Nuprl Lemma : q-constraint-sum
∀[x,x':ℕ ⟶ ℚ]. ∀[r1,r2:ℤ]. ∀[k:ℕ]. ∀[y:ℚ List].
(q-rel(q-rel-lub(r1;r2);q-linear(k;j.(x j) + (x' j);y))) supposing
(q-rel(r2;q-linear(k;j.x' j;y)) and
q-rel(r1;q-linear(k;j.x j;y)) and
(k ≤ ||y||))
Proof
Definitions occuring in Statement :
q-rel-lub: q-rel-lub(r1;r2),
q-rel: q-rel(r;x),
q-linear: q-linear(k;i.X[i];y),
qadd: r + s,
rationals: ℚ,
length: ||as||,
list: T List,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ
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,
q-rel: q-rel(r;x),
ifthenelse: if b then t else f fi ,
all: ∀x:A. B[x],
bool: 𝔹,
nat: ℕ,
q-rel-lub: q-rel-lub(r1;r2),
uiff: uiff(P;Q),
band: p ∧b q,
btrue: tt,
not: ¬A,
false: False,
eq_int: (i =z j),
bfalse: ff,
unit: Unit,
it: ⋅
Lemmas referenced :
q-rel_wf,
squash_wf,
true_wf,
rationals_wf,
q-rel-lub_wf,
q-linear-sum,
nat_wf,
subtype_rel_self,
iff_weakening_equal,
equal-wf-base-T,
q-linear_wf,
qadd_wf,
qle_witness,
int-subtype-rationals,
qle_wf,
qless_witness,
qless_wf,
equal_wf,
ifthenelse_wf,
eq_int_wf,
le_wf,
length_wf,
list_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
qadd_preserves_qle,
qadd_preserves_qless,
qle_transitivity_qorder,
qless_transitivity_1_qorder,
qless_transitivity_2_qorder,
qless_transitivity,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
qadd_comm_q,
mon_ident_q,
qle_weakening_eq_qorder
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,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
independent_functionElimination,
because_Cache,
lambdaFormation,
unionElimination,
axiomEquality,
dependent_functionElimination,
isect_memberEquality,
setElimination,
rename,
functionEquality,
baseApply,
closedConclusion,
applyLambdaEquality,
voidElimination,
equalityElimination,
independent_pairFormation
Latex:
\mforall{}[x,x':\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[r1,r2:\mBbbZ{}]. \mforall{}[k:\mBbbN{}]. \mforall{}[y:\mBbbQ{} List].
(q-rel(q-rel-lub(r1;r2);q-linear(k;j.(x j) + (x' j);y))) supposing
(q-rel(r2;q-linear(k;j.x' j;y)) and
q-rel(r1;q-linear(k;j.x j;y)) and
(k \mleq{} ||y||))
Date html generated:
2019_10_16-PM-00_33_30
Last ObjectModification:
2018_08_25-AM-10_26_47
Theory : rationals
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