Nuprl Lemma : weighted-sum-linear
∀[a,b:ℚ]. ∀[p:ℚ List]. ∀[F,G:ℕ||p|| ─→ ℚ].
(weighted-sum(p;λx.((a * (F x)) + (b * (G x)))) = ((a * weighted-sum(p;F)) + (b * weighted-sum(p;G))) ∈ ℚ)
Proof
Definitions occuring in Statement :
weighted-sum: weighted-sum(p;F),
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
natural_number: $n,
equal: s = t ∈ T,
qmul: r * s,
qadd: r + s,
rationals: ℚ
Lemmas :
rationals_wf,
int_seg_wf,
length_wf,
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,
decidable__equal_int,
subtype_rel-int_seg,
le_weakening,
int_seg_properties,
list-cases,
product_subtype_list,
decidable__lt,
not-equal-2,
le-add-cancel-alt,
lelt_wf,
not-le-2,
sq_stable__le,
add-mul-special,
zero-mul,
nat_wf,
iff_weakening_equal,
Error :mon_ident_q,
Error :qmul_zero_qrng,
true_wf,
squash_wf,
qadd_wf,
int-subtype-rationals,
ws_nil_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_wf,
nil_wf,
qmul_wf,
length_nil,
non_neg_length,
length_wf_nil,
length_wf_nat,
length_cons,
length_append,
subtype_rel_list,
top_wf,
append_wf,
subtype_rel_dep_function,
subtype_rel_self,
weighted-sum-split,
ws_single_lemma,
length_of_cons_lemma,
length_of_nil_lemma,
Error :qadd_ac_1_q,
Error :qadd_comm_q,
Error :mon_assoc_q,
Error :qmul_ac_1_qrng,
Error :qmul_assoc_qrng,
Error :qmul_over_plus_qrng,
weighted-sum_wf,
equal_wf,
le-add-cancel2,
minus-zero
\mforall{}[a,b:\mBbbQ{}]. \mforall{}[p:\mBbbQ{} List]. \mforall{}[F,G:\mBbbN{}||p|| {}\mrightarrow{} \mBbbQ{}].
(weighted-sum(p;\mlambda{}x.((a * (F x)) + (b * (G x))))
= ((a * weighted-sum(p;F)) + (b * weighted-sum(p;G))))
Date html generated:
2015_07_17-AM-07_58_20
Last ObjectModification:
2015_07_16-AM-09_52_22
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