Nuprl Lemma : set-equal-cons

`∀[T:Type]`
`  ∀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]))`

Proof

Definitions occuring in Statement :  set-equal: `set-equal(T;x;y)` no_repeats: `no_repeats(T;l)` append: `as @ bs` cons: `[a / b]` list: `T List` uimplies: `b supposing a` 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]` uimplies: `b supposing a` member: `t ∈ T` implies: `P `` Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` prop: `ℙ` rev_implies: `P `` 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: `A 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

Latex:
\mforall{}[T:Type]
\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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