Nuprl Lemma : rprod-is-zero
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. rprod(n;m;k.x[k]) = r0 supposing ∃k:{n..m + 1-}. (x[k] = r0)
Proof
Definitions occuring in Statement :
rprod: rprod(n;m;k.x[k]),
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
exists: ∃x:A. B[x],
int_seg: {i..j-},
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
rprod-split,
int_seg_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
req_witness,
rprod_wf,
int_seg_wf,
int-to-real_wf,
req_wf,
real_wf,
rmul_wf,
decidable__lt,
istype-le,
istype-less_than,
req_functionality,
req_weakening,
subtract_wf,
subtract-add-cancel,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
rmul_functionality,
rprod-split-last,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
setElimination,
rename,
independent_isectElimination,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
because_Cache,
applyEquality,
productIsType,
functionIsType,
inhabitedIsType,
dependent_set_memberEquality_alt,
closedConclusion
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. rprod(n;m;k.x[k]) = r0 supposing \mexists{}k:\{n..m + 1\msupminus{}\}. (x[k] = r0)
Date html generated:
2019_10_29-AM-10_19_10
Last ObjectModification:
2019_01_19-AM-11_34_58
Theory : reals
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