Nuprl Lemma : bag-summation-reindex
∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
∀[T,A:Type]. ∀[g:T ⟶ A]. ∀[h:A ⟶ T]. ∀[f:T ⟶ R].
∀[b:bag(T)]. (Σ(x∈b). f[x] = Σ(x∈bag-map(g;b)). f[h x] ∈ R) supposing ∀x:T. (x = (h (g x)) ∈ T)
supposing Comm(R;add) ∧ Assoc(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-map: bag-map(f;bs),
bag: bag(T),
comm: Comm(T;op),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
top: Top,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
uimplies: b supposing a,
cand: A c∧ B,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
bag-summation-map,
bag-subtype-list,
bag-summation_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag_wf,
all_wf,
comm_wf,
assoc_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
hypothesisEquality,
applyEquality,
dependent_functionElimination,
hypothesis,
lambdaEquality,
imageElimination,
because_Cache,
independent_isectElimination,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
functionExtensionality,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
functionEquality,
productEquality,
isect_memberFormation,
axiomEquality
Latex:
\mforall{}[R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R].
\mforall{}[T,A:Type]. \mforall{}[g:T {}\mrightarrow{} A]. \mforall{}[h:A {}\mrightarrow{} T]. \mforall{}[f:T {}\mrightarrow{} R].
\mforall{}[b:bag(T)]. (\mSigma{}(x\mmember{}b). f[x] = \mSigma{}(x\mmember{}bag-map(g;b)). f[h x]) supposing \mforall{}x:T. (x = (h (g x)))
supposing Comm(R;add) \mwedge{} Assoc(R;add)
Date html generated:
2017_10_01-AM-08_51_12
Last ObjectModification:
2017_07_26-PM-04_33_10
Theory : bags
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