Nuprl Lemma : cubic_converge2_wf
∀a:ℕ+. ∀b:{a + 1...}. ∀k:{k:ℕ| (2 * a^3^k) ≤ b^3^k} . ∀m:ℕ. (cubic_converge2(a;b;k;m) ∈ {n:ℕ| (a^3^n * m) ≤ b^3^n} )
Proof
Definitions occuring in Statement :
cubic_converge2: cubic_converge2(a;b;k;m),
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
nat_plus: ℕ+,
int_upper: {i...},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
squash: ↓T,
cubic_converge2: cubic_converge2(a;b;k;m),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
has-value: (a)↓,
exp: i^n,
primrec: primrec(n;b;c),
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
int_upper_properties,
nat_plus_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
sq_stable__le,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
set_wf,
exp_wf2,
exp_wf4,
int_upper_wf,
nat_plus_wf,
exp0_lemma,
squash_wf,
true_wf,
exp1,
iff_weakening_equal,
int_subtype_base,
subtype_rel_sets,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
iroot-property,
iroot_wf,
value-type-has-value,
int-value-type,
add_nat_wf,
exp-one,
set_subtype_base,
member-less_than,
subtract-add-cancel,
exp_preserves_lt,
le_weakening2,
exp_wf_nat_plus,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
set-value-type,
exp_preserves_le,
mul-non-neg1,
zero-le-nat,
int_upper_subtype_nat,
exp-of-mul,
exp_mul,
exp_add,
nat_plus_subtype_nat,
le_functionality,
multiply_functionality_wrt_le,
le_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
addEquality,
productElimination,
unionElimination,
applyEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
multiplyEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
universeEquality,
int_eqReduceFalseSq,
setEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
callbyvalueReduce,
productEquality,
minusEquality,
sqequalIntensionalEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}a:\mBbbN{}\msupplus{}. \mforall{}b:\{a + 1...\}. \mforall{}k:\{k:\mBbbN{}| (2 * a\^{}3\^{}k) \mleq{} b\^{}3\^{}k\} . \mforall{}m:\mBbbN{}.
(cubic\_converge2(a;b;k;m) \mmember{} \{n:\mBbbN{}| (a\^{}3\^{}n * m) \mleq{} b\^{}3\^{}n\} )
Date html generated:
2017_10_04-PM-10_25_09
Last ObjectModification:
2017_07_28-AM-08_49_00
Theory : reals_2
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