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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