Nuprl Lemma : groupoid-edges-commute1
∀G:Groupoid. ∀I:Cname List. ∀J:nameset(I) List. ∀j:nameset(I).
((j ∈ J)
⇒ (∀j'∈J.j' = j ∈ Cname)
⇒ (∀x:nameset(I). ∀i:ℕ2. ∀box:open_box(cubical-nerve(fst(G));I;J;x;i).
edge-arrows-commute(fst(G);I;λc.nerve_box_label(box;c);λy,c. nerve_box_edge1(G;box;x;i;j;c;y))))
Proof
Definitions occuring in Statement :
nerve_box_edge1: nerve_box_edge1(G;box;x;i;j;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_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
int_seg: {i..j-},
pi1: fst(t),
all: ∀x:A. B[x],
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),
not: ¬A,
member: t ∈ T,
false: False,
name-morph: name-morph(I;J),
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
respects-equality: respects-equality(S;T),
groupoid: Groupoid,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
nerve_box_edge1: nerve_box_edge1(G;box;x;i;j;c;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
coordinate_name: Cname,
int_upper: {i...},
sq_type: SQType(T),
guard: {T},
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
groupoid-cat: cat(G),
top: Top,
decidable: Dec(P),
true: True,
sq_stable: SqStable(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
name-morph-flip: flip(f;y),
open_box: open_box(X;I;J;x;i),
cand: A c∧ B,
pi2: snd(t),
pi1: fst(t),
face-name: face-name(f),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
eq_int: (i =z j),
nequal: a ≠ b ∈ T ,
cat-square-commutes: x_y1 o y1_z = x_y2 o y2_z,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
nerve_box_edge': nerve_box_edge'(box; c; y),
less_than': less_than'(a;b)
Lemmas referenced :
istype-void,
extd-nameset-respects-equality,
nil_wf,
coordinate_name_wf,
int_seg_wf,
name-morph_wf,
open_box_wf,
cubical-nerve_wf,
pi1_wf_top,
small-category_wf,
subtype_rel_list,
nameset_wf,
l_all_wf2,
equal_wf,
l_member_wf,
list_wf,
groupoid_wf,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
eq_int_wf,
extd-nameset_subtype_int,
name-morph-flip_wf,
subtype_rel-equal,
cat-arrow_wf,
groupoid-cat_wf,
nerve_box_label_wf,
decidable__assert,
null_wf3,
top_wf,
member-implies-null-eq-bfalse,
assert_elim,
btrue_neq_bfalse,
squash_wf,
true_wf,
istype-universe,
or_wf,
equal-wf-T-base,
not_wf,
name-morph-flips-commute,
subtype_rel_self,
iff_weakening_equal,
extd-nameset-nil,
bor-bfalse,
bor-btrue,
assert_of_eq_int,
neg_assert_of_eq_int,
l_exists_wf,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
itermConstant_wf,
intformle_wf,
decidable__le,
sq_stable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
int_seg_properties,
l_member_subtype,
same-face-edge-arrows-commute0,
equal-wf-base,
nameset_subtype_base,
get_face-wf,
pi2_wf,
I-face_wf,
member_wf,
bnot_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
bool_cases,
iff_transitivity,
assert_of_bnot,
and_wf,
name-morph-flip-flip,
extd-nameset_wf,
name-morph-ext,
cat-square-commutes_wf,
set_wf,
nerve_box_edge_wf,
groupoid-square1_wf,
cat-ob_wf,
cat-comp_wf,
nerve_box_edge_wf2,
l_exists_iff,
subtract_wf,
groupoid-square2_wf,
int_seg_subtype_special,
int_seg_cases,
istype-le,
istype-less_than,
nequal_wf,
istype-assert,
nsub2-flip,
groupoid-cube,
cat-square-commutes-sym,
get_face_wf,
same-face-square-commutes2,
face-dimension_wf,
face-direction_wf,
nerve_box_edge'_wf,
false_wf,
equal-wf-base-T,
same-face-edge-arrows-commute4,
face-name_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalRule,
cut,
rename,
setElimination,
thin,
hypothesis,
functionIsType,
equalityIstype,
because_Cache,
hypothesisEquality,
sqequalHypSubstitution,
introduction,
extract_by_obid,
inhabitedIsType,
setIsType,
applyEquality,
baseClosed,
isectElimination,
productElimination,
imageElimination,
dependent_functionElimination,
independent_functionElimination,
sqequalBase,
equalitySymmetry,
universeIsType,
instantiate,
independent_pairEquality,
Error :memTop,
independent_isectElimination,
lambdaEquality_alt,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
cumulativity,
intEquality,
voidElimination,
dependent_pairFormation_alt,
promote_hyp,
isect_memberEquality_alt,
inlFormation_alt,
inrFormation_alt,
universeEquality,
imageMemberEquality,
setEquality,
productEquality,
dependent_set_memberEquality,
int_eqEquality,
approximateComputation,
applyLambdaEquality,
dependent_pairFormation,
independent_pairFormation,
voidEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
inrFormation,
hyp_replacement,
inlFormation,
hypothesis_subsumption,
dependent_set_memberEquality_alt,
productIsType,
closedConclusion,
equalityIsType1,
equalityIsType2
Latex:
\mforall{}G:Groupoid. \mforall{}I:Cname List. \mforall{}J:nameset(I) List. \mforall{}j:nameset(I).
((j \mmember{} J)
{}\mRightarrow{} (\mforall{}j'\mmember{}J.j' = j)
{}\mRightarrow{} (\mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2. \mforall{}box:open\_box(cubical-nerve(fst(G));I;J;x;i).
edge-arrows-commute(fst(G);I;\mlambda{}c.nerve\_box\_label(box;c);\mlambda{}y,c. ...)))
Date html generated:
2020_05_21-AM-10_56_56
Last ObjectModification:
2019_12_10-PM-06_28_11
Theory : cubical!sets
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