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)
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],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
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],
top: Top,
less_than: a < b,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
sum-as-primrec,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
subtract_wf,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
le_wf,
add-member-int_seg1,
primrec_add,
subtract-add-cancel,
primrec_wf,
nat_wf,
equal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
ge_wf,
less_than_wf,
primrec0_lemma,
decidable__equal_int,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
add-member-int_seg2,
intformimplies_wf,
int_formual_prop_imp_lemma,
primrec-unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
addEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomEquality,
functionEquality,
applyLambdaEquality,
intWeakElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
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)))
Date html generated:
2017_04_14-AM-09_21_15
Last ObjectModification:
2017_02_27-PM-03_57_55
Theory : int_2
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