Nuprl Lemma : p-minus-int-eventually
∀p:{2...}. ∀k:ℕ+. ∃n:ℕ+. ∀m:{n...}. ((-k(p) m) = (p^m - k) ∈ ℤ)
Proof
Definitions occuring in Statement :
p-int: k(p),
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
subtract: n - m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
top: Top,
less_than': less_than'(a;b),
true: True,
nat_plus: ℕ+,
guard: {T},
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
nat: ℕ,
less_than: a < b,
squash: ↓T,
p-int: k(p),
p-reduce: i mod(p^n),
p-adics: p-adics(p),
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k
Lemmas referenced :
nat_plus_wf,
int_upper_wf,
log-property,
subtype_rel_sets,
le_wf,
less_than_wf,
decidable__lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
nat_plus_subtype_nat,
nat_plus_properties,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
log_wf,
nat_wf,
add_nat_plus,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
equal_wf,
exp_wf2,
all_wf,
p-int_wf,
zero-add,
le-add-cancel,
p-adics_wf,
less_than_transitivity1,
int_seg_wf,
upper_subtype_nat,
sq_stable__le,
le_weakening2,
subtract_wf,
exp-nondecreasing,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
mod_bounds,
less_than_transitivity2,
modulus_base_neg,
itermMinus_wf,
int_term_value_minus_lemma,
lelt_wf,
decidable__equal_int,
itermSubtract_wf,
int_term_value_subtract_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
isect_memberEquality,
voidEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
dependent_set_memberEquality,
addEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
minusEquality,
imageElimination
Latex:
\mforall{}p:\{2...\}. \mforall{}k:\mBbbN{}\msupplus{}. \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\{n...\}. ((-k(p) m) = (p\^{}m - k))
Date html generated:
2018_05_21-PM-03_19_12
Last ObjectModification:
2018_05_19-AM-08_10_18
Theory : rings_1
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