Nuprl Lemma : rv-partial-sum-monotone
∀[p:FinProbSpace]. ∀[f:ℕ ⟶ ℕ]. ∀[X:n:ℕ ⟶ RandomVariable(p;f[n])].
(∀[m:ℕ]. ∀[n:ℕm + 1]. rv-partial-sum(n;i.X[i]) ≤ rv-partial-sum(m;i.X[i])) supposing
((∀n:ℕ. 0 ≤ X[n]) and
(∀n:ℕ. ∀i:ℕn. f[i] < f[n]))
Proof
Definitions occuring in Statement :
rv-partial-sum: rv-partial-sum(n;i.X[i]),
rv-le: X ≤ Y,
rv-const: a,
random-variable: RandomVariable(p;n),
finite-prob-space: FinProbSpace,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
and: P ∧ Q,
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
rv-le: X ≤ Y,
rv-partial-sum: rv-partial-sum(n;i.X[i]),
random-variable: RandomVariable(p;n),
p-outcome: Outcome,
istype: istype(T),
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
subtract: n - m,
rv-const: a,
rv-add: X + Y,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b,
squash: ↓T
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
nat_properties,
decidable__le,
le_wf,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_weakening2,
int_seg_subtype_nat,
istype-false,
rv-partial-sum_wf,
subtype_rel-random-variable,
decidable__lt,
intformand_wf,
intformless_wf,
intformeq_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
less_than_wf,
rv-le_witness,
int_seg_wf,
nat_wf,
rv-le_wf,
rv-const_wf,
int-subtype-rationals,
random-variable_wf,
finite-prob-space_wf,
complete_nat_ind,
all_wf,
qsum_wf,
subtype_rel_dep_function,
p-outcome_wf,
int_seg_subtype,
le_weakening,
qle_weakening_eq_qorder,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtract-add-cancel,
rv-partial-sum-unroll,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
add-swap,
zero-add,
le-add-cancel,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-commutes,
le-add-cancel2,
subtype_rel_function,
subtype_rel_self,
qle_wf,
less_than_transitivity2,
qadd_wf,
qle_functionality_wrt_implies,
qadd_preserves_qle,
qmul_wf,
squash_wf,
true_wf,
rationals_wf,
qadd_comm_q,
qadd_ac_1_q,
qinverse_q,
qadd_inv_assoc_q,
iff_weakening_equal
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
because_Cache,
hypothesis,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesisEquality,
productElimination,
applyEquality,
dependent_set_memberEquality_alt,
universeIsType,
sqequalRule,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
lambdaFormation_alt,
productIsType,
isect_memberFormation_alt,
functionIsType,
inhabitedIsType,
closedConclusion,
minusEquality,
equalityIsType1,
promote_hyp,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[p:FinProbSpace]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbN{}]. \mforall{}[X:n:\mBbbN{} {}\mrightarrow{} RandomVariable(p;f[n])].
(\mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}m + 1]. rv-partial-sum(n;i.X[i]) \mleq{} rv-partial-sum(m;i.X[i])) supposing
((\mforall{}n:\mBbbN{}. 0 \mleq{} X[n]) and
(\mforall{}n:\mBbbN{}. \mforall{}i:\mBbbN{}n. f[i] < f[n]))
Date html generated:
2019_10_16-PM-00_39_31
Last ObjectModification:
2018_10_11-PM-10_38_17
Theory : randomness
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