Nuprl Lemma : ws-monotone
∀[p:ℚ List]. ∀[F,G:ℕ||p|| ⟶ ℚ].
(weighted-sum(p;F) ≤ weighted-sum(p;G)) supposing ((∀x:ℕ||p||. ((F x) ≤ (G x))) and (∀q:ℚ. ((q ∈ p) ⇒ (0 ≤ q))))
Proof
Definitions occuring in Statement :
weighted-sum: weighted-sum(p;F),
qle: r ≤ s,
rationals: ℚ,
l_member: (x ∈ l),
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
qle: r ≤ s,
grp_leq: a ≤ b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
cons: [a / b],
le: A ≤ B,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
subtract: n - m,
less_than: a < b,
less_than': less_than'(a;b),
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
qle_witness,
weighted-sum_wf,
int_seg_wf,
length_wf,
rationals_wf,
all_wf,
qle_wf,
l_member_wf,
int-subtype-rationals,
list_wf,
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,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
length_wf_nat,
list-cases,
length_of_nil_lemma,
ws_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_wf,
nil_wf,
ws_single_lemma,
squash_wf,
true_wf,
weighted-sum-split,
iff_weakening_equal,
add-is-int-iff,
false_wf,
add-member-int_seg2,
decidable__lt,
lelt_wf,
cons_member,
equal_wf,
qmul_com,
add_nat_plus,
nat_plus_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
qmul_wf,
decidable__qless,
qmul_preserves_qle,
qless_complement_qorder,
qle_antisymmetry,
qmul_zero_qrng,
qadd_wf,
mon_assoc_q,
mon_ident_q,
grp_op_preserves_le_qorder,
qadd_com,
qle_transitivity_qorder
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
extract_by_obid,
isectElimination,
functionExtensionality,
applyEquality,
natural_numberEquality,
sqequalRule,
lambdaEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_set_memberEquality,
addEquality,
inrFormation,
applyLambdaEquality,
hyp_replacement,
inlFormation
Latex:
\mforall{}[p:\mBbbQ{} List]. \mforall{}[F,G:\mBbbN{}||p|| {}\mrightarrow{} \mBbbQ{}].
(weighted-sum(p;F) \mleq{} weighted-sum(p;G)) supposing
((\mforall{}x:\mBbbN{}||p||. ((F x) \mleq{} (G x))) and
(\mforall{}q:\mBbbQ{}. ((q \mmember{} p) {}\mRightarrow{} (0 \mleq{} q))))
Date html generated:
2018_05_22-AM-00_34_27
Last ObjectModification:
2017_07_26-PM-06_59_48
Theory : randomness
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