Nuprl Lemma : open_box_image_wf
∀[X:CubicalSet]. ∀[I,J,K:Cname List]. ∀[f:name-morph(I;K)]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[bx:open_box(X;I;J;x;i)]. (open_box_image(X;I;K;f;bx) ∈ open_box(X;K;map(f;J);f x;i))
supposing nameset([x / J]) ⊆r name-morph-domain(f;I)
Proof
Definitions occuring in Statement :
open_box_image: open_box_image(X;I;K;f;bx)
,
open_box: open_box(X;I;J;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)
,
open_box: open_box(X;I;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
,
I-face: I-face(X;I)
,
true: True
,
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
,
open_box_image: open_box_image(X;I;K;f;bx)
,
adjacent-compatible: adjacent-compatible(X;I;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)
,
face-name: face-name(f)
,
face-image: face-image(X;I;K;f;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,
open_box_wf,
subtype_rel_wf,
int_seg_wf,
name-morph_wf,
cubical-set_wf,
lelt_wf,
length_wf,
I-face_wf,
not_wf,
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,
squash_wf,
true_wf,
pi1_wf_top,
top_wf,
subtype_rel_product,
I-cube_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
iff_weakening_equal,
sq_stable__assert,
face-image_wf,
pairwise-map,
face-compatible_wf,
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,
face-name-image,
product_subtype_base,
pi2_wf,
and_wf,
map-length,
less_than_wf,
isname-nameset,
assert_of_tt,
equal_functionality_wrt_subtype_rel2,
extd-nameset_wf,
map_cons_lemma,
length_of_cons_lemma,
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
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I,J,K:Cname List]. \mforall{}[f:name-morph(I;K)]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[bx:open\_box(X;I;J;x;i)]. (open\_box\_image(X;I;K;f;bx) \mmember{} open\_box(X;K;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_22_26
Last ObjectModification:
2017_07_28-AM-11_21_37
Theory : cubical!sets
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