Nuprl Lemma : bag-mapfilter-mapfilter
∀[A,B,C:Type]. ∀[b:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]} ⟶ B]. ∀[Q:B ⟶ 𝔹]. ∀[g:{x:B| ↑Q[x]} ⟶ C].
(bag-mapfilter(g;Q;bag-mapfilter(f;P;b)) = bag-mapfilter(g o f;λx.(P[x] ∧b Q[f[x]]);b) ∈ bag(C))
Proof
Definitions occuring in Statement :
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
compose: f o g,
band: p ∧b q,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
prop: ℙ,
so_apply: x[s],
bag-mapfilter: bag-mapfilter(f;P;bs),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
true: True,
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
all: ∀x:A. B[x],
compose: f o g,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
top: Top,
decidable: Dec(P),
not: ¬A
Lemmas referenced :
bag_wf,
bool_wf,
assert_wf,
istype-universe,
bag-filter_wf,
bag-map_wf,
bfalse_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
iff_weakening_equal,
subtype_rel_self,
bag-filter-map2,
true_wf,
squash_wf,
equal_wf,
istype-void,
bag-map-map,
decidable__assert,
istype-assert,
iff_imp_equal_bool,
btrue_wf,
istype-true,
bag-filter-filter2,
subtype_rel_sets,
subtype_rel_bag
Rules used in proof :
universeEquality,
inhabitedIsType,
because_Cache,
axiomEquality,
isect_memberEquality_alt,
applyEquality,
universeIsType,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
setIsType,
functionIsType,
hypothesis,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
closedConclusion,
setEquality,
natural_numberEquality,
voidElimination,
independent_functionElimination,
cumulativity,
instantiate,
dependent_functionElimination,
promote_hyp,
equalityIsType1,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
rename,
setElimination,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation_alt,
lambdaEquality_alt,
baseClosed,
imageMemberEquality,
imageElimination,
independent_pairFormation,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[b:bag(A)]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:A| \muparrow{}P[x]\} {}\mrightarrow{} B]. \mforall{}[Q:B {}\mrightarrow{} \mBbbB{}].
\mforall{}[g:\{x:B| \muparrow{}Q[x]\} {}\mrightarrow{} C].
(bag-mapfilter(g;Q;bag-mapfilter(f;P;b)) = bag-mapfilter(g o f;\mlambda{}x.(P[x] \mwedge{}\msubb{} Q[f[x]]);b))
Date html generated:
2020_05_20-AM-08_01_36
Last ObjectModification:
2020_01_24-PM-06_25_28
Theory : bags
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