Nuprl Lemma : nim-sum-rec
∀[x,y:ℕ].  (nim-sum(x;y) ~ (2 * nim-sum(x ÷ 2;y ÷ 2)) + if x rem 2=y rem 2 then 0 else 1)
Proof
Definitions occuring in Statement : 
nim-sum: nim-sum(x;y)
, 
nat: ℕ
, 
uall: ∀[x:A]. B[x]
, 
int_eq: if a=b then c else d
, 
remainder: n rem m
, 
divide: n ÷ m
, 
multiply: n * m
, 
add: n + m
, 
natural_number: $n
, 
sqequal: s ~ t
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
int_nzero: ℤ-o
, 
true: True
, 
nequal: a ≠ b ∈ T 
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
uimplies: b supposing a
, 
sq_type: SQType(T)
, 
all: ∀x:A. B[x]
, 
guard: {T}
, 
false: False
, 
prop: ℙ
, 
top: Top
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
and: P ∧ Q
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
Lemmas referenced : 
div_rem_sum, 
nim-sum_wf, 
subtype_base_sq, 
int_subtype_base, 
equal-wf-base, 
true_wf, 
nequal_wf, 
mul-commutes, 
nim-sum-div2, 
divide_wf, 
less_than_wf, 
nim-sum-rem2, 
nat_wf, 
set_subtype_base, 
le_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
lambdaEquality, 
setElimination, 
rename, 
because_Cache, 
sqequalRule, 
dependent_set_memberEquality, 
natural_numberEquality, 
addLevel, 
lambdaFormation, 
instantiate, 
cumulativity, 
intEquality, 
independent_isectElimination, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
voidElimination, 
baseClosed, 
addEquality, 
divideEquality, 
isect_memberEquality, 
voidEquality, 
multiplyEquality, 
independent_pairFormation, 
imageMemberEquality, 
int_eqEquality, 
remainderEquality, 
hyp_replacement, 
applyLambdaEquality, 
sqequalIntensionalEquality, 
baseApply, 
closedConclusion
Latex:
\mforall{}[x,y:\mBbbN{}].    (nim-sum(x;y)  \msim{}  (2  *  nim-sum(x  \mdiv{}  2;y  \mdiv{}  2))  +  if  x  rem  2=y  rem  2  then  0  else  1)
Date html generated:
2018_05_21-PM-09_11_29
Last ObjectModification:
2018_05_19-PM-05_13_47
Theory : general
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