Nuprl Lemma : gcd_wf
∀a,b:ℤ. (gcd(a;b) ∈ ℤ)
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
all: ∀x:A. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o
Lemmas referenced :
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
absval_wf,
nat_wf,
absval-non-neg,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
iff_weakening_uiff,
assert_of_bnot,
rem_bounds_z,
nequal_wf,
decidable__lt,
not-lt-2,
not-le-2,
le_wf,
add-mul-special,
zero-mul
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
intEquality,
productElimination,
because_Cache,
unionElimination,
independent_pairFormation,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
baseClosed,
impliesFunctionality,
remainderEquality,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}a,b:\mBbbZ{}. (gcd(a;b) \mmember{} \mBbbZ{})
Date html generated:
2017_04_14-AM-07_18_48
Last ObjectModification:
2017_02_27-PM-02_52_44
Theory : arithmetic
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