Nuprl Lemma : radd-approx
∀[a,b:ℝ]. ∀[n:ℕ+]. ((a + b) n ~ ((a (4 * n)) + (b (4 * n))) ÷ 4)
Proof
Definitions occuring in Statement :
radd: a + b,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
divide: n ÷ m,
multiply: n * m,
add: n + m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
radd: a + b,
reg-seq-list-add: reg-seq-list-add(L),
accelerate: accelerate(k;f),
cbv_list_accum: cbv_list_accum(x,a.f[x; a];y;L),
has-value: (a)↓,
nat_plus: ℕ+,
cons: [a / b],
real: ℝ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
prop: ℙ,
all: ∀x:A. B[x],
so_lambda: λ2x y.t[x; y],
top: Top,
so_apply: x[s1;s2],
nil: [],
it: ⋅,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
value-type-has-value,
int-value-type,
mul_nat_plus,
less_than_wf,
list_wf,
real_wf,
list-value-type,
cons_wf,
nil_wf,
spread_cons_lemma,
add-associates,
zero-add,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
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_formula_prop_wf,
decidable__lt,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
equal-wf-base,
true_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
callbyvalueReduce,
sqleReflexivity,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
because_Cache,
addEquality,
applyEquality,
dependent_set_memberEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
divideEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
addLevel,
lambdaFormation,
independent_functionElimination,
sqequalAxiom
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a + b) n \msim{} ((a (4 * n)) + (b (4 * n))) \mdiv{} 4)
Date html generated:
2017_01_09-AM-08_54_58
Last ObjectModification:
2016_11_21-PM-03_00_30
Theory : reals
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