Nuprl Lemma : remove-repeats-mapfilter-with-fun
∀[T,U:Type]. ∀[eq:EqDecider(U)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[L:T List]. ∀[P:T ⟶ 𝔹]. ∀[f:{x:T| ↑(P x)} ⟶ U].
(remove-repeats(eq;mapfilter(f;P;L))
= mapfilter(f;λa.((P a) ∧b (¬b(∃x∈L.(P x) ∧b R[x;a] ∧b (eq (f x) (f a)))_b));L)
∈ (U List)) supposing
(sorted-by(λx,y. (↑R[x;y]);L) and
StAntiSym(T;x,y.↑R[x;y]) and
Irrefl(T;x,y.↑R[x;y]))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
bl-exists: (∃x∈L.P[x])_b,
sorted-by: sorted-by(R;L),
mapfilter: mapfilter(f;P;L),
list: T List,
deq: EqDecider(T),
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
band: p ∧b q,
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
mapfilter: mapfilter(f;P;L),
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
so_apply: x[s1;s2],
deq: EqDecider(T),
bfalse: ff,
or: P ∨ Q,
assert: ↑b,
true: True,
false: False,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
irrefl: Irrefl(T;x,y.E[x; y]),
st_anti_sym: StAntiSym(T;x,y.R[x; y]),
top: Top
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
list_wf,
remove-repeats-fun-map2,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
subtype_rel_self,
set_wf,
map_wf,
assert_wf,
eqtt_to_assert,
bnot_wf,
bl-exists_wf,
subtype_rel_sets,
bool_cases_sqequal,
filter_type,
iff_weakening_equal,
sorted-by_wf,
st_anti_sym_wf,
irrefl_wf,
deq_wf,
member_filter_2,
remove-repeats-fun-as-filter,
sorted-by-filter,
filter-filter,
bl-exists-filter,
mapfilter_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
cumulativity,
sqequalRule,
setEquality,
independent_isectElimination,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
functionExtensionality,
dependent_set_memberEquality,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
voidElimination,
imageMemberEquality,
baseClosed,
universeEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
voidEquality
Latex:
\mforall{}[T,U:Type]. \mforall{}[eq:EqDecider(U)]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}].
\mforall{}[f:\{x:T| \muparrow{}(P x)\} {}\mrightarrow{} U].
(remove-repeats(eq;mapfilter(f;P;L))
= mapfilter(f;\mlambda{}a.((P a) \mwedge{}\msubb{} (\mneg{}\msubb{}(\mexists{}x\mmember{}L.(P x) \mwedge{}\msubb{} R[x;a] \mwedge{}\msubb{} (eq (f x) (f a)))\_b));L)) supposing
(sorted-by(\mlambda{}x,y. (\muparrow{}R[x;y]);L) and
StAntiSym(T;x,y.\muparrow{}R[x;y]) and
Irrefl(T;x,y.\muparrow{}R[x;y]))
Date html generated:
2017_04_17-AM-09_12_38
Last ObjectModification:
2017_02_27-PM-05_20_00
Theory : decidable!equality
Home
Index