Nuprl Lemma : rv-partial-sum-unroll
∀[m:ℕ+]. ∀[X:Top].  (rv-partial-sum(m;i.X[i]) ~ rv-partial-sum(m - 1;i.X[i]) + X[m - 1])
Proof
Definitions occuring in Statement : 
rv-partial-sum: rv-partial-sum(n;i.X[i])
, 
rv-add: X + Y
, 
nat_plus: ℕ+
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
so_apply: x[s]
, 
subtract: n - m
, 
natural_number: $n
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
rv-partial-sum: rv-partial-sum(n;i.X[i])
, 
rv-add: X + Y
, 
qsum: Σa ≤ j < b. E[j]
, 
rng_sum: rng_sum, 
mon_itop: Π lb ≤ i < ub. E[i]
, 
add_grp_of_rng: r↓+gp
, 
grp_op: *
, 
pi2: snd(t)
, 
pi1: fst(t)
, 
grp_id: e
, 
qrng: <ℚ+*>
, 
rng_plus: +r
, 
rng_zero: 0
, 
itop: Π(op,id) lb ≤ i < ub. E[i]
, 
ycomb: Y
, 
nat_plus: ℕ+
, 
infix_ap: x f y
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
all: ∀x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
guard: {T}
Lemmas referenced : 
top_wf, 
nat_plus_wf, 
lt_int_wf, 
bool_wf, 
equal-wf-T-base, 
assert_wf, 
less_than_wf, 
le_int_wf, 
le_wf, 
bnot_wf, 
nat_plus_properties, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformle_wf, 
itermVar_wf, 
itermConstant_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_le_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
uiff_transitivity, 
eqtt_to_assert, 
assert_of_lt_int, 
eqff_to_assert, 
assert_functionality_wrt_uiff, 
bnot_of_lt_int, 
assert_of_le_int, 
equal_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
sqequalRule, 
hypothesis, 
sqequalAxiom, 
extract_by_obid, 
sqequalHypSubstitution, 
isect_memberEquality, 
isectElimination, 
thin, 
hypothesisEquality, 
because_Cache, 
natural_numberEquality, 
setElimination, 
rename, 
equalityTransitivity, 
equalitySymmetry, 
baseClosed, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
lambdaFormation, 
unionElimination, 
equalityElimination, 
independent_functionElimination, 
productElimination
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}].  \mforall{}[X:Top].    (rv-partial-sum(m;i.X[i])  \msim{}  rv-partial-sum(m  -  1;i.X[i])  +  X[m  -  1])
Date html generated:
2018_05_22-AM-00_38_04
Last ObjectModification:
2017_07_26-PM-07_00_48
Theory : randomness
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