Nuprl Lemma : exp-divides-exp2
∀x,y:ℤ. (x | y ⇐⇒ ∃n:ℕ+. (x^n | y^n))
Proof
Definitions occuring in Statement :
divides: b | a,
exp: i^n,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
exp: i^n,
top: Top,
divides: b | a,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
uiff: uiff(P;Q),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
so_lambda: λ2x.t[x],
so_apply: x[s],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T
Lemmas referenced :
divides_wf,
nat_plus_wf,
exp_wf2,
nat_plus_subtype_nat,
istype-int,
istype-less_than,
primrec1_lemma,
istype-void,
mul-commutes,
one-mul,
divides-iff-gcd,
gcd_is_divisor_2,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
nat_plus_properties,
multiply-is-int-iff,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
false_wf,
gcd_is_divisor_1,
divides_transitivity,
gcd_wf,
gcd-exp,
assoced_elim,
equal_wf,
squash_wf,
true_wf,
istype-universe,
exp-of-mul,
subtype_rel_self,
iff_weakening_equal,
set_subtype_base,
less_than_wf,
mul_cancel_in_eq,
exp_wf3,
nequal_wf,
exp-equal-one,
intformless_wf,
int_formula_prop_less_lemma,
minus-is-int-iff,
itermMinus_wf,
int_term_value_minus_lemma,
exp-equal-minusone
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
independent_pairFormation,
Error :universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
Error :productIsType,
applyEquality,
because_Cache,
Error :inhabitedIsType,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberEquality_alt,
closedConclusion,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
setElimination,
rename,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
approximateComputation,
Error :lambdaEquality_alt,
int_eqEquality,
minusEquality,
Error :equalityIstype,
sqequalBase,
imageElimination,
universeEquality,
Error :functionIsType
Latex:
\mforall{}x,y:\mBbbZ{}. (x | y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. (x\^{}n | y\^{}n))
Date html generated:
2019_06_20-PM-02_32_51
Last ObjectModification:
2018_11_28-PM-07_19_16
Theory : num_thy_1
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