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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