Thm* n: , a,b: . sum(a+b i | i < n) = ((n (a+a+b (n-1))) 2) | [sum_arith] |
Thm* n: , a,b: . sum(a+b i | i < n) 2 = n (a+a+b (n-1)) | [sum_arith1] |
Thm* k: , f,g:( k  ), p:( k  ). sum(if p(i) f(i)+g(i) else f(i) fi | i < k) = sum(f(i) | i < k)+sum(if p(i) g(i) else 0 fi | i < k) | [sum-ite] |
Thm* k,b: , f:( k  ). ( x: k. b f(x))  b k sum(f(x) | x < k) | [sum_lower_bound] |
Thm* k,b: , f:( k  ). ( x: k. f(x) b)  sum(f(x) | x < k) b k | [sum_bound] |
Thm* k: , f,g:( k  ). ( x: k. f(x) g(x))  sum(f(x) | x < k) sum(g(x) | x < k) | [sum_le] |