Nuprl Lemma : div-positive-1
∀n:ℕ+. ∀i:{1..n + 1-}. 0 < n ÷ i
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
divide: n ÷ m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
top: Top,
true: True,
subtract: n - m,
int_seg: {i..j-},
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
so_apply: x[s],
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
sq_type: SQType(T),
sq_stable: SqStable(P),
squash: ↓T
Lemmas referenced :
sq_stable_from_decidable,
less-iff-le,
zero-mul,
le_antisymmetry_iff,
rem_bounds_1,
int_subtype_base,
subtype_base_sq,
equal_wf,
or_wf,
le-add-cancel2,
minus-one-mul-top,
add-swap,
minus-one-mul,
not-le-2,
decidable__le,
nequal_wf,
less_than_wf,
le_wf,
and_wf,
subtype_rel_sets,
div_rem_sum,
nat_plus_wf,
int_seg_wf,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
decidable__lt,
decidable__int_equal,
false_wf,
int_seg_subtype_nat_plus,
nat_plus_subtype_nat,
div_bounds_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
natural_numberEquality,
addEquality,
setElimination,
rename,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
productElimination,
unionElimination,
voidElimination,
independent_functionElimination,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
setEquality,
inlFormation,
inrFormation,
addLevel,
orFunctionality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
introduction,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}i:\{1..n + 1\msupminus{}\}. 0 < n \mdiv{} i
Date html generated:
2016_05_13-PM-03_35_39
Last ObjectModification:
2016_01_14-PM-06_39_48
Theory : arithmetic
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