Nuprl Lemma : orbit-decomp2
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ finite-type(T)
⇒ (∀f:T ⟶ T
∃orbits:T List List
((∀o∈orbits.orbit(T;f;o))
∧ (∀a:T. (∃orbit∈orbits. (a ∈ orbit)))
∧ (∀o1,o2∈orbits. l_disjoint(T;o1;o2))
∧ no_repeats(T List;orbits))
supposing Inj(T;T;f)))
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
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),
list: T List,
inject: Inj(A;B;f),
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,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uimplies: b supposing a,
inject: Inj(A;B;f),
prop: ℙ,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
l_all: (∀x∈L.P[x]),
orbit: orbit(T;f;L),
pairwise: (∀x,y∈L. P[x; y]),
no_repeats: no_repeats(T;l),
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
less_than': less_than'(a;b),
cons: [a / b],
l_disjoint: l_disjoint(T;l1;l2)
Lemmas referenced :
orbit-decomp,
equal_wf,
l_all_wf,
list_wf,
l_member_wf,
orbit_wf,
all_wf,
l_exists_wf,
pairwise_wf2,
l_disjoint_wf,
no_repeats_wf,
inject_wf,
finite-type_wf,
decidable_wf,
int_seg_wf,
length_wf,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
not_wf,
nat_wf,
less_than_wf,
decidable__lt,
lelt_wf,
squash_wf,
true_wf,
iff_weakening_equal,
intformless_wf,
intformeq_wf,
int_formula_prop_less_lemma,
int_formula_prop_eq_lemma,
decidable__equal_int,
le_wf,
equal-wf-base,
int_subtype_base,
list-cases,
length_of_nil_lemma,
nil_wf,
product_subtype_list,
length_of_cons_lemma,
cons_wf,
cons_member
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
axiomEquality,
cumulativity,
applyEquality,
functionExtensionality,
rename,
independent_isectElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation,
productEquality,
setElimination,
because_Cache,
setEquality,
universeEquality,
instantiate,
functionEquality,
natural_numberEquality,
voidElimination,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
promote_hyp,
hypothesis_subsumption,
addEquality,
inlFormation
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{}o\mmember{}orbits.orbit(T;f;o))
\mwedge{} (\mforall{}a:T. (\mexists{}orbit\mmember{}orbits. (a \mmember{} orbit)))
\mwedge{} (\mforall{}o1,o2\mmember{}orbits. l\_disjoint(T;o1;o2))
\mwedge{} no\_repeats(T List;orbits))
supposing Inj(T;T;f)))
Date html generated:
2017_04_17-AM-08_16_45
Last ObjectModification:
2017_02_27-PM-04_41_07
Theory : list_1
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