Nuprl Lemma : filter_before
∀[A:Type]. ∀L:A List. ∀P:A ⟶ 𝔹. ∀x,y:A. (x before y ∈ filter(P;L) ⇐⇒ (↑(P x)) ∧ (↑(P y)) ∧ x before y ∈ L)
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
top: Top,
implies: P ⇒ Q,
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
so_apply: x[s],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
false: False,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
or: P ∨ Q,
cand: A c∧ B,
istype: istype(T),
not: ¬A
Lemmas referenced :
filter_cons_lemma,
filter_nil_lemma,
list_wf,
assert_wf,
set_wf,
subtype_rel_self,
l_member_wf,
subtype_rel_dep_function,
filter_wf5,
l_before_wf,
iff_wf,
bool_wf,
all_wf,
list_induction,
nil_wf,
nil_before,
false_wf,
assert_of_bnot,
eqff_to_assert,
uiff_transitivity,
eqtt_to_assert,
not_wf,
bnot_wf,
equal-wf-T-base,
cons_wf,
cons_before,
equal_wf,
or_wf,
assert_witness,
member_filter,
member_filter_2,
assert_elim,
not_assert_elim,
btrue_neq_bfalse
Rules used in proof :
universeEquality,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
independent_functionElimination,
productEquality,
rename,
setElimination,
independent_isectElimination,
setEquality,
because_Cache,
applyEquality,
hypothesis,
functionEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
promote_hyp,
productElimination,
independent_pairFormation,
natural_numberEquality,
equalityElimination,
unionElimination,
inrFormation,
baseClosed,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
hyp_replacement,
inlFormation_alt,
lambdaEquality_alt,
inhabitedIsType,
setIsType,
universeIsType,
lambdaFormation_alt,
inrFormation_alt,
productIsType,
equalityIstype,
dependent_set_memberEquality_alt
Latex:
\mforall{}[A:Type]
\mforall{}L:A List. \mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:A. (x before y \mmember{} filter(P;L) \mLeftarrow{}{}\mRightarrow{} (\muparrow{}(P x)) \mwedge{} (\muparrow{}(P y)) \mwedge{} x before y \mmember{} L)
Date html generated:
2019_10_15-AM-10_21_49
Last ObjectModification:
2019_08_20-PM-05_00_31
Theory : list_1
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