Nuprl Lemma : lsum-upto
∀[k:ℕ]. ∀[f:ℕk ⟶ ℤ]. (Σ(f[x] | x ∈ upto(k)) = Σ(f[x] | x < k) ∈ ℤ)
Proof
Definitions occuring in Statement :
lsum: Σ(f[x] | x ∈ L),
upto: upto(n),
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
so_lambda: λ2x.t[x],
so_apply: x[s],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
true: True,
guard: {T},
rev_implies: P ⇐ Q,
le: A ≤ B,
less_than: a < b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than': less_than'(a;b),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑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,
lsum_nil_lemma,
int_seg_wf,
subtract-1-ge-0,
istype-nat,
upto_decomp1,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
lsum-append,
upto_wf,
subtract_wf,
subtype_rel_list,
cons_wf,
nil_wf,
l_member_wf,
append_wf,
member_append,
from-upto-member,
member_singleton,
decidable__le,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
istype-le,
sum_wf,
subtype_rel_self,
iff_weakening_equal,
lsum_wf,
int_seg_properties,
add_functionality_wrt_eq,
sum-unroll,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
lsum_cons_lemma,
istype-top,
decidable__equal_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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
because_Cache,
functionIsType,
dependent_set_memberEquality_alt,
unionElimination,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
instantiate,
universeEquality,
intEquality,
productElimination,
productIsType,
setIsType,
imageMemberEquality,
baseClosed,
closedConclusion,
equalityElimination,
lessCases,
axiomSqEquality,
equalityIstype,
promote_hyp,
cumulativity
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[f:\mBbbN{}k {}\mrightarrow{} \mBbbZ{}]. (\mSigma{}(f[x] | x \mmember{} upto(k)) = \mSigma{}(f[x] | x < k))
Date html generated:
2020_05_19-PM-09_47_52
Last ObjectModification:
2019_11_27-AM-10_12_19
Theory : list_1
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