Nuprl Lemma : sum-as-bag-accum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. (Σ(f[x] | x < n) ~ bag-accum(x,y.x + y;0;[x∈bag-map(λx.f[x];upto(n))|¬b(x =z 0)]))
Proof
Definitions occuring in Statement :
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
upto: upto(n),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
bnot: ¬bb,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
bag-map: bag-map(f;bs),
bag-filter: [x∈b|p[x]],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T}
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
sum-as-accum-filter,
int_seg_wf,
list_accum_wf,
filter_wf5,
map_wf,
upto_wf,
l_member_wf,
bnot_wf,
eq_int_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesis,
sqequalRule,
hypothesisEquality,
lambdaEquality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
intEquality,
lambdaFormation,
setEquality,
addEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
sqequalAxiom,
functionEquality,
isect_memberEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(\mSigma{}(f[x] | x < n) \msim{} bag-accum(x,y.x + y;0;[x\mmember{}bag-map(\mlambda{}x.f[x];upto(n))|\mneg{}\msubb{}(x =\msubz{} 0)]))
Date html generated:
2016_05_15-PM-02_30_14
Last ObjectModification:
2015_12_27-AM-09_48_54
Theory : bags
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