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) - 1)))
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 :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uimplies: b supposing a,
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
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,
subtype_rel: A ⊆r B,
nat: ℕ,
sq_stable: SqStable(P),
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
guard: {T},
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
reg_seq_mul: reg_seq_mul(x;y),
less_than: a < b,
cand: A c∧ B,
ge: i ≥ j ,
regular-int-seq: k-regular-seq(f),
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q)
Lemmas referenced :
sq_stable__le,
absval_wf,
subtract_wf,
reg_seq_mul_wf,
int_upper_properties,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformle_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_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
istype-less_than,
mul_cancel_in_le,
absval_nat_plus,
int_entire_a,
subtype_base_sq,
int_subtype_base,
mul_nzero,
subtype_rel_sets_simple,
le_wf,
nequal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
istype-le,
squash_wf,
true_wf,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
istype-int_upper,
le_witness_for_triv,
nat_plus_wf,
real_wf,
decidable__equal_int,
itermMultiply_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_term_value_subtract_lemma,
rounding-div_wf,
rounding-div-property,
nat_wf,
set_subtype_base,
absval-non-neg,
absval_pos,
multiply_nat_wf,
decidable__le,
upper_subtype_nat,
le_weakening2,
nat_properties,
mul_preserves_le,
absval-diff-symmetry,
itermAdd_wf,
int_term_value_add_lemma,
le_functionality,
le_transitivity,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
absval-diff-product-bound2,
multiply_functionality_wrt_le,
add_functionality_wrt_eq,
multiply-is-int-iff,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
multiplyEquality,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_set_memberEquality_alt,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
because_Cache,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
addEquality,
instantiate,
cumulativity,
intEquality,
equalityIstype,
baseClosed,
sqequalBase,
imageElimination,
imageMemberEquality,
universeEquality,
productElimination,
functionIsTypeImplies,
functionIsType,
isectIsTypeImplies,
applyLambdaEquality,
closedConclusion,
pointwiseFunctionality,
promote_hyp,
baseApply
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) - 1)))
Date html generated:
2019_10_16-PM-03_06_11
Last ObjectModification:
2019_02_15-AM-10_33_27
Theory : reals
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