Nuprl Lemma : ws-constant
∀[a:ℚ]. ∀[p:FinProbSpace]. ∀[F:Outcome ⟶ ℚ].  weighted-sum(p;F) = a ∈ ℚ supposing ∀x:Outcome. ((F x) = a ∈ ℚ)
Proof
Definitions occuring in Statement : 
weighted-sum: weighted-sum(p;F)
, 
p-outcome: Outcome
, 
finite-prob-space: FinProbSpace
, 
rationals: ℚ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
all: ∀x:A. B[x]
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
weighted-sum: weighted-sum(p;F)
, 
p-outcome: Outcome
, 
finite-prob-space: FinProbSpace
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
prop: ℙ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
and: P ∧ Q
, 
guard: {T}
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
all: ∀x:A. B[x]
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
top: Top
, 
less_than: a < b
, 
squash: ↓T
, 
subtype_rel: A ⊆r B
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
select: L[n]
, 
nil: []
, 
it: ⋅
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
uiff: uiff(P;Q)
, 
qeq: qeq(r;s)
, 
callbyvalueall: callbyvalueall, 
evalall: evalall(t)
, 
qsum: Σa ≤ j < b. E[j]
, 
rng_sum: rng_sum, 
mon_itop: Π lb ≤ i < ub. E[i]
, 
itop: Π(op,id) lb ≤ i < ub. E[i]
, 
ycomb: Y
, 
ifthenelse: if b then t else f fi 
, 
lt_int: i <z j
, 
bfalse: ff
, 
grp_id: e
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
add_grp_of_rng: r↓+gp
, 
rng_zero: 0
, 
qrng: <ℚ+*>
, 
btrue: tt
, 
eq_int: (i =z j)
, 
assert: ↑b
, 
cons: [a / b]
, 
nat: ℕ
, 
le: A ≤ B
, 
cand: A c∧ B
, 
subtract: n - m
, 
less_than': less_than'(a;b)
Lemmas referenced : 
all_wf, 
int_seg_wf, 
length_wf, 
rationals_wf, 
equal_wf, 
set_wf, 
list_wf, 
equal-wf-T-base, 
qsum_wf, 
select_wf, 
int_seg_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
l_all_wf2, 
l_member_wf, 
qle_wf, 
int-subtype-rationals, 
squash_wf, 
true_wf, 
qmul_wf, 
iff_weakening_equal, 
prod_sum_l_q, 
le_weakening2, 
list-cases, 
length_of_nil_lemma, 
stuck-spread, 
base_wf, 
assert-qeq, 
product_subtype_list, 
length_of_cons_lemma, 
length_wf_nat, 
nat_wf, 
false_wf, 
not-lt-2, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
zero-add, 
minus-one-mul-top, 
add-commutes, 
add_functionality_wrt_le, 
add-associates, 
add-zero, 
le-add-cancel, 
qmul_one_qrng
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
isect_memberFormation, 
introduction, 
cut, 
hypothesis, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
natural_numberEquality, 
setElimination, 
rename, 
hypothesisEquality, 
lambdaEquality, 
applyEquality, 
functionExtensionality, 
because_Cache, 
isect_memberEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
productEquality, 
independent_isectElimination, 
productElimination, 
dependent_functionElimination, 
unionElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
imageElimination, 
baseClosed, 
lambdaFormation, 
setEquality, 
universeEquality, 
imageMemberEquality, 
independent_functionElimination, 
promote_hyp, 
hypothesis_subsumption, 
addEquality, 
minusEquality, 
hyp_replacement, 
applyLambdaEquality
Latex:
\mforall{}[a:\mBbbQ{}].  \mforall{}[p:FinProbSpace].  \mforall{}[F:Outcome  {}\mrightarrow{}  \mBbbQ{}].
    weighted-sum(p;F)  =  a  supposing  \mforall{}x:Outcome.  ((F  x)  =  a)
Date html generated:
2018_05_22-AM-00_34_31
Last ObjectModification:
2017_07_26-PM-06_59_49
Theory : randomness
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