Nuprl Lemma : length-filter-bnot
∀[T:Type]. ∀[L:T List]. ∀[P:{x:T| (x ∈ L)} ⟶ 𝔹]. (||filter(λa.(¬bP[a]);L)|| = (||L|| - ||filter(λa.P[a];L)||) ∈ ℤ)
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
length: ||as||,
filter: filter(P;l),
list: T List,
bnot: ¬bb,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
all: ∀x:A. B[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
subtract: n - m,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
decidable: Dec(P),
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
sq_type: SQType(T),
assert: ↑b
Lemmas referenced :
list_induction,
uall_wf,
l_member_wf,
bool_wf,
equal_wf,
length_wf,
filter_wf5,
bnot_wf,
subtract_wf,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
nil_wf,
filter_cons_lemma,
length_of_cons_lemma,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
subtype_rel_self,
set_wf,
eqtt_to_assert,
decidable__equal_int,
subtract-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
setEquality,
cumulativity,
because_Cache,
hypothesis,
lambdaFormation,
setElimination,
rename,
intEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
inrFormation,
inlFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
dependent_pairFormation,
int_eqEquality,
independent_pairFormation,
computeAll,
instantiate,
axiomEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[P:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbB{}].
(||filter(\mlambda{}a.(\mneg{}\msubb{}P[a]);L)|| = (||L|| - ||filter(\mlambda{}a.P[a];L)||))
Date html generated:
2017_04_17-AM-08_58_52
Last ObjectModification:
2017_02_27-PM-05_15_43
Theory : list_1
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