Nuprl Lemma : sum_split_first
∀[n:ℕ+]. ∀[f:ℕn ⟶ ℤ]. (Σ(f[x] | x < n) = (f[0] + Σ(f[x + 1] | x < 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,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
nat_plus: 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],
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b
Lemmas referenced :
nat_plus_wf,
int_seg_wf,
sum1,
lelt_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
false_wf,
nat_plus_subtype_nat,
sum_split
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
setElimination,
rename,
dependent_functionElimination,
addEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
because_Cache,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(f[x] | x < n) = (f[0] + \mSigma{}(f[x + 1] | x < n - 1)))
Date html generated:
2016_05_14-AM-07_33_30
Last ObjectModification:
2016_01_14-PM-09_54_22
Theory : int_2
Home
Index