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
Home
Index