Nuprl Lemma : series-sum-constant
∀x:ℝ. Σi.if (i =z 0) then x else r0 fi = x
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
int-to-real: r(n),
real: ℝ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
all: ∀x:A. B[x],
natural_number: $n
Definitions unfolded in proof :
series-sum: Σn.x[n] = a,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
so_apply: x[s],
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
constant-limit,
req_weakening,
real_wf,
nat_wf,
rsum_wf,
ifthenelse_wf,
eq_int_wf,
int-to-real_wf,
int_seg_wf,
converges-to_functionality,
radd_wf,
subtract_wf,
false_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
rsum-split-shift,
req_functionality,
radd_functionality,
rsum-single,
rsum_functionality,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
le_wf,
rmul_wf,
req_wf,
rsum-constant,
uiff_transitivity,
rmul-zero-both,
radd_comm,
radd-zero-both
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
independent_functionElimination,
isectElimination,
independent_isectElimination,
hypothesis,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
addEquality,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
setEquality,
applyEquality,
baseClosed,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}x:\mBbbR{}. \mSigma{}i.if (i =\msubz{} 0) then x else r0 fi = x
Date html generated:
2017_10_03-AM-09_17_49
Last ObjectModification:
2017_07_28-AM-07_43_11
Theory : reals
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