(* $Id: ex.thy,v 1.3 2005/06/29 07:38:01 kleing Exp$ Author: Tobias Nipkow *) header {* Power, Sum *} (*<*) theory ex imports Main begin (*>*) subsubsection {* Power *} text {* Define a primitive recursive function $pow~x~n$ that computes $x^n$ on natural numbers. *} consts pow :: "nat => nat => nat" text {* Prove the well known equation $x^{m \cdot n} = (x^m)^n$: *} theorem pow_mult: "pow x (m * n) = pow (pow x m) n" (*<*) oops (*>*) text {* Hint: prove a suitable lemma first. If you need to appeal to associativity and commutativity of multiplication: the corresponding simplification rules are named @{text mult_ac}. *} subsubsection {* Summation *} text {* Define a (primitive recursive) function $sum~ns$ that sums a list of natural numbers: $sum [n_1, \dots, n_k] = n_1 + \cdots + n_k$. *} consts sum :: "nat list => nat" text {* Show that $sum$ is compatible with $rev$. You may need a lemma. *} theorem sum_rev: "sum (rev ns) = sum ns" (*<*) oops (*>*) text {* Define a function $Sum~f~k$ that sums $f$ from $0$ up to $k-1$: $Sum~f~k = f~0 + \cdots + f(k - 1)$. *} consts Sum :: "(nat => nat) => nat => nat" text {* Show the following equations for the pointwise summation of functions. Determine first what the expression @{text whatever} should be. *}; theorem "Sum (%i. f i + g i) k = Sum f k + Sum g k" (*<*) oops (*>*) theorem "Sum f (k + l) = Sum f k + Sum whatever l" (*<*) oops (*>*) text {* What is the relationship between @{term sum} and @{text Sum}? Prove the following equation, suitably instantiated. *} theorem "Sum f k = sum whatever" (*<*) oops (*>*) text {* Hint: familiarize yourself with the predefined functions @{term map} and @{text"[i..*)