Nuprl Lemma : lift-id-faces_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[alpha:X(I)]. ∀[box:A-open-box(X;(Id_A a b);I;alpha;J;x;i)].
(lift-id-faces(X;A;I;alpha;box) ∈ A-open-box(X;A;[fresh-cname(I) / I];iota(fresh-cname(I))(alpha);J;x;i))
Proof
Definitions occuring in Statement :
lift-id-faces: lift-id-faces(X;A;I;alpha;box),
cubical-identity: (Id_A a b),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
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),
nameset: nameset(L),
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,
iota': iota'(I),
add-fresh-cname: I+,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
has-value: (a)↓
Lemmas referenced :
lift-id-faces-wf,
A-open-box_wf,
cubical-identity_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
I-cube_wf,
int_seg_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
value-type-has-value,
not_wf,
l_member_wf,
set-value-type,
coordinate_name-value-type,
fresh-cname_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
natural_numberEquality,
setEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List].
\mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[alpha:X(I)]. \mforall{}[box:A-open-box(X;(Id\_A a b);I;alpha;J;x;i)].
(lift-id-faces(X;A;I;alpha;box) \mmember{} A-open-box(X;A;[fresh-cname(I) /
I];iota(fresh-cname(I))(alpha);J;x;i))
Date html generated:
2016_06_16-PM-07_51_07
Last ObjectModification:
2015_12_28-PM-04_12_16
Theory : cubical!sets
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