Nuprl Lemma : gcd-non-neg
∀[y,x:ℕ]. (0 ≤ gcd(x;y))
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat: ℕ,
prop: ℙ,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
gcd: gcd(a;b),
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
btrue: tt,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
less_than': less_than'(a;b),
nequal: a ≠ b ∈ T ,
int_upper: {i...},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T
Lemmas referenced :
decidable__equal_int,
less_than'_wf,
gcd_wf,
nat_wf,
subtype_base_sq,
int_subtype_base,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_upper_subtype_nat,
false_wf,
le_wf,
nequal-le-implies,
zero-add,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
zero-rem,
subtype_rel_sets,
nequal_wf,
int_upper_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
gcd-positive,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
because_Cache,
natural_numberEquality,
hypothesis,
unionElimination,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
hypothesisEquality,
isectElimination,
setElimination,
rename,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
voidElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
voidEquality,
independent_pairFormation,
computeAll,
hypothesis_subsumption,
dependent_set_memberEquality,
lambdaFormation,
equalityElimination,
promote_hyp,
applyEquality,
setEquality,
applyLambdaEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[y,x:\mBbbN{}]. (0 \mleq{} gcd(x;y))
Date html generated:
2017_04_17-AM-09_45_50
Last ObjectModification:
2017_02_27-PM-05_40_34
Theory : num_thy_1
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