Nuprl Lemma : rsum-difference2
∀[k,n,m:ℤ]. ∀[x,y:{k..m + 1-} ⟶ ℝ].
((Σ{x[i] | k≤i≤m} - Σ{y[i] | k≤i≤n}) = Σ{x[i] | n + 1≤i≤m}) supposing
((∀i:{k..n + 1-}. (x[i] = y[i])) and
(n ≤ m) and
(k ≤ n))
Proof
Definitions occuring in Statement :
rsum: Σ{x[k] | n≤k≤m},
rsub: x - y,
req: x = y,
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
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,
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,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rsub: x - y,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermVar: vvar,
itermMinus: "-"num
Lemmas referenced :
rsum-split,
req_witness,
rsub_wf,
rsum_wf,
int_seg_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
decidable__le,
all_wf,
req_wf,
le_wf,
real_wf,
radd_wf,
rminus_wf,
req_functionality,
rsub_functionality,
req_weakening,
uiff_transitivity,
req_inversion,
radd-assoc,
radd-ac,
real_term_polynomial,
itermSubtract_wf,
itermMinus_wf,
int-to-real_wf,
req-iff-rsub-is-0,
radd_functionality,
rminus_functionality,
rsum_functionality2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
because_Cache,
addEquality,
natural_numberEquality,
setElimination,
rename,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaFormation,
lemma_by_obid
Latex:
\mforall{}[k,n,m:\mBbbZ{}]. \mforall{}[x,y:\{k..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}].
((\mSigma{}\{x[i] | k\mleq{}i\mleq{}m\} - \mSigma{}\{y[i] | k\mleq{}i\mleq{}n\}) = \mSigma{}\{x[i] | n + 1\mleq{}i\mleq{}m\}) supposing
((\mforall{}i:\{k..n + 1\msupminus{}\}. (x[i] = y[i])) and
(n \mleq{} m) and
(k \mleq{} n))
Date html generated:
2017_10_03-AM-08_58_52
Last ObjectModification:
2017_07_28-AM-07_38_30
Theory : reals
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