What holds in a context?
In order to establish that ℒ[A] = ℒ[B] follows from a set of assumptions Γ often one provesA =B and then invokes the principle of substitution of equals for equals. It has been observed that in the ancillary proof ofA =B one is allowed to use, in addition to those assumptions of Γ which are free for ℒ, certain (open) sentencesP which may not be part of Γ and may not follow from Γ, but are related to the context ℒ. We show that in an appropriate formal system there is a closed form solution to the problem of determining precisely what sentencesP can be used. We say that those sentenceshold in the context ℒ under the set of assumptions Γ. We suggest how the solution could be exploited in an interactive theorem prover.
Key wordsContext sensitive substitution interactive theorem proving rewriting typed lambda calculus higher order logic
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