Nuprl Lemma : cycle-injection
∀[n:ℕ]. ∀[L:ℕn List].  Inj(ℕn;ℕn;cycle(L)) supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement : 
cycle: cycle(L)
, 
no_repeats: no_repeats(T;l)
, 
list: T List
, 
inject: Inj(A;B;f)
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
natural_number: $n
Definitions unfolded in proof : 
or: P ∨ Q
, 
decidable: Dec(P)
, 
nat: ℕ
, 
prop: ℙ
, 
implies: P 
⇒ Q
, 
all: ∀x:A. B[x]
, 
inject: Inj(A;B;f)
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
cand: A c∧ B
, 
exists: ∃x:A. B[x]
, 
l_member: (x ∈ l)
, 
guard: {T}
, 
sq_type: SQType(T)
, 
so_apply: x[s]
, 
so_lambda: λ2x.t[x]
, 
int_seg: {i..j-}
, 
bfalse: ff
, 
rev_implies: P 
⇐ Q
, 
btrue: tt
, 
ifthenelse: if b then t else f fi 
, 
uiff: uiff(P;Q)
, 
iff: P 
⇐⇒ Q
, 
squash: ↓T
, 
top: Top
, 
not: ¬A
, 
false: False
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
subtype_rel: A ⊆r B
, 
ge: i ≥ j 
, 
true: True
, 
le: A ≤ B
, 
and: P ∧ Q
, 
lelt: i ≤ j < k
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
no_repeats: no_repeats(T;l)
Lemmas referenced : 
decidable__equal_int_seg, 
decidable__l_member, 
nat_wf, 
list_wf, 
no_repeats_wf, 
cycle_wf, 
int_seg_wf, 
equal_wf, 
int_subtype_base, 
lelt_wf, 
set_subtype_base, 
subtype_base_sq, 
assert_of_bnot, 
iff_weakening_uiff, 
iff_transitivity, 
eqff_to_assert, 
assert_of_eq_int, 
eqtt_to_assert, 
bool_subtype_base, 
bool_wf, 
bool_cases, 
iff_weakening_equal, 
apply-cycle-member, 
true_wf, 
squash_wf, 
not_wf, 
bnot_wf, 
assert_wf, 
int_formula_prop_wf, 
int_term_value_var_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_not_lemma, 
itermVar_wf, 
intformeq_wf, 
intformnot_wf, 
satisfiable-full-omega-tt, 
select_wf, 
nat_properties, 
int_seg_properties, 
subtract_wf, 
eq_int_wf, 
length_wf, 
equal-wf-base-T, 
int_term_value_subtract_lemma, 
itermSubtract_wf, 
int_formula_prop_less_lemma, 
intformless_wf, 
decidable__lt, 
int_term_value_add_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_and_lemma, 
itermAdd_wf, 
itermConstant_wf, 
intformle_wf, 
intformand_wf, 
decidable__le, 
le_wf, 
false_wf, 
decidable__equal_int, 
less_than_wf, 
cycle-closed, 
istype-int, 
full-omega-unsat, 
istype-void, 
istype-universe, 
apply-cycle-non-member, 
subtype_rel_self, 
istype-le, 
istype-less_than, 
l_member_wf
Rules used in proof : 
unionElimination, 
independent_functionElimination, 
equalitySymmetry, 
equalityTransitivity, 
isect_memberEquality, 
axiomEquality, 
dependent_functionElimination, 
lambdaEquality, 
sqequalRule, 
hypothesisEquality, 
applyEquality, 
because_Cache, 
rename, 
setElimination, 
natural_numberEquality, 
thin, 
isectElimination, 
sqequalHypSubstitution, 
extract_by_obid, 
hypothesis, 
lambdaFormation, 
cut, 
introduction, 
isect_memberFormation, 
sqequalReflexivity, 
computationStep, 
sqequalTransitivity, 
sqequalSubstitution, 
productElimination, 
intEquality, 
independent_isectElimination, 
cumulativity, 
instantiate, 
impliesFunctionality, 
baseClosed, 
imageMemberEquality, 
universeEquality, 
imageElimination, 
computeAll, 
voidEquality, 
voidElimination, 
int_eqEquality, 
dependent_pairFormation, 
applyLambdaEquality, 
independent_pairFormation, 
dependent_set_memberEquality, 
levelHypothesis, 
equalityUniverse, 
addEquality, 
productEquality, 
Error :lambdaEquality_alt, 
approximateComputation, 
Error :dependent_pairFormation_alt, 
Error :isect_memberEquality_alt, 
Error :universeIsType, 
Error :dependent_set_memberEquality_alt, 
Error :inhabitedIsType, 
Error :productIsType, 
Error :lambdaFormation_alt, 
hyp_replacement, 
Error :equalityIstype
Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[L:\mBbbN{}n  List].    Inj(\mBbbN{}n;\mBbbN{}n;cycle(L))  supposing  no\_repeats(\mBbbN{}n;L)
Date html generated:
2019_06_20-PM-01_40_24
Last ObjectModification:
2019_01_13-PM-02_03_53
Theory : list_1
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