Strong negation in the theory of computable functionals TCF

We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and G\"{o}del's system $\mathbf{T}$, by defining simultaneously the strong negation $A^{\mathbf{N}}$ of a formula $A$ and the strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get the strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of the double negation elimination for strong negation in TCF, and we study the so-called tight formulas of TCF i.e., formulas implied from the weak negation of their strong negation. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.