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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