Nuprl Lemma : rsum_int
∀[n:ℕ]. ∀[y:ℕn ⟶ ℤ]. (Σ{r(y[k]) | 0≤k≤n - 1} = r(Σ(y[k] | k < n)))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
int-to-real: r(n),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
subtract: n - m,
less_than': less_than'(a;b),
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
req_witness,
int_seg_wf,
rsum_wf,
subtract_wf,
int-to-real_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
istype-le,
sum_wf,
subtract-1-ge-0,
istype-nat,
rsum-empty,
req_weakening,
radd_wf,
req_functionality,
rsum-split-last,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
radd_functionality,
radd-int,
subtype_base_sq,
int_subtype_base,
sum_split1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
applyEquality,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
unionElimination,
productIsType,
addEquality,
because_Cache,
isectIsTypeImplies,
minusEquality,
imageMemberEquality,
baseClosed,
intEquality,
multiplyEquality,
instantiate,
cumulativity,
equalitySymmetry,
equalityTransitivity
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[y:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}\{r(y[k]) | 0\mleq{}k\mleq{}n - 1\} = r(\mSigma{}(y[k] | k < n)))
Date html generated:
2019_10_29-AM-10_12_21
Last ObjectModification:
2019_06_17-PM-06_09_47
Theory : reals
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