Nuprl Lemma : member_filter2
∀[T:Type]. ∀L:T List. ∀P:ℕ||L|| ⟶ 𝔹. ∀x:T. ((x ∈ filter2(P;L)) ⇐⇒ ∃i:ℕ||L||. ((x = L[i] ∈ T) ∧ (↑(P i))))
Proof
Definitions occuring in Statement :
filter2: filter2(P;L),
l_member: (x ∈ l),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
assert: ↑b,
bool: 𝔹,
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,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
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,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
cand: A c∧ B,
nat: ℕ,
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
nat_plus: ℕ+,
true: True,
uiff: uiff(P;Q),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
subtract: n - m,
cons: [a / b]
Lemmas referenced :
list_induction,
all_wf,
int_seg_wf,
length_wf,
bool_wf,
iff_wf,
l_member_wf,
filter2_wf,
exists_wf,
equal_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,
assert_wf,
istype-universe,
length_of_nil_lemma,
filter2_nil_lemma,
stuck-spread,
istype-base,
length_of_cons_lemma,
list_wf,
nat_properties,
nat_wf,
less_than_wf,
istype-false,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
le_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
cons_filter2,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
non_neg_length,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
add-member-int_seg2,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
select_cons_tl,
select-cons-tl,
add-subtract-cancel,
select-cons-hd,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
natural_numberEquality,
hypothesis,
because_Cache,
productEquality,
setElimination,
rename,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
imageElimination,
applyEquality,
functionIsType,
baseClosed,
inhabitedIsType,
productIsType,
equalityIsType1,
universeEquality,
dependent_set_memberEquality_alt,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
equalityElimination,
instantiate,
cumulativity,
intEquality
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}. \mforall{}x:T. ((x \mmember{} filter2(P;L)) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}||L||. ((x = L[i]) \mwedge{} (\muparrow{}(P i))))
Date html generated:
2019_10_15-AM-10_55_05
Last ObjectModification:
2018_10_09-AM-10_21_29
Theory : list!
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