Nuprl Lemma : triangular-reciprocal-series-sum
Σn.(r1/r(t(n + 1))) = r(2)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
series-sum: Σn.x[n] = a,
rdiv: (x/y),
int-to-real: r(n),
triangular-num: t(n),
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
converges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
le: A ≤ B,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
triangular-num: t(n),
divide: n ÷ m,
series-sum: Σn.x[n] = a,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rsub: x - y,
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermConstant: "const",
rtermDivide: num "/" denom,
rtermVar: rtermVar(var),
pi1: fst(t),
rtermMinus: rtermMinus(num),
pi2: snd(t)
Lemmas referenced :
rsub-limit,
int-to-real_wf,
nat_wf,
rdiv_wf,
rless-int,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rless_wf,
constant-limit,
req_weakening,
nat_plus_wf,
nat_plus_properties,
decidable__le,
itermMultiply_wf,
int_term_value_mul_lemma,
le_wf,
all_wf,
rleq_wf,
rabs_wf,
rsub_wf,
rmul_wf,
rinv_wf2,
rleq-int-fractions2,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
rleq-int-fractions,
uiff_transitivity,
rleq_functionality,
rabs_functionality,
real_term_polynomial,
itermSubtract_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rinv-as-rdiv,
rabs-of-nonneg,
subtract_wf,
converges-to_functionality,
rsub-int,
triangular-num-le,
full-omega-unsat,
istype-int,
istype-void,
istype-le,
istype-nat,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
rsum_wf,
triangular-num_wf,
int_seg_properties,
rneq-int,
lelt_wf,
int_seg_wf,
int_seg_subtype_nat,
istype-false,
rsum_functionality,
radd_wf,
istype-less_than,
mul_bounds_1b,
req_functionality,
radd-int-fractions,
mul_nzero,
nequal_wf,
req-int-fractions,
decidable__equal_int,
iff_weakening_equal,
subtype_rel_self,
twice-triangular,
istype-universe,
true_wf,
squash_wf,
equal_wf,
mul-distributes,
mul-distributes-right,
mul-associates,
mul-commutes,
one-mul,
add-swap,
add-mul-special,
zero-mul,
req_transitivity,
rsum_linearity1,
subtract-add-cancel,
radd_functionality,
rsum-split-first,
rsum-split-last,
radd_assoc,
rminus_wf,
req_inversion,
rsum-shift,
radd-rdiv,
rdiv_functionality,
radd-int,
real_polynomial_null,
rsum-zero,
assert-rat-term-eq2,
rtermMinus_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rtermAdd_wf,
rmul_preserves_req,
rmul-int,
rmul_functionality,
rinv1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
isectElimination,
natural_numberEquality,
hypothesis,
addEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
productElimination,
independent_functionElimination,
hypothesisEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
lambdaFormation,
dependent_set_memberFormation,
dependent_set_memberEquality,
multiplyEquality,
functionEquality,
applyEquality,
minusEquality,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
universeIsType,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
inrFormation_alt,
imageElimination,
inhabitedIsType,
imageMemberEquality,
universeEquality,
instantiate
Latex:
\mSigma{}n.(r1/r(t(n + 1))) = r(2)
Date html generated:
2019_10_29-AM-10_25_26
Last ObjectModification:
2019_04_02-AM-10_01_11
Theory : reals
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