### Nuprl Lemma : bag-combine-size-bound

`∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[L:A List]. ∀[a:A].  #(f[a]) ≤ #(⋃a∈L.f[a]) supposing (a ∈ L)`

Proof

Definitions occuring in Statement :  bag-combine: `⋃x∈bs.f[x]` bag-size: `#(bs)` bag: `bag(T)` l_member: `(x ∈ l)` list: `T List` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` subtype_rel: `A ⊆r B` squash: `↓T` prop: `ℙ` so_apply: `x[s]` nat: `ℕ` so_lambda: `λ2x.t[x]` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` le: `A ≤ B` bag-size: `#(bs)` bag-sum: `bag-sum(ba;x.f[x])` less_than': `less_than'(a;b)` not: `¬A` false: `False` all: `∀x:A. B[x]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` or: `P ∨ Q` ge: `i ≥ j ` decidable: `Dec(P)` uiff: `uiff(P;Q)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` cons: `[a / b]` colength: `colength(L)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b`
Lemmas referenced :  list-subtype-bag le_wf squash_wf true_wf istype-int bag-size_wf bag-combine-size istype-nat subtype_rel_self iff_weakening_equal le_witness_for_triv l_member_wf list_wf bag_wf istype-universe istype-void istype-le list_induction nat_wf list_accum_wf list_accum_nil_lemma null_nil_lemma btrue_wf member-implies-null-eq-bfalse nil_wf btrue_neq_bfalse list_accum_cons_lemma cons_wf cons_member add_nat_wf nat_properties decidable__le add-is-int-iff full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf false_wf intformless_wf int_formula_prop_less_lemma ge_wf istype-less_than list-cases product_subtype_list colength-cons-not-zero colength_wf_list subtract-1-ge-0 subtype_base_sq set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf int_term_value_subtract_lemma add-swap add-commutes
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut hypothesisEquality applyEquality extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache independent_isectElimination lambdaEquality_alt universeIsType hypothesis sqequalRule imageElimination equalityTransitivity equalitySymmetry inhabitedIsType setElimination rename natural_numberEquality imageMemberEquality baseClosed instantiate universeEquality productElimination independent_functionElimination isect_memberEquality_alt isectIsTypeImplies functionIsType dependent_set_memberEquality_alt independent_pairFormation lambdaFormation_alt voidElimination equalityIstype dependent_functionElimination functionEquality intEquality addEquality Error :memTop,  unionElimination applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion approximateComputation dependent_pairFormation_alt int_eqEquality hyp_replacement intWeakElimination functionIsTypeImplies hypothesis_subsumption sqequalBase

Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[L:A  List].  \mforall{}[a:A].    \#(f[a])  \mleq{}  \#(\mcup{}a\mmember{}L.f[a])  supposing  (a  \mmember{}  L)

Date html generated: 2020_05_20-AM-08_01_40
Last ObjectModification: 2019_12_31-PM-06_30_47

Theory : bags

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