Nuprl Lemma : sum-equal-terms
∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ]. ∀[m:ℕ]. ∀[b:ℕm ⟶ ℤ].
Σ(a[i] | i < n) = Σ(b[j] | j < m) ∈ ℤ
supposing permutation(ℤ;filter(λx.(¬b(x =z 0));map(λi.a[i];upto(n)));filter(λx.(¬b(x =z 0));map(λj.b[j];upto(m))))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
upto: upto(n),
sum: Σ(f[x] | x < k),
filter: filter(P;l),
map: map(f;as),
int_seg: {i..j-},
nat: ℕ,
bnot: ¬bb,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
nat: ℕ
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
sum-l_sum,
int_seg_wf,
subtype_rel_self,
iff_weakening_equal,
l_sum_filter0,
map_wf,
upto_wf,
l_sum_wf,
l_sum_functionality_wrt_permutation,
filter_wf5,
bnot_wf,
eq_int_wf,
istype-int,
l_member_wf,
permutation_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
intEquality,
sqequalRule,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
setElimination,
rename,
setIsType,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsType
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}]. \mforall{}[b:\mBbbN{}m {}\mrightarrow{} \mBbbZ{}].
\mSigma{}(a[i] | i < n) = \mSigma{}(b[j] | j < m)
supposing permutation(\mBbbZ{};filter(\mlambda{}x.(\mneg{}\msubb{}(x =\msubz{} 0));map(\mlambda{}i.a[i];upto(n)));
filter(\mlambda{}x.(\mneg{}\msubb{}(x =\msubz{} 0));map(\mlambda{}j.b[j];upto(m))))
Date html generated:
2020_05_20-AM-08_15_56
Last ObjectModification:
2020_01_04-PM-11_11_57
Theory : general
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