Nuprl Lemma : cycle-conjugate
∀[n:ℕ]. ∀[L:ℕn List].
∀[f,g:ℕn ⟶ ℕn].
((g o (cycle(L) o f)) = cycle(map(g;L)) ∈ (ℕn ⟶ ℕn)) supposing
((∀a:ℕn. ((f (g a)) = a ∈ ℕn)) and
(∀a:ℕn. ((g (f a)) = a ∈ ℕn)))
supposing no_repeats(ℕn;L)
Proof
Definitions occuring in Statement :
cycle: cycle(L),
no_repeats: no_repeats(T;l),
map: map(f;as),
list: T List,
compose: f o g,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
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,
compose: f o g,
all: ∀x:A. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
top: Top,
lelt: i ≤ j < k,
and: P ∧ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
not: ¬A,
no_repeats: no_repeats(T;l),
le: A ≤ B
Lemmas referenced :
int_seg_wf,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
no_repeats_wf,
list_wf,
nat_wf,
decidable__l_member,
decidable__equal_int_seg,
subtype_base_sq,
select-map,
istype-void,
subtype_rel_list,
top_wf,
le_wf,
less_than_wf,
length_wf,
equal_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
map_wf,
map-length,
eqtt_to_assert,
assert_of_eq_int,
map_select,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
apply-cycle-member,
not_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_seg_properties,
nat_properties,
select_wf,
length-map,
apply-cycle-non-member,
l_member_wf,
member_map
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :functionExtensionality_alt,
sqequalRule,
Error :universeIsType,
because_Cache,
hypothesis,
Error :functionIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
Error :equalityIsType3,
Error :inhabitedIsType,
applyEquality,
intEquality,
Error :lambdaEquality_alt,
independent_isectElimination,
Error :isect_memberEquality_alt,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
Error :lambdaFormation_alt,
unionElimination,
productElimination,
instantiate,
cumulativity,
voidElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
Error :productIsType,
imageElimination,
universeEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
equalityElimination,
Error :dependent_pairFormation_alt,
Error :equalityIsType1,
promote_hyp,
addEquality,
computeAll,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
applyLambdaEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
isect_memberFormation,
dependent_set_memberEquality,
levelHypothesis,
equalityUniverse
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[L:\mBbbN{}n List].
\mforall{}[f,g:\mBbbN{}n {}\mrightarrow{} \mBbbN{}n].
((g o (cycle(L) o f)) = cycle(map(g;L))) supposing
((\mforall{}a:\mBbbN{}n. ((f (g a)) = a)) and
(\mforall{}a:\mBbbN{}n. ((g (f a)) = a)))
supposing no\_repeats(\mBbbN{}n;L)
Date html generated:
2019_06_20-PM-01_40_07
Last ObjectModification:
2018_10_04-PM-02_28_48
Theory : list_1
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