = Plus(); let flag = false; const coefs = {}; for(let i=1; i { const r = Times(); let flag = false; const coefs = {}; for(let i=1; i { const r = Times(); let flag = false; const coefs = {}; for(let i=1; i Integer(1)); addRule(Power(1, _('_')), () => Integer(1)); addRule(Power(_('a_'), 1), ({a}) => a); addRule(Power(_('a_Integer'),_('b_Integer')), ({a,b}) => { if(b[1]<0) return Rational(1, Math.pow(a[1], -b[1])); else return Integer(Math.pow(a[1], b[1])); }); addRule(Power(_('a_Rational'),_('b_Integer')), ({a,b}) => { if(b[1]<0) return Divide(Math.pow(a[2], -b[1]), Math.pow(a[1], -b[1])); else return Rational(Math.pow(a[1], b[1]), Math.pow(a[2], b[1])); }); const rootContent = (fact, p, q) => { let [u, v] = [1, 1]; for(let i = 0; i { if(a[1]<0) return null; const fact = factorization(a[1]); if(b[1]>0) { const [u, v] = rootContent(fact, b[1], b[2]); if(u==1 && b[1]==1) return null; else return Times(u, Power(v, Rational(1, b[2]))); } else { const b1 = -b[1]; const k = Math.floor(b1 / b[2]); const r = b1 - k * b[2]; const [u, v] = rootContent(fact, b[2] - r, b[2]); return Times(Divide(u, Math.pow(a[1], (k + 1))), Power(v, Rational(1, b[2]))); } }); addRule(Power(_('a_Rational'),_('b_Rational')), ({a,b}) => Times(Power(a[1], b), Power(a[2], Rational(-b[1], b[2]))) ); addRule(Power(Power(_('a_'), _('b_')), _('c_Integer')), ({a,b,c}) => Power(a, Times(b, c))); addRule(Power(Times(_('a__')), _('b_Integer')), ({a,b}) => { const r = Times(); for(let i = 1; i Power(a, Rational(1,2))); addRule(LaTeX(), ({a}) => Literal('')); addRule(LaTeX(_('a_Integer')), ({a}) => Literal(`${a[1]}`)); addRule(LaTeX(_('a_Literal')), ({a}) => a); addRule(LaTeX(_('a_Rational')), ({a}) => { if(a[1]<0) return Literal(`-\\frac{${-a[1]}}{${a[2]}}`); else return Literal(`\\frac{${a[1]}}{${a[2]}}`); }); addRule(LaTeX(Plus(_('c__'))), ({c}) => { let r = ''; for(let i=1; i { const r = '-'+Eval(LaTeX(Times(c)))[1]; return Literal(r); }); const parenthesisPlus = (c) => { let r = ''; for(let i=1; i { if(c[1]==1 && c[2]==2) { let r = Eval(LaTeX(Power(b, c)))[1]; let s = parenthesisPlus(d); //if(s==='1') s = ''; if(a[1]!=1) r = `${a[1]}` + r; r = `\\frac{${r}}{${a[2]}}` return Literal(r+s); } return null; }); addRule(LaTeX(Times(_('c__'))), ({c}) => { return Literal(parenthesisPlus(c)); }); addRule(LaTeX(Power(_('a_'), _('b_Rational'))), ({a,b}) => { if(b[1]==1 && b[2]==2) { let s = Eval(LaTeX(a))[1]; return Literal(`\\sqrt{${s}}`); } return null; }); addRule(LaTeX(Power(_('a_'), _('b_'))), ({a,b}) => { let r = ''; let s = Eval(LaTeX(a))[1]; if(!(same(a, Literal()) || same(a, Integer()))) { s = '(' + s + ')'; } r = r + s +'^{'+Eval(LaTeX(b))[1]+'}'; return Literal(r); }); Ex.toString = toString; Ex.Eval = Eval; return Ex; }34:T1e86,primes = [ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 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Mello","avatar_url":"https://avatars-next.observableusercontent.com/b5d4671aa488d8bcb984f575f985332403cde5f63942ea883a6a1a630941ec05","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"Expression 2","cells":[{"id":0,"value":"md`# Expression 2`","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"$33","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":249,"value":"{ \n const {Eval, toString, Power, LaTeX, List, Plus, Times, Divide, Rational, Function, Cat, Apply, Map} = Ex;\n return tex`${(Eval(LaTeX(Times(Power(81,Rational(1,3)),Power(12, Rational(-1,2)),Plus(1,'x'))))[1])}`;\n// return toString(Eval((Times(Power(3, Rational(-1,2)),'x'))));\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":137,"value":"function gcd(a, b) { \n return (b==0) ? a : gcd(b, a % b);\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":197,"value":"function factorization(n) {\n const f = [];\n for(let p of primes) {\n let e = 0;\n while (n % p == 0) {\n n = (n / p);\n e = e + 1;\n }\n if(e != 0) f.push([p, e]);\n if(n == 1) break;\n }\n if(n == 1) return f;\n else throw \"Factorization fail.\";\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":189,"value":"$34","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W35","stdlib":"1","titling":"auto"}} 32:"$31:body"