Nuprl Lemma : mod_bounds
∀[a:ℤ]. ∀[n:ℕ+]. ((0 ≤ (a mod n)) ∧ a mod n < n)
Proof
Definitions occuring in Statement :
modulus: a mod n,
nat_plus: ℕ+,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
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},
and: P ∧ Q,
squash: ↓T,
le: A ≤ B,
true: True,
iff: P ⇐⇒ Q
Lemmas referenced :
nat_plus_wf,
iff_weakening_equal,
nat_plus_subtype_nat,
absval_pos,
true_wf,
squash_wf,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
nequal_wf,
less_than_wf,
subtype_rel_sets,
mod_bounds_1
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
intEquality,
because_Cache,
lambdaEquality,
natural_numberEquality,
independent_isectElimination,
setElimination,
rename,
setEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
independent_pairFormation,
productElimination,
promote_hyp,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((0 \mleq{} (a mod n)) \mwedge{} a mod n < n)
Date html generated:
2016_05_13-PM-03_37_33
Last ObjectModification:
2016_01_14-PM-06_38_58
Theory : arithmetic
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