Nuprl Lemma : real-vec-sum-split
∀[j,n,m:ℤ]. ∀[k:ℕ]. ∀[x:{n..m + 1-} ⟶ ℝ^k].
(req-vec(k;Σ{x[i] | n≤i≤m};Σ{x[i] | n≤i≤j} + Σ{x[i] | j + 1≤i≤m})) supposing ((j ≤ m) and (n ≤ j))
Proof
Definitions occuring in Statement :
real-vec-sum: Σ{x[k] | n≤k≤m},
real-vec-add: X + Y,
req-vec: req-vec(n;x;y),
real-vec: ℝ^n,
int_seg: {i..j-},
nat: ℕ,
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,
req-vec: req-vec(n;x;y),
all: ∀x:A. B[x],
real-vec-sum: Σ{x[k] | n≤k≤m},
real-vec-add: X + Y,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
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,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
req_witness,
real-vec-sum_wf,
subtype_rel_self,
real_wf,
real-vec-add_wf,
int_seg_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-le,
istype-less_than,
decidable__le,
real-vec_wf,
istype-nat,
rsum_wf,
radd_wf,
req_weakening,
req_functionality,
req_inversion,
rsum-split
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalRule,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
lambdaEquality_alt,
dependent_functionElimination,
applyEquality,
addEquality,
functionEquality,
productElimination,
imageElimination,
because_Cache,
dependent_set_memberEquality_alt,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
productIsType,
closedConclusion,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
functionIsType
Latex:
\mforall{}[j,n,m:\mBbbZ{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}\^{}k].
(req-vec(k;\mSigma{}\{x[i] | n\mleq{}i\mleq{}m\};\mSigma{}\{x[i] | n\mleq{}i\mleq{}j\} + \mSigma{}\{x[i] | j + 1\mleq{}i\mleq{}m\})) supposing ((j \mleq{} m) and (n \mleq{} j))
Date html generated:
2019_10_30-AM-08_02_21
Last ObjectModification:
2019_09_17-PM-05_23_03
Theory : reals
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