Nuprl Lemma : absval_le_zero
∀[i:ℤ]. uiff(|i| ≤ 0;i = 0 ∈ ℤ)
Proof
Definitions occuring in Statement :
absval: |i|,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
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,
le: A ≤ B,
cand: A c∧ B,
guard: {T},
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
decidable: Dec(P),
subtract: n - m
Lemmas referenced :
absval_unfold2,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
less_than_transitivity1,
less_than_irreflexivity,
istype-le,
le_weakening,
le_witness_for_triv,
int_subtype_base,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
istype-less_than,
istype-assert,
istype-int,
add_functionality_wrt_le,
le_reflexive,
decidable__int_equal,
istype-false,
not-equal-2,
condition-implies-le,
minus-zero,
add-zero,
add-associates,
minus-add,
minus-minus,
minus-one-mul,
zero-add,
minus-one-mul-top,
two-mul,
add-commutes,
mul-distributes-right,
one-mul,
le-add-cancel,
not-lt-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
lessCases,
axiomSqEquality,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
Error :equalityIstype,
applyEquality,
sqequalBase,
Error :dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
Error :functionIsType,
Error :universeIsType,
independent_pairEquality,
axiomEquality,
minusEquality,
addEquality,
multiplyEquality,
intEquality
Latex:
\mforall{}[i:\mBbbZ{}]. uiff(|i| \mleq{} 0;i = 0)
Date html generated:
2019_06_20-AM-11_24_30
Last ObjectModification:
2019_02_12-PM-01_59_51
Theory : arithmetic
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