materias/conteudos/8. prova _de_corretude_de_algoritmos_recursivos.svg

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  <rect x="0" y="0" width="927.1468433630628" height="1264.0680313585442" fill="#ffffff"></rect><g transform="translate(170.79723398806277 10) rotate(0 373.1748046875 20.69999999999999)"><text x="0" y="0" font-family="Helvetica, Segoe UI Emoji" font-size="36px" fill="#1864ab" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">8. Prova de corretude de algoritmos recursivos</text></g><g transform="translate(23.217672561634117 134.3249377185312) rotate(0 59.76597595214844 17.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="28px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Exemplos</text></g><g transform="translate(100.19225636695114 247.1082607315903) rotate(0 288.1797790527344 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Provando a corretudo do algoritmo que calcula o fatorial</text></g><g stroke-linecap="round" transform="translate(10 220.8258350388258) rotate(0 34 34.5)"><path d="M14.39 6.44 C19.37 1.93, 27.94 -0.18, 34.45 -0.26 C40.95 -0.35, 48.11 1.96, 53.41 5.94 C58.71 9.91, 64.26 16.76, 66.24 23.59 C68.23 30.42, 67.4 40.7, 65.33 46.91 C63.27 53.12, 59.2 57.23, 53.83 60.85 C48.46 64.46, 40.01 68.29, 33.11 68.62 C26.21 68.94, 17.69 66.44, 12.45 62.79 C7.2 59.13, 3.12 53.49, 1.62 46.68 C0.13 39.86, 0.45 29.06, 3.5 21.9 C6.55 14.75, 16.65 7.03, 19.94 3.74 C23.22 0.44, 23.12 1.92, 23.22 2.13 M38.32 1.91 C44.57 1.39, 52.22 2.06, 57.3 6.88 C62.39 11.69, 67.16 23.92, 68.84 30.83 C70.53 37.74, 70.47 42.22, 67.41 48.34 C64.34 54.47, 56.56 64.53, 50.45 67.58 C44.34 70.63, 37.32 67.73, 30.76 66.64 C24.2 65.54, 16.22 64.88, 11.09 61.03 C5.95 57.19, 1.2 50.7, -0.07 43.56 C-1.33 36.43, 0.32 25.03, 3.48 18.21 C6.64 11.39, 13.56 5.62, 18.89 2.66 C24.22 -0.3, 32.54 0.64, 35.45 0.47 C38.35 0.29, 35.85 0.87, 36.31 1.62" stroke="#d9480f" stroke-width="2" fill="none"></path></g><g transform="translate(40.16437072139564 237.93065108788988) rotate(0 3.7939987182617188 17.5)"><text x="3.7939987182617188" y="0" font-family="Virgil, Segoe UI Emoji" font-size="28px" fill="#d9480f" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">1</text></g><g transform="translate(188.03201511998805 325.903230897064) rotate(0 187.5 60)"><text x="0" y="0" font-family="Cascadia, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(n)</text><text x="0" y="24" font-family="Cascadia, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">    if n == 0</text><text x="0" y="48" font-family="Cascadia, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">        return 1</text><text x="0" y="72" font-family="Cascadia, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">    else</text><text x="0" y="96" font-family="Cascadia, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">        return n * fatorial(n-1)</text></g><g stroke-linecap="round" transform="translate(174.27788729229735 305.55437135852736) rotate(0 210.61922613289067 83.93095805972303)"><path d="M32 0 M32 0 C106.15 0.26, 180.93 -0.21, 389.24 0 M32 0 C113.07 -1.52, 193 -1.18, 389.24 0 M389.24 0 C411.19 -1.77, 419.74 10.58, 421.24 32 M389.24 0 C410.8 0.26, 423.15 9.45, 421.24 32 M421.24 32 C420.79 59.07, 420.83 90.19, 421.24 135.86 M421.24 32 C419.71 72.53, 421.2 112.24, 421.24 135.86 M421.24 135.86 C420.99 158.09, 409.41 168.32, 389.24 167.86 M421.24 135.86 C421.28 158.65, 409.27 166.41, 389.24 167.86 M389.24 167.86 C261.57 167.56, 135.32 168.08, 32 167.86 M389.24 167.86 C276.83 170.5, 164.28 169.95, 32 167.86 M32 167.86 C10.35 166.19, 1.5 156.06, 0 135.86 M32 167.86 C12.49 169.18, -0.53 156.92, 0 135.86 M0 135.86 C2.3 111.1, -0.87 86.33, 0 32 M0 135.86 C0.45 96.38, -1.25 57.51, 0 32 M0 32 C-1.94 8.75, 10.77 -0.01, 32 0 M0 32 C-0.08 11.27, 12.64 1.09, 32 0" stroke="#000000" stroke-width="1" fill="none"></path></g><g stroke-linecap="round" transform="translate(175.96053853365004 543.2389126826392) rotate(0 24.5 24.5)"><path d="M43.47 10.15 C46.2 14.06, 48.18 20.57, 47.76 25.78 C47.34 30.99, 44.28 37.3, 40.94 41.41 C37.6 45.52, 32.47 50.01, 27.72 50.43 C22.97 50.86, 17.01 47.04, 12.45 43.95 C7.88 40.86, 2.01 36.94, 0.33 31.9 C-1.36 26.87, 0.13 18.56, 2.34 13.73 C4.54 8.9, 8.97 5.4, 13.55 2.91 C18.14 0.43, 24.53 -3.34, 29.86 -1.18 C35.19 0.98, 42.6 12.66, 45.55 15.87 C48.49 19.09, 47.76 18.05, 47.54 18.11 M23.9 0.94 C29.01 0.81, 37.53 0.3, 41.9 3.82 C46.27 7.35, 49.68 16.75, 50.13 22.09 C50.59 27.44, 47.65 31.78, 44.62 35.89 C41.59 40, 36.77 45.08, 31.98 46.73 C27.18 48.39, 20.3 47.77, 15.88 45.83 C11.45 43.89, 8.28 39.1, 5.42 35.09 C2.56 31.08, -1.94 26.82, -1.28 21.75 C-0.61 16.68, 5.3 8.06, 9.42 4.67 C13.54 1.29, 20.57 2.51, 23.43 1.43 C26.29 0.34, 26.22 -1.87, 26.6 -1.85" stroke="#000000" stroke-width="2" fill="none"></path></g><g transform="translate(197.92642331010694 555.4147965435689) rotate(0 2.7099990844726562 12.5)"><text x="2.7099990844726562" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">1</text></g><g transform="translate(234.78654143580786 555.2812497485493) rotate(0 52.71995544433594 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Conjectura</text></g><g transform="translate(299.97190669345923 605.5125411877091) rotate(0 142.21994018554688 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(n) = n!, para n &gt;= 0</text></g><g stroke-linecap="round" transform="translate(184.4958974560152 656.6764565300234) rotate(0 24.5 24.5)"><path d="M27.68 0.96 C32.38 1.13, 37.68 3.92, 41.55 7.79 C45.43 11.66, 51.01 18.82, 50.94 24.2 C50.87 29.57, 44.58 36.08, 41.15 40.04 C37.71 44.01, 35.23 46.76, 30.34 47.98 C25.45 49.2, 16.5 49.59, 11.81 47.36 C7.12 45.13, 3.93 40, 2.23 34.58 C0.52 29.17, -0.44 19.98, 1.57 14.89 C3.58 9.8, 6.49 3.18, 14.27 4.04 C22.05 4.9, 44.25 13.1, 48.24 20.05 C52.23 26.99, 38.56 45.7, 38.24 45.73 M5.2 8.78 C8.25 5.53, 16.87 3.04, 22.62 1.96 C28.38 0.89, 35.4 -0.61, 39.74 2.31 C44.07 5.22, 47.55 13.94, 48.62 19.45 C49.69 24.95, 48.1 30.5, 46.17 35.32 C44.23 40.14, 42.02 46.52, 36.99 48.36 C31.97 50.21, 21.74 48.03, 16 46.4 C10.27 44.77, 4.97 42.14, 2.6 38.59 C0.24 35.03, 0.92 30.47, 1.8 25.07 C2.69 19.68, 6.94 8.7, 7.91 6.22 C8.87 3.74, 7.62 10.07, 7.59 10.19" stroke="#000000" stroke-width="2" fill="none"></path></g><g transform="translate(202.05178619975726 668.852340390953) rotate(0 7.1199951171875 12.5)"><text x="7.1199951171875" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">2</text></g><g transform="translate(243.32190035817302 668.7187935959334) rotate(0 51.729957580566406 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Caso base</text></g><g transform="translate(308.5072656158245 718.9500850350935) rotate(0 213.80987548828125 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Se n = 0, então fatorial(n) = 1, pois 0! = 1</text></g><g stroke-linecap="round" transform="translate(193.4150949758241 879.3730383790248) rotate(0 24.5 24.5)"><path d="M20.34 -1.32 C24.76 -3.14, 30.94 -1.77, 35.44 1.22 C39.93 4.2, 45.07 11.29, 47.31 16.6 C49.55 21.91, 50.46 28, 48.86 33.08 C47.27 38.15, 42.41 44.19, 37.75 47.04 C33.09 49.89, 26.41 51.39, 20.9 50.19 C15.4 49, 8.22 44.38, 4.73 39.89 C1.24 35.41, -0.41 28.76, -0.04 23.29 C0.34 17.81, 3.37 10.74, 6.98 7.02 C10.59 3.3, 18.68 1.93, 21.62 0.97 C24.56 0, 24.35 0.64, 24.63 1.22 M21.62 2.09 C27.01 1.96, 35.19 4.88, 39.57 7.11 C43.95 9.34, 47.12 11.25, 47.88 15.47 C48.65 19.68, 46.07 27.01, 44.17 32.4 C42.27 37.8, 40.8 45.44, 36.48 47.83 C32.17 50.23, 23.18 48.64, 18.28 46.79 C13.38 44.94, 10.45 40.42, 7.08 36.74 C3.7 33.06, -2.16 29.18, -1.97 24.71 C-1.78 20.24, 4.4 14.2, 8.2 9.91 C12 5.61, 18.21 0.68, 20.84 -1.06 C23.46 -2.79, 24.08 -1.49, 23.97 -0.52" stroke="#000000" stroke-width="2" fill="none"></path></g><g transform="translate(211.2809812781599 891.5489222399544) rotate(0 6.80999755859375 12.5)"><text x="6.80999755859375" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">3</text></g><g transform="translate(252.24109787798216 891.4153754449349) rotate(0 83.159912109375 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Hipótese indutiva</text></g><g transform="translate(312.9416651624283 937.5059889193751) rotate(0 142.81991577148438 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(k) = k!, para k &gt;= 0</text></g><g stroke-linecap="round" transform="translate(198.0301758995746 994.5577107259237) rotate(0 24.5 24.5)"><path d="M29.66 0.18 C35.1 0.82, 41.06 3.27, 44.47 7.55 C47.87 11.84, 50.21 20.1, 50.09 25.87 C49.96 31.64, 47.16 38.43, 43.72 42.17 C40.28 45.91, 34.66 47.85, 29.43 48.3 C24.2 48.75, 16.76 47.67, 12.34 44.89 C7.93 42.1, 4.84 36.59, 2.94 31.59 C1.04 26.59, -0.66 19.86, 0.94 14.88 C2.54 9.9, 7.21 4.01, 12.51 1.72 C17.82 -0.57, 28.83 0.79, 32.78 1.14 C36.72 1.49, 36.41 3.37, 36.2 3.84 M25.31 -2.74 C30.25 -2.6, 39.48 5.44, 42.97 10.03 C46.45 14.61, 46.32 19.85, 46.2 24.77 C46.08 29.68, 44.85 35.37, 42.23 39.53 C39.6 43.69, 35.11 48.83, 30.45 49.71 C25.78 50.6, 18.53 47.47, 14.25 44.82 C9.96 42.17, 7.14 37.94, 4.73 33.81 C2.33 29.68, -1.46 25.03, -0.18 20.04 C1.09 15.04, 7.57 6.82, 12.39 3.86 C17.22 0.89, 26.45 2.49, 28.79 2.24 C31.12 1.99, 26.93 2.19, 26.41 2.37" stroke="#000000" stroke-width="2" fill="none"></path></g><g transform="translate(216.30606586401979 1006.7335945868533) rotate(0 6.399993896484375 12.5)"><text x="6.399993896484375" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">4</text></g><g transform="translate(256.85617880173265 1006.6000477918337) rotate(0 98.84991455078125 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Devemos provar que</text></g><g transform="translate(314.21687013042015 1175.9627248596612) rotate(0 163.46987915039062 37.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(k+1) = fatorial(k) * (k+1)</text><text x="0" y="25" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">             = k! * (k+1)</text><text x="0" y="50" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">             = (k+1)!</text></g><g transform="translate(316.39842599057204 1065.2539916504777) rotate(0 100.35993194580078 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(k+1) = (k+1)!</text></g><g stroke-linecap="round" transform="translate(205.2398723547467 1111.2906518614348) rotate(0 24.5 24.5)"><path d="M7.69 8.44 C10.86 4.99, 17.68 2.26, 22.41 1.37 C27.15 0.48, 31.97 0.5, 36.1 3.09 C40.22 5.68, 45.09 12.03, 47.16 16.92 C49.22 21.8, 50.16 27.75, 48.5 32.39 C46.84 37.04, 42.32 42.32, 37.18 44.81 C32.05 47.29, 23.41 48.43, 17.7 47.3 C12 46.18, 6.22 42.41, 2.96 38.07 C-0.31 33.73, -2.97 26.64, -1.89 21.25 C-0.8 15.85, 6.57 8.88, 9.47 5.7 C12.37 2.52, 14.95 2.3, 15.51 2.17 M42.75 10.77 C45.65 15.15, 50.61 22.3, 50.36 27.79 C50.11 33.29, 45.55 40.31, 41.23 43.72 C36.92 47.12, 29.9 48.56, 24.48 48.22 C19.06 47.88, 12.9 44.75, 8.71 41.7 C4.53 38.65, 0.49 34.17, -0.61 29.92 C-1.72 25.67, -0.57 20.6, 2.07 16.21 C4.7 11.82, 10.32 5.89, 15.18 3.59 C20.04 1.28, 26.06 1.48, 31.25 2.38 C36.44 3.28, 44.48 7.24, 46.31 9 C48.15 10.76, 42.83 12.31, 42.27 12.92" stroke="#000000" stroke-width="2" fill="none"></path></g><g transform="translate(223.735763539895 1123.4665357223644) rotate(0 6.17999267578125 12.5)"><text x="6.17999267578125" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="middle" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">5</text></g><g transform="translate(264.06587525690475 1123.3329889273448) rotate(0 71.51992797851562 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#c92a2a" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">Passo indutivo</text></g><g transform="translate(344.4037082375977 822.4230774436953) rotate(0 71.7999496459961 12.5)"><text x="0" y="0" font-family="Virgil, Segoe UI Emoji" font-size="20px" fill="#000000" text-anchor="start" style="white-space: pre;" direction="ltr" dominant-baseline="text-before-edge">fatorial(0) = 1</text></g><g transform="translate(502.14849815663604 911.9885074273052) rotate(0 12.321888133203288 -85.41817824861164)" stroke="none"><path fill="#5c940d" d="M 1.51,-83.17 Q 1.51,-83.17 2.87,-81.94 4.23,-80.70 5.55,-79.29 6.87,-77.88 7.99,-76.60 9.11,-75.32 9.86,-74.43 10.61,-73.54 11.29,-72.71 11.97,-71.88 12.14,-71.63 12.32,-71.37 12.44,-71.09 12.56,-70.81 12.61,-70.51 12.66,-70.21 12.64,-69.90 12.61,-69.60 12.52,-69.31 12.42,-69.02 12.26,-68.77 12.10,-68.51 11.88,-68.30 11.66,-68.08 11.40,-67.93 11.13,-67.78 10.84,-67.69 10.55,-67.61 10.24,-67.59 9.94,-67.58 9.64,-67.64 9.34,-67.70 9.07,-67.83 8.79,-67.96 8.56,-68.15 8.32,-68.34 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