Nuprl Lemma : pairwise-filter
∀[T:Type]. ∀L:T List. ∀[P:T ⟶ T ⟶ ℙ']. ((∀x,y∈L. P[x;y]) ⇒ (∀Q:T ⟶ 𝔹. (∀x,y∈filter(Q;L). P[x;y])))
Proof
Definitions occuring in Statement :
pairwise: (∀x,y∈L. P[x; y]),
filter: filter(P;l),
list: T List,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
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,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a
Lemmas referenced :
bool_wf,
pairwise_wf2,
list_wf,
pairwise-sublist,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
set_wf,
filter_is_sublist
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
functionEquality,
hypothesisEquality,
cut,
lemma_by_obid,
hypothesis,
thin,
instantiate,
sqequalHypSubstitution,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
applyEquality,
universeEquality,
dependent_functionElimination,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
independent_functionElimination
Latex:
\mforall{}[T:Type]
\mforall{}L:T List. \mforall{}[P:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}']. ((\mforall{}x,y\mmember{}L. P[x;y]) {}\mRightarrow{} (\mforall{}Q:T {}\mrightarrow{} \mBbbB{}. (\mforall{}x,y\mmember{}filter(Q;L). P[x;y])))
Date html generated:
2016_05_14-PM-01_50_15
Last ObjectModification:
2015_12_26-PM-05_36_45
Theory : list_1
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