### Nuprl Lemma : cycle-append

`∀[n:ℕ]. ∀[as,bs:ℕn List].  cycle(as @ bs) = cycle(bs @ as) ∈ (ℕn ⟶ ℕn) supposing no_repeats(ℕn;as @ bs)`

Proof

Definitions occuring in Statement :  cycle: `cycle(L)` no_repeats: `no_repeats(T;l)` append: `as @ bs` list: `T List` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` decidable: `Dec(P)` or: `P ∨ Q` l_member: `(x ∈ l)` exists: `∃x:A. B[x]` cand: `A c∧ B` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` top: `Top` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` sq_type: `SQType(T)` less_than: `a < b` no_repeats: `no_repeats(T;l)` subtract: `n - m` le: `A ≤ B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` less_than': `less_than'(a;b)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` cons: `[a / b]` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` nil: `[]` select: `L[n]` so_apply: `x[s1;s2;s3]` so_lambda: `so_lambda(x,y,z.t[x; y; z])` append: `as @ bs` assert: `↑b` bnot: `¬bb` nequal: `a ≠ b ∈ T `
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut Error :functionExtensionality_alt,  Error :universeIsType,  extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename because_Cache hypothesis hypothesisEquality sqequalRule Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  dependent_functionElimination independent_functionElimination Error :lambdaFormation_alt,  unionElimination productElimination applyEquality Error :lambdaEquality_alt,  imageElimination equalityTransitivity equalitySymmetry instantiate universeEquality imageMemberEquality baseClosed independent_isectElimination closedConclusion independent_pairFormation Error :dependent_set_memberEquality_alt,  voidElimination applyLambdaEquality approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality Error :productIsType,  cumulativity intEquality addEquality computeAll lambdaEquality dependent_pairFormation baseApply promote_hyp pointwiseFunctionality voidEquality isect_memberEquality lambdaFormation dependent_set_memberEquality Error :equalityIsType4,  Error :functionIsType,  equalityElimination Error :equalityIsType1,  hypothesis_subsumption minusEquality multiplyEquality comment impliesFunctionality productEquality inlFormation inrFormation

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[as,bs:\mBbbN{}n  List].    cycle(as  @  bs)  =  cycle(bs  @  as)  supposing  no\_repeats(\mBbbN{}n;as  @  bs)

Date html generated: 2019_06_20-PM-01_41_06
Last ObjectModification: 2018_10_18-AM-11_47_44

Theory : list_1

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