Nuprl Lemma : csm-id-fiber-center
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(id-fiber-center(K;(A)tau) = (id-fiber-center(G;A))tau+ ∈ {K.(A)tau ⊢ _:Fiber((cubical-id-fun(K))p;q)})
Proof
Definitions occuring in Statement :
id-fiber-center: id-fiber-center(X;T),
cubical-fiber: Fiber(w;a),
cubical-id-fun: cubical-id-fun(X),
csm+: tau+,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
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,
id-fiber-center: id-fiber-center(X;T),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
cc-snd: q,
csm+: tau+,
csm-ap-term: (t)s,
cc-fst: p,
csm-ap-type: (AF)s,
csm-comp: G o F,
csm-adjoin: (s;u),
csm-ap: (s)x,
pi2: snd(t),
rev_implies: P ⇐ Q,
csm-id: 1(X),
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),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g
Lemmas referenced :
csm-cubical-pair,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
cubical-refl_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
csm-ap-type_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cc-snd_wf,
cubical-fiber-id-fun,
csm-cubical-id-fun,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-fiber_wf,
cubical-term_wf,
cubical-fun_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-pair_wf,
csm-id-adjoin_wf,
path-type_wf,
csm-ap-term_wf,
csm-path-type,
csm_id_adjoin_fst_type_lemma,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
csm-ap-id-term,
csm-cubical-refl,
csm+_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
Error :memTop,
hypothesis,
universeIsType,
hypothesisEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
hyp_replacement
Latex:
\mforall{}[G,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} G]. \mforall{}[A:\{G \mvdash{} \_\}]. (id-fiber-center(K;(A)tau) = (id-fiber-center(G;A))tau+)
Date html generated:
2020_05_20-PM-03_30_50
Last ObjectModification:
2020_04_08-AM-11_48_56
Theory : cubical!type!theory
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