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

`∀[T:Type]. ∀[bs:bag(T)].  uiff(↓∃x:T. x ↓∈ bs;0 < #(bs))`

Proof

Definitions occuring in Statement :  bag-member: `x ↓∈ bs` bag-size: `#(bs)` bag: `bag(T)` less_than: `a < b` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` exists: `∃x:A. B[x]` squash: `↓T` natural_number: `\$n` universe: `Type`
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` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` nat: `ℕ` exists: `∃x:A. B[x]` bag-member: `x ↓∈ bs` bag-size: `#(bs)` all: `∀x:A. B[x]` or: `P ∨ Q` iff: `P `⇐⇒` Q` implies: `P `` Q` false: `False` cons: `[a / b]` top: `Top` guard: `{T}` le: `A ≤ B` decidable: `Dec(P)` not: `¬A` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` bag: `bag(T)` quotient: `x,y:A//B[x; y]` less_than: `a < b` length: `||as||` list_ind: list_ind nil: `[]` it: `⋅` cand: `A c∧ B`
Lemmas referenced :  squash_wf exists_wf bag-member_wf less_than_wf bag-size_wf nat_wf member-less_than bag_wf list-cases length_of_nil_lemma nil_member product_subtype_list length_of_cons_lemma length_wf_nat decidable__lt false_wf not-lt-2 condition-implies-le minus-add minus-one-mul zero-add minus-one-mul-top add-commutes add_functionality_wrt_le add-associates add-zero le-add-cancel equal_wf equal-wf-base list_wf permutation_wf length_wf list-subtype-bag cons_wf l_member_wf cons_member
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut independent_pairFormation sqequalHypSubstitution imageElimination hypothesis extract_by_obid isectElimination thin cumulativity hypothesisEquality sqequalRule lambdaEquality imageMemberEquality baseClosed natural_numberEquality applyEquality setElimination rename productElimination independent_pairEquality isect_memberEquality because_Cache independent_isectElimination equalityTransitivity equalitySymmetry universeEquality dependent_functionElimination unionElimination independent_functionElimination voidElimination promote_hyp hypothesis_subsumption voidEquality lambdaFormation addEquality intEquality minusEquality hyp_replacement applyLambdaEquality pointwiseFunctionalityForEquality functionEquality pertypeElimination productEquality dependent_pairFormation inlFormation

Latex:
\mforall{}[T:Type].  \mforall{}[bs:bag(T)].    uiff(\mdownarrow{}\mexists{}x:T.  x  \mdownarrow{}\mmember{}  bs;0  <  \#(bs))

Date html generated: 2017_10_01-AM-08_53_21
Last ObjectModification: 2017_07_26-PM-04_35_04

Theory : bags

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