Nuprl Lemma : gcd_ex_n
∀b:ℕ. ∀a:ℤ. (∃y:{ℤ| GCD(a;b;y)})
Proof
Definitions occuring in Statement :
gcd_p: GCD(a;b;y),
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_exists: ∃x:{A| B[x]},
less_than: a < b,
ge: i ≥ j ,
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
uiff: uiff(P;Q),
subtract: n - m
Lemmas referenced :
gcd_p_shift,
add_com,
gcd_p_sym,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
quot_rem_exists,
gcd_p_zero,
iff_weakening_equal,
true_wf,
squash_wf,
int_term_value_add_lemma,
itermAdd_wf,
nat_properties,
nat_wf,
primrec-wf2,
less_than_wf,
set_wf,
decidable__lt,
gcd_p_wf,
sq_exists_wf,
guard_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,
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
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
introduction,
addEquality,
dependent_set_memberFormation,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
minusEquality,
multiplyEquality
Latex:
\mforall{}b:\mBbbN{}. \mforall{}a:\mBbbZ{}. (\mexists{}y:\{\mBbbZ{}| GCD(a;b;y)\})
Date html generated:
2016_05_14-PM-04_19_08
Last ObjectModification:
2016_01_14-PM-11_41_52
Theory : num_thy_1
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