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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