Nuprl Lemma : q-constraint-times
∀[x:ℕ ⟶ ℚ]. ∀[r:ℤ]. ∀[a:ℚ]. ∀[k:ℕ]. ∀[y:ℚ List].
(q-rel(r;q-linear(k;j.a * (x j);y))) supposing
(q-rel(r;q-linear(k;j.x j;y)) and
((r = 0 ∈ ℤ) ∨ 0 < a ∨ ((0 ≤ a) ∧ (r = 1 ∈ ℤ))) and
(k ≤ ||y||))
Proof
Definitions occuring in Statement :
q-rel: q-rel(r;x),
q-linear: q-linear(k;i.X[i];y),
qle: r ≤ s,
qless: r < s,
qmul: r * s,
rationals: ℚ,
length: ||as||,
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
or: P ∨ Q,
sq_type: SQType(T),
all: ∀x:A. B[x],
q-rel: q-rel(r;x),
ifthenelse: if b then t else f fi ,
bool: 𝔹,
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
cand: A c∧ B,
not: ¬A,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff
Lemmas referenced :
q-rel_wf,
squash_wf,
true_wf,
rationals_wf,
q-linear-times,
nat_wf,
iff_weakening_equal,
subtype_base_sq,
int_subtype_base,
equal-wf-base-T,
q-linear_wf,
qmul_wf,
qle_witness,
int-subtype-rationals,
qle_wf,
qless_witness,
qless_wf,
equal_wf,
ifthenelse_wf,
eq_int_wf,
or_wf,
equal-wf-base,
le_wf,
length_wf,
list_wf,
bool_wf,
assert_wf,
qmul_comm_qrng,
bnot_wf,
not_wf,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
qmul_preserves_qle2,
qle_weakening_lt_qorder,
qmul-positive,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
qmul_zero_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
intEquality,
sqequalRule,
functionExtensionality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
unionElimination,
instantiate,
cumulativity,
dependent_functionElimination,
because_Cache,
lambdaFormation,
axiomEquality,
isect_memberEquality,
productEquality,
setElimination,
rename,
functionEquality,
applyLambdaEquality,
voidElimination,
promote_hyp,
dependent_pairFormation,
voidEquality,
computeAll,
baseApply,
closedConclusion,
inlFormation,
independent_pairFormation,
minusEquality,
equalityElimination,
impliesFunctionality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[r:\mBbbZ{}]. \mforall{}[a:\mBbbQ{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[y:\mBbbQ{} List].
(q-rel(r;q-linear(k;j.a * (x j);y))) supposing
(q-rel(r;q-linear(k;j.x j;y)) and
((r = 0) \mvee{} 0 < a \mvee{} ((0 \mleq{} a) \mwedge{} (r = 1))) and
(k \mleq{} ||y||))
Date html generated:
2018_05_22-AM-00_19_08
Last ObjectModification:
2017_07_26-PM-06_53_53
Theory : rationals
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