Nuprl Lemma : exp-ratio-property
∀a:ℕ. ∀b:{a + 1...}. ∀k:ℕ. (exp-ratio(a;b;0;k;1) ∈ {n:ℕ| k * a^n < b^n} )
Proof
Definitions occuring in Statement :
exp-ratio: exp-ratio(a;b;n;p;q),
exp: i^n,
int_upper: {i...},
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
nat_plus: ℕ+,
implies: P ⇒ Q,
prop: ℙ,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
cand: A c∧ B,
sq_type: SQType(T),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
mul-one,
exp-ratio_wf,
mul_bounds_1b,
subtract-add-cancel,
le_weakening,
less_than_functionality,
minus-zero,
not-equal-2,
mul_nat_plus,
le-add-cancel,
zero-add,
add-zero,
add_functionality_wrt_le,
minus-one-mul-top,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
exp_wf_nat_plus,
decidable__lt,
exp-zero,
less_than_wf,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
mul_bounds_1a,
add-swap,
add-commutes,
one-mul,
mul-swap,
mul-commutes,
add-associates,
mul-associates,
mul-distributes,
mul-distributes-right,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
intformeq_wf,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
multiply_nat_wf,
exp_wf4,
add_nat_wf,
int_upper_subtype_nat,
mul_preserves_le,
exp_add,
add-subtract-cancel,
iff_weakening_equal,
exp1,
add_functionality_wrt_eq,
true_wf,
squash_wf,
int_term_value_add_lemma,
itermAdd_wf,
false_wf,
exp0_lemma,
nat_plus_wf,
nat_plus_subtype_nat,
primrec-wf-nat-plus,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_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_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
int_upper_properties,
nat_properties,
exp_wf2,
le_wf,
nat_plus_properties,
int_upper_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
hypothesis,
lemma_by_obid,
isectElimination,
thin,
addEquality,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
multiplyEquality,
because_Cache,
applyEquality,
introduction,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
setEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
equalityEquality,
minusEquality,
instantiate,
cumulativity
Latex:
\mforall{}a:\mBbbN{}. \mforall{}b:\{a + 1...\}. \mforall{}k:\mBbbN{}. (exp-ratio(a;b;0;k;1) \mmember{} \{n:\mBbbN{}| k * a\^{}n < b\^{}n\} )
Date html generated:
2016_05_15-PM-04_07_57
Last ObjectModification:
2016_01_16-AM-11_07_12
Theory : general
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