Nuprl Lemma : csm-Kan-cubical-type_wf
∀[Delta,Gamma:CubicalSet]. ∀[s:Delta ⟶ Gamma]. ∀[AK:{Gamma ⊢ _(Kan)}]. ((AK)s ∈ {Delta ⊢ _(Kan)})
Proof
Definitions occuring in Statement :
csm-Kan-cubical-type: (AK)s,
Kan-cubical-type: {X ⊢ _(Kan)},
cube-set-map: A ⟶ B,
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-type: {X ⊢ _(Kan)},
csm-Kan-cubical-type: (AK)s,
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
and: P ∧ Q,
uimplies: b supposing a,
nameset: nameset(L),
Kan-A-filler: Kan-A-filler(X;A;filler),
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
implies: P ⇒ Q,
name-morph: name-morph(I;J),
prop: ℙ,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
sq_stable: SqStable(P),
squash: ↓T,
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]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
less_than: a < b,
A-face: A-face(X;A;I;alpha),
is-A-face: is-A-face(X;A;I;alpha;bx;f),
spreadn: spread3,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
nat: ℕ,
ge: i ≥ j ,
uiff: uiff(P;Q),
name-morph-domain: name-morph-domain(f;I),
pi1: fst(t),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
Kan-cubical-type_wf,
cube-set-map_wf,
cubical-set_wf,
csm-ap-type_wf,
csm-type-at,
istype-void,
csm-ap_wf,
csm-A-open-box,
A-open-box_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
I-cube_wf,
cubical-type-at_wf,
istype-assert,
isname_wf,
l_member_wf,
name-morph_wf,
fills-A-open-box_wf,
sq_stable__l_subset,
decidable__equal-coordinate_name,
select_wf,
A-face_wf,
int_seg_properties,
length_wf,
sq_stable__l_member,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
csm-ap-restriction,
subtype_rel_self,
iff_weakening_equal,
csm-cubical-type-ap-morph,
cubical-type-ap-morph_wf,
subtype_rel-equal,
is-A-face_wf,
list-subtype,
map_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
nat_properties,
assert-isname,
sq_stable__assert,
subtype_rel_dep_function,
cons_member,
member_filter_2,
filter_wf5,
A-open-box-equal,
A-open-box-image_wf,
subtype_rel_set,
A-adjacent-compatible_wf,
not_wf,
l_subset_wf,
all_wf,
l_exists_wf,
A-face-name_wf,
nameset_subtype,
l_all_wf2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
istype-less_than,
pairwise_wf2,
subtype_rel_universe1,
pi1_wf_top,
assert_elim,
bool_wf,
bool_subtype_base,
name-morph_subtype_remove1,
cubical-type_wf,
name-comp_wf,
face-map-comp,
cube-set-restriction-comp,
Kan-A-filler_wf,
uniform-Kan-A-filler_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_set_memberEquality_alt,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_pairEquality_alt,
functionExtensionality,
dependent_functionElimination,
voidElimination,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
natural_numberEquality,
functionIsType,
because_Cache,
independent_pairFormation,
promote_hyp,
lambdaFormation_alt,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
equalityIsType1,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
hyp_replacement,
instantiate,
universeEquality,
productIsType,
cumulativity,
intEquality,
closedConclusion,
applyLambdaEquality,
functionEquality,
setIsType,
productEquality,
independent_pairEquality,
setEquality
Latex:
\mforall{}[Delta,Gamma:CubicalSet]. \mforall{}[s:Delta {}\mrightarrow{} Gamma]. \mforall{}[AK:\{Gamma \mvdash{} \_(Kan)\}]. ((AK)s \mmember{} \{Delta \mvdash{} \_(Kan)\})
Date html generated:
2019_11_05-PM-00_29_47
Last ObjectModification:
2018_11_14-AM-11_19_16
Theory : cubical!sets
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