Nuprl Lemma : isOdd-isEven-add
∀[n,m:ℤ]. (uiff(↑isOdd(n + m);¬↑same-parity(n;m)) ∧ uiff(↑isEven(n + m);↑same-parity(n;m)))
Proof
Definitions occuring in Statement :
same-parity: same-parity(n;m),
isEven: isEven(n),
isOdd: isOdd(n),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
cand: A c∧ B,
all: ∀x:A. B[x],
guard: {T},
top: Top,
subtract: n - m,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
same-parity: same-parity(n;m),
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
isOdd: isOdd(n),
eq_int: (i =z j),
modulus: a mod n,
isEven: isEven(n),
absval: |i|
Lemmas referenced :
assert_wf,
same-parity_wf,
isOdd_wf,
assert_witness,
not_wf,
isEven_wf,
odd-implies,
general_add_assoc,
same-parity-implies,
add-subtract-cancel,
not-same-parity-implies,
even-implies,
add-associates,
add-swap,
add-commutes,
zero-add,
and_wf,
equal_wf,
assert_elim,
subtype_base_sq,
bool_wf,
bool_subtype_base,
add-zero,
eqtt_to_assert,
even-iff-not-odd,
eqff_to_assert,
bool_cases_sqequal,
assert-bnot,
bool_cases,
assert_of_bnot,
odd-iff-not-even,
subtract_wf,
subtract-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
intWeakElimination,
sqequalHypSubstitution,
thin,
rename,
hypothesis,
productElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
isectElimination,
hypothesisEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
extract_by_obid,
addEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
natural_numberEquality,
because_Cache,
intEquality,
independent_pairFormation,
lambdaFormation,
independent_isectElimination,
voidEquality,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
instantiate,
cumulativity,
hyp_replacement,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
callbyvalueReduce,
sqleReflexivity,
minusEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. (uiff(\muparrow{}isOdd(n + m);\mneg{}\muparrow{}same-parity(n;m)) \mwedge{} uiff(\muparrow{}isEven(n + m);\muparrow{}same-parity(n;m)))
Date html generated:
2017_04_17-AM-09_43_48
Last ObjectModification:
2017_02_27-PM-05_38_43
Theory : num_thy_1
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