Nuprl Lemma : greatest-p-zero-property
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ.
((greatest-p-zero(n;a) ≤ n)
∧ (∀i:ℕ+n + 1. (((i ≤ greatest-p-zero(n;a)) ⇒ ((a i) = 0 ∈ ℤ)) ∧ (greatest-p-zero(n;a) < i ⇒ (¬((a i) = 0 ∈ ℤ))))))
Proof
Definitions occuring in Statement :
greatest-p-zero: greatest-p-zero(n;a),
p-adics: p-adics(p),
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
le: A ≤ B,
p-adics: p-adics(p),
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
greatest-p-zero: greatest-p-zero(n;a),
primrec: primrec(n;b;c),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
less_than_wf,
less_than'_wf,
greatest-p-zero_wf,
nat_plus_properties,
le_wf,
int_seg_properties,
itermAdd_wf,
int_term_value_add_lemma,
equal-wf-T-base,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
exp_wf2,
int_seg_subtype_nat,
false_wf,
sq_stable__le,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
zero-add,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__lt,
not-lt-2,
le-add-cancel,
nat_wf,
p-adics_wf,
nat_plus_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
subtract-add-cancel,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
eq_int_wf,
assert_of_eq_int,
add-zero,
neg_assert_of_eq_int,
not-equal-2,
lelt_wf,
primrec-unroll,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
p-adic-non-decreasing,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
productElimination,
independent_pairEquality,
because_Cache,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
dependent_set_memberEquality,
baseClosed,
imageMemberEquality,
imageElimination,
minusEquality,
applyLambdaEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}.
((greatest-p-zero(n;a) \mleq{} n)
\mwedge{} (\mforall{}i:\mBbbN{}\msupplus{}n + 1
(((i \mleq{} greatest-p-zero(n;a)) {}\mRightarrow{} ((a i) = 0))
\mwedge{} (greatest-p-zero(n;a) < i {}\mRightarrow{} (\mneg{}((a i) = 0))))))
Date html generated:
2018_05_21-PM-03_22_12
Last ObjectModification:
2018_05_19-AM-08_21_03
Theory : rings_1
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