### Nuprl Lemma : l_member_decomp

`∀[T:Type]. ∀l:T List. ∀x:T.  ((x ∈ l) `⇐⇒` ∃l1,l2:T List. (l = (l1 @ [x] @ l2) ∈ (T List)))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` append: `as @ bs` cons: `[a / b]` nil: `[]` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` member: `t ∈ T` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` false: `False` prop: `ℙ` rev_implies: `P `` Q` exists: `∃x:A. B[x]` or: `P ∨ Q` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` cons: `[a / b]` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` true: `True` not: `¬A` uiff: `uiff(P;Q)` append: `as @ bs` squash: `↓T` cand: `A c∧ B` nat: `ℕ` ge: `i ≥ j ` decidable: `Dec(P)` sq_stable: `SqStable(P)` subtract: `n - m` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)`
Lemmas referenced :  list_induction all_wf iff_wf l_member_wf exists_wf list_wf equal_wf append_wf cons_wf nil_wf false_wf equal-wf-base-T nil_member or_wf cons_member equal-wf-T-base list-cases list_ind_nil_lemma product_subtype_list list_ind_cons_lemma subtype_base_sq int_subtype_base null_nil_lemma btrue_wf null_cons_lemma bfalse_wf and_wf null_wf btrue_neq_bfalse iff_transitivity iff_weakening_uiff append_is_nil squash_wf true_wf hd_wf length_of_nil_lemma cons_neq_nil length_of_cons_lemma length_wf_nat nat_wf decidable__le not-ge-2 sq_stable__le condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-associates add-commutes add_functionality_wrt_le add-zero le-add-cancel2 reduce_hd_cons_lemma tl_wf reduce_tl_cons_lemma member_append
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality sqequalRule lambdaEquality cumulativity hypothesis independent_functionElimination independent_pairFormation voidElimination productElimination baseClosed because_Cache addLevel allFunctionality impliesFunctionality dependent_functionElimination rename universeEquality equalitySymmetry equalityTransitivity productEquality applyLambdaEquality unionElimination isect_memberEquality voidEquality natural_numberEquality promote_hyp hypothesis_subsumption instantiate intEquality independent_isectElimination dependent_set_memberEquality setElimination dependent_pairFormation applyEquality imageElimination imageMemberEquality addEquality minusEquality inlFormation inrFormation hyp_replacement

Latex:
\mforall{}[T:Type].  \mforall{}l:T  List.  \mforall{}x:T.    ((x  \mmember{}  l)  \mLeftarrow{}{}\mRightarrow{}  \mexists{}l1,l2:T  List.  (l  =  (l1  @  [x]  @  l2)))

Date html generated: 2017_04_14-AM-08_41_14
Last ObjectModification: 2017_02_27-PM-03_31_54

Theory : list_0

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