Nuprl Lemma : l_before_filter_set_type
∀[T:Type]. ∀l:T List. ∀P:T ⟶ 𝔹. ∀x,y:{x:T| ↑(P x)} . (x before y ∈ l ⇒ x before y ∈ filter(P;l))
Proof
Definitions occuring in Statement :
l_before: x before y ∈ l,
filter: filter(P;l),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: 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],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
top: Top
Lemmas referenced :
sublist_wf,
cons_wf,
nil_wf,
set_wf,
assert_wf,
bool_wf,
list_wf,
sublist_filter_set_type,
l_all_cons,
assert_elim,
subtype_base_sq,
bool_subtype_base,
l_all_nil
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
lambdaEquality,
applyEquality,
functionEquality,
Error :universeIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
independent_isectElimination,
setEquality,
productElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}P:T {}\mrightarrow{} \mBbbB{}. \mforall{}x,y:\{x:T| \muparrow{}(P x)\} . (x before y \mmember{} l {}\mRightarrow{} x before y \mmember{} filter(P;l))
Date html generated:
2019_06_20-PM-01_25_30
Last ObjectModification:
2018_09_26-PM-05_29_01
Theory : list_1
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