Nuprl Lemma : sublist_filter_2
∀[T:Type]. ∀L1,L2:T List. ∀P:{x:T| (x ∈ L1)} ⟶ 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]),
l_member: (x ∈ l),
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,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
sublist: L1 ⊆ L2,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
nat: ℕ,
cand: A c∧ B,
true: True
Lemmas referenced :
l_member_wf,
bool_wf,
list_wf,
sublist_filter,
list-subtype,
filter_wf2,
subtype_rel_list,
member_sublist,
member_filter_2,
subtype_rel_list_set,
increasing_wf,
length_wf_nat,
int_seg_wf,
length_wf,
select_wf,
int_seg_properties,
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,
non_neg_length,
le_wf,
less_than_wf,
nat_properties,
select_member,
equal_wf,
sublist_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
l_all_wf,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
Error :functionIsType,
Error :setIsType,
Error :universeIsType,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
Error :inhabitedIsType,
universeEquality,
setEquality,
dependent_functionElimination,
applyEquality,
independent_isectElimination,
Error :lambdaEquality_alt,
setElimination,
rename,
sqequalRule,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
because_Cache,
Error :dependent_pairFormation_alt,
Error :productIsType,
functionExtensionality,
natural_numberEquality,
Error :equalityIsType1,
unionElimination,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
imageElimination,
Error :dependent_set_memberEquality_alt,
applyLambdaEquality,
promote_hyp,
hyp_replacement,
imageMemberEquality,
baseClosed,
instantiate
Latex:
\mforall{}[T:Type]. \mforall{}L1,L2:T List. \mforall{}P:\{x:T| (x \mmember{} L1)\} {}\mrightarrow{} \mBbbB{}. (L2 \msubseteq{} filter(P;L1) \mLeftarrow{}{}\mRightarrow{} L2 \msubseteq{} L1 \mwedge{} (\mforall{}x\mmember{}L2.\muparrow{}(P x))\000C)
Date html generated:
2019_06_20-PM-01_26_25
Last ObjectModification:
2018_10_15-PM-05_47_45
Theory : list_1
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