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),
qmul: r * s,
qadd: r + s,
rationals: ℚ,
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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
cons: [a / b],
less_than: a < b,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
subtract: n - m
Lemmas referenced :
rationals_wf,
int_seg_wf,
length_wf,
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,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
product_subtype_list,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
ws_nil_lemma,
int-subtype-rationals,
equal_wf,
squash_wf,
true_wf,
qadd_wf,
qmul_zero_qrng,
mon_ident_q,
iff_weakening_equal,
list_ind_cons_lemma,
list_ind_nil_lemma,
cons_wf,
nil_wf,
qmul_wf,
length_of_cons_lemma,
length_nil,
non_neg_length,
length_cons,
append_wf,
length_append,
subtype_rel_list,
top_wf,
length-append,
length_of_nil_lemma,
subtype_rel_dep_function,
length-singleton,
subtype_rel_self,
weighted-sum-split,
ws_single_lemma,
qmul_over_plus_qrng,
qmul_assoc_qrng,
qmul_ac_1_qrng,
mon_assoc_q,
qadd_comm_q,
qadd_ac_1_q,
le_weakening2,
add-member-int_seg2,
add-is-int-iff,
weighted-sum_wf,
length_wf_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
because_Cache,
extract_by_obid,
functionEquality,
natural_numberEquality,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
productElimination,
unionElimination,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
addEquality,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
functionExtensionality,
hyp_replacement,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\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:
2018_05_22-AM-00_34_18
Last ObjectModification:
2017_07_26-PM-06_59_47
Theory : randomness
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