Nuprl Lemma : path-type-at-subtype
∀X:j⊢. ∀A:{X ⊢ _}. ∀a,b:{X ⊢ _:A}. ∀I:fset(ℕ). ∀rho:X(I). ((Path_A a b)(rho) ⊆r Path(A)(rho))
Proof
Definitions occuring in Statement :
path-type: (Path_A a b),
pathtype: Path(A),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
path-type: (Path_A a b),
cubical-subset: cubical-subset,
uall: ∀[x:A]. B[x]
Lemmas referenced :
cubical_type_at_pair_lemma,
istype-cubical-type-at,
path-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
I_cube_wf,
cubical-term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
Error :memTop,
hypothesis,
sqequalRule,
setElimination,
rename,
hypothesisEquality,
instantiate,
isectElimination,
applyEquality,
universeIsType,
because_Cache,
inhabitedIsType
Latex:
\mforall{}X:j\mvdash{}. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}a,b:\{X \mvdash{} \_:A\}. \mforall{}I:fset(\mBbbN{}). \mforall{}rho:X(I). ((Path\_A a b)(rho) \msubseteq{}r Path(A)(rho))
Date html generated:
2020_05_20-PM-03_14_47
Last ObjectModification:
2020_04_06-PM-05_35_37
Theory : cubical!type!theory
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