Nuprl Lemma : csm-fiber-path
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[p:{X ⊢ _:Fiber(w;a)}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((fiber-path(p))s = fiber-path((p)s) ∈ {H ⊢ _:(Path_(A)s (a)s app((w)s; fiber-member((p)s)))})
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),
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
cubical-fiber: Fiber(w;a),
fiber-path: fiber-path(p),
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
true: True,
prop: ℙ,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cc-fst: p,
csm-id-adjoin: [u],
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
pi1: fst(t),
csm-ap-term: (t)s,
fiber-member: fiber-member(p),
cubical-fst: p.1,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
csm-ap-cubical-snd,
cc-snd_wf,
cubical_set_cumulativity-i-j,
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical-fun_wf,
cc-fst_wf,
csm-cubical-fun,
cubical-term-eqcd,
cubical-app_wf_fun,
csm-ap-type_wf,
cube_set_map_wf,
istype-cubical-term,
cubical-fiber_wf,
cubical-type_wf,
cubical_set_wf,
cubical-term_wf,
csm-cubical-fiber,
fiber-member_wf,
path-type_wf,
squash_wf,
true_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-id-adjoin_wf,
cubical-fst_wf,
csm_id_adjoin_fst_type_lemma,
csm-id_wf,
csm-cubical-app,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
csm_id_ap_term_lemma,
equal_wf,
istype-universe,
csm-path-type,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
introduction,
extract_by_obid,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
instantiate,
applyEquality,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
Error :memTop,
universeEquality,
independent_functionElimination
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)\}]. \mforall{}[H:j\mvdash{}].
\mforall{}[s:H j{}\mrightarrow{} X].
((fiber-path(p))s = fiber-path((p)s))
Date html generated:
2020_05_20-PM-03_24_54
Last ObjectModification:
2020_04_20-AM-10_03_03
Theory : cubical!type!theory
Home
Index