Implementing IsLawfulNum as part of Issue #108
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This commit adds the new type class IsLawfulNum that contains all the laws the Haskell Num typeclass should satisfy. There currently is one law we cannot formalize here because toInteger is not yet part of our Prelude. This commit is part of issue agda#108.
These functions were formerly private. They had to made accessible to other parts of the library to allow proofs, in particular for the IsLawfulNum instances. Both functions are still "hidden" in an "Internal" module though to signal these functions as not being intended to be used from any Haskell code. This change also redefines subNat based on pattern-matching instead of an if-else-expression with ltNat, to simplify proofs.
pmbittner
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Jun 13, 2024
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to make it visible for proofs
Haskell.Law.Num should publicly import all available instances as well as the LawfulNum class. To avoid cyclic dependencies, the functions for concluding number laws were move to Haskell.Law.Num.Def.
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jespercockx
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Jul 9, 2024
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This pull-request adds the laws for the
Numtypeclass in Haskell as part of issue #108.Progress:
NatinstanceIntinstanceIntegerinstanceWordinstanceDoubleinstanceThere was one law we cannot formalize yet, which is
fromInteger (toInteger i) == ibecause there is notoIntegerfunction available yet. I documented this law as missing.For the
Natinstance, I mostly copied code from the Agda standard library for comparability, consistency, and also because it is easy 😅.For
IntegerI had to come up with new proofs because the definitions for this type are slightly different from those in the standard library. I oriented on the proofs from the standard library where possible though. Also I tried to favor equational reasoning in favor ofrewritefor better readability.Arithmetics of
Wordis mostly relying on arithmetics ofNatand arithmetics ofIntis defined solely in terms of arithmetics onWord. I hence opted to reuse the laws from the other instances, respectively. In fact, reducing the arithmetics for aNumtype in terms of anotherNuminstance is feasible as long as this reduction is homomorphic.To formalize this observation, I added definitions for homomorphisms, embeddings, as well as some other common predicates in a new module
Haskell.Prim.Functionwith respective proofs inHaskell.Law.Function. I thought that such definitions might be useful in other contexts as well, if desired. Using these definitions, we can then define the circumstances under which we may conclude anILawfulNuminstance from another one.The question remains whether such abstractions are desired. On the one hand, these are quite common concepts. On the other hand, they introduce another layer of indirection and abstractions, making some of the proofs harder to comprehend. Also, should we reuse the definitions such as
Associative,Commutative, and so on in the definition ofIsLawfulNumor should we inline those (as it currently is) for readability?Moreover, I had to introduce six new axioms to prove lawfulness of
Word, all of which are stated inlib/Haskell/Law/Num/Word.agda. I documented the new axioms and explained why they should be reasonable. It would be good if these could be reviewed. I am not 100% sure that they are correct and minimal.Also, the
Doubleinstance cannot be proven as of now. It is solely based on primitive functions for which no axioms exist yet. So I left that out for now.