Nuprl Lemma : groupoid-nerve-filler_wf
∀[G:Groupoid]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(cat(G));I;J;x;i)].
(groupoid-nerve-filler(G;I;J;x;i;box) ∈ cubical-nerve(cat(G))(I))
Proof
Definitions occuring in Statement :
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
I-cube: X(I),
nameset: nameset(L),
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
groupoid-nerve-filler: groupoid-nerve-filler(G;I;J;x;i;box),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
ifthenelse: if b then t else f fi ,
rev_uimplies: rev_uimplies(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
open_box: open_box(X;I;J;x;i),
guard: {T},
cand: A c∧ B,
nameset: nameset(L)
Lemmas referenced :
null_wf3,
subtype_rel_list,
top_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
list_wf,
nameset_wf,
eqtt_to_assert,
assert_of_null,
groupoid-nerve-filler0_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
lt_int_wf,
length_wf,
I-face_wf,
cubical-nerve_wf,
groupoid-cat_wf,
less_than_wf,
assert_of_lt_int,
groupoid-nerve-filler2_wf,
le_int_wf,
le_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
groupoid-nerve-filler1_wf,
equal_wf,
open_box_wf,
coordinate_name_wf,
int_seg_wf,
groupoid_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
productElimination,
independent_pairFormation,
impliesFunctionality,
natural_numberEquality,
setElimination,
rename,
addLevel,
dependent_functionElimination,
axiomEquality
Latex:
\mforall{}[G:Groupoid]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(cat(G));I;J;x;i)].
(groupoid-nerve-filler(G;I;J;x;i;box) \mmember{} cubical-nerve(cat(G))(I))
Date html generated:
2017_10_05-PM-03_43_51
Last ObjectModification:
2017_07_28-AM-11_27_14
Theory : cubical!sets
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