Nuprl Lemma : nim-sum-rem2
∀[x,y:ℕ]. (nim-sum(x;y) rem 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,
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
remainder: n rem m,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
nim-sum: nim-sum(x;y),
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
has-value: (a)↓,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
label: ...$L... t,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
top: Top
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
nim_sum0_lemma,
div_rem_sum,
nequal_wf,
rem_bounds_1,
divide_wfa,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
false_wf,
divide_wf,
value-type-has-value,
int-value-type,
remainder_wfa,
nat_wf,
set-value-type,
le_wf,
nim-sum_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
has-value_wf_base,
is-exception_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-nat,
int_seg_subtype_special,
int_seg_cases,
remainder_wf,
squash_wf,
true_wf,
nat_plus_wf,
nim-sum-0,
iff_weakening_equal,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
add-zero,
rem-exact,
equal_wf,
and_wf,
rem_invariant,
less_than_wf,
add-commutes,
mul-commutes,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomSqEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
promote_hyp,
hypothesis_subsumption,
cumulativity,
intEquality,
equalityIstype,
baseClosed,
sqequalBase,
int_eqReduceFalseSq,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
callbyvalueReduce,
imageMemberEquality,
equalityElimination,
int_eqReduceTrueSq,
divergentSqle,
sqleReflexivity,
addEquality,
sqequalIntensionalEquality,
multiplyEquality,
remainderEquality,
hyp_replacement,
dependent_set_memberEquality,
lambdaFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
addLevel
Latex:
\mforall{}[x,y:\mBbbN{}]. (nim-sum(x;y) rem 2 \msim{} if x rem 2=y rem 2 then 0 else 1)
Date html generated:
2020_05_20-AM-08_21_18
Last ObjectModification:
2019_12_26-PM-04_52_09
Theory : general
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