Nuprl Lemma : fiber-path_wf
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}].
(fiber-path(p) ∈ {X ⊢ _:(Path_A a app(w; fiber-member(p)))})
Proof
Definitions occuring in Statement :
fiber-path: fiber-path(p),
fiber-member: fiber-member(p),
cubical-fiber: Fiber(w;a),
path-type: (Path_A a b),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-fiber: Fiber(w;a),
subtype_rel: A ⊆r B,
fiber-path: fiber-path(p),
fiber-member: fiber-member(p),
all: ∀x:A. B[x],
and: P ∧ Q,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-term_wf,
cubical-fiber_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
cubical-type_wf,
cubical_set_wf,
cubical-snd_wf,
path-type_wf,
cube-context-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cc-snd_wf,
csm-cubical-fun,
cubical-app_wf_fun,
squash_wf,
true_wf,
equal_wf,
istype-universe,
csm-path-type,
csm-id-adjoin_wf,
fiber-member_wf,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
csm-ap-id-type,
csm-ap-id-term,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-cubical-app,
cc_snd_csm_id_adjoin_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
Error :memTop
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[p:\{X \mvdash{} \_:Fiber(w;a)\}].
(fiber-path(p) \mmember{} \{X \mvdash{} \_:(Path\_A a app(w; fiber-member(p)))\})
Date html generated:
2020_05_20-PM-03_24_42
Last ObjectModification:
2020_04_07-PM-04_05_51
Theory : cubical!type!theory
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