Nuprl Lemma : Kan-discrete_wf
∀[X:CubicalSet]. ∀[T:Type]. (Kan-discrete(T) ∈ {X ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
Kan-discrete: Kan-discrete(T),
Kan-cubical-type: {X ⊢ _(Kan)},
cubical-set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
Kan-discrete: Kan-discrete(T),
Kan-cubical-type: {X ⊢ _(Kan)},
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
implies: P ⇒ Q,
or: P ∨ Q,
cons: [a / b],
top: Top,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
guard: {T},
int_seg: {i..j-},
false: False,
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
A-face: A-face(X;A;I;alpha),
pi2: snd(t),
discrete-cubical-type: discr(T),
cubical-type-at: A(a),
pi1: fst(t),
Kan-A-filler: Kan-A-filler(X;A;filler),
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]),
is-A-face: is-A-face(X;A;I;alpha;bx;f),
cubical-type-ap-morph: (u a f),
decidable: Dec(P),
sq_type: SQType(T),
ge: i ≥ j ,
sq_stable: SqStable(P),
squash: ↓T,
less_than: a < b,
le: A ≤ B,
less_than': less_than'(a;b),
spreadn: spread3,
so_lambda: λ2x.t[x],
so_apply: x[s],
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
pairwise: (∀x,y∈L. P[x; y]),
A-face-name: A-face-name(f),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
name-morph: name-morph(I;J),
A-face-image: A-face-image(X;A;I;K;f;alpha;face)
Lemmas referenced :
discrete-cubical-type_wf,
A-open-box_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
I-cube_wf,
cubical-type-at_wf,
Kan-A-filler_wf,
uniform-Kan-A-filler_wf,
cubical-set_wf,
hd_wf,
A-face_wf,
equal_wf,
list-cases,
product_subtype_list,
length_cons_ge_one,
top_wf,
length_of_nil_lemma,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
length_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
select0,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
select_wf,
false_wf,
decidable__lt,
pi2_wf,
not_wf,
set_subtype_base,
le_wf,
lelt_wf,
decidable__equal_int_seg,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
l_member_wf,
equal-wf-base,
pi1_wf_top,
subtype_rel_product,
nameset_subtype_base,
squash_wf,
true_wf,
iff_weakening_equal,
map_nil_lemma,
reduce_hd_cons_lemma,
map_cons_lemma,
assert_wf,
isname_wf,
all_wf,
name-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
dependent_set_memberEquality,
dependent_pairEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
lambdaEquality,
dependent_functionElimination,
because_Cache,
applyEquality,
independent_isectElimination,
setElimination,
rename,
sqequalRule,
natural_numberEquality,
functionEquality,
independent_pairFormation,
productElimination,
productEquality,
functionExtensionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
isect_memberEquality,
lambdaFormation,
independent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
voidElimination,
voidEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
instantiate,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
addLevel,
levelHypothesis,
baseApply,
closedConclusion
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[T:Type]. (Kan-discrete(T) \mmember{} \{X \mvdash{} \_(Kan)\})
Date html generated:
2017_10_05-AM-10_26_13
Last ObjectModification:
2017_07_28-AM-11_22_45
Theory : cubical!sets
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