Nuprl Lemma : orbit-exists
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ finite-type(T)
⇒ (∀f:T ⟶ T. ∀a:T.
∃L:T List
(no_repeats(T;L) ∧ (∀i:ℕ||L||. (L[i] = (f^i a) ∈ T)) ∧ (∀b:T. ((b ∈ L) ⇐⇒ ∃n:ℕ. (b = (f^n a) ∈ T))))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
prop: ℙ,
so_apply: x[s],
finite-type: finite-type(T),
top: Top,
surject: Surj(A;B;f),
pi1: fst(t),
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
label: ...$L... t,
sq_type: SQType(T),
nat_plus: ℕ+,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
decidable__exists_int_seg,
equal_wf,
fun_exp_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
int_seg_wf,
istype-nat,
finite-type_wf,
decidable_wf,
istype-universe,
int_term_value_add_lemma,
istype-void,
itermAdd_wf,
decidable__equal_int_seg,
not-inject,
inject_wf,
injection_le,
istype-false,
int_seg_subtype_nat,
int_formula_prop_less_lemma,
intformless_wf,
mu-dec-property,
mu-dec_wf,
exists_wf,
map_wf,
upto_wf,
length_wf,
l_member_wf,
nat_wf,
subtype_rel_list,
member_map,
no_repeats_wf,
select_wf,
decidable__lt,
istype-less_than,
before-upto,
before-map,
no_repeats_iff,
not_wf,
l_before_wf,
iff_weakening_uiff,
length-map,
length_upto,
iff_weakening_equal,
subtype_rel_self,
map_select,
true_wf,
squash_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
select_upto,
member_upto,
subtype_base_sq,
int_subtype_base,
subtract-add-cancel,
fun_exp_add_sq,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
fun_exp-rem,
rem_bounds_1,
remainder_wfa,
nequal_wf,
remainder_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
isectElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
dependent_set_memberEquality_alt,
productElimination,
imageElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
functionIsType,
universeEquality,
equalityIstype,
isect_memberEquality_alt,
addEquality,
inhabitedIsType,
promote_hyp,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
productIsType,
applyLambdaEquality,
hyp_replacement,
productEquality,
imageMemberEquality,
baseClosed,
closedConclusion,
isectIsTypeImplies,
functionIsTypeImplies,
isectEquality,
functionEquality,
cumulativity,
intEquality,
baseApply,
sqequalBase
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} finite-type(T)
{}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. \mforall{}a:T.
\mexists{}L:T List
(no\_repeats(T;L)
\mwedge{} (\mforall{}i:\mBbbN{}||L||. (L[i] = (f\^{}i a)))
\mwedge{} (\mforall{}b:T. ((b \mmember{} L) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. (b = (f\^{}n a)))))))
Date html generated:
2020_05_19-PM-09_44_35
Last ObjectModification:
2020_01_01-AM-10_06_05
Theory : list_1
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