Nuprl Lemma : bag-maximals-not-max
∀[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x,y:T].
(¬↑(R x y)) supposing
(y ↓∈ bag-maximals(b;R) and
x ↓∈ bag-maximals(b;R) and
(∀x,y:T. ((↑(R x y)) ⇒ (↑(R y x)) ⇒ (x = y ∈ T))) and
(∀x:T. (¬↑(R x x))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-maximals: bag-maximals(bg;R),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
not: ¬A,
implies: P ⇒ Q,
bag-maximals: bag-maximals(bg;R),
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
false: False,
exists: ∃x:A. B[x],
prop: ℙ,
bag-member: x ↓∈ bs,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
bag-maximal?: bag-maximal?(bg;x;R),
bag-accum: bag-accum(v,x.f[v; x];init;bs),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
true: True,
guard: {T},
subtype_rel: A ⊆r B,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
assert: ↑b,
rev_implies: P ⇐ Q,
sq_type: SQType(T)
Lemmas referenced :
bag-member-filter,
bag-maximal?_wf,
bag_to_squash_list,
assert_wf,
bag-member_wf,
l_member_decomp,
assert_functionality_wrt_uiff,
list_accum_wf,
bool_wf,
btrue_wf,
band_wf,
append_wf,
cons_wf,
nil_wf,
squash_wf,
true_wf,
list_wf,
eqtt_to_assert,
equal_wf,
subtype_rel_list,
top_wf,
list_accum_nil_lemma,
decidable__assert,
bag-maximals_wf,
all_wf,
not_wf,
bag_wf,
list_accum_append,
list_accum_cons,
iff_imp_equal_bool,
bfalse_wf,
false_wf,
subtype_base_sq,
bool_subtype_base,
band-bfalse,
list_accum_invariant,
not_assert_elim,
and_wf,
assert_elim,
btrue_neq_bfalse
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
hypothesis,
productElimination,
independent_isectElimination,
imageElimination,
promote_hyp,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
rename,
dependent_functionElimination,
independent_functionElimination,
equalityTransitivity,
functionEquality,
universeEquality,
unionElimination,
equalityElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
isect_memberFormation,
independent_pairFormation,
instantiate,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x,y:T].
(\mneg{}\muparrow{}(R x y)) supposing
(y \mdownarrow{}\mmember{} bag-maximals(b;R) and
x \mdownarrow{}\mmember{} bag-maximals(b;R) and
(\mforall{}x,y:T. ((\muparrow{}(R x y)) {}\mRightarrow{} (\muparrow{}(R y x)) {}\mRightarrow{} (x = y))) and
(\mforall{}x:T. (\mneg{}\muparrow{}(R x x))))
Date html generated:
2017_10_01-AM-08_58_46
Last ObjectModification:
2017_07_26-PM-04_40_37
Theory : bags
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