Nuprl Lemma : rprod-of-negative
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ.
(((m - n rem 2) = 1 ∈ ℤ) ⇒ (r0 < rprod(n;m;k.x[k]))) ∧ (((m - n rem 2) = 0 ∈ ℤ) ⇒ (rprod(n;m;k.x[k]) < r0))
supposing (∀k:{n..m + 1-}. (x[k] < r0)) ∧ (n ≤ m)
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
rless: x < y,
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],
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
remainder: n rem m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
so_apply: x[s],
and: P ∧ Q,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
guard: {T},
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
false: False,
not: ¬A,
top: Top,
nat: ℕ,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
ifthenelse: if b then t else f fi ,
ge: i ≥ j ,
int_seg: {i..j-},
lelt: i ≤ j < k,
eq_int: (i =z j),
cand: A c∧ B
Lemmas referenced :
rprod-of-positive,
rminus_wf,
int_seg_wf,
rmul_reverses_rless_iff,
int-to-real_wf,
rless-int,
rless_wf,
istype-le,
real_wf,
istype-int,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
rless_functionality,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
istype-void,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rprod_wf,
rnexp_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermAdd_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
subtype_base_sq,
int_subtype_base,
eq_int_wf,
ifthenelse_wf,
btrue_wf,
bfalse_wf,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
req_weakening,
rprod-rminus,
rmul_functionality,
req_inversion,
rnexp-add,
rnexp1,
rnexp-minus-one,
rem_bounds_1,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-less_than,
decidable__equal_int,
int_seg_properties,
rless-implies-rless,
int_seg_subtype_special,
int_seg_cases,
rsub_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
applyEquality,
hypothesis,
universeIsType,
addEquality,
natural_numberEquality,
independent_isectElimination,
productElimination,
because_Cache,
minusEquality,
independent_functionElimination,
independent_pairFormation,
imageMemberEquality,
baseClosed,
productIsType,
functionIsType,
inhabitedIsType,
approximateComputation,
int_eqEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
setElimination,
rename,
unionElimination,
dependent_pairFormation_alt,
equalityIstype,
remainderEquality,
closedConclusion,
instantiate,
cumulativity,
intEquality,
sqequalBase,
equalityElimination,
applyLambdaEquality,
hypothesis_subsumption
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}.
(((m - n rem 2) = 1) {}\mRightarrow{} (r0 < rprod(n;m;k.x[k])))
\mwedge{} (((m - n rem 2) = 0) {}\mRightarrow{} (rprod(n;m;k.x[k]) < r0))
supposing (\mforall{}k:\{n..m + 1\msupminus{}\}. (x[k] < r0)) \mwedge{} (n \mleq{} m)
Date html generated:
2019_10_29-AM-10_17_54
Last ObjectModification:
2019_01_15-PM-01_16_04
Theory : reals
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