Nuprl Lemma : absval_unfold3
∀[x:ℤ]. (|x| ~ if x <z 0 then -x else x fi )
Proof
Definitions occuring in Statement :
absval: |i|,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
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,
ifthenelse: if b then t else f fi ,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
decidable: Dec(P),
le: A ≤ B,
subtract: n - m
Lemmas referenced :
value-type-has-value,
int-value-type,
subtype_base_sq,
int_subtype_base,
istype-int,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
istype-void,
less_than_transitivity2,
le_weakening2,
less_than_irreflexivity,
eqff_to_assert,
bool_cases_sqequal,
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,
decidable__int_equal,
istype-false,
not-equal-2,
not-lt-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
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,
natural_numberEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
lessCases,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
independent_pairFormation,
voidElimination,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
Error :functionIsType,
Error :universeIsType,
minusEquality,
addEquality
Latex:
\mforall{}[x:\mBbbZ{}]. (|x| \msim{} if x <z 0 then -x else x fi )
Date html generated:
2019_06_20-AM-11_24_22
Last ObjectModification:
2019_01_31-AM-09_41_21
Theory : arithmetic
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