Nuprl Lemma : filter_is_interleaving
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L:T List. interleaving(T;filter(λx.(¬b(P x));L);filter(P;L);L)
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
filter: filter(P;l),
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
l_all: (∀x∈L.P[x]),
so_apply: x[s],
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2)
Lemmas referenced :
interleaving_split,
assert_wf,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
decidable__assert,
list_wf,
istype-universe,
bool_wf,
occurence_implies_interleaving,
interleaving_as_filter_2,
interleaving_symmetry,
filter_trivial2,
nil_wf,
filter_is_nil,
not_wf,
interleaving_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
because_Cache,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
imageElimination,
functionIsType,
universeEquality,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. interleaving(T;filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L);filter(P;L);L)
Date html generated:
2019_10_15-AM-10_57_29
Last ObjectModification:
2018_10_09-AM-09_58_35
Theory : list!
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