Calculational HoTT
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Based on a loose correspondence between, on one hand, a first order version of intuitionistic logic with preeminence of equality and equivalence over implication, and on the other hand, homotopic equivalence properties of identity, Π, Σ and coproduct types, we formally restate homotopic type theory (HoTT) with equality and homotopic equivalence playing a preeminent role. In addition to this, we exhibit a calculational way of writing effective and elegant formal proofs based on appropriate notations and formats, as well as algebraic identities and inference rules involving the homotopic equivalences with which we restate HoTT.
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