### Nuprl Lemma : single-valued-bag-is-rep

`∀[A:Type]. ∀[as:bag(A)].  ∀a:A. (a ↓∈ as `` (as = bag-rep(#(as);a) ∈ bag(A))) supposing single-valued-bag(as;A)`

Proof

Definitions occuring in Statement :  single-valued-bag: `single-valued-bag(b;T)` bag-member: `x ↓∈ bs` bag-rep: `bag-rep(n;x)` bag-size: `#(bs)` bag: `bag(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` implies: `P `` Q` squash: `↓T` exists: `∃x:A. B[x]` prop: `ℙ` bag-size: `#(bs)` subtype_rel: `A ⊆r B` top: `Top` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` and: `P ∧ Q` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` single-valued-list: `single-valued-list(L;T)`
Lemmas referenced :  bag_to_squash_list bag-member_wf single-valued-bag_wf bag-member-sq-list-member single-valued-bag-list list_extensionality bag-rep_wf length_wf_nat bag-size-rep bag-size_wf list-subtype-bag nat_wf equal_wf squash_wf true_wf select_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_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_formula_prop_wf select-bag-rep lelt_wf length_wf iff_weakening_equal select_member less_than_wf bag_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality imageElimination productElimination promote_hyp hypothesis equalitySymmetry hyp_replacement applyLambdaEquality cumulativity rename dependent_functionElimination independent_isectElimination sqequalRule applyEquality lambdaEquality isect_memberEquality voidElimination voidEquality setElimination equalityTransitivity natural_numberEquality unionElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll dependent_set_memberEquality imageMemberEquality baseClosed universeEquality independent_functionElimination axiomEquality

Latex:
\mforall{}[A:Type].  \mforall{}[as:bag(A)].
\mforall{}a:A.  (a  \mdownarrow{}\mmember{}  as  {}\mRightarrow{}  (as  =  bag-rep(\#(as);a)))  supposing  single-valued-bag(as;A)

Date html generated: 2017_10_01-AM-08_55_35
Last ObjectModification: 2017_07_26-PM-04_37_44

Theory : bags

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