Nuprl Lemma : absval-positive
∀[x:ℤ]. uiff(0 < |x|;x ≠ 0)
Proof
Definitions occuring in Statement :
absval: |i|,
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
nequal: a ≠ b ∈ T ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
nequal: a ≠ b ∈ T ,
guard: {T},
subtype_rel: A ⊆r B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
subtract: n - m,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
nat: ℕ,
gt: i > j
Lemmas referenced :
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
equal-wf-base,
int_subtype_base,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-commutes,
zero-add,
le-add-cancel,
condition-implies-le,
minus-add,
minus-zero,
add-zero,
nequal_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
decidable__le,
le_wf,
not-le-2,
or_wf,
absval_wf,
nat_wf,
member-less_than,
not-gt-2,
subtract_wf,
le_reflexive,
minus-one-mul-top,
minus-le,
le-add-cancel-alt,
minus-one-mul,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
intEquality,
applyEquality,
lambdaEquality,
addEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
inlFormation,
inrFormation,
addLevel,
orFunctionality,
independent_pairEquality,
setElimination,
rename,
multiplyEquality
Latex:
\mforall{}[x:\mBbbZ{}]. uiff(0 < |x|;x \mneq{} 0)
Date html generated:
2017_04_14-AM-07_16_57
Last ObjectModification:
2017_02_27-PM-02_51_48
Theory : arithmetic
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