Nuprl Lemma : orbit-of-involution
∀[T:Type]. ∀[f:T ⟶ T].
∀o:T List. (||o|| = 1 ∈ ℤ) ∨ (||o|| = 2 ∈ ℤ) supposing orbit(T;f;o) supposing ∀x:T. ((f (f x)) = x ∈ T)
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
cons: [a / b],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
false: False,
orbit: orbit(T;f;L),
and: P ∧ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
top: Top,
decidable: Dec(P),
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
ge: i ≥ j ,
le: A ≤ B,
no_repeats: no_repeats(T;l),
nat: ℕ,
select: L[n],
subtract: n - m,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
sq_type: SQType(T),
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list-cases,
product_subtype_list,
orbit_wf,
list_wf,
all_wf,
equal_wf,
length_of_nil_lemma,
length_of_cons_lemma,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
equal-wf-base,
non_neg_length,
decidable__le,
intformand_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
equal-wf-T-base,
length_wf,
false_wf,
le_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
nat_properties,
nat_wf,
lelt_wf,
eq_int_wf,
subtract_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
uiff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base-T,
assert_of_bnot,
not_functionality_wrt_uiff,
itermSubtract_wf,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionEquality,
universeEquality,
imageElimination,
voidElimination,
isect_memberEquality,
voidEquality,
inlFormation,
natural_numberEquality,
because_Cache,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
intEquality,
baseClosed,
rename,
inrFormation,
int_eqEquality,
independent_pairFormation,
addEquality,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
instantiate,
cumulativity,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T].
\mforall{}o:T List. (||o|| = 1) \mvee{} (||o|| = 2) supposing orbit(T;f;o) supposing \mforall{}x:T. ((f (f x)) = x)
Date html generated:
2019_06_20-PM-01_38_21
Last ObjectModification:
2018_09_22-PM-10_50_40
Theory : list_1
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