Nuprl Lemma : qrep-coprime
∀[r:ℚ]. (|gcd(fst(qrep(r));snd(qrep(r)))| = 1 ∈ ℤ)
Proof
Definitions occuring in Statement :
qrep: qrep(r),
rationals: ℚ,
gcd: gcd(a;b),
absval: |i|,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
bfalse: ff,
pi2: snd(t),
pi1: fst(t),
it: ⋅,
unit: Unit,
bool: 𝔹,
spreadn: spread3,
btrue: tt,
ifthenelse: if b then t else f fi ,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
top: Top,
implies: P ⇒ Q,
qmul: r * s,
qinv: 1/r,
qrep: qrep(r),
qdiv: (r/s),
prop: ℙ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
not: ¬A,
cand: A c∧ B,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
gcd_p: GCD(a;b;y),
coprime: CoPrime(a,b),
assoced: a ~ b,
absval: |i|,
rev_implies: P ⇐ Q,
true: True,
less_than': less_than'(a;b),
le: A ≤ B,
squash: ↓T,
iff: P ⇐⇒ Q,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
le_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
gcd_reduce_wf,
gcd_reduce_property,
evalall-sqequal,
product-valueall-type,
evalall-reduce,
int-valueall-type,
valueall-type-has-valueall,
nat_wf,
nat_plus_wf,
pi2_wf,
equal_wf,
pi1_wf_top,
qrep_wf,
gcd_wf,
absval_wf,
equal-wf-T-base,
int_subtype_base,
rationals_wf,
equal-wf-base,
not_wf,
qeq_wf2,
assert_wf,
int-subtype-rationals,
assert-qeq,
nat_plus_properties,
q-elim,
one_divs_any,
gcd_is_divisor_2,
gcd_is_divisor_1,
iff_weakening_equal,
false_wf,
absval_pos,
assoced_elim,
coprime_bezout_id1,
coprime_bezout_id2,
nat_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermMultiply_wf,
itermMinus_wf,
itermVar_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf
Rules used in proof :
cumulativity,
instantiate,
promote_hyp,
dependent_pairFormation,
equalityElimination,
unionElimination,
multiplyEquality,
closedConclusion,
baseApply,
productEquality,
isintReduceTrue,
callbyvalueReduce,
lambdaEquality,
independent_functionElimination,
equalityTransitivity,
voidEquality,
voidElimination,
isect_memberEquality,
independent_pairEquality,
intEquality,
lambdaFormation,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
baseClosed,
because_Cache,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
applyEquality,
impliesFunctionality,
addLevel,
rename,
setElimination,
hypothesis,
isectElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_pairFormation,
imageMemberEquality,
dependent_set_memberEquality,
levelHypothesis,
equalityUniverse,
imageElimination,
minusEquality,
dependent_pairFormation_alt,
approximateComputation,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
universeIsType,
equalityIstype,
inhabitedIsType,
sqequalBase,
productIsType
Latex:
\mforall{}[r:\mBbbQ{}]. (|gcd(fst(qrep(r));snd(qrep(r)))| = 1)
Date html generated:
2020_05_20-AM-09_13_18
Last ObjectModification:
2020_02_01-AM-11_26_41
Theory : rationals
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