Nuprl Lemma : l_before-filter
∀[T:Type]
∀l:T List. ∀P:{x:T| (x ∈ l)} ⟶ 𝔹. ∀x,y:T. (x before y ∈ filter(P;l) ⇐⇒ x before y ∈ l ∧ (↑(P x)) ∧ (↑(P y)))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
l_member: (x ∈ 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,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
l_before: x before y ∈ l,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T},
rev_implies: P ⇐ Q,
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B
Lemmas referenced :
l_member_wf,
bool_wf,
list_wf,
filter_wf2,
subtype_rel_list,
member_sublist,
cons_wf,
nil_wf,
member_filter_2,
cons_member,
assert_witness,
equal_wf,
sublist_wf,
assert_wf,
sublist_filter_2,
l_all_cons,
l_all_single
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
hypothesisEquality,
functionEquality,
setEquality,
cumulativity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
universeEquality,
functionExtensionality,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
independent_pairFormation,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
allFunctionality,
productElimination,
promote_hyp,
inlFormation,
dependent_set_memberEquality,
inrFormation,
productEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]
\mforall{}l:T List. \mforall{}P:\{x:T| (x \mmember{} l)\} {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:T.
(x before y \mmember{} filter(P;l) \mLeftarrow{}{}\mRightarrow{} x before y \mmember{} l \mwedge{} (\muparrow{}(P x)) \mwedge{} (\muparrow{}(P y)))
Date html generated:
2017_04_17-AM-07_28_19
Last ObjectModification:
2017_02_27-PM-04_06_10
Theory : list_1
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