Nuprl Lemma : nim-sum-com
∀[x,y:ℕ]. (nim-sum(x;y) = nim-sum(y;x) ∈ ℤ)
Proof
Definitions occuring in Statement :
nim-sum: nim-sum(x;y),
nat: ℕ,
uall: ∀[x:A]. B[x],
int: ℤ,
equal: s = t ∈ 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],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
nim-sum: nim-sum(x;y),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
true: True,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
has-value: (a)↓
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
less_than_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
lelt_wf,
subtype_rel_self,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
div_rem_sum,
equal-wf-base,
true_wf,
nequal_wf,
rem_bounds_1,
itermAdd_wf,
int_term_value_add_lemma,
add-is-int-iff,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
false_wf,
divide_wf,
value-type-has-value,
int-value-type,
set-value-type,
nim-sum_wf,
has-value_wf_base,
is-exception_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
axiomEquality,
because_Cache,
productElimination,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
cumulativity,
equalityElimination,
int_eqReduceTrueSq,
promote_hyp,
int_eqReduceFalseSq,
callbyvalueReduce,
sqleReflexivity,
addLevel,
baseClosed,
imageMemberEquality,
addEquality,
divideEquality,
imageElimination,
pointwiseFunctionality,
baseApply,
closedConclusion,
remainderEquality,
divergentSqle
Latex:
\mforall{}[x,y:\mBbbN{}]. (nim-sum(x;y) = nim-sum(y;x))
Date html generated:
2018_05_21-PM-09_10_26
Last ObjectModification:
2018_05_19-PM-05_12_25
Theory : general
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