Nuprl Lemma : sum-in-vs-split-shift
∀[k,n,m:ℤ]. ∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[f:{n..m + 1-} ⟶ Point(vs)].
Σ{f[i] | n≤i≤m} = Σ{f[i] | n≤i≤k} + Σ{f[k + i + 1] | 0≤i≤m - k + 1} ∈ Point(vs) supposing (n ≤ k) ∧ (k ≤ m)
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vs-add: x + y,
vector-space: VectorSpace(K),
vs-point: Point(vs),
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
and: P ∧ Q,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
and: P ∧ Q,
rng: Rng,
all: ∀x:A. B[x],
top: Top,
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,
prop: ℙ,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
sum-in-vs-split,
istype-le,
int_seg_wf,
vs-point_wf,
vector-space_wf,
rng_wf,
istype-int,
istype-void,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
add-swap,
sum-in-vs-shift
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
addEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
inhabitedIsType,
isect_memberEquality_alt,
voidElimination,
productElimination,
instantiate,
cumulativity,
intEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:\{n..m + 1\msupminus{}\} {}\mrightarrow{} Point(vs)].
\mSigma{}\{f[i] | n\mleq{}i\mleq{}m\} = \mSigma{}\{f[i] | n\mleq{}i\mleq{}k\} + \mSigma{}\{f[k + i + 1] | 0\mleq{}i\mleq{}m - k + 1\} supposing (n \mleq{} k) \mwedge{} (k \mleq{} m)
Date html generated:
2019_10_31-AM-06_26_20
Last ObjectModification:
2019_08_08-PM-00_32_17
Theory : linear!algebra
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