Nuprl Lemma : mod_bounds_1
∀[a:ℤ]. ∀[n:ℤ-o]. ((0 ≤ (a mod n)) ∧ a mod n < |n|)
Proof
Definitions occuring in Statement :
modulus: a mod n,
absval: |i|,
int_nzero: ℤ-o,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
int_nzero: ℤ-o,
uimplies: b supposing a,
modulus: a mod n,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
gt: i > j
Lemmas referenced :
less_than'_wf,
modulus_wf,
nat_wf,
member-less_than,
absval_wf,
int_nzero_wf,
zero-le-nat,
value-type-has-value,
int-value-type,
equal_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
add_functionality_wrt_lt,
le_reflexive,
zero-add,
add-commutes,
le_wf,
absval_pos,
not-gt-2,
iff_weakening_equal,
rem_bounds_z,
less_than_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
extract_by_obid,
isectElimination,
applyEquality,
setElimination,
rename,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
isect_memberEquality,
voidElimination,
intEquality,
callbyvalueReduce,
remainderEquality,
lambdaFormation,
independent_functionElimination,
unionElimination,
equalityElimination,
lessCases,
sqequalAxiom,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. ((0 \mleq{} (a mod n)) \mwedge{} a mod n < |n|)
Date html generated:
2018_05_21-PM-00_02_02
Last ObjectModification:
2018_05_19-AM-07_13_07
Theory : arithmetic
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