Nuprl Lemma : A-open-box-equal
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀I:Cname List. ∀alpha:X(I).
∀[J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[bx1:A-open-box(X;A;I;alpha;J;x;i)]. ∀[bx2:A-face(X;A;I;alpha) List].
bx1 = bx2 ∈ A-open-box(X;A;I;alpha;J;x;i) supposing bx1 = bx2 ∈ (A-face(X;A;I;alpha) List)
Proof
Definitions occuring in Statement :
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
A-face: A-face(X;A;I;alpha),
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
and: P ∧ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
nameset: nameset(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
A-face: A-face(X;A;I;alpha),
pi1: fst(t),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
cubical-set_wf,
cubical-type_wf,
I-cube_wf,
A-open-box_wf,
list_wf,
pairwise_wf2,
cons_wf,
lelt_wf,
decidable__lt,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
sq_stable__le,
decidable__equal-coordinate_name,
sq_stable__l_member,
int_seg_properties,
subtract_wf,
l_all_wf2,
nameset_subtype,
A-face-name_wf,
equal_wf,
A-face_wf,
l_exists_wf,
int_seg_wf,
nameset_wf,
all_wf,
l_subset_wf,
coordinate_name_wf,
l_member_wf,
not_wf,
A-adjacent-compatible_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
productElimination,
productEquality,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
cumulativity,
universeEquality,
sqequalRule,
because_Cache,
natural_numberEquality,
independent_pairEquality,
independent_isectElimination,
setEquality,
independent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
instantiate,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}I:Cname List. \mforall{}alpha:X(I).
\mforall{}[J:Cname List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[bx1:A-open-box(X;A;I;alpha;J;x;i)].
\mforall{}[bx2:A-face(X;A;I;alpha) List].
bx1 = bx2 supposing bx1 = bx2
Date html generated:
2016_06_16-PM-05_56_26
Last ObjectModification:
2016_01_18-PM-04_53_20
Theory : cubical!sets
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