Nuprl Lemma : rprod-empty
∀[n,m:ℤ]. ∀[x:Top].  rprod(n;m;k.x[k]) ~ r1 supposing m < n
Proof
Definitions occuring in Statement : 
rprod: rprod(n;m;k.x[k])
, 
int-to-real: r(n)
, 
less_than: a < b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
top: Top
, 
so_apply: x[s]
, 
natural_number: $n
, 
int: ℤ
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
rprod: rprod(n;m;k.x[k])
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
and: P ∧ Q
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
exists: ∃x:A. B[x]
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
assert: ↑b
, 
false: False
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
prop: ℙ
Lemmas referenced : 
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, 
full-omega-unsat, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
intformnot_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_wf, 
istype-less_than, 
istype-top
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
inhabitedIsType, 
lambdaFormation_alt, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
independent_isectElimination, 
because_Cache, 
sqequalRule, 
dependent_pairFormation_alt, 
equalityIstype, 
promote_hyp, 
dependent_functionElimination, 
instantiate, 
cumulativity, 
independent_functionElimination, 
voidElimination, 
natural_numberEquality, 
approximateComputation, 
lambdaEquality_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
independent_pairFormation, 
universeIsType, 
axiomSqEquality, 
isectIsTypeImplies
Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x:Top].    rprod(n;m;k.x[k])  \msim{}  r1  supposing  m  <  n
Date html generated:
2019_10_29-AM-10_16_42
Last ObjectModification:
2019_01_15-AM-10_03_40
Theory : reals
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