Nuprl Lemma : p-adic-bounds
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+. ((0 ≤ ((a (n + 1)) - a n)) ∧ (((a (n + 1)) - a n) ≤ (p^(n + 1) - p^n)))
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
exp: i^n,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
p-adics: p-adics(p),
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
uimplies: b supposing a,
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
sq_stable: SqStable(P),
eqmod: a ≡ b mod m,
divides: b | a,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
squash: ↓T,
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
lelt: i ≤ j < k
Lemmas referenced :
sq_stable__le,
subtract_wf,
nat_plus_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
less_than_wf,
int_seg_wf,
exp_wf2,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
nat_plus_subtype_nat,
p-adics_wf,
mul_bounds_1a,
exp_wf4,
mul_preserves_le,
mul-non-neg1,
exp_wf_nat_plus,
multiply-is-int-iff,
int_seg_properties,
itermMultiply_wf,
intformeq_wf,
itermSubtract_wf,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
squash_wf,
true_wf,
exp_add,
subtype_rel_self,
iff_weakening_equal,
exp1,
left_mul_subtract_distrib,
mul-one,
equal_wf,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
introduction,
extract_by_obid,
isectElimination,
natural_numberEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
addEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
minusEquality,
lambdaEquality,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidEquality,
baseApply,
closedConclusion,
applyLambdaEquality,
universeEquality,
multiplyEquality
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}\msupplus{}.
((0 \mleq{} ((a (n + 1)) - a n)) \mwedge{} (((a (n + 1)) - a n) \mleq{} (p\^{}(n + 1) - p\^{}n)))
Date html generated:
2018_05_21-PM-03_19_43
Last ObjectModification:
2018_05_19-AM-08_10_59
Theory : rings_1
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