Nuprl Lemma : sum_split+
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn + 1].
((Σ(f[x] | x < n) = (Σ(f[x] | x < m) + Σ(f[x + m] | x < n - m)) ∈ ℤ)
∧ (Σ(f[x] | x < m) ∈ ℤ)
∧ (Σ(f[x + m] | x < n - m) ∈ ℤ))
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
le: A ≤ B,
less_than: a < b,
uiff: uiff(P;Q)
Lemmas referenced :
nat_wf,
add-member-int_seg1,
le_wf,
int_term_value_subtract_lemma,
int_formula_prop_le_lemma,
itermSubtract_wf,
intformle_wf,
decidable__le,
subtract_wf,
int_seg_wf,
lelt_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
int_seg_properties,
int_seg_subtype_nat,
sum_wf,
sum_split
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
applyEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
independent_isectElimination,
introduction,
because_Cache,
sqequalRule,
lambdaEquality,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}n + 1].
((\mSigma{}(f[x] | x < n) = (\mSigma{}(f[x] | x < m) + \mSigma{}(f[x + m] | x < n - m)))
\mwedge{} (\mSigma{}(f[x] | x < m) \mmember{} \mBbbZ{})
\mwedge{} (\mSigma{}(f[x + m] | x < n - m) \mmember{} \mBbbZ{}))
Date html generated:
2016_05_14-AM-07_33_27
Last ObjectModification:
2016_01_14-PM-09_54_37
Theory : int_2
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