Nuprl Lemma : member-rat-complex-boundary-n
∀n,k:ℕ. ∀K:n-dim-complex. ∀f:ℚCube(k). ((f ∈ ∂(K)) ⇐⇒ (↑in-complex-boundary(k;f;K)) ∧ (dim(f) = (n - 1) ∈ ℤ))
Proof
Definitions occuring in Statement :
rat-complex-boundary: ∂(K),
in-complex-boundary: in-complex-boundary(k;f;K),
rational-cube-complex: n-dim-complex,
rat-cube-dimension: dim(c),
rational-cube: ℚCube(k),
l_member: (x ∈ l),
nat: ℕ,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
sq_type: SQType(T),
rat-cube-dimension: dim(c),
nat_plus: ℕ+,
cand: A c∧ B,
select: L[n],
less_than': less_than'(a;b),
le: A ≤ B,
l_member: (x ∈ l),
assert: ↑b,
remainder: n rem m,
modulus: a mod n,
eq_int: (i =z j),
isOdd: isOdd(n),
cons: [a / b],
in-complex-boundary: in-complex-boundary(k;f;K),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
true: True,
squash: ↓T,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
rat-complex-boundary: ∂(K),
bfalse: ff,
so_apply: x[s],
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
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,
uall: ∀[x:A]. B[x],
member: t ∈ T,
face-complex: face-complex(k;L),
rational-cube-complex: n-dim-complex,
all: ∀x:A. B[x]
Lemmas referenced :
assert_of_bnot,
eqff_to_assert,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases,
assert-is-rat-cube-face,
isOdd_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
select_wf,
length_wf,
false_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
nat_plus_properties,
istype-less_than,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
length_wf_nat,
add_nat_plus,
istype-le,
length_of_cons_lemma,
product_subtype_list,
length_of_nil_lemma,
list-cases,
is-rat-cube-face_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
true_wf,
squash_wf,
equal_wf,
l_all_iff,
istype-nat,
rational-cube-complex_wf,
filter_wf5,
member_filter,
member-face-complex,
member-rat-cube-faces,
le_wf,
istype-assert,
l_member_wf,
in-complex-boundary_wf,
assert_witness,
nil_wf,
subtract_wf,
int_subtype_base,
istype-int,
lelt_wf,
set_subtype_base,
rat-cube-dimension_wf,
equal-wf-base,
rat-cube-face_wf,
subtype_rel_list,
rat-cube-faces_wf,
eqtt_to_assert,
inhabited-rat-cube_wf,
list_wf,
map_wf,
concat_wf,
rc-deq_wf,
rational-cube_wf,
remove-repeats_wf
Rules used in proof :
cumulativity,
pointwiseFunctionality,
applyLambdaEquality,
dependent_set_memberEquality_alt,
hypothesis_subsumption,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
imageMemberEquality,
universeEquality,
instantiate,
imageElimination,
promote_hyp,
dependent_pairFormation_alt,
baseClosed,
closedConclusion,
baseApply,
independent_pairFormation,
independent_functionElimination,
dependent_functionElimination,
sqequalBase,
equalityIstype,
universeIsType,
productIsType,
setIsType,
equalitySymmetry,
equalityTransitivity,
addEquality,
natural_numberEquality,
minusEquality,
intEquality,
productEquality,
setEquality,
applyEquality,
independent_isectElimination,
productElimination,
equalityElimination,
unionElimination,
inhabitedIsType,
lambdaEquality_alt,
because_Cache,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
introduction,
sqequalRule,
cut,
rename,
thin,
setElimination,
sqequalHypSubstitution,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n,k:\mBbbN{}. \mforall{}K:n-dim-complex. \mforall{}f:\mBbbQ{}Cube(k).
((f \mmember{} \mpartial{}(K)) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}in-complex-boundary(k;f;K)) \mwedge{} (dim(f) = (n - 1)))
Date html generated:
2019_10_29-AM-07_58_54
Last ObjectModification:
2019_10_21-AM-10_23_00
Theory : rationals
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