Nuprl Lemma : qv-convex-lower-half
∀[r:ℚ]. ∀[v:ℚ List]. qv-convex(p.(dimension(p) = dimension(v) ∈ ℤ) ∧ (qdot(v;p) ≤ r))
Proof
Definitions occuring in Statement :
qv-convex: qv-convex(p.S[p]),
qv-dim: dimension(as),
qdot: qdot(as;bs),
qle: r ≤ s,
rationals: ℚ,
list: T List,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
qv-convex: qv-convex(p.S[p]),
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
qsub: r - s,
qge: a ≥ b
Lemmas referenced :
dim-qv-add,
qv-mul_wf,
subtype_rel_list,
rationals_wf,
top_wf,
istype-void,
qsub_wf,
dim-qv-mul,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
qle_wf,
qdot-linear,
qv-dim_wf,
set_subtype_base,
le_wf,
int_subtype_base,
qdot_wf,
istype-int,
qle_witness,
qv-add_wf,
qadd_wf,
qdot-linear2,
qmul_preserves_qle2,
qmul_wf,
int-subtype-rationals,
qadd_preserves_qle,
qadd_comm_q,
qadd_ac_1_q,
qinverse_q,
mon_ident_q,
qle_functionality_wrt_implies,
qadd_functionality_wrt_qle,
qle_weakening_eq_qorder,
qle_reflexivity,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_one_qrng,
qmul_comm_qrng
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
isectElimination,
hypothesisEquality,
because_Cache,
hypothesis,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
inhabitedIsType,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
intEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation,
productIsType,
equalityIstype,
sqequalBase,
dependent_functionElimination,
independent_pairEquality,
axiomEquality,
functionIsTypeImplies,
isectIsTypeImplies,
promote_hyp,
minusEquality
Latex:
\mforall{}[r:\mBbbQ{}]. \mforall{}[v:\mBbbQ{} List]. qv-convex(p.(dimension(p) = dimension(v)) \mwedge{} (qdot(v;p) \mleq{} r))
Date html generated:
2020_05_20-AM-09_27_46
Last ObjectModification:
2019_11_27-PM-01_44_57
Theory : rationals
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