Nuprl Lemma : rng_lsum-partition
∀[k:ℕ]. ∀[A:Type]. ∀[p:A ⟶ ℕk]. ∀[r:Rng]. ∀[f:A ⟶ |r|]. ∀[as:A List].
(Σ{r} x ∈ as. f[x] = (Σ(r) 0 ≤ i < k. Σ{r} x ∈ filter(λa.(p a =z i);as). f[x]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
filter: filter(P;l),
list: T List,
int_seg: {i..j-},
nat: ℕ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
rng: Rng,
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
prop: ℙ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
squash: ↓T,
infix_ap: x f y,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
true: True,
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
list_induction,
equal_wf,
rng_car_wf,
rng_lsum_wf,
rng_sum_wf,
filter_wf5,
eq_int_wf,
int_seg_wf,
l_member_wf,
list_wf,
rng_lsum_nil_lemma,
filter_nil_lemma,
rng_lsum_cons_lemma,
filter_cons_lemma,
rng_plus_wf,
ifthenelse_wf,
cons_wf,
iff_weakening_equal,
rng_wf,
nat_wf,
rng_zero_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rng_sum_0,
decidable__lt,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_subtype_nat,
false_wf,
ge_wf,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
rng_sum_unroll_lo,
subtype_rel_self,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
rng_plus_zero,
rng_sum_shift,
rng_sum_unroll_hi,
itermAdd_wf,
int_term_value_add_lemma,
add-subtract-cancel,
decidable__equal_int,
rng_plus_comm,
rng_sum_plus
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesis,
applyEquality,
natural_numberEquality,
setEquality,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaFormation,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
axiomEquality,
functionEquality,
universeEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
applyLambdaEquality,
intWeakElimination,
instantiate,
equalityElimination,
promote_hyp,
cumulativity,
hyp_replacement,
addEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[A:Type]. \mforall{}[p:A {}\mrightarrow{} \mBbbN{}k]. \mforall{}[r:Rng]. \mforall{}[f:A {}\mrightarrow{} |r|]. \mforall{}[as:A List].
(\mSigma{}\{r\} x \mmember{} as. f[x] = (\mSigma{}(r) 0 \mleq{} i < k. \mSigma{}\{r\} x \mmember{} filter(\mlambda{}a.(p a =\msubz{} i);as). f[x]))
Date html generated:
2018_05_21-PM-09_33_07
Last ObjectModification:
2018_05_19-PM-04_22_03
Theory : matrices
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