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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