Nuprl Lemma : sublist_filter
∀[T:Type]. ∀L1,L2:T List. ∀P:T ⟶ 𝔹. (L2 ⊆ filter(P;L1) ⇐⇒ L2 ⊆ L1 ∧ (∀x∈L2.↑(P x)))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
top: Top,
it: ⋅,
nil: [],
list_ind: list_ind,
reduce: reduce(f;k;as),
filter: filter(P;l),
implies: P ⇒ Q,
and: P ∧ Q,
istype: istype(T),
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
true: True,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
sq_type: SQType(T),
assert: ↑b
Lemmas referenced :
istype-universe,
filter_cons_lemma,
cons_wf,
istype-void,
filter_nil_lemma,
nil_wf,
assert_wf,
l_all_wf,
l_member_wf,
subtype_rel_dep_function,
filter_wf5,
sublist_wf,
iff_wf,
bool_wf,
list_wf,
list_induction,
l_all_nil,
assert_witness,
select_wf,
length_of_nil_lemma,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
length_wf,
l_all_wf_nil,
false_wf,
cons_sublist_nil,
true_wf,
ifthenelse_wf,
subtype_rel_self,
set_wf,
nil_sublist,
not_wf,
bnot_wf,
equal-wf-T-base,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
l_all_cons,
or_wf,
equal_wf,
cons_sublist_cons,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base,
istype-assert
Rules used in proof :
universeEquality,
instantiate,
inhabitedIsType,
productIsType,
functionIsType,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
independent_functionElimination,
productEquality,
rename,
setElimination,
independent_isectElimination,
setIsType,
setEquality,
universeIsType,
because_Cache,
applyEquality,
hypothesis,
functionEquality,
lambdaEquality_alt,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
lambdaFormation,
independent_pairFormation,
isect_memberEquality,
voidEquality,
lambdaEquality,
functionExtensionality,
cumulativity,
natural_numberEquality,
productElimination,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairEquality,
promote_hyp,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
equalityElimination,
equalityIstype,
imageElimination,
inlFormation,
inrFormation,
dependent_set_memberEquality,
applyLambdaEquality,
inrFormation_alt,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. (L2 \msubseteq{} filter(P;L1) \mLeftarrow{}{}\mRightarrow{} L2 \msubseteq{} L1 \mwedge{} (\mforall{}x\mmember{}L2.\muparrow{}(P x)))
Date html generated:
2019_10_15-AM-10_22_07
Last ObjectModification:
2019_08_05-PM-02_12_08
Theory : list_1
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