Nuprl Lemma : sum_aux_wf
∀[v,i,k:ℤ]. ∀[f:{i..k-} ⟶ ℤ].  sum_aux(k;v;i;x.f[x]) ∈ ℤ supposing i ≤ k
Proof
Definitions occuring in Statement : 
sum_aux: sum_aux(k;v;i;x.f[x])
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
le: A ≤ B
, 
member: t ∈ T
, 
function: x:A ⟶ B[x]
, 
int: ℤ
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
nat: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
sum_aux: sum_aux(k;v;i;x.f[x])
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
true: True
, 
squash: ↓T
, 
bfalse: ff
, 
or: P ∨ Q
, 
sq_type: SQType(T)
, 
guard: {T}
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
decidable: Dec(P)
, 
has-value: (a)↓
, 
so_apply: x[s]
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
so_lambda: λ2x.t[x]
Lemmas referenced : 
nat_properties, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_less_lemma, 
int_formula_prop_wf, 
ge_wf, 
less_than_wf, 
int_seg_wf, 
lt_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
top_wf, 
itermAdd_wf, 
int_term_value_add_lemma, 
eqff_to_assert, 
equal_wf, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_subtype_base, 
assert-bnot, 
decidable__le, 
subtract_wf, 
intformnot_wf, 
itermSubtract_wf, 
int_formula_prop_not_lemma, 
int_term_value_subtract_lemma, 
nat_wf, 
value-type-has-value, 
int-value-type, 
lelt_wf, 
int_subtype_base, 
decidable__equal_int, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
decidable__lt, 
le_wf
Rules used in proof : 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
sqequalRule, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
functionEquality, 
addEquality, 
isect_memberFormation, 
unionElimination, 
equalityElimination, 
productElimination, 
lessCases, 
sqequalAxiom, 
because_Cache, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
promote_hyp, 
instantiate, 
cumulativity, 
callbyvalueReduce, 
applyEquality, 
functionExtensionality, 
dependent_set_memberEquality
Latex:
\mforall{}[v,i,k:\mBbbZ{}].  \mforall{}[f:\{i..k\msupminus{}\}  {}\mrightarrow{}  \mBbbZ{}].    sum\_aux(k;v;i;x.f[x])  \mmember{}  \mBbbZ{}  supposing  i  \mleq{}  k
Date html generated:
2017_04_14-AM-09_19_31
Last ObjectModification:
2017_02_27-PM-03_56_17
Theory : int_2
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