Nuprl Lemma : path-type-at
∀[X,A,a,b,I,alpha:Top].
((Path_A a b)(alpha) ~ {p:{w:J:fset(ℕ) ⟶ f:J ⟶ I ⟶ u:Point(dM(J)) ⟶ A(f(alpha))|
∀J,K:fset(ℕ). ∀f:J ⟶ I. ∀g:K ⟶ J. ∀u:Point(dM(J)).
((w J f u f(alpha) g) = (w K f ⋅ g (u f(alpha) g)) ∈ A(g(f(alpha))))} |
((p I 1 0) = a(alpha) ∈ A(alpha)) ∧ ((p I 1 1) = b(alpha) ∈ A(alpha))} )
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
interval-type: 𝕀,
cubical-term-at: u(a),
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cube-set-restriction: f(s),
nh-comp: g ⋅ f,
nh-id: 1,
names-hom: I ⟶ J,
dM1: 1,
dM0: 0,
dM: dM(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
top: Top,
all: ∀x:A. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
path-type: (Path_A a b),
pathtype: Path(A),
cubical-subset: {t:T | ∀I,alpha. psi[I; alpha; t]},
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
uall: ∀[x:A]. B[x],
interval-presheaf: 𝕀
Lemmas referenced :
cubical_type_at_pair_lemma,
interval-type-at,
I_cube_pair_redex_lemma,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
because_Cache,
isect_memberFormation,
sqequalAxiom,
hypothesisEquality
Latex:
\mforall{}[X,A,a,b,I,alpha:Top].
((Path\_A a b)(alpha) \msim{} \{p:\{w:J:fset(\mBbbN{}) {}\mrightarrow{} f:J {}\mrightarrow{} I {}\mrightarrow{} u:Point(dM(J)) {}\mrightarrow{} A(f(alpha))|
\mforall{}J,K:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I. \mforall{}g:K {}\mrightarrow{} J. \mforall{}u:Point(dM(J)).
((w J f u f(alpha) g) = (w K f \mcdot{} g (u f(alpha) g)))\} |
((p I 1 0) = a(alpha)) \mwedge{} ((p I 1 1) = b(alpha))\} )
Date html generated:
2017_01_10-AM-08_53_25
Last ObjectModification:
2016_12_19-AM-11_17_09
Theory : cubical!type!theory
Home
Index