Nuprl Lemma : same-face-edge-arrows-commute
∀C:SmallCategory. ∀I:Cname List. ∀f:name-morph(I;[]). ∀a,b:nameset(I). ∀v:I-face(cubical-nerve(C);I).
(((f a) = 0 ∈ ℕ2)
⇒ ((f b) = 0 ∈ ℕ2)
⇒ (¬(a = b ∈ nameset(I)))
⇒ (¬(dimension(v) = a ∈ Cname))
⇒ (¬(dimension(v) = b ∈ Cname))
⇒ ((cat-comp(C) (ob(cube(v)) f) (ob(cube(v)) flip(f;a)) (ob(cube(v)) flip(flip(f;a);b))
(arrow(cube(v)) f flip(f;a) (λx.Ax))
(arrow(cube(v)) flip(f;a) flip(flip(f;a);b) (λx.Ax)))
= (cat-comp(C) (ob(cube(v)) f) (ob(cube(v)) flip(f;b)) (ob(cube(v)) flip(flip(f;b);a))
(arrow(cube(v)) f flip(f;b) (λx.Ax))
(arrow(cube(v)) flip(f;b) flip(flip(f;b);a) (λx.Ax)))
∈ (cat-arrow(C) (ob(cube(v)) f) (ob(cube(v)) flip(flip(f;a);b)))))
Proof
Definitions occuring in Statement :
cubical-nerve: cubical-nerve(X),
face-cube: cube(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,
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
nil: [],
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
axiom: Ax
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
I-face: I-face(X;I),
face-cube: cube(f),
pi2: snd(t),
face-dimension: dimension(f),
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
cat-functor: Functor(C1;C2),
functor-arrow: arrow(F),
functor-ob: ob(F),
prop: ℙ,
subtype_rel: A ⊆r B,
nameset: nameset(L),
name-morph: name-morph(I;J),
cat-comp: cat-comp(C),
poset-cat: poset-cat(J),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
cat-id: cat-id(C),
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
name-morph-flip: flip(f;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
coordinate_name: Cname,
int_upper: {i...},
sq_type: SQType(T),
int_seg: {i..j-},
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
false: False,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
sq_stable: SqStable(P),
squash: ↓T,
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
label: ...$L... t
Lemmas referenced :
cubical-nerve-I-cube,
not_wf,
equal_wf,
coordinate_name_wf,
face-dimension_wf,
cubical-nerve_wf,
nameset_wf,
equal-wf-T-base,
int_seg_wf,
extd-nameset-nil,
I-face_wf,
name-morph_wf,
nil_wf,
list_wf,
small-category_wf,
all_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
le_wf,
assert_witness,
le_int_wf,
extd-nameset_subtype_int,
assert_of_le_int,
name-morph_subtype,
nameset_subtype,
list-diff-subset,
name-morph-flip_wf,
eq-cname_wf,
bool_wf,
eqtt_to_assert,
assert-eq-cname,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
lelt_wf,
false_wf,
subtract_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
int_seg_properties,
decidable__le,
l_member_wf,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
intformand_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
le_reflexive,
squash_wf,
true_wf,
cat-arrow_wf,
cat-ob_wf,
poset-cat_wf,
subtype_rel_self,
extd-nameset_wf,
assert_wf,
isname_wf,
name-morph-flips-commute,
iff_weakening_equal,
nameset_subtype_base,
le_weakening,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
applyEquality,
lambdaEquality,
because_Cache,
natural_numberEquality,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
instantiate,
cumulativity,
intEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
imageMemberEquality,
imageElimination,
applyLambdaEquality,
dependent_set_memberEquality,
int_eqEquality,
computeAll,
hyp_replacement,
universeEquality,
functionExtensionality,
setEquality,
functionEquality
Latex:
\mforall{}C:SmallCategory. \mforall{}I:Cname List. \mforall{}f:name-morph(I;[]). \mforall{}a,b:nameset(I).
\mforall{}v:I-face(cubical-nerve(C);I).
(((f a) = 0)
{}\mRightarrow{} ((f b) = 0)
{}\mRightarrow{} (\mneg{}(a = b))
{}\mRightarrow{} (\mneg{}(dimension(v) = a))
{}\mRightarrow{} (\mneg{}(dimension(v) = b))
{}\mRightarrow{} ((cat-comp(C) (ob(cube(v)) f) (ob(cube(v)) flip(f;a)) (ob(cube(v)) flip(flip(f;a);b))
(arrow(cube(v)) f flip(f;a) (\mlambda{}x.Ax))
(arrow(cube(v)) flip(f;a) flip(flip(f;a);b) (\mlambda{}x.Ax)))
= (cat-comp(C) (ob(cube(v)) f) (ob(cube(v)) flip(f;b)) (ob(cube(v)) flip(flip(f;b);a))
(arrow(cube(v)) f flip(f;b) (\mlambda{}x.Ax))
(arrow(cube(v)) flip(f;b) flip(flip(f;b);a) (\mlambda{}x.Ax)))))
Date html generated:
2017_10_05-PM-03_37_25
Last ObjectModification:
2017_07_28-AM-11_25_34
Theory : cubical!sets
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