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

Home Index