Nuprl Lemma : rsum-split-shift
∀[k,n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].
(Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[k + i + 1] | 0≤i≤m - k + 1})) supposing ((k ≤ m) and (n ≤ k))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
prop: ℙ,
top: Top,
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,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
rsum-shift,
add-swap,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
itermConstant_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
real_wf,
int_seg_wf,
le_wf,
rsum-split
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
functionEquality,
addEquality,
natural_numberEquality,
because_Cache,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
instantiate,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \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[k + i + 1] | 0\mleq{}i\mleq{}m - k + 1\})) supposing
((k \mleq{} m) and
(n \mleq{} k))
Date html generated:
2016_05_18-AM-07_45_50
Last ObjectModification:
2016_01_17-AM-02_08_25
Theory : reals
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