Nuprl Lemma : path-type-subtype
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ({X ⊢ _:(Path_A a b)} ⊆r {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,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-term: {X ⊢ _:A},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
cubical-type-ap-morph: (u a f),
pi2: snd(t),
pathtype: Path(A),
cubical-fun: (A ⟶ B),
path-type: (Path_A a b),
cubical-subset: cubical-subset
Lemmas referenced :
subtype_rel_dep_function,
I_cube_wf,
cubical-type-at_wf,
path-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
pathtype_wf,
path-type-at-subtype,
fset_wf,
nat_wf,
cube-set-restriction_wf,
names-hom_wf,
istype-cubical-type-at,
cubical-type-ap-morph_wf,
cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
instantiate,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
universeIsType,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
inhabitedIsType,
functionIsType,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. (\{X \mvdash{} \_:(Path\_A a b)\} \msubseteq{}r \{X \mvdash{} \_:Path(A)\})
Date html generated:
2020_05_20-PM-03_14_58
Last ObjectModification:
2020_04_06-PM-05_36_40
Theory : cubical!type!theory
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