Nuprl Lemma : reg-seq-inv_wf
∀[b:ℕ+]. ∀[x:{f:ℕ+ ⟶ ℤ| b-regular-seq(f)} ]. ∀[k:ℕ+].
  reg-seq-inv(x) ∈ {f:ℕ+ ⟶ ℤ| b * ((k * k) + 1)-regular-seq(f)}  supposing ∀m:ℕ+. ((2 * m) ≤ (k * |x m|))
Proof
Definitions occuring in Statement : 
reg-seq-inv: reg-seq-inv(x)
, 
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]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
apply: f a
, 
function: x:A ⟶ B[x]
, 
multiply: n * m
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
uimplies: b supposing a
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
nat: ℕ
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
squash: ↓T
, 
prop: ℙ
, 
nat_plus: ℕ+
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
true: True
, 
subtype_rel: A ⊆r B
, 
guard: {T}
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
sq_type: SQType(T)
, 
reg-seq-inv: reg-seq-inv(x)
, 
nequal: a ≠ b ∈ T 
, 
regular-int-seq: k-regular-seq(f)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
less_than: a < b
, 
less_than': less_than'(a;b)
, 
bfalse: ff
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
le: A ≤ B
, 
int_nzero: ℤ-o
, 
sq_stable: SqStable(P)
, 
rev_uimplies: rev_uimplies(P;Q)
, 
ge: i ≥ j 
, 
subtract: n - m
, 
absval: |i|
Lemmas referenced : 
decidable__not, 
decidable__equal_int, 
nat_plus_wf, 
subtype_base_sq, 
nat_wf, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
absval-non-neg, 
equal_wf, 
squash_wf, 
true_wf, 
absval_pos, 
nat_plus_properties, 
decidable__le, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformeq_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_term_value_var_lemma, 
int_formula_prop_eq_lemma, 
int_formula_prop_wf, 
iff_weakening_equal, 
itermMultiply_wf, 
intformless_wf, 
int_term_value_mul_lemma, 
int_formula_prop_less_lemma, 
equal-wf-T-base, 
regular-int-seq_wf, 
all_wf, 
absval_wf, 
set_wf, 
absval_unfold, 
lt_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
top_wf, 
less_than_wf, 
decidable__lt, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
assert-bnot, 
itermMinus_wf, 
int_term_value_minus_lemma, 
mul_cancel_in_le, 
subtract_wf, 
absval_nat_plus, 
int_entire_a, 
absval_mul, 
multiply_nat_plus, 
add_nat_plus, 
multiply_nat_wf, 
nat_plus_subtype_nat, 
left_mul_subtract_distrib, 
div_rem_sum2, 
nequal_wf, 
rem_bounds_absval, 
sq_stable__less_than, 
le_functionality, 
le_weakening, 
add_functionality_wrt_le, 
int-triangle-inequality, 
mul-distributes, 
minus-add, 
add-associates, 
minus-one-mul, 
mul-swap, 
mul-commutes, 
mul-associates, 
one-mul, 
add-swap, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
add_functionality_wrt_eq, 
mul_bounds_1a, 
multiply_functionality_wrt_le, 
false_wf, 
itermAdd_wf, 
int_term_value_add_lemma, 
multiply-is-int-iff, 
absval_sym, 
nat_properties
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
setElimination, 
thin, 
rename, 
lambdaFormation, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
because_Cache, 
independent_functionElimination, 
dependent_functionElimination, 
applyEquality, 
functionExtensionality, 
hypothesisEquality, 
hypothesis, 
natural_numberEquality, 
unionElimination, 
instantiate, 
cumulativity, 
independent_isectElimination, 
sqequalRule, 
intEquality, 
lambdaEquality, 
dependent_set_memberEquality, 
imageElimination, 
equalityTransitivity, 
equalitySymmetry, 
universeEquality, 
dependent_pairFormation, 
int_eqEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
imageMemberEquality, 
baseClosed, 
productElimination, 
divideEquality, 
multiplyEquality, 
addEquality, 
axiomEquality, 
functionEquality, 
minusEquality, 
equalityElimination, 
lessCases, 
sqequalAxiom, 
promote_hyp, 
remainderEquality, 
applyLambdaEquality, 
baseApply, 
closedConclusion
Latex:
\mforall{}[b:\mBbbN{}\msupplus{}].  \mforall{}[x:\{f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}|  b-regular-seq(f)\}  ].  \mforall{}[k:\mBbbN{}\msupplus{}].
    reg-seq-inv(x)  \mmember{}  \{f:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}|  b  *  ((k  *  k)  +  1)-regular-seq(f)\}    supposing  \mforall{}m:\mBbbN{}\msupplus{}.  ((2  *  m)  \mleq{}  (k  *  |\000Cx  m|))
Date html generated:
2017_10_02-PM-07_16_26
Last ObjectModification:
2017_07_28-AM-07_20_56
Theory : reals
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