Nuprl Lemma : gcd_reduce_property
∀p,q:ℤ. let g,a,b = gcd_reduce(p;q) in (p = (a * g) ∈ ℤ) ∧ (q = (b * g) ∈ ℤ) ∧ CoPrime(a,b) ∧ ((p * b) = (a * q) ∈ ℤ)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
gcd_reduce: gcd_reduce(p;q),
coprime: CoPrime(a,b),
spreadn: spread3,
all: ∀x:A. B[x],
and: P ∧ Q,
multiply: n * m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
guard: {T},
true: True,
squash: ↓T,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
spreadn: spread3,
spreadn: spread4,
implies: P ⇒ Q,
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
gcd_reduce: gcd_reduce(p;q),
all: ∀x:A. B[x]
Lemmas referenced :
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
equal_wf,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__equal_int,
nat_properties,
coprime_bezout_id,
istype-int,
le_wf,
set_subtype_base,
int_subtype_base,
equal-wf-base,
nat_wf,
subtype_rel_self,
gcd-reduce-ext
Rules used in proof :
imageMemberEquality,
multiplyEquality,
universeEquality,
imageElimination,
Error :productIsType,
sqequalBase,
Error :universeIsType,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
rename,
setElimination,
Error :dependent_pairFormation_alt,
independent_functionElimination,
dependent_functionElimination,
equalitySymmetry,
equalityTransitivity,
Error :equalityIstype,
independent_pairFormation,
productElimination,
because_Cache,
independent_isectElimination,
Error :inhabitedIsType,
natural_numberEquality,
Error :lambdaEquality_alt,
baseClosed,
closedConclusion,
baseApply,
hypothesisEquality,
productEquality,
intEquality,
functionEquality,
isectElimination,
sqequalHypSubstitution,
introduction,
sqequalRule,
hypothesis,
extract_by_obid,
instantiate,
thin,
applyEquality,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}p,q:\mBbbZ{}.
let g,a,b = gcd\_reduce(p;q) in
(p = (a * g)) \mwedge{} (q = (b * g)) \mwedge{} CoPrime(a,b) \mwedge{} ((p * b) = (a * q))
Date html generated:
2019_06_20-PM-02_27_19
Last ObjectModification:
2019_06_19-PM-02_32_40
Theory : num_thy_1
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