Nuprl Lemma : singleton-orbit
∀[T:Type]. ∀[f:T ⟶ T]. ∀[o:T List]. (o ∈ {x:T| (f x) = x ∈ T} List) supposing (orbit(T;f;o) and (||o|| = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
orbit: orbit(T;f;L),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
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],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
false: False,
cons: [a / b],
top: Top,
prop: ℙ,
ge: i ≥ j ,
le: A ≤ B,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
nat: ℕ,
less_than': less_than'(a;b),
compose: f o g
Lemmas referenced :
orbit-closed,
list-cases,
length_of_nil_lemma,
subtype_base_sq,
int_subtype_base,
product_subtype_list,
length_of_cons_lemma,
cons_wf,
equal_wf,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
nil_wf,
list-set-type2,
orbit_wf,
equal-wf-T-base,
length_wf,
list_wf,
l_all_cons,
all_wf,
nat_wf,
l_member_wf,
fun_exp_wf,
false_wf,
le_wf,
fun_exp1_lemma,
member_singleton
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
independent_isectElimination,
hypothesis,
unionElimination,
sqequalRule,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
voidElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidEquality,
setEquality,
because_Cache,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
rename,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
independent_pairFormation,
computeAll,
axiomEquality,
baseClosed,
functionEquality,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[o:T List].
(o \mmember{} \{x:T| (f x) = x\} List) supposing (orbit(T;f;o) and (||o|| = 1))
Date html generated:
2017_04_17-AM-08_14_31
Last ObjectModification:
2017_02_27-PM-04_39_29
Theory : list_1
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