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,
guard: {T},
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
prop: ℙ,
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),
sq_stable: SqStable(P),
cand: A c∧ B,
sq_type: SQType(T),
less_than: a < b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_upper_wf,
nat_wf,
nat_plus_properties,
le_wf,
exp_wf2,
nat_properties,
int_upper_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
primrec-wf-nat-plus,
nat_plus_subtype_nat,
nat_plus_wf,
exp0_lemma,
istype-false,
itermAdd_wf,
int_term_value_add_lemma,
squash_wf,
true_wf,
add_functionality_wrt_eq,
exp1,
subtype_rel_self,
iff_weakening_equal,
add-subtract-cancel,
exp_add,
mul_preserves_le,
upper_subtype_nat,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
add_nat_wf,
exp_wf4,
multiply_nat_wf,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
intformeq_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
false_wf,
mul-distributes-right,
mul-distributes,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
exp_step,
int_subtype_base,
set_subtype_base,
istype-less_than,
mul_bounds_1a,
decidable__equal_int,
subtype_base_sq,
less_than_wf,
exp-zero,
zero-mul,
exp-positive,
decidable__lt,
not-lt-2,
mul_nat_plus,
not-equal-2,
minus-zero,
subtract_nat_wf,
less_than_functionality,
le_weakening,
subtract-add-cancel,
equal_wf,
istype-universe,
exp_wf_nat_plus,
mul_bounds_1b,
exp-ratio_wf,
mul-one
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
sqequalHypSubstitution,
hypothesis,
Error :inhabitedIsType,
hypothesisEquality,
Error :universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
addEquality,
setElimination,
rename,
natural_numberEquality,
Error :dependent_set_memberEquality_alt,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
Error :isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
multiplyEquality,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
closedConclusion,
minusEquality,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
Error :equalityIsType1,
intEquality,
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:
2019_06_20-PM-02_30_44
Last ObjectModification:
2019_01_27-PM-06_37_12
Theory : num_thy_1
Home
Index