Nuprl Lemma : qabs-qsum-qle
∀[a,b:ℤ]. ∀[E:{a..b-} ⟶ ℚ]. ∀[x:ℚ].
|Σa ≤ j < b. E[j]| ≤ ((b - a) * x) supposing (a ≤ b) ∧ (∀j:ℤ. ((a ≤ j) ⇒ j < b ⇒ (|E[j]| ≤ x)))
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qabs: |r|,
qle: r ≤ s,
qmul: r * s,
rationals: ℚ,
int_seg: {i..j-},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
subtract: n - m,
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
qabs: |r|,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
ifthenelse: if b then t else f fi ,
qpositive: qpositive(r),
btrue: tt,
lt_int: i <z j,
bfalse: ff,
qmul: r * s,
qle: r ≤ s,
grp_leq: a ≤ b,
assert: ↑b,
infix_ap: x f y,
grp_le: ≤b,
pi1: fst(t),
pi2: snd(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
bor: p ∨bq,
qsub: r - s,
qadd: r + s,
qeq: qeq(r;s),
eq_int: (i =z j),
le: A ≤ B,
less_than': less_than'(a;b),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q),
qge: a ≥ b,
uiff: uiff(P;Q),
subtract: n - m
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,
qle_witness,
qabs_wf,
qsum_wf,
int_seg_wf,
qmul_wf,
int-subtype-rationals,
all_wf,
qle_wf,
rationals_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
squash_wf,
true_wf,
sum_unroll_base_q,
iff_weakening_equal,
qmul_zero_qrng,
sum_unroll_hi_q,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
decidable__lt,
lelt_wf,
qadd_wf,
qle_functionality_wrt_implies,
qle_transitivity_qorder,
q-triangle-inequality,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
subtract-add-cancel,
qadd-add,
qmul_over_plus_qrng,
qmul_one_qrng,
le_wf,
add-member-int_seg1,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
sum_shift_q,
equal_wf,
decidable__equal_int,
intformeq_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
minus-minus,
add-commutes
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
isect_memberFormation,
because_Cache,
unionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
dependent_set_memberEquality,
addEquality,
productEquality,
minusEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[E:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[x:\mBbbQ{}].
|\mSigma{}a \mleq{} j < b. E[j]| \mleq{} ((b - a) * x) supposing (a \mleq{} b) \mwedge{} (\mforall{}j:\mBbbZ{}. ((a \mleq{} j) {}\mRightarrow{} j < b {}\mRightarrow{} (|E[j]| \mleq{} x)))
Date html generated:
2018_05_22-AM-00_26_20
Last ObjectModification:
2017_07_26-PM-06_56_18
Theory : rationals
Home
Index