Nuprl Lemma : rsum-split
∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ]. ∀[k:ℤ].
(Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤k} + Σ{x[i] | k + 1≤i≤m})) supposing ((k ≤ m) and (n ≤ k))
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],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rsum: Σ{x[k] | n≤k≤m},
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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
radd-list-append,
req_inversion,
req_functionality,
req_weakening,
from-upto-split,
map_append_sq,
append_wf,
subtype_rel_self,
list-subtype-bag,
radd-list_wf-bag,
valueall-type-real-list,
evalall-reduce,
from-upto_wf,
less_than_wf,
and_wf,
map_wf,
real-valueall-type,
list-valueall-type,
list_wf,
valueall-type-has-valueall,
int-value-type,
value-type-has-value,
real_wf,
le_wf,
decidable__le,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
radd_wf,
int_seg_wf,
rsum_wf,
req_witness
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
addEquality,
natural_numberEquality,
hypothesis,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
setEquality,
callbyvalueReduce,
productEquality,
lambdaFormation
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. \mforall{}[k:\mBbbZ{}].
(\mSigma{}\{x[i] | n\mleq{}i\mleq{}m\} = (\mSigma{}\{x[i] | n\mleq{}i\mleq{}k\} + \mSigma{}\{x[i] | k + 1\mleq{}i\mleq{}m\})) supposing ((k \mleq{} m) and (n \mleq{} k))
Date html generated:
2016_05_18-AM-07_45_36
Last ObjectModification:
2016_01_17-AM-02_08_00
Theory : reals
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