Nuprl Lemma : path-point_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (path-point(pth) ∈ {X.𝕀 ⊢ _:(A)p})
Proof
Definitions occuring in Statement :
path-point: path-point(pth),
path-type: (Path_A a b),
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,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
path-point: path-point(pth),
subtype_rel: A ⊆r B,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X)
Lemmas referenced :
cubical-path-app_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cc-fst_wf,
csm-ap-term_wf,
path-type_wf,
cubical-term_wf,
csm-path-type,
cubical-type-cumulativity2,
cc-snd_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_A a b)\}].
(path-point(pth) \mmember{} \{X.\mBbbI{} \mvdash{} \_:(A)p\})
Date html generated:
2020_05_20-PM-03_27_45
Last ObjectModification:
2020_04_06-PM-06_46_27
Theory : cubical!type!theory
Home
Index