Nuprl Lemma : lift-id-face_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[a,b:{X ⊢ _:A}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[face:A-face(X;(Id_A a b);I;alpha)].
(lift-id-face(X;A;I;alpha;face) ∈ A-face(X;A;I+;iota'(I)(alpha)))
Proof
Definitions occuring in Statement :
lift-id-face: lift-id-face(X;A;I;alpha;face),
cubical-identity: (Id_A a b),
A-face: A-face(X;A;I;alpha),
cubical-term: {X ⊢ _:AF},
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
iota': iota'(I),
add-fresh-cname: I+,
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
A-face: A-face(X;A;I;alpha),
lift-id-face: lift-id-face(X;A;I;alpha;face),
spreadn: spread3,
subtype_rel: A ⊆r B,
nameset: nameset(L),
not: ¬A,
false: False,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
bor: p ∨bq,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
eq-cname: eq-cname(x;y),
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
true: True,
prop: ℙ,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
I-path: I-path(X;A;a;b;I;alpha),
named-path: named-path(X;A;a;b;I;alpha;z),
pi1: fst(t),
pi2: snd(t),
iota': iota'(I),
add-fresh-cname: I+,
has-value: (a)↓
Lemmas referenced :
subtype-add-fresh-cname,
cubical-type-at_wf,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
add-fresh-cname_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
iota'_wf,
A-face_wf,
cubical-identity_wf,
I-cube_wf,
list_wf,
cubical-term_wf,
cubical-type_wf,
cubical-set_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert-eq-cname,
eqtt_to_assert,
bool_wf,
eq-cname_wf,
deq_member_nil_lemma,
deq_member_cons_lemma,
iff_weakening_equal,
list-diff-cons,
l_member_wf,
not_wf,
fresh-cname_wf,
true_wf,
squash_wf,
equal_wf,
fresh-cname-not-member2,
int_subtype_base,
le_wf,
set_subtype_base,
set-path-name_wf,
subtype_rel_sets_simple,
member-list-diff,
istype-void,
istype-int,
value-type-has-value,
set-value-type,
coordinate_name-value-type,
cube-set-restriction-comp,
iota_wf,
istype-universe,
subtype_rel-equal,
length_wf,
length_of_cons_lemma,
subtype_rel_self,
iota-face-map,
fresh-cname-not-equal2,
name-morph_subtype,
nameset_subtype,
l_subset_wf,
l_subset_refl
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
dependent_pairEquality_alt,
hypothesisEquality,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
rename,
universeIsType,
setElimination,
because_Cache,
productIsType,
inhabitedIsType,
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
equalityElimination,
unionElimination,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
setEquality,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
lambdaEquality,
intEquality,
lambdaEquality_alt,
lambdaFormation_alt,
functionIsType,
equalityIsType1,
hyp_replacement,
closedConclusion,
callbyvalueReduce,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
isect_memberEquality_alt
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[a,b:\{X \mvdash{} \_:A\}]. \mforall{}[I:Cname List]. \mforall{}[alpha:X(I)].
\mforall{}[face:A-face(X;(Id\_A a b);I;alpha)].
(lift-id-face(X;A;I;alpha;face) \mmember{} A-face(X;A;I+;iota'(I)(alpha)))
Date html generated:
2019_11_06-PM-00_39_31
Last ObjectModification:
2018_11_08-AM-10_07_56
Theory : cubical!sets
Home
Index