Nuprl Lemma : filter2_wf
∀[T:Type]. ∀[L:T List]. ∀[P:ℕ||L|| ⟶ 𝔹]. (filter2(P;L) ∈ T List)
Proof
Definitions occuring in Statement :
filter2: filter2(P;L),
length: ||as||,
list: T List,
int_seg: {i..j-},
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
filter2: filter2(P;L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
reduce2_wf,
list_wf,
nil_wf,
false_wf,
le_wf,
int_seg_wf,
length_wf,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
lelt_wf,
bool_wf,
eqtt_to_assert,
cons_wf,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
lambdaFormation,
lambdaEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
productElimination,
dependent_functionElimination,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
imageElimination,
baseClosed,
baseApply,
closedConclusion,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
independent_functionElimination,
addEquality,
axiomEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\mBbbN{}||L|| {}\mrightarrow{} \mBbbB{}]. (filter2(P;L) \mmember{} T List)
Date html generated:
2017_10_01-AM-08_35_05
Last ObjectModification:
2017_07_26-PM-04_25_41
Theory : list!
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