Nuprl Lemma : cu-box-in-box_wf
∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[d:ℕ2]. ∀[box:open_box(c𝕌;I;J;x;d)].
(cu-box-in-box(I;box) ∈ Type)
Proof
Definitions occuring in Statement :
cu-box-in-box: cu-box-in-box(I;box),
cubical-universe: c𝕌,
open_box: open_box(X;I;J;x;i),
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
open_box: open_box(X;I;J;x;i),
cu-box-in-box: cu-box-in-box(I;box),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
and: P ∧ Q,
int_seg: {i..j-},
guard: {T},
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,
less_than: a < b,
I-face: I-face(X;I),
face-dimension: dimension(f),
face-cube: cube(f),
pi2: snd(t),
pi1: fst(t),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
adjacent-compatible: adjacent-compatible(X;I;L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
face-compatible: face-compatible(X;I;f1;f2),
spreadn: spread3,
face-direction: direction(f),
id-morph: 1,
face-map: (x:=i),
name-comp: (f o g),
compose: f o g,
uext: uext(g),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
isname: isname(z),
le_int: i ≤z j,
lt_int: i <z j,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T
Lemmas referenced :
open_box_wf,
cubical-universe_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
length_wf,
I-face_wf,
select_wf,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
full-omega-unsat,
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,
equal_wf,
cu-cube-family_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
id-morph_wf,
not_wf,
face-dimension_wf,
istype-int,
istype-void,
face-cube_wf,
face-map_wf2,
face-direction_wf,
name-morph_wf,
name-comp_wf,
subtype_rel_wf,
squash_wf,
true_wf,
istype-universe,
list-diff2,
iff_weakening_equal,
subtype_rel_self,
cu-cube-restriction_wf,
subtype_rel-equal,
select_member,
pairwise-implies,
face-compatible_wf,
cubical-set_wf,
face-compatible-symmetry,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-le,
I-cube_wf,
cu-cube-family-comp,
name-morph-ext,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
int_seg_subtype_special,
int_seg_cases,
isname-name,
nsub2_subtype_extd-nameset,
member-list-diff,
member_singleton,
set_subtype_base,
le_wf,
l_member_wf,
cons_member,
nameset_subtype_extd-nameset,
list_subtype_base,
list-diff2-sym
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
natural_numberEquality,
setEquality,
functionEquality,
productElimination,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
lambdaFormation_alt,
dependent_pairFormation_alt,
universeEquality,
cumulativity,
functionIsType,
equalityIsType1,
applyLambdaEquality,
dependent_set_memberEquality_alt,
equalityElimination,
equalityIsType3,
promote_hyp,
hypothesis_subsumption,
equalityIsType2,
closedConclusion,
inlFormation_alt,
inrFormation_alt,
unionIsType
Latex:
\mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[d:\mBbbN{}2]. \mforall{}[box:open\_box(c\mBbbU{};I;J;x;d)].
(cu-box-in-box(I;box) \mmember{} Type)
Date html generated:
2019_11_06-PM-00_54_10
Last ObjectModification:
2018_11_12-PM-05_53_07
Theory : cubical!sets
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