Nuprl Lemma : term-A-face_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[i:ℕ2].
(term-A-face(a;I;alpha;i) ∈ A-face(X;A;[fresh-cname(I) / I];iota(fresh-cname(I))(alpha)))
Proof
Definitions occuring in Statement :
term-A-face: term-A-face(a;I;alpha;i),
A-face: A-face(X;A;I;alpha),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
iota: iota(x),
fresh-cname: fresh-cname(I),
coordinate_name: Cname,
cons: [a / b],
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
term-A-face: term-A-face(a;I;alpha;i),
A-face: A-face(X;A;I;alpha),
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
or: P ∨ Q,
prop: ℙ,
nameset: nameset(L),
squash: ↓T,
uimplies: b supposing a,
guard: {T},
true: True,
iota': iota'(I)
Lemmas referenced :
cons_member,
fresh-cname_wf,
l_member_wf,
coordinate_name_wf,
cons_wf,
not_wf,
cubical-type-at_wf,
add-remove-fresh-sq,
equal_wf,
I-cube_wf,
cube-set-restriction-comp,
list-diff_wf,
cname_deq_wf,
nil_wf,
iota_wf,
face-map_wf2,
subtype_rel-equal,
iff_weakening_equal,
cubical-term-at_wf,
cube-set-restriction_wf,
int_seg_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
face-map_wf,
iota'-identity,
squash_wf,
true_wf,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
productElimination,
independent_functionElimination,
inlFormation,
dependent_set_memberEquality,
lambdaEquality,
setElimination,
rename,
setEquality,
imageElimination,
independent_isectElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
hyp_replacement,
productEquality,
axiomEquality,
isect_memberEquality,
applyLambdaEquality,
universeEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[alpha:X(I)]. \mforall{}[i:\mBbbN{}2].
(term-A-face(a;I;alpha;i) \mmember{} A-face(X;A;[fresh-cname(I) / I];iota(fresh-cname(I))(alpha)))
Date html generated:
2017_10_05-PM-04_05_01
Last ObjectModification:
2017_07_28-AM-11_28_57
Theory : cubical!sets
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