Nuprl Lemma : csm-A-open-box
∀Delta,Gamma:CubicalSet. ∀s:Delta ⟶ Gamma. ∀A:{Gamma ⊢ _}. ∀I:Cname List. ∀alpha:Delta(I). ∀J:nameset(I) List.
∀x:nameset(I). ∀i:ℕ2.
(A-open-box(Delta;(A)s;I;alpha;J;x;i) ⊆r A-open-box(Gamma;A;I;(s)alpha;J;x;i))
Proof
Definitions occuring in Statement :
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-ap: (s)x,
I-cube: X(I),
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
uimplies: b supposing a,
A-face: A-face(X;A;I;alpha),
nameset: nameset(L),
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
pairwise: (∀x,y∈L. P[x; y]),
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
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,
less_than: a < b,
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
spreadn: spread3,
cubical-type: {X ⊢ _},
csm-ap: (s)x,
cubical-type-ap-morph: (u a f),
csm-ap-type: (AF)s,
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
pi1: fst(t),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
A-open-box_wf,
csm-ap-type_wf,
int_seg_wf,
nameset_wf,
list_wf,
I-cube_wf,
coordinate_name_wf,
cubical-type_wf,
cube-set-map_wf,
cubical-set_wf,
subtype_rel_list,
A-face_wf,
csm-ap_wf,
subtype_rel-equal,
cubical-type-at_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
csm-type-at,
equal_wf,
squash_wf,
true_wf,
csm-ap-restriction,
iff_weakening_equal,
length_wf,
select_wf,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
A-face-compatible_wf,
not_wf,
name-comp_wf,
name-morph_wf,
subtype_rel_self,
subtype_rel_wf,
list-diff2-sym,
list-diff2,
cubical-type-ap-morph_wf,
cube-set-restriction-comp,
face-maps-commute,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
int_subtype_base,
subtype_rel_weakening,
ext-eq_weakening,
A-adjacent-compatible_wf,
l_member_wf,
l_subset_wf,
all_wf,
l_exists_wf,
A-face-name_wf,
nameset_subtype,
l_all_wf2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
pairwise_wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
lambdaEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
sqequalRule,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_isectElimination,
productElimination,
dependent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
productEquality,
independent_pairFormation,
promote_hyp,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
addLevel,
hyp_replacement,
levelHypothesis,
applyLambdaEquality,
cumulativity,
independent_pairEquality,
setEquality
Latex:
\mforall{}Delta,Gamma:CubicalSet. \mforall{}s:Delta {}\mrightarrow{} Gamma. \mforall{}A:\{Gamma \mvdash{} \_\}. \mforall{}I:Cname List. \mforall{}alpha:Delta(I).
\mforall{}J:nameset(I) List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2.
(A-open-box(Delta;(A)s;I;alpha;J;x;i) \msubseteq{}r A-open-box(Gamma;A;I;(s)alpha;J;x;i))
Date html generated:
2017_10_05-AM-10_22_03
Last ObjectModification:
2017_07_28-AM-11_21_32
Theory : cubical!sets
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