Nuprl Lemma : qsum-const2
∀[a,b:ℤ]. ∀[q:ℚ]. (Σa ≤ i < b. q = (if a ≤z b then b - a else 0 fi * q) ∈ ℚ)
Proof
Definitions occuring in Statement :
qsum: Σa ≤ j < b. E[j],
qmul: r * s,
rationals: ℚ,
le_int: i ≤z j,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
guard: {T},
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtract: n - m,
squash: ↓T,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
true: True,
iff: P ⇐⇒ Q,
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,
infix_ap: x f y
Lemmas referenced :
rationals_wf,
le_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
le_wf,
lt_int_wf,
less_than_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_le_int,
assert_of_lt_int,
equal_wf,
sum_shift_q,
int_seg_wf,
add-inverse,
squash_wf,
true_wf,
qsum-const,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
iff_weakening_equal,
intformless_wf,
int_formula_prop_less_lemma,
qmul_zero_qrng,
bnot_of_lt_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
lambdaFormation,
unionElimination,
equalityElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
lambdaEquality,
minusEquality,
imageElimination,
universeEquality,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[q:\mBbbQ{}]. (\mSigma{}a \mleq{} i < b. q = (if a \mleq{}z b then b - a else 0 fi * q))
Date html generated:
2018_05_22-AM-00_02_09
Last ObjectModification:
2017_07_26-PM-06_50_37
Theory : rationals
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