Nuprl Lemma : filter_is_singleton
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. ∀[x:T].
(filter(P;L) = [x] ∈ (T List)) supposing ((∀y∈L.(↑P[y]) ⇒ (y = x ∈ T)) and (↑P[x]) and (x ∈! L))
Proof
Definitions occuring in Statement :
l_member!: (x ∈! l),
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
cons: [a / b],
nil: [],
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
nat: ℕ,
and: P ∧ Q,
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
select: L[n],
cand: A c∧ B,
exists: ∃x:A. B[x],
l_member!: (x ∈! l),
iff: P ⇐⇒ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
true: True,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
l_all_wf_nil,
nil_wf,
cons_wf,
set_wf,
subtype_rel_self,
bool_wf,
subtype_rel_dep_function,
filter_wf5,
list_wf,
l_member_wf,
equal_wf,
l_all_wf,
assert_wf,
l_member!_wf,
isect_wf,
uall_wf,
list_induction,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
filter_nil_lemma,
length_of_nil_lemma,
base_wf,
stuck-spread,
l_all_cons,
cons_member!,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
filter_cons_lemma,
true_wf,
squash_wf,
filter_is_nil,
not_wf,
l_all_iff,
and_wf,
iff_weakening_equal
Rules used in proof :
universeEquality,
Error :functionIsType,
dependent_functionElimination,
Error :inhabitedIsType,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
isect_memberEquality,
voidEquality,
voidElimination,
Error :universeIsType,
independent_functionElimination,
lambdaFormation,
independent_isectElimination,
setEquality,
because_Cache,
rename,
setElimination,
functionEquality,
applyEquality,
hypothesis,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
thin,
cut,
introduction,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
natural_numberEquality,
baseClosed,
productElimination,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
unionElimination,
imageMemberEquality,
imageElimination,
applyLambdaEquality,
dependent_set_memberEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[x:T].
(filter(P;L) = [x]) supposing ((\mforall{}y\mmember{}L.(\muparrow{}P[y]) {}\mRightarrow{} (y = x)) and (\muparrow{}P[x]) and (x \mmember{}! L))
Date html generated:
2019_06_20-PM-01_26_07
Last ObjectModification:
2019_01_28-PM-10_41_08
Theory : list_1
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