Nuprl Lemma : reg-seq-mul-regular-eventually
∀[x,y:ℝ].
∀B,b:ℕ+.
∀n,m:{b...}. (|(m * (reg-seq-mul(x;y) n)) - n * (reg-seq-mul(x;y) m)| ≤ ((2 * B) * (n + m)))
supposing ∀n,m:{b...}. ((2 * ((m * |x n|) + (n * |y m|))) ≤ ((n * m) * ((4 * B) - 2)))
Proof
Definitions occuring in Statement :
reg-seq-mul: reg-seq-mul(x;y),
real: ℝ,
absval: |i|,
int_upper: {i...},
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
le: A ≤ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
guard: {T},
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
true: True,
sq_stable: SqStable(P),
nat: ℕ,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
nat_plus: ℕ+,
int_upper: {i...},
real: ℝ,
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
reg-seq-mul: reg-seq-mul(x;y),
subtract: n - m,
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
regular-int-seq: k-regular-seq(f),
less_than: a < b,
cand: A c∧ B,
less_than': less_than'(a;b),
absval: |i|,
uiff: uiff(P;Q)
Latex:
\mforall{}[x,y:\mBbbR{}].
\mforall{}B,b:\mBbbN{}\msupplus{}.
\mforall{}n,m:\{b...\}. (|(m * (reg-seq-mul(x;y) n)) - n * (reg-seq-mul(x;y) m)| \mleq{} ((2 * B) * (n + m)))
supposing \mforall{}n,m:\{b...\}. ((2 * ((m * |x n|) + (n * |y m|))) \mleq{} ((n * m) * ((4 * B) - 2)))
Date html generated:
2020_05_20-AM-10_53_08
Last ObjectModification:
2020_03_19-PM-05_00_34
Theory : reals
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