Nuprl Lemma : same-face-square-commutes2
∀[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,g,h,k:name-morph(I;[])].
∀a,b:nameset(I).
(nerve_box_edge(box;f;a) o nerve_box_edge(box;g;b) = nerve_box_edge(box;f;b) o nerve_box_edge(box;h;a)) supposing
(((((¬(a = b ∈ nameset(I))) ∧ ((f a) = 0 ∈ ℕ2))
∧ ((f b) = 0 ∈ ℕ2)
∧ (g = flip(f;a) ∈ name-morph(I;[]))
∧ (h = flip(f;b) ∈ name-morph(I;[]))
∧ (k = flip(flip(f;a);b) ∈ name-morph(I;[])))
∧ (∃v:I-face(cubical-nerve(C);I)
((v ∈ box)
∧ (¬(dimension(v) = b ∈ Cname))
∧ (¬(dimension(v) = a ∈ Cname))
∧ (direction(v) = (f dimension(v)) ∈ ℕ2)))) and
(((∃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-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
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],
exists: ∃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,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
and: P ∧ Q,
cand: A c∧ B,
exists: ∃x:A. B[x],
prop: ℙ,
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
so_apply: x[s],
top: Top,
int_seg: {i..j-},
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
l_exists: (∃x∈L. P[x]),
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
lelt: i ≤ j < k,
false: False,
coordinate_name: Cname,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
true: True,
cons: [a / b],
bfalse: ff,
squash: ↓T,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
nerve_box_edge': nerve_box_edge'(box; c; y),
name-morph-flip: flip(f;y),
uiff: uiff(P;Q),
sq_type: SQType(T)
Lemmas referenced :
same-face-edge-arrows-commute0,
not_wf,
equal_wf,
equal-wf-T-base,
int_seg_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
name-morph-flip_wf,
exists_wf,
I-face_wf,
cubical-nerve_wf,
l_member_wf,
face-dimension_wf,
nameset_wf,
face-direction_wf,
or_wf,
l_exists_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
extd-nameset_subtype_int,
open_box_wf,
list_wf,
small-category_wf,
list-cases,
null_nil_lemma,
stuck-spread,
base_wf,
length_of_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,
squash_wf,
true_wf,
cat-arrow_wf,
nerve_box_label_wf,
decidable__assert,
extd-nameset-nil,
cat-comp_wf,
cat-ob_wf,
iff_weakening_equal,
nerve_box_edge_wf2,
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,
subtype_rel-equal,
nerve_box_edge'_wf,
set_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqff_to_assert,
eq-cname_wf,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
assert_of_bnot,
assert-eq-cname,
set_subtype_base,
le_wf,
int_subtype_base,
name-morph-flips-commute
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
lambdaFormation,
productElimination,
independent_isectElimination,
independent_pairFormation,
productEquality,
because_Cache,
natural_numberEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
baseClosed,
lambdaEquality,
setEquality,
isect_memberEquality,
voidElimination,
voidEquality,
intEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption,
hyp_replacement,
imageElimination,
universeEquality,
inlFormation,
inrFormation,
imageMemberEquality,
dependent_set_memberEquality,
instantiate,
cumulativity,
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,g,h,k:name-morph(I;[])].
\mforall{}a,b:nameset(I).
(nerve\_box\_edge(box;f;a) o nerve\_box\_edge(box;g;b)
= nerve\_box\_edge(box;f;b) o nerve\_box\_edge(box;h;a)) supposing
(((((\mneg{}(a = b)) \mwedge{} ((f a) = 0))
\mwedge{} ((f b) = 0)
\mwedge{} (g = flip(f;a))
\mwedge{} (h = flip(f;b))
\mwedge{} (k = flip(flip(f;a);b)))
\mwedge{} (\mexists{}v:I-face(cubical-nerve(C);I)
((v \mmember{} box)
\mwedge{} (\mneg{}(dimension(v) = b))
\mwedge{} (\mneg{}(dimension(v) = a))
\mwedge{} (direction(v) = (f dimension(v)))))) and
(((\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_39_45
Last ObjectModification:
2017_07_28-AM-11_26_32
Theory : cubical!sets
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