The idea of a \emph{context lemma} spans a range of programming-language models: from Milner's original through the CIU theorem to `CIU-like' results for multiple language features. Each shows that to prove observational equivalence between program terms it is enough to test only some restricted class of contexts: applicative, evaluation, reduction, \textit{etc.}

We formally reconstruct a distinctive proof method for context lemmas based on cyclic inclusion of three program approximations: by triangulating between applicative' andlogical' relations we prove that both match the observational notion, while being simpler to compute. Moreover, the observational component of the triangle condenses a series of approximations covering variation in the literature around what variable-capturing structure qualifies as a `context'.

Although entirely concrete, our approach involves no term dissection or inspection of reduction sequences; instead we draw on previous context lemmas using operational logical relations and biorthogonality. We demonstrate the method for a fine-grained call-by-value presentation of the simply-typed lambda-calculus, and extend to a CIU result formulated with frame stacks.

All this is formalised and proved in Agda: building on work of Allais et al., we exploit dependent types to specify lambda-calculus terms as well-typed and well-scoped by construction. By doing so, we seek to dispel any lingering anxieties about the manipulation of concrete contexts when reasoning about bound variables, capturing substitution, and observational equivalences.