Nuprl Lemma : not-same-parity-implies
∀[n,m:ℤ]. ((¬↑same-parity(n;m)) ⇒ {(↑same-parity(n;m - 1)) ∧ (↑same-parity(n;m + 1))})
Proof
Definitions occuring in Statement :
same-parity: same-parity(n;m),
assert: ↑b,
uall: ∀[x:A]. B[x],
guard: {T},
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
guard: {T},
and: P ∧ Q,
same-parity: same-parity(n;m),
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
sq_type: SQType(T),
bfalse: ff,
not: ¬A,
false: False,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
isEven_wf,
bool_wf,
eqtt_to_assert,
bool_cases,
subtype_base_sq,
bool_subtype_base,
eqff_to_assert,
assert_of_bnot,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
equal_wf,
same-parity_wf,
assert_witness,
subtract_wf,
odd-iff-not-even,
odd-implies,
even-iff-not-odd,
even-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
lambdaEquality,
independent_pairEquality,
natural_numberEquality,
addEquality,
intEquality,
isect_memberEquality
Latex:
\mforall{}[n,m:\mBbbZ{}]. ((\mneg{}\muparrow{}same-parity(n;m)) {}\mRightarrow{} \{(\muparrow{}same-parity(n;m - 1)) \mwedge{} (\muparrow{}same-parity(n;m + 1))\})
Date html generated:
2017_04_17-AM-09_43_33
Last ObjectModification:
2017_02_27-PM-05_38_19
Theory : num_thy_1
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