Nuprl Lemma : sum_split1
∀[n:ℕ+]. ∀[f:ℕn ⟶ ℤ]. (Σ(f[x] | x < n) = (Σ(f[x] | x < n - 1) + f[n - 1]) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat_plus: 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,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
sum-as-primrec,
nat_plus_subtype_nat,
int_seg_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
decidable__lt,
lelt_wf,
primrec-unroll,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
primrec_wf,
decidable__equal_int,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
lambdaEquality,
functionExtensionality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
productElimination,
lambdaFormation,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
addEquality,
functionEquality,
axiomEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(f[x] | x < n) = (\mSigma{}(f[x] | x < n - 1) + f[n - 1]))
Date html generated:
2017_04_14-AM-09_21_19
Last ObjectModification:
2017_02_27-PM-03_57_23
Theory : int_2
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