Nuprl Lemma : csm-cubical-fiber
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[a:{X ⊢ _:A}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X].
((Fiber(w;a))s = Z ⊢ Fiber((w)s;(a)s) ∈ {Z ⊢ _})
Proof
Definitions occuring in Statement :
cubical-fiber: Fiber(w;a),
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],
member: t ∈ T,
subtype_rel: A ⊆r B,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
cubical-fiber: Fiber(w;a),
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
cubical-type: {X ⊢ _},
cc-snd: q,
csm-ap-type: (AF)s,
cc-fst: p,
csm-comp: G o F,
csm-ap: (s)x,
csm-adjoin: (s;u),
csm-ap-term: (t)s
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-fun_wf,
cc-fst_wf,
cubical-term_wf,
csm-cubical-fun,
csm-ap-type_wf,
equal_wf,
cubical-type_wf,
csm-cubical-sigma,
path-type_wf,
cubical-app_wf_fun,
cc-snd_wf,
cubical-sigma_wf,
equal_functionality_wrt_subtype_rel2,
iff_weakening_equal,
cube_set_map_wf,
cubical_set_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
csm-ap-comp-type,
csm-path-type,
csm-adjoin_wf,
csm-comp_wf,
csm-comp-term,
csm_ap_term_fst_adjoin_lemma,
csm-ap-comp-term-sq2,
csm-ap-cubical-app-fun
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
lambdaEquality_alt,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
universeIsType,
cumulativity,
independent_isectElimination,
independent_functionElimination,
productElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
setElimination,
rename,
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{}[Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} X].
((Fiber(w;a))s = Z \mvdash{} Fiber((w)s;(a)s))
Date html generated:
2020_05_20-PM-03_23_54
Last ObjectModification:
2020_04_07-PM-05_20_52
Theory : cubical!type!theory
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