Nuprl Lemma : p-int-eventually-constant
∀p:{2...}. ∀k:ℕ. ∃n:ℕ+. ∀m:{n...}. ((k(p) m) = k ∈ ℤ)
Proof
Definitions occuring in Statement :
p-int: k(p),
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
apply: f a,
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,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
exists: ∃x:A. B[x],
nat_plus: ℕ+,
int_upper: {i...},
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
false: False,
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
le: A ≤ B,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
squash: ↓T,
rev_uimplies: rev_uimplies(P;Q),
p-int: k(p),
p-reduce: i mod(p^n),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
sq_stable: SqStable(P)
Lemmas referenced :
istype-nat,
istype-int_upper,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_properties,
int_upper_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
exp-positive,
intformand_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
exp_wf2,
nat_plus_properties,
decidable__le,
istype-le,
log_wf,
add_nat_plus,
subtype_rel_sets_simple,
le_wf,
less_than_wf,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
log-property,
squash_wf,
true_wf,
exp_add,
subtype_rel_self,
iff_weakening_equal,
exp1,
upper_subtype_nat,
less_than_functionality,
le_weakening,
multiply_functionality_wrt_le,
mul_preserves_lt,
itermMultiply_wf,
int_term_value_mul_lemma,
set_subtype_base,
modulus_base,
exp_wf_nat_plus,
exp-nondecreasing,
nat_plus_subtype_nat,
sq_stable__le,
le_weakening2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
dependent_functionElimination,
setElimination,
rename,
hypothesisEquality,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
equalityTransitivity,
equalitySymmetry,
int_eqEquality,
independent_pairFormation,
addEquality,
applyEquality,
inhabitedIsType,
productElimination,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
equalityIstype,
imageElimination,
imageMemberEquality,
universeEquality,
multiplyEquality,
functionIsType,
sqequalBase,
productIsType
Latex:
\mforall{}p:\{2...\}. \mforall{}k:\mBbbN{}. \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\{n...\}. ((k(p) m) = k)
Date html generated:
2019_10_15-AM-10_34_25
Last ObjectModification:
2019_02_11-PM-02_03_22
Theory : rings_1
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