Nuprl Lemma : rsum-telescopes2
∀[n:ℤ]. ∀[m:{n...}]. ∀[x,y:{n..m + 1-} ⟶ ℝ].
Σ{x[k] - y[k] | n≤k≤m} = (x[n] - y[m]) supposing ∀i:{n..m-}. (x[i + 1] = y[i])
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rsub: x - y,
req: x = y,
real: ℝ,
int_upper: {i...},
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
rsub: x - y,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
prop: ℙ,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
all: ∀x:A. B[x],
int_upper: {i...},
so_apply: x[s],
so_lambda: λ2x.t[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
uiff_transitivity,
rminus-rminus,
radd_functionality,
radd_comm,
le_wf,
radd_wf,
rsum_functionality,
rminus_functionality,
req_functionality,
req_weakening,
int_upper_wf,
real_wf,
add-subtract-cancel,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
add-member-int_seg2,
req_wf,
all_wf,
lelt_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
itermAdd_wf,
intformless_wf,
intformand_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
int_upper_properties,
rsub_wf,
rsum_wf,
req_witness,
int_seg_wf,
rminus_wf,
rsum-telescopes
Rules used in proof :
impliesLevelFunctionality,
levelHypothesis,
impliesFunctionality,
addLevel,
functionEquality,
equalitySymmetry,
equalityTransitivity,
productElimination,
independent_functionElimination,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
dependent_set_memberEquality,
lambdaFormation,
independent_isectElimination,
natural_numberEquality,
hypothesis,
because_Cache,
rename,
setElimination,
addEquality,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbZ{}]. \mforall{}[m:\{n...\}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
\mSigma{}\{x[k] - y[k] | n\mleq{}k\mleq{}m\} = (x[n] - y[m]) supposing \mforall{}i:\{n..m\msupminus{}\}. (x[i + 1] = y[i])
Date html generated:
2016_11_08-AM-09_00_22
Last ObjectModification:
2016_11_05-PM-07_18_17
Theory : reals
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