Nuprl Lemma : rsum'-eq-rsum
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. (rsum'(n;m;k.x[k]) = Σ{x[k] | n≤k≤m} ∈ (ℕ+ ⟶ ℤ))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rsum': rsum'(n;m;k.x[k]),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rsum: Σ{x[k] | n≤k≤m},
rsum': rsum'(n;m;k.x[k]),
uimplies: b supposing a,
has-value: (a)↓,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
radd-list: radd-list(L),
satisfiable_int_formula: satisfiable_int_formula(fmla),
eq_int: (i =z j),
subtype_rel: A ⊆r B,
real: ℝ,
nat_plus: ℕ+,
accelerate: accelerate(k;f),
nat: ℕ,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
so_lambda: λ2x.t[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
le: A ≤ B,
compose: f o g,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x])
Lemmas referenced :
int_seg_wf,
real_wf,
value-type-has-value,
int-value-type,
subtract_wf,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
real-valueall-type,
map_wf,
le_wf,
less_than_wf,
from-upto_wf,
evalall-reduce,
valueall-type-real-list,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
length-map,
length-from-upto,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
int-to-real_wf,
nat_plus_wf,
eq_int_wf,
assert_of_eq_int,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
intformnot_wf,
int_formula_prop_not_lemma,
squash_wf,
true_wf,
sum_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
add-member-int_seg1,
decidable__lt,
lelt_wf,
mul_nat_plus,
false_wf,
not-lt-2,
not-equal-2,
condition-implies-le,
add-associates,
minus-one-mul,
add-commutes,
minus-one-mul-top,
add-swap,
zero-add,
minus-add,
minus-minus,
add_functionality_wrt_le,
le-add-cancel2,
itermMultiply_wf,
int_term_value_mul_lemma,
equal-wf-base,
int_subtype_base,
reg-seq-list-add-as-l_sum,
decidable__equal_int,
mul_bounds_1b,
iff_weakening_equal,
map-map,
l_sum-sum,
l_member_wf,
set_wf,
general_arith_equation1,
nat_wf,
select-from-upto,
int_seg_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
addEquality,
natural_numberEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
intEquality,
independent_isectElimination,
callbyvalueReduce,
setEquality,
productEquality,
lambdaEquality,
lambdaFormation,
setElimination,
rename,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
int_eqEquality,
computeAll,
sqleReflexivity,
multiplyEquality,
universeEquality,
divideEquality,
minusEquality,
baseApply,
closedConclusion
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (rsum'(n;m;k.x[k]) = \mSigma{}\{x[k] | n\mleq{}k\mleq{}m\})
Date html generated:
2017_10_03-AM-08_57_06
Last ObjectModification:
2017_07_28-AM-07_37_16
Theory : reals
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