Nuprl Lemma : rsum-one
∀[n,m:ℤ]. (Σ{r1 | n≤k≤m} = if m <z n then r0 else r((m - n) + 1) fi )
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
int-to-real: r(n),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rsum_wf,
int-to-real_wf,
int_seg_wf,
ifthenelse_wf,
lt_int_wf,
real_wf,
subtract_wf,
rmul_wf,
eqtt_to_assert,
assert_of_lt_int,
rmul-zero,
eqff_to_assert,
equal_wf,
bool_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
rmul-identity1,
req_functionality,
rsum-constant2,
req_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
natural_numberEquality,
hypothesis,
addEquality,
independent_functionElimination,
intEquality,
isect_memberEquality,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
voidElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. (\mSigma{}\{r1 | n\mleq{}k\mleq{}m\} = if m <z n then r0 else r((m - n) + 1) fi )
Date html generated:
2017_10_03-AM-08_59_53
Last ObjectModification:
2017_07_28-AM-07_39_15
Theory : reals
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