Nuprl Lemma : l_all_filter_iff
∀[T:Type]. ∀[Q:T ⟶ ℙ]. ∀P:T ⟶ 𝔹. ∀L:T List. ((∀x∈filter(P;L).Q[x]) ⇐⇒ ∀x:T. ((x ∈ L) ⇒ (↑(P x)) ⇒ Q[x]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
cand: A c∧ B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
assert_wf,
l_member_wf,
all_wf,
and_wf,
l_all_iff,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
subtype_rel_self,
set_wf,
member_filter,
l_all_wf,
iff_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
independent_pairFormation,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
independent_functionElimination,
lemma_by_obid,
isectElimination,
applyEquality,
sqequalRule,
lambdaEquality,
functionEquality,
productElimination,
addLevel,
because_Cache,
setEquality,
independent_isectElimination,
setElimination,
rename,
allFunctionality,
cumulativity,
impliesFunctionality,
levelHypothesis,
allLevelFunctionality,
impliesLevelFunctionality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[Q:T {}\mrightarrow{} \mBbbP{}].
\mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}L:T List. ((\mforall{}x\mmember{}filter(P;L).Q[x]) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. ((x \mmember{} L) {}\mRightarrow{} (\muparrow{}(P x)) {}\mRightarrow{} Q[x]))
Date html generated:
2016_05_14-AM-06_52_17
Last ObjectModification:
2015_12_26-PM-00_21_42
Theory : list_0
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