Nuprl Lemma : cycle-decomp
∀n:ℕ. ∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} .
∃cycles:ℕn List List
(no_repeats(ℕn List;cycles)
∧ (∀c1∈cycles.(∀c2∈cycles.(c1 = c2 ∈ (ℕn List)) ∨ l_disjoint(ℕn;c1;c2)))
∧ (∀c∈cycles.0 < ||c|| ∧ no_repeats(ℕn;c))
∧ (f = reduce(λc,g. (cycle(c) o g);λx.x;cycles) ∈ (ℕn ⟶ ℕn)))
Proof
Definitions occuring in Statement :
cycle: cycle(L),
l_disjoint: l_disjoint(T;l1;l2),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
length: ||as||,
reduce: reduce(f;k;as),
list: T List,
inject: Inj(A;B;f),
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_stable: SqStable(P),
squash: ↓T,
exists: ∃x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
prop: ℙ,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
le: A ≤ B,
less_than': less_than'(a;b),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
no_repeats: no_repeats(T;l),
pairwise: (∀x,y∈L. P[x; y]),
l_disjoint: l_disjoint(T;l1;l2),
so_lambda: λ2x y.t[x; y],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
l_exists: (∃x∈L. P[x]),
l_member: (x ∈ l),
true: True,
l_all: (∀x∈L.P[x]),
compose: f o g
Lemmas referenced :
orbit-decomp,
int_seg_wf,
decidable__equal_int_seg,
finite-type-int_seg,
sq_stable__inject,
l_all_iff,
list_wf,
l_member_wf,
less_than_wf,
length_wf,
no_repeats_wf,
all_wf,
equal_wf,
select_wf,
int_seg_properties,
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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
eq_int_wf,
subtract_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
itermAdd_wf,
int_term_value_add_lemma,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
l_all_wf,
nat_wf,
fun_exp_wf,
or_wf,
l_disjoint_wf,
reduce_wf,
compose_wf,
cycle_wf,
set_wf,
inject_wf,
not_wf,
lelt_wf,
list_subtype_base,
set_subtype_base,
int_subtype_base,
select_member,
decidable__equal_int,
le_wf,
pairwise-implies,
squash_wf,
true_wf,
apply-cycle-member,
iff_weakening_equal,
apply-cycle-non-member,
cycle-closed,
list_induction,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
reduce_cons_lemma,
cons_member,
cons_wf,
and_wf,
no_repeats_cons,
reduce_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_functionElimination,
dependent_functionElimination,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_pairFormation,
lambdaEquality,
productEquality,
applyEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
addEquality,
setEquality,
addLevel,
functionEquality,
functionExtensionality,
isect_memberFormation,
dependent_set_memberEquality,
universeEquality,
inlFormation,
inrFormation,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}n:\mBbbN{}. \mforall{}f:\{f:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n| Inj(\mBbbN{}n;\mBbbN{}n;f)\} .
\mexists{}cycles:\mBbbN{}n List List
(no\_repeats(\mBbbN{}n List;cycles)
\mwedge{} (\mforall{}c1\mmember{}cycles.(\mforall{}c2\mmember{}cycles.(c1 = c2) \mvee{} l\_disjoint(\mBbbN{}n;c1;c2)))
\mwedge{} (\mforall{}c\mmember{}cycles.0 < ||c|| \mwedge{} no\_repeats(\mBbbN{}n;c))
\mwedge{} (f = reduce(\mlambda{}c,g. (cycle(c) o g);\mlambda{}x.x;cycles)))
Date html generated:
2017_04_17-AM-08_19_29
Last ObjectModification:
2017_02_27-PM-04_45_06
Theory : list_1
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