Nuprl Lemma : absval_unfold
∀[x:ℤ]. (|x| ~ if (-1) < (x) then x else (-x))
Proof
Definitions occuring in Statement :
absval: |i|,
uall: ∀[x:A]. B[x],
less: if (a) < (b) then c else d,
minus: -n,
natural_number: $n,
int: ℤ,
sqequal: s ~ t
Definitions unfolded in proof :
absval: |i|,
uall: ∀[x:A]. B[x],
member: t ∈ T,
has-value: (a)↓,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
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,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtype_rel: A ⊆r B,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m
Lemmas referenced :
value-type-has-value,
int-value-type,
subtype_base_sq,
int_subtype_base,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
uiff_transitivity,
assert_wf,
bnot_wf,
not_wf,
assert_of_bnot,
not_functionality_wrt_uiff,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-associates,
zero-add,
add-commutes,
le-add-cancel2,
decidable__int_equal,
false_wf,
not-equal-2,
add-swap,
add-zero,
le-add-cancel,
condition-implies-le,
minus-add,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
introduction,
cut,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
hypothesisEquality,
instantiate,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
axiomSqEquality,
Error :universeIsType,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
lessCases,
Error :inhabitedIsType,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
because_Cache,
dependent_pairFormation,
promote_hyp,
addEquality,
applyEquality,
lambdaEquality
Latex:
\mforall{}[x:\mBbbZ{}]. (|x| \msim{} if (-1) < (x) then x else (-x))
Date html generated:
2019_06_20-AM-11_24_18
Last ObjectModification:
2018_09_26-AM-10_58_21
Theory : arithmetic
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