Nuprl Lemma : interleaved_split
∀[T:Type]
∀L:T List
∀[P:T ⟶ ℙ]
((∀x:T. Dec(P[x]))
⇒ (∃L1,L2:T List
(interleaving(T;L1;L2;L)
∧ (∀x:T. ((x ∈ L1) ⇐⇒ (x ∈ L) ∧ P[x]))
∧ (∀x:T. ((x ∈ L2) ⇐⇒ (x ∈ L) ∧ (¬P[x]))))))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
cand: A c∧ B,
false: False,
not: ¬A,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
guard: {T}
Lemmas referenced :
not_wf,
subtype_rel_self,
l_member_wf,
iff_wf,
interleaving_wf,
list_wf,
exists_wf,
decidable_wf,
all_wf,
uall_wf,
list_induction,
nil_wf,
false_wf,
nil_member,
interleaving_of_nil,
cons_wf,
equal_wf,
and_wf,
or_wf,
cons_member,
cons_interleaving,
interleaving_symmetry
Rules used in proof :
universeIsType,
dependent_functionElimination,
because_Cache,
rename,
independent_functionElimination,
productEquality,
hypothesis,
applyEquality,
universeEquality,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
cumulativity,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
instantiate,
thin,
cut,
lambdaFormation,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
voidElimination,
productElimination,
independent_pairEquality,
dependent_pairFormation,
functionIsType,
promote_hyp,
unionElimination,
inhabitedIsType,
inlFormation,
inrFormation,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:T {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:T. Dec(P[x]))
{}\mRightarrow{} (\mexists{}L1,L2:T List
(interleaving(T;L1;L2;L)
\mwedge{} (\mforall{}x:T. ((x \mmember{} L1) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} P[x]))
\mwedge{} (\mforall{}x:T. ((x \mmember{} L2) \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} (\mneg{}P[x]))))))
Date html generated:
2019_10_15-AM-10_57_03
Last ObjectModification:
2018_09_27-AM-10_28_26
Theory : list!
Home
Index