Nuprl Lemma : orbit-transitive
∀[T:Type]. ∀f:T ⟶ T. ∀L:T List. (∀a∈L.(∀b∈L.∃n:ℕ. ((f^n a) = b ∈ T))) supposing orbit(T;f;L)
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
l_all: (∀x∈L.P[x]),
list: T List,
fun_exp: f^n,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃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,
so_lambda: λ2x.t[x],
prop: ℙ,
exists: ∃x:A. B[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
squash: ↓T,
less_than: a < b,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
subtract: n - m,
nat_plus: ℕ+
Lemmas referenced :
member-less_than,
length_wf,
no_repeats_witness,
l_all_iff,
l_all_wf,
nat_wf,
equal_wf,
fun_exp_wf,
l_member_wf,
exists_wf,
select_wf,
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,
orbit_wf,
list_wf,
istype-universe,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
intformless_wf,
itermAdd_wf,
bnot_wf,
less_than_wf,
lt_int_wf,
istype-le,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
assert_wf,
int_subtype_base,
le_wf,
set_subtype_base,
bool_wf,
equal-wf-base,
le_int_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
squash_wf,
true_wf,
orbit-iterates,
istype-less_than,
subtype_rel_self,
iff_weakening_equal,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
rem_base_case,
decidable__lt,
rem_bounds_1,
trivial-equal,
zero-add,
add-associates,
rem_rec_case
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,
productEquality,
applyEquality,
setElimination,
Error :setIsType,
Error :universeIsType,
because_Cache,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
Error :functionIsType,
instantiate,
universeEquality,
imageElimination,
addEquality,
Error :dependent_set_memberEquality_alt,
intEquality,
baseClosed,
closedConclusion,
baseApply,
equalityElimination,
Error :equalityIsType1,
equalityTransitivity,
Error :productIsType,
imageMemberEquality,
Error :equalityIstype
Latex:
\mforall{}[T:Type]. \mforall{}f:T {}\mrightarrow{} T. \mforall{}L:T List. (\mforall{}a\mmember{}L.(\mforall{}b\mmember{}L.\mexists{}n:\mBbbN{}. ((f\^{}n a) = b))) supposing orbit(T;f;L)
Date html generated:
2019_06_20-PM-01_38_31
Last ObjectModification:
2019_03_06-AM-10_52_02
Theory : list_1
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