Nuprl Lemma : rsum-split2
∀[n,m:ℤ]. ∀[k:{n..m + 1-}]. ∀[x:{n..m + 1-} ⟶ ℝ]. (Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[i] | k + 1≤i≤m}))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ
Lemmas referenced :
rsum-split,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
req_witness,
rsum_wf,
radd_wf,
decidable__lt,
istype-le,
istype-less_than,
int_seg_wf,
real_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
applyEquality,
inhabitedIsType,
setElimination,
rename,
independent_isectElimination,
productElimination,
imageElimination,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
dependent_set_memberEquality_alt,
addEquality,
productIsType,
closedConclusion,
functionIsType,
isectIsTypeImplies
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[k:\{n..m + 1\msupminus{}\}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
(\mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} = (\mSigma{}\{x[i] | n\mleq{}i\mleq{}k\} + \mSigma{}\{x[i] | k + 1\mleq{}i\mleq{}m\}))
Date html generated:
2019_10_29-AM-10_11_52
Last ObjectModification:
2019_06_25-PM-04_43_55
Theory : reals
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