Nuprl Lemma : rsum-split-first-shift
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. Σ{x[i] | n≤i≤m} = (x[n] + Σ{x[i + 1] | n≤i≤m - 1}) supposing n ≤ m
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
rsum-split-first,
int_seg_wf,
real_wf,
req_witness,
radd_wf,
decidable__le,
full-omega-unsat,
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,
intformand_wf,
intformless_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
lelt_wf,
rsum_wf,
subtract_wf,
add-member-int_seg2,
subtract-add-cancel,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
le_wf,
req_wf,
iff_weakening_uiff,
req_functionality,
req_weakening,
radd_functionality,
rsum-shift
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionEquality,
addEquality,
natural_numberEquality,
intEquality,
independent_isectElimination,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
because_Cache,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
setElimination,
rename,
productElimination,
cumulativity,
isectEquality,
addLevel
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} = (x[n] + \mSigma{}\{x[i + 1] | n\mleq{}i\mleq{}m - 1\}) supposing n \mleq{} m
Date html generated:
2018_05_22-PM-01_51_47
Last ObjectModification:
2017_10_25-PM-03_41_35
Theory : reals
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