Nuprl Lemma : equal-paths-eta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[p,q:{X ⊢ _:Path(A)}].
p = q ∈ {X ⊢ _:Path(A)} supposing path-eta(p) = path-eta(q) ∈ {X.𝕀 ⊢ _:(A)p}
Proof
Definitions occuring in Statement :
path-eta: path-eta(pth),
pathtype: Path(A),
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
pathtype: Path(A),
cubical-lam: cubical-lam(X;b),
term-to-path: <>(a),
cubicalpath-app: pth @ r,
path-eta: path-eta(pth)
Lemmas referenced :
cubical-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
cubical-type-cumulativity2,
path-eta_wf,
pathtype_wf,
cubical-type_wf,
cubical_set_wf,
term-to-path_wf,
cubical-fun-eta
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
equalityIstype,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
applyLambdaEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[p,q:\{X \mvdash{} \_:Path(A)\}]. p = q supposing path-eta(p) = path-eta(q)
Date html generated:
2020_05_20-PM-03_19_05
Last ObjectModification:
2020_04_06-PM-06_35_07
Theory : cubical!type!theory
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