Nuprl Lemma : mod2-cases
∀x:ℤ. (((x mod 2) = 0 ∈ ℤ) ∨ ((x mod 2) = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
modulus: a mod n,
all: ∀x:A. B[x],
or: P ∨ Q,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtract: n - m,
top: Top,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
le: A ≤ B,
or: P ∨ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
false: False,
guard: {T},
sq_type: SQType(T),
uimplies: b supposing a,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
prop: ℙ,
and: P ∧ Q,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
nat_plus: ℕ+,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
less-iff-le,
or_wf,
le_antisymmetry_iff,
add-swap,
le-add-cancel2,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
false_wf,
equal_wf,
nequal_wf,
true_wf,
equal-wf-base,
int_subtype_base,
subtype_base_sq,
modulus_wf,
decidable__int_equal,
less_than_wf,
mod_bounds
Rules used in proof :
orFunctionality,
minusEquality,
voidEquality,
isect_memberEquality,
lambdaEquality,
addEquality,
inrFormation,
inlFormation,
unionElimination,
because_Cache,
applyEquality,
voidElimination,
independent_functionElimination,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
intEquality,
cumulativity,
instantiate,
addLevel,
productElimination,
dependent_functionElimination,
hypothesis,
baseClosed,
imageMemberEquality,
independent_pairFormation,
sqequalRule,
natural_numberEquality,
dependent_set_memberEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}x:\mBbbZ{}. (((x mod 2) = 0) \mvee{} ((x mod 2) = 1))
Date html generated:
2020_05_19-PM-09_35_40
Last ObjectModification:
2019_12_16-PM-02_26_13
Theory : arithmetic
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