### Nuprl Lemma : bag-double-summation

`∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].`
`  ∀[T,A,B:Type]. ∀[f:T ⟶ bag(A)]. ∀[g:T ⟶ B]. ∀[h:B ⟶ A ⟶ R]. ∀[b:bag(T)].`
`    (Σ(x∈b). Σ(y∈f[x]). h[g[x];y] = Σ(p∈⋃x∈b.bag-map(λy.<g[x], y>;f[x])). h[fst(p);snd(p)] ∈ R) `
`  supposing IsMonoid(R;add;zero) ∧ Comm(R;add)`

Proof

Definitions occuring in Statement :  bag-summation: `Σ(x∈b). f[x]` bag-combine: `⋃x∈bs.f[x]` bag-map: `bag-map(f;bs)` bag: `bag(T)` comm: `Comm(T;op)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` pi1: `fst(t)` pi2: `snd(t)` and: `P ∧ Q` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` pair: `<a, b>` universe: `Type` equal: `s = t ∈ T` monoid_p: `IsMonoid(T;op;id)`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` bag: `bag(T)` quotient: `x,y:A//B[x; y]` all: `∀x:A. B[x]` implies: `P `` Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` nat: `ℕ` false: `False` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` prop: `ℙ` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` empty-bag: `{}` cons: `[a / b]` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` int_iseg: `{i...j}` cand: `A c∧ B` single-bag: `{x}` bag-append: `as + bs` pi1: `fst(t)` pi2: `snd(t)` true: `True` infix_ap: `x f y` monoid_p: `IsMonoid(T;op;id)`
Lemmas referenced :  quotient-member-eq list_wf permutation_wf permutation-equiv nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 int_seg_properties int_seg_wf subtract-1-ge-0 decidable__equal_int subtract_wf subtype_base_sq set_subtype_base lelt_wf int_subtype_base intformnot_wf intformeq_wf itermSubtract_wf int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__le decidable__lt istype-le subtype_rel_self non_neg_length length_wf decidable__assert null_wf list-cases product_subtype_list null_cons_lemma last-lemma-sq pos_length iff_transitivity not_wf equal-wf-T-base assert_wf bnot_wf iff_weakening_uiff assert_of_null istype-assert nil_wf length_of_nil_lemma assert_of_bnot firstn_wf length_firstn itermAdd_wf int_term_value_add_lemma istype-nat length_wf_nat equal_wf bag-summation_wf bag-combine_wf list-subtype-bag bag-map_wf bag_wf monoid_p_wf comm_wf bag-append_wf single-bag_wf last_wf pi1_wf pi2_wf infix_ap_wf bag-summation-empty bag-combine-empty-left squash_wf true_wf bag-combine-append-left iff_weakening_equal bag-summation-append bag-subtype-list bag-summation-map istype-universe bag-summation-single bag-combine-single-left
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution productElimination thin pointwiseFunctionalityForEquality hypothesisEquality hypothesis sqequalRule pertypeElimination promote_hyp equalityTransitivity equalitySymmetry inhabitedIsType lambdaFormation_alt rename extract_by_obid isectElimination lambdaEquality_alt independent_isectElimination dependent_functionElimination independent_functionElimination setElimination intWeakElimination natural_numberEquality approximateComputation dependent_pairFormation_alt int_eqEquality isect_memberEquality_alt voidElimination independent_pairFormation universeIsType axiomEquality functionIsTypeImplies unionElimination applyEquality instantiate cumulativity intEquality applyLambdaEquality dependent_set_memberEquality_alt because_Cache productIsType hypothesis_subsumption imageElimination baseClosed functionIsType equalityIstype addEquality hyp_replacement productEquality independent_pairEquality sqequalBase isectIsTypeImplies closedConclusion imageMemberEquality universeEquality

Latex:
\mforall{}[R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].
\mforall{}[T,A,B:Type].  \mforall{}[f:T  {}\mrightarrow{}  bag(A)].  \mforall{}[g:T  {}\mrightarrow{}  B].  \mforall{}[h:B  {}\mrightarrow{}  A  {}\mrightarrow{}  R].  \mforall{}[b:bag(T)].
(\mSigma{}(x\mmember{}b).  \mSigma{}(y\mmember{}f[x]).  h[g[x];y]  =  \mSigma{}(p\mmember{}\mcup{}x\mmember{}b.bag-map(\mlambda{}y.<g[x],  y>f[x])).  h[fst(p);snd(p)])
supposing  IsMonoid(R;add;zero)  \mwedge{}  Comm(R;add)

Date html generated: 2019_10_15-AM-11_00_57
Last ObjectModification: 2019_08_08-PM-06_16_03

Theory : bags

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