Nuprl Lemma : rprod-of-positive
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. r0 < rprod(n;m;k.x[k]) supposing ∀k:{n..m + 1-}. (r0 < x[k])
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],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_stable: SqStable(P),
rprod: rprod(n;m;k.x[k]),
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,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
nat: ℕ,
ge: i ≥ j ,
subtract: n - m
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
rprod_wf,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
int_seg_wf,
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,
rmul-is-positive,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__le,
rless-int,
rless_wf,
nat_plus_properties,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
primrec-wf2,
le_wf,
nat_properties,
istype-nat,
real_wf,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
addEquality,
because_Cache,
closedConclusion,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
productElimination,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
universeIsType,
productIsType,
inhabitedIsType,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
inlFormation_alt,
imageMemberEquality,
baseClosed,
imageElimination,
intEquality,
functionIsType,
setIsType,
functionEquality
Latex:
\mforall{}n,m:\mBbbZ{}. \mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}. r0 < rprod(n;m;k.x[k]) supposing \mforall{}k:\{n..m + 1\msupminus{}\}. (r0 < x[k])
Date html generated:
2019_10_29-AM-10_17_25
Last ObjectModification:
2019_01_15-AM-09_45_19
Theory : reals
Home
Index