Nuprl Lemma : ireal-approx-rmul2
∀[x,y:ℝ]. ∀[j,M:ℕ+]. ∀[a,b:ℤ].
∀k:ℕ+. ((|x| ≤ (r1/r(2))) ⇒ (|b| ≤ k) ⇒ j-approx(x;k;a) ⇒ j-approx(y;2 * M;b) ⇒ j-approx(x * y;M;(a * b) ÷ 4 * k))
Proof
Definitions occuring in Statement :
ireal-approx: j-approx(x;M;z),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rmul: a * b,
int-to-real: r(n),
real: ℝ,
absval: |i|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
divide: n ÷ m,
multiply: n * m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
ireal-approx: j-approx(x;M;z),
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
uiff: uiff(P;Q),
sq_type: SQType(T),
int_nzero: ℤ-o,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
sq_stable: SqStable(P)
Lemmas referenced :
rabs-diff-rmul,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
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,
rless_wf,
ireal-approx_wf,
nat_plus_subtype_nat,
mul_nat_plus,
less_than_wf,
le_wf,
absval_wf,
nat_wf,
rleq_wf,
rabs_wf,
nat_plus_wf,
less_than'_wf,
rsub_wf,
rmul_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
int_subtype_base,
real_wf,
rleq_functionality_wrt_implies,
radd_wf,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rmul_preserves_rleq,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rless_functionality,
req_weakening,
rabs-of-nonneg,
rleq_functionality,
req_inversion,
rabs-rmul,
rmul_preserves_req,
subtract_wf,
rinv_wf2,
rneq_functionality,
rmul-int,
rneq-int,
equal-wf-T-base,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
subtype_base_sq,
decidable__equal_int,
true_wf,
nequal_wf,
rminus_wf,
itermMinus_wf,
rmul-one,
req_functionality,
req_transitivity,
rmul_functionality,
rsub_functionality,
rinv_functionality2,
rinv-of-rmul,
radd_functionality,
rmul-rinv,
rmul-rinv3,
int-rinv-cancel,
rminus-int,
radd-int,
rsub-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
rabs_functionality,
mul-commutes,
equal_wf,
squash_wf,
rem_to_div,
iff_weakening_equal,
rem_bounds_absval_le,
rless_transitivity1,
rleq_weakening,
rabs-rdiv,
rneq_wf,
rabs-int,
set_subtype_base,
absval-non-neg,
absval_pos,
sq_stable__less_than,
mul_preserves_le,
rleq-int-fractions,
rleq_weakening_rless,
rdiv_functionality,
rabs-abs,
rmul-int-rdiv,
rleq-int-fractions3,
rmul_preserves_rleq2,
rmul-nonneg-case1,
false_wf,
int_term_value_add_lemma,
uiff_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
multiplyEquality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
applyEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
independent_pairEquality,
divideEquality,
baseApply,
closedConclusion,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
instantiate,
cumulativity,
addLevel,
imageElimination,
universeEquality,
remainderEquality,
equalityUniverse,
levelHypothesis,
promote_hyp
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[j,M:\mBbbN{}\msupplus{}]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}k:\mBbbN{}\msupplus{}
((|x| \mleq{} (r1/r(2)))
{}\mRightarrow{} (|b| \mleq{} k)
{}\mRightarrow{} j-approx(x;k;a)
{}\mRightarrow{} j-approx(y;2 * M;b)
{}\mRightarrow{} j-approx(x * y;M;(a * b) \mdiv{} 4 * k))
Date html generated:
2018_05_22-PM-02_00_36
Last ObjectModification:
2017_10_25-PM-09_25_53
Theory : reals
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