Nuprl Lemma : coprime-exp1
∀a,b:ℤ. (CoPrime(a,b) ⇒ (∀n:ℕ. CoPrime(a,b^n)))
Proof
Definitions occuring in Statement :
coprime: CoPrime(a,b),
exp: i^n,
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
top: Top,
prop: ℙ,
uall: ∀[x:A]. B[x],
nat: ℕ,
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,
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
exp: i^n,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
exp0_lemma,
coprime_wf,
exp_wf2,
decidable__le,
subtract_wf,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
coprime_bezout_id,
decidable__equal_int,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
equal-wf-base,
int_subtype_base,
exists_wf,
primrec-unroll,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
coprime_prod
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
rename,
setElimination,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality,
because_Cache,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
productElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}a,b:\mBbbZ{}. (CoPrime(a,b) {}\mRightarrow{} (\mforall{}n:\mBbbN{}. CoPrime(a,b\^{}n)))
Date html generated:
2018_05_21-PM-00_55_51
Last ObjectModification:
2018_05_19-AM-06_34_03
Theory : num_thy_1
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