Nuprl Lemma : exp-ratio_wf2
∀b:{2...}. ∀k:ℕ. ∀M:ℕ+.  (exp-ratio(1;b;0;k;M) ∈ {n:ℕ| k < M * b^n} )
Proof
Definitions occuring in Statement : 
exp-ratio: exp-ratio(a;b;n;p;q)
, 
exp: i^n
, 
int_upper: {i...}
, 
nat_plus: ℕ+
, 
nat: ℕ
, 
less_than: a < b
, 
all: ∀x:A. B[x]
, 
member: t ∈ T
, 
set: {x:A| B[x]} 
, 
multiply: n * m
, 
natural_number: $n
Definitions unfolded in proof : 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
nat: ℕ
, 
so_lambda: λ2x.t[x]
, 
nat_plus: ℕ+
, 
int_upper: {i...}
, 
prop: ℙ
, 
so_apply: x[s]
, 
all: ∀x:A. B[x]
, 
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
, 
exp-ratio: exp-ratio(a;b;n;p;q)
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
guard: {T}
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
subtract: n - m
, 
le: A ≤ B
, 
has-value: (a)↓
, 
sq_type: SQType(T)
, 
subtype_rel: A ⊆r B
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
less_than': less_than'(a;b)
, 
true: True
, 
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced : 
le_wf, 
nat_wf, 
set_wf, 
less_than_wf, 
exp_wf2, 
nat_plus_wf, 
int_upper_wf, 
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, 
decidable__le, 
subtract_wf, 
intformnot_wf, 
itermSubtract_wf, 
int_formula_prop_not_lemma, 
int_term_value_subtract_lemma, 
lt_int_wf, 
sq_stable__less_than, 
nat_plus_properties, 
int_upper_properties, 
bool_wf, 
equal-wf-T-base, 
assert_wf, 
le_int_wf, 
bnot_wf, 
uiff_transitivity, 
eqtt_to_assert, 
assert_of_lt_int, 
eqff_to_assert, 
assert_functionality_wrt_uiff, 
bnot_of_lt_int, 
assert_of_le_int, 
equal_wf, 
minus-zero, 
add-zero, 
multiply-is-int-iff, 
itermMultiply_wf, 
int_term_value_mul_lemma, 
false_wf, 
value-type-has-value, 
int-value-type, 
subtype_base_sq, 
int_subtype_base, 
decidable__equal_int, 
intformeq_wf, 
itermAdd_wf, 
int_formula_prop_eq_lemma, 
int_term_value_add_lemma, 
one-mul, 
mul-swap, 
exp_step, 
decidable__lt, 
add-subtract-cancel, 
set_subtype_base, 
exp_wf_nat_plus, 
squash_wf, 
true_wf, 
minus-add, 
minus-minus, 
minus-one-mul, 
add-associates, 
minus-one-mul-top, 
add-mul-special, 
zero-mul, 
zero-add, 
not-lt-2, 
add_functionality_wrt_le, 
add-commutes, 
le-add-cancel, 
primrec-wf-nat-plus, 
nat_plus_subtype_nat, 
exp1, 
iff_weakening_equal, 
mul_preserves_le, 
exp_add, 
int_upper_subtype_nat, 
le_functionality, 
le_weakening, 
multiply_functionality_wrt_le, 
exp0_lemma, 
not-equal-2, 
condition-implies-le, 
mul-one
Rules used in proof : 
cut, 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
sqequalRule, 
lambdaEquality, 
multiplyEquality, 
because_Cache, 
setEquality, 
natural_numberEquality, 
isect_memberFormation, 
lambdaFormation, 
dependent_functionElimination, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
isect_memberEquality, 
intWeakElimination, 
independent_isectElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
unionElimination, 
dependent_set_memberEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
equalityElimination, 
productElimination, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
callbyvalueReduce, 
addEquality, 
instantiate, 
cumulativity, 
applyEquality, 
minusEquality, 
universeEquality
Latex:
\mforall{}b:\{2...\}.  \mforall{}k:\mBbbN{}.  \mforall{}M:\mBbbN{}\msupplus{}.    (exp-ratio(1;b;0;k;M)  \mmember{}  \{n:\mBbbN{}|  k  <  M  *  b\^{}n\}  )
Date html generated:
2018_05_21-PM-01_03_50
Last ObjectModification:
2018_01_28-PM-02_12_55
Theory : num_thy_1
Home
Index