Nuprl Lemma : exp-ratio_wf
∀[a:ℕ]. ∀[b:{a + 1...}]. ∀[k:ℕ].
∀c:{n:ℕ| k * a^n < b^n} . ∀n:ℕ. ((n ≤ c) ⇒ (exp-ratio(a;b;n;k * a^n;b^n) ∈ {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,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
int_upper: {i...},
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
exp-ratio: exp-ratio(a;b;n;p;q),
decidable: Dec(P),
or: P ∨ Q,
sq_stable: SqStable(P),
squash: ↓T,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
le: A ≤ B,
subtract: n - m,
has-value: (a)↓,
sq_type: SQType(T),
nat_plus: ℕ+
Lemmas referenced :
le_wf,
nat_wf,
set_wf,
less_than_wf,
exp_wf2,
int_upper_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_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,
ge_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
lt_int_wf,
sq_stable__less_than,
int_upper_properties,
bool_wf,
equal-wf-T-base,
assert_wf,
le_int_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
minus-zero,
add-zero,
value-type-has-value,
int-value-type,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
mul-swap,
exp_step,
decidable__lt,
add-subtract-cancel,
Error :trivial-int-eq1
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
multiplyEquality,
because_Cache,
setEquality,
addEquality,
natural_numberEquality,
isect_memberFormation,
lambdaFormation,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
intWeakElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
unionElimination,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
imageElimination,
equalityElimination,
productElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
callbyvalueReduce,
instantiate,
cumulativity
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[b:\{a + 1...\}]. \mforall{}[k:\mBbbN{}].
\mforall{}c:\{n:\mBbbN{}| k * a\^{}n < b\^{}n\} . \mforall{}n:\mBbbN{}. ((n \mleq{} c) {}\mRightarrow{} (exp-ratio(a;b;n;k * a\^{}n;b\^{}n) \mmember{} \{n:\mBbbN{}| k * a\^{}n < b\^{}n\} \000C))
Date html generated:
2018_05_21-PM-01_03_11
Last ObjectModification:
2018_01_28-PM-02_12_52
Theory : num_thy_1
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