Nuprl Lemma : mod2-add1
∀n:ℤ. (((n + 1) mod 2) = if (n mod 2 =z 0) then 1 else 0 fi  ∈ ℤ)
Proof
Definitions occuring in Statement : 
modulus: a mod n
, 
ifthenelse: if b then t else f fi 
, 
eq_int: (i =z j)
, 
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]
, 
member: t ∈ T
, 
and: P ∧ Q
, 
uall: ∀[x:A]. B[x]
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
true: True
, 
prop: ℙ
, 
exists: ∃x:A. B[x]
, 
int_nzero: ℤ-o
, 
nequal: a ≠ b ∈ T 
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
uimplies: b supposing a
, 
sq_type: SQType(T)
, 
guard: {T}
, 
false: False
, 
subtype_rel: A ⊆r B
, 
top: Top
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
subtract: n - m
, 
or: P ∨ Q
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
bnot: ¬bb
, 
assert: ↑b
, 
decidable: Dec(P)
, 
le: A ≤ B
Lemmas referenced : 
div_floor_mod_sum, 
less_than_wf, 
subtract_wf, 
div_floor_wf, 
subtype_base_sq, 
int_subtype_base, 
equal-wf-base, 
true_wf, 
nequal_wf, 
modulus_wf, 
equal_wf, 
two-mul, 
mul-commutes, 
squash_wf, 
add_functionality_wrt_eq, 
subtype_rel_self, 
iff_weakening_equal, 
add-associates, 
minus-one-mul, 
add-swap, 
minus-one-mul-top, 
add-mul-special, 
zero-mul, 
add-zero, 
add-commutes, 
mul-distributes-right, 
minus-add, 
rem-exact, 
mod2-cases, 
eq_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
iff_transitivity, 
assert_wf, 
bnot_wf, 
not_wf, 
iff_weakening_uiff, 
assert_of_bnot, 
decidable__int_equal, 
false_wf, 
not-equal-2, 
le_antisymmetry_iff, 
condition-implies-le, 
minus-zero, 
zero-add, 
add_functionality_wrt_le, 
le-add-cancel, 
minus-minus, 
mul-associates, 
le-add-cancel2
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
hypothesis, 
intEquality, 
independent_pairFormation, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
addEquality, 
hypothesisEquality, 
natural_numberEquality, 
dependent_set_memberEquality, 
sqequalRule, 
imageMemberEquality, 
baseClosed, 
because_Cache, 
dependent_pairFormation, 
addLevel, 
instantiate, 
cumulativity, 
independent_isectElimination, 
dependent_functionElimination, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
voidElimination, 
baseApply, 
closedConclusion, 
applyEquality, 
lambdaEquality, 
isect_memberEquality, 
voidEquality, 
multiplyEquality, 
equalityUniverse, 
levelHypothesis, 
imageElimination, 
universeEquality, 
productElimination, 
minusEquality, 
promote_hyp, 
unionElimination, 
equalityElimination, 
impliesFunctionality
Latex:
\mforall{}n:\mBbbZ{}.  (((n  +  1)  mod  2)  =  if  (n  mod  2  =\msubz{}  0)  then  1  else  0  fi  )
Date html generated:
2018_07_25-PM-01_27_40
Last ObjectModification:
2018_06_07-AM-10_00_47
Theory : arithmetic
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