Nuprl Lemma : ireal-approx-rmul
∀[x,y:ℝ]. ∀[j,M:ℕ+]. ∀[a,b:ℤ].
∀k:ℕ+
((|x| ≤ (r1/r(4)))
⇒ ((2 * |b|) ≤ (k * M))
⇒ j-approx(x;k * M;a)
⇒ j-approx(y;M;b)
⇒ j-approx(x * y;M;(a * b) ÷ 2 * k * M))
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: ℕ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
rneq: x ≠ y,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
rge: x ≥ y,
rless: x < y,
sq_exists: ∃x:A [B[x]],
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
real: ℝ,
sq_stable: SqStable(P),
ge: i ≥ j ,
top: Top
Latex:
\mforall{}[x,y:\mBbbR{}]. \mforall{}[j,M:\mBbbN{}\msupplus{}]. \mforall{}[a,b:\mBbbZ{}].
\mforall{}k:\mBbbN{}\msupplus{}
((|x| \mleq{} (r1/r(4)))
{}\mRightarrow{} ((2 * |b|) \mleq{} (k * M))
{}\mRightarrow{} j-approx(x;k * M;a)
{}\mRightarrow{} j-approx(y;M;b)
{}\mRightarrow{} j-approx(x * y;M;(a * b) \mdiv{} 2 * k * M))
Date html generated:
2020_05_20-AM-11_16_30
Last ObjectModification:
2020_01_06-PM-00_56_44
Theory : reals
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