### Nuprl Lemma : empty-bag-union

`∀T:Type. ∀bs:bag(bag(T)).  ((bag-union(bs) = {} ∈ bag(T)) `` (bs ~ bag-rep(#(bs);{})))`

Proof

Definitions occuring in Statement :  bag-rep: `bag-rep(n;x)` bag-union: `bag-union(bbs)` bag-size: `#(bs)` empty-bag: `{}` bag: `bag(T)` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type` sqequal: `s ~ t` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` bag-rep: `bag-rep(n;x)` concat: `concat(ll)` cons-bag: `x.b` empty-bag: `{}` nat: `ℕ` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` subtype_rel: `A ⊆r B` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` le: `A ≤ B` bfalse: `ff` bnot: `¬bb` assert: `↑b` bag-size: `#(bs)` bag: `bag(T)` quotient: `x,y:A//B[x; y]` true: `True` bag-union: `bag-union(bbs)` bag-append: `as + bs` iff: `P `⇐⇒` Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` rev_implies: `P `` Q`
Lemmas referenced :  equal-wf-T-base bag_wf bag-union_wf nil_wf top_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma 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 less_than_wf list_wf reduce_wf append_wf nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases reduce_nil_lemma length_of_nil_lemma primrec0_lemma equal-wf-base product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf equal_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int reduce_cons_lemma length_of_cons_lemma primrec-unroll length_wf eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int non_neg_length eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int append_is_nil null_nil_lemma btrue_wf and_wf null_wf null_cons_lemma bfalse_wf btrue_neq_bfalse add-subtract-cancel permutation_wf permutation_weakening subtype_rel_list bag-subtype-list member_wf squash_wf true_wf permutation-length list_extensionality l_member_wf bag-append_wf empty-bag_wf member-implies-null-eq-bfalse cons_wf bag-append-is-empty cons_member equal-empty-bag equal-nil-sq-nil select_wf select_member lelt_wf decidable__lt member-permutation l_member_subtype
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut hypothesis introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality baseClosed universeEquality sqequalRule setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination sqequalAxiom applyEquality because_Cache unionElimination promote_hyp hypothesis_subsumption productElimination equalityTransitivity equalitySymmetry applyLambdaEquality dependent_set_memberEquality addEquality instantiate imageElimination equalityElimination pointwiseFunctionality pertypeElimination productEquality imageMemberEquality comment hyp_replacement

Latex:
\mforall{}T:Type.  \mforall{}bs:bag(bag(T)).    ((bag-union(bs)  =  \{\})  {}\mRightarrow{}  (bs  \msim{}  bag-rep(\#(bs);\{\})))

Date html generated: 2017_10_01-AM-08_52_10
Last ObjectModification: 2017_07_26-PM-04_33_50

Theory : bags

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