### Nuprl Lemma : sv-bag-only-append

`∀[A:Type]. ∀[as,bs:bag(A)].`
`  (sv-bag-only(as + bs) = sv-bag-only(as) ∈ A) supposing `
`     (similar-bags(A;as;bs) and `
`     0 < #(as) and `
`     single-valued-bag(as;A) and `
`     single-valued-bag(bs;A))`

Proof

Definitions occuring in Statement :  sv-bag-only: `sv-bag-only(b)` single-valued-bag: `single-valued-bag(b;T)` similar-bags: `similar-bags(A;as;bs)` bag-size: `#(bs)` bag-append: `as + bs` bag: `bag(T)` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` squash: `↓T` exists: `∃x:A. B[x]` implies: `P `` Q` prop: `ℙ` subtype_rel: `A ⊆r B` nat: `ℕ` true: `True` and: `P ∧ Q` cand: `A c∧ B` top: `Top` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` sv-bag-only: `sv-bag-only(b)` bag-append: `as + bs` bag-size: `#(bs)` listp: `A List+` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  bag_to_squash_list similar-bags_wf less_than_wf bag-size_wf single-valued-bag_wf bag_wf nat_wf single-valued-bag-append bag-size-append decidable__lt nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf intformless_wf itermAdd_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_add_lemma int_formula_prop_wf hd-append-sq subtype_rel_list top_wf length_wf hd_wf listp_properties list-subtype-bag squash_wf true_wf equal_wf sv-bag-only_wf bag-append_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality imageElimination productElimination independent_functionElimination hypothesis cumulativity sqequalRule isect_memberEquality axiomEquality because_Cache equalityTransitivity equalitySymmetry natural_numberEquality applyEquality lambdaEquality setElimination rename universeEquality independent_isectElimination independent_pairFormation voidElimination voidEquality dependent_functionElimination addEquality unionElimination applyLambdaEquality dependent_pairFormation int_eqEquality intEquality computeAll lambdaFormation dependent_set_memberEquality functionEquality imageMemberEquality baseClosed productEquality

Latex:
\mforall{}[A:Type].  \mforall{}[as,bs:bag(A)].
(sv-bag-only(as  +  bs)  =  sv-bag-only(as))  supposing
(similar-bags(A;as;bs)  and
0  <  \#(as)  and
single-valued-bag(as;A)  and
single-valued-bag(bs;A))

Date html generated: 2017_10_01-AM-08_55_52
Last ObjectModification: 2017_07_26-PM-04_37_57

Theory : bags

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