### Nuprl Lemma : bag-drop-property

`∀[T:Type]`
`  ∀eq:EqDecider(T). ∀x:T. ∀bs:bag(T).`
`    ((bs = ({x} + bag-drop(eq;bs;x)) ∈ bag(T)) ∨ ((¬x ↓∈ bs) ∧ (bs = bag-drop(eq;bs;x) ∈ bag(T))))`

Proof

Definitions occuring in Statement :  bag-drop: `bag-drop(eq;bs;a)` bag-member: `x ↓∈ bs` bag-append: `as + bs` single-bag: `{x}` bag: `bag(T)` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` not: `¬A` or: `P ∨ Q` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` bag-drop: `bag-drop(eq;bs;a)` implies: `P `` Q` or: `P ∨ Q` exists: `∃x:A. B[x]` and: `P ∧ Q` outl: `outl(x)` prop: `ℙ` uimplies: `b supposing a` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` true: `True` not: `¬A` false: `False` guard: `{T}` cand: `A c∧ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  bag-remove1-property bag-remove1_wf bag_wf unit_wf2 and_wf equal_wf outl_wf assert_wf isl_wf bag-append_wf single-bag_wf btrue_wf bfalse_wf btrue_neq_bfalse not_wf bag-member_wf or_wf exists_wf equal-wf-T-base deq_wf
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation hypothesis sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaFormation dependent_functionElimination unionEquality unionElimination inlFormation productElimination sqequalRule equalitySymmetry dependent_set_memberEquality independent_pairFormation equalityTransitivity applyLambdaEquality setElimination rename independent_isectElimination promote_hyp hyp_replacement natural_numberEquality independent_functionElimination voidElimination productEquality inrFormation lambdaEquality inlEquality baseClosed universeEquality

Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T).  \mforall{}x:T.  \mforall{}bs:bag(T).
((bs  =  (\{x\}  +  bag-drop(eq;bs;x)))  \mvee{}  ((\mneg{}x  \mdownarrow{}\mmember{}  bs)  \mwedge{}  (bs  =  bag-drop(eq;bs;x))))

Date html generated: 2019_10_16-AM-11_31_19
Last ObjectModification: 2018_08_21-PM-01_59_20

Theory : bags_2

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