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Equational Reasoning in Algebraic Structures: a Complete Tactic
"... We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete fo ..."
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We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete for groups and rings. Completeness means that the method succeeds in proving an equality if and only if that equality is provable from the the group/ring axioms. Finally we characterize in what way our method is incomplete for fields.
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"... We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete fo ..."
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We present rational, a Coq tactic for equational reasoning in abelian groups, commutative rings, and fields. We give an mathematical description of the method that this tactic uses, which abstracts from Coq specifics. We prove that the method that rational uses is correct, and that it is complete for groups and rings. Completeness means that the method succeeds in proving an equality if and only if that equality is provable from the the group/ring axioms. Finally we characterize in what way our method is incomplete for fields.
A Decision Procedure for Equational Reasoning in Commutative Algebraic Structures
"... Abstract. We present a decision procedure for equational reasoning in abelian groups, commutative rings and fields that checks whether a given equality can be proven from the axioms of these structures. This has been implemented as a tactic in Coq; here we give a mathematical description of the deci ..."
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Abstract. We present a decision procedure for equational reasoning in abelian groups, commutative rings and fields that checks whether a given equality can be proven from the axioms of these structures. This has been implemented as a tactic in Coq; here we give a mathematical description of the decision procedure that abstracts from Coq specifics, making the work in principle adaptable to other theorem provers. Within Coq we prove that this decision procedure is correct. On the metalevel we analyse its completeness, showing that it is complete for groups and rings in the sense that the tactic succeeds in finding a proof of an equality if and only if that equality is provable from the group/ring axioms without any hypotheses. Finally we characterize in what way our method is incomplete for fields.