Nuprl Lemma : paths-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[p:{X ⊢ _:(Path_A a b)}]. ∀[q:{X ⊢ _:Path(A)}].
p = q ∈ {X ⊢ _:(Path_A a b)} supposing p = q ∈ {X ⊢ _:Path(A)}
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
pathtype: Path(A),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-path-app: pth @ r,
guard: {T},
prop: ℙ
Lemmas referenced :
cubicalpath-app_wf,
interval-0_wf,
interval-1_wf,
path-type-subtype,
cubical-term_wf,
pathtype_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
path-type_wf,
cubical-type_wf,
cubical_set_wf,
cubical-path-app-0,
cubical-path-app-1,
path-type-ext-eq,
subtype_rel_weakening,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
hypothesis,
independent_pairFormation,
sqequalRule,
productIsType,
equalityIstype,
because_Cache,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
universeIsType,
instantiate,
setEquality,
productEquality,
independent_isectElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[p:\{X \mvdash{} \_:(Path\_A a b)\}]. \mforall{}[q:\{X \mvdash{} \_:Path(A)\}].
p = q supposing p = q
Date html generated:
2020_05_20-PM-03_17_51
Last ObjectModification:
2020_04_07-PM-00_59_12
Theory : cubical!type!theory
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