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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