By Bob Stat
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Additional info for Altogether Fun and Absolutely Understanding Rubik's Pocket Cube
TYPES 2008, LNCS 5497, pp. 32–48, 2009. c Springer-Verlag Berlin Heidelberg 2009 A New Elimination Rule for the Calculus of Inductive Constructions 33 precise typing. vector (S n) → vector n, thus ensuring that it cannot be applied to an empty vector. In Coquand’s setting, we could write the tail function as tail n (cons k x t) = t . Note the missing case for nil. This deﬁnition is accepted because the type system can ensure that the vector argument, being a term of type vector (S n), cannot reduce to nil.
The substitution lemma is still valid with the new rule. Lemma 1. If Γ t : T and σ : Γ → Δ, then Δ tσ : T σ. The following lemmas formally state the intuitive meaning of the uniﬁcation judgment. If a uniﬁcation succeeds positively, then the result is a uniﬁer (Lemma 2); moreover, in a sense, it is a most general uniﬁer (Lemma 3). If a uniﬁcation succeeds negatively, then there is no uniﬁer (Lemma 4). σ be a unification judgment, Lemma 2. Let Γ ; Δ, ζ [u = v : Θ] → Δ , ζ with Γ Δ u : Θ, and Γ Δ v : Θ.
We also deﬁne a translation of the uniﬁcation judgment, that takes as input the sequence of equalities H, and returns a sequence of terms whose type corresponds to the substitution of the branch (if the uniﬁcation succeeds positively); or a proof of contradiction (if the uniﬁcation succeeds negatively). In the latter case, we are done, while in the former, we use the returned sequence of terms to rewrite in the translation of the body proper, and obtain a term of the right return type. We can prove that typing is preserved by this translation: Lemma 6.
Altogether Fun and Absolutely Understanding Rubik's Pocket Cube by Bob Stat