Nuprl Lemma : mod2-is-one
∀x:ℤ. ((x mod 2) = 1 ∈ ℤ ⇐⇒ ∃n:ℤ. (x = ((2 * n) + 1) ∈ ℤ))
Proof
Definitions occuring in Statement :
modulus: a mod n,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
decidable: Dec(P),
exists: ∃x:A. B[x],
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
less_than': less_than'(a;b),
le: A ≤ B,
uiff: uiff(P;Q),
top: Top,
subtract: n - m,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
prop: ℙ,
false: False,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
true: True,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
iff_wf,
mod2-is-zero,
exists_wf,
minus-minus,
mul-associates,
minus-add,
not-equal-2,
false_wf,
decidable__int_equal,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
eqff_to_assert,
assert_of_eq_int,
eqtt_to_assert,
bool_subtype_base,
bool_wf,
bool_cases,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
minus-one-mul-top,
minus-one-mul,
condition-implies-le,
le_antisymmetry_iff,
not_wf,
bnot_wf,
assert_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
nequal_wf,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
modulus_wf,
eq_int_wf,
subtract_wf,
mod2-add1
Rules used in proof :
promote_hyp,
multiplyEquality,
dependent_pairFormation,
impliesFunctionality,
unionElimination,
productElimination,
addEquality,
minusEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
closedConclusion,
baseApply,
independent_pairFormation,
sqequalRule,
because_Cache,
applyEquality,
baseClosed,
voidElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
cumulativity,
instantiate,
addLevel,
dependent_set_memberEquality,
hypothesis,
natural_numberEquality,
hypothesisEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
intEquality,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x:\mBbbZ{}. ((x mod 2) = 1 \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbZ{}. (x = ((2 * n) + 1)))
Date html generated:
2018_07_25-PM-01_27_57
Last ObjectModification:
2018_06_27-PM-05_54_42
Theory : arithmetic
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