Nuprl Lemma : div_4_to_1
∀[a:ℕ]. ∀[b:{...-1}]. ((a ÷ b) = (-(a ÷ -b)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_lower: {...i},
nat: ℕ,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
int_lower: {...i},
nat: ℕ,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
subtract: n - m,
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
prop: ℙ,
ge: i ≥ j ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
guard: {T},
squash: ↓T,
sq_type: SQType(T)
Lemmas referenced :
int_lower_properties,
nat_properties,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
int_lower_wf,
istype-nat,
istype-void,
not-equal-2,
minus-one-mul-top,
decidable__le,
istype-le,
istype-false,
not-le-2,
condition-implies-le,
minus-one-mul,
add-commutes,
minus-add,
minus-minus,
add-swap,
add-associates,
zero-add,
le-add-cancel2,
add-mul-special,
zero-mul,
add-zero,
div_unique3,
minus-zero,
le-add-cancel,
nequal_wf,
rem-zero,
div_rem_sum,
rem_bounds_1,
decidable__lt,
not-lt-2,
minus-le,
le-add-cancel-alt,
istype-less_than,
mul-associates,
mul-commutes,
mul-swap,
less_than_transitivity1,
less_than_irreflexivity,
absval_wf,
int_subtype_base,
less_than_wf,
squash_wf,
true_wf,
istype-int,
absval_pos,
absval_neg,
decidable__int_equal,
subtype_base_sq,
le_weakening,
nequal-le-implies
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
Error :universeIsType,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
voidElimination,
multiplyEquality,
productElimination,
addEquality,
unionElimination,
Error :inlFormation_alt,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :inrFormation_alt,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
intEquality,
divideEquality,
equalityTransitivity,
equalitySymmetry,
applyEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
remainderEquality,
Error :productIsType,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
Error :functionIsType,
hyp_replacement,
imageElimination,
imageMemberEquality,
instantiate,
cumulativity
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[b:\{...-1\}]. ((a \mdiv{} b) = (-(a \mdiv{} -b)))
Date html generated:
2019_06_20-AM-11_25_04
Last ObjectModification:
2019_01_03-PM-04_06_01
Theory : arithmetic
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