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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