Nuprl Lemma : summand-qle-sum
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. ∀[i:{a..b-}]. (E[i] ≤ Σa ≤ j < b. E[j]) supposing ∀j:{a..b-}. (0 ≤ E[j])
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qle: r ≤ s,
rationals: ℚ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_seg: {i..j-},
so_apply: x[s],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
and: P ∧ Q,
all: ∀x:A. B[x],
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
delta: delta(i;j),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
squash: ↓T
Lemmas referenced :
qsum-delta,
qle_witness,
int_seg_wf,
qsum_wf,
all_wf,
qle_wf,
int-subtype-rationals,
rationals_wf,
band_wf,
le_int_wf,
lt_int_wf,
assert_wf,
le_wf,
less_than_wf,
qsum-qle,
qmul_wf,
delta_wf,
subtype_rel_set,
lelt_wf,
bnot_wf,
not_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_le_int,
assert_of_lt_int,
eqff_to_assert,
assert_of_bnot,
eq_int_wf,
equal-wf-T-base,
int_subtype_base,
qle_reflexivity,
uiff_transitivity,
assert_of_eq_int,
equal_wf,
squash_wf,
true_wf,
qmul_one_qrng,
iff_weakening_equal,
qmul_zero_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
applyEquality,
functionExtensionality,
sqequalRule,
lambdaEquality,
independent_functionElimination,
isect_memberEquality,
because_Cache,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
intEquality,
productEquality,
productElimination,
independent_isectElimination,
lambdaFormation,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
instantiate,
cumulativity,
impliesFunctionality,
baseClosed,
dependent_set_memberEquality,
equalityElimination,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}].
\mforall{}[i:\{a..b\msupminus{}\}]. (E[i] \mleq{} \mSigma{}a \mleq{} j < b. E[j]) supposing \mforall{}j:\{a..b\msupminus{}\}. (0 \mleq{} E[j])
Date html generated:
2018_05_22-AM-00_00_05
Last ObjectModification:
2017_07_26-PM-06_49_08
Theory : rationals
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