Nuprl Lemma : reg-seq-mul-regular
∀[x,y:ℝ]. ∀k:ℕ+. k + 1-regular-seq(reg-seq-mul(x;y)) supposing ∀n:ℕ+. ((|x n| ≤ (n * k)) ∧ (|y n| ≤ (n * k)))
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],
and: P ∧ Q,
apply: f a,
multiply: 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],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
regular-int-seq: k-regular-seq(f),
subtype_rel: A ⊆r B,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
real: ℝ,
nat: ℕ,
less_than': less_than'(a;b),
guard: {T},
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
reg-seq-mul-regular-eventually,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
istype-int_upper,
subtype_rel_sets_simple,
less_than_wf,
le_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_witness_for_triv,
istype-le,
absval_wf,
nat_plus_wf,
real_wf,
int_upper_properties,
mul_preserves_le,
upper_subtype_nat,
istype-false,
subtract_wf,
add_nat_wf,
multiply_nat_wf,
nat_properties,
itermMultiply_wf,
intformeq_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
itermSubtract_wf,
int_term_value_subtract_lemma,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
add_functionality_wrt_le
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,
addEquality,
setElimination,
rename,
hypothesis,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
inhabitedIsType,
applyEquality,
intEquality,
because_Cache,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
functionIsType,
productIsType,
multiplyEquality,
isectIsTypeImplies,
applyLambdaEquality,
equalityIstype
Latex:
\mforall{}[x,y:\mBbbR{}].
\mforall{}k:\mBbbN{}\msupplus{}
k + 1-regular-seq(reg-seq-mul(x;y)) supposing \mforall{}n:\mBbbN{}\msupplus{}. ((|x n| \mleq{} (n * k)) \mwedge{} (|y n| \mleq{} (n * k)))
Date html generated:
2019_10_16-PM-03_06_27
Last ObjectModification:
2019_02_14-PM-06_37_56
Theory : reals
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