Nuprl Lemma : orbit-closed
∀[T:Type]. ∀f:T ⟶ T. ∀L:T List. (∀a∈L.∀n:ℕ. (f^n a ∈ L)) supposing orbit(T;f;L)
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
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
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,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
less_than: a < b
Lemmas referenced :
member-less_than,
length_wf,
no_repeats_witness,
l_all_iff,
nat_wf,
l_member_wf,
fun_exp_wf,
orbit_wf,
list_wf,
istype-universe,
squash_wf,
true_wf,
orbit-iterates,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
subtype_rel_self,
iff_weakening_equal,
select_member,
remainder_wfa,
intformeq_wf,
intformless_wf,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
length_wf_nat,
set_subtype_base,
le_wf,
int_subtype_base,
nequal_wf,
rem_bounds_1,
itermAdd_wf,
int_term_value_add_lemma,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
natural_numberEquality,
hypothesisEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
Error :lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
rename,
functionEquality,
applyEquality,
setElimination,
Error :setIsType,
Error :universeIsType,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
because_Cache,
Error :functionIsType,
instantiate,
universeEquality,
imageElimination,
equalityTransitivity,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
Error :productIsType,
imageMemberEquality,
baseClosed,
addEquality,
Error :equalityIstype,
intEquality,
sqequalBase
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}L:T List. (\mforall{}a\mmember{}L.\mforall{}n:\mBbbN{}. (f\^{}n a \mmember{} L)) supposing orbit(T;f;L)
Date html generated:
2019_06_20-PM-01_38_09
Last ObjectModification:
2019_03_06-AM-10_51_34
Theory : list_1
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