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