cubical
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The Internal n-Cubes
This PR contains a mutual recursive definition of n-cubes, their boundaries and n-cubes with fixed boundary, as family of types parametrized over ℕ
. They're strictly isomorphic to external ones as, for example, (i j : I) → A
, (i j : I) → Partial (i ∨ ~ i ∨ j ∨ ~ j) A
and (i j : I) → A [ (i ∨ ~ i ∨ j ∨ ~ j) ↦ u i j ]
. Unless one fixes a canonical natural number n
as dimension, these isomorphisms cannot be described internally, so for more general results one must resort to macros or 2LTT.
My aim is to prove that a type has h-level n
if and only if n-boundaries can always be filled into n-cubes, generalizing the special cases about isSet'
and isGroupoid'
already existing in library.