Nuprl Lemma : interleaving_filter2
∀[T:Type]
∀L,L1,L2:T List.
(interleaving(T;L1;L2;L)
⇐⇒ ∃P:ℕ||L|| ⟶ 𝔹. ((L1 = filter2(P;L) ∈ (T List)) ∧ (L2 = filter2(λi.(¬b(P i));L) ∈ (T List))))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
filter2: filter2(P;L),
length: ||as||,
list: T List,
int_seg: {i..j-},
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
top: Top,
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
uiff: uiff(P;Q),
cand: A c∧ B,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
lelt: i ≤ j < k,
nat: ℕ,
int_seg: {i..j-},
guard: {T},
interleaving: interleaving(T;L1;L2;L),
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
eq_int: (i =z j),
subtract: n - m,
bnot: ¬bb,
sq_type: SQType(T),
assert: ↑b,
nequal: a ≠ b ∈ T ,
true: True,
select: L[n],
nil: [],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than': less_than'(a;b),
cons: [a / b],
colength: colength(L),
nat_plus: ℕ+
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iff_wf,
interleaving_wf,
exists_wf,
int_seg_wf,
length_wf,
bool_wf,
equal_wf,
filter2_wf,
bnot_wf,
istype-universe,
equal-wf-T-base,
nil_wf,
filter2_nil_lemma,
length_of_nil_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
decidable__equal_int,
non_neg_length,
length_zero,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
int_seg_properties,
nil_interleaving,
nat_wf,
length_wf_nat,
length_of_cons_lemma,
istype-void,
cons_wf,
le_wf,
less_than_wf,
interleaving_of_cons,
tl_wf,
eq_int_wf,
equal-wf-base,
set_subtype_base,
lelt_wf,
istype-int,
int_subtype_base,
assert_wf,
btrue_wf,
not_wf,
subtract_wf,
decidable__le,
full-omega-unsat,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
add-is-int-iff,
false_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
cons_filter2,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtype_rel_self,
iff_weakening_equal,
bfalse_wf,
ge_wf,
list-cases,
stuck-spread,
istype-base,
reduce_tl_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
subtract-1-ge-0,
spread_cons_lemma,
reduce_tl_cons_lemma,
add-subtract-cancel,
add_nat_plus,
nat_plus_properties,
add-member-int_seg2,
cons_interleaving,
interleaving_symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
hypothesis,
because_Cache,
functionEquality,
natural_numberEquality,
productEquality,
applyLambdaEquality,
functionIsType,
universeIsType,
inhabitedIsType,
independent_functionElimination,
rename,
productIsType,
equalityIsType1,
applyEquality,
dependent_functionElimination,
universeEquality,
baseClosed,
lambdaEquality,
cumulativity,
independent_pairFormation,
lambdaFormation,
voidEquality,
voidElimination,
isect_memberEquality,
unionElimination,
computeAll,
intEquality,
int_eqEquality,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
setElimination,
dependent_pairFormation,
productElimination,
hyp_replacement,
dependent_set_memberEquality,
isect_memberEquality_alt,
addEquality,
dependent_set_memberEquality_alt,
dependent_pairFormation_alt,
baseApply,
closedConclusion,
equalityIsType4,
approximateComputation,
pointwiseFunctionality,
promote_hyp,
imageElimination,
equalityElimination,
equalityIsType2,
instantiate,
imageMemberEquality,
intWeakElimination,
axiomEquality,
functionIsTypeImplies,
hypothesis_subsumption,
functionExtensionality_alt
Latex:
\mforall{}[T:Type]
\mforall{}L,L1,L2:T List.
(interleaving(T;L1;L2;L)
\mLeftarrow{}{}\mRightarrow{} \mexists{}P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}. ((L1 = filter2(P;L)) \mwedge{} (L2 = filter2(\mlambda{}i.(\mneg{}\msubb{}(P i));L))))
Date html generated:
2019_10_15-AM-10_56_35
Last ObjectModification:
2018_10_09-AM-10_11_44
Theory : list!
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