Nuprl Lemma : A-open-box-image_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[I,J,K:Cname List]. ∀[alpha:X(I)]. ∀[f:name-morph(I;K)]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[bx:A-open-box(X;A;I;alpha;J;x;i)]
(A-open-box-image(X;A;I;K;f;alpha;bx) ∈ A-open-box(X;A;K;f(alpha);map(f;J);f x;i))
supposing nameset([x / J]) ⊆r name-morph-domain(f;I)
Proof
Definitions occuring in Statement :
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph-domain: name-morph-domain(f;I),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
map: map(f;as),
cons: [a / b],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
name-morph-domain: name-morph-domain(f;I),
nameset: nameset(L),
name-morph: name-morph(I;J),
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
rev_implies: P ⇐ Q,
or: P ∨ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
coordinate_name: Cname,
int_upper: {i...},
sq_type: SQType(T),
nat: ℕ,
squash: ↓T,
sq_stable: SqStable(P),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
A-face: A-face(X;A;I;alpha),
pi1: fst(t),
isname: isname(z),
le_int: i ≤z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
true: True,
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
pairwise: (∀x,y∈L. P[x; y]),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
l_subset: l_subset(T;as;bs),
l_exists: (∃x∈L. P[x]),
pi2: snd(t),
A-face-name: A-face-name(f),
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
spreadn: spread3
Lemmas referenced :
name-morph-domain_wf,
member_filter_2,
coordinate_name_wf,
isname_wf,
l_member_wf,
assert-isname,
subtype_rel_sets,
filter_wf5,
equal_wf,
nameset_wf,
cons_wf,
nameset_subtype,
l_subset_right_cons_trivial,
list-subtype,
list_wf,
map_wf,
cons_member,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
select_wf,
sq_stable__le,
nat_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,
subtype_rel_list,
A-open-box_wf,
subtype_rel_wf,
int_seg_wf,
name-morph_wf,
I-cube_wf,
cubical-type_wf,
cubical-set_wf,
lelt_wf,
length_wf,
A-face_wf,
not_wf,
A-face-name_wf,
subtract_wf,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__lt,
sq_stable__assert,
squash_wf,
true_wf,
pi1_wf_top,
top_wf,
subtype_rel_product,
cubical-type-at_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
iff_weakening_equal,
A-face-image_wf,
pairwise-map,
A-face-compatible_wf,
A-face-compatible-image,
select_member,
member-map,
assert_of_le_int,
nameset_subtype_extd-nameset,
nameset_subtype_base,
l_member_subtype,
assert_wf,
eq-cname_wf,
decidable__l_exists_better-extract,
decidable__assert,
assert-eq-cname,
length-map,
select-map,
A-face-name-image,
product_subtype_base,
pi2_wf,
and_wf,
isname-nameset,
assert_of_tt,
equal_functionality_wrt_subtype_rel2,
extd-nameset_wf,
map_cons_lemma,
length_of_cons_lemma,
length-map-sq,
A-adjacent-compatible_wf,
l_subset_wf,
all_wf,
l_exists_wf,
l_all_wf2,
pairwise_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
thin,
extract_by_obid,
isectElimination,
setElimination,
rename,
dependent_functionElimination,
lambdaEquality,
setEquality,
productElimination,
independent_functionElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
promote_hyp,
functionExtensionality,
voidElimination,
voidEquality,
inlFormation,
dependent_set_memberEquality,
instantiate,
cumulativity,
intEquality,
natural_numberEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
independent_pairFormation,
computeAll,
axiomEquality,
hyp_replacement,
productEquality,
independent_pairEquality,
universeEquality,
equalityElimination,
addEquality,
comment
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[I,J,K:Cname List]. \mforall{}[alpha:X(I)]. \mforall{}[f:name-morph(I;K)].
\mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[bx:A-open-box(X;A;I;alpha;J;x;i)]
(A-open-box-image(X;A;I;K;f;alpha;bx) \mmember{} A-open-box(X;A;K;f(alpha);map(f;J);f x;i))
supposing nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
Date html generated:
2017_10_05-AM-10_23_11
Last ObjectModification:
2017_07_28-AM-11_21_46
Theory : cubical!sets
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