Nuprl Lemma : not-same-parity-implies-even-odd
∀n,m:ℤ. ((¬↑same-parity(n;m)) ⇒ (((↑isEven(n)) ∧ (↑isOdd(m))) ∨ ((↑isOdd(n)) ∧ (↑isEven(m)))))
Proof
Definitions occuring in Statement :
same-parity: same-parity(n;m),
isEven: isEven(n),
isOdd: isOdd(n),
assert: ↑b,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
same-parity: same-parity(n;m),
member: t ∈ T,
uall: ∀[x:A]. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
false: False,
cand: A c∧ B,
true: True,
not: ¬A
Lemmas referenced :
isEven_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
odd-iff-not-even,
even-iff-not-odd,
false_wf,
assert_wf,
isOdd_wf,
not_wf,
same-parity_wf,
bool_cases,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
inrFormation,
independent_pairFormation,
productEquality,
intEquality,
inlFormation,
natural_numberEquality
Latex:
\mforall{}n,m:\mBbbZ{}. ((\mneg{}\muparrow{}same-parity(n;m)) {}\mRightarrow{} (((\muparrow{}isEven(n)) \mwedge{} (\muparrow{}isOdd(m))) \mvee{} ((\muparrow{}isOdd(n)) \mwedge{} (\muparrow{}isEven(m)))))
Date html generated:
2017_04_17-AM-09_43_40
Last ObjectModification:
2017_02_27-PM-05_38_16
Theory : num_thy_1
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