Nuprl Lemma : bag-summation_wf
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[f:T ⟶ R]. ∀[b:bag(T)].
Σ(x∈b). f[x] ∈ R supposing Assoc(R;add) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
comm: Comm(T;op),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag-summation: Σ(x∈b). f[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
prop: ℙ,
true: True,
infix_ap: x f y,
guard: {T},
assoc: Assoc(T;op),
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
comm: Comm(T;op)
Lemmas referenced :
bag-accum_wf,
assoc_wf,
comm_wf,
bag_wf,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
independent_isectElimination,
lambdaFormation,
hypothesis,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productEquality,
isect_memberEquality,
functionEquality,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[T,R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[f:T {}\mrightarrow{} R]. \mforall{}[b:bag(T)].
\mSigma{}(x\mmember{}b). f[x] \mmember{} R supposing Assoc(R;add) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-08_48_29
Last ObjectModification:
2017_07_26-PM-04_32_39
Theory : bags
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