Nuprl Lemma : qsum-int
∀[i,j:ℤ]. ∀[X:{i..j-} ⟶ ℤ]. (Σi ≤ x < j. X[x] ∈ ℤ)
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
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,
qsum: Σa ≤ j < b. E[j],
rng_sum: rng_sum,
mon_itop: Π lb ≤ i < ub. E[i],
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
qrng: <ℚ+*>,
rng_plus: +r,
rng_zero: 0,
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
infix_ap: x f y
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,
int_seg_wf,
le_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
lt_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
infix_ap_wf,
itermAdd_wf,
int_term_value_add_lemma,
itop_wf,
lelt_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
qadd-add,
decidable__lt
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
addEquality,
unionElimination,
because_Cache,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
productElimination,
equalityElimination,
isect_memberFormation
Latex:
\mforall{}[i,j:\mBbbZ{}]. \mforall{}[X:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}i \mleq{} x < j. X[x] \mmember{} \mBbbZ{})
Date html generated:
2018_05_21-PM-11_59_36
Last ObjectModification:
2017_07_26-PM-06_48_50
Theory : rationals
Home
Index