cubetac
Installation
The solver is packaged up with stack, a simple stack build should install and
compile everything.
Usage
The solver can be called on .cube files, some examples can be found in
examples/.
A .cube file consists of a context and a number of goals. Example:
zero : []
one : []
seg : [(zero , one)]
---
app1 = [(seg 1, seg 1) , (zero , one)]
inv = [(one , zero)]
The cube has three constructors. The 0-paths zero and one have an empty
boundary. The 1-path seg has one pair of faces as its boundary. The marker
--- ends the definition of the context and starts the goal section. Each goal
consists of an identifier, we have app1 and inv in this example, and a
boundary prescription after an = sign.
The boundary of app1 consists of 4 faces, so we are describing the boundary of
a square. Each face is implicitly abstracted over a interval variable, which may
be used. For instance, seg 1 means that we apply that interval variable to the
seg path, while zero is just a constant path.
A .cube file can be called with a single goal to be solved. For instance,
stack exec cubetac-exe examples/interval.cube app1
will load the above example and search for a proof of app1. The prover can be
asked to give more debugging information by appending a verbose as follows:
stack exec cubetac-exe examples/interval.cube app1 verbose
Next steps
Introduce constructor terms
Integrate with Agda
Making representations of types and terms more general
- Extension types
- Partially specified boundaries
- Reversal of paths: Translation between formulas with reversals and partially specified cubes
Kan composition solver
- Implement some basic heuristics:
- Open squares on top
- have depth limit
- congruence closure of hcomps with constructors, search with depth limit (unifier?)
Dependent cubes? Match hosting cube
Integrate with proof search in dependent type theory (Agsy rescuable?)
Examples
- Loopspaces and simple filling problems in arbitrary dimensions with even number of interval substitutions
- S1 = Susp Bool for constructors around hcomps
- https://github.com/agda/cubical/blob/master/Cubical/HITs/James/Inductive/Coherence.agda
- Smash product:
- ...
- pentagon!