Nuprl Lemma : modulus_base
∀[m:ℕ+]. ∀[a:ℕm].  (a mod m ~ a)
Proof
Definitions occuring in Statement : 
modulus: a mod n
, 
int_seg: {i..j-}
, 
nat_plus: ℕ+
, 
uall: ∀[x:A]. B[x]
, 
natural_number: $n
, 
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)↓
, 
nat_plus: ℕ+
, 
int_seg: {i..j-}
, 
nequal: a ≠ b ∈ T 
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
false: False
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
guard: {T}
, 
all: ∀x:A. B[x]
, 
prop: ℙ
, 
le: A ≤ B
, 
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)
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
subtype_rel: A ⊆r B
, 
int_nzero: ℤ-o
, 
decidable: Dec(P)
, 
subtract: n - m
Lemmas referenced : 
subtype_base_sq, 
nat_wf, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
value-type-has-value, 
int-value-type, 
less_than_transitivity1, 
le_weakening, 
less_than_irreflexivity, 
equal_wf, 
rem_bounds_1, 
lt_int_wf, 
bool_wf, 
eqtt_to_assert, 
assert_of_lt_int, 
top_wf, 
less_than_wf, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
iff_transitivity, 
assert_wf, 
bnot_wf, 
not_wf, 
iff_weakening_uiff, 
assert_of_bnot, 
int_seg_wf, 
nat_plus_wf, 
div_bounds_1, 
int_seg_subtype_nat, 
false_wf, 
div_rem_sum, 
subtype_rel_sets, 
nequal_wf, 
equal-wf-base, 
decidable__int_equal, 
zero-mul, 
zero-add, 
decidable__le, 
not-le-2, 
not-equal-2, 
add_functionality_wrt_le, 
add-associates, 
add-zero, 
le-add-cancel, 
condition-implies-le, 
add-commutes, 
minus-add, 
minus-zero, 
minus-one-mul, 
add-swap, 
minus-one-mul-top, 
le-add-cancel2, 
mul_preserves_le, 
nat_plus_subtype_nat, 
le_reflexive, 
multiply-is-int-iff, 
add-is-int-iff, 
mul-commutes, 
one-mul
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, 
because_Cache, 
setElimination, 
rename, 
lambdaFormation, 
productElimination, 
dependent_functionElimination, 
independent_functionElimination, 
voidElimination, 
unionElimination, 
equalityElimination, 
equalityTransitivity, 
equalitySymmetry, 
lessCases, 
sqequalAxiom, 
isect_memberEquality, 
independent_pairFormation, 
voidEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
dependent_pairFormation, 
promote_hyp, 
impliesFunctionality, 
applyEquality, 
setEquality, 
dependent_set_memberEquality, 
addEquality, 
minusEquality, 
multiplyEquality, 
baseApply, 
closedConclusion
Latex:
\mforall{}[m:\mBbbN{}\msupplus{}].  \mforall{}[a:\mBbbN{}m].    (a  mod  m  \msim{}  a)
Date html generated:
2017_04_14-AM-07_19_10
Last ObjectModification:
2017_02_27-PM-02_53_28
Theory : arithmetic
Home
Index