Nuprl Lemma : csm-is-cubical-equiv
∀[X:j⊢]. ∀[T,A:{X ⊢ _}]. ∀[w:{X ⊢ _:(T ⟶ A)}]. ∀[Z:j⊢]. ∀[s:Z j⟶ X].
((IsEquiv(T;A;w))s = IsEquiv((T)s;(A)s;(w)s) ∈ {Z ⊢ _})
Proof
Definitions occuring in Statement :
is-cubical-equiv: IsEquiv(T;A;w),
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,
is-cubical-equiv: IsEquiv(T;A;w),
squash: ↓T,
prop: ℙ,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
true: True,
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,
cubical-type: {X ⊢ _},
cc-snd: q,
csm-ap-type: (AF)s,
cc-fst: p,
csm-comp: G o F,
csm-ap: (s)x
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type_wf,
csm-cubical-pi,
contractible-type_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-fiber_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
cubical-fun_wf,
csm-cubical-fun,
cubical-term-eqcd,
cc-snd_wf,
cubical-pi_wf,
subtype_rel_self,
iff_weakening_equal,
cube_set_map_wf,
istype-cubical-term,
cubical_set_wf,
csm-contractible-type,
cubical-term_wf,
csm-adjoin_wf,
csm-comp_wf,
csm-cubical-fiber,
csm-ap-type-fst-adjoin,
csm_ap_term_fst_adjoin_lemma,
csm-ap-comp-type,
csm-ap-comp-term,
csm-ap-term-snd-adjoin
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
instantiate,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
universeEquality,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_isectElimination,
hyp_replacement,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
setElimination,
rename,
Error :memTop
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T,A:\{X \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[Z:j\mvdash{}]. \mforall{}[s:Z j{}\mrightarrow{} X].
((IsEquiv(T;A;w))s = IsEquiv((T)s;(A)s;(w)s))
Date html generated:
2020_05_20-PM-03_25_31
Last ObjectModification:
2020_04_19-PM-01_56_00
Theory : cubical!type!theory
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