Nuprl Lemma : poset_functor_extend_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)))].
∀[c1,c2:name-morph(I;[])].
poset_functor_extend(C;I;L;E;c1;c2) ∈ cat-arrow(C) (L c1) (L c2) supposing ∀x:nameset(I). ((c1 x) ≤ (c2 x))
Proof
Definitions occuring in Statement :
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-}
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
le: A ≤ B
,
all: ∀x:A. B[x]
,
member: t ∈ T
,
set: {x:A| B[x]}
,
apply: f a
,
function: x:A ⟶ B[x]
,
natural_number: $n
,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
name-morph: name-morph(I;J)
,
int_seg: {i..j-}
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
implies: P
⇒ Q
,
uimplies: b supposing a
,
respects-equality: respects-equality(S;T)
,
all: ∀x:A. B[x]
,
nat: ℕ
,
false: False
,
ge: i ≥ j
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
top: Top
,
and: P ∧ Q
,
prop: ℙ
,
guard: {T}
,
lelt: i ≤ j < k
,
decidable: Dec(P)
,
or: P ∨ Q
,
subtype_rel: A ⊆r B
,
sq_type: SQType(T)
,
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2)
,
has-value: (a)↓
,
nameset: nameset(L)
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
ifthenelse: if b then t else f fi
,
bfalse: ff
,
bnot: ¬bb
,
assert: ↑b
,
rev_implies: P
⇐ Q
,
iff: P
⇐⇒ Q
,
band: p ∧b q
,
sq_stable: SqStable(P)
,
squash: ↓T
,
l_member: (x ∈ l)
,
cand: A c∧ B
,
coordinate_name: Cname
,
int_upper: {i...}
,
le: A ≤ B
,
less_than': less_than'(a;b)
,
true: True
,
cons: [a / b]
,
subtract: n - m
,
name-morph-flip: flip(f;y)
,
eq_int: (i =z j)
,
less_than: a < b
Lemmas referenced :
nameset_wf,
int_seg_wf,
respects-equality-set,
extd-nameset_wf,
nil_wf,
coordinate_name_wf,
lelt_wf,
istype-int,
subtype-respects-equality,
extd-nameset_subtype_int,
cat-arrow_wf,
name-morph-flip_wf,
name-morph_wf,
cat-ob_wf,
list_wf,
small-category_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
value-type-has-value,
list-value-type,
filter_wf5,
band_wf,
eq_int_wf,
l_member_wf,
null_wf3,
filter_wf_top,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal_wf,
length_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
name-morph-ext,
cat-id_wf,
list-subtype,
filter_is_nil_implies,
bool_cases,
btrue_wf,
bfalse_wf,
l_all_iff,
not_wf,
assert_of_eq_int,
sq_stable__l_member,
decidable__equal-coordinate_name,
le_wf,
select_wf,
sq_stable__le,
respects-equality-set-trivial2,
and_wf,
istype-assert,
iff_transitivity,
assert_of_band,
extd-nameset-nil,
int_seg_subtype_special,
int_seg_cases,
nsub2_subtype_extd-nameset,
filter_type,
hd_wf,
set_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
istype-false,
not-ge-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2,
subtype_rel_list,
cat-comp_wf,
list_subtype_base,
filter-filter,
squash_wf,
true_wf,
istype-universe,
eq-cname_wf,
assert-eq-cname,
neg_assert_of_eq_int,
filter-less,
member_filter_2,
decidable__equal_set,
l_member-settype,
iff_imp_equal_bool,
iff_functionality_wrt_iff,
iff_weakening_equal,
subtract_nat_wf,
non_neg_length,
subtype_rel_universe1,
nameset_subtype_base,
all_wf,
length-filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setIsType,
because_Cache,
equalityIstype,
natural_numberEquality,
applyEquality,
setElimination,
rename,
baseClosed,
sqequalRule,
intEquality,
lambdaEquality_alt,
independent_functionElimination,
independent_isectElimination,
dependent_functionElimination,
sqequalBase,
equalitySymmetry,
lambdaFormation_alt,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
unionElimination,
instantiate,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
callbyvalueReduce,
equalityElimination,
promote_hyp,
cumulativity,
addEquality,
hyp_replacement,
closedConclusion,
imageMemberEquality,
imageElimination,
setEquality,
minusEquality,
universeEquality,
functionExtensionality,
isect_memberEquality,
lambdaEquality,
isect_memberFormation
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{}[c1,c2:name-morph(I;[])].
poset\_functor\_extend(C;I;L;E;c1;c2) \mmember{} cat-arrow(C) (L c1) (L c2)
supposing \mforall{}x:nameset(I). ((c1 x) \mleq{} (c2 x))
Date html generated:
2019_11_05-PM-00_32_05
Last ObjectModification:
2018_12_10-AM-11_02_42
Theory : cubical!sets
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