Nuprl Lemma : poset-functor-extends_wf
∀[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)))].
∀[F:Functor(poset-cat(I);C)].
(poset-functor-extends(C;I;L;E;F) ∈ ℙ)
Proof
Definitions occuring in Statement :
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
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-},
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
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],
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
prop: ℙ,
and: P ∧ Q,
member: t ∈ T,
name-morph: name-morph(I;J),
respects-equality: respects-equality(S;T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
poset-cat: poset-cat(J),
pi1: fst(t),
cat-ob: cat-ob(C),
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
cat-functor: Functor(C1;C2),
functor-arrow: arrow(F),
pi2: snd(t),
cat-arrow: cat-arrow(C),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
name-morph-flip: flip(f;y),
nameset: nameset(L),
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),
guard: {T},
int_seg: {i..j-},
squash: ↓T,
sq_stable: SqStable(P),
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtract: n - m,
false: False,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
functor-ob: ob(F)
Lemmas referenced :
cat-functor_wf,
poset-cat_wf,
nameset_wf,
int_seg_wf,
extd-nameset-respects-equality,
nil_wf,
coordinate_name_wf,
cat-arrow_wf,
name-morph-flip_wf,
name-morph_wf,
cat-ob_wf,
list_wf,
small-category_wf,
isname_wf,
assert_wf,
all_wf,
extd-nameset_wf,
subtype_rel_self,
subtype_rel_dep_function,
functor-ob_wf,
equal_wf,
equal-wf-T-base,
extd-nameset-nil,
assert_witness,
le_int_wf,
extd-nameset_subtype_int,
assert_of_le_int,
eq-cname_wf,
eqtt_to_assert,
assert-eq-cname,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
lelt_wf,
sq_stable__le,
sq_stable__l_member,
decidable__equal-coordinate_name,
int_seg_properties,
decidable__le,
subtract_wf,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
istype-le,
l_member_wf,
itermVar_wf,
int_term_value_var_lemma,
subtype_rel-equal,
subtype_rel_set
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
productEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionIsType,
setIsType,
because_Cache,
equalityIstype,
natural_numberEquality,
applyEquality,
setElimination,
rename,
baseClosed,
dependent_functionElimination,
independent_functionElimination,
sqequalBase,
equalitySymmetry,
lambdaFormation,
functionExtensionality,
setEquality,
independent_isectElimination,
lambdaEquality,
sqequalRule,
functionEquality,
productElimination,
lambdaEquality_alt,
inhabitedIsType,
equalityTransitivity,
lambdaFormation_alt,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
intEquality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
approximateComputation,
dependent_pairFormation_alt,
Error :memTop,
voidElimination,
dependent_set_memberEquality_alt,
int_eqEquality,
independent_pairFormation,
productIsType
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)))].
\mforall{}[F:Functor(poset-cat(I);C)].
(poset-functor-extends(C;I;L;E;F) \mmember{} \mBbbP{})
Date html generated:
2020_05_21-AM-10_52_54
Last ObjectModification:
2019_12_08-PM-07_05_43
Theory : cubical!sets
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