Nuprl Lemma : log-property
∀[b:{i:ℤ| 1 < i} ]. ∀[x:ℕ].  (b^log(b;x) ≤ x) ∧ x < b^log(b;x) + 1 supposing 0 < x
Proof
Definitions occuring in Statement : 
log: log(b;n)
, 
exp: i^n
, 
nat: ℕ
, 
less_than: a < b
, 
uimplies: b supposing a
, 
uall: ∀[x:A]. B[x]
, 
le: A ≤ B
, 
and: P ∧ Q
, 
set: {x:A| B[x]} 
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
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}
, 
le: A ≤ B
, 
int_seg: {i..j-}
, 
less_than: a < b
, 
lelt: i ≤ j < k
, 
subtype_rel: A ⊆r B
, 
int_nzero: ℤ-o
, 
nequal: a ≠ b ∈ T 
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
less_than': less_than'(a;b)
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
log: log(b;n)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
cand: A c∧ B
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
bfalse: ff
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
nat_plus: ℕ+
, 
sq_type: SQType(T)
, 
exp: i^n
Lemmas referenced : 
mul-one, 
primrec1_lemma, 
exp_add, 
mul_preserves_lt, 
mul_preserves_le, 
int_term_value_mul_lemma, 
itermMultiply_wf, 
int_subtype_base, 
subtype_base_sq, 
div_bounds_1, 
le-add-cancel, 
zero-add, 
add-associates, 
add-commutes, 
add-swap, 
add_functionality_wrt_le, 
less-iff-le, 
not-lt-2, 
rem_bounds_1, 
nequal_wf, 
subtype_rel_sets, 
div_rem_sum, 
set_wf, 
decidable__lt, 
int_term_value_add_lemma, 
itermAdd_wf, 
add-is-int-iff, 
nat_wf, 
add_nat_wf, 
assert_of_le_int, 
bnot_of_lt_int, 
assert_functionality_wrt_uiff, 
eqff_to_assert, 
bnot_wf, 
le_int_wf, 
bfalse_wf, 
iff_weakening_equal, 
exp1, 
true_wf, 
squash_wf, 
exp0_lemma, 
assert_of_lt_int, 
eqtt_to_assert, 
assert_wf, 
btrue_wf, 
equal_wf, 
uiff_transitivity, 
bool_wf, 
lt_int_wf, 
le_wf, 
int_formula_prop_eq_lemma, 
intformeq_wf, 
sq_stable__less_than, 
lelt_wf, 
false_wf, 
int_seg_subtype, 
decidable__equal_int, 
int_term_value_subtract_lemma, 
int_formula_prop_not_lemma, 
itermSubtract_wf, 
intformnot_wf, 
subtract_wf, 
decidable__le, 
int_seg_wf, 
int_nzero_properties, 
less_than_irreflexivity, 
less_than_transitivity1, 
exp_wf3, 
int_seg_properties, 
member-less_than, 
log_wf, 
exp_wf2, 
less_than'_wf, 
less_than_wf, 
ge_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
nat_properties
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
thin, 
lambdaFormation, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
hypothesisEquality, 
hypothesis, 
setElimination, 
rename, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
dependent_functionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
sqequalRule, 
independent_pairFormation, 
computeAll, 
independent_functionElimination, 
productElimination, 
independent_pairEquality, 
axiomEquality, 
equalityTransitivity, 
equalitySymmetry, 
applyEquality, 
because_Cache, 
unionElimination, 
setEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
hypothesis_subsumption, 
dependent_set_memberEquality, 
equalityElimination, 
universeEquality, 
equalityEquality, 
addEquality, 
pointwiseFunctionality, 
promote_hyp, 
baseApply, 
closedConclusion, 
instantiate, 
cumulativity, 
divideEquality, 
multiplyEquality
Latex:
\mforall{}[b:\{i:\mBbbZ{}|  1  <  i\}  ].  \mforall{}[x:\mBbbN{}].    (b\^{}log(b;x)  \mleq{}  x)  \mwedge{}  x  <  b\^{}log(b;x)  +  1  supposing  0  <  x
Date html generated:
2016_05_15-PM-04_49_24
Last ObjectModification:
2016_01_16-AM-11_40_48
Theory : general
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