### Nuprl Lemma : bag-summation-single-non-zero-no-repeats

[T,R:Type]. ∀[eq:EqDecider(T)]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f:T ⟶ R].
∀z:T
(x∈b). f[x] f[z] ∈ R) supposing
((bag-no-repeats(T;b) ∧ z ↓∈ b) and
(∀x:T. (x ↓∈  ((x z ∈ T) ∨ (f[x] zero ∈ R)))))

Proof

Definitions occuring in Statement :  bag-member: x ↓∈ bs bag-no-repeats: bag-no-repeats(T;bs) bag-summation: Σ(x∈b). f[x] bag: bag(T) deq: EqDecider(T) comm: Comm(T;op) uimplies: supposing a uall: [x:A]. B[x] so_apply: x[s] all: x:A. B[x] implies:  Q or: P ∨ Q and: P ∧ Q function: x:A ⟶ B[x] universe: Type equal: t ∈ T monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] and: P ∧ Q squash: T prop: so_apply: x[s] true: True subtype_rel: A ⊆B guard: {T} iff: ⇐⇒ Q rev_implies:  Q implies:  Q so_lambda: λ2x.t[x] deq: EqDecider(T) cand: c∧ B uiff: uiff(P;Q) bag-member: x ↓∈ bs or: P ∨ Q monoid_p: IsMonoid(T;op;id) eqof: eqof(d) rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :  bag-summation-single-non-zero equal_wf squash_wf true_wf iff_weakening_equal bag-extensionality-no-repeats decidable-equal-deq bag-filter_wf subtype_rel_bag assert_wf single-bag_wf bag-single-no-repeats bag-member-single bag-member_wf bag-summation-single bag-summation_wf bag-no-repeats_wf all_wf or_wf monoid_p_wf comm_wf bag_wf deq_wf bag-filter-no-repeats bag-member-filter safe-assert-deq and_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination lambdaFormation dependent_functionElimination productElimination applyEquality lambdaEquality imageElimination equalityTransitivity equalitySymmetry universeEquality functionExtensionality cumulativity natural_numberEquality sqequalRule imageMemberEquality baseClosed independent_functionElimination because_Cache setElimination rename setEquality independent_pairFormation hyp_replacement applyLambdaEquality productEquality isect_memberEquality axiomEquality functionEquality dependent_set_memberEquality

Latex:
\mforall{}[T,R:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].  \mforall{}[b:bag(T)].  \mforall{}[f:T  {}\mrightarrow{}  R].
\mforall{}z:T
(\mSigma{}(x\mmember{}b).  f[x]  =  f[z])  supposing
((bag-no-repeats(T;b)  \mwedge{}  z  \mdownarrow{}\mmember{}  b)  and
(\mforall{}x:T.  (x  \mdownarrow{}\mmember{}  b  {}\mRightarrow{}  ((x  =  z)  \mvee{}  (f[x]  =  zero)))))