Nuprl Lemma : div_floor_wf
∀[a:ℤ]. ∀[n:ℤ-o]. (a ÷↓ n ∈ ℤ)
Proof
Definitions occuring in Statement :
div_floor: a ÷↓ n,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
div_floor: a ÷↓ n,
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,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
guard: {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
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
value-type-has-value,
int-value-type,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
subtract_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,
int_nzero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
lambdaFormation,
unionElimination,
equalityElimination,
because_Cache,
productElimination,
independent_isectElimination,
lessCases,
axiomSqEquality,
isect_memberEquality,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
callbyvalueReduce,
intEquality,
remainderEquality,
equalitySymmetry,
dependent_functionElimination,
equalityTransitivity,
divideEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
axiomEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (a \mdiv{}\mdownarrow{} n \mmember{} \mBbbZ{})
Date html generated:
2019_06_20-AM-11_25_38
Last ObjectModification:
2018_08_20-PM-09_28_22
Theory : arithmetic
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