Nuprl Lemma : csm-pathtype
∀X,Delta:j⊢. ∀s:Delta j⟶ X. ∀A:{X ⊢ _}. ((Path(A))s = Path((A)s) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
pathtype: Path(A),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
pathtype: Path(A),
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
csm-cubical-fun,
interval-type_wf,
cubical-fun_wf,
csm-ap-type_wf,
subtype_rel_self,
iff_weakening_equal,
csm-interval-type,
cube_set_map_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :memTop,
because_Cache,
inhabitedIsType
Latex:
\mforall{}X,Delta:j\mvdash{}. \mforall{}s:Delta j{}\mrightarrow{} X. \mforall{}A:\{X \mvdash{} \_\}. ((Path(A))s = Path((A)s))
Date html generated:
2020_05_20-PM-03_15_09
Last ObjectModification:
2020_04_06-PM-05_36_05
Theory : cubical!type!theory
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