Nuprl Lemma : sigma-box-snd_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)]. ∀[cbA:Kan-type(A)(alpha)].
sigma-box-snd(bx) ∈ A-open-box(X.Kan-type(A);Kan-type(B);I;(alpha;cbA);J;x;i)
supposing fills-A-open-box(X;Kan-type(A);I;alpha;sigma-box-fst(bx);cbA)
Proof
Definitions occuring in Statement :
sigma-box-snd: sigma-box-snd(bx),
sigma-box-fst: sigma-box-fst(bx),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
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-sigma: Σ A B,
cc-adjoin-cube: (v;u),
cube-context-adjoin: X.A,
cubical-type-at: A(a),
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nameset: nameset(L),
sigma-box-fst: sigma-box-fst(bx),
sigma-box-snd: sigma-box-snd(bx),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
A-face: A-face(X;A;I;alpha),
top: Top,
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
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]),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
guard: {T},
sq_stable: SqStable(P),
coordinate_name: Cname,
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
cubical-type-at: A(a),
cubical-sigma: Σ A B,
is-A-face: is-A-face(X;A;I;alpha;bx;f),
spreadn: spread3,
pairwise: (∀x,y∈L. P[x; y]),
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
le: A ≤ B,
less_than: a < b,
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
cubical-type-ap-morph: (u a f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
A-face-name: A-face-name(f),
l_exists: (∃x∈L. P[x]),
so_apply: x[s],
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
cubical-type-at_wf,
Kan-type_wf,
A-open-box_wf,
cubical-sigma_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,
sigma-box-fst_wf,
list-subtype,
A-face_wf,
map_wf,
l_member_wf,
cc-adjoin-cube_wf,
cubical-sigma-at,
istype-void,
cc-adjoin-cube-restriction,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
cubical-type-ap-morph_wf,
length-map,
nat_properties,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
istype-le,
istype-less_than,
length_wf,
select-map,
top_wf,
is-A-face_wf,
pi1_wf_top,
subtype_rel_self,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
satisfiable-full-omega-tt,
decidable__lt,
equal_wf,
A-face-compatible_wf,
select_wf,
not_wf,
true_wf,
squash_wf,
cubical-type_wf,
name-morph_wf,
name-comp_wf,
subtype_rel_wf,
list-diff2-sym,
iff_weakening_equal,
cubical-type-ap-morph-comp,
cube-set-restriction-comp,
subtype_rel-equal,
list-diff2,
trivial-equal,
face-maps-commute,
ext-eq_weakening,
subtype_rel_weakening,
l_all_map,
A-face-name_wf,
pairwise-map,
length-map-sq,
A-adjacent-compatible_wf,
l_subset_wf,
l_exists_wf,
l_all_wf2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
pairwise_wf2,
fills-A-open-box_wf,
sq_stable__l_subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
sqequalRule,
natural_numberEquality,
promote_hyp,
dependent_set_memberEquality_alt,
productElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaFormation_alt,
setEquality,
functionExtensionality,
isect_memberEquality_alt,
voidElimination,
dependent_pairEquality_alt,
productIsType,
equalityIsType1,
independent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
hyp_replacement,
independent_pairFormation,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
applyLambdaEquality,
independent_pairEquality,
spreadEquality,
productEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
computeAll,
intEquality,
dependent_pairFormation,
dependent_set_memberEquality,
lambdaEquality,
dependent_pairEquality,
equalityElimination,
universeEquality,
instantiate,
cumulativity,
functionIsType,
setIsType,
closedConclusion
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)]. \mforall{}[cbA:Kan-type(A)(alpha)].
sigma-box-snd(bx) \mmember{} A-open-box(X.Kan-type(A);Kan-type(B);I;(alpha;cbA);J;x;i)
supposing fills-A-open-box(X;Kan-type(A);I;alpha;sigma-box-fst(bx);cbA)
Date html generated:
2019_11_05-PM-00_30_15
Last ObjectModification:
2018_11_10-PM-02_40_20
Theory : cubical!sets
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