Nuprl Lemma : gcd-exp
∀x,y:ℤ. ∀n:ℕ. (gcd(x^n;y^n) ~ gcd(x;y)^n)
Proof
Definitions occuring in Statement :
assoced: a ~ b,
exp: i^n,
gcd: gcd(a;b),
nat: ℕ,
all: ∀x:A. B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
gcd_p: GCD(a;b;y),
and: P ∧ Q,
cand: A c∧ B,
prop: ℙ,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
top: Top,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
true: True,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
squash: ↓T,
rev_implies: P ⇐ Q
Lemmas referenced :
gcd_unique,
exp_wf2,
gcd_wf,
gcd_sat_pred,
exp-divides,
gcd_is_divisor_1,
gcd_is_divisor_2,
divides_wf,
istype-nat,
istype-int,
gcd-property,
coprime_wf,
int_subtype_base,
coprime-exp,
coprime_bezout_id,
istype-void,
subtype_base_sq,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul-distributes,
mul-swap,
mul-commutes,
one-mul,
squash_wf,
true_wf,
add_functionality_wrt_eq,
exp-of-mul,
subtype_rel_self,
iff_weakening_equal,
divides-add,
divides-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
independent_functionElimination,
independent_pairFormation,
because_Cache,
productElimination,
sqequalRule,
Error :productIsType,
Error :universeIsType,
Error :inhabitedIsType,
Error :equalityIstype,
applyEquality,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
multiplyEquality,
Error :isect_memberEquality_alt,
voidElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
imageElimination,
imageMemberEquality,
universeEquality,
Error :inrFormation_alt
Latex:
\mforall{}x,y:\mBbbZ{}. \mforall{}n:\mBbbN{}. (gcd(x\^{}n;y\^{}n) \msim{} gcd(x;y)\^{}n)
Date html generated:
2019_06_20-PM-02_32_44
Last ObjectModification:
2018_11_28-PM-07_17_30
Theory : num_thy_1
Home
Index