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Then reverse (reverse xs) = ⊥, both in the lazy and in the strict case. However, xs = ⊥ in the strict case, while it is unequal to ⊥ in the lazy case. 3 it will be shown how mathematical conditions such as finite xs and ∀x [g x = ⊥] can be expressed within the Sparkle framework. In principle, all invalidated properties can be fixed this way. The definedness conditions to be added can be obtained by carefully considering the consequences of components of quantified variables to be undefined. Such an analysis is far from easy, however, and it is easy to forget certain conditions.

Moreover, it only allows a single non-recursive expression to be shared at a time; this keeps the strict let as simple as possible, yet still sufficiently powerful. In the base set of expressions, we will include basic values (b ∈ BasicV alue), constructors (c ∈ Constructor) and case distinctions in the same manner as in [12]. Furthermore, we will also include a constant expression ‘⊥’ that denotes the undefined computation. This ⊥ can simply be regarded as an abbreviation for let x = x in x. Adding the strict let to this set of expressions leads to: e ∈ Exp ::= λ x.

In other words: the introduction of explicit strictness causes a previously valid reduction to become invalid. This in turn causes proof steps that depend on it to become invalid. That in turn causes the proof as a whole to become invalid. This effect is illustrated in the following basic example: Property: ∀x [id x = x]. Proof: Introduce x. Reduce (id x) to x. Use reflexivity. QED. Validity: This proof is only valid if the first argument of id is not explicitly marked as strict. If it is, then the strictness annotation forces x to be reduced to weak head normal form before the application (id x) may be expanded.

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A Survey of Free Math Fonts for TEX and LATEX by Stephen G.Hartke

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