Nuprl Lemma : nerve_box_edge1_wf
∀[G:Groupoid]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[j:nameset(J)]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(cat(G));I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].
nerve_box_edge1(G;box;x;i;j;c;y) ∈ cat-arrow(cat(G)) nerve_box_label(box;c) nerve_box_label(box;flip(c;y))
supposing (∀j'∈J.j' = j ∈ Cname)
Proof
Definitions occuring in Statement :
nerve_box_edge1: nerve_box_edge1(G;box;x;i;j;c;y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
cat-arrow: cat-arrow(C),
l_all: (∀x∈L.P[x]),
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
nameset: nameset(L),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
false: False,
coordinate_name: Cname,
int_upper: {i...},
guard: {T},
int_seg: {i..j-},
squash: ↓T,
lelt: i ≤ j < k,
and: P ∧ Q,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
cons: [a / b],
subtype_rel: A ⊆r B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
iff: P ⇐⇒ Q,
null: null(as),
rev_implies: P ⇐ Q,
true: True,
nerve_box_edge1: nerve_box_edge1(G;box;x;i;j;c;y),
name-morph: name-morph(I;J),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
bor: p ∨bq,
sq_type: SQType(T),
bnot: ¬bb,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
sq_stable: SqStable(P),
open_box: open_box(X;I;J;x;i),
eq_int: (i =z j),
le: A ≤ B,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
less_than: a < b,
name-morph-flip: flip(f;y),
subtract: n - m,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x])
Lemmas referenced :
nameset_wf,
list-cases,
null_nil_lemma,
btrue_wf,
stuck-spread,
base_wf,
length_of_nil_lemma,
int_seg_properties,
nat_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,
btrue_neq_bfalse,
product_subtype_list,
assert_elim,
null_wf3,
cons_wf,
subtype_rel_list,
top_wf,
equal_wf,
bool_wf,
ppcc-problem,
iff_imp_equal_bool,
bfalse_wf,
true_wf,
false_wf,
iff_weakening_equal,
assert_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
extd-nameset_subtype_int,
nil_wf,
coordinate_name_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
l_all_wf2,
l_member_wf,
set_wf,
name-morph_wf,
equal-wf-T-base,
int_seg_wf,
extd-nameset-nil,
open_box_wf,
cubical-nerve_wf,
groupoid-cat_wf,
list_wf,
groupoid_wf,
nerve_box_edge_wf,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
decidable__lt,
lelt_wf,
l_exists_wf,
not_wf,
eq-cname_wf,
assert-eq-cname,
set_subtype_base,
le_wf,
int_subtype_base,
int_seg_subtype,
int_seg_cases,
null_cons_lemma,
equal-wf-base,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
name-morph-ext,
name-morph-flip_wf,
squash_wf,
extd-nameset_wf,
name-morph-flips-commute,
name-morph-flip-flip,
groupoid-square1_wf,
nerve_box_label_wf,
decidable__assert,
cat-arrow_wf,
cat-square-commutes_wf,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
not-lt-2,
not-equal-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
length_wf,
select_wf,
nerve_box_edge_wf2,
subtype_rel-equal,
or_wf,
small-category_wf,
groupoid-square2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
setElimination,
rename,
productElimination,
baseClosed,
independent_isectElimination,
lambdaFormation,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
addLevel,
applyEquality,
because_Cache,
levelHypothesis,
inlEquality,
equalityElimination,
instantiate,
cumulativity,
axiomEquality,
setEquality,
dependent_set_memberEquality,
inrFormation,
addEquality,
hyp_replacement,
universeEquality,
inlFormation,
minusEquality,
productEquality
Latex:
\mforall{}[G:Groupoid]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[j:nameset(J)]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(cat(G));I;J;x;i)]. \mforall{}[y:nameset(I)].
\mforall{}[c:\{c:name-morph(I;[])| (c y) = 0\} ].
nerve\_box\_edge1(G;box;x;i;j;c;y) \mmember{} cat-arrow(cat(G)) nerve\_box\_label(box;c)
nerve\_box\_label(box;flip(c;y))
supposing (\mforall{}j'\mmember{}J.j' = j)
Date html generated:
2017_10_05-PM-03_41_31
Last ObjectModification:
2017_07_28-AM-11_26_47
Theory : cubical!sets
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