Nuprl Lemma : sum_aux-as-primrec
∀[v,i,k:ℤ]. ∀[f:{i..k-} ⟶ ℤ].  sum_aux(k;v;i;x.f[x]) ~ primrec(k - i;v;λj,x. (x + f[i + j])) supposing i ≤ k
Proof
Definitions occuring in Statement : 
sum_aux: sum_aux(k;v;i;x.f[x])
, 
primrec: primrec(n;b;c)
, 
int_seg: {i..j-}
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
le: A ≤ B
, 
lambda: λx.A[x]
, 
function: x:A ⟶ B[x]
, 
subtract: n - m
, 
add: n + m
, 
int: ℤ
, 
sqequal: s ~ t
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
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
sum_aux: sum_aux(k;v;i;x.f[x])
, 
lt_int: i <z j
, 
subtract: n - m
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
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
, 
assert: ↑b
, 
rev_implies: P 
⇐ Q
, 
iff: P 
⇐⇒ Q
, 
has-value: (a)↓
, 
so_apply: x[s]
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
decidable: Dec(P)
, 
so_lambda: λ2x.t[x]
, 
le: A ≤ B
Lemmas referenced : 
nat_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
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, 
istype-less_than, 
primrec-unroll, 
lt_int_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
itermAdd_wf, 
int_term_value_add_lemma, 
eqff_to_assert, 
bool_cases_sqequal, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
iff_weakening_uiff, 
assert_wf, 
less_than_wf, 
istype-top, 
int_seg_wf, 
subtract-1-ge-0, 
value-type-has-value, 
int-value-type, 
decidable__le, 
intformnot_wf, 
int_formula_prop_not_lemma, 
decidable__lt, 
istype-le, 
istype-nat, 
int_subtype_base, 
decidable__equal_int, 
intformeq_wf, 
itermSubtract_wf, 
int_formula_prop_eq_lemma, 
int_term_value_subtract_lemma, 
int_seg_properties, 
subtract_wf, 
primrec0_lemma, 
add-commutes, 
add-zero, 
lelt_wf, 
le_wf
Rules used in proof : 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
Error :dependent_pairFormation_alt, 
Error :lambdaEquality_alt, 
int_eqEquality, 
dependent_functionElimination, 
Error :isect_memberEquality_alt, 
voidElimination, 
sqequalRule, 
independent_pairFormation, 
Error :universeIsType, 
axiomSqEquality, 
Error :isectIsTypeImplies, 
Error :inhabitedIsType, 
Error :functionIsTypeImplies, 
Error :isect_memberFormation_alt, 
because_Cache, 
addEquality, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
productElimination, 
lessCases, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
Error :equalityIstype, 
promote_hyp, 
instantiate, 
cumulativity, 
Error :functionIsType, 
callbyvalueReduce, 
intEquality, 
applyEquality, 
Error :dependent_set_memberEquality_alt, 
Error :productIsType, 
functionEquality, 
functionExtensionality, 
dependent_set_memberEquality, 
voidEquality, 
isect_memberEquality, 
lambdaEquality, 
dependent_pairFormation, 
isect_memberFormation
Latex:
\mforall{}[v,i,k:\mBbbZ{}].  \mforall{}[f:\{i..k\msupminus{}\}  {}\mrightarrow{}  \mBbbZ{}].
    sum\_aux(k;v;i;x.f[x])  \msim{}  primrec(k  -  i;v;\mlambda{}j,x.  (x  +  f[i  +  j]))  supposing  i  \mleq{}  k
Date html generated:
2019_06_20-PM-01_17_48
Last ObjectModification:
2019_02_06-PM-03_57_03
Theory : int_2
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