Nuprl Lemma : member-mapfilter-witness_wf
member-mapfilter-witness() ∈ ∀[T:Type]
                               ∀L:T List. ∀P:{x:T| (x ∈ L)}  ⟶ 𝔹.
                                 ∀[T':Type]
                                   ∀f:{x:T| (x ∈ L) c∧ (↑(P x))}  ⟶ T'. ∀x:T'.
                                     ((x ∈ mapfilter(f;P;L)) 
⇐⇒ ∃y:T. ((y ∈ L) ∧ ((↑(P y)) c∧ (x = (f y) ∈ T'))))
Proof
Definitions occuring in Statement : 
member-mapfilter-witness: member-mapfilter-witness()
, 
mapfilter: mapfilter(f;P;L)
, 
l_member: (x ∈ l)
, 
list: T List
, 
assert: ↑b
, 
bool: 𝔹
, 
uall: ∀[x:A]. B[x]
, 
cand: A c∧ B
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
universe: Type
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
member: t ∈ T
, 
member-mapfilter-witness: member-mapfilter-witness()
, 
member-mapfilter-univ, 
member_map_filter, 
uall: ∀[x:A]. B[x]
, 
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w])
, 
so_apply: x[s1;s2;s3;s4]
, 
so_lambda: λ2x y.t[x; y]
, 
top: Top
, 
so_apply: x[s1;s2]
, 
uimplies: b supposing a
, 
member-map, 
member_map, 
select: L[n]
, 
subtract: n - m
, 
filter: filter(P;l)
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
ifthenelse: if b then t else f fi 
, 
cons: [a / b]
, 
nil: []
, 
it: ⋅
, 
strict4: strict4(F)
, 
and: P ∧ Q
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
has-value: (a)↓
, 
prop: ℙ
, 
guard: {T}
, 
or: P ∨ Q
, 
squash: ↓T
, 
member_filter, 
list_induction, 
cons_member, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
decidable__assert, 
eq_int: (i =z j)
, 
btrue: tt
, 
bfalse: ff
, 
iff_weakening_equal, 
uiff_transitivity, 
nil_member, 
l_member-settype, 
assert_of_bnot, 
l_member_set2, 
select_member
Lemmas referenced : 
member-mapfilter-univ, 
lifting-strict-spread, 
strict4-spread, 
has-value_wf_base, 
base_wf, 
is-exception_wf, 
lifting-strict-decide, 
top_wf, 
equal_wf, 
lifting-strict-int_eq, 
member_map_filter, 
member-map, 
member_map, 
member_filter, 
list_induction, 
cons_member, 
decidable__assert, 
iff_weakening_equal, 
uiff_transitivity, 
nil_member, 
l_member-settype, 
assert_of_bnot, 
l_member_set2, 
select_member
Rules used in proof : 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
instantiate, 
extract_by_obid, 
hypothesis, 
sqequalRule, 
thin, 
sqequalHypSubstitution, 
equalityTransitivity, 
equalitySymmetry, 
introduction, 
isectElimination, 
baseClosed, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_isectElimination, 
independent_pairFormation, 
lambdaFormation, 
callbyvalueApply, 
baseApply, 
closedConclusion, 
hypothesisEquality, 
applyExceptionCases, 
inrFormation, 
imageMemberEquality, 
imageElimination, 
exceptionSqequal, 
inlFormation, 
callbyvalueDecide, 
unionEquality, 
unionElimination, 
sqleReflexivity, 
dependent_functionElimination, 
independent_functionElimination, 
decideExceptionCases, 
because_Cache
Latex:
member-mapfilter-witness()  \mmember{}  \mforall{}[T:Type]
                                                              \mforall{}L:T  List.  \mforall{}P:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbB{}.
                                                                  \mforall{}[T':Type]
                                                                      \mforall{}f:\{x:T|  (x  \mmember{}  L)  c\mwedge{}  (\muparrow{}(P  x))\}    {}\mrightarrow{}  T'.  \mforall{}x:T'.
                                                                          ((x  \mmember{}  mapfilter(f;P;L))
                                                                          \mLeftarrow{}{}\mRightarrow{}  \mexists{}y:T.  ((y  \mmember{}  L)  \mwedge{}  ((\muparrow{}(P  y))  c\mwedge{}  (x  =  (f  y)))))
Date html generated:
2017_04_17-AM-07_25_54
Last ObjectModification:
2017_02_27-PM-04_04_53
Theory : list_1
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