Nuprl Lemma : expectation-rv-sample
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[i:ℕn]. ∀[X:Outcome ─→ ℚ]. (E(n;X@i) = weighted-sum(p;X) ∈ ℚ)
Proof
Definitions occuring in Statement :
expectation: E(n;F),
rv-sample: X@i,
weighted-sum: weighted-sum(p;F),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
rationals: ℚ
Lemmas :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
p-outcome_wf,
rationals_wf,
int_seg_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
finite-prob-space_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
decidable__equal_int,
le_weakening2,
rv-sample_wf,
rv-shift_wf,
le_wf,
not-le-2,
expectation-constant,
true_wf,
squash_wf,
weighted-sum_wf2,
subtype_base_sq,
equal_wf,
expectation_wf,
ws-constant,
length_wf,
lelt_wf,
le-add-cancel-alt,
decidable__lt,
minus-zero,
sq_stable__le,
not-equal-2,
equal-wf-T-base,
bool_wf,
eq_int_wf
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}n]. \mforall{}[X:Outcome {}\mrightarrow{} \mBbbQ{}]. (E(n;X@i) = weighted-sum(p;X))
Date html generated:
2015_07_17-AM-07_59_03
Last ObjectModification:
2015_07_16-AM-09_52_18
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