Nuprl Lemma : cycle-append
∀[n:ℕ]. ∀[as,bs:ℕn List]. cycle(as @ bs) = cycle(bs @ as) ∈ (ℕn ⟶ ℕn) supposing no_repeats(ℕn;as @ bs)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
no_repeats: no_repeats(T;l),
append: as @ bs,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
top: Top,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
sq_type: SQType(T),
less_than: a < b,
no_repeats: no_repeats(T;l),
subtract: n - m,
le: A ≤ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
cons: [a / b],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
nil: [],
select: L[n],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs,
assert: ↑b,
bnot: ¬bb,
nequal: a ≠ b ∈ T
Lemmas referenced :
int_seg_wf,
no_repeats_wf,
append_wf,
istype-nat,
decidable__l_member,
decidable__equal_int_seg,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cycle_wf,
subtype_rel_self,
iff_weakening_equal,
no_repeats-append,
l_disjoint-symmetry,
length-append,
istype-void,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
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,
istype-le,
istype-less_than,
length_wf,
apply-cycle-member,
decidable__lt,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
int_formula_prop_less_lemma,
intformless_wf,
le_wf,
false_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
intformeq_wf,
itermAdd_wf,
satisfiable-full-omega-tt,
add-is-int-iff,
nat_wf,
length_wf_nat,
add_nat_wf,
lelt_wf,
select_append_back,
add-associates,
minus-one-mul,
add-mul-special,
zero-mul,
add-zero,
select_append_front,
eq_int_wf,
subtract_wf,
equal-wf-base,
bool_wf,
list_subtype_base,
set_subtype_base,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
le_weakening2,
select_wf,
istype-false,
non_neg_length,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_list,
add-comm,
add-swap,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
less_than_wf,
list_ind_cons_lemma,
length_of_cons_lemma,
product_subtype_list,
base_wf,
stuck-spread,
list_ind_nil_lemma,
length_of_nil_lemma,
list-cases,
select-cons-hd,
top_wf,
length_append,
cons_wf,
equal-wf-T-base,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
add-commutes,
two-mul,
list_wf,
int_term_value_mul_lemma,
itermMultiply_wf,
multiply-is-int-iff,
nequal_wf,
l_member_wf,
member_append,
apply-cycle-non-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :functionExtensionality_alt,
Error :universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
dependent_functionElimination,
independent_functionElimination,
Error :lambdaFormation_alt,
unionElimination,
productElimination,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
closedConclusion,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
voidElimination,
applyLambdaEquality,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :productIsType,
cumulativity,
intEquality,
addEquality,
computeAll,
lambdaEquality,
dependent_pairFormation,
baseApply,
promote_hyp,
pointwiseFunctionality,
voidEquality,
isect_memberEquality,
lambdaFormation,
dependent_set_memberEquality,
Error :equalityIsType4,
Error :functionIsType,
equalityElimination,
Error :equalityIsType1,
hypothesis_subsumption,
minusEquality,
multiplyEquality,
comment,
impliesFunctionality,
productEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[as,bs:\mBbbN{}n List]. cycle(as @ bs) = cycle(bs @ as) supposing no\_repeats(\mBbbN{}n;as @ bs)
Date html generated:
2019_06_20-PM-01_41_06
Last ObjectModification:
2018_10_18-AM-11_47_44
Theory : list_1
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