Nuprl Lemma : bag-summation-hom
∀[r,s:Rng]. ∀[f:|r| ⟶ |s|].
∀[A:Type]. ∀[g:A ⟶ |r|]. ∀[b:bag(A)]. (Σ(x∈b). f[g[x]] = f[Σ(x∈b). g[x]] ∈ |s|) supposing rng_hom_p(r;s;f)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
rng_hom_p: rng_hom_p(r;s;f),
rng: Rng,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
monoid_p: IsMonoid(T;op;id),
squash: ↓T,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
rng: Rng,
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
implies: P ⇒ Q,
empty-bag: {},
top: Top,
all: ∀x:A. B[x],
cons-bag: x.b,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rng_hom_p: rng_hom_p(r;s;f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
infix_ap: x f y
Lemmas referenced :
rng_all_properties,
bag_to_squash_list,
list_induction,
equal_wf,
rng_car_wf,
bag-summation_wf,
rng_plus_wf,
rng_zero_wf,
list-subtype-bag,
rng_plus_comm2,
list_wf,
bag-summation-empty,
bag_wf,
rng_hom_p_wf,
rng_wf,
squash_wf,
true_wf,
bag-summation-cons,
iff_weakening_equal,
rng_hom_zero,
infix_ap_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
hypothesis,
because_Cache,
promote_hyp,
imageElimination,
rename,
sqequalRule,
lambdaEquality,
setElimination,
cumulativity,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
axiomEquality,
functionEquality,
universeEquality,
equalityTransitivity,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[r,s:Rng]. \mforall{}[f:|r| {}\mrightarrow{} |s|].
\mforall{}[A:Type]. \mforall{}[g:A {}\mrightarrow{} |r|]. \mforall{}[b:bag(A)]. (\mSigma{}(x\mmember{}b). f[g[x]] = f[\mSigma{}(x\mmember{}b). g[x]])
supposing rng\_hom\_p(r;s;f)
Date html generated:
2017_10_01-AM-08_51_28
Last ObjectModification:
2017_07_26-PM-04_33_19
Theory : bags
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