Nuprl Lemma : same-face-edge-arrows-commute2
∀[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. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
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),
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
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],
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nameset: nameset(L),
prop: ℙ,
open_box: open_box(X;I;J;x;i),
name-morph: name-morph(I;J),
so_lambda: λ2x.t[x],
so_apply: x[s],
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b
Lemmas referenced :
same-face-edge-arrows-commute1,
nameset_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
open_box_wf,
cubical-nerve_wf,
subtype_rel_list,
int_seg_wf,
l_member_wf,
I-face_wf,
not_wf,
equal_wf,
face-dimension_wf,
equal-wf-T-base,
extd-nameset-nil,
face-direction_wf,
l_exists_wf,
list_wf,
small-category_wf,
decidable__equal-coordinate_name,
select_wf,
sq_stable__l_member,
sq_stable__le,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
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,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
promote_hyp,
lambdaFormation,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
applyEquality,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productEquality,
setEquality,
productElimination,
independent_functionElimination,
imageMemberEquality,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
applyLambdaEquality,
dependent_set_memberEquality
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. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2)))
Date html generated:
2017_10_05-PM-03_39_00
Last ObjectModification:
2017_07_28-AM-11_26_12
Theory : cubical!sets
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