Nuprl Lemma : cycle-flip-lemma
∀[n:ℕ]. ∀[L:ℕn List].
(cycle(L) = ((hd(L), hd(tl(L))) o cycle(tl(L))) ∈ (ℕn ⟶ ℕn)) supposing (1 < ||L|| and no_repeats(ℕn;L))
Proof
Definitions occuring in Statement :
cycle: cycle(L),
flip: (i, j),
no_repeats: no_repeats(T;l),
length: ||as||,
tl: tl(l),
hd: hd(l),
list: T List,
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
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 :
cycle: cycle(L),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
let: let,
subtract: n - m,
flip: (i, j),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uiff: uiff(P;Q),
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
select: L[n],
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
compose: f o g,
implies: P ⇒ Q,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
all: ∀x:A. B[x],
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
cons: [a / b],
top: Top,
prop: ℙ
Lemmas referenced :
l_member_wf,
null_cons_lemma,
list_ind_cons_lemma,
null_nil_lemma,
hd_wf,
list_ind_wf,
ifthenelse_wf,
le_wf,
length_wf_nat,
apply-cycle-non-member,
istype-void,
select_member,
full-omega-unsat,
istype-int,
istype-le,
istype-less_than,
cons_member,
assert_elim,
btrue_neq_bfalse,
select_cons_tl,
add-associates,
add-swap,
add-commutes,
zero-add,
non_neg_length,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
select_wf,
decidable__le,
add-is-int-iff,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
false_wf,
eq_int_wf,
subtract_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
select-cons-tl,
int_seg_properties,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
add-subtract-cancel,
no_repeats_cons,
equal_wf,
squash_wf,
true_wf,
cycle_wf,
flip_wf,
apply-cycle-member,
iff_weakening_equal,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
cons_wf,
decidable__l_member,
decidable__equal_int_seg,
int_seg_wf,
list-cases,
length_of_nil_lemma,
reduce_tl_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
less_than_wf,
length_wf,
no_repeats_wf,
list_wf,
nat_wf
Rules used in proof :
inlFormation,
hyp_replacement,
Error :isect_memberEquality_alt,
Error :dependent_set_memberEquality_alt,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :universeIsType,
Error :productIsType,
inrFormation,
productEquality,
addLevel,
levelHypothesis,
pointwiseFunctionality,
baseApply,
closedConclusion,
equalityElimination,
impliesFunctionality,
addEquality,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
applyEquality,
universeEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
instantiate,
cumulativity,
independent_isectElimination,
intEquality,
lambdaEquality,
functionExtensionality,
independent_functionElimination,
lambdaFormation,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_functionElimination,
unionElimination,
sqequalRule,
imageElimination,
productElimination,
voidElimination,
promote_hyp,
hypothesis_subsumption,
isect_memberEquality,
voidEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:\mBbbN{}n List].
(cycle(L) = ((hd(L), hd(tl(L))) o cycle(tl(L)))) supposing (1 < ||L|| and no\_repeats(\mBbbN{}n;L))
Date html generated:
2019_06_20-PM-02_13_03
Last ObjectModification:
2019_06_20-PM-02_08_21
Theory : list_1
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