Nuprl Lemma : even-iff-not-odd
∀[n:ℤ]. uiff(↑isEven(n);¬↑isOdd(n))
Proof
Definitions occuring in Statement :
isEven: isEven(n),
isOdd: isOdd(n),
assert: ↑b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
not: ¬A,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False,
all: ∀x:A. B[x],
guard: {T},
prop: ℙ,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
assert: ↑b,
btrue: tt,
true: True
Lemmas referenced :
even-implies,
assert_wf,
isOdd_wf,
isEven_wf,
odd-or-even,
not_assert_elim,
and_wf,
equal_wf,
bool_wf,
bor_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base,
assert_witness,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
thin,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
hypothesis,
productElimination,
voidElimination,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
independent_isectElimination,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyEquality,
setElimination,
rename,
setEquality,
instantiate,
cumulativity,
natural_numberEquality,
independent_pairEquality,
isect_memberEquality,
intEquality
Latex:
\mforall{}[n:\mBbbZ{}]. uiff(\muparrow{}isEven(n);\mneg{}\muparrow{}isOdd(n))
Date html generated:
2016_05_14-PM-04_24_04
Last ObjectModification:
2015_12_26-PM-08_19_25
Theory : num_thy_1
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