Nuprl Lemma : cycle-as-flips
∀n:ℕ. ∀L:ℕn List. ∃flips:(ℕn × ℕn) List. (cycle(L) = compose-flips(flips) ∈ (ℕn ⟶ ℕn)) supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement :
compose-flips: compose-flips(flips),
cycle: cycle(L),
no_repeats: no_repeats(T;l),
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
product: 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,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cycle: cycle(L),
ifthenelse: if b then t else f fi ,
null: null(as),
nil: [],
it: ⋅,
btrue: tt,
compose-flips: compose-flips(flips),
reduce: reduce(f;k;as),
list_ind: list_ind,
map: map(f;as),
or: P ∨ Q,
cons: [a / b],
top: Top,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bfalse: ff,
let: let,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
and: P ∧ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
int_seg_wf,
isect_wf,
no_repeats_wf,
exists_wf,
list_wf,
equal_wf,
cycle_wf,
compose-flips_wf,
no_repeats_witness,
nil_wf,
cons_wf,
nat_wf,
equal-wf-base-T,
list-cases,
product_subtype_list,
null_cons_lemma,
reduce_hd_cons_lemma,
list_ind_cons_lemma,
null_nil_lemma,
list_ind_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
eqtt_to_assert,
assert_of_eq_int,
int_seg_properties,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
eqff_to_assert,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
cycle-flip-lemma,
length_of_cons_lemma,
non_neg_length,
length_wf,
itermAdd_wf,
int_term_value_add_lemma,
reduce_tl_cons_lemma,
no_repeats_cons,
map_cons_lemma,
reduce_cons_lemma,
squash_wf,
true_wf,
compose_wf,
flip_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
lambdaEquality,
hypothesisEquality,
productEquality,
functionEquality,
dependent_functionElimination,
independent_functionElimination,
isect_memberFormation,
dependent_pairFormation,
functionExtensionality,
baseClosed,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityElimination,
independent_isectElimination,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
instantiate,
addEquality,
independent_pairEquality,
applyEquality,
imageElimination,
universeEquality,
cumulativity,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}L:\mBbbN{}n List.
\mexists{}flips:(\mBbbN{}n \mtimes{} \mBbbN{}n) List. (cycle(L) = compose-flips(flips)) supposing no\_repeats(\mBbbN{}n;L)
Date html generated:
2017_04_17-AM-08_20_33
Last ObjectModification:
2017_02_27-PM-04_43_22
Theory : list_1
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