Nuprl Lemma : same-face-edge-arrows-commute0
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)].
∀f:name-morph(I;[]). ∀a,b:nameset(I).
∀[v:I-face(cubical-nerve(C);I)]
((cat-comp(C) nerve_box_label(box;f) nerve_box_label(box;flip(f;a)) nerve_box_label(box;flip(flip(f;a);b))
nerve_box_edge(box;f;a)
nerve_box_edge(box;flip(f;a);b))
= (cat-comp(C) nerve_box_label(box;f) nerve_box_label(box;flip(f;b)) nerve_box_label(box;flip(flip(f;b);a))
nerve_box_edge(box;f;b)
nerve_box_edge(box;flip(f;b);a))
∈ (cat-arrow(C) nerve_box_label(box;f) nerve_box_label(box;flip(flip(f;a);b)))) supposing
((v ∈ box) and
(¬(dimension(v) = b ∈ Cname)) and
(¬(dimension(v) = a ∈ Cname)) and
(¬(a = b ∈ nameset(I))) and
((f b) = 0 ∈ ℕ2) and
(((f a) = 0 ∈ ℕ2) ∧ (direction(v) = (f dimension(v)) ∈ ℕ2)))
supposing ((∃j1∈J. ¬(j1 = a ∈ Cname)) ∧ (∃j2∈J. ¬(j2 = b ∈ Cname)))
∨ ((¬↑null(J)) ∧ ((f x) = i ∈ ℤ) ∧ ((flip(f;a) x) = i ∈ ℤ) ∧ ((flip(f;b) x) = i ∈ ℕ2))
Proof
Definitions occuring in Statement :
nerve_box_edge: nerve_box_edge(box;c;y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
null: null(as),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
and: P ∧ Q,
or: P ∨ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
nameset: nameset(L),
false: False,
coordinate_name: Cname,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
prop: ℙ,
cons: [a / b],
bfalse: ff,
subtype_rel: A ⊆r B,
decidable: Dec(P),
name-morph: name-morph(I;J),
name-morph-flip: flip(f;y),
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
open_box: open_box(X;I;J;x;i),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_stable: SqStable(P),
label: ...$L... t
Lemmas referenced :
nameset_wf,
list-cases,
stuck-spread,
base_wf,
length_of_nil_lemma,
null_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,
product_subtype_list,
null_cons_lemma,
false_wf,
nerve_box_label_same,
name-morph-flip_wf,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
equal_wf,
int_seg_wf,
eq-cname_wf,
face-dimension_wf,
cubical-nerve_wf,
bool_wf,
eqtt_to_assert,
assert-eq-cname,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
coordinate_name_wf,
l_member_wf,
I-face_wf,
not_wf,
equal-wf-T-base,
extd-nameset-nil,
face-direction_wf,
or_wf,
l_exists_wf,
assert_wf,
extd-nameset_subtype_int,
nil_wf,
name-morph_wf,
open_box_wf,
list_wf,
small-category_wf,
same-face-edge-arrows-commute,
squash_wf,
true_wf,
cat-ob_wf,
small-category-subtype,
cat-arrow_wf,
iff_weakening_equal,
cat-comp_wf,
nerve_box_edge_same1,
subtype_rel-equal,
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,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
set_subtype_base,
le_wf,
int_subtype_base,
nerve_box_label_wf,
name-morph-flips-commute
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
unionElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
sqequalRule,
baseClosed,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
promote_hyp,
hypothesis_subsumption,
independent_functionElimination,
because_Cache,
applyEquality,
inlFormation,
inrFormation,
equalityElimination,
instantiate,
cumulativity,
axiomEquality,
productEquality,
setEquality,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
functionEquality,
dependent_set_memberEquality,
impliesFunctionality
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(C);I;J;x;i)].
\mforall{}f:name-morph(I;[]). \mforall{}a,b:nameset(I).
\mforall{}[v:I-face(cubical-nerve(C);I)]
((cat-comp(C) nerve\_box\_label(box;f) nerve\_box\_label(box;flip(f;a))
nerve\_box\_label(box;flip(flip(f;a);b))
nerve\_box\_edge(box;f;a)
nerve\_box\_edge(box;flip(f;a);b))
= (cat-comp(C) nerve\_box\_label(box;f) nerve\_box\_label(box;flip(f;b))
nerve\_box\_label(box;flip(flip(f;b);a))
nerve\_box\_edge(box;f;b)
nerve\_box\_edge(box;flip(f;b);a))) supposing
((v \mmember{} box) and
(\mneg{}(dimension(v) = b)) and
(\mneg{}(dimension(v) = a)) and
(\mneg{}(a = b)) and
((f b) = 0) and
(((f a) = 0) \mwedge{} (direction(v) = (f dimension(v)))))
supposing ((\mexists{}j1\mmember{}J. \mneg{}(j1 = a)) \mwedge{} (\mexists{}j2\mmember{}J. \mneg{}(j2 = b)))
\mvee{} ((\mneg{}\muparrow{}null(J)) \mwedge{} ((f x) = i) \mwedge{} ((flip(f;a) x) = i) \mwedge{} ((flip(f;b) x) = i))
Date html generated:
2017_10_05-PM-03_38_41
Last ObjectModification:
2017_07_28-AM-11_25_57
Theory : cubical!sets
Home
Index