Nuprl Lemma : groupoid-nerve-functor-flip
∀[G:Groupoid]. ∀[I:Cname List]. ∀[u:nameset(I)]. ∀[K:Cname List]. ∀[f:name-morph(I;K)]. ∀[f1:name-morph(K;[])].
∀x1:nameset(I)
∀[F:Functor(poset-cat(I-[x1]);cat(G))]
(F (f o f1) flip((f o f1);u) (λx.Ax))
= (f(F) f1 flip(f1;f u) (λx.Ax))
∈ (cat-arrow(cat(G)) (F (f o f1)) (F (f o flip(f1;f u))))
supposing (↑isname(f u)) ∧ ((f1 (f u)) = 0 ∈ ℕ2)
Proof
Definitions occuring in Statement :
cubical-nerve: cubical-nerve(X),
poset-cat: poset-cat(J),
cube-set-restriction: f(s),
name-morph-flip: flip(f;y),
name-comp: (f o g),
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
groupoid-cat: cat(G),
groupoid: Groupoid,
functor-arrow: arrow(F),
functor-ob: ob(F),
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T,
axiom: Ax
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
nameset: nameset(L),
and: P ∧ Q,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
cubical-nerve: cubical-nerve(X),
cube-set-restriction: f(s),
pi2: snd(t),
poset-functor: poset-functor(J;K;f),
functor-comp: functor-comp(F;G),
functor-arrow: arrow(F),
functor-ob: ob(F),
top: Top,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
pi1: fst(t),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
ifthenelse: if b then t else f fi ,
btrue: tt,
int_seg: {i..j-},
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False
Lemmas referenced :
cat-functor_wf,
poset-cat_wf,
list-diff_wf,
coordinate_name_wf,
cname_deq_wf,
cons_wf,
nil_wf,
groupoid-cat_wf,
name-morph_wf,
nameset_wf,
list_wf,
groupoid_wf,
assert-isname,
ob_pair_lemma,
istype-void,
arrow_pair_lemma,
name-morph-ext,
name-comp_wf,
name-morph-flip_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
extd-nameset_wf,
name-comp-flip,
subtype_rel_self,
iff_weakening_equal,
arrow_mk_functor_lemma,
functor-arrow_wf,
name-morph_subtype,
nameset_subtype,
list-diff-subset,
member-poset-cat-arrow,
poset-cat-arrow-flip,
cat-ob_wf,
isname-name,
int_seg_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
poset-cat-arrow_subtype,
subtype_rel_wf,
cat-arrow_wf,
istype-assert,
isname_wf,
int_seg_wf,
extd-nameset-nil
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
setElimination,
rename,
because_Cache,
productElimination,
applyEquality,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
equalitySymmetry,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
inhabitedIsType,
instantiate,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
applyLambdaEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
hyp_replacement,
productIsType,
equalityIsType3
Latex:
\mforall{}[G:Groupoid]. \mforall{}[I:Cname List]. \mforall{}[u:nameset(I)]. \mforall{}[K:Cname List]. \mforall{}[f:name-morph(I;K)].
\mforall{}[f1:name-morph(K;[])].
\mforall{}x1:nameset(I)
\mforall{}[F:Functor(poset-cat(I-[x1]);cat(G))]
(F (f o f1) flip((f o f1);u) (\mlambda{}x.Ax)) = (f(F) f1 flip(f1;f u) (\mlambda{}x.Ax))
supposing (\muparrow{}isname(f u)) \mwedge{} ((f1 (f u)) = 0)
Date html generated:
2019_11_05-PM-00_39_25
Last ObjectModification:
2018_11_10-PM-03_22_44
Theory : cubical!sets
Home
Index