Nuprl Lemma : equiv-path_wf
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. (EquivPath(G;A;B;f) ∈ {G ⊢ _:(Path_c𝕌 A B)})
Proof
Definitions occuring in Statement :
equiv-path: EquivPath(G;A;B;f),
universe-decode: decode(t),
cubical-universe: c𝕌,
cubical-equiv: Equiv(T;A),
path-type: (Path_A 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],
equiv-path: EquivPath(G;A;B;f),
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a
Lemmas referenced :
term-to-path-wf,
cubical-universe_wf,
csm-cubical-universe,
equiv_path_wf,
equiv_path-1,
equiv_path-0,
istype-cubical-term,
cubical-equiv_wf,
universe-decode_wf,
istype-cubical-universe-term,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :memTop,
dependent_functionElimination,
independent_isectElimination,
universeIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A,B:\{G \mvdash{} \_:c\mBbbU{}\}]. \mforall{}[f:\{G \mvdash{} \_:Equiv(decode(A);decode(B))\}].
(EquivPath(G;A;B;f) \mmember{} \{G \mvdash{} \_:(Path\_c\mBbbU{} A B)\})
Date html generated:
2020_05_20-PM-07_30_30
Last ObjectModification:
2020_04_28-PM-10_11_18
Theory : cubical!type!theory
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