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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