diff --git a/engine-proto/Engine.jl b/engine-proto/engine.jl
similarity index 76%
rename from engine-proto/Engine.jl
rename to engine-proto/engine.jl
index a632581..2f7294a 100644
--- a/engine-proto/Engine.jl
+++ b/engine-proto/engine.jl
@@ -1,5 +1,3 @@
-include("HittingSet.jl")
-
 module Engine
 
 export Construction, mprod
@@ -8,24 +6,6 @@ import Subscripts
 using LinearAlgebra
 using AbstractAlgebra
 using Groebner
-using ..HittingSet
-
-# --- commutative algebra ---
-
-# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
-# polynomial rings when the coefficients are integers. we use Groebner to extend
-# support to rationals and to finite fields of prime order
-Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
-  Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
-
-function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}}
-  leading = [exponent_vector(f, 1) for f in gens(I)]
-  targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading]
-  length(HittingSet.solve(HittingSetProblem(targets), maxdepth))
-end
-
-dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
-  length(gens(base_ring(I))) - codimension(I, maxdepth)
 
 # --- primitve elements ---
 
@@ -43,6 +23,8 @@ mutable struct Point{T} <: Element{T}
   ) where T = new(coords, vec, nothing)
 end
 
+##coordnames(_::Point) = [:xₚ, :yₚ, :zₚ]
+
 function buildvec!(pt::Point)
   coordring = parent(pt.coords[1])
   pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
@@ -61,6 +43,8 @@ mutable struct Sphere{T} <: Element{T}
   ) where T = new(coords, vec, rel)
 end
 
+##coordnames(_::Sphere) = [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
+
 function buildvec!(sph::Sphere)
   coordring = parent(sph.coords[1])
   sph.vec = sph.coords
@@ -146,6 +130,10 @@ function realize(ctx::Construction{T}) where T
     end
   end
   
+  display(collect(elemenum))
+  display(coordnamelist)
+  println()
+  
   # construct coordinate ring
   coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
   
@@ -162,14 +150,16 @@ function realize(ctx::Construction{T}) where T
   # construct coordinate vectors
   for (_, elem) in elemenum
     buildvec!(elem)
+    display(elem.coords)
+    display(elem.vec)
+    println()
   end
   
   # turn relations into equations
-  eqns = vcat(
+  vcat(
     equation.(ctx.relations),
     [elem.rel for elem in ctx.elements if !isnothing(elem.rel)]
   )
-  Generic.Ideal(coordring, eqns)
 end
 
 end
@@ -182,26 +172,22 @@ a = Engine.Point{CoeffType}()
 s = Engine.Sphere{CoeffType}()
 a_on_s = Engine.LiesOn{CoeffType}(a, s)
 ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
-ideal_a_s = Engine.realize(ctx)
-println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freeom")
+eqns_a_s = Engine.realize(ctx)
 
 b = Engine.Point{CoeffType}()
 b_on_s = Engine.LiesOn{CoeffType}(b, s)
 Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
-ideal_ab_s = Engine.realize(ctx)
-println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freeom")
+eqns_ab_s = Engine.realize(ctx)
 
 spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 tangencies = [
   Engine.AlignsWithBy{CoeffType}(
     spheres[n],
     spheres[mod1(n+1, length(spheres))],
-    CoeffType(-1//1)
+    -1//1
   )
   for n in 1:3
 ]
-ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
-ideal_tan_sph = Engine.realize(ctx_tan_sph)
-println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freeom")
\ No newline at end of file
+ctx_chain = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
\ No newline at end of file
diff --git a/engine-proto/HittingSet.jl b/engine-proto/hitting-set.jl
similarity index 89%
rename from engine-proto/HittingSet.jl
rename to engine-proto/hitting-set.jl
index 347c4d2..e9aacf4 100644
--- a/engine-proto/HittingSet.jl
+++ b/engine-proto/hitting-set.jl
@@ -1,15 +1,13 @@
 module HittingSet
 
-export HittingSetProblem, solve
-
 HittingSetProblem{T} = Pair{Set{T}, Vector{Pair{T, Set{Set{T}}}}}
 
-# `targets` should be a collection of Set objects
-function HittingSetProblem(targets, chosen = Set())
-  wholeset = union(targets...)
+# `subsets` should be a collection of Set objects
+function HittingSetProblem(subsets, chosen = Set())
+  wholeset = union(subsets...)
   T = eltype(wholeset)
   unsorted_moves = [
-    elt => Set(filter(s -> elt ∉ s, targets))
+    elt => Set(filter(s -> elt ∉ s, subsets))
     for elt in wholeset
   ]
   moves = sort(unsorted_moves, by = pair -> length(pair.second))
@@ -34,6 +32,7 @@ end
 
 function solve(pblm::HittingSetProblem{T}, maxdepth = Inf) where T
   problems = Dict(pblm)
+  println(typeof(problems))
   while length(first(problems).first) < maxdepth
     subproblems = typeof(problems)()
     for (chosen, moves) in problems
@@ -57,7 +56,7 @@ end
 
 function test(n = 1)
   T = [Int64, Int64, Symbol, Symbol][n]
-  targets = Set{T}.([
+  subsets = Set{T}.([
     [
       [1, 3, 5],
       [2, 3, 4],
@@ -99,7 +98,7 @@ function test(n = 1)
       [:b, :z, :t14]
     ]
   ][n])
-  problem = HittingSetProblem(targets)
+  problem = HittingSetProblem(subsets)
   if isa(problem, HittingSetProblem{T})
     println("Correct type")
   else