Nuprl Lemma : permutation-split2
∀[A:Type]
∀p,q:A ⟶ 𝔹. ((∀a:A. (↑q[a] ⇐⇒ ¬↑p[a])) ⇒ (∀L:A List. permutation(A;filter(λx.p[x];L) @ filter(λx.q[x];L);L)))
Proof
Definitions occuring in Statement :
permutation: permutation(T;L1;L2),
filter: filter(P;l),
append: as @ bs,
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
squash: ↓T,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
true: True,
so_lambda: λ2x.t[x],
guard: {T}
Lemmas referenced :
permutation-split,
permutation_wf,
squash_wf,
true_wf,
list_wf,
append_wf,
filter_wf5,
l_member_wf,
iff_imp_equal_bool,
bnot_wf,
assert_wf,
not_wf,
assert_of_bnot,
iff_wf,
all_wf,
bool_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
universeEquality,
because_Cache,
cumulativity,
setElimination,
rename,
functionExtensionality,
setEquality,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
voidElimination,
addLevel,
productElimination,
impliesFunctionality,
imageMemberEquality,
baseClosed,
natural_numberEquality,
functionEquality
Latex:
\mforall{}[A:Type]
\mforall{}p,q:A {}\mrightarrow{} \mBbbB{}.
((\mforall{}a:A. (\muparrow{}q[a] \mLeftarrow{}{}\mRightarrow{} \mneg{}\muparrow{}p[a]))
{}\mRightarrow{} (\mforall{}L:A List. permutation(A;filter(\mlambda{}x.p[x];L) @ filter(\mlambda{}x.q[x];L);L)))
Date html generated:
2016_10_21-AM-10_24_58
Last ObjectModification:
2016_07_12-AM-05_37_41
Theory : list_1
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