Nuprl Lemma : div_floor_mod_sum
∀[a:ℤ]. ∀[n:ℕ+]. (a = (((a ÷↓ n) * n) + (a mod n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
div_floor: a ÷↓ n,
modulus: a mod n,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
modulus: a mod n,
div_floor: a ÷↓ n,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
sq_type: SQType(T),
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,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
has-value: (a)↓,
subtract: n - m
Lemmas referenced :
div_rem_sum,
subtype_rel_sets,
less_than_wf,
nequal_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
equal-wf-base,
int_subtype_base,
rem_bounds_z,
subtype_base_sq,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_transitivity2,
le_weakening2,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
iff_weakening_uiff,
assert_of_bnot,
equal-wf-base-T,
absval_wf,
nat_wf,
nat_plus_wf,
value-type-has-value,
int-value-type,
subtract_wf,
nat_plus_subtype_nat,
mul-commutes,
add-commutes,
mul-distributes-right,
add-associates,
minus-one-mul-top,
add-swap,
add-mul-special,
zero-mul,
zero-add,
squash_wf,
true_wf,
add_functionality_wrt_eq,
absval_pos,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
hypothesis,
independent_isectElimination,
setElimination,
rename,
setEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
baseClosed,
remainderEquality,
divideEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
unionElimination,
equalityElimination,
productElimination,
lessCases,
sqequalAxiom,
isect_memberEquality,
independent_pairFormation,
voidEquality,
imageMemberEquality,
imageElimination,
dependent_pairFormation,
promote_hyp,
impliesFunctionality,
addEquality,
multiplyEquality,
axiomEquality,
callbyvalueReduce,
equalityUniverse,
levelHypothesis,
minusEquality,
universeEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (a = (((a \mdiv{}\mdownarrow{} n) * n) + (a mod n)))
Date html generated:
2017_04_14-AM-07_19_33
Last ObjectModification:
2017_02_27-PM-02_53_18
Theory : arithmetic
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