Nuprl Lemma : interleaving_as_filter_2
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L,L1,L2:T List].
({(L2 = filter(P;L) ∈ (T List)) ∧ (L1 = filter(λx.(¬b(P x));L) ∈ (T List))}) supposing
((filter(P;L1) = [] ∈ (T List)) and
(L2 = filter(P;L2) ∈ (T List)) and
interleaving(T;L1;L2;L))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
filter: filter(P;l),
nil: [],
list: T List,
bnot: ¬bb,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
equal-wf-T-base,
list_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
equal_wf,
interleaving_wf,
filter_interleaving,
nil_interleaving,
bnot_wf,
filter_trivial,
l_all_iff,
not_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
and_wf,
null_wf,
btrue_neq_bfalse,
assert_wf,
not_functionality_wrt_iff,
member_filter,
assert_of_bnot,
filter_is_nil,
nil_interleaving2,
not_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
axiomEquality,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaFormation,
baseClosed,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
dependent_set_memberEquality,
voidElimination,
productEquality,
cumulativity
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L,L1,L2:T List].
(\{(L2 = filter(P;L)) \mwedge{} (L1 = filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L))\}) supposing
((filter(P;L1) = []) and
(L2 = filter(P;L2)) and
interleaving(T;L1;L2;L))
Date html generated:
2019_10_15-AM-10_56_52
Last ObjectModification:
2018_09_17-PM-07_00_26
Theory : list!
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