Nuprl Lemma : open_box-nil
∀[X:CubicalSet]. ∀[I:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
open_box(X;I;[];x;i) ≡ {L:I-face(X;I) List| (||L|| = 1 ∈ ℤ) ∧ (face-name(hd(L)) = <x, i> ∈ (nameset(I) × ℕ2))}
Proof
Definitions occuring in Statement :
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,
length: ||as||,
hd: hd(l),
nil: [],
list: T List,
int_seg: {i..j-},
ext-eq: A ≡ B,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
open_box: open_box(X;I;J;x;i),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
ge: i ≥ j ,
guard: {T},
int_seg: {i..j-},
nameset: nameset(L),
lelt: i ≤ j < k,
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
pi1: fst(t),
I-face: I-face(X;I),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
cons: [a / b],
l_exists: (∃x∈L. P[x]),
sq_type: SQType(T),
select: L[n],
l_all: (∀x∈L.P[x]),
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
subtract: n - m,
pairwise: (∀x,y∈L. P[x; y]),
less_than: a < b,
true: True,
face-name: face-name(f),
pi2: snd(t),
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
adjacent-compatible: adjacent-compatible(X;I;L)
Lemmas referenced :
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
face-name_wf,
hd_wf,
int_seg_properties,
sq_stable__l_member,
coordinate_name_wf,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
open_box_wf,
nil_wf,
adjacent-compatible_wf,
l_member_wf,
l_subset_wf,
nameset_wf,
l_exists_wf,
I-face_wf,
equal_wf,
int_seg_wf,
l_all_wf2,
not_wf,
subtract_wf,
itermSubtract_wf,
intformless_wf,
int_term_value_subtract_lemma,
int_formula_prop_less_lemma,
decidable__lt,
istype-le,
istype-less_than,
cons_wf,
pairwise_wf2,
list_wf,
cubical-set_wf,
list-cases,
product_subtype_list,
length_of_nil_lemma,
length_of_cons_lemma,
reduce_hd_cons_lemma,
subtype_base_sq,
lelt_wf,
decidable__equal_int,
istype-false,
non_neg_length,
length_wf,
itermAdd_wf,
int_term_value_add_lemma,
member_singleton,
pi1_wf_top,
decidable__equal_int_seg,
le_antisymmetry_iff,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
l_subset_nil_left,
bool_wf,
ppcc-problem,
unit_wf2,
iff_weakening_equal,
nameset_subtype,
l_all_cons,
cons_member,
pairwise-singleton,
l_all_single,
l_all_nil,
select_wf,
product_subtype_base,
nameset_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
hypothesisEquality,
sqequalRule,
productIsType,
equalityIsType4,
because_Cache,
productElimination,
extract_by_obid,
isectElimination,
hypothesis,
applyEquality,
intEquality,
closedConclusion,
natural_numberEquality,
independent_isectElimination,
equalityIsType1,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation_alt,
inhabitedIsType,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
independent_pairEquality,
functionIsType,
productEquality,
setIsType,
instantiate,
cumulativity,
axiomEquality,
isectIsTypeImplies,
promote_hyp,
hypothesis_subsumption,
addEquality,
applyLambdaEquality,
unionEquality,
inlEquality_alt,
inlFormation_alt
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I:Cname List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
open\_box(X;I;[];x;i) \mequiv{} \{L:I-face(X;I) List| (||L|| = 1) \mwedge{} (face-name(hd(L)) = <x, i>)\}
Date html generated:
2019_11_05-PM-00_28_11
Last ObjectModification:
2018_11_08-PM-01_16_01
Theory : cubical!sets
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