Nuprl Lemma : partition-sum_functionality
∀I:Interval
(icompact(I)
⇒ (∀p:partition(I). ∀q:ℝ List.
((||q|| = ||p|| ∈ ℤ)
⇒ (∀i:ℕ||q||. (q[i] = p[i]))
⇒ (∀f:I ⟶ℝ. ∀x:partition-choice(full-partition(I;p)).
(S(f;full-partition(I;p)) = S(f;full-partition(I;q)))))))
Proof
Definitions occuring in Statement :
partition-sum: S(f;p),
partition-choice: partition-choice(p),
full-partition: full-partition(I;p),
partition: partition(I),
icompact: icompact(I),
rfun: I ⟶ℝ,
interval: Interval,
req: x = y,
real: ℝ,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
cons: [a / b],
select: L[n],
less_than': less_than'(a;b),
le: A ≤ B,
rfun: I ⟶ℝ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
icompact: icompact(I),
subtype_rel: A ⊆r B,
ge: i ≥ j ,
nat: ℕ,
so_apply: x[s],
partition: partition(I),
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
guard: {T},
sq_type: SQType(T),
prop: ℙ,
and: P ∧ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
top: Top,
member: t ∈ T,
full-partition: full-partition(I;p),
partition-sum: S(f;p),
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
rsub_functionality,
rmul_functionality,
req_functionality,
select-append,
select-cons-tl,
less_than_wf,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
req_weakening,
lelt_wf,
false_wf,
int_seg_subtype_nat,
rsub_wf,
partition-choice-member,
rmul_wf,
rsum_functionality,
nat_wf,
subtract_wf,
le_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
top_wf,
subtype_rel_list,
length_append,
length_cons,
non_neg_length,
length_nil,
nat_properties,
nil_wf,
right-endpoint_wf,
append_wf,
left-endpoint_wf,
cons_wf,
interval_wf,
icompact_wf,
partition_wf,
list_wf,
length_wf,
equal_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
int_seg_properties,
real_wf,
select_wf,
req_wf,
int_seg_wf,
all_wf,
rfun_wf,
full-partition_wf,
partition-choice_wf,
int_subtype_base,
subtype_base_sq,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
length_of_nil_lemma,
length-append,
length_of_cons_lemma
Rules used in proof :
promote_hyp,
equalityElimination,
dependent_set_memberEquality,
addEquality,
applyEquality,
productElimination,
rename,
setElimination,
cumulativity,
instantiate,
independent_pairFormation,
intEquality,
hypothesisEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
unionElimination,
because_Cache,
isectElimination,
hypothesis,
voidEquality,
voidElimination,
isect_memberEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
sqequalRule,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}I:Interval
(icompact(I)
{}\mRightarrow{} (\mforall{}p:partition(I). \mforall{}q:\mBbbR{} List.
((||q|| = ||p||)
{}\mRightarrow{} (\mforall{}i:\mBbbN{}||q||. (q[i] = p[i]))
{}\mRightarrow{} (\mforall{}f:I {}\mrightarrow{}\mBbbR{}. \mforall{}x:partition-choice(full-partition(I;p)).
(S(f;full-partition(I;p)) = S(f;full-partition(I;q)))))))
Date html generated:
2018_05_22-PM-02_07_59
Last ObjectModification:
2018_05_21-AM-00_21_10
Theory : reals
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