Nuprl Lemma : interleaving_as_filter
∀[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
((∀x∈L1.¬↑(P x)) and
(∀x∈L2.↑(P x)) and
interleaving(T;L1;L2;L))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
list: T List,
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
not: ¬A,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
l_all_wf,
not_wf,
assert_wf,
l_member_wf,
interleaving_wf,
list_wf,
bool_wf,
filter_interleaving,
filter_trivial,
filter_is_nil,
nil_interleaving,
filter_wf5,
subtype_rel_dep_function,
subtype_rel_self,
set_wf,
bnot_wf,
assert_of_bnot,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
not_functionality_wrt_uiff,
false_wf,
nil_interleaving2
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,
lambdaEquality,
applyEquality,
setElimination,
rename,
setEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
lambdaFormation,
functionExtensionality,
cumulativity,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
imageElimination
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
((\mforall{}x\mmember{}L1.\mneg{}\muparrow{}(P x)) and
(\mforall{}x\mmember{}L2.\muparrow{}(P x)) and
interleaving(T;L1;L2;L))
Date html generated:
2019_10_15-AM-10_56_47
Last ObjectModification:
2018_09_17-PM-06_33_10
Theory : list!
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