Nuprl Lemma : mod2-add2
∀n:ℤ. (((n + 2) mod 2) = (n mod 2) ∈ ℤ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
all: ∀x:A. B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
true: True,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
prop: ℙ,
or: P ∨ Q,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
bfalse: ff,
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
add-commutes,
add-associates,
add-swap,
modulus_wf,
equal-wf-base,
true_wf,
nequal_wf,
mod2-cases,
equal_wf,
squash_wf,
mod2-add1,
ifthenelse_wf,
bool_wf,
eq_int_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
sqequalRule,
hypothesisEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
addEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberEquality,
addLevel,
baseClosed,
unionElimination,
imageElimination,
universeEquality,
imageMemberEquality,
productElimination
Latex:
\mforall{}n:\mBbbZ{}. (((n + 2) mod 2) = (n mod 2))
Date html generated:
2018_07_25-PM-01_27_43
Last ObjectModification:
2018_06_07-PM-04_56_45
Theory : arithmetic
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