Nuprl Lemma : rsum-constant2
∀[n,m:ℤ]. ∀[a:ℝ]. (Σ{a | n≤k≤m} = (a * 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,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
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 :
subtract: n - m,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
false: False,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
all: ∀x:A. B[x],
top: Top,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
callbyvalueall: callbyvalueall,
prop: ℙ,
has-value: (a)↓,
rsum: Σ{x[k] | n≤k≤m},
rev_uimplies: rev_uimplies(P;Q),
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
add-commutes,
add-swap,
minus-one-mul,
add-associates,
radd-list-one,
rmul_functionality,
int_formula_prop_not_lemma,
intformnot_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
length-from-upto,
subtype_rel_list,
length_wf,
list-subtype-bag,
top_wf,
map_wf_bag,
radd-list_wf-bag,
int_subtype_base,
evalall-sqequal,
from-upto_wf,
less_than_wf,
le_wf,
map_wf,
real-list-has-valueall,
int-value-type,
value-type-has-value,
req_weakening,
rsum-constant,
req_functionality,
subtract_wf,
int-to-real_wf,
real_wf,
lt_int_wf,
ifthenelse_wf,
rmul_wf,
int_seg_wf,
rsum_wf,
req_witness
Rules used in proof :
minusEquality,
multiplyEquality,
cumulativity,
instantiate,
promote_hyp,
independent_pairFormation,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation,
approximateComputation,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
lambdaFormation,
voidEquality,
voidElimination,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
productEquality,
setEquality,
callbyvalueReduce,
productElimination,
independent_isectElimination,
intEquality,
because_Cache,
isect_memberEquality,
independent_functionElimination,
hypothesis,
natural_numberEquality,
addEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[a:\mBbbR{}]. (\mSigma{}\{a | n\mleq{}k\mleq{}m\} = (a * if m <z n then r0 else r((m - n) + 1) fi ))
Date html generated:
2018_05_22-PM-01_52_06
Last ObjectModification:
2018_05_21-AM-00_12_35
Theory : reals
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