Nuprl Lemma : ws-lower-bound
∀[p:FinProbSpace]. ∀[F:Outcome ⟶ ℚ]. ∀[q:ℚ]. q ≤ weighted-sum(p;F) supposing ∀x:Outcome. (q ≤ (F x))
Proof
Definitions occuring in Statement :
weighted-sum: weighted-sum(p;F),
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
qle: r ≤ s,
rationals: ℚ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
p-outcome: Outcome,
finite-prob-space: FinProbSpace,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
squash: ↓T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
true: True
Lemmas referenced :
ws-monotone,
int_seg_wf,
length_wf,
rationals_wf,
sq_stable_from_decidable,
qle_wf,
decidable__qle,
l_all_iff,
l_member_wf,
qle_witness,
weighted-sum_wf2,
all_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,
int-subtype-rationals,
squash_wf,
true_wf,
ws-constant,
p-outcome_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
lambdaEquality,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
lambdaFormation,
applyEquality,
independent_functionElimination,
dependent_functionElimination,
productElimination,
setEquality,
imageMemberEquality,
baseClosed,
imageElimination,
functionExtensionality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
productEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
hyp_replacement
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[F:Outcome {}\mrightarrow{} \mBbbQ{}]. \mforall{}[q:\mBbbQ{}].
q \mleq{} weighted-sum(p;F) supposing \mforall{}x:Outcome. (q \mleq{} (F x))
Date html generated:
2016_10_26-AM-06_49_35
Last ObjectModification:
2016_07_12-AM-08_04_12
Theory : randomness
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