Nuprl Lemma : bag-double-summation1

[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
  ∀[T:Type]. ∀[A:T ⟶ Type]. ∀[f:x:T ⟶ bag(A[x])]. ∀[h:x:T ⟶ A[x] ⟶ R]. ∀[b:bag(T)].
    (x∈b). Σ(y∈f[x]). h[x;y] = Σ(p∈⋃x∈b.bag-map(λy.<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: 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: t ∈ T monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a and: P ∧ Q bag: bag(T) quotient: x,y:A//B[x; y] all: x:A. B[x] implies:  Q so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] nat: false: False ge: i ≥  not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top prop: guard: {T} int_seg: {i..j-} lelt: i ≤ j < k le: A ≤ B decidable: Dec(P) or: P ∨ Q subtype_rel: A ⊆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 then else fi  bfalse: ff iff: ⇐⇒ Q rev_implies:  Q uiff: uiff(P;Q) int_iseg: {i...j} cand: c∧ B single-bag: {x} bag-append: as bs true: True monoid_p: IsMonoid(T;op;id) pi1: fst(t) pi2: snd(t) infix_ap: y
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 list-subtype-bag equal_wf squash_wf true_wf bag-summation-append bag-summation_wf single-bag_wf last_wf pi1_wf pi2_wf bag-combine_wf bag-append_wf bag-map_wf iff_weakening_equal itermAdd_wf int_term_value_add_lemma istype-nat length_wf_nat bag_wf monoid_p_wf comm_wf bag-summation-empty bag-combine-empty-left bag-combine-append-left istype-universe infix_ap_wf bag-combine-single-left bag-summation-map bag-subtype-list bag-summation-single
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 productEquality dependent_pairEquality_alt imageMemberEquality addEquality hyp_replacement sqequalBase isectIsTypeImplies universeEquality

Latex:
\mforall{}[R:Type].  \mforall{}[add:R  {}\mrightarrow{}  R  {}\mrightarrow{}  R].  \mforall{}[zero:R].
    \mforall{}[T:Type].  \mforall{}[A:T  {}\mrightarrow{}  Type].  \mforall{}[f:x:T  {}\mrightarrow{}  bag(A[x])].  \mforall{}[h:x:T  {}\mrightarrow{}  A[x]  {}\mrightarrow{}  R].  \mforall{}[b:bag(T)].
        (\mSigma{}(x\mmember{}b).  \mSigma{}(y\mmember{}f[x]).  h[x;y]  =  \mSigma{}(p\mmember{}\mcup{}x\mmember{}b.bag-map(\mlambda{}y.<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_50
Last ObjectModification: 2019_08_08-PM-05_58_55

Theory : bags


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