### Nuprl Lemma : cycle-transitive

`∀n:ℕ. ∀L:ℕn List.  ∀a,b:ℕ||L||.  ∃m:ℕ||L||. ((cycle(L)^m L[a]) = L[b] ∈ ℕn) supposing no_repeats(ℕn;L)`

Proof

Definitions occuring in Statement :  cycle: `cycle(L)` no_repeats: `no_repeats(T;l)` select: `L[n]` length: `||as||` list: `T List` fun_exp: `f^n` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` apply: `f a` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` implies: `P `` Q` int_seg: `{i..j-}` decidable: `Dec(P)` or: `P ∨ Q` prop: `ℙ` exists: `∃x:A. B[x]` lelt: `i ≤ j < k` and: `P ∧ Q` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` le: `A ≤ B` subtype_rel: `A ⊆r B` less_than': `less_than'(a;b)` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` eq_int: `(i =z j)` subtract: `n - m` ifthenelse: `if b then t else f fi ` bfalse: `ff` compose: `f o g` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  no_repeats_witness int_seg_wf decidable__le length_wf no_repeats_wf list_wf nat_wf subtract_wf int_seg_properties nat_properties satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf non_neg_length fun_exp_wf le_wf cycle_wf select_wf length_wf_nat equal_wf squash_wf true_wf cycle-transitive1 int_seg_subtype_nat false_wf iff_weakening_equal fun_exp_unroll fun_exp0_lemma apply-cycle-member eq_int_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int decidable__equal_int intformeq_wf int_formula_prop_eq_lemma itermAdd_wf int_term_value_add_lemma fun_exp_add_apply set_subtype_base int_subtype_base
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation isect_memberFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality hypothesis independent_functionElimination dependent_functionElimination because_Cache unionElimination dependent_pairFormation dependent_set_memberEquality independent_pairFormation productElimination independent_isectElimination lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule computeAll imageElimination applyEquality equalityTransitivity equalitySymmetry applyLambdaEquality universeEquality imageMemberEquality baseClosed equalityElimination promote_hyp instantiate cumulativity addEquality

Latex:
\mforall{}n:\mBbbN{}.  \mforall{}L:\mBbbN{}n  List.    \mforall{}a,b:\mBbbN{}||L||.    \mexists{}m:\mBbbN{}||L||.  ((cycle(L)\^{}m  L[a])  =  L[b])  supposing  no\_repeats(\mBbbN{}n;L)

Date html generated: 2017_04_17-AM-08_19_02
Last ObjectModification: 2017_02_27-PM-04_43_45

Theory : list_1

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