Nuprl Lemma : set-equal-cons

  ∀u:T. ∀v,bs:T List.
    (set-equal(T;[u v];bs) ⇐⇒ ∃cs,ds:T List. ((bs (cs [u ds]) ∈ (T List)) ∧ set-equal(T;v;cs ds))) supposing 
       (no_repeats(T;bs) and 
       no_repeats(T;[u v]))


Definitions occuring in Statement :  set-equal: set-equal(T;x;y) no_repeats: no_repeats(T;l) append: as bs cons: [a b] list: List uimplies: supposing a uall: [x:A]. B[x] all: x:A. B[x] exists: x:A. B[x] iff: ⇐⇒ Q and: P ∧ Q universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] all: x:A. B[x] uimplies: supposing a member: t ∈ T implies:  Q iff: ⇐⇒ Q and: P ∧ Q prop: rev_implies:  Q so_lambda: λ2x.t[x] top: Top so_apply: x[s] set-equal: set-equal(T;x;y) or: P ∨ Q exists: x:A. B[x] append: as bs so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] cand: c∧ B guard: {T} uiff: uiff(P;Q) not: ¬A false: False
Lemmas referenced :  no_repeats_witness cons_wf set-equal_wf exists_wf list_wf equal_wf append_wf length_wf length-append no_repeats_wf cons_member l_member_wf l_member_decomp list_ind_cons_lemma list_ind_nil_lemma member_append or_wf iff_wf no_repeats_cons no_repeats-append l_disjoint_cons length_wf_nat nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality cumulativity hypothesis independent_functionElimination rename because_Cache independent_pairFormation sqequalRule lambdaEquality productEquality applyLambdaEquality isect_memberEquality voidElimination voidEquality universeEquality dependent_functionElimination productElimination inlFormation dependent_pairFormation equalitySymmetry hyp_replacement promote_hyp addLevel orFunctionality impliesFunctionality inrFormation unionElimination independent_isectElimination dependent_set_memberEquality setElimination

    \mforall{}u:T.  \mforall{}v,bs:T  List.
        (set-equal(T;[u  /  v];bs)
              \mLeftarrow{}{}\mRightarrow{}  \mexists{}cs,ds:T  List.  ((bs  =  (cs  @  [u  /  ds]))  \mwedge{}  set-equal(T;v;cs  @  ds)))  supposing 
              (no\_repeats(T;bs)  and 
              no\_repeats(T;[u  /  v]))

Date html generated: 2017_04_17-AM-07_37_04
Last ObjectModification: 2017_02_27-PM-04_12_23

Theory : list_1

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