Nuprl Lemma : map-path_wf
∀[G:j⊢]. ∀[S,T:{G ⊢ _}]. ∀[f:{G ⊢ _:(S ⟶ T)}]. ∀[a,b:{G ⊢ _:S}]. ∀[pth:{G ⊢ _:(Path_S a b)}].
(map-path(G;f;pth) ∈ {G ⊢ _:(Path_T app(f; a) app(f; b))})
Proof
Definitions occuring in Statement :
map-path: map-path(G;f;pth),
path-type: (Path_A a b),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
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,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
map-path: map-path(G;f;pth),
cubical-path-app: pth @ r
Lemmas referenced :
cubical-term_wf,
squash_wf,
true_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-cubical-fun,
cubical-type-cumulativity2,
cc-fst_wf,
csm-ap-term_wf,
cubical-fun_wf,
term-to-path_wf,
cubical-app_wf_fun,
csm-ap-type_wf,
path-eta_wf,
path-type-subtype,
path-type_wf,
csm-cubical-app,
csm_id_adjoin_fst_term_lemma,
csm-ap-id-term,
path-eta-0,
cubical-path-app-0,
path-eta-1,
cubical-path-app-1,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
because_Cache,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
Error :memTop,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[S,T:\{G \mvdash{} \_\}]. \mforall{}[f:\{G \mvdash{} \_:(S {}\mrightarrow{} T)\}]. \mforall{}[a,b:\{G \mvdash{} \_:S\}]. \mforall{}[pth:\{G \mvdash{} \_:(Path\_S a b)\}].
(map-path(G;f;pth) \mmember{} \{G \mvdash{} \_:(Path\_T app(f; a) app(f; b))\})
Date html generated:
2020_05_20-PM-03_20_47
Last ObjectModification:
2020_04_07-PM-03_25_45
Theory : cubical!type!theory
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