Nuprl Lemma : l_before_mapfilter
∀[A,B:Type].
∀l:A List. ∀P:A ⟶ 𝔹. ∀f:{a:A| ↑(P a)} ⟶ B. ∀x,y:B.
(x before y ∈ mapfilter(f;P;l)
⇐⇒ ∃u,v:A. ((↑(P u)) ∧ (↑(P v)) ∧ (x = (f u) ∈ B) ∧ (y = (f v) ∈ B) ∧ u before v ∈ l))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_before: x before y ∈ l,
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃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,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
mapfilter: mapfilter(f;P;L),
member: t ∈ T,
prop: ℙ,
exists: ∃x:A. B[x],
cand: A c∧ B,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
l_before_map,
assert_wf,
filter_type,
assert_elim,
subtype_base_sq,
bool_subtype_base,
l_before_filter_subtype,
l_before_filter,
equal_wf,
l_before_wf,
exists_wf,
mapfilter_wf,
bool_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
because_Cache,
dependent_functionElimination,
productElimination,
independent_functionElimination,
dependent_pairFormation,
setElimination,
rename,
independent_isectElimination,
instantiate,
natural_numberEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type].
\mforall{}l:A List. \mforall{}P:A {}\mrightarrow{} \mBbbB{}. \mforall{}f:\{a:A| \muparrow{}(P a)\} {}\mrightarrow{} B. \mforall{}x,y:B.
(x before y \mmember{} mapfilter(f;P;l)
\mLeftarrow{}{}\mRightarrow{} \mexists{}u,v:A. ((\muparrow{}(P u)) \mwedge{} (\muparrow{}(P v)) \mwedge{} (x = (f u)) \mwedge{} (y = (f v)) \mwedge{} u before v \mmember{} l))
Date html generated:
2017_04_17-AM-07_26_51
Last ObjectModification:
2017_02_27-PM-04_04_56
Theory : list_1
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