Nuprl Lemma : absval_lbound
∀i:ℤ. ∀n:ℕ. (|i| > n ⇐⇒ i < -n ∨ (i > n))
Proof
Definitions occuring in Statement :
absval: |i|,
nat: ℕ,
less_than: a < b,
gt: i > j,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
minus: -n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
iff: P ⇐⇒ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
rev_implies: P ⇐ Q,
gt: i > j,
guard: {T},
subtype_rel: A ⊆r B,
le: A ≤ B,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
ge: i ≥ j ,
subtract: n - m,
sq_stable: SqStable(P)
Lemmas referenced :
absval_unfold2,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
decidable__lt,
gt_wf,
false_wf,
not-lt-2,
minus-le,
not-gt-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-commutes,
le-add-cancel,
or_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
nat_wf,
add_functionality_wrt_lt,
le_reflexive,
minus-one-mul,
add-mul-special,
zero-mul,
nat_properties,
condition-implies-le,
minus-add,
add-swap,
minus-one-mul-top,
add-zero,
minus-zero,
zero-add,
two-mul,
subtract_wf,
sq_stable_from_decidable,
le_wf,
decidable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
isect_memberFormation,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
minusEquality,
setElimination,
rename,
inlFormation,
inrFormation,
addEquality,
applyEquality,
addLevel,
orFunctionality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
intEquality,
multiplyEquality,
lambdaEquality
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}n:\mBbbN{}. (|i| > n \mLeftarrow{}{}\mRightarrow{} i < -n \mvee{} (i > n))
Date html generated:
2017_04_14-AM-07_17_12
Last ObjectModification:
2017_02_27-PM-02_52_09
Theory : arithmetic
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