Nuprl Lemma : interleaving_split
∀[T:Type]
∀L:T List
∀[P:ℕ||L|| ⟶ ℙ]
((∀x:ℕ||L||. Dec(P x))
⇒ (∃L1,L2:T List
∃f1:ℕ||L1|| ⟶ ℕ||L||
∃f2:ℕ||L2|| ⟶ ℕ||L||
(interleaving_occurence(T;L1;L2;L;f1;f2)
∧ ((∀i:ℕ||L1||. (P (f1 i))) ∧ (∀i:ℕ||L2||. (¬(P (f2 i)))))
∧ (∀i:ℕ||L||
(((P i) ⇒ (∃j:ℕ||L1||. ((f1 j) = i ∈ ℤ))) ∧ ∃j:ℕ||L2||. ((f2 j) = i ∈ ℤ) supposing ¬(P i))))))
Proof
Definitions occuring in Statement :
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
length: ||as||,
list: T List,
int_seg: {i..j-},
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_type: SQType(T),
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
sublist_occurence: sublist_occurence(T;L1;L2;f),
interleaving_occurence: interleaving_occurence(T;L1;L2;L;f1;f2),
true: True,
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
increasing_split,
length_wf_nat,
all_wf,
int_seg_wf,
length_wf,
decidable_wf,
list_wf,
range_sublist,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
subtype_rel_self,
interleaving_occurence_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf,
not_wf,
exists_wf,
equal_wf,
int_seg_properties,
disjoint_increasing_onto,
itermAdd_wf,
int_term_value_add_lemma,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
productElimination,
natural_numberEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionIsType,
universeIsType,
universeEquality,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
instantiate,
cumulativity,
intEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
dependent_set_memberEquality,
functionEquality,
productEquality,
functionExtensionality,
isectEquality,
promote_hyp,
lambdaFormation_alt,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbP{}]
((\mforall{}x:\mBbbN{}||L||. Dec(P x))
{}\mRightarrow{} (\mexists{}L1,L2:T List
\mexists{}f1:\mBbbN{}||L1|| {}\mrightarrow{} \mBbbN{}||L||
\mexists{}f2:\mBbbN{}||L2|| {}\mrightarrow{} \mBbbN{}||L||
(interleaving\_occurence(T;L1;L2;L;f1;f2)
\mwedge{} ((\mforall{}i:\mBbbN{}||L1||. (P (f1 i))) \mwedge{} (\mforall{}i:\mBbbN{}||L2||. (\mneg{}(P (f2 i)))))
\mwedge{} (\mforall{}i:\mBbbN{}||L||
(((P i) {}\mRightarrow{} (\mexists{}j:\mBbbN{}||L1||. ((f1 j) = i)))
\mwedge{} \mexists{}j:\mBbbN{}||L2||. ((f2 j) = i) supposing \mneg{}(P i))))))
Date html generated:
2019_10_15-AM-10_57_17
Last ObjectModification:
2018_09_27-AM-09_57_58
Theory : list!
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