Nuprl Lemma : poset_functor_extend_id
∀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))).
∀x:cat-ob(poset-cat(I)).
(poset_functor_extend(C;I;L;E;x;x) = (cat-id(C) (L x)) ∈ (cat-arrow(C) (L x) (L x)))
Proof
Definitions occuring in Statement :
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,
cat-id: cat-id(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
nil: [],
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
name-morph: name-morph(I;J),
subtype_rel: A ⊆r B,
prop: ℙ,
poset-cat: poset-cat(J),
cat-ob: cat-ob(C),
pi1: fst(t),
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
nameset: nameset(L),
uimplies: b supposing a,
nil: [],
it: ⋅,
ifthenelse: if b then t else f fi ,
null: null(as),
btrue: tt,
cat-id: cat-id(C),
pi2: snd(t),
l_all: (∀x∈L.P[x]),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q
Lemmas referenced :
cat-ob_wf,
poset-cat_wf,
nameset_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
equal-wf-T-base,
int_seg_wf,
extd-nameset-nil,
cat-arrow_wf,
name-morph-flip_wf,
list_wf,
small-category_wf,
filter_is_nil,
band_wf,
eq_int_wf,
extd-nameset_subtype_int,
l_member_wf,
list-subtype,
cat-id_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
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,
extd-nameset_wf,
equal_wf,
length_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
not_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionEquality,
setEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
baseClosed,
functionExtensionality,
because_Cache,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
callbyvalueReduce,
sqleReflexivity,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
productEquality,
addLevel,
impliesFunctionality
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{}x:cat-ob(poset-cat(I)).
(poset\_functor\_extend(C;I;L;E;x;x) = (cat-id(C) (L x)))
Date html generated:
2017_10_05-AM-10_30_51
Last ObjectModification:
2017_07_28-AM-11_24_32
Theory : cubical!sets
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