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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