Nuprl Lemma : filter_fseg
∀[T:Type]. ∀P:T ⟶ 𝔹. ∀L2,L1:T List. (fseg(T;L1;L2) ⇒ fseg(T;filter(P;L1);filter(P;L2)))
Proof
Definitions occuring in Statement :
fseg: fseg(T;L1;L2),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
fseg: fseg(T;L1;L2),
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
length_wf_nat,
equal_wf,
nat_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
filter_append,
append_wf,
list_wf,
exists_wf,
fseg_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
dependent_set_memberEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
dependent_pairFormation,
applyEquality,
sqequalRule,
lambdaEquality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
equalityTransitivity,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L2,L1:T List. (fseg(T;L1;L2) {}\mRightarrow{} fseg(T;filter(P;L1);filter(P;L2)))
Date html generated:
2016_10_25-AM-10_45_27
Last ObjectModification:
2016_07_12-AM-06_55_17
Theory : general
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