Nuprl Lemma : sigma-box-fst_wf
∀X:CubicalSet. ∀A:{X ⊢ _(Kan)}. ∀B:{X.Kan-type(A) ⊢ _(Kan)}. ∀I:Cname List. ∀alpha:X(I). ∀J:nameset(I) List.
∀x:nameset(I). ∀i:ℕ2. ∀bx:A-open-box(X;Σ Kan-type(A) Kan-type(B);I;alpha;J;x;i).
(sigma-box-fst(bx) ∈ A-open-box(X;Kan-type(A);I;alpha;J;x;i))
Proof
Definitions occuring in Statement :
sigma-box-fst: sigma-box-fst(bx),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
sigma-box-fst: sigma-box-fst(bx),
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
and: P ∧ Q,
A-face: A-face(X;A;I;alpha),
top: Top,
pi1: fst(t),
pi2: snd(t),
cubical-type-at: A(a),
cubical-sigma: Σ A B,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
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,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
pairwise: (∀x,y∈L. P[x; y]),
less_than: a < b,
le: A ≤ B,
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
spreadn: spread3,
cubical-type-ap-morph: (u a f),
l_exists: (∃x∈L. P[x]),
A-face-name: A-face-name(f),
l_all: (∀x∈L.P[x]),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
A-open-box_wf,
cubical-sigma_wf,
Kan-type_wf,
cube-context-adjoin_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
I-cube_wf,
Kan-cubical-type_wf,
cubical-set_wf,
map_wf,
A-face_wf,
pi1_wf_top,
subtype_rel_self,
cubical-type-at_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
cc-adjoin-cube_wf,
A-adjacent-compatible_wf,
not_wf,
l_member_wf,
l_subset_wf,
all_wf,
l_exists_wf,
equal_wf,
A-face-name_wf,
nameset_subtype,
l_all_wf2,
subtract_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,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
lelt_wf,
pairwise_wf2,
length-map,
length_wf,
top_wf,
select_wf,
cubical-sigma-at,
A-face-compatible_wf,
select-map,
pairwise-map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
introduction,
extract_by_obid,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
productElimination,
dependent_pairEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
spreadEquality,
productEquality,
independent_pairFormation,
cumulativity,
universeEquality,
setEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_(Kan)\}. \mforall{}B:\{X.Kan-type(A) \mvdash{} \_(Kan)\}. \mforall{}I:Cname List. \mforall{}alpha:X(I).
\mforall{}J:nameset(I) List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2. \mforall{}bx:A-open-box(X;\mSigma{} Kan-type(A) Kan-type(B);I;alpha;J;x;i).
(sigma-box-fst(bx) \mmember{} A-open-box(X;Kan-type(A);I;alpha;J;x;i))
Date html generated:
2017_10_05-AM-10_24_18
Last ObjectModification:
2017_07_28-AM-11_22_19
Theory : cubical!sets
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