Nuprl Lemma : ml-gcd-sq
∀[y,x:ℤ]. (ml-gcd(x;y) ~ better-gcd(x;y))
Proof
Definitions occuring in Statement :
ml-gcd: ml-gcd(a;b),
better-gcd: better-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,
all: ∀x:A. B[x],
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
false: False,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
less_than': less_than'(a;b),
not: ¬A,
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
true: True,
sq_type: SQType(T),
ml-gcd: ml-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],
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
better-gcd: better-gcd(a;b),
has-value: (a)↓,
int_nzero: ℤ-o
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
absval-non-neg,
less_than_transitivity1,
less_than_irreflexivity,
absval_wf,
less_than_wf,
istype-int,
subtract_wf,
better-gcd_wf,
primrec-wf2,
all_wf,
equal-wf-base,
nat_wf,
add_nat_wf,
istype-void,
le_wf,
sq_stable__le,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
add-zero,
add-associates,
add-commutes,
le-add-cancel,
eq_int_wf,
bool_wf,
assert_wf,
ml_apply-sq,
int-valueall-type,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bnot_wf,
not_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
value-type-has-value,
int-value-type,
rem_bounds_z,
nequal_wf,
not-equal-2,
minus-minus,
less-iff-le,
add_functionality_wrt_le
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,
Error :lambdaFormation_alt,
hypothesisEquality,
productElimination,
natural_numberEquality,
applyEquality,
because_Cache,
sqequalRule,
independent_functionElimination,
voidElimination,
Error :inhabitedIsType,
Error :universeIsType,
rename,
setElimination,
Error :functionIsType,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
Error :setIsType,
Error :lambdaEquality_alt,
functionEquality,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
addEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
imageMemberEquality,
imageElimination,
Error :equalityIsType1,
unionElimination,
Error :isect_memberEquality_alt,
minusEquality,
multiplyEquality,
axiomSqEquality,
equalityElimination,
Error :dependent_pairFormation_alt,
promote_hyp,
callbyvalueReduce,
remainderEquality
Latex:
\mforall{}[y,x:\mBbbZ{}]. (ml-gcd(x;y) \msim{} better-gcd(x;y))
Date html generated:
2019_06_20-PM-00_44_29
Last ObjectModification:
2018_10_03-PM-06_59_02
Theory : ML
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