Nuprl Lemma : orbit-decomp
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ finite-type(T)
⇒ (∀f:T ⟶ T
∃orbits:T List List
((∀orbit∈orbits.0 < ||orbit||
∧ no_repeats(T;orbit)
∧ (∀i:ℕ||orbit||. ((f orbit[i]) = if (i =z ||orbit|| - 1) then orbit[0] else orbit[i + 1] fi ∈ T))
∧ (∀x∈orbit.∀n:ℕ. (f^n x ∈ orbit)))
∧ (∀a:T. (∃orbit∈orbits. (a ∈ orbit)))
∧ (∀o1,o2∈orbits. l_disjoint(T;o1;o2)))
supposing Inj(T;T;f)))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
fun_exp: f^n,
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
less_than: a < b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
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],
uimplies: b supposing a,
member: t ∈ T,
inject: Inj(A;B;f),
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
int_seg: {i..j-},
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
l_disjoint: l_disjoint(T;l1;l2),
pairwise: (∀x,y∈L. P[x; y]),
ge: i ≥ j ,
nat: ℕ,
true: True,
subtract: n - m,
uiff: uiff(P;Q),
cons: [a / b],
compose: f o g,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
l_all: (∀x∈L.P[x]),
sq_type: SQType(T),
cand: A c∧ B,
l_member: (x ∈ l),
no_repeats: no_repeats(T;l),
last: last(L),
nequal: a ≠ b ∈ T ,
bnot: ¬bb,
listp: A List+,
rev_uimplies: rev_uimplies(P;Q),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
l_contains: A ⊆ B,
l_exists: (∃x∈L. P[x])
Lemmas referenced :
orbit-cover,
l_all_iff,
list_wf,
l_member_wf,
less_than_wf,
length_wf,
no_repeats_wf,
all_wf,
int_seg_wf,
equal_wf,
select_wf,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
fun_exp_wf,
int_seg_subtype_nat,
istype-false,
iff_wf,
exists_wf,
nat_wf,
istype-nat,
l_all_wf,
l_contains_wf,
inject_wf,
finite-type_wf,
decidable_wf,
istype-universe,
l_disjoint_wf,
le_wf,
select_member,
decidable__l_member,
subtract-1-ge-0,
ge_wf,
nat_properties,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
length_wf_nat,
length_of_cons_lemma,
product_subtype_list,
nil_member,
length_of_nil_lemma,
list-cases,
fun_exp0_lemma,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
not_wf,
bnot_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_wf,
int_subtype_base,
bool_wf,
equal-wf-base,
eq_int_wf,
fun_exp_unroll,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
iff_weakening_equal,
subtype_rel_self,
true_wf,
squash_wf,
fun_exp_add1,
int_term_value_add_lemma,
itermAdd_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
null_wf,
assert_elim,
last_wf,
null_cons_lemma,
null_nil_lemma,
last_member,
subtype_base_sq,
decidable__equal_int,
add-swap,
subtract-add-cancel,
equal-wf-T-base,
set_subtype_base,
lelt_wf,
istype-assert,
add-member-int_seg2,
istype-le,
istype-less_than,
fun_exp_apply_add1,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
assert_of_null,
listp_properties,
length_of_not_nil,
fun_exp_add_sq,
pairwise_wf2,
l_exists_wf,
fun_exp_add_apply,
l_disjoint-representatives,
pairwise-implies
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,
extract_by_obid,
isectElimination,
independent_functionElimination,
productElimination,
Error :universeIsType,
setElimination,
productEquality,
closedConclusion,
natural_numberEquality,
because_Cache,
independent_isectElimination,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
imageElimination,
applyEquality,
Error :setIsType,
functionEquality,
Error :functionIsType,
instantiate,
universeEquality,
Error :inrFormation_alt,
Error :unionIsType,
Error :inlFormation_alt,
Error :productIsType,
Error :dependent_set_memberEquality_alt,
cumulativity,
intWeakElimination,
functionExtensionality,
minusEquality,
addEquality,
hypothesis_subsumption,
promote_hyp,
Error :equalityIsType1,
equalitySymmetry,
equalityTransitivity,
Error :equalityIsType4,
intEquality,
baseClosed,
baseApply,
equalityElimination,
imageMemberEquality,
hyp_replacement,
applyLambdaEquality,
Error :equalityIstype,
sqequalBase
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} finite-type(T)
{}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T
\mexists{}orbits:T List List
((\mforall{}orbit\mmember{}orbits.0 < ||orbit||
\mwedge{} no\_repeats(T;orbit)
\mwedge{} (\mforall{}i:\mBbbN{}||orbit||
((f orbit[i]) = if (i =\msubz{} ||orbit|| - 1) then orbit[0] else orbit[i + 1] fi ))
\mwedge{} (\mforall{}x\mmember{}orbit.\mforall{}n:\mBbbN{}. (f\^{}n x \mmember{} orbit)))
\mwedge{} (\mforall{}a:T. (\mexists{}orbit\mmember{}orbits. (a \mmember{} orbit)))
\mwedge{} (\mforall{}o1,o2\mmember{}orbits. l\_disjoint(T;o1;o2)))
supposing Inj(T;T;f)))
Date html generated:
2019_06_20-PM-01_39_37
Last ObjectModification:
2018_11_22-PM-10_09_09
Theory : list_1
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