Nuprl Lemma : abs-val_wf
∀[x:ℤ]. (|x| ∈ ℕ)
Proof
Definitions occuring in Statement :
abs-val: |x|,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
abs-val: |x|,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
nat: ℕ,
prop: ℙ,
bfalse: ff,
guard: {T},
top: Top,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
true: True,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-base,
int_subtype_base,
assert_wf,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
le_weakening2,
minus-one-mul,
le_wf,
le_int_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
not-lt-2,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul-top,
minus-zero,
add-zero,
one-mul,
zero-add,
add-commutes,
add-mul-special,
zero-mul,
less-iff-le,
false_wf,
add-associates,
add-swap,
omega-shadow,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
lambdaFormation,
unionElimination,
equalityElimination,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
independent_functionElimination,
because_Cache,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
minusEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
intEquality,
multiplyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
independent_pairFormation,
imageMemberEquality
Latex:
\mforall{}[x:\mBbbZ{}]. (|x| \mmember{} \mBbbN{})
Date html generated:
2019_06_20-PM-00_25_54
Last ObjectModification:
2018_08_07-PM-04_55_34
Theory : int_1
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