Nuprl Lemma : rat-complex-boundary-remove1
∀k,n:ℕ. ∀K:n-dim-complex. ∀c:ℚCube(k).
((c ∈ K)
⇒ (∀f:ℚCube(k)
((f ∈ ∂(rat-cube-sub-complex(λa.(¬brceq(k;a;c));K)))
⇐⇒ ((f ∈ ∂(K)) ∧ (¬f ≤ c)) ∨ ((¬(f ∈ ∂(K))) ∧ f ≤ c ∧ (dim(f) = (dim(c) - 1) ∈ ℤ)))))
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
rational-cube-complex: n-dim-complex,
rat-cube-dimension: dim(c),
rat-cube-face: c ≤ d,
rceq: rceq(k;a;b),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
bnot: ¬bb,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
lambda: λx.A[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
l_member: (x ∈ l),
nat_plus: ℕ+,
select: L[n],
less_than': less_than'(a;b),
le: A ≤ B,
no_repeats: no_repeats(T;l),
remainder: n rem m,
modulus: a mod n,
eq_int: (i =z j),
isEven: isEven(n),
cons: [a / b],
rat-cube-dimension: dim(c),
band: p ∧b q,
true: True,
squash: ↓T,
subtract: n - m,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
top: Top,
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
in-complex-boundary: in-complex-boundary(k;f;K),
cand: A c∧ B,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
exists: ∃x:A. B[x],
uimplies: b supposing a,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
false: False,
not: ¬A,
or: P ∨ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
rational-cube-complex: n-dim-complex,
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
even-implies,
select_wf,
non_neg_length,
nat_plus_properties,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
false_wf,
add-is-int-iff,
length_wf_nat,
add_nat_wf,
add_nat_plus,
assert_elim,
length_of_nil_lemma,
isEven_wf,
nil_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
null_nil_lemma,
product_subtype_list,
list-cases,
assert_of_band,
iff_transitivity,
bfalse_wf,
btrue_wf,
band_wf,
bool_cases,
filter-sq,
filter-filter,
int_term_value_subtract_lemma,
itermSubtract_wf,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
true_wf,
squash_wf,
le_wf,
equal-wf-base,
l_all_iff,
zero-add,
add-swap,
add-commutes,
add-associates,
odd-implies,
even-iff-not-odd,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
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,
istype-le,
decidable__equal_int,
nat_properties,
length_of_cons_lemma,
filter-cons,
permutation-length,
decidable__equal_rc,
member_wf,
no_repeats_filter,
no_repeats_cons,
cons_member,
permutation-when-no_repeats,
cons_wf,
filter_functionality_wrt_permutation,
list_wf,
length_wf,
isOdd_wf,
no_repeats_wf,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert-is-rat-cube-face,
eqtt_to_assert,
is-rat-cube-face_wf,
member-rat-complex-boundary,
filter_wf5,
in-complex-boundary_wf,
inhabited-rat-cube_wf,
istype-assert,
assert-rceq,
equal_wf,
assert_of_bnot,
not_wf,
assert_wf,
iff_weakening_uiff,
member_filter,
istype-nat,
rational-cube-complex_wf,
subtract_wf,
int_subtype_base,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
istype-int,
istype-void,
rat-cube-face_wf,
rceq_wf,
bnot_wf,
rat-cube-sub-complex_wf,
rat-complex-boundary_wf,
rational-cube_wf,
l_member_wf
Rules used in proof :
closedConclusion,
baseApply,
pointwiseFunctionality,
voidEquality,
hypothesis_subsumption,
productEquality,
baseClosed,
imageMemberEquality,
universeEquality,
imageElimination,
int_eqEquality,
approximateComputation,
isect_memberEquality_alt,
applyLambdaEquality,
dependent_set_memberEquality_alt,
hyp_replacement,
cumulativity,
instantiate,
equalityElimination,
inrFormation_alt,
inlFormation_alt,
dependent_pairFormation_alt,
unionElimination,
voidElimination,
promote_hyp,
independent_functionElimination,
dependent_functionElimination,
productElimination,
sqequalBase,
independent_isectElimination,
addEquality,
natural_numberEquality,
minusEquality,
intEquality,
equalityIstype,
because_Cache,
functionIsType,
productIsType,
unionIsType,
sqequalRule,
equalitySymmetry,
equalityTransitivity,
applyEquality,
inhabitedIsType,
setIsType,
rename,
setElimination,
lambdaEquality_alt,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
universeIsType,
independent_pairFormation,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}k,n:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}c:\mBbbQ{}Cube(k).
((c \mmember{} K)
{}\mRightarrow{} (\mforall{}f:\mBbbQ{}Cube(k)
((f \mmember{} \mpartial{}(rat-cube-sub-complex(\mlambda{}a.(\mneg{}\msubb{}rceq(k;a;c));K)))
\mLeftarrow{}{}\mRightarrow{} ((f \mmember{} \mpartial{}(K)) \mwedge{} (\mneg{}f \mleq{} c)) \mvee{} ((\mneg{}(f \mmember{} \mpartial{}(K))) \mwedge{} f \mleq{} c \mwedge{} (dim(f) = (dim(c) - 1))))))
Date html generated:
2019_10_29-AM-07_59_06
Last ObjectModification:
2019_10_28-PM-01_57_27
Theory : rationals
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