Nuprl Lemma : get_face_wf
∀[X:CubicalSet]. ∀[I,J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)]. ∀[y:nameset(J)]. ∀[c:ℕ2].
(get_face(y;c;box) ∈ {f:I-face(X;I)| (f ∈ box) ∧ (face-name(f) = <y, c> ∈ (nameset(I) × ℕ2))} )
Proof
Definitions occuring in Statement :
get_face: get_face(y;c;box),
open_box: open_box(X;I;J;x;i),
face-name: face-name(f),
I-face: I-face(X;I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
l_member: (x ∈ l),
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
open_box: open_box(X;I;J;x;i),
subtype_rel: A ⊆r B,
nameset: nameset(L),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
get_face: get_face(y;c;box),
prop: ℙ,
I-face: I-face(X;I),
pi1: fst(t),
coordinate_name: Cname,
int_upper: {i...},
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
pi2: snd(t),
int_seg: {i..j-},
bfalse: ff,
face-direction: direction(f),
face-dimension: dimension(f),
so_lambda: λ2x.t[x],
so_apply: x[s],
or: P ∨ Q,
not: ¬A,
false: False,
cons: [a / b],
top: Top,
guard: {T},
nat: ℕ,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
listp: A List+,
cand: A c∧ B,
face-name: face-name(f),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T)
Lemmas referenced :
non-trivial-open-box,
list_wf,
coordinate_name_wf,
cubical-set_wf,
int_seg_wf,
nameset_wf,
open_box_wf,
list-subtype,
subtype_rel_list,
nameset_subtype,
filter_type,
I-face_wf,
l_member_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
hd_wf,
assert_wf,
face-dimension_wf,
face-direction_wf,
listp_properties,
band_wf,
set_wf,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
length_wf_nat,
nat_wf,
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,
not_wf,
nil_wf,
less_than_wf,
length_wf,
face-name_wf,
int_seg_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
subtype_base_sq,
int_subtype_base
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
because_Cache,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
setElimination,
rename,
applyEquality,
productElimination,
independent_isectElimination,
dependent_functionElimination,
setEquality,
lambdaEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
voidEquality,
addEquality,
independent_pairFormation,
intEquality,
minusEquality,
dependent_set_memberEquality,
productEquality,
independent_pairEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I,J:Cname List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[box:open\_box(X;I;J;x;i)].
\mforall{}[y:nameset(J)]. \mforall{}[c:\mBbbN{}2].
(get\_face(y;c;box) \mmember{} \{f:I-face(X;I)| (f \mmember{} box) \mwedge{} (face-name(f) = <y, c>)\} )
Date html generated:
2017_10_05-AM-10_20_52
Last ObjectModification:
2017_07_28-AM-11_21_07
Theory : cubical!sets
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