Nuprl Lemma : quotient-is-zero
∀[a,n:ℕ]. (a ÷ n) = 0 ∈ ℤ supposing a < n
Proof
Definitions occuring in Statement :
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtract: n - m,
true: True,
less_than': less_than'(a;b),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_type: SQType(T),
top: Top,
uiff: uiff(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
nat_plus: ℕ+,
all: ∀x:A. B[x],
guard: {T},
false: False,
implies: P ⇒ Q,
not: ¬A,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
nat: ℕ,
prop: ℙ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
not_wf,
le-add-cancel2,
minus-zero,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
condition-implies-le,
not-le-2,
le-add-cancel,
add-zero,
not-equal-2,
false_wf,
decidable__int_equal,
div_bounds_1,
add-commutes,
one-mul,
zero-add,
mul-commutes,
subtype_base_sq,
le_reflexive,
int_subtype_base,
le_wf,
set_subtype_base,
multiply-is-int-iff,
add_functionality_wrt_le,
mul_preserves_le,
decidable__le,
less_than_transitivity2,
rem_bounds_1,
nequal_wf,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
div_rem_sum,
nat_wf,
less_than_wf
Rules used in proof :
minusEquality,
independent_pairFormation,
addEquality,
voidEquality,
cumulativity,
instantiate,
lambdaEquality,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
remainderEquality,
multiplyEquality,
unionElimination,
divideEquality,
productElimination,
intEquality,
voidElimination,
independent_functionElimination,
dependent_functionElimination,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
dependent_set_memberEquality,
equalitySymmetry,
equalityTransitivity,
because_Cache,
axiomEquality,
isect_memberEquality,
sqequalRule,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
hypothesis,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,n:\mBbbN{}]. (a \mdiv{} n) = 0 supposing a < n
Date html generated:
2017_09_29-PM-05_47_15
Last ObjectModification:
2017_09_06-PM-00_34_27
Theory : arithmetic
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