Nuprl Lemma : constant-Kan-type_wf
∀[X:KanCubicalSet]. ∀[Gamma:CubicalSet]. (constant-Kan-type(X) ∈ {Gamma ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
constant-Kan-type: constant-Kan-type(X),
Kan-cubical-set: KanCubicalSet,
Kan-cubical-type: {X ⊢ _(Kan)},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Kan-cubical-set: KanCubicalSet,
Kan-cubical-type: {X ⊢ _(Kan)},
constant-Kan-type: constant-Kan-type(X),
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
uimplies: b supposing a,
top: Top,
nameset: nameset(L),
I-cube: X(I),
functor-ob: ob(F),
pi1: fst(t),
cubical-type-at: A(a),
constant-cubical-type: (X),
cubical-set: CubicalSet,
cand: A c∧ B,
prop: ℙ,
Kan-filler: Kan-filler(X;filler),
Kan-A-filler: Kan-A-filler(X;A;filler),
fills-open_box: fills-open_box(X;I;bx;cube),
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube),
fills-faces: fills-faces(X;I;bx;L),
fills-A-faces: fills-A-faces(X;A;I;alpha;bx;L),
is-A-face: is-A-face(X;A;I;alpha;bx;f),
cubical-type-ap-morph: (u a f),
pi2: snd(t),
is-face: is-face(X;I;bx;f),
l_all: (∀x∈L.P[x]),
open_box: open_box(X;I;J;x;i),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
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,
less_than: a < b,
I-face: I-face(X;I),
face-cube: cube(f),
face-direction: direction(f),
face-dimension: dimension(f),
spreadn: spread3,
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
uniform-Kan-filler: uniform-Kan-filler(X;filler),
open_box_image: open_box_image(X;I;K;f;bx),
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
face-image: face-image(X;I;K;f;face),
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
name-morph: name-morph(I;J)
Lemmas referenced :
constant-cubical-type_wf,
list_wf,
coordinate_name_wf,
subtype_rel_dep_function,
nameset_wf,
int_seg_wf,
open_box_wf,
I-cube_wf,
A-open-box_wf,
cubical-type-at_wf,
subtype_rel-equal,
constant-A-open-box,
subtype_rel_list,
subtype_rel_self,
Kan-A-filler_wf,
uniform-Kan-A-filler_wf,
cubical-set_wf,
Kan-cubical-set_wf,
select_wf,
I-face_wf,
int_seg_properties,
length_wf,
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,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
assert_wf,
isname_wf,
all_wf,
l_member_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
dependent_set_memberEquality,
sqequalRule,
dependent_pairEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
spreadEquality,
independent_pairFormation,
productEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[X:KanCubicalSet]. \mforall{}[Gamma:CubicalSet]. (constant-Kan-type(X) \mmember{} \{Gamma \mvdash{} \_(Kan)\})
Date html generated:
2017_10_05-AM-10_26_26
Last ObjectModification:
2017_07_28-AM-11_22_53
Theory : cubical!sets
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