Nuprl Lemma : sorted-filter
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. sorted(filter(P;L)) supposing sorted(L) supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
sorted: sorted(L),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
top: Top,
sorted: sorted(L),
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
sorted_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
list_wf,
filter_nil_lemma,
filter_cons_lemma,
nil_wf,
sorted-cons,
cons_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
equal_wf,
l_all_iff,
le_wf,
member_filter
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
independent_isectElimination,
hypothesis,
applyEquality,
because_Cache,
setEquality,
setElimination,
rename,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
independent_pairFormation,
baseClosed,
unionElimination,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. sorted(filter(P;L)) supposing sorted(L) supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2017_04_14-AM-08_52_48
Last ObjectModification:
2017_02_27-PM-03_38_25
Theory : list_0
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