Nuprl Lemma : path-point-1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. ((path-point(pth))[1(𝕀)] = b ∈ {X ⊢ _:A})
Proof
Definitions occuring in Statement :
path-point: path-point(pth),
path-type: (Path_A a b),
interval-1: 1(𝕀),
csm-id-adjoin: [u],
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
path-point: path-point(pth),
all: ∀x:A. B[x],
interval-1: 1(𝕀),
csm-id: 1(X),
csm-ap-term: (t)s,
subtype_rel: A ⊆r B,
uimplies: b supposing a
Lemmas referenced :
csm-cubical-path-app,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
cubical-path-app-1,
csm-ap-term_wf,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
csm-ap-id-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
dependent_functionElimination,
universeIsType,
instantiate,
hypothesisEquality,
applyEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
independent_isectElimination
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))[1(\mBbbI{})] = b)
Date html generated:
2020_05_20-PM-03_28_08
Last ObjectModification:
2020_04_06-PM-06_47_00
Theory : cubical!type!theory
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