Nuprl Lemma : absval_neg
∀[i:{...0}]. (|i| = (-i) ∈ ℤ)
Proof
Definitions occuring in Statement :
absval: |i|,
int_lower: {...i},
uall: ∀[x:A]. B[x],
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_lower: {...i},
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,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B,
decidable: Dec(P),
subtype_rel: A ⊆r B,
subtract: n - m
Lemmas referenced :
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
less_than_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,
int_lower_wf,
int_lower_properties,
int_subtype_base,
decidable__int_equal,
false_wf,
not-equal-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
add-swap,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
impliesFunctionality,
cumulativity,
intEquality,
addEquality,
applyEquality,
lambdaEquality
Latex:
\mforall{}[i:\{...0\}]. (|i| = (-i))
Date html generated:
2017_04_14-AM-07_16_49
Last ObjectModification:
2017_02_27-PM-02_51_38
Theory : arithmetic
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