cabal-version: 2.2 name: natural-arithmetic version: 0.1.1.0 synopsis: Arithmetic of natural numbers description: A search for terms like `arithmetic` and `natural` on hackage reveals no shortage of libraries for handling the arithmetic of natural numbers. How is this library any different some of the others? It has a particular purpose: providing a foundation on top on which other libraries may define types indexed by sizes. This uses GHC's non-inductively-defined `GHC.TypeNats.Nat`. As a rule, this does not use `unsafeCoerce` internally anywhere. . Perhaps the most direct competitor to `natural-arithmetic` is a typechecker plugin like . The big difference is that `type-nat-solver` can really only be used in application code, not in library code. This is because libraries should not require the presence of typechecker plugins. Technically, they can (you could document it), but many developers will not use libraries that have unusual install procedures like this. . This library, in places, requires users to use the 'TypeApplications` language extension. This is done when a number is only need at the type level (without a runtime witness). . This library uses a non-minimal core, providing redundant primitives in `Arithmetic.Lt` and `Arithmetic.Lte`. This is done in the interest of making it easy for user to assemble proofs. Recall that proof assembly is done by hand rather than by an SMT solver, so removing some tediousness from this is helpful to users. . This library provides left and variants variants of several functions. For example, `Arithmetic.Lte` provides both `substituteL` and `substituteR`. This is only done when there are two variants of a function. For substitution, this is the case because we have `b = c, a ≤ b ==> a ≤ c` and `a = c, a ≤ b ==> c ≤ b`. So, we provide both `substituteL` and `substituteR`. However, for addition of inequalities, we have four possible variants: `a ≤ b, c ≤ d ==> a + c ≤ b + d`, `a ≤ b, c ≤ d ==> c + a ≤ b + d`, `a ≤ b, c ≤ d ==> a + c ≤ d + b`, `a ≤ b, c ≤ d ==> c + a ≤ d + b`. Consequently, we only provide a single `plus` function, and users must use `Arithmetic.Plus.commutative` to further manipulate the inequality. . Here are the proof-manipulation vocabulary used by this library. Many of these terms are not standard, but we try to be consistent in this library: . * Weaken: Increase an upper bound without changing the bounded value . * Increment: Increase an upper bound along with the bounded value . * Decrement: Decrease an upper bound along with the bounded value . * Substitute: Replace a number with an equal number homepage: https://github.com/andrewthad/natural-arithmetic bug-reports: https://github.com/andrewthad/natural-arithmetic/issues license: BSD-3-Clause license-file: LICENSE author: Andrew Martin maintainer: andrew.thaddeus@gmail.com copyright: 2019 Andrew Martin category: Math extra-source-files: CHANGELOG.md library exposed-modules: Arithmetic.Fin Arithmetic.Equal Arithmetic.Lt Arithmetic.Lte Arithmetic.Nat Arithmetic.Types Arithmetic.Unsafe Arithmetic.Plus build-depends: base>=4.11 && <5 hs-source-dirs: src default-language: Haskell2010 ghc-options: -Wall -O2