Nuprl Lemma : boundary-of-0-dim-is-nil
∀[k:ℕ]. ∀[K:ℚCube(k) List]. ∂(K) ~ [] supposing (∀c∈K.dim(c) = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
l_all: (∀x∈L.P[x]),
nil: [],
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
int: ℤ,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
concat: concat(ll),
nequal: a ≠ b ∈ T ,
less_than': less_than'(a;b),
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
rat-cube-dimension: dim(c),
le: A ≤ B,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
less_than: a < b,
lelt: i ≤ j < k,
rational-cube: ℚCube(k),
mapfilter: mapfilter(f;P;L),
rat-cube-faces: rat-cube-faces(k;c),
rev_implies: P ⇐ Q,
guard: {T},
true: True,
squash: ↓T,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
bfalse: ff,
and: P ∧ Q,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
face-complex: face-complex(k;L),
prop: ℙ,
so_apply: x[s],
nat: ℕ,
int_seg: {i..j-},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
top: Top,
all: ∀x:A. B[x],
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
rat-complex-boundary: ∂(K),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
reduce_nil_lemma,
map_nil_lemma,
int_term_value_add_lemma,
itermAdd_wf,
sum_wf,
neg_assert_of_eq_int,
assert-bnot,
ifthenelse_wf,
non_neg_sum,
assert_of_eq_int,
le_wf,
istype-false,
int_seg_subtype_nat,
Error :isolate_summand2,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
int_formula_prop_eq_lemma,
intformeq_wf,
member_filter,
istype-less_than,
istype-le,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
rat-interval-dimension_wf,
eq_int_wf,
upto_wf,
filter_wf5,
int_seg_wf,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
true_wf,
squash_wf,
equal_wf,
l_all_iff,
istype-assert,
member-face-complex,
nil_wf,
subtract_wf,
rat-cube-face_wf,
subtype_rel_list,
rat-cube-faces_wf,
eqtt_to_assert,
inhabited-rat-cube_wf,
map_wf,
concat_wf,
rc-deq_wf,
remove-repeats_wf,
no-member-sq-nil,
istype-nat,
list_wf,
l_member_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
rational-cube_wf,
l_all_wf2,
istype-void,
filter_nil_lemma
Rules used in proof :
applyLambdaEquality,
cumulativity,
promote_hyp,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
independent_pairFormation,
dependent_set_memberEquality_alt,
imageMemberEquality,
universeEquality,
instantiate,
imageElimination,
independent_functionElimination,
sqequalBase,
equalityIstype,
productIsType,
equalitySymmetry,
equalityTransitivity,
productEquality,
setEquality,
productElimination,
equalityElimination,
unionElimination,
lambdaFormation_alt,
because_Cache,
inhabitedIsType,
isectIsTypeImplies,
setIsType,
baseClosed,
independent_isectElimination,
addEquality,
natural_numberEquality,
minusEquality,
applyEquality,
rename,
setElimination,
intEquality,
lambdaEquality_alt,
hypothesisEquality,
isectElimination,
universeIsType,
axiomSqEquality,
hypothesis,
voidElimination,
isect_memberEquality_alt,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:\mBbbQ{}Cube(k) List]. \mpartial{}(K) \msim{} [] supposing (\mforall{}c\mmember{}K.dim(c) = 0)
Date html generated:
2019_10_29-AM-07_58_25
Last ObjectModification:
2019_10_19-PM-10_28_06
Theory : rationals
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