Nuprl Lemma : path-trans_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[p:{G ⊢ _:(Path_c𝕌 A B)}]. (PathTransport(p) ∈ {G ⊢ _:(decode(A) ⟶ decode(B))})
Proof
Definitions occuring in Statement :
path-trans: PathTransport(p),
universe-decode: decode(t),
cubical-universe: c𝕌,
path-type: (Path_A a b),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
path-trans: PathTransport(p),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
cubical-path-app: pth @ r
Lemmas referenced :
istype-cubical-term,
path-type_wf,
cubical-universe_wf,
istype-cubical-universe-term,
cubical_set_wf,
cubical-fun_wf,
squash_wf,
true_wf,
cubical-type_wf,
csm-universe-decode,
path-eta-0,
path-eta-1,
cubical-term-eqcd,
cubical-path-app-0,
universe-decode_wf,
subset-cubical-type,
sub_cubical_set_self,
cubical-path-app-1,
path-type-subtype,
path-eta_wf,
csm-cubical-universe,
univ-trans_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
lambdaEquality_alt,
hyp_replacement,
hypothesisEquality,
universeIsType,
sqequalHypSubstitution,
sqequalRule,
axiomEquality,
thin,
instantiate,
extract_by_obid,
isectElimination,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
imageElimination,
Error :memTop,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
applyLambdaEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[p:\{G \mvdash{} \_:(Path\_c\mBbbU{} A B)\}].
(PathTransport(p) \mmember{} \{G \mvdash{} \_:(decode(A) {}\mrightarrow{} decode(B))\})
Date html generated:
2020_05_20-PM-07_32_30
Last ObjectModification:
2020_04_29-PM-11_14_19
Theory : cubical!type!theory
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