Nuprl Lemma : length-filter-le
∀[T:Type]. ∀[P1,P2:T ⟶ 𝔹]. ∀[L:T List]. ||filter(P1;L)|| ≤ ||filter(P2;L)|| supposing (∀x∈L.(↑(P1 x)) ⇒ (↑(P2 x)))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
length: ||as||,
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
isect_wf,
l_all_wf,
l_member_wf,
assert_wf,
le_wf,
length_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
list_wf,
filter_nil_lemma,
length_of_nil_lemma,
false_wf,
less_than'_wf,
l_all_wf_nil,
filter_cons_lemma,
l_all_cons,
eqtt_to_assert,
length_of_cons_lemma,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
lambdaFormation,
hypothesis,
setElimination,
rename,
functionEquality,
applyEquality,
functionExtensionality,
because_Cache,
setEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
natural_numberEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
addEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
promote_hyp,
instantiate,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P1,P2:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List].
||filter(P1;L)|| \mleq{} ||filter(P2;L)|| supposing (\mforall{}x\mmember{}L.(\muparrow{}(P1 x)) {}\mRightarrow{} (\muparrow{}(P2 x)))
Date html generated:
2017_04_17-AM-07_48_24
Last ObjectModification:
2017_02_27-PM-04_22_52
Theory : list_1
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