Nuprl Lemma : rprod-rminus
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. rprod(n;m;k.-(x[k])) = (r(-1)^(m - n) + 1 * rprod(n;m;k.x[k])) supposing n ≤ m
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
rnexp: x^k1,
req: x = y,
rmul: a * b,
rminus: -(x),
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rprod: rprod(n;m;k.x[k]),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
req_witness,
rprod_wf,
rminus_wf,
decidable__lt,
intformnot_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
istype-le,
int_seg_wf,
rmul_wf,
rnexp_wf,
decidable__le,
int-to-real_wf,
subtract-1-ge-0,
subtract-add-cancel,
istype-nat,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
trivial-int-eq1,
real_wf,
add-zero,
rminus-as-rmul,
req_functionality,
rprod-single,
rmul_functionality,
req_weakening,
rnexp1,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
itermMultiply_wf,
itermMinus_wf,
req-iff-rsub-is-0,
req_inversion,
rnexp-add,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
addEquality,
because_Cache,
closedConclusion,
applyEquality,
dependent_set_memberEquality_alt,
productElimination,
unionElimination,
productIsType,
minusEquality,
functionIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
intEquality
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}.
rprod(n;m;k.-(x[k])) = (r(-1)\^{}(m - n) + 1 * rprod(n;m;k.x[k])) supposing n \mleq{} m
Date html generated:
2019_10_29-AM-10_17_39
Last ObjectModification:
2019_01_15-PM-01_06_28
Theory : reals
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