Nuprl Lemma : extend-A-open-box_wf
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀I:Cname List. ∀alpha:X(I).
∀[J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[bx:A-open-box(X;A;I;alpha;J;x;i)]. ∀[f1,f2:A-face(X;A;I;alpha)].
∀[z:nameset(I)].
extend-A-open-box(bx;f1;f2) ∈ A-open-box(X;A;I;alpha;[z / J];x;i)
supposing ((¬(z ∈ J))
∧ (A-face-name(f1) = <z, 0> ∈ (nameset(I) × ℕ2))
∧ (A-face-name(f2) = <z, 1> ∈ (nameset(I) × ℕ2)))
∧ (¬(x = z ∈ Cname))
∧ (∀f∈bx.A-face-compatible(X;A;I;alpha;f1;f) ∧ A-face-compatible(X;A;I;alpha;f2;f))
Proof
Definitions occuring in Statement :
extend-A-open-box: extend-A-open-box(bx;f1;f2),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
A-face-name: A-face-name(f),
A-face: A-face(X;A;I;alpha),
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
cons: [a / b],
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
and: P ∧ Q,
member: t ∈ T,
pair: <a, b>,
product: 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,
and: P ∧ Q,
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
nameset: nameset(L),
extend-A-open-box: extend-A-open-box(bx;f1;f2),
A-adjacent-compatible: A-adjacent-compatible(X;A;I;alpha;L),
pairwise: (∀x,y∈L. P[x; y]),
not: ¬A,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
false: False,
prop: ℙ,
rev_implies: P ⇐ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_apply: x[s],
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
A-face: A-face(X;A;I;alpha),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
respects-equality: respects-equality(S;T),
sq_type: SQType(T),
guard: {T},
top: Top,
sq_stable: SqStable(P),
subtract: n - m,
cons: [a / b],
select: L[n],
spreadn: spread3,
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
pi2: snd(t),
pi1: fst(t),
A-face-name: A-face-name(f),
ge: i ≥ j ,
l_all: (∀x∈L.P[x]),
uiff: uiff(P;Q),
nat_plus: ℕ+,
nat: ℕ,
less_than': less_than'(a;b),
l_exists: (∃x∈L. P[x]),
true: True,
l_member: (x ∈ l)
Lemmas referenced :
cons_wf,
A-face_wf,
cons_member,
coordinate_name_wf,
l_member_wf,
l_subset_cons,
A-adjacent-compatible_wf,
istype-void,
l_subset_wf,
nameset_wf,
l_exists_wf,
equal_wf,
int_seg_wf,
A-face-name_wf,
nameset_subtype,
l_all_wf2,
not_wf,
subtract_wf,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
decidable__lt,
istype-le,
istype-less_than,
pi1_wf_top,
subtype_rel_product,
cubical-type-at_wf,
list-diff_wf,
cname_deq_wf,
nil_wf,
cube-set-restriction_wf,
face-map_wf2,
top_wf,
pairwise_wf2,
respects-equality-product,
respects-equality-trivial,
subtype-base-respects-equality,
int_subtype_base,
istype-base,
A-face-compatible_wf,
A-open-box_wf,
I-cube_wf,
list_wf,
cubical-type_wf,
cubical-set_wf,
length_wf,
decidable__equal_int,
subtype_base_sq,
length_of_cons_lemma,
decidable__equal-coordinate_name,
sq_stable__l_member,
sq_stable__le,
int_formula_prop_eq_lemma,
intformeq_wf,
lelt_wf,
set_subtype_base,
le_wf,
nameset_subtype_base,
subtype_rel_universe1,
pi2_wf,
int_term_value_add_lemma,
itermAdd_wf,
non_neg_length,
select-cons-tl,
l_all_cons,
int_seg_cases,
int_seg_subtype_special,
select_wf,
false_wf,
add-is-int-iff,
nat_plus_properties,
length_wf_nat,
add_nat_wf,
add_nat_plus,
istype-false,
zero-add,
add-commutes,
add-swap,
add-associates,
respects-equality-set-trivial2,
respects-equality-set,
add-member-int_seg2,
squash_wf,
true_wf,
istype-universe,
select_cons_tl,
subtype_rel_self,
iff_weakening_equal,
nat_properties,
add-subtract-cancel,
less_than_wf,
and_wf,
product_subtype_base,
pairwise-cons,
subtype_rel_list
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
setElimination,
rename,
dependent_set_memberEquality_alt,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
independent_pairFormation,
promote_hyp,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
voidElimination,
universeIsType,
sqequalRule,
productIsType,
functionIsType,
because_Cache,
lambdaEquality_alt,
productEquality,
imageElimination,
independent_pairEquality,
applyEquality,
independent_isectElimination,
setIsType,
inhabitedIsType,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
instantiate,
cumulativity,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
intEquality,
sqequalBase,
isect_memberEquality_alt,
isectIsTypeImplies,
addEquality,
baseClosed,
imageMemberEquality,
applyLambdaEquality,
baseApply,
closedConclusion,
hypothesis_subsumption,
pointwiseFunctionality,
universeEquality,
inlFormation_alt,
inrFormation_alt,
hyp_replacement
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{}[bx:A-open-box(X;A;I;alpha;J;x;i)].
\mforall{}[f1,f2:A-face(X;A;I;alpha)]. \mforall{}[z:nameset(I)].
extend-A-open-box(bx;f1;f2) \mmember{} A-open-box(X;A;I;alpha;[z / J];x;i)
supposing ((\mneg{}(z \mmember{} J)) \mwedge{} (A-face-name(f1) = <z, 0>) \mwedge{} (A-face-name(f2) = <z, 1>))
\mwedge{} (\mneg{}(x = z))
\mwedge{} (\mforall{}f\mmember{}bx.A-face-compatible(X;A;I;alpha;f1;f) \mwedge{} A-face-compatible(X;A;I;alpha;f2;f))
Date html generated:
2020_05_21-AM-10_52_01
Last ObjectModification:
2020_01_05-AM-00_09_15
Theory : cubical!sets
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