Nuprl Lemma : rng_lsum-split
∀[A:Type]. ∀[p:A ⟶ 𝔹]. ∀[r:Rng]. ∀[f:A ⟶ |r|]. ∀[as:A List].
(Σ{r} x ∈ as. f[x] = (Σ{r} x ∈ filter(p;as). f[x] +r Σ{r} x ∈ filter(λa.(¬b(p a));as). f[x]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
filter: filter(P;l),
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
false: False,
assert: ↑b,
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
infix_ap: x f y,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
and: P ∧ Q,
top: Top,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
rng: Rng,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rng_wf,
rng_plus_ac_1,
true_wf,
squash_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
rng_plus_assoc,
iff_weakening_equal,
eqtt_to_assert,
filter_cons_lemma,
rng_lsum_cons_lemma,
rng_zero_wf,
rng_plus_zero,
filter_nil_lemma,
rng_lsum_nil_lemma,
list_wf,
bnot_wf,
set_wf,
subtype_rel_self,
l_member_wf,
bool_wf,
subtype_rel_dep_function,
filter_wf5,
rng_plus_wf,
infix_ap_wf,
rng_lsum_wf,
rng_car_wf,
equal_wf,
list_induction
Rules used in proof :
functionEquality,
axiomEquality,
universeEquality,
instantiate,
promote_hyp,
dependent_pairFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
levelHypothesis,
equalityUniverse,
imageElimination,
equalityTransitivity,
equalityElimination,
unionElimination,
productElimination,
equalitySymmetry,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation,
independent_isectElimination,
setEquality,
functionExtensionality,
applyEquality,
cumulativity,
hypothesis,
because_Cache,
rename,
setElimination,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A:Type]. \mforall{}[p:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[r:Rng]. \mforall{}[f:A {}\mrightarrow{} |r|]. \mforall{}[as:A List].
(\mSigma{}\{r\} x \mmember{} as. f[x] = (\mSigma{}\{r\} x \mmember{} filter(p;as). f[x] +r \mSigma{}\{r\} x \mmember{} filter(\mlambda{}a.(\mneg{}\msubb{}(p a));as). f[x]))
Date html generated:
2018_05_21-PM-09_33_01
Last ObjectModification:
2017_12_11-PM-05_31_14
Theory : matrices
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