Nuprl Lemma : int-ineq-constraint-factor
∀[a:ℤ]. ∀[g:ℕ+]. ∀[xs,L:ℤ List]. uiff(0 ≤ [1 / xs] ⋅ [a / g * L];0 ≤ [1 / xs] ⋅ [a ÷↓ g / L])
Proof
Definitions occuring in Statement :
int-vec-mul: a * as,
integer-dot-product: as ⋅ bs,
cons: [a / b],
list: T List,
div_floor: a ÷↓ n,
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
guard: {T}
Lemmas referenced :
one-mul,
multiply-is-int-iff,
set_subtype_base,
list_subtype_base,
int_subtype_base,
add-is-int-iff,
div_reduce_inequality,
int-dot-mul-right,
nat_plus_wf,
list_wf,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
nequal_wf,
less_than_wf,
subtype_rel_sets,
div_floor_wf,
int-vec-mul_wf,
cons_wf,
integer-dot-product_wf,
le_wf,
less_than'_wf,
int_dot_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
sqequalRule,
lemma_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
productElimination,
independent_pairEquality,
lambdaEquality,
hypothesisEquality,
because_Cache,
isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
intEquality,
setElimination,
rename,
applyEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
independent_functionElimination,
multiplyEquality,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[g:\mBbbN{}\msupplus{}]. \mforall{}[xs,L:\mBbbZ{} List]. uiff(0 \mleq{} [1 / xs] \mcdot{} [a / g * L];0 \mleq{} [1 / xs] \mcdot{} [a \mdiv{}\mdownarrow{} g / L])
Date html generated:
2016_05_14-AM-06_57_19
Last ObjectModification:
2016_01_14-PM-08_44_29
Theory : omega
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