By Klaus Weich
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Extra resources for Improving proof search in intuitionistic propositional logic
If G, 1, ° is a category, then Gop, 1, ° op is a category, too, and by the left-to-right direction of Lemma 1, we obtain (1) and (2). 60 Suppose now (1) and (2). Then, by the right-to-left direction of Lemma 1, it follows from (1) that (cat 1 left) holds in G, 1, ° , and from (2) that (cat 1 left) holds in Gop, 1, ° op . But the arrow 1A ° op f : B A of Gop is the arrow f ° 1A : A B of G; so (cat 1 right) holds in G, 1, ° . That (cat 2) holds in G, 1, ° follows from the rightto-left direction of Lemma 1 and either (1) or (2).
An equivalence of categories where these natural isomorphisms are identities boils down to isomorphism of categories as we have defined it in the preceding section. e. the category whose objects are these graphs) with graph-morphisms as arrows is isomorphic to the category of F graphs with graphmorphisms as arrows. Hence, these categories are also equivalent. This justifies our saying that the two notions of graph are equivalent. In general, two notions are to be called equivalent iff they cover objects of two categories that are equivalent.
If f = g is an instance of (cat 1 right), which means that it is of the form g ° 1A = g, then the n-th cut of f is linked to the n-th cut of g, provided there are at least n cuts in g. e. the main ° of f, displayed in g ° 1A, is not linked to any cut of g. If f = g is an instance of (cat 1 left), which means that it is of the form 1B ° g = g, then the n+1-th cut of f is linked to the n-th cut of g, provided there are at least n cuts in g. e. the main ° of 1B ° g, is not linked to any cut of g. If f = g is an instance of (cat 2), then the n-th cut of f is linked to the n-th cut of g.