Nuprl Lemma : poset_functor_extend-extends
∀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))).
poset-functor-extends(C;I;L;E;<L, λc1,c2,p. poset_functor_extend(C;I;L;E;c1;c2)>)
Proof
Definitions occuring in Statement :
poset-functor-extends: poset-functor-extends(C;I;L;E;F),
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
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,
lambda: λx.A[x],
function: x:A ⟶ B[x],
pair: <a, b>,
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-functor-extends: poset-functor-extends(C;I;L;E;F),
and: P ∧ Q,
functor-ob: ob(F),
pi1: fst(t),
functor-arrow: arrow(F),
pi2: snd(t),
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
so_lambda: λ2x.t[x],
so_apply: x[s],
nameset: nameset(L),
has-value: (a)↓,
uimplies: b supposing a,
implies: P ⇒ Q,
or: P ∨ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
top: Top,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
guard: {T},
int_seg: {i..j-},
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
name-morph-flip: flip(f;y),
rev_uimplies: rev_uimplies(P;Q),
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
subtract: n - m,
sq_stable: SqStable(P),
bool: 𝔹,
unit: Unit,
it: ⋅,
band: p ∧b q,
cat-ob: cat-ob(C),
poset-cat: poset-cat(J)
Lemmas referenced :
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,
cat-ob_wf,
list_wf,
small-category_wf,
set_wf,
filter_type,
band_wf,
eq_int_wf,
extd-nameset_subtype_int,
list-subtype,
value-type-has-value,
assert_wf,
list-value-type,
list-cases,
null_nil_lemma,
product_subtype_list,
null_cons_lemma,
reduce_hd_cons_lemma,
equal_wf,
member_filter,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_eq_int,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
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,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eq-cname_wf,
assert-eq-cname,
nsub2-flip,
false_wf,
lelt_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
set_subtype_base,
int_subtype_base,
l_member-settype,
l_member_wf,
btrue_wf,
member-implies-null-eq-bfalse,
subtype_rel_list,
and_wf,
null_wf3,
top_wf,
btrue_neq_bfalse,
list_subtype_base,
nameset_subtype_base,
bnot_wf,
not_wf,
bool_cases,
eqff_to_assert,
assert_of_bnot,
le_wf,
squash_wf,
true_wf,
cat-comp_wf,
poset_functor_extend_id,
subtype_rel_self,
all_wf,
extd-nameset_wf,
iff_weakening_equal,
cat-comp-ident2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setEquality,
natural_numberEquality,
applyEquality,
setElimination,
rename,
sqequalRule,
baseClosed,
functionExtensionality,
because_Cache,
independent_pairFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
callbyvalueReduce,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
productEquality,
intEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
instantiate,
cumulativity,
dependent_set_memberEquality,
imageMemberEquality,
imageElimination,
equalityElimination,
universeEquality,
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))).
poset-functor-extends(C;I;L;E;<L, \mlambda{}c1,c2,p. poset\_functor\_extend(C;I;L;E;c1;c2)>)
Date html generated:
2017_10_05-AM-10_31_15
Last ObjectModification:
2017_07_28-AM-11_24_45
Theory : cubical!sets
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