Nuprl Lemma : permutation-split
∀[A:Type]. ∀p:A ⟶ 𝔹. ∀L:A List. permutation(A;filter(λx.p[x];L) @ filter(λx.(¬bp[x]);L);L)
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
filter: filter(P;l),
append: as @ bs,
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
top: Top,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
permutation_wf,
append_wf,
filter_wf5,
l_member_wf,
bnot_wf,
list_wf,
filter_nil_lemma,
list_ind_nil_lemma,
permutation-nil,
filter_cons_lemma,
bool_wf,
eqtt_to_assert,
list_ind_cons_lemma,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
equal_wf,
cons_wf,
nil_wf,
permutation_weakening,
permutation_functionality_wrt_permutation,
append_functionality_wrt_permutation,
permutation-rotate
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
applyEquality,
functionExtensionality,
setElimination,
rename,
hypothesis,
setEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
functionEquality,
universeEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
baseClosed
Latex:
\mforall{}[A:Type]. \mforall{}p:A {}\mrightarrow{} \mBbbB{}. \mforall{}L:A List. permutation(A;filter(\mlambda{}x.p[x];L) @ filter(\mlambda{}x.(\mneg{}\msubb{}p[x]);L);L)
Date html generated:
2017_04_17-AM-08_25_15
Last ObjectModification:
2017_02_27-PM-04_46_25
Theory : list_1
Home
Index