Nuprl Lemma : vs-lift-bfs-reduce
∀[K:Rng]. ∀[S:Type]. ∀[as,bs:basic-formal-sum(K;S)].
∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. (vs-lift(vs;f;as) = vs-lift(vs;f;bs) ∈ Point(vs))
supposing bfs-reduce(K;S;as;bs)
Proof
Definitions occuring in Statement :
vs-lift: vs-lift(vs;f;fs),
bfs-reduce: bfs-reduce(K;S;as;bs),
basic-formal-sum: basic-formal-sum(K;S),
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng: Rng
Definitions unfolded in proof :
basic-formal-sum: basic-formal-sum(K;S),
exists: ∃x:A. B[x],
or: P ∨ Q,
bfs-reduce: bfs-reduce(K;S;as;bs),
comm: Comm(T;op),
cand: A c∧ B,
ident: Ident(T;op;id),
rng: Rng,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
prop: ℙ,
squash: ↓T,
infix_ap: x f y,
assoc: Assoc(T;op),
and: P ∧ Q,
monoid_p: IsMonoid(T;op;id),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
quotient: x,y:A//B[x; y],
bag: bag(T),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-summation: Σ(x∈b). f[x],
top: Top,
vs-bag-add: Σ{f[b] | b ∈ bs},
bag-map: bag-map(f;bs),
vs-lift: vs-lift(vs;f;fs),
formal-sum-mul: k * x,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
bag-append: as + bs,
single-bag: {x}
Lemmas referenced :
infix_ap_wf,
rng_car_wf,
bag-append_wf,
rng_plus_wf,
formal-sum-mul_wf1,
zero-bfs_wf,
vs-lift_wf,
basic-formal-sum_wf,
bfs-reduce_wf,
vs-0_wf,
vs-add-comm,
vs-mon_ident,
iff_weakening_equal,
vs-add_wf,
vs-mon_assoc,
vs-point_wf,
true_wf,
squash_wf,
equal_wf,
vs-lift-append,
vector-space_wf,
rng_sig_wf,
vs-lift-zero-bfs,
permutation_wf,
list_wf,
equal-wf-base,
rng_wf,
bag_wf,
subtype_rel_self,
permutation-equiv,
quotient-member-eq,
list-subtype-bag,
list_induction,
vs-zero-add,
list_accum_nil_lemma,
map_nil_lemma,
map_wf,
rng_times_wf,
cons_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
map_cons_lemma,
single-bag_wf,
list_accum_cons_lemma,
vs-mul_wf,
rng_times_over_plus,
vs-mul-add,
vs-ac_1
Rules used in proof :
productEquality,
functionExtensionality,
cumulativity,
functionEquality,
unionElimination,
independent_pairEquality,
axiomEquality,
isect_memberEquality,
rename,
setElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
because_Cache,
universeEquality,
equalitySymmetry,
hypothesis,
equalityTransitivity,
hypothesisEquality,
isectElimination,
extract_by_obid,
imageElimination,
sqequalHypSubstitution,
lambdaEquality,
thin,
applyEquality,
sqequalRule,
independent_pairFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
dependent_functionElimination,
pertypeElimination,
pointwiseFunctionalityForEquality,
applyLambdaEquality,
hyp_replacement,
lambdaFormation,
voidEquality,
voidElimination,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[K:Rng]. \mforall{}[S:Type]. \mforall{}[as,bs:basic-formal-sum(K;S)].
\mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. (vs-lift(vs;f;as) = vs-lift(vs;f;bs))
supposing bfs-reduce(K;S;as;bs)
Date html generated:
2018_05_22-PM-09_44_57
Last ObjectModification:
2018_01_09-AM-11_04_22
Theory : linear!algebra
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