Nuprl Lemma : better-gcd-gcd
∀[y,x:ℤ]. (better-gcd(x;y) ~ gcd(x;y))
Proof
Definitions occuring in Statement :
better-gcd: better-gcd(a;b),
gcd: gcd(a;b),
uall: ∀[x:A]. B[x],
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
nat: ℕ,
false: False,
ge: i ≥ j ,
prop: ℙ,
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
better-gcd: better-gcd(a;b),
int_nzero: ℤ-o,
squash: ↓T
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
absval_wf,
decidable__le,
subtract_wf,
istype-false,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
istype-void,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
nat_wf,
absval-non-neg,
eq_int_wf,
equal-wf-base,
bool_wf,
assert_wf,
bnot_wf,
not_wf,
value-type-has-value,
int-value-type,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
rem_bounds_z,
nequal_wf,
equal_wf,
squash_wf,
true_wf,
decidable__lt,
not-lt-2,
gcd_wf,
subtype_rel_self,
iff_weakening_equal,
zero-mul,
add-mul-special,
false_wf,
le_wf,
not-le-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomSqEquality,
Error :inhabitedIsType,
hypothesisEquality,
sqequalRule,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :lambdaFormation_alt,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
voidElimination,
Error :lambdaEquality_alt,
axiomEquality,
applyEquality,
because_Cache,
unionElimination,
independent_pairFormation,
productElimination,
addEquality,
minusEquality,
baseApply,
closedConclusion,
baseClosed,
Error :equalityIsType3,
callbyvalueReduce,
remainderEquality,
equalityElimination,
Error :equalityIsType1,
Error :dependent_set_memberEquality_alt,
imageElimination,
universeEquality,
imageMemberEquality,
multiplyEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
lambdaFormation,
dependent_set_memberEquality
Latex:
\mforall{}[y,x:\mBbbZ{}]. (better-gcd(x;y) \msim{} gcd(x;y))
Date html generated:
2019_06_20-AM-11_25_31
Last ObjectModification:
2018_09_28-AM-11_57_29
Theory : arithmetic
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