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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