We introduce a new way of formalizing the intensional identity type based on the fact that a entity known as computational paths can be interpreted as terms of the identity type. Our approach enjoys the fact that our elimination rule is easy to understand and use. We make this point clear constructing terms of some relevant types using our proposed elimination rule. We also show that the identity type, as defined by our approach, induces a groupoid structure. This result is on par with the fact that the traditional identity type induces a groupoid, as exposed by Hofmann & Streicher (1994).
Abstract. In this paper we present and evaluate the skill of the EC-Earth3.3 decadal prediction system contributing to the Decadal Climate Prediction Project - Component A (DCPP-A). This prediction system is capable of skilfully simulating past global mean surface temperature variations at interannual and decadal forecast times as well as the local surface temperature in regions such as the Tropical Atlantic, the Indian Ocean and most of the continental areas, although most of the skill comes from the representation of the externally forced trends. A benefit of initialisation in the predictive skill is evident in some areas of the Tropical Pacific and North Atlantic Oceans in the first forecast years, an added value that gets mostly confined to the south-east Tropical Pacific and the eastern Subpolar North Atlantic at the longest forecast times (6–10 years). The central Subpolar North Atlantic shows poor predictive skill and a detrimental effect of the initialisation due to the occurrence of an initialisation shock, itself related to a collapse in Labrador Sea convection by the third forecast year that leads to a rapid weakening of the Atlantic Meridional Overturning Circulation (AMOC) and excessive local sea ice growth. The shutdown in Labrador Sea convection responds to a gradual increase in the local density stratification in the first years of the forecast, ultimately related to the different paces at which surface and subsurface temperature and salinity drift towards their preferred mean state. This transition happens rapidly in the surface and more slowly in the subsurface, where, by the tenth forecast year, the model is still far from the typical mean states in the corresponding ensemble of historical simulations with EC-Earth3. Our study thus highlights the importance of the Labrador Sea for initialisation, the relevance of reducing model bias by model tuning or, preferably, model improvement when using full-field initialisation, and the need to identify optimal initialisation strategies.
The treatment of equality as a type in type theory gives rise to an interesting type-theoretic structure known as 'identity type'. The idea is that, given terms a, b of a type A, one may form the type IdA(a, b), whose elements are proofs that a and b are equal elements of type A. A term of this type, p : IdA(a, b), makes up for the grounds (or proof) that establishes that a is indeed equal to b. Based on that, a proof of equality can be seen as a sequence of substitutions and rewrites, also known as a 'computational path'. One interesting fact is that it is possible to rewrite computational paths using a set of reduction rules arising from an analysis of redundancies in paths. These rules were mapped by De Oliveira in 1994 in a term rewrite system known as LN DEQ −T RS. Here we use computational paths and this term rewrite system to develop the main foundations of homotopy type theory, i.e., we develop the lemmas and theorems connected to the main types of this theory, types such as products, coproducts, identity type, transport and many others. We also show that it is possible to directly construct path spaces through computational paths. To show this, we construct the natural numbers and the fundamental group of the circle, showing results connected to these structures. Keywords. Type theory, computational paths, homotopy type theory, Identity type, fundamental group of the circle, path space of natural numbers, term rewriting systems.
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