Nuprl Lemma : l_all_filter
∀[T:Type]. ∀[P:T ⟶ 𝔹]. ∀[L:T List]. (∀x∈filter(P;L).↑(P x))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T
Lemmas referenced :
list_wf,
length_wf,
int_seg_wf,
sq_stable__le,
select_wf,
assert_witness,
member_filter,
assert_wf,
set_wf,
subtype_rel_self,
l_member_wf,
bool_wf,
subtype_rel_dep_function,
filter_wf5,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
hypothesis,
setEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
productElimination,
independent_functionElimination,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
isect_memberEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. (\mforall{}x\mmember{}filter(P;L).\muparrow{}(P x))
Date html generated:
2016_05_14-AM-06_52_12
Last ObjectModification:
2016_01_14-PM-08_14_23
Theory : list_0
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