Nuprl Lemma : log_wf

[b:{i:ℤ1 < i} ]. ∀[x:ℤ].  (log(b;x) ∈ ℕ)


Proof




Definitions occuring in Statement :  log: log(b;n) nat: less_than: a < b uall: [x:A]. B[x] member: t ∈ T set: {x:A| B[x]}  natural_number: $n int:
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B decidable: Dec(P) or: P ∨ Q sq_stable: SqStable(P) squash: T so_lambda: λ2x.t[x] so_apply: x[s] sq_type: SQType(T) guard: {T} uiff: uiff(P;Q) ifthenelse: if then else fi  btrue: tt iff: ⇐⇒ Q rev_implies:  Q bfalse: ff log: log(b;n) bool: 𝔹 unit: Unit it: le: A ≤ B less_than': less_than'(a;b) nequal: a ≠ b ∈  int_nzero: -o nat_plus: + true: True less_than: a < b
Lemmas referenced :  int_term_value_mul_lemma itermMultiply_wf mul_preserves_lt set_subtype_base equal-wf-base div_bounds_1 le-add-cancel zero-add add-associates add-commutes add-swap add_functionality_wrt_le less-iff-le not-lt-2 decidable__lt rem_bounds_1 nequal_wf subtype_rel_sets div_rem_sum int_term_value_add_lemma itermAdd_wf add_nat_wf assert_of_le_int bnot_of_lt_int assert_functionality_wrt_uiff le_int_wf bfalse_wf false_wf btrue_wf equal_wf uiff_transitivity int_term_value_minus_lemma int_formula_prop_eq_lemma itermMinus_wf intformeq_wf decidable__equal_int int_subtype_base assert_of_bnot iff_weakening_uiff not_wf bnot_wf assert_wf iff_transitivity eqff_to_assert assert_of_lt_int eqtt_to_assert bool_subtype_base bool_wf subtype_base_sq bool_cases lt_int_wf absval_ifthenelse set_wf sq_stable__less_than int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf subtract_wf decidable__le nat_wf absval_wf le_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 lambdaFormation lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry applyEquality because_Cache unionElimination imageMemberEquality baseClosed imageElimination instantiate cumulativity productElimination impliesFunctionality equalityElimination dependent_set_memberEquality equalityEquality divideEquality addEquality setEquality baseApply closedConclusion minusEquality

Latex:
\mforall{}[b:\{i:\mBbbZ{}|  1  <  i\}  ].  \mforall{}[x:\mBbbZ{}].    (log(b;x)  \mmember{}  \mBbbN{})



Date html generated: 2016_05_15-PM-04_49_10
Last ObjectModification: 2016_01_16-AM-11_27_41

Theory : general


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