Nuprl Lemma : sq_stable_fills-A-open-box
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[cube:A(alpha)].
  ∀bx:A-open-box(X;A;I;alpha;J;x;i). SqStable(fills-A-open-box(X;A;I;alpha;bx;cube)) supposing l_subset(Cname;J;I)
Proof
Definitions occuring in Statement : 
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube)
, 
A-open-box: A-open-box(X;A;I;alpha;J;x;i)
, 
cubical-type-at: A(a)
, 
cubical-type: {X ⊢ _}
, 
I-cube: X(I)
, 
cubical-set: CubicalSet
, 
nameset: nameset(L)
, 
coordinate_name: Cname
, 
l_subset: l_subset(T;as;bs)
, 
list: T List
, 
int_seg: {i..j-}
, 
sq_stable: SqStable(P)
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
prop: ℙ
, 
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube)
, 
fills-A-faces: fills-A-faces(X;A;I;alpha;bx;L)
, 
l_all: (∀x∈L.P[x])
, 
A-open-box: A-open-box(X;A;I;alpha;J;x;i)
, 
so_lambda: λ2x.t[x]
, 
int_seg: {i..j-}
, 
guard: {T}
, 
nameset: nameset(L)
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
coordinate_name: Cname
, 
int_upper: {i...}
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
not: ¬A
, 
top: Top
, 
less_than: a < b
, 
so_apply: x[s]
, 
A-face: A-face(X;A;I;alpha)
, 
is-A-face: is-A-face(X;A;I;alpha;bx;f)
, 
spreadn: spread3
Lemmas referenced : 
l_subset_wf, 
coordinate_name_wf, 
A-open-box_wf, 
cubical-type-at_wf, 
int_seg_wf, 
nameset_wf, 
list_wf, 
I-cube_wf, 
cubical-type_wf, 
cubical-set_wf, 
sq_stable__all, 
length_wf, 
A-face_wf, 
is-A-face_wf, 
select_wf, 
int_seg_properties, 
sq_stable__l_member, 
decidable__equal-coordinate_name, 
sq_stable__le, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
equal_wf, 
sq_stable__equal, 
list-diff_wf, 
cname_deq_wf, 
cons_wf, 
nil_wf, 
cube-set-restriction_wf, 
face-map_wf2, 
cubical-type-ap-morph_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
lambdaFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesis, 
hypothesisEquality, 
dependent_functionElimination, 
natural_numberEquality, 
setElimination, 
rename, 
sqequalRule, 
lambdaEquality, 
because_Cache, 
independent_isectElimination, 
productElimination, 
independent_functionElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
equalityTransitivity, 
equalitySymmetry
Latex:
\mforall{}[X:CubicalSet].  \mforall{}[A:\{X  \mvdash{}  \_\}].  \mforall{}[I:Cname  List].  \mforall{}[alpha:X(I)].  \mforall{}[J:Cname  List].  \mforall{}[x:nameset(I)].
\mforall{}[i:\mBbbN{}2].  \mforall{}[cube:A(alpha)].
    \mforall{}bx:A-open-box(X;A;I;alpha;J;x;i)
        SqStable(fills-A-open-box(X;A;I;alpha;bx;cube))  supposing  l\_subset(Cname;J;I)
Date html generated:
2017_10_05-AM-10_23_18
Last ObjectModification:
2017_07_28-AM-11_21_50
Theory : cubical!sets
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