### Nuprl Lemma : permutation-cons

`∀[A:Type]`
`  ∀x:A. ∀L1,L2:A List.`
`    (permutation(A;[x / L1];L2) `⇐⇒` ∃as,bs:A List. ((L2 = (as @ [x / bs]) ∈ (A List)) ∧ permutation(A;L1;as @ bs)))`

Proof

Definitions occuring in Statement :  permutation: `permutation(T;L1;L2)` append: `as @ bs` cons: `[a / b]` list: `T List` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` rev_implies: `P `` Q` exists: `∃x:A. B[x]` so_lambda: `λ2x.t[x]` top: `Top` so_apply: `x[s]` l_contains: `A ⊆ B` l_all: `(∀x∈L.P[x])` int_seg: `{i..j-}` lelt: `i ≤ j < k` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` nat_plus: `ℕ+` less_than: `a < b` squash: `↓T` true: `True` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` uiff: `uiff(P;Q)` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` select: `L[n]` cons: `[a / b]` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` cand: `A c∧ B`
Lemmas referenced :  permutation_wf cons_wf exists_wf list_wf equal_wf append_wf length_wf length-append permutation_inversion permutation-contains length_of_cons_lemma false_wf add_nat_plus length_wf_nat less_than_wf nat_plus_wf nat_plus_properties decidable__lt add-is-int-iff satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf itermAdd_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_term_value_add_lemma int_formula_prop_eq_lemma int_formula_prop_wf lelt_wf l_member_decomp list_ind_cons_lemma list_ind_nil_lemma permutation_transitivity permutation-rotate nil_wf permutation_weakening append_functionality_wrt_permutation cons_cancel_wrt_permutation
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation independent_pairFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin cumulativity hypothesisEquality hypothesis productElimination sqequalRule lambdaEquality productEquality applyLambdaEquality isect_memberEquality voidElimination voidEquality universeEquality dependent_functionElimination independent_functionElimination because_Cache dependent_set_memberEquality natural_numberEquality imageMemberEquality baseClosed equalityTransitivity equalitySymmetry setElimination rename unionElimination pointwiseFunctionality promote_hyp baseApply closedConclusion independent_isectElimination dependent_pairFormation int_eqEquality intEquality computeAll addEquality hyp_replacement

Latex:
\mforall{}[A:Type]
\mforall{}x:A.  \mforall{}L1,L2:A  List.
(permutation(A;[x  /  L1];L2)
\mLeftarrow{}{}\mRightarrow{}  \mexists{}as,bs:A  List.  ((L2  =  (as  @  [x  /  bs]))  \mwedge{}  permutation(A;L1;as  @  bs)))

Date html generated: 2017_04_17-AM-08_23_42
Last ObjectModification: 2017_02_27-PM-04_45_25

Theory : list_1

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