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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