Nuprl Lemma : decidable-filter
∀[T:Type]
∀L:T List
∀[P:{x:T| (x ∈ L)} ⟶ ℙ]. ((∀x∈L.Dec(P[x])) ⇒ (∃L':T List. (L' ⊆ L ∧ (∀x:T. ((x ∈ L') ⇐⇒ (x ∈ L) ∧ P[x])))))
Proof
Definitions occuring in Statement :
sublist: L1 ⊆ L2,
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
list: T List,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
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],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
top: Top,
uimplies: b supposing a,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
guard: {T},
cons: [a / b],
true: True,
sublist: L1 ⊆ L2,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
le: A ≤ B,
nat: ℕ,
l_member: (x ∈ l)
Lemmas referenced :
list_induction,
uall_wf,
l_all_wf,
l_member_wf,
decidable_wf,
exists_wf,
list_wf,
sublist_wf,
all_wf,
iff_wf,
l_all_wf_nil,
cons_wf,
nil_wf,
nil-sublist,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
l_all_cons,
cons_member,
cons_sublist_cons,
equal_wf,
and_wf,
list-cases,
product_subtype_list,
nil_sublist,
list-subtype,
subtype_rel_list,
int_seg_wf,
length_wf,
increasing_wf,
length_wf_nat,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
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,
non_neg_length,
lelt_wf,
nat_properties,
l_member-settype,
set_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
because_Cache,
universeEquality,
hypothesis,
setElimination,
rename,
applyEquality,
functionExtensionality,
setEquality,
productEquality,
independent_functionElimination,
voidElimination,
voidEquality,
dependent_functionElimination,
dependent_pairFormation,
isect_memberEquality,
independent_pairFormation,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
unionElimination,
inlFormation,
inrFormation,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
promote_hyp,
hypothesis_subsumption,
natural_numberEquality,
addLevel,
levelHypothesis,
int_eqEquality,
intEquality,
computeAll,
imageElimination
Latex:
\mforall{}[T:Type]
\mforall{}L:T List
\mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbP{}]
((\mforall{}x\mmember{}L.Dec(P[x])) {}\mRightarrow{} (\mexists{}L':T List. (L' \msubseteq{} L \mwedge{} (\mforall{}x:T. ((x \mmember{} L') \mLeftarrow{}{}\mRightarrow{} (x \mmember{} L) \mwedge{} P[x])))))
Date html generated:
2017_10_01-AM-09_13_07
Last ObjectModification:
2017_07_26-PM-04_48_38
Theory : general
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