Nuprl Lemma : concat-map-map-decide
∀[T:Type]. ∀[g:Top]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)} ⟶ (Top + Top)].
(concat(map(λx.map(λy.g[x;y];case f[x] of inl(m) => [m] | inr(x) => []);L)) ~ mapfilter(λx.g[x;outl(f[x])];
λx.isl(f[x]);
L))
Proof
Definitions occuring in Statement :
mapfilter: mapfilter(f;P;L),
l_member: (x ∈ l),
concat: concat(ll),
map: map(f;as),
cons: [a / b],
nil: [],
list: T List,
outl: outl(x),
isl: isl(x),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
so_apply: x[s],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
union: left + right,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
mapfilter: mapfilter(f;P;L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
concat: concat(ll),
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,
isl: isl(x),
outl: outl(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
bfalse: ff,
true: True,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q)
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,
top_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
map_nil_lemma,
filter_nil_lemma,
reduce_nil_lemma,
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,
map_cons_lemma,
filter_cons_lemma,
list_wf,
nil_wf,
subtype_rel_dep_function,
cons_wf,
subtype_rel_sets,
cons_member,
subtype_rel_self,
set_wf,
reduce_cons_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
true_wf,
false_wf,
not_wf,
isl_wf,
bool_wf,
assert_wf,
bnot_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
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,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
functionEquality,
setEquality,
cumulativity,
unionEquality,
applyEquality,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
universeEquality,
inrFormation,
functionExtensionality,
inlFormation,
equalityElimination
Latex:
\mforall{}[T:Type]. \mforall{}[g:Top]. \mforall{}[L:T List]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} (Top + Top)].
(concat(map(\mlambda{}x.map(\mlambda{}y.g[x;y];case f[x] of inl(m) => [m] | inr(x) => []);L))
\msim{} mapfilter(\mlambda{}x.g[x;outl(f[x])];\mlambda{}x.isl(f[x]);L))
Date html generated:
2018_05_21-PM-07_35_48
Last ObjectModification:
2017_07_26-PM-05_09_54
Theory : general
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