Two important requirements when aggregating the preferences of multiple agents are that the outcome should be economically efficient and the aggregation mechanism should not be manipulable. In this paper, we provide a computer-aided proof of a sweeping impossibility using these two conditions for randomized aggregation mechanisms. More precisely, we show that every efficient aggregation mechanism can be manipulated for all expected utility representations of the agents' preferences. This settles an open problem and strengthens a number of existing theorems, including statements that were shown within the special domain of assignment. Our proof is obtained by formulating the claim as a satisfiability problem over predicates from real-valued arithmetic, which is then checked using an SMT (satisfiability modulo theories) solver. In order to verify the correctness of the result, a minimal unsatisfiable set of constraints returned by the SMT solver was translated back into a proof in higher-order logic, which was automatically verified by an interactive theorem prover. To the best of our knowledge, this is the first application of SMT solvers in computational social choice.1 Alternative proofs for this important theorem were provided by Duggan [1996], Nandeibam [1997], andTanaka [2003]. R 1

Sturm sequences are a method for computing the number of real roots of a univariate real polynomial inside a given interval efficiently. In this paper, this fact and a number of methods to construct Sturm sequences efficiently have been formalised with the interactive theorem prover Isabelle/HOL. Building upon this, an Isabelle/HOL proof method was then implemented to prove interesting statements about the number of real roots of a univariate real polynomial and related properties such as non-negativity and monotonicity.

Bhat et al. developed an inductive compiler that computes density functions for probability spaces described by programs in a simple probabilistic functional language. In this work, we implement such a compiler for a modified version of this language within the theorem prover Isabelle and give a formal proof of its soundness w.r.t. the semantics of the source and target language. Together with Isabelle's code generation for inductive predicates, this yields a fully verified, executable density compiler. The proof is done in two steps, using a standard refinement approach: first, an abstract compiler working with abstract functions modelled directly in the theorem prover's logic is defined and proven sound. Then, this compiler is refined to a concrete version that returns a target-language expression.Comment: Presented at ESOP 201

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