Nuprl Lemma : sum-as-accum2
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].
(Σ(f[x] | x < n) ~ accumulate (with value x and list item y):
x + f[y]
over list:
upto(n)
with starting value:
0))
Proof
Definitions occuring in Statement :
upto: upto(n),
sum: Σ(f[x] | x < k),
list_accum: list_accum,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
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,
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[s],
so_apply: x[s1;s2],
sq_type: SQType(T),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k
Lemmas referenced :
nat_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
lelt_wf,
decidable__lt,
list_accum_nil_lemma,
list_accum_cons_lemma,
subtype_rel_list,
list_accum_append,
upto_decomp1,
subtype_rel_self,
int_seg_subtype,
subtype_rel_dep_function,
top_wf,
sum-unroll,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
upto_wf,
list_accum_wf,
int_subtype_base,
subtype_base_sq,
int_seg_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_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,
because_Cache,
addEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
lessCases,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
dependent_set_memberEquality
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 + f[y]
over list:
upto(n)
with starting value:
0))
Date html generated:
2016_05_14-PM-02_34_29
Last ObjectModification:
2016_01_15-AM-07_44_03
Theory : list_1
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