Nuprl Lemma : poset_functor_extend-face-map1
∀[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)))].
∀[y:nameset(I)]. ∀[a:ℕ2]. ∀[c1,c2:name-morph(I;[])].
poset_functor_extend(C;I;L;E;((y:=a) o c1);((y:=a) o c2))
= poset_functor_extend(C;I-[y];L o (λf.((y:=a) o f));λz,f. (E z ((y:=a) o f));c1;c2)
∈ (cat-arrow(C) (L ((y:=a) o c1)) (L ((y:=a) o 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-comp: (f o g),
face-map: (x:=i),
name-morph: name-morph(I;J),
nameset: nameset(L),
cname_deq: CnameDeq,
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
small-category: SmallCategory,
list-diff: as-bs,
cons: [a / b],
nil: [],
list: T List,
compose: f o g,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
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},
nameset: nameset(L),
coordinate_name: Cname,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T),
poset_functor_extend: poset_functor_extend(C;I;L;E;c1;c2),
uiff: uiff(P;Q),
bfalse: ff,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
cand: A c∧ B,
l_member: (x ∈ l),
eq_int: (i =z j),
le: A ≤ B,
less_than': less_than'(a;b),
cons: [a / b],
subtract: n - m,
name-morph-flip: flip(f;y),
less_than: a < b
Lemmas referenced :
int_seg_wf,
nameset_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,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
length_wf,
filter_wf5,
band_wf,
eq_int_wf,
l_member_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
list-subtype,
cname_deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set2,
name-comp_wf,
face-map_wf,
name-morph_subtype,
cons_wf,
nameset_subtype,
l_subset_right_cons_trivial,
filter-list-diff,
bool_cases,
bool_subtype_base,
eqtt_to_assert,
btrue_wf,
bfalse_wf,
filter_cons_lemma,
filter_nil_lemma,
equal-wf-T-base,
bool_wf,
assert_wf,
assert_of_eq_int,
istype-assert,
subtype_rel_universe1,
list_subtype_base,
nameset_subtype_base,
bor_wf,
bnot_wf,
or_wf,
not_wf,
equal_wf,
list-diff-disjoint,
l_disjoint_nil2,
iff_weakening_equal,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
bnot_thru_band,
eqff_to_assert,
assert_of_bor,
assert_of_bnot,
int_seg_subtype_special,
int_seg_cases,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
filter-sq,
list-diff_wf,
member-list-diff,
squash_wf,
true_wf,
istype-universe,
uext-ap-name,
uext_wf,
subtype_rel_set,
nameset_subtype_extd-nameset,
and_wf,
member_singleton,
value-type-has-value,
list-value-type,
null_wf3,
subtype_rel_list,
top_wf,
assert_of_null,
filter_wf_top,
cat-id_wf,
name-morph-ext,
nsub2_subtype_extd-nameset,
member_filter_2,
le_wf,
select_wf,
nil_member,
null_nil_lemma,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
extd-nameset-nil,
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,
filter-filter,
eq-cname_wf,
assert-eq-cname,
filter-less,
member_filter,
hd_member,
cons_neq_nil,
filter_wf3,
non_neg_length,
l_member_subtype,
name-morph-flip-face-map,
cat-comp_wf,
subtract_nat_wf,
poset_functor_extend_wf,
all_wf,
subtype_rel-equal,
length-filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesis,
because_Cache,
functionIsType,
hypothesisEquality,
setIsType,
equalityIstype,
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,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
unionElimination,
instantiate,
equalityTransitivity,
applyLambdaEquality,
dependent_set_memberEquality_alt,
imageMemberEquality,
imageElimination,
productIsType,
hypothesis_subsumption,
addEquality,
setEquality,
closedConclusion,
productEquality,
unionIsType,
cumulativity,
equalityElimination,
promote_hyp,
inlFormation_alt,
inrFormation_alt,
universeEquality,
callbyvalueReduce,
hyp_replacement,
minusEquality,
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{}[y:nameset(I)]. \mforall{}[a:\mBbbN{}2]. \mforall{}[c1,c2:name-morph(I;[])].
poset\_functor\_extend(C;I;L;E;((y:=a) o c1);((y:=a) o c2))
= poset\_functor\_extend(C;I-[y];L o (\mlambda{}f.((y:=a) o f));\mlambda{}z,f. (E z ((y:=a) o f));c1;c2)
supposing \mforall{}x:nameset(I). ((c1 x) \mleq{} (c2 x))
Date html generated:
2019_11_05-PM-00_32_42
Last ObjectModification:
2018_12_10-PM-00_12_21
Theory : cubical!sets
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