Nuprl Lemma : p-adic-property
∀p:ℕ+. ∀a:p-adics(p). ∀n:ℕ+. ∀m:{n...}. ((a m) ≡ (a n) mod p^n)
Proof
Definitions occuring in Statement :
p-adics: p-adics(p),
eqmod: a ≡ b mod m,
exp: i^n,
int_upper: {i...},
nat_plus: ℕ+,
all: ∀x:A. B[x],
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
p-adics: p-adics(p),
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,
int_seg: {i..j-},
nat: ℕ,
guard: {T},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
so_apply: x[s],
ge: i ≥ j ,
int_upper: {i...},
subtract: n - m,
sq_type: SQType(T),
sq_stable: SqStable(P),
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
p-reduce: i mod(p^n)
Lemmas referenced :
eqmod_wf,
exp_wf2,
nat_plus_subtype_nat,
subtract_wf,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
itermSubtract_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_formula_prop_wf,
less_than_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
le_wf,
set_wf,
primrec-wf2,
nat_properties,
nat_wf,
int_upper_properties,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
subtype_base_sq,
int_subtype_base,
int_upper_wf,
nat_plus_wf,
p-adics_wf,
eqmod_weakening,
sq_stable__eqmod,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
equal_wf,
modulus-equal-iff-eqmod,
exp_wf_nat_plus,
p-reduce-eqmod-exp,
eqmod_inversion,
eqmod_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
rename,
setElimination,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
because_Cache,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
productElimination,
hyp_replacement,
instantiate,
functionExtensionality,
cumulativity,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}p:\mBbbN{}\msupplus{}. \mforall{}a:p-adics(p). \mforall{}n:\mBbbN{}\msupplus{}. \mforall{}m:\{n...\}. ((a m) \mequiv{} (a n) mod p\^{}n)
Date html generated:
2018_05_21-PM-03_19_37
Last ObjectModification:
2018_05_19-AM-08_12_17
Theory : rings_1
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