Nuprl Lemma : qsum-linearity2
∀[a,b:ℤ]. ∀[X,Y:{a..b-} ⟶ ℚ]. (Σa ≤ i < b. X[i] + Y[i] = (Σa ≤ i < b. X[i] + Σa ≤ i < b. Y[i]) ∈ ℚ)
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qadd: r + s,
rationals: ℚ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
true: True,
so_lambda: λ2x.t[x],
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
rationals_wf,
le_wf,
subtract_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
qadd_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int-subtype-rationals,
decidable__lt,
lelt_wf,
qsum_wf,
qadd_assoc,
iff_weakening_equal,
qadd_ac_1_q,
qsum_unroll,
squash_wf,
true_wf
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,
axiomEquality,
functionEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberFormation,
because_Cache,
unionElimination,
applyEquality,
functionExtensionality,
equalityElimination,
productElimination,
promote_hyp,
instantiate,
cumulativity,
dependent_set_memberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[X,Y:\{a..b\msupminus{}\} {}\mrightarrow{} \mBbbQ{}]. (\mSigma{}a \mleq{} i < b. X[i] + Y[i] = (\mSigma{}a \mleq{} i < b. X[i] + \mSigma{}a \mleq{} i < b. Y[i]))
Date html generated:
2018_05_22-AM-00_02_23
Last ObjectModification:
2017_07_26-PM-06_50_48
Theory : rationals
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