Nuprl Lemma : reduce-mapfilter
∀[f1,x:Top]. ∀[T,A:Type]. ∀[as:T List]. ∀[P:{a:T| (a ∈ as)} ⟶ 𝔹]. ∀[f2:{a:T| (a ∈ as) ∧ (↑(P a))} ⟶ A].
(reduce(f1;x;mapfilter(f2;P;as)) ~ reduce(λu,z. if P u then f1 (f2 u) z else z fi ;x;as))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
reduce: reduce(f;k;as),
list: T List,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
mapfilter: mapfilter(f;P;L),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
l_member_wf,
assert_wf,
bool_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
reduce_nil_lemma,
filter_nil_lemma,
map_nil_lemma,
nil_wf,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
reduce_cons_lemma,
filter_cons_lemma,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
subtype_rel_self,
set_wf,
eqtt_to_assert,
map_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_wf,
top_wf
Rules used in proof :
cut,
thin,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
functionEquality,
setEquality,
cumulativity,
productEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
inrFormation,
inlFormation,
equalityElimination,
universeEquality,
isect_memberFormation
Latex:
\mforall{}[f1,x:Top]. \mforall{}[T,A:Type]. \mforall{}[as:T List]. \mforall{}[P:\{a:T| (a \mmember{} as)\} {}\mrightarrow{} \mBbbB{}].
\mforall{}[f2:\{a:T| (a \mmember{} as) \mwedge{} (\muparrow{}(P a))\} {}\mrightarrow{} A].
(reduce(f1;x;mapfilter(f2;P;as)) \msim{} reduce(\mlambda{}u,z. if P u then f1 (f2 u) z else z fi ;x;as))
Date html generated:
2017_04_17-AM-07_30_26
Last ObjectModification:
2017_02_27-PM-04_08_28
Theory : list_1
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