def test_graft(self):
with self.assertRaises(DerivationError):
UnnamedOpetopicSet.graft(
UnnamedOpetopicSet.Sequent(),
UnnamedOpetopicSet.PastingDiagram.degeneratePastingDiagram(
UnnamedOpetope.OpetopicInteger(0), "x"))
with self.assertRaises(DerivationError):
UnnamedOpetopicSet.graft(
UnnamedOpetopicSet.Sequent(),
UnnamedOpetopicSet.PastingDiagram.nonDegeneratePastingDiagram(
UnnamedOpetope.Arrow(),
{UnnamedOpetope.Address.epsilon(0): "x"}))
# Incorrect grafting: ab on top of cd
with self.assertRaises(DerivationError):
UnnamedOpetopicSet.graft(
self.seq,
UnnamedOpetopicSet.PastingDiagram.nonDegeneratePastingDiagram(
UnnamedOpetope.OpetopicInteger(2), {
UnnamedOpetope.Address.epsilon(1): "cd",
UnnamedOpetope.Address.epsilon(0).shift(): "ab"
}))
# Correct grafting: ab on top of bc
UnnamedOpetopicSet.graft(
self.seq,
UnnamedOpetopicSet.PastingDiagram.nonDegeneratePastingDiagram(
UnnamedOpetope.OpetopicInteger(2), {
UnnamedOpetope.Address.epsilon(1): "bc",
UnnamedOpetope.Address.epsilon(0).shift(): "ab"
}))
def test_shift(self):
s = UnnamedOpetopicSet.graft(
self.seq,
UnnamedOpetopicSet.PastingDiagram.nonDegeneratePastingDiagram(
UnnamedOpetope.OpetopicInteger(1),
{UnnamedOpetope.Address.epsilon(1): "ac"}))
with self.assertRaises(DerivationError):
UnnamedOpetopicSet.shift(s, "ab", "A")
with self.assertRaises(DerivationError):
UnnamedOpetopicSet.shift(s, "bc", "A")
UnnamedOpetopicSet.shift(s, "ac", "A")
def suniv(seq: UnnamedOpetopicSet.Sequent, suCellName: str, cellName: str,
addr: UnnamedOpetope.Address, factorizationName: str,
fillerName: str) -> UnnamedOpetopicSet.Sequent:
"""
From
* an address :math:`[p]` (argument ``addr``);
* a cell :math:`\\alpha : \\forall_{[p]} \\mathbf{P} \\longrightarrow u`
(with name ``suCellName``);
* a cell :math:`\\beta : \\mathbf{P'} \\longrightarrow u` (with name
``cellName``), where :math:`\\mathbf{P'}` is :math:`\\mathbf{P}` except
at address :math:`[p]` where it is :math:`s`;
applies the source universal property of :math:`\\alpha` at :math:`[p]`
over :math:`\\beta`, thus creating
* a factorization cell :math:`\\xi : s \\longrightarrow \\mathsf{s}_{[p]}
\\mathbf{P}`;
* a filler :math:`A`, target universal, and source universal at
:math:`\\xi`, i.e. at address :math:`[[p]]`.
"""
# Inits
alphatype = seq.context[suCellName].type
betatype = seq.context[cellName].type
P = alphatype.source
Q = betatype.source
u = alphatype.target
# Checks & inits
if seq.pastingDiagram is not None:
raise DerivationError(
"Apply source univ. prop.",
"Sequent expected to not type a pasting diagram")
elif u is None:
raise RuntimeError("[Apply source univ. prop.] Source universal cell "
"{sucell} is a point. In valid derivations, this "
"should not happen".format(sucell=suCellName))
elif P.nodes is None:
raise DerivationError(
"Apply source univ. prop.",
"Source universal cell {sucell} cannot be degenerate",
sucell=suCellName)
elif Q.nodes is None:
raise DerivationError("Apply source univ. prop.",
"Cell {cell} cannot be degenerate",
cell=cellName)
elif addr not in P.nodes.keys():
raise DerivationError("Apply source univ. prop.",
"Address {addr} not in source of {sucell}",
addr=addr,
sucell=suCellName)
elif betatype.target != u:
raise DerivationError(
"Apply source univ. prop.",
"Cells {sucell} and {cell} are not compatible: targets differ",
cell=cellName,
sucell=suCellName)
elif P.nodes.keys() != Q.nodes.keys():
raise DerivationError(
"Apply source univ. prop.",
"Cells {sucell} and {cell} are not compatible: source pasting "
"diagrams do not have the same addresses",
cell=cellName,
sucell=suCellName)
for a in P.nodes.keys():
if a != addr and P.nodes[a] != Q.nodes[a]:
raise DerivationError(
"Apply source univ. prop.",
"Cells {sucell} and {cell} are not compatible: source pasting "
"diagrams do not agree on address {a}",
cell=cellName,
sucell=suCellName,
a=a)
# Derive xi
xishapeproof = seq.context[Q.source(addr)].type.source.shapeProof
res = UnnamedOpetopicSet.graft(
seq,
UnnamedOpetopicSet.pastingDiagram(UnnamedOpetope.Shift(xishapeproof), {
UnnamedOpetope.address([], Q.shape.dimension - 1):
Q.source(addr)
}))
res = UnnamedOpetopicSet.shift(res, P.source(addr), factorizationName)
# Derive A
omega = UnnamedOpetope.Graft(UnnamedOpetope.Shift(P.shapeProof),
UnnamedOpetope.Shift(xishapeproof),
addr.shift())
res = UnnamedOpetopicSet.graft(
res,
UnnamedOpetopicSet.pastingDiagram(
omega, {
UnnamedOpetope.address([], P.shape.dimension): suCellName,
addr.shift(): factorizationName
}))
res = UnnamedOpetopicSet.shift(res, cellName, fillerName)
# Mark A as source universal at xi and target universal
rawFillerType = res.context[fillerName].type
fillerType = Type(rawFillerType.source, rawFillerType.target)
fillerType.targetUniversal = True
fillerType.sourceUniversal.add(addr.shift())
res.context[fillerName].type = fillerType
# Done
return res
def tuniv(seq: UnnamedOpetopicSet.Sequent, tuCell: str, cell: str,
factorizationName: str,
fillerName: str) -> UnnamedOpetopicSet.Sequent:
"""
From a target universal cell :math:`\\alpha : \\mathbf{P} \\longrightarrow
t` (whose name is ``tuCell``), and another cell :math:`\\beta : \\mathbf{P}
\\longrightarrow u`, creates the universal factorization.
"""
# Inits
typealpha = seq.context[tuCell].type
typebeta = seq.context[cell].type
P = typealpha.source
targetalpha = typealpha.target
targetbeta = typebeta.target
# Checks
if seq.pastingDiagram is not None:
raise DerivationError("Apply target univ. prop.",
"Sequent cannot type a pasting diagram")
elif not isTargetUniversal(typealpha):
raise DerivationError("Apply target univ. prop.",
"First cell is expected to be target universal")
elif typebeta.source != P:
raise DerivationError(
"Apply target univ. prop.",
"Cells are expected to have the same source pasting diagram")
elif targetalpha is None or targetbeta is None:
raise RuntimeError(
"[Apply target univ. prop.] Target universal cell is a point. In "
"valid derivations, this should not happen")
# Derive the factorization cell
n = targetalpha.shape.dimension
res = UnnamedOpetopicSet.graft(
deepcopy(seq),
UnnamedOpetopicSet.pastingDiagram(
UnnamedOpetope.Shift(targetalpha.shapeProof),
{UnnamedOpetope.address([], n): targetalpha.name}))
res = UnnamedOpetopicSet.shift(res, targetbeta.name, factorizationName)
# Derive the filler
res = UnnamedOpetopicSet.graft(
res,
UnnamedOpetopicSet.pastingDiagram(
UnnamedOpetope.Graft(
UnnamedOpetope.Shift(
UnnamedOpetope.Shift(targetalpha.shapeProof)),
P.shapeProof, UnnamedOpetope.address([[]], n + 1)), {
UnnamedOpetope.address([], n + 1): factorizationName,
UnnamedOpetope.address([[]], n + 1): tuCell
}))
res = UnnamedOpetopicSet.shift(res, cell, fillerName)
# Mark the filler as target universal and source universal at the facto.
rawFillerType = res.context[fillerName].type
fillerType = Type(rawFillerType.source, rawFillerType.target)
fillerType.targetUniversal = True
fillerType.sourceUniversal.add(UnnamedOpetope.address([], n + 1))
res.context[fillerName].type = fillerType
# Done
return res
def tfill(seq: UnnamedOpetopicSet.Sequent, targetName: str,
fillerName: str) -> UnnamedOpetopicSet.Sequent:
"""
This function takes a :class:`opetopy.UnnamedOpetopicSet.Sequent`, (recall
that the context of a sequent derivable in :math:`\\textbf{OptSet${}^?$}`
is a finite opetopic set) typing a pasting diagram :math:`\\mathbf{P}`, and
solves the Kan filler problem by adding
* a new cell :math:`t` with name ``targetName``;
* a new cell :math:`\\alpha : \\mathbf{P} \\longrightarrow t` with name
``fillerName``.
"""
if seq.pastingDiagram is None:
raise DerivationError(
"Kan filling, target",
"Argument sequent expecting to type a pasting diagram")
# Source of alpha
P = seq.pastingDiagram
tPshapeProof = UnnamedOpetope.ProofTree(P.shapeTarget().toDict())
# Start deriving
res = deepcopy(seq)
res.pastingDiagram = None
# Derive t
if P.shape.dimension - 1 == 0:
# t is a point
res = UnnamedOpetopicSet.point(res, targetName)
else:
# Set u, target of t
if P.shape.isDegenerate:
u = P.degeneracyVariable()
else:
u = seq.context.target(
P.source(UnnamedOpetope.address([], P.shape.dimension - 1)))
# Derive Q, source of t
if P.shapeTarget().isDegenerate:
Q = UnnamedOpetopicSet.pastingDiagram(tPshapeProof,
seq.context.target(u))
else:
nodes = {} # type: Dict[UnnamedOpetope.Address, str]
if P.shape.isDegenerate:
nodes[UnnamedOpetope.address([], P.shape.dimension - 2)] = \
P.degeneracyVariable()
else:
readdress = P.shapeProof.eval().context
for l in P.shape.leafAddresses():
p, q = l.edgeDecomposition()
nodes[readdress(l)] = seq.context.source(P[p], q)
Q = UnnamedOpetopicSet.pastingDiagram(tPshapeProof, nodes)
if Q.shape.isDegenerate:
res = UnnamedOpetopicSet.degen(res, Q.degeneracyVariable())
else:
res = UnnamedOpetopicSet.graft(res, Q)
# Derive t, target of alpha
res = UnnamedOpetopicSet.shift(res, u, targetName)
# Derive P, source of alpha
if P.shape.isDegenerate:
res = UnnamedOpetopicSet.degen(res, u)
else:
res = UnnamedOpetopicSet.graft(res, P)
# Derive alpha
res = UnnamedOpetopicSet.shift(res, targetName, fillerName)
# Mark t as universal in the type of alpha
rawFillerType = res.context[fillerName].type
fillerType = Type(rawFillerType.source, rawFillerType.target)
fillerType.targetUniversal = True
res.context[fillerName].type = fillerType
# Done
return res