Nuprl Lemma : get_face-wf
∀[X:CubicalSet]. ∀[I,J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)].
(get_face(x;i;box) ∈ {f:I-face(X;I)| (f ∈ box) ∧ (face-name(f) = <x, i> ∈ (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),
and: P ∧ Q,
cand: A c∧ B,
get_face: get_face(y;c;box),
prop: ℙ,
all: ∀x:A. B[x],
I-face: I-face(X;I),
pi1: fst(t),
nameset: nameset(L),
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),
uimplies: b supposing a,
pi2: snd(t),
int_seg: {i..j-},
bfalse: ff,
subtype_rel: A ⊆r B,
face-name: face-name(f),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
less_than: a < b
Lemmas referenced :
filter_type,
I-face_wf,
l_member_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
equal_wf,
list-subtype,
list_wf,
assert_wf,
hd_wf,
face-name_wf,
open_box_wf,
int_seg_wf,
nameset_wf,
coordinate_name_wf,
cubical-set_wf,
subtype_rel_list,
band_wf,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
subtype_base_sq,
int_subtype_base,
set_wf,
list-cases,
product_subtype_list,
sqequal-nil,
nil_wf,
filter_is_nil_implies,
select_wf,
length_wf,
pi1_wf_top,
set_subtype_base,
nameset_subtype_base,
length_cons_ge_one,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
introduction,
extract_by_obid,
isectElimination,
setEquality,
hypothesisEquality,
hypothesis,
lambdaEquality,
lambdaFormation,
sqequalRule,
unionElimination,
equalityElimination,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
productEquality,
independent_pairEquality,
natural_numberEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
intEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
imageMemberEquality,
baseClosed,
imageElimination,
instantiate,
cumulativity,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality
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)].
(get\_face(x;i;box) \mmember{} \{f:I-face(X;I)| (f \mmember{} box) \mwedge{} (face-name(f) = <x, i>)\} )
Date html generated:
2017_10_05-AM-10_21_03
Last ObjectModification:
2017_07_28-AM-11_21_11
Theory : cubical!sets
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