Nuprl Lemma : csm-cubical-id-is-equiv
∀[G,K:j⊢]. ∀[tau:K j⟶ G]. ∀[A:{G ⊢ _}].
(cubical-id-is-equiv(K;(A)tau) = (cubical-id-is-equiv(G;A))tau ∈ {K ⊢ _:IsEquiv((A)tau;(A)tau;cubical-id-fun(K))})
Proof
Definitions occuring in Statement :
cubical-id-is-equiv: cubical-id-is-equiv(X;T),
is-cubical-equiv: IsEquiv(T;A;w),
cubical-id-fun: cubical-id-fun(X),
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,
cubical-id-is-equiv: cubical-id-is-equiv(X;T),
is-cubical-equiv: IsEquiv(T;A;w),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
true: True,
uimplies: b supposing a,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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,
prop: ℙ,
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,
contr-witness: contr-witness(X;c;p),
contractible-type: Contractible(A),
csm-id: 1(X),
csm-id-adjoin: [u],
cubical-fiber: Fiber(w;a),
cubical-sigma: Σ A B,
cc-adjoin-cube: (v;u),
csm+: tau+,
cubical-term: {X ⊢ _:A},
I_cube: A(I),
functor-ob: ob(F),
cube-context-adjoin: X.A,
cubical-type-at: A(a),
path-type: (Path_A a b),
cubical-subset: cubical-subset,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
cubical-term-at: u(a),
cubical-app: app(w; u),
cubical-id-fun: cubical-id-fun(X),
cubical-lam: cubical-lam(X;b),
cubical-lambda: (λb),
cubical-type-ap-morph: (u a f)
Lemmas referenced :
csm-cubical-id-fun,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cc-fst_wf,
cubical-id-fun_wf,
cubical-fiber-id-fun,
csm-ap-type_wf,
cc-snd_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
csm-ap-cubical-lambda,
contractible-type_wf,
cubical-fiber_wf,
csm-ap-term_wf,
cubical-fun_wf,
contr-witness_wf,
id-fiber-center_wf,
id-fiber-contraction_wf,
subtype_rel-equal,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity,
equal_wf,
iff_weakening_equal,
cubical-sigma_wf,
subtype_rel_self,
csm-cubical-fun,
squash_wf,
true_wf,
istype-universe,
cubical-pi_wf,
csm-cubical-pi,
csm-contractible-type,
csm-adjoin_wf,
csm-comp_wf,
csm-cubical-fiber,
cubical-lambda_wf,
csm-cubical-pair,
cubical-pair_wf,
csm-id-adjoin_wf,
csm-id-fiber-center,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
csm-cubical-lambda,
csm+_wf,
csm-id_wf,
csm-cubical-sigma,
path-type-q-csm-adjoin,
csm-comp-term,
q-csm+,
csm-comp-type,
p-csm+-type,
csm-ap-id-type,
csm-path-type,
csm_ap_term_fst_adjoin_lemma,
csm+-p-type,
csm+-p-term,
equal_functionality_wrt_subtype_rel2,
subset-cubical-term2,
sub_cubical_set_self,
p-csm+-term,
csm-ap-term-snd-adjoin,
csm-id-fiber-contraction
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,
dependent_functionElimination,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
setElimination,
rename,
hyp_replacement,
universeEquality,
Error :memTop,
cumulativity,
lambdaFormation_alt
Latex:
\mforall{}[G,K:j\mvdash{}]. \mforall{}[tau:K j{}\mrightarrow{} G]. \mforall{}[A:\{G \mvdash{} \_\}].
(cubical-id-is-equiv(K;(A)tau) = (cubical-id-is-equiv(G;A))tau)
Date html generated:
2020_05_20-PM-03_33_58
Last ObjectModification:
2020_04_08-PM-11_34_28
Theory : cubical!type!theory
Home
Index