Nuprl Lemma : filter_is_singleton2
∀[T:Type]
∀P:T ⟶ 𝔹. ∀L:T List.
(||filter(P;L)|| = 1 ∈ ℤ ⇐⇒ ∃i:ℕ||L||. ((↑(P L[i])) ∧ (∀j:ℕ||L||. i = j ∈ ℤ supposing ↑(P L[j]))))
Proof
Definitions occuring in Statement :
select: L[n],
length: ||as||,
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
sq_type: SQType(T),
true: True,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
cons: [a / b],
cand: A c∧ B,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m
Lemmas referenced :
bool_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
list_induction,
iff_wf,
exists_wf,
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,
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,
all_wf,
isect_wf,
equal_wf,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
filter_cons_lemma,
length_of_cons_lemma,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
cons_wf,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
decidable__assert,
null_wf,
not_functionality_wrt_uiff,
assert_of_null,
pos_length,
intformeq_wf,
int_formula_prop_eq_lemma,
filter_is_empty,
false_wf,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
lelt_wf,
decidable__equal_int,
assert_functionality_wrt_uiff,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select_cons_tl,
add-member-int_seg2,
select-cons-tl,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
applyEquality,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
equalityTransitivity,
equalitySymmetry,
sqequalHypSubstitution,
isectElimination,
thin,
baseClosed,
isect_memberFormation,
lambdaFormation,
sqequalRule,
lambdaEquality,
because_Cache,
natural_numberEquality,
cumulativity,
productEquality,
functionExtensionality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
instantiate,
equalityElimination,
functionEquality,
universeEquality,
addEquality,
setEquality,
dependent_set_memberEquality,
imageMemberEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
hyp_replacement
Latex:
\mforall{}[T:Type]
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List.
(||filter(P;L)|| = 1 \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. ((\muparrow{}(P L[i])) \mwedge{} (\mforall{}j:\mBbbN{}||L||. i = j supposing \muparrow{}(P L[j]))))
Date html generated:
2019_06_20-PM-01_26_15
Last ObjectModification:
2018_09_17-PM-06_29_20
Theory : list_1
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