Nuprl Lemma : get_face_image
∀[X:CubicalSet]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[bx:open_box(X;I;J;x;i)].
∀[K:Cname List]. ∀[f:name-morph(I;K)]. ∀[c:ℕ2]. ∀[y:nameset(J)].
get_face(f y;c;open_box_image(X;I;K;f;bx)) = face-image(X;I;K;f;get_face(y;c;bx)) ∈ I-face(X;K)
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),
get_face: get_face(y;c;box),
open_box: open_box(X;I;J;x;i),
face-image: face-image(X;I;K;f;face),
I-face: I-face(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,
cons: [a / b],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
get_face: get_face(y;c;box),
all: ∀x:A. B[x],
face-direction: direction(f),
face-dimension: dimension(f),
nameset: nameset(L),
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
name-morph-domain: name-morph-domain(f;I),
prop: ℙ,
name-morph: name-morph(I;J),
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
isname: isname(z),
le_int: i ≤z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
squash: ↓T,
open_box: open_box(X;I;J;x;i),
uiff: uiff(P;Q),
cand: A c∧ B,
I-face: I-face(X;I),
top: Top,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
l_all: (∀x∈L.P[x]),
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
pi1: fst(t),
sq_type: SQType(T),
rev_implies: P ⇐ Q,
or: P ∨ Q,
coordinate_name: Cname,
int_upper: {i...},
open_box_image: open_box_image(X;I;K;f;bx),
compose: f o g,
spreadn: spread3,
pi2: snd(t),
face-image: face-image(X;I;K;f;face),
false: False,
assert: ↑b,
bfalse: ff,
band: p ∧b q,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
true: True,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
nequal: a ≠ b ∈ T ,
label: ...$L... t,
listp: A List+,
cons: [a / b],
decidable: Dec(P),
subtract: n - m,
less_than': less_than'(a;b),
respects-equality: respects-equality(S;T),
ge: i ≥ j
Lemmas referenced :
non-trivial-open-box,
subtype_rel_wf,
nameset_wf,
cons_wf,
coordinate_name_wf,
subtype_rel_list,
name-morph-domain_wf,
name-morph_wf,
open_box_wf,
int_seg_wf,
list_wf,
cubical-set_wf,
equal_wf,
sq_stable__assert,
member_filter_2,
isname_wf,
l_member_wf,
subtype_rel_sets,
filter_wf5,
list-subtype,
assert-isname,
nameset_subtype,
l_subset_right_cons_trivial,
I-face_wf,
pi1_wf_top,
lelt_wf,
length_wf,
assert_wf,
and_wf,
subtype_rel_product,
I-cube_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
top_wf,
subtype_base_sq,
nameset_subtype_base,
map_wf,
cons_member,
list_subtype_base,
sq_stable__l_member,
decidable__equal-coordinate_name,
set_subtype_base,
le_wf,
int_subtype_base,
select_member,
member-map,
open_box_image_wf,
filter-map,
istype-void,
istype-universe,
bool_wf,
true_wf,
squash_wf,
filter_wf2,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
extd-nameset_subtype_int,
assert_of_eq_int,
eqtt_to_assert,
eq_int_wf,
iff_weakening_equal,
subtype_rel_self,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-int,
itermVar_wf,
intformeq_wf,
intformnot_wf,
full-omega-unsat,
int_seg_properties,
nequal_wf,
bfalse_wf,
hd_map,
istype-less_than,
face-image_wf,
equal-wf-T-base,
face-dimension_wf,
face-direction_wf,
sqequal-nil,
length_wf_nat,
nat_wf,
set_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
not_wf,
filter_type,
bool_cases,
band_wf,
btrue_wf,
respects-equality-list,
respects-equality-set-trivial2,
hd_wf,
ge_wf,
listp_properties,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
hypothesis,
universeIsType,
setElimination,
rename,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
inhabitedIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
because_Cache,
natural_numberEquality,
lambdaFormation,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
productElimination,
imageMemberEquality,
baseClosed,
imageElimination,
functionExtensionality,
independent_pairFormation,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
productEquality,
instantiate,
cumulativity,
inlFormation,
dependent_pairFormation,
intEquality,
universeEquality,
setIsType,
functionIsType,
promote_hyp,
equalityIsType1,
dependent_pairFormation_alt,
equalityElimination,
unionElimination,
lambdaFormation_alt,
int_eqEquality,
approximateComputation,
dependent_pairEquality,
dependent_set_memberEquality_alt,
hypothesis_subsumption,
addEquality,
minusEquality,
equalityIstype
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[bx:open\_box(X;I;J;x;i)]. \mforall{}[K:Cname List]. \mforall{}[f:name-morph(I;K)]. \mforall{}[c:\mBbbN{}2]. \mforall{}[y:nameset(J)].
get\_face(f y;c;open\_box\_image(X;I;K;f;bx)) = face-image(X;I;K;f;get\_face(y;c;bx))
supposing nameset([x / J]) \msubseteq{}r name-morph-domain(f;I)
Date html generated:
2019_11_05-PM-00_29_19
Last ObjectModification:
2018_12_10-AM-10_00_04
Theory : cubical!sets
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