Nuprl Lemma : nerve_box_edge_same1
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].
∀f:I-face(cubical-nerve(C);I)
(cube(f) c flip(c;y) (λx.Ax))
= nerve_box_edge(box;c;y)
∈ (cat-arrow(C) nerve_box_label(box;c) nerve_box_label(box;flip(c;y)))
supposing (f ∈ box) ∧ (direction(f) = (c dimension(f)) ∈ ℕ2) ∧ (direction(f) = (flip(c;y) dimension(f)) ∈ ℕ2)
supposing (∃j∈J. ¬(j = y ∈ Cname)) ∨ (((c x) = i ∈ ℕ2) ∧ (¬↑null(J)))
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-cube: cube(f),
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,
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,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
axiom: Ax,
functor-arrow: arrow(F),
cat-arrow: cat-arrow(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
nerve_box_edge: nerve_box_edge(box;c;y),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
name-morph: name-morph(I;J),
so_apply: x[s],
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
implies: 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},
nameset: nameset(L),
false: False,
coordinate_name: Cname,
int_upper: {i...},
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
prop: ℙ,
cons: [a / b],
bfalse: ff,
open_box: open_box(X;I;J;x;i),
respects-equality: respects-equality(S;T),
decidable: Dec(P),
sq_type: SQType(T),
face-name: face-name(f),
pi2: snd(t),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
true: True,
uiff: uiff(P;Q),
istype: istype(T),
sq_stable: SqStable(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bnot: ¬bb,
subtract: n - m,
less_than': less_than'(a;b),
unit: Unit,
bool: 𝔹,
name-morph-flip: flip(f;y),
cand: A c∧ B,
adjacent-compatible: adjacent-compatible(X;I;L),
face-compatible: face-compatible(X;I;f1;f2),
spreadn: spread3,
face-cube: cube(f),
cubical-nerve: cubical-nerve(X),
cube-set-restriction: f(s),
I-cube: X(I),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda3,
functor-comp: functor-comp(F;G),
poset-functor: poset-functor(J;K;f),
lt_int: i <z j,
le_int: i ≤z j,
isname: isname(z),
functor-ob: ob(F),
cat-functor: Functor(C1;C2)
Lemmas referenced :
nerve_box_label_same,
nerve-box-common-face_wf2,
subtype_rel_set,
name-morph_wf,
nil_wf,
coordinate_name_wf,
cat-ob_wf,
poset-cat_wf,
equal-wf-T-base,
int_seg_wf,
subtype_rel_self,
nameset_wf,
list-cases,
stuck-spread,
istype-base,
istype-void,
length_of_nil_lemma,
null_nil_lemma,
int_seg_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
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,
name-morph-flip_wf,
subtype_rel-equal,
cat-arrow_wf,
l_member_wf,
I-face_wf,
cubical-nerve_wf,
face-direction_wf,
face-dimension_wf,
extd-nameset-nil,
l_exists_wf,
not_wf,
equal_wf,
istype-assert,
null_wf3,
subtype_rel_list,
top_wf,
extd-nameset-respects-equality,
open_box_wf,
list_wf,
small-category_wf,
decidable__equal-coordinate_name,
lelt_wf,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
face-name_wf,
pairwise-implies,
cubical-nerve-I-cube,
cubical-set_wf,
true_wf,
squash_wf,
face-cube_wf,
list-diff-subset,
nameset_subtype,
cons_wf,
cname_deq_wf,
list-diff_wf,
name-morph_subtype,
poset-cat-arrow-iff,
member-poset-cat-arrow,
int_formula_prop_not_lemma,
intformnot_wf,
istype-le,
subtype_rel_dep_function,
isname_wf,
all_wf,
extd-nameset_wf,
decidable__le,
sq_stable__le,
sq_stable__l_member,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_wf,
bool_cases_sqequal,
bool_subtype_base,
eqff_to_assert,
istype-false,
assert-eq-cname,
eqtt_to_assert,
eq-cname_wf,
functor-ob_wf,
functor-arrow_wf,
sq_stable__face-compatible,
face-compatible_wf,
sq_stable__pairwise,
face-compatible-symmetry,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
ob_pair_lemma,
ob_mk_functor_lemma,
face-map-comp-id,
int_seg_cases,
int_seg_subtype_special,
cat-functor_wf,
istype-universe,
le_reflexive,
I-cube_wf,
nameset_subtype_base,
arrow_mk_functor_lemma,
arrow_pair_lemma,
cat_ob_pair_lemma,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
extd-nameset_subtype_int
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation_alt,
productElimination,
applyEquality,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
imageElimination,
because_Cache,
baseClosed,
universeIsType,
independent_isectElimination,
inhabitedIsType,
unionElimination,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
promote_hyp,
hypothesis_subsumption,
inrFormation_alt,
equalityIstype,
dependent_set_memberEquality_alt,
productIsType,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
unionIsType,
setIsType,
functionIsType,
sqequalBase,
independent_pairEquality,
intEquality,
instantiate,
cumulativity,
closedConclusion,
productEquality,
imageMemberEquality,
functionEquality,
equalityIsType1,
equalityIsType3,
equalityElimination,
hyp_replacement,
universeEquality,
dependent_pairEquality_alt
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{}[y:nameset(I)]. \mforall{}[c:\{c:name-morph(I;[])| (c y) = 0\} ].
\mforall{}f:I-face(cubical-nerve(C);I)
(cube(f) c flip(c;y) (\mlambda{}x.Ax)) = nerve\_box\_edge(box;c;y)
supposing (f \mmember{} box)
\mwedge{} (direction(f) = (c dimension(f)))
\mwedge{} (direction(f) = (flip(c;y) dimension(f)))
supposing (\mexists{}j\mmember{}J. \mneg{}(j = y)) \mvee{} (((c x) = i) \mwedge{} (\mneg{}\muparrow{}null(J)))
Date html generated:
2020_05_21-AM-10_55_18
Last ObjectModification:
2019_12_08-PM-07_05_29
Theory : cubical!sets
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