Nuprl Lemma : sum_arith
∀[n:ℕ]. ∀[a,b:ℤ]. (Σ(a + (b * i) | i < n) = ((n * (a + a + (b * (n - 1)))) ÷ 2) ∈ ℤ)
Proof
Definitions occuring in Statement :
sum: Σ(f[x] | x < k),
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s],
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
guard: {T},
false: False,
prop: ℙ,
int_nzero: ℤ-o,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B
Lemmas referenced :
sum_arith1,
nat_wf,
mul_cancel_in_eq,
sum_wf,
int_seg_wf,
subtype_base_sq,
int_subtype_base,
equal-wf-base,
nequal_wf,
div_rem_sum,
nat_properties,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
false_wf,
mul-commutes,
rem-exact
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
Error :inhabitedIsType,
sqequalRule,
isect_memberEquality,
axiomEquality,
intEquality,
Error :universeIsType,
because_Cache,
lambdaEquality,
addEquality,
multiplyEquality,
setElimination,
rename,
natural_numberEquality,
divideEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
instantiate,
cumulativity,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
baseClosed,
dependent_set_memberEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
productElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidEquality,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
applyEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a,b:\mBbbZ{}]. (\mSigma{}(a + (b * i) | i < n) = ((n * (a + a + (b * (n - 1)))) \mdiv{} 2))
Date html generated:
2019_06_20-PM-01_18_12
Last ObjectModification:
2018_09_26-PM-02_38_30
Theory : int_2
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