Formalization of Succinctness #98
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This is the first definition has come to my mind. It works at least for ADT < CCC intuitively.
I expect `2CC < CCC` so `ADT < CCC` should follow using transitivity.
As ≤Size is not total, <Size is not transitive without requiring ≤Size. Moreover, ≱Size is not antisymmetric. Note that ≱Size is ¬ ≤Size with the arguments flipped and the negation moved inside.
This makes it easier to use because the artifact type doesn't need to be applied when invoking `≤Size`. Furthermore, this enables proofs of `≱Size` to fix a single artifact type, for example the natural numbers, and automatically have the inhabitants it needs. The order between the quantifier over `n` and `A` doesn't have a big impact. On the one hand, the chosen order allows `≱Size` to use different artifact types for each `n`. However, it doesn't change the relation inhabitants if they are swapped because there exists a type with enough elements (i.e., union of all `A` ranging all `n`s) that can be fixed and then only a subset of the artifacts can be used for a specific `n`. On the other hand, `≤Size` is a `Set` and, thus, can't be inspected if the order is changed. This specific order is chosen purely as it's more convenient for pattern matching (e.g., one less `with` clause in case of `≤Size`).
Previously, these where not inferred correctly, but now it works™.
This reduces duplication and allows refactoring of 𝔸.
This allows to easily add more fields.
The designed succinctness definition includes a translatable constraint that the old definition was missing. This gets rid of the unfortunate `¬Compiler→¬≤` and `¬Compiler→≤` properties. A drawback of this new definition is that it breaks transitivity. Consider some languages L1 and L3 that are complete and a language L2 that is incomplete. There is an expression e in L1 that cannot be translated to L2. If we have L1 <= L2 and L2 <= L3 we cannot conclude L1 <= L3 because we know nothing about the size of e translated to L3 because we just proved that there exists no translation to L2. Note that the order of `∀ (A : 𝔸)` and `Σ[ m ∈ ℕ ]` was changed. Due to parametricity (type parameters cannot be inspected) this does not change the actual semantics of the definitions. However, it does simplify the proofs by being friendlier to pattern matching and `with` clauses avoiding additional helper functions in many cases.
This makes the names consistent with the symbol that is used now.
This merge introduces succinctness as explored in my master thesis "On the Succinctness of Languages for Static Variability."
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This is the formalization of succinctness as advertised in my master thesis "On the Succinctness of
Languages for Static Variability".
Note that there are merge conflicts between
thesis_bmandmain. Hence, this branch already includes the merge tomainand should be merged using a fast-forward merge.