Nuprl Lemma : odd-plus-odd
∀[n,m:ℤ]. ↑isEven(n + m) supposing (↑isOdd(n)) ∧ (↑isOdd(m))
Proof
Definitions occuring in Statement :
isEven: isEven(n),
isOdd: isOdd(n),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
add: n + m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
same-parity: same-parity(n;m),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q
Lemmas referenced :
isEven_wf,
eqtt_to_assert,
even-iff-not-odd,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
assert_witness,
same-parity_wf,
istype-assert,
isOdd_wf,
istype-int,
iff_weakening_uiff,
assert_wf,
isEven-add
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
sqequalRule,
independent_functionElimination,
voidElimination,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
because_Cache,
productIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
addEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. \muparrow{}isEven(n + m) supposing (\muparrow{}isOdd(n)) \mwedge{} (\muparrow{}isOdd(m))
Date html generated:
2020_05_19-PM-10_01_14
Last ObjectModification:
2019_11_13-AM-10_38_17
Theory : num_thy_1
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