Nuprl Lemma : orbit-size-divides-order
∀[T:Type]. ∀f:T ⟶ T. ∀n:ℕ. ∀L:T List. ||L|| | n supposing orbit(T;f;L) supposing ∀x:T. ((f^n x) = x ∈ T)
Proof
Definitions occuring in Statement :
divides: b | a,
orbit: orbit(T;f;L),
length: ||as||,
list: T List,
fun_exp: f^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
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,
orbit: orbit(T;f;L),
and: P ∧ Q,
implies: P ⇒ Q,
int_seg: {i..j-},
nat: ℕ,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
false: False,
less_than: a < b,
squash: ↓T,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
divides: b | a,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
guard: {T},
iff: P ⇐⇒ Q,
no_repeats: no_repeats(T;l),
nat_plus: ℕ+
Lemmas referenced :
member-less_than,
length_wf,
no_repeats_witness,
orbit-iterates,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
istype-le,
istype-less_than,
decidable__equal_int,
remainder_wfa,
intformeq_wf,
int_formula_prop_eq_lemma,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
nequal_wf,
orbit_wf,
list_wf,
fun_exp_wf,
istype-nat,
istype-universe,
div_rem_sum,
false_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
itermAdd_wf,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
equal_wf,
squash_wf,
true_wf,
select_wf,
istype-false,
subtype_rel_self,
iff_weakening_equal,
zero-add,
remainder_wf,
rem_bounds_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
Error :lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
productElimination,
independent_pairEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
setElimination,
independent_pairFormation,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :universeIsType,
imageElimination,
int_eqEquality,
Error :productIsType,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :equalityIstype,
applyEquality,
intEquality,
baseClosed,
sqequalBase,
Error :functionIsType,
instantiate,
universeEquality,
closedConclusion,
multiplyEquality,
baseApply,
promote_hyp,
pointwiseFunctionality,
divideEquality,
imageMemberEquality,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T. \mforall{}n:\mBbbN{}. \mforall{}L:T List. ||L|| | n supposing orbit(T;f;L) supposing \mforall{}x:T. ((f\^{}n x) = x)
Date html generated:
2019_06_20-PM-02_20_41
Last ObjectModification:
2019_03_06-AM-10_53_46
Theory : num_thy_1
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