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),
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,
rationals: ℚ
Lemmas :
Error :qle_witness,
weighted-sum_wf,
all_wf,
int_seg_wf,
length_wf,
rationals_wf,
Error :qle_wf,
l_member_wf,
int-subtype-rationals,
list_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
le_wf,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
length_wf_nat,
non_neg_length,
length_of_cons_lemma,
product_subtype_list,
ws_nil_lemma,
length_of_nil_lemma,
list-cases,
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,
not-le-2,
sq_stable__le,
decidable__lt,
le-add-cancel2,
lelt_wf,
cons_member,
equal_wf,
qmul_com,
qmul_wf,
Error :decidable__qless,
Error :qmul_preserves_qle,
Error :qle_antisymmetry,
Error :qless_complement_qorder,
Error :qmul_zero_qrng,
Error :mon_ident_q,
Error :mon_assoc_q,
qadd_wf,
Error :qadd_com,
Error :grp_op_preserves_le_qorder,
Error :qle_transitivity_qorder
\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:
2015_07_17-AM-07_58_25
Last ObjectModification:
2015_07_16-PM-00_36_59
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