### Nuprl Lemma : list-diff_functionality

`∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs,cs:T List].`
`  as-bs = as-cs ∈ (T List) supposing ∀x:T. ((x ∈ as) `` ((x ∈ bs) `⇐⇒` (x ∈ cs)))`

Proof

Definitions occuring in Statement :  list-diff: `as-bs` l_member: `(x ∈ l)` list: `T List` deq: `EqDecider(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` 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` list-diff: `as-bs` all: `∀x:A. B[x]` implies: `P `` Q` prop: `ℙ` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` squash: `↓T` subtype_rel: `A ⊆r B` not: `¬A` false: `False` true: `True` guard: `{T}`
Lemmas referenced :  list-subtype l_member_wf filter_wf5 squash_wf true_wf bool_wf list_wf istype-universe subtype_rel_list iff_imp_equal_bool bnot_wf deq-member_wf istype-void iff_transitivity assert_wf not_wf iff_weakening_uiff assert_of_bnot assert-deq-member istype-assert
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis inhabitedIsType lambdaFormation_alt equalityIstype equalityTransitivity equalitySymmetry dependent_functionElimination independent_functionElimination sqequalRule functionIsType universeIsType productIsType because_Cache isect_memberEquality_alt axiomEquality isectIsTypeImplies applyEquality lambdaEquality_alt imageElimination setIsType instantiate universeEquality setEquality independent_isectElimination setElimination rename independent_pairFormation voidElimination productElimination promote_hyp natural_numberEquality imageMemberEquality baseClosed hyp_replacement dependent_set_memberEquality_alt applyLambdaEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[as,bs,cs:T  List].
as-bs  =  as-cs  supposing  \mforall{}x:T.  ((x  \mmember{}  as)  {}\mRightarrow{}  ((x  \mmember{}  bs)  \mLeftarrow{}{}\mRightarrow{}  (x  \mmember{}  cs)))

Date html generated: 2020_05_19-PM-09_52_33
Last ObjectModification: 2020_01_04-PM-08_00_00

Theory : decidable!equality

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