Nuprl Lemma : term-to-pathtype-beta
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀r:{G ⊢ _:𝕀}. ∀a:{G.𝕀 ⊢ _:(A)p}. (<>a @ r = (a)[r] ∈ {G ⊢ _:A})
Proof
Definitions occuring in Statement :
term-to-pathtype: <>a,
cubicalpath-app: pth @ r,
interval-type: 𝕀,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
term-to-pathtype: <>a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
term-to-path-beta,
cubical-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
lambdaFormation_alt,
hypothesis,
universeIsType,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}r:\{G \mvdash{} \_:\mBbbI{}\}. \mforall{}a:\{G.\mBbbI{} \mvdash{} \_:(A)p\}. (<>a @ r = (a)[r])
Date html generated:
2020_05_20-PM-03_20_35
Last ObjectModification:
2020_04_06-PM-06_37_30
Theory : cubical!type!theory
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