Nuprl Lemma : rsum_single
∀[n:ℤ]. ∀[x:{n..n + 1-} ⟶ ℝ]. (Σ{x[k] | n≤k≤n} = x[n])
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rsum_wf,
int_seg_wf,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
lelt_wf,
real_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
int-to-real_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
radd_wf,
subtract_wf,
subtract-add-cancel,
intformand_wf,
int_formula_prop_and_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
req_weakening,
req_functionality,
rsum_unroll
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
hypothesis,
because_Cache,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
functionEquality,
lambdaFormation,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
setElimination,
rename
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[x:\{n..n + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (\mSigma{}\{x[k] | n\mleq{}k\mleq{}n\} = x[n])
Date html generated:
2017_10_03-AM-08_57_48
Last ObjectModification:
2017_07_28-AM-07_37_44
Theory : reals
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