Nuprl Lemma : isolate_summand
∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn]. (Σ(f[x] | x < n) = (f[m] + Σ(if (x =z m) then 0 else f[x] fi | x < n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
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,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
sq_type: SQType(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
squash: ↓T,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
subtype_rel_function,
int_seg_subtype,
false_wf,
not-le-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-commutes,
le-add-cancel2,
subtype_rel_self,
nat_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
eq_int_wf,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
not_functionality_wrt_uiff,
assert_wf,
sum-unroll,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
sum_functionality,
le_wf,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
functionEquality,
because_Cache,
productElimination,
Error :universeIsType,
Error :functionIsType,
Error :inhabitedIsType,
unionElimination,
applyEquality,
addEquality,
minusEquality,
multiplyEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
equalityElimination,
lessCases,
axiomSqEquality,
imageMemberEquality,
baseClosed,
imageElimination,
promote_hyp,
dependent_set_memberEquality,
functionExtensionality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[f:\mBbbN{}n {}\mrightarrow{} \mBbbZ{}]. \mforall{}[m:\mBbbN{}n].
(\mSigma{}(f[x] | x < n) = (f[m] + \mSigma{}(if (x =\msubz{} m) then 0 else f[x] fi | x < n)))
Date html generated:
2019_06_20-PM-01_18_23
Last ObjectModification:
2018_09_26-PM-02_40_38
Theory : int_2
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