Nuprl Lemma : filter_interleaving_occurence
∀[T:Type]
∀P:T ⟶ 𝔹. ∀L:T List.
∃f1:ℕ||filter(λx.(¬b(P x));L)|| ⟶ ℕ||L||
∃f2:ℕ||filter(P;L)|| ⟶ ℕ||L||
(interleaving_occurence(T;filter(λx.(¬b(P x));L);filter(P;L);L;f1;f2)
∧ ((∀i:ℕ||L||. ∃k:ℕ||filter(P;L)||. ((i = (f2 k) ∈ ℤ) ∧ (L[i] = filter(P;L)[k] ∈ T)) supposing ↑(P L[i]))
∧ (∀i:ℕ||L||
∃k:ℕ||filter(λx.(¬b(P x));L)||. ((i = (f1 k) ∈ ℤ) ∧ (L[i] = filter(λx.(¬b(P x));L)[k] ∈ T))
supposing ¬↑(P L[i])))
∧ (∀i:ℕ||filter(λx.(¬b(P x));L)||. (¬↑(P L[f1 i])))
∧ (∀i:ℕ||filter(P;L)||. (↑(P L[f2 i]))))
Proof
Definitions occuring in Statement :
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
cand: A c∧ B,
and: P ∧ Q,
exists: ∃x:A. B[x],
istype: istype(T),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
less_than': less_than'(a;b),
l_member: (x ∈ l),
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
uiff: uiff(P;Q),
sq_type: SQType(T),
btrue: tt,
ifthenelse: if b then t else f fi ,
bnot: ¬bb
Lemmas referenced :
bool_wf,
list_wf,
filter_is_interleaving,
set_wf,
subtype_rel_self,
subtype_rel_dep_function,
l_member_wf,
bnot_wf,
filter_wf5,
interleaving_implies_occurence,
equal_wf,
exists_wf,
isect_wf,
all_wf,
interleaving_occurence_wf,
nat_properties,
length_wf_nat,
lelt_wf,
non_neg_length,
not_wf,
int_seg_wf,
assert_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
assert_witness,
interleaving_occurence_onto,
int_formula_prop_eq_lemma,
intformeq_wf,
iff_weakening_equal,
less_than_wf,
le_wf,
true_wf,
squash_wf,
false_wf,
int_seg_subtype_nat,
member_filter,
assert_of_bnot,
int_subtype_base,
subtype_base_sq,
btrue_neq_bfalse,
and_wf,
bfalse_wf,
assert_elim
Rules used in proof :
universeEquality,
functionEquality,
hypothesis,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
setEquality,
rename,
setElimination,
functionExtensionality,
applyEquality,
lambdaEquality,
dependent_functionElimination,
productEquality,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
imageElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
unionElimination,
natural_numberEquality,
independent_pairFormation,
dependent_pairFormation,
productElimination,
lambdaFormation_alt,
universeIsType,
setIsType,
lambdaEquality_alt,
baseClosed,
imageMemberEquality,
instantiate,
levelHypothesis,
addLevel
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List.
\mexists{}f1:\mBbbN{}||filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L)|| {}\mrightarrow{} \mBbbN{}||L||
\mexists{}f2:\mBbbN{}||filter(P;L)|| {}\mrightarrow{} \mBbbN{}||L||
(interleaving\_occurence(T;filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L);filter(P;L);L;f1;f2)
\mwedge{} ((\mforall{}i:\mBbbN{}||L||
\mexists{}k:\mBbbN{}||filter(P;L)||. ((i = (f2 k)) \mwedge{} (L[i] = filter(P;L)[k])) supposing \muparrow{}(P L[i]))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
\mexists{}k:\mBbbN{}||filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L)||. ((i = (f1 k)) \mwedge{} (L[i] = filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L)[k]))
supposing \mneg{}\muparrow{}(P L[i])))
\mwedge{} (\mforall{}i:\mBbbN{}||filter(\mlambda{}x.(\mneg{}\msubb{}(P x));L)||. (\mneg{}\muparrow{}(P L[f1 i])))
\mwedge{} (\mforall{}i:\mBbbN{}||filter(P;L)||. (\muparrow{}(P L[f2 i]))))
Date html generated:
2020_05_20-AM-07_48_54
Last ObjectModification:
2020_01_25-AM-09_00_33
Theory : list!
Home
Index