def setUp(self):
self.a = UnnamedOpetopicSet.Variable("a", UnnamedOpetope.Arrow())
self.b = UnnamedOpetopicSet.Variable("b", UnnamedOpetope.Arrow())
self.i1 = UnnamedOpetopicSet.Variable(
"i1", UnnamedOpetope.OpetopicInteger(1))
self.i2 = UnnamedOpetopicSet.Variable(
"i2", UnnamedOpetope.OpetopicInteger(2))
self.i3 = UnnamedOpetopicSet.Variable(
"i3", UnnamedOpetope.OpetopicInteger(3))
self.c = UnnamedOpetopicSet.Variable(
"c",
UnnamedOpetope.Graft(
UnnamedOpetope.Shift(UnnamedOpetope.OpetopicInteger(2)),
UnnamedOpetope.OpetopicInteger(2),
UnnamedOpetope.Address.fromList([['*']], 2)))
def test_OpetopicTree(self):
self.assertEqual(
UnnamedOpetope.OpetopicTree(None).eval(),
UnnamedOpetope.Degen(UnnamedOpetope.Arrow()).eval())
for i in range(5):
self.assertEqual(
UnnamedOpetope.OpetopicTree([None] * i).eval(),
UnnamedOpetope.Shift(UnnamedOpetope.OpetopicInteger(i)).eval())
for i in range(5):
tree = [None] * i + [[None]] + [None] * (4 - i)
self.assertEqual(
UnnamedOpetope.OpetopicTree(tree).eval(),
UnnamedOpetope.Graft(
UnnamedOpetope.Shift(UnnamedOpetope.OpetopicInteger(5)),
UnnamedOpetope.OpetopicInteger(1),
UnnamedOpetope.address([['*'] * i], 2)).eval())
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
