Nuprl Lemma : sum-in-vs-single
∀[n:ℤ]. ∀[K:Rng]. ∀[vs:VectorSpace(K)]. ∀[f:{n..n + 1-} ⟶ Point(vs)]. (Σ{f[i] | n≤i≤n} = f[n] ∈ Point(vs))
Proof
Definitions occuring in Statement :
sum-in-vs: Σ{f[i] | n≤i≤m},
vector-space: VectorSpace(K),
vs-point: Point(vs),
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,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sum-in-vs: Σ{f[i] | n≤i≤m},
from-upto: [n, m),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
has-value: (a)↓,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
vs-bag-add: Σ{f[b] | b ∈ bs},
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
cons: [a / b],
nil: [],
vs-add: x + y,
record-select: r.x,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
rng: Rng
Lemmas referenced :
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
value-type-has-value,
int-value-type,
full-omega-unsat,
intformless_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_less_lemma,
istype-void,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-less_than,
intformnot_wf,
int_formula_prop_not_lemma,
vs-mon_ident,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
istype-le,
int_seg_wf,
vs-point_wf,
vector-space_wf,
rng_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
callbyvalueReduce,
intEquality,
because_Cache,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
universeIsType,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
applyEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
functionIsType,
setElimination,
rename,
axiomEquality,
isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[K:Rng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:\{n..n + 1\msupminus{}\} {}\mrightarrow{} Point(vs)]. (\mSigma{}\{f[i] | n\mleq{}i\mleq{}n\} = f[n])
Date html generated:
2019_10_31-AM-06_26_23
Last ObjectModification:
2019_08_08-PM-00_18_59
Theory : linear!algebra
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