Nuprl Lemma : div_rem_sum
∀[a:ℤ]. ∀[n:ℤ-o]. (a = (((a ÷ n) * n) + (a rem n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
remainder: n rem m,
divide: n ÷ m,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
nequal: a ≠ b ∈ T ,
or: P ∨ Q,
guard: {T},
le: A ≤ B,
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
bfalse: ff,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
prop: ℙ
Lemmas referenced :
int_nzero_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
not-equal-2,
le_antisymmetry_iff,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
istype-void,
minus-add,
minus-zero,
eqff_to_assert,
set_subtype_base,
nequal_wf,
int_subtype_base,
bool_subtype_base,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
iff_weakening_uiff,
assert_of_bnot,
false_wf
Rules used in proof :
intEquality,
because_Cache,
axiomEquality,
hypothesisEquality,
thin,
isectElimination,
isect_memberEquality,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
divideRemainderSum,
setElimination,
rename,
natural_numberEquality,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
int_eqReduceTrueSq,
dependent_functionElimination,
addEquality,
independent_functionElimination,
voidElimination,
minusEquality,
applyEquality,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
Error :dependent_pairFormation_alt,
equalityTransitivity,
equalitySymmetry,
Error :equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
promote_hyp,
instantiate,
cumulativity,
independent_pairFormation,
Error :functionIsType,
int_eqReduceFalseSq,
Error :equalityIsType1
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (a = (((a \mdiv{} n) * n) + (a rem n)))
Date html generated:
2019_06_20-AM-11_23_57
Last ObjectModification:
2018_10_15-PM-00_14_22
Theory : arithmetic
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