Nuprl Lemma : bag-summation-from-upto
∀[a,b:ℤ]. ∀[f:{a..b-} ⟶ ℤ]. (Σ(i∈[a, b)). f[i] = Σ(f[j + a] | j < b - a) ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
from-upto: [n, m),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
nil: [],
list_accum: list_accum,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-summation: Σ(x∈b). f[x],
squash: ↓T,
true: True,
less_than': less_than'(a;b),
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
subtype_rel: A ⊆r B,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
from-upto: [n, m),
sum_aux: sum_aux(k;v;i;x.f[x]),
sum: Σ(f[x] | x < k),
decidable: Dec(P),
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
subtract: n - m,
bag-append: as + bs,
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
infix_ap: x f y,
ident: Ident(T;op;id),
comm: Comm(T;op),
le: A ≤ B,
bag-map: bag-map(f;bs),
has-value: (a)↓,
single-bag: {x},
cons: [a / b],
empty-bag: {}
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_wf,
subtract_wf,
subtract-1-ge-0,
istype-nat,
istype-top,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
int_subtype_base,
eqff_to_assert,
int_term_value_subtract_lemma,
itermSubtract_wf,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
decidable__lt,
sum_split1,
intformnot_wf,
int_formula_prop_not_lemma,
add-member-int_seg2,
decidable__le,
istype-le,
general_arith_equation2,
itermAdd_wf,
int_term_value_add_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_properties,
add-member-int_seg1,
zero-add,
add-commutes,
add-swap,
add-associates,
le_wf,
sum_wf,
from-upto-split,
bag-summation-append,
equal_wf,
squash_wf,
true_wf,
istype-universe,
from-upto_wf,
list-subtype-bag,
subtype_rel_sets_simple,
lelt_wf,
istype-false,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add_functionality_wrt_le,
le-add-cancel2,
not-le-2,
minus-minus,
le-add-cancel,
subtype_rel_self,
iff_weakening_equal,
bag-summation_wf,
from-upto-shift,
subtract-add-cancel,
top_wf,
subtype_rel_list,
bag-summation-map,
subtype_rel_sets,
add-subtract-cancel,
bag_wf,
comm_wf,
assoc_wf,
value-type-has-value,
int-value-type,
list_wf,
list_subtype_base,
cons_wf,
nil_wf,
bag-summation-single,
bag-summation-empty,
satisfiable-full-omega-tt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
isect_memberFormation_alt,
functionIsType,
because_Cache,
equalityIsType1,
imageElimination,
imageMemberEquality,
axiomSqEquality,
lessCases,
cumulativity,
instantiate,
promote_hyp,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIsType2,
sqleReflexivity,
callbyvalueReduce,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
intEquality,
dependent_set_memberEquality_alt,
productIsType,
addEquality,
independent_pairEquality,
universeEquality,
setEquality,
productEquality,
minusEquality,
setIsType,
equalityIstype,
isect_memberFormation,
functionEquality,
isect_memberEquality,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaEquality,
voidEquality,
computeAll,
functionExtensionality,
lambdaFormation
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[f:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(i\mmember{}[a, b)). f[i] = \mSigma{}(f[j + a] | j < b - a))
Date html generated:
2019_10_15-AM-11_03_47
Last ObjectModification:
2018_11_27-AM-00_30_14
Theory : bags
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