Nuprl Lemma : rng_lsum-from-upto
∀[a,b:ℤ]. ∀[r:Rng]. ∀[f:{a..b-} ⟶ |r|]. (Σ{r} x ∈ [a, b). f[x] = (Σ(r) a ≤ i < b. f[i]) ∈ |r|)
Proof
Definitions occuring in Statement :
rng_lsum: Σ{r} x ∈ as. f[x],
from-upto: [n, m),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
int: ℤ,
equal: s = t ∈ T,
rng_sum: rng_sum,
rng: Rng,
rng_car: |r|
Definitions unfolded in proof :
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
lelt: i ≤ j < k,
int_seg: {i..j-},
rng: Rng,
guard: {T},
prop: ℙ,
and: P ∧ Q,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
infix_ap: x f y,
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
from-upto: [n, m),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
so_apply: x[s],
so_lambda: λ2x.t[x],
squash: ↓T,
ycomb: Y,
itop: Π(op,id) lb ≤ i < ub. E[i],
grp_id: e,
pi1: fst(t),
pi2: snd(t),
grp_op: *,
add_grp_of_rng: r↓+gp,
mon_itop: Π lb ≤ i < ub. E[i],
rng_sum: rng_sum
Lemmas referenced :
nat_wf,
int_term_value_add_lemma,
itermAdd_wf,
lelt_wf,
decidable__lt,
int_formula_prop_eq_lemma,
intformeq_wf,
false_wf,
int_seg_subtype,
decidable__equal_int,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
int_seg_properties,
le_wf,
rng_wf,
rng_car_wf,
int_seg_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
rng_lsum_nil_lemma,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
rng_plus_wf,
int-value-type,
value-type-has-value,
rng_lsum_cons_lemma,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
iff_weakening_equal,
rng_sum_unroll_lo,
from-upto_wf,
rng_lsum_wf,
true_wf,
squash_wf,
equal_wf,
rng_sum_wf,
rng_zero_wf,
imax_ub,
imax_wf,
assert_of_le_int,
le_int_wf,
ifthenelse_wf,
add_functionality_wrt_eq,
imax_unfold
Rules used in proof :
cut,
isect_memberFormation,
dependent_set_memberEquality,
hypothesis_subsumption,
applyLambdaEquality,
applyEquality,
unionElimination,
productElimination,
because_Cache,
equalitySymmetry,
equalityTransitivity,
addEquality,
functionEquality,
axiomEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cumulativity,
instantiate,
promote_hyp,
callbyvalueReduce,
equalityElimination,
baseClosed,
imageMemberEquality,
functionExtensionality,
productEquality,
setEquality,
universeEquality,
imageElimination,
inlFormation
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[r:Rng]. \mforall{}[f:\{a..b\msupminus{}\} {}\mrightarrow{} |r|]. (\mSigma{}\{r\} x \mmember{} [a, b). f[x] = (\mSigma{}(r) a \mleq{} i < b. f[i]))
Date html generated:
2018_05_21-PM-09_33_12
Last ObjectModification:
2017_12_15-AM-10_09_36
Theory : matrices
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