Nuprl Lemma : vs-lift-unique
∀[S:Type]. ∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[f:S ⟶ Point(vs)]. ∀[h:free-vs(K;S) ⟶ vs].
h = (λx.vs-lift(vs;f;x)) ∈ free-vs(K;S) ⟶ vs supposing ∀s:S. ((h <s>) = (f s) ∈ Point(vs))
Proof
Definitions occuring in Statement :
free-vs-inc: <s>,
free-vs: free-vs(K;S),
vs-lift: vs-lift(vs;f;fs),
vs-map: A ⟶ B,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
vs-map: A ⟶ B,
and: P ∧ Q,
free-vs: free-vs(K;S),
vs-point: Point(vs),
mk-vs: mk-vs,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
basic-formal-sum: basic-formal-sum(K;S),
subtype_rel: A ⊆r B,
formal-sum: formal-sum(K;S),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-bag: {},
prop: ℙ,
squash: ↓T,
true: True,
vs-mul: a * x,
record-select: r.x,
record-update: r[x := v],
formal-sum-mul: k * x,
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
nil: [],
it: ⋅,
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
bag: bag(T),
nat: ℕ,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
or: P ∨ Q,
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
decidable: Dec(P),
vs-lift: vs-lift(vs;f;fs),
vs-bag-add: Σ{f[b] | b ∈ bs},
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
vs-add: x + y,
formal-sum-add: x + y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
free-vs-inc: <s>,
equiv_rel: EquivRel(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y])
Lemmas referenced :
vs-point_wf,
free-vs-inc_wf,
vs-map_wf,
free-vs_wf,
vector-space_wf,
crng_wf,
istype-universe,
vs-add_wf,
rng_car_wf,
vs-mul_wf,
rec_select_update_lemma,
istype-void,
empty-bag_wf,
subtype_quotient,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
rng_zero_wf,
equal_wf,
squash_wf,
true_wf,
vs-mul-zero,
list_wf,
permutation_wf,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
list-cases,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
list_accum_nil_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
single-bag_wf,
subtype_rel_self,
bag_qinc,
vs-lift-append,
list-subtype-bag,
iff_weakening_equal,
rng_sig_wf,
list_accum_cons_lemma,
bag_map_single_lemma,
rng_times_one,
vs-0_wf,
vs-mon_ident,
vs-lift_wf2,
formal-sum_wf,
quotient-member-eq,
equal_functionality_wrt_subtype_rel2,
quotient_wf,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
hypothesis,
sqequalRule,
functionIsType,
universeIsType,
hypothesisEquality,
equalityIstype,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
applyEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
dependent_functionElimination,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
productElimination,
functionExtensionality,
productIsType,
because_Cache,
voidElimination,
productEquality,
lambdaEquality_alt,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
pointwiseFunctionalityForEquality,
pertypeElimination,
promote_hyp,
sqequalBase,
lambdaFormation_alt,
independent_functionElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
applyLambdaEquality,
baseApply,
closedConclusion,
intEquality,
independent_pairEquality
Latex:
\mforall{}[S:Type]. \mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[f:S {}\mrightarrow{} Point(vs)]. \mforall{}[h:free-vs(K;S) {}\mrightarrow{} vs].
h = (\mlambda{}x.vs-lift(vs;f;x)) supposing \mforall{}s:S. ((h <s>) = (f s))
Date html generated:
2019_10_31-AM-06_29_39
Last ObjectModification:
2019_07_31-PM-04_20_27
Theory : linear!algebra
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