Nuprl Lemma : filter_interleaving
∀[T:Type]
∀P:T ⟶ 𝔹. ∀L,L1,L2:T List. (interleaving(T;L1;L2;L) ⇒ interleaving(T;filter(P;L1);filter(P;L2);filter(P;L)))
Proof
Definitions occuring in Statement :
interleaving: interleaving(T;L1;L2;L),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
top: Top,
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
not: ¬A,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
le: A ≤ B,
less_than': less_than'(a;b),
select: L[n],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than: a < b,
squash: ↓T,
cons: [a / b],
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
colength: colength(L),
guard: {T},
decidable: Dec(P),
sq_type: SQType(T)
Lemmas referenced :
list_induction,
all_wf,
list_wf,
interleaving_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
interleaving_of_nil,
filter_nil_lemma,
nil_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
and_wf,
equal_wf,
null_wf,
btrue_neq_bfalse,
nil_interleaving,
interleaving_of_cons,
cons_wf,
less_than_wf,
length_wf,
select_wf,
false_wf,
tl_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
reduce_tl_nil_lemma,
length_of_cons_lemma,
reduce_tl_cons_lemma,
cons_interleaving,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
filter_cons_lemma,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
cons_interleaving2,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
nat_wf,
colength_wf_list,
int_subtype_base,
list-cases,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
decidable__equal_int,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
hypothesis,
functionEquality,
applyEquality,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination,
dependent_functionElimination,
universeEquality,
productElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
unionElimination,
natural_numberEquality,
cumulativity,
baseClosed,
imageElimination,
addEquality,
equalityElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
axiomEquality,
promote_hyp,
hypothesis_subsumption,
instantiate
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L,L1,L2:T List.
(interleaving(T;L1;L2;L) {}\mRightarrow{} interleaving(T;filter(P;L1);filter(P;L2);filter(P;L)))
Date html generated:
2019_10_15-AM-10_56_43
Last ObjectModification:
2018_09_17-PM-06_39_19
Theory : list!
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