Nuprl Lemma : groupoid-edges-commute
∀G:Groupoid. ∀I:Cname List. ∀J:nameset(I) List. ∀x:nameset(I). ∀i:ℕ2. ∀box:open_box(cubical-nerve(fst(G));I;J;x;i).
((∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname)))
⇒ edge-arrows-commute(fst(G);I;λc.nerve_box_label(box;c);λy,c. nerve_box_edge(box;c;y)))
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),
edge-arrows-commute: edge-arrows-commute(C;I;L;E),
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
coordinate_name: Cname,
l_exists: (∃x∈L. P[x]),
list: T List,
int_seg: {i..j-},
pi1: fst(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
groupoid: Groupoid
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
edge-arrows-commute: edge-arrows-commute(C;I;L;E),
uall: ∀[x:A]. B[x],
member: t ∈ T,
groupoid: Groupoid,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
nameset: nameset(L),
so_apply: x[s],
uimplies: b supposing a,
int_seg: {i..j-},
name-morph: name-morph(I;J),
open_box: open_box(X;I;J;x;i),
not: ¬A,
false: False,
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
respects-equality: respects-equality(S;T),
top: Top,
pi1: fst(t),
iff: P ⇐⇒ Q,
groupoid-cat: cat(G),
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
guard: {T},
sq_stable: SqStable(P),
coordinate_name: Cname,
int_upper: {i...},
satisfiable_int_formula: satisfiable_int_formula(fmla),
I-face: I-face(X;I),
face-dimension: dimension(f),
face-direction: direction(f),
face-name: face-name(f),
pi2: snd(t),
sq_type: SQType(T),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
name-morph-flip: flip(f;y),
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
label: ...$L... t,
l_subset: l_subset(T;as;bs),
subtract: n - m,
cons: [a / b]
Lemmas referenced :
decidable__l_exists_better-extract,
I-face_wf,
cubical-nerve_wf,
pi1_wf_top,
small-category_wf,
not_wf,
equal_wf,
coordinate_name_wf,
face-dimension_wf,
equal-wf-base-T,
face-direction_wf,
subtype_rel_product,
cat-ob_wf,
cat-arrow_wf,
cat-comp_wf,
cat-id_wf,
top_wf,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
decidable__cand,
istype-void,
decidable__not,
decidable__equal-coordinate_name,
decidable__equal_int_seg,
extd-nameset-respects-equality,
nil_wf,
int_seg_wf,
name-morph_wf,
l_exists_wf,
nameset_wf,
l_member_wf,
open_box_wf,
subtype_rel_list,
list_wf,
groupoid_wf,
l_exists_iff,
same-face-edge-arrows-commute3,
groupoid-cat_wf,
l_all_iff,
or_wf,
decidable__or,
l_member_subtype,
extd-nameset-nil,
select_wf,
int_seg_properties,
length_wf,
sq_stable__l_member,
sq_stable__le,
decidable__le,
full-omega-unsat,
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,
subtype_base_sq,
nameset_subtype_base,
subtype_rel_universe1,
all_wf,
intformeq_wf,
intformor_wf,
int_formula_prop_eq_lemma,
int_formula_prop_or_lemma,
decidable__equal_int,
istype-le,
int_seg_subtype_nat,
istype-false,
istype-less_than,
nat_properties,
respects-equality-set-trivial2,
not_functionality_wrt_implies,
equal-wf-base,
base_wf,
equal_functionality_wrt_subtype_rel2,
iff_weakening_equal,
squash_wf,
true_wf,
subtype_rel_self,
pi2_wf,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
bool_wf,
bool_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
iff_weakening_uiff,
assert_wf,
iff_imp_equal_bool,
istype-assert,
groupoid-cube-lemma2,
nerve_box_label_wf,
assert_elim,
null_wf3,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
name-morph-flip_wf,
decidable__assert,
nerve_box_edge_wf,
int_seg_subtype_special,
int_seg_cases,
cat-square-commutes_wf,
subtype_rel-equal,
istype-universe,
equal-wf-T-base,
name-morph-flips-commute,
same-face-square-commutes,
select_member,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
member_wf,
btrue_wf,
iff_functionality_wrt_iff,
name-morph-flip-flip,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
face-name_wf,
decidable__equal_product
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
rename,
setElimination,
thin,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
instantiate,
hypothesis,
productElimination,
independent_pairEquality,
hypothesisEquality,
Error :memTop,
lambdaEquality_alt,
productEquality,
applyEquality,
because_Cache,
cumulativity,
setEquality,
functionEquality,
universeIsType,
independent_isectElimination,
setIsType,
functionIsType,
productIsType,
equalityIstype,
intEquality,
natural_numberEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberEquality_alt,
imageElimination,
unionElimination,
inhabitedIsType,
baseClosed,
sqequalBase,
equalitySymmetry,
voidElimination,
equalityTransitivity,
closedConclusion,
independent_pairFormation,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
imageMemberEquality,
applyLambdaEquality,
approximateComputation,
int_eqEquality,
equalityIsType1,
inrFormation_alt,
equalityIsType4,
unionIsType,
inlFormation_alt,
equalityIsType2,
universeEquality,
hyp_replacement,
equalityElimination,
promote_hyp,
hypothesis_subsumption,
equalityIsType3
Latex:
\mforall{}G:Groupoid. \mforall{}I:Cname List. \mforall{}J:nameset(I) List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2.
\mforall{}box:open\_box(cubical-nerve(fst(G));I;J;x;i).
((\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2)))
{}\mRightarrow{} edge-arrows-commute(fst(G);I;\mlambda{}c.nerve\_box\_label(box;c);\mlambda{}y,c. nerve\_box\_edge(box;c;y)))
Date html generated:
2020_05_21-AM-10_55_54
Last ObjectModification:
2020_01_03-AM-00_21_29
Theory : cubical!sets
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