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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