Nuprl Lemma : rsub-approx
∀[a,b:ℝ]. ∀[n:ℕ+]. ((a - b) n ~ ((a (4 * n)) - b (4 * n)) ÷ 4)
Proof
Definitions occuring in Statement :
rsub: x - y,
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
apply: f a,
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
rsub: x - y,
divide: n ÷ m,
rminus: -(x),
subtract: n - m,
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
radd-approx,
rminus_wf,
divide_wfa,
subtract_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
nequal_wf,
nat_plus_wf,
real_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
hypothesisEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality_alt,
multiplyEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
because_Cache,
lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
equalityIstype,
baseClosed,
sqequalBase,
axiomSqEquality,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[a,b:\mBbbR{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a - b) n \msim{} ((a (4 * n)) - b (4 * n)) \mdiv{} 4)
Date html generated:
2019_10_16-PM-03_07_46
Last ObjectModification:
2019_05_23-PM-05_22_49
Theory : reals
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