Nuprl Lemma : cyclic-map-transitive
∀n:ℕ. ∀f:cyclic-map(ℕn). ∀x,y:ℕn. ∃m:ℕn. ((f^m x) = y ∈ ℕn)
Proof
Definitions occuring in Statement :
cyclic-map: cyclic-map(T),
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
cyclic-map: cyclic-map(T),
injection: A →⟶ B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
squash: ↓T,
uimplies: b supposing a,
l_exists: (∃x∈L. P[x]),
l_all: (∀x∈L.P[x]),
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
uiff: uiff(P;Q),
le: A ≤ B,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
l_member: (x ∈ l),
eq_int: (i =z j),
subtract: n - m,
compose: f o g,
nat_plus: ℕ+,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
sq_stable_from_decidable,
exists_wf,
int_seg_wf,
equal_wf,
fun_exp_wf,
le_wf,
decidable__exists_int_seg,
decidable__equal_int_seg,
cyclic-map_wf,
nat_wf,
orbit-decomp,
decidable__equal-int_seg,
finite-type-int_seg,
injection_le,
length_wf_nat,
select_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
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,
decidable__lt,
length_wf,
list_wf,
intformless_wf,
int_formula_prop_less_lemma,
no_repeats_inject,
inject_wf,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
less_than_wf,
all_wf,
eq_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
l_member_wf,
set_subtype_base,
int_subtype_base,
ge_wf,
fun_exp_unroll,
non_neg_length,
squash_wf,
true_wf,
rem_base_case,
less_than_transitivity2,
subtype_rel_self,
iff_weakening_equal,
decidable__equal_int,
equal-wf-T-base,
rem_bounds_1,
rem_add1,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
rem_rec_case,
fun_exp-rem,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
dependent_set_memberEquality,
productElimination,
hypothesisEquality,
independent_functionElimination,
instantiate,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_isectElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
functionExtensionality,
addEquality,
productEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
cumulativity,
intWeakElimination,
axiomEquality,
applyLambdaEquality,
universeEquality,
remainderEquality,
impliesFunctionality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:cyclic-map(\mBbbN{}n). \mforall{}x,y:\mBbbN{}n. \mexists{}m:\mBbbN{}n. ((f\^{}m x) = y)
Date html generated:
2018_05_21-PM-08_25_27
Last ObjectModification:
2018_05_19-PM-05_01_17
Theory : general
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