Nuprl Lemma : orbit-iterates
∀[T:Type]. ∀[f:T ⟶ T]. ∀[L:T List]. ∀[i:ℕ||L||]. ∀[n:ℕ]. ((f^n L[i]) = L[i + n rem ||L||] ∈ T) supposing orbit(T;f;L)
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
select: L[n],
length: ||as||,
list: T List,
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
remainder: n rem m,
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
orbit: orbit(T;f;L),
and: P ∧ Q,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
cand: A c∧ B,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
compose: f o g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than': less_than'(a;b)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
fun_exp0_lemma,
subtract-1-ge-0,
istype-nat,
int_seg_wf,
length_wf,
orbit_wf,
list_wf,
istype-universe,
select_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
istype-le,
rem_bounds_1,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
squash_wf,
true_wf,
le_wf,
less_than_wf,
rem_base_case,
subtype_rel_self,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
rem_add1,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
not_wf,
bnot_wf,
assert_wf,
equal-wf-base,
bool_wf,
eq_int_wf,
satisfiable-full-omega-tt,
fun_exp_unroll,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
false_wf,
equal-wf-T-base,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
Error :functionIsTypeImplies,
Error :inhabitedIsType,
Error :isectIsTypeImplies,
Error :functionIsType,
because_Cache,
instantiate,
universeEquality,
imageElimination,
unionElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
applyEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
intEquality,
closedConclusion,
baseApply,
computeAll,
voidEquality,
isect_memberEquality,
lambdaEquality,
dependent_pairFormation,
dependent_set_memberEquality,
lambdaFormation,
equalityElimination,
impliesFunctionality,
functionExtensionality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[L:T List].
\mforall{}[i:\mBbbN{}||L||]. \mforall{}[n:\mBbbN{}]. ((f\^{}n L[i]) = L[i + n rem ||L||]) supposing orbit(T;f;L)
Date html generated:
2019_06_20-PM-01_38_00
Last ObjectModification:
2019_03_06-AM-11_06_13
Theory : list_1
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