Nuprl Lemma : path-type-p
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ((X.B ⊢ Path_(A)p (a)p (b)p) = ((Path_A a b))p ∈ {X.B ⊢ _})
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B
Lemmas referenced :
csm-path-type,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-term_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
equalitySymmetry,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
isectElimination,
because_Cache,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A,B:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. ((X.B \mvdash{} Path\_(A)p (a)p (b)p) = ((Path\_A a b))p)
Date html generated:
2020_05_20-PM-03_16_07
Last ObjectModification:
2020_04_06-PM-05_57_58
Theory : cubical!type!theory
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