Nuprl Lemma : bezout_ident_n
∀b:ℕ. ∀a:ℤ. ∃u,v:ℤ. GCD(a;b;(u * a) + (v * b))
Proof
Definitions occuring in Statement :
gcd_p: GCD(a;b;y),
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
int: ℤ
Definitions unfolded in proof :
ge: i ≥ j ,
less_than: a < b,
nat: ℕ,
so_apply: x[s],
so_lambda: λ2x.t[x],
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
prop: ℙ,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
guard: {T},
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtract: n - m,
true: True,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
squash: ↓T
Lemmas referenced :
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
nat_wf,
primrec-wf2,
less_than_wf,
decidable__lt,
gcd_p_wf,
exists_wf,
all_wf,
le_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
intformeq_wf,
itermSubtract_wf,
intformnot_wf,
decidable__le,
lelt_wf,
set_wf,
false_wf,
int_seg_subtype,
subtract_wf,
decidable__equal_int,
int_seg_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
satisfiable-full-omega-tt,
int_seg_properties,
add-zero,
one-mul,
gcd_p_zero,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
quot_rem_exists,
istype-void,
gcd_p_sym,
gcd_p_shift,
mul-commutes,
mul-distributes,
mul-distributes-right,
minus-one-mul,
mul-swap,
mul-associates,
add-swap,
add-mul-special,
zero-mul,
squash_wf,
true_wf,
istype-int,
add_functionality_wrt_eq,
mul_com,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
multiplyEquality,
addEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
levelHypothesis,
equalitySymmetry,
equalityTransitivity,
applyEquality,
addLevel,
unionElimination,
computeAll,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
productElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
because_Cache,
natural_numberEquality,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
thin,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
Error :dependent_pairFormation_alt,
hyp_replacement,
applyLambdaEquality,
Error :universeIsType,
Error :productIsType,
Error :inhabitedIsType,
minusEquality,
independent_functionElimination,
Error :isect_memberEquality_alt,
Error :lambdaEquality_alt,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality
Latex:
\mforall{}b:\mBbbN{}. \mforall{}a:\mBbbZ{}. \mexists{}u,v:\mBbbZ{}. GCD(a;b;(u * a) + (v * b))
Date html generated:
2019_06_20-PM-02_22_20
Last ObjectModification:
2019_01_11-AM-09_03_44
Theory : num_thy_1
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