Nuprl Lemma : open-expectation-monotone
∀[p:FinProbSpace]. ∀[n,m:ℕ]. ∀[C:p-open(p)]. (E(n;λs.(C <n, s>)) ≤ E(m;λs.(C <m, s>))) supposing n ≤ m
Proof
Definitions occuring in Statement :
p-open: p-open(p),
expectation: E(n;F),
finite-prob-space: FinProbSpace,
qle: r ≤ s,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
lambda: λx.A[x],
pair: <a, b>
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
random-variable: RandomVariable(p;n),
p-open: p-open(p),
subtype_rel: A ⊆r B,
guard: {T},
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
p-outcome: Outcome,
decidable: Dec(P),
or: P ∨ Q,
expectation: E(n;F),
ycomb: Y,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
less_than': less_than'(a;b),
rv-le: X ≤ Y,
uiff: uiff(P;Q),
sq_type: SQType(T),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
nequal: a ≠ b ∈ T ,
int_upper: {i...},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
rv-shift: rv-shift(x;X),
pi1: fst(t),
pi2: snd(t),
cons-seq: cons-seq(x;s)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
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,
less_than_wf,
qle_witness,
expectation_wf,
p-open_wf,
le_wf,
int_seg_properties,
length_wf,
rationals_wf,
int_seg_wf,
p-outcome_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
finite-prob-space_wf,
expectation-constant,
false_wf,
null-seq_wf,
expectation-monotone,
qle_wf,
qle-int,
subtype_base_sq,
int_subtype_base,
sq_stable__le,
subtype_rel_dep_function,
int_seg_subtype,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
lelt_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
natural_number_wf_p-outcome,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
decidable__lt,
eq_int_wf,
equal-wf-base,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
uiff_transitivity,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
ws-monotone,
rv-shift_wf,
less_than_transitivity1,
sq_stable_from_decidable,
int-subtype-rationals,
decidable__qle,
l_all_iff,
l_member_wf,
pi1_wf_top,
add_nat_wf,
add-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
cons-seq_wf,
int_seg_subtype_nat,
subtype_rel_self,
neg_assert_of_eq_int,
int_upper_properties,
all_wf,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality,
applyEquality,
dependent_pairEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
functionEquality,
unionElimination,
hyp_replacement,
instantiate,
cumulativity,
productEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
promote_hyp,
hypothesis_subsumption,
baseApply,
closedConclusion,
impliesFunctionality,
setEquality,
addEquality,
independent_pairEquality,
pointwiseFunctionality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[n,m:\mBbbN{}]. \mforall{}[C:p-open(p)]. (E(n;\mlambda{}s.(C <n, s>)) \mleq{} E(m;\mlambda{}s.(C <m, s>))) supposing n \000C\mleq{} m
Date html generated:
2018_05_22-AM-00_36_13
Last ObjectModification:
2017_07_26-PM-07_00_24
Theory : randomness
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