Nuprl Lemma : absval_ubound
∀[i:ℤ]. ∀[n:ℕ]. uiff(|i| ≤ n;((-n) ≤ i) ∧ (i ≤ n))
Proof
Definitions occuring in Statement :
absval: |i|,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
minus: -n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
uimplies: b supposing a,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
not: ¬A,
false: False,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
subtype_rel: A ⊆r B,
top: Top
Lemmas referenced :
absval_unfold,
nat_properties,
add_functionality_wrt_le,
le_reflexive,
le_witness_for_triv,
istype-nat,
istype-int,
minus-one-mul,
zero-add,
add-mul-special,
zero-mul,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
decidable__le,
minus-one-mul-top,
istype-false,
not-le-2,
less-iff-le,
condition-implies-le,
minus-add,
add-swap,
add-commutes,
minus-minus,
mul-associates,
one-mul,
add-associates,
le-add-cancel,
istype-le,
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-void,
not-lt-2,
le-add-cancel2,
le_functionality,
le_weakening,
minus-le,
le-iff-nonneg,
add-is-int-iff,
int_subtype_base,
subtract_wf,
add-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
minusEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
isectIsTypeImplies,
inhabitedIsType,
multiplyEquality,
Error :memTop,
lambdaFormation_alt,
unionElimination,
equalityElimination,
lessCases,
axiomSqEquality,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
voidElimination,
independent_functionElimination,
addEquality,
productIsType,
dependent_pairFormation_alt,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
functionIsType,
universeIsType,
baseApply,
closedConclusion,
applyEquality,
intEquality,
voidEquality,
isect_memberEquality,
lambdaEquality
Latex:
\mforall{}[i:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. uiff(|i| \mleq{} n;((-n) \mleq{} i) \mwedge{} (i \mleq{} n))
Date html generated:
2020_05_19-PM-09_35_23
Last ObjectModification:
2020_01_04-PM-07_56_38
Theory : arithmetic
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