### Nuprl Lemma : bag-member-iff-hd

`∀[T:Type]. ∀[bs:bag(T)]. ∀[x:T].  uiff(x ↓∈ bs;↓∃L:T List. (bs = [x / L] ∈ bag(T)))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag: `bag(T)` cons: `[a / b]` list: `T List` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` squash: `↓T` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` squash: `↓T` prop: `ℙ` bag-member: `x ↓∈ bs` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` so_apply: `x[s]` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` true: `True` bag: `bag(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` rev_implies: `P `` Q` cand: `A c∧ B` or: `P ∨ Q`
Lemmas referenced :  bag-member_wf squash_wf exists_wf list_wf equal_wf bag_wf cons_wf list-subtype-bag l_member_decomp append_wf true_wf quotient-member-eq permutation_wf permutation-equiv nil_wf list_ind_cons_lemma list_ind_nil_lemma permutation_functionality_wrt_permutation cons_functionality_wrt_permutation permutation-rotate permutation_weakening cons_member l_member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation hypothesis sqequalHypSubstitution imageElimination sqequalRule imageMemberEquality hypothesisEquality thin baseClosed extract_by_obid isectElimination cumulativity lambdaEquality applyEquality because_Cache independent_isectElimination productElimination independent_pairEquality isect_memberEquality equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination independent_functionElimination dependent_pairFormation hyp_replacement natural_numberEquality applyLambdaEquality voidElimination voidEquality inlFormation productEquality

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].  \mforall{}[x:T].    uiff(x  \mdownarrow{}\mmember{}  bs;\mdownarrow{}\mexists{}L:T  List.  (bs  =  [x  /  L]))

Date html generated: 2017_10_01-AM-08_53_49
Last ObjectModification: 2017_07_26-PM-04_35_30

Theory : bags

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