数学归纳分析程序正确性

{"type":"doc","content":[{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"1 数学归纳","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"先看个数学证明问题:","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"    s(n) = 1+2+3+...n = ((n+1) * n) / 2, 证明公式对任意整数n(n >= 0)都成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"证明：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"设置断言 G(n): 0到n的整数之和s(n) = ((n + 1) * n) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"步骤1： 基底的证明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"s(0) = ((0+1)*0) / 2 = 0，G(0)成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"步骤2：归纳的证明 ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"先假设 G(k)成立，可得到：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    s(k) = 0+1+2...+k = ((k+1) * k) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"则：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    s(k+1) = s(k) + k+1 = ((k+1) * k) / 2 + ((k+1) * 2) / 2 = ((k+1) * (k+2)) / 2","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"等式成立，推出G(k) 成立，则G(k+1)也成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"通过步骤1，步骤2证得断言G(n)是正确的","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"上面求证过程，就运用到了","attrs":{}},{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"数学归纳","attrs":{}},{"type":"text","text":"思想。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"2 运用数学归纳分析计算机程序正确性","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"求fibonacci数列([0, 1 , 1,  2,  3,....ai, a(i+1), a(i)+a(i+1)])第i项(i >= 0)","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"常规写法如下","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"static int fib(int i) {\n    if (i == 0) {\n      return 0;\n    }\n    if (i == 1) {\n      return 1;\n    }\n\n    int tmp;\n    int a = 0;\n    int b = 1;\n    int $i = 2;\n \n    while ($i <= i) {\n      $i++;\n      tmp = a;\n      a = b;\n      b = tmp +b;\n    }\n\n    return b;\n}","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"思考：对任意的输入i, 怎么证明代码输出结果都是正确的? ......","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"把上面代码结构调整为下面这样,只要水为了方便分析，逻辑和上面代码等效","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":" static int fib(int i) {\n     int[] L = new int[] {0, 0, 1};\n     if (i == 0) {\n       return L[1];\n     }\n     if (i == 1) {\n       return L[2];\n     }\n     while (L[0] <= i) {\n       next(L)；\n     }\n     return L[2];\n  }\n\n  static int next(int[] L) {\n     int tmp;\n     int a = L[1];\n     int b = L[2];\n     tmp = a;\n     a = b;\n     b = b + tmp;\n     // 计算下一次循环需要的变量\n     L[0] = L[0] + 1;\n     L[1] = a;\n     L[2] = b;\n \n     return L[2];\n  }","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"用上面的","attrs":{}},{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"数学归纳思想","attrs":{}},{"type":"text","text":"，分析下程序是否正确。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":" 证明：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    设置断言 M(i): L(i) = [i, fib(i-2), fib(i-1)]（i > 2，fib(i) 表示fibonacci第i项）","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    步骤1： 基底的证明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        i = 2时， L(2) = [2, 0, 1]；由fibonacci前3项为[0, 1, 1], 可知， L(2)成立 ;","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"    步骤2：归纳的证明","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        由程序代码可知：L(k+1) = [k+1, L(k)[2], L(k)[1]+L(k)[2]],   假设M(k)成立，则：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":1,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" L(k) = [k, fib(k-2), fib(k-1)]，所以：","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"        L(k+1)=[k+1, fib(k-1), fib(k-1)+fib(k-2)] = [k+1, fib(k-1), fib(k)], ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":" 推出M(k)成立时，M(k+1)成立","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"       ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"       所以断言正确","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"通过上面证明，可以得出，输入为i, 当循环结束时 $i = i + 1, 此时 L($i) = [i+1, fib(i-1), fib(i)], ","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","marks":[{"type":"strong","attrs":{}}],"text":"所以L($i)[2] 为fibonacci第i项, 输出正确 。","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null},"content":[{"type":"text","text":"问题：证明下面二分查找程序是否正确？","attrs":{}}]},{"type":"codeblock","attrs":{"lang":"java"},"content":[{"type":"text","text":"static boolean search(int[] a, int target) {\n    int left = 0;\n    int right = a.length - 1;\n    while (left < right) {\n      int mid = (left + right) / 2;\n      if (a[mid] > target) {\n        right = mid + 1;\n      }\n      if(a[mid] < target) {\n        left = mid + 1;\n      } else {\n        return true;\n      }\n    }\n    return false;\n  }","attrs":{}}]},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}},{"type":"paragraph","attrs":{"indent":0,"number":0,"align":null,"origin":null}}]}