Nuprl Lemma : sum-as-accum
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].
(Σ(f[x] | x < n) ~ accumulate (with value x and list item y):
x + y
over list:
map(λx.f[x];upto(n))
with starting value:
0))
Proof
Definitions occuring in Statement :
upto: upto(n),
sum: Σ(f[x] | x < k),
map: map(f;as),
list_accum: list_accum,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
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,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
list_accum: list_accum,
map: map(f;as),
list_ind: list_ind,
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
le: A ≤ B,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
bnot: ¬bb,
assert: ↑b
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,
subtype_base_sq,
int_subtype_base,
list_accum_wf,
nil_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
sum-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
subtype_rel_dep_function,
int_seg_subtype,
false_wf,
subtype_rel_self,
upto_decomp1,
map_append_sq,
list_accum_append,
map_wf,
decidable__lt,
lelt_wf,
upto_wf,
subtype_rel_list,
map_cons_lemma,
map_nil_lemma,
list_accum_cons_lemma,
list_accum_nil_lemma,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
functionEquality,
instantiate,
cumulativity,
because_Cache,
addEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
productElimination,
lessCases,
imageMemberEquality,
baseClosed,
imageElimination,
applyEquality,
dependent_set_memberEquality,
functionExtensionality,
promote_hyp
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}].
(\mSigma{}(f[x] | x < n) \msim{} accumulate (with value x and list item y):
x + y
over list:
map(\mlambda{}x.f[x];upto(n))
with starting value:
0))
Date html generated:
2017_04_17-AM-08_25_28
Last ObjectModification:
2017_02_27-PM-04_48_11
Theory : list_1
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