Nuprl Lemma : rsum_linearity-rsub
∀[n,m:ℤ]. ∀[x,y:{n..m + 1-} ⟶ ℝ]. (Σ{x[k] - y[k] | n≤k≤m} = (Σ{x[k] | n≤k≤m} - Σ{y[k] | n≤k≤m}))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rsub: x - y,
req: x = y,
real: ℝ,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
rsum: Σ{x[k] | n≤k≤m},
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
rsub: x - y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
top: Top,
nat: ℕ,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
less_than: a < b,
squash: ↓T
Lemmas referenced :
req_witness,
rsum_wf,
rsub_wf,
int_seg_wf,
real_wf,
value-type-has-value,
int-value-type,
from-upto_wf,
list_wf,
le_wf,
less_than_wf,
valueall-type-has-valueall,
list-valueall-type,
real-valueall-type,
map_wf,
evalall-reduce,
valueall-type-real-list,
equal_wf,
radd-list_wf-bag,
radd_wf,
rminus_wf,
list-subtype-bag,
subtype_rel_self,
rmul_wf,
int-to-real_wf,
req_weakening,
req_functionality,
radd-list-linearity1,
radd_functionality,
rminus-as-rmul,
req_inversion,
radd-list-linearity2,
radd-list_functionality,
map-length,
length_wf_nat,
nat_wf,
length_wf,
length-map,
subtype_rel_list,
top_wf,
select-map,
req_wf,
select_wf,
int_seg_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,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
uiff_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
addEquality,
natural_numberEquality,
hypothesis,
independent_functionElimination,
functionEquality,
isect_memberEquality,
because_Cache,
intEquality,
independent_isectElimination,
setEquality,
productEquality,
lambdaFormation,
callbyvalueReduce,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
minusEquality,
productElimination,
voidElimination,
voidEquality,
setElimination,
rename,
independent_pairFormation,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageElimination
Latex:
\mforall{}[n,m:\mBbbZ{}]. \mforall{}[x,y:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}]. (\mSigma{}\{x[k] - y[k] | n\mleq{}k\mleq{}m\} = (\mSigma{}\{x[k] | n\mleq{}k\mleq{}m\} - \mSigma{}\{y[k] | n\mleq{}k\mleq{}m\}))
Date html generated:
2017_10_03-AM-08_59_32
Last ObjectModification:
2017_07_28-AM-07_39_00
Theory : reals
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