Nuprl Lemma : log-property
∀[b:{i:ℤ| 1 < i} ]. ∀[x:ℤ]. (x ≤ b^log(b;x))
Proof
Definitions occuring in Statement :
log: log(b;n),
exp: i^n,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
log: log(b;n),
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
nat: ℕ,
sq_stable: SqStable(P),
le: A ≤ B
Lemmas referenced :
exp-as-genfact,
log_wf,
genfact-inv_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-less_than,
sq_stable__le,
genfact_wf,
le_witness_for_triv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
Error :lambdaEquality_alt,
because_Cache,
Error :universeIsType,
independent_isectElimination,
Error :lambdaFormation_alt,
dependent_functionElimination,
unionElimination,
natural_numberEquality,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
closedConclusion,
imageMemberEquality,
baseClosed,
Error :inhabitedIsType,
imageElimination,
Error :equalityIstype,
equalityTransitivity,
equalitySymmetry,
productElimination,
Error :isectIsTypeImplies,
Error :setIsType
Latex:
\mforall{}[b:\{i:\mBbbZ{}| 1 < i\} ]. \mforall{}[x:\mBbbZ{}]. (x \mleq{} b\^{}log(b;x))
Date html generated:
2019_06_20-PM-02_32_37
Last ObjectModification:
2019_02_11-AM-11_59_13
Theory : num_thy_1
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