Nuprl Lemma : radd-list-linearity
∀[T:Type]. ∀[x,y:T ⟶ ℝ]. ∀[a,b:ℝ]. ∀[L:T List].
(radd-list(map(λk.((a * x[k]) + (b * y[k]));L)) = ((a * radd-list(map(λk.x[k];L))) + (b * radd-list(map(λk.y[k];L)))))
Proof
Definitions occuring in Statement :
req: x = y,
rmul: a * b,
radd: a + b,
radd-list: radd-list(L),
real: ℝ,
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
and: P ∧ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
list_induction,
req_wf,
radd-list_wf-bag,
map_wf,
real_wf,
radd_wf,
rmul_wf,
list-subtype-bag,
subtype_rel_self,
list_wf,
map_nil_lemma,
radd_list_nil_lemma,
map_cons_lemma,
req_witness,
int-to-real_wf,
req_weakening,
cons_wf,
uiff_transitivity,
req_functionality,
radd_functionality,
rmul-zero-both,
radd-zero-both,
req_transitivity,
radd-list-cons,
rmul_functionality,
rmul-distrib,
rmul_comm,
req_inversion,
radd-assoc,
radd-ac,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
rename,
functionEquality,
universeEquality,
natural_numberEquality,
productElimination
Latex:
\mforall{}[T:Type]. \mforall{}[x,y:T {}\mrightarrow{} \mBbbR{}]. \mforall{}[a,b:\mBbbR{}]. \mforall{}[L:T List].
(radd-list(map(\mlambda{}k.((a * x[k]) + (b * y[k]));L))
= ((a * radd-list(map(\mlambda{}k.x[k];L))) + (b * radd-list(map(\mlambda{}k.y[k];L)))))
Date html generated:
2017_10_02-PM-07_15_48
Last ObjectModification:
2017_07_28-AM-07_20_41
Theory : reals
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