Intuitionistic theory of algebraic systems and the Heyting-valued analysis (Q1191160)

Given a ring \(f\) with a metric and a formula \(\varphi\Rightarrow\psi\) in the language of rings, we construct a translation \(\varphi^ +\Rightarrow \psi^ +\) such that the (classical) truth of the inference \(f\models (\varphi\Rightarrow \psi)\) (i.e., the inference \(\varphi_ f\Rightarrow \psi_ f\)) in the classical theory will imply, in a certain sense, the (intuitionistic) truth of the inference \(f\models (\varphi^ +\Rightarrow \psi^ +)\) [i.e., the inference \((\varphi^ +)_ f\Rightarrow (\psi^ +)_ f]\) in intuitionistic set theory.