Nuprl Lemma : modulus_wf
∀[a:ℤ]. ∀[n:ℤ-o]. (a mod n ∈ ℕ)
Proof
Definitions occuring in Statement :
modulus: a mod n,
int_nzero: ℤ-o,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
modulus: a mod n,
has-value: (a)↓,
uimplies: b supposing a,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
nat: ℕ,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
gt: i > j,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
decidable: Dec(P),
subtract: n - m,
subtype_rel: A ⊆r B,
le: A ≤ B
Lemmas referenced :
value-type-has-value,
int-value-type,
equal_wf,
less_than_wf,
not-gt-2,
le_wf,
int_nzero_wf,
rem_bounds_z,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_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,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
minus-minus,
minus-one-mul-top,
add_functionality_wrt_le,
add-associates,
zero-add,
add-swap,
add-commutes,
add-zero,
le-add-cancel,
absval_unfold2,
add_functionality_wrt_lt,
le_reflexive,
less_than_transitivity2,
le_weakening2,
add-mul-special,
zero-mul,
subtract_wf,
absval_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
independent_isectElimination,
hypothesis,
remainderEquality,
because_Cache,
setElimination,
rename,
lambdaFormation,
independent_functionElimination,
voidElimination,
hypothesisEquality,
natural_numberEquality,
lessCases,
independent_pairFormation,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
isect_memberEquality,
voidEquality,
imageElimination,
productElimination,
dependent_set_memberEquality,
dependent_functionElimination,
axiomEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
minusEquality,
applyEquality,
lambdaEquality,
addEquality,
multiplyEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (a mod n \mmember{} \mBbbN{})
Date html generated:
2019_06_20-AM-11_25_35
Last ObjectModification:
2018_08_20-PM-09_28_29
Theory : arithmetic
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