Nuprl Lemma : modulus-is-rem
∀[a:ℕ]. ∀[n:ℤ-o]. (a mod n ~ a rem n)
Proof
Definitions occuring in Statement :
modulus: a mod n,
int_nzero: ℤ-o,
nat: ℕ,
uall: ∀[x:A]. B[x],
remainder: n rem m,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
modulus: a mod n,
has-value: (a)↓,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
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,
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
le: A ≤ B,
guard: {T},
int_lower: {...i},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
gt: i > j
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
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,
decidable__lt,
rem_bounds_1,
less_than_transitivity1,
less_than_irreflexivity,
rem_bounds_4,
decidable__le,
false_wf,
not-le-2,
not-equal-2,
condition-implies-le,
minus-zero,
add-zero,
minus-add,
minus-minus,
add-swap,
add-commutes,
add-associates,
zero-add,
add_functionality_wrt_le,
le-add-cancel,
not-lt-2,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
not-gt-2,
int_nzero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
callbyvalueReduce,
remainderEquality,
setElimination,
rename,
because_Cache,
lambdaFormation,
independent_functionElimination,
voidElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_functionElimination,
dependent_set_memberEquality,
minusEquality,
addEquality,
applyEquality,
dependent_pairFormation,
promote_hyp,
impliesFunctionality
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (a mod n \msim{} a rem n)
Date html generated:
2017_04_14-AM-07_19_04
Last ObjectModification:
2017_02_27-PM-02_53_16
Theory : arithmetic
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