Nuprl Lemma : expectation-monotone
∀[p:FinProbSpace]. ∀[n:ℕ]. ∀[X,Y:RandomVariable(p;n)]. E(n;X) ≤ E(n;Y) supposing X ≤ Y
Proof
Definitions occuring in Statement :
rv-le: X ≤ Y,
expectation: E(n;F),
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
qle: r ≤ s,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
qle: r ≤ s,
grp_leq: a ≤ b,
expectation: E(n;F),
ycomb: Y,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
infix_ap: x f y,
subtype_rel: A ⊆r B,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
rationals: ℚ,
quotient: x,y:A//B[x; y],
grp_car: |g|,
pi1: fst(t),
qadd_grp: <ℚ+>,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
p-outcome: Outcome,
int_seg: {i..j-},
nat_plus: ℕ+,
lelt: i ≤ j < k,
rv-le: X ≤ Y,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
rv-shift: rv-shift(x;X),
cand: A c∧ B,
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
assert_witness,
grp_le_wf,
qadd_grp_wf2,
null-seq_wf,
int_seg_wf,
length_wf,
rationals_wf,
subtype_rel_self,
grp_car_wf,
rv-le_wf,
istype-le,
random-variable_wf,
expectation_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
ocgrp_subtype_ocmon,
subtype_rel_transitivity,
ocgrp_wf,
ocmon_wf,
abdmonoid_wf,
dmon_wf,
mon_wf,
grp_sig_wf,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-nat,
finite-prob-space_wf,
p-outcome_wf,
eq_int_wf,
equal-wf-base,
bool_wf,
int_subtype_base,
assert_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
istype-assert,
weighted-sum_wf2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
rv-shift_wf,
decidable__lt,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
ws-monotone,
int_seg_properties,
sq_stable_from_decidable,
qle_wf,
int-subtype-rationals,
decidable__qle,
l_all_iff,
l_member_wf,
cons-seq_wf,
subtype_rel_function,
int_seg_subtype,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
minus-minus,
add-associates,
add-commutes,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
applyEquality,
because_Cache,
dependent_set_memberEquality_alt,
instantiate,
unionElimination,
baseApply,
closedConclusion,
baseClosed,
intEquality,
equalityIstype,
sqequalBase,
equalitySymmetry,
functionIsType,
productElimination,
equalityElimination,
equalityTransitivity,
imageElimination,
setIsType,
imageMemberEquality,
addEquality,
minusEquality,
multiplyEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n:\mBbbN{}]. \mforall{}[X,Y:RandomVariable(p;n)]. E(n;X) \mleq{} E(n;Y) supposing X \mleq{} Y
Date html generated:
2020_05_20-AM-09_31_29
Last ObjectModification:
2019_11_27-PM-04_57_38
Theory : randomness
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