Nuprl Lemma : reg_seq_mul-regular
∀[x,y:ℝ].
∀B:ℕ+
B-regular-seq(reg_seq_mul(x;y)) supposing ∀n,m:ℕ+. ((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,
regular-int-seq: k-regular-seq(f),
absval: |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,
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],
top: Top,
prop: ℙ,
false: False,
regular-int-seq: k-regular-seq(f),
subtype_rel: A ⊆r B,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
le: A ≤ B,
real: ℝ,
nat: ℕ,
guard: {T}
Lemmas referenced :
reg_seq_mul-regular-eventually,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
istype-int_upper,
subtype_rel_sets_simple,
less_than_wf,
le_wf,
decidable__le,
intformand_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
le_witness_for_triv,
istype-le,
absval_wf,
subtract_wf,
nat_plus_wf,
real_wf,
int_upper_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
hypothesis,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
inhabitedIsType,
applyEquality,
intEquality,
int_eqEquality,
independent_pairFormation,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
functionIsType,
multiplyEquality,
addEquality,
isectIsTypeImplies
Latex:
\mforall{}[x,y:\mBbbR{}].
\mforall{}B:\mBbbN{}\msupplus{}
B-regular-seq(reg\_seq\_mul(x;y))
supposing \mforall{}n,m:\mBbbN{}\msupplus{}. ((2 * ((m * |x n|) + (n * |y m|))) \mleq{} ((n * m) * ((4 * B) - 1)))
Date html generated:
2019_10_16-PM-03_06_16
Last ObjectModification:
2019_02_14-PM-06_18_44
Theory : reals
Home
Index