Nuprl Lemma : cubical-interval-non-trivial-box
∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀bx:open_box(cubical-interval();I;J;x;i). ∀h:name-morph(I;[]).
((¬(J = [] ∈ (nameset(I) List)))
⇒ (¬(filter(λf.(h (fst(f)) =z fst(snd(f)));bx)
= []
∈ ({x:{f:I-face(cubical-interval();I)| (f ∈ bx)} | ↑(h (fst(x)) =z fst(snd(x)))} List))))
Proof
Definitions occuring in Statement :
open_box: open_box(X;I;J;x;i),
I-face: I-face(X;I),
cubical-interval: cubical-interval(),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
l_member: (x ∈ l),
filter: filter(P;l),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
open_box: open_box(X;I;J;x;i),
and: P ∧ Q,
name-morph: name-morph(I;J),
I-face: I-face(X;I),
pi1: fst(t),
pi2: snd(t),
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
top: Top,
or: P ∨ Q,
cons: [a / b],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_exists: (∃x∈L. P[x]),
exists: ∃x:A. B[x],
l_all: (∀x∈L.P[x]),
guard: {T},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
squash: ↓T,
coordinate_name: Cname,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
face-name: face-name(f),
true: True
Lemmas referenced :
equal-wf-T-base,
list_wf,
filter_type,
list-subtype,
not_wf,
nameset_wf,
name-morph_wf,
nil_wf,
coordinate_name_wf,
open_box_wf,
cubical-interval_wf,
subtype_rel_list,
int_seg_wf,
I-face_wf,
l_member_wf,
assert_wf,
eq_int_wf,
l_all_wf2,
null-filter2,
null_wf3,
filter_wf5,
top_wf,
assert_of_null,
list-cases,
product_subtype_list,
cons_member,
cons_wf,
extd-nameset-nil,
assert_of_eq_int,
select_wf,
int_seg_properties,
length_wf,
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,
pi1_wf_top,
equal_wf,
squash_wf,
true_wf,
extd-nameset_subtype_int,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
baseClosed,
hypothesisEquality,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
isect_memberEquality,
productElimination,
setEquality,
addLevel,
impliesFunctionality,
voidEquality,
independent_pairFormation,
unionElimination,
promote_hyp,
hypothesis_subsumption,
inlFormation,
dependent_set_memberEquality,
functionExtensionality,
applyLambdaEquality,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_pairEquality,
universeEquality
Latex:
\mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}bx:open\_box(cubical-interval();I;J;x;i). \mforall{}h:name-morph(I;[]).
((\mneg{}(J = [])) {}\mRightarrow{} (\mneg{}(filter(\mlambda{}f.(h (fst(f)) =\msubz{} fst(snd(f)));bx) = [])))
Date html generated:
2017_10_05-AM-10_26_36
Last ObjectModification:
2017_07_28-AM-11_22_57
Theory : cubical!sets
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