Nuprl Lemma : poset_functor_extend-is-functor
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[L:name-morph(I;[]) ⟶ cat-ob(C)]. ∀[E:i:nameset(I)
⟶ c:{c:name-morph(I;[])| (c i) = 0 ∈ ℕ2}
⟶ (cat-arrow(C) (L c) (L flip(c;i)))].
<L, λc1,c2,p. poset_functor_extend(C;I;L;E;c1;c2)> ∈ Functor(poset-cat(I);C) supposing edge-arrows-commute(C;I;L;E)
Proof
Definitions occuring in Statement :
edge-arrows-commute: edge-arrows-commute(C;I;L;E),
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
poset-cat: poset-cat(J),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
nil: [],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
natural_number: $n,
equal: s = t ∈ T,
cat-functor: Functor(C1;C2),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cat-functor: Functor(C1;C2),
subtype_rel: A ⊆r B,
name-morph: name-morph(I;J),
cat-ob: cat-ob(C),
pi1: fst(t),
poset-cat: poset-cat(J),
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
respects-equality: respects-equality(S;T),
implies: P ⇒ Q,
cat-arrow: cat-arrow(C),
pi2: snd(t),
uiff: uiff(P;Q),
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
ge: i ≥ j ,
guard: {T},
le: A ≤ B,
top: Top,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nameset: nameset(L),
sq_stable: SqStable(P),
coordinate_name: Cname,
int_upper: {i...},
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
has-value: (a)↓,
sq_type: SQType(T),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
assert: ↑b,
btrue: tt,
eq_int: (i =z j),
name-morph-flip: flip(f;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
rev_uimplies: rev_uimplies(P;Q),
l_member: (x ∈ l),
select: L[n],
cons: [a / b],
nat_plus: ℕ+,
subtract: n - m,
edge-arrows-commute: edge-arrows-commute(C;I;L;E)
Lemmas referenced :
subtype_rel_self,
cat-ob_wf,
poset-cat_wf,
poset_functor_extend_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
cat-arrow_wf,
poset_functor_extend_id,
cat-id_wf,
cat-comp_wf,
edge-arrows-commute_wf,
nameset_wf,
int_seg_wf,
extd-nameset-respects-equality,
name-morph-flip_wf,
list_wf,
small-category_wf,
assert_of_le_int,
extd-nameset_wf,
istype-assert,
isname_wf,
extd-nameset_subtype_int,
poset-cat-dist_wf,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_properties,
intformand_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
ge_wf,
istype-less_than,
istype-le,
subtract-1-ge-0,
istype-nat,
poset-cat-dist-add,
intformeq_wf,
itermAdd_wf,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
poset-cat-dist-zero,
equal_wf,
squash_wf,
true_wf,
istype-universe,
poset_functor_extend_same,
subtype_rel-equal,
iff_weakening_equal,
cat-comp-ident,
decidable__equal-poset-cat-ob,
equal_functionality_wrt_subtype_rel2,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
l_member_wf,
value-type-has-value,
list-value-type,
filter_wf5,
eq_int_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
band_wf,
btrue_wf,
assert_of_eq_int,
bfalse_wf,
name-morph-ext,
le_wf,
extd-nameset-nil,
int_seg_properties,
decidable__equal_int,
decidable__lt,
set_subtype_base,
lelt_wf,
int_subtype_base,
poset-cat-dist-non-zero,
int_seg_subtype_special,
int_seg_cases,
non_null_filter_iff,
list-subtype,
l_exists_iff,
assert_wf,
equal-wf-T-base,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
respects-equality-set-trivial2,
respects-equality-list,
length_of_nil_lemma,
filter_nil_lemma,
filter_wf2,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
null_nil_lemma,
or_wf,
length_of_cons_lemma,
length_wf,
cons_wf,
exists_wf,
equal-wf-base,
all_wf,
list_induction,
filter_cons_lemma,
cons_member,
int_formual_prop_imp_lemma,
intformimplies_wf,
subtype_rel_dep_function,
subtype_rel_sets_simple,
subtype_rel_sets,
subtype_rel_list,
list_subtype_base,
reduce_hd_cons_lemma,
hd-wf-not-null,
assert_of_ff,
member_filter_2,
nameset_subtype_base,
hd_member,
eq-cname_wf,
assert-eq-cname,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
cat-comp-assoc,
poset-cat-arrow-flip,
add-is-int-iff,
false_wf,
poset-cat-dist-flip,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
select_wf,
poset-cat-arrow-iff,
name-morph-flips-commute,
poset_functor_extend-flip
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
hypothesisEquality,
applyEquality,
thin,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
functionEquality,
hypothesis,
lambdaEquality_alt,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
universeIsType,
inhabitedIsType,
functionIsType,
dependent_functionElimination,
independent_pairFormation,
productElimination,
productIsType,
equalityIstype,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
isectIsTypeImplies,
setIsType,
natural_numberEquality,
setElimination,
rename,
baseClosed,
independent_functionElimination,
sqequalBase,
equalityIsType1,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
voidElimination,
intWeakElimination,
functionIsTypeImplies,
applyLambdaEquality,
imageElimination,
instantiate,
universeEquality,
imageMemberEquality,
hyp_replacement,
callbyvalueReduce,
cumulativity,
intEquality,
closedConclusion,
equalityIsType4,
hypothesis_subsumption,
equalityIsType3,
productEquality,
promote_hyp,
functionExtensionality,
unionIsType,
setEquality,
inlFormation_alt,
inrFormation_alt,
equalityElimination,
pointwiseFunctionality,
baseApply
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[L:name-morph(I;[]) {}\mrightarrow{} cat-ob(C)].
\mforall{}[E:i:nameset(I) {}\mrightarrow{} c:\{c:name-morph(I;[])| (c i) = 0\} {}\mrightarrow{} (cat-arrow(C) (L c) (L flip(c;i)))].
<L, \mlambda{}c1,c2,p. poset\_functor\_extend(C;I;L;E;c1;c2)> \mmember{} Functor(poset-cat(I);C)
supposing edge-arrows-commute(C;I;L;E)
Date html generated:
2020_05_21-AM-10_54_36
Last ObjectModification:
2019_12_09-PM-07_32_21
Theory : cubical!sets
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