Nuprl Lemma : bag-maximal?-iff
∀[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x:T]. uiff(↑bag-maximal?(b;x;R);∀y:T. (y ↓∈ b ⇒ (↑(R x y))))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-maximal?: bag-maximal?(bg;x;R),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
or: 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],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
band: p ∧b q,
true: True,
sq_stable: SqStable(P),
cons-bag: x.b,
rev_uimplies: rev_uimplies(P;Q),
cand: A c∧ B,
sq_or: a ↓∨ b
Lemmas referenced :
bag-maximal?-max,
bag-member_wf,
assert_witness,
assert_wf,
bag-maximal?_wf,
all_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
list-subtype-bag,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
list_accum_nil_lemma,
true_wf,
nil_wf,
product_subtype_list,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_accum_cons_lemma,
list_accum_wf,
bool_wf,
band_wf,
cons_wf,
member_wf,
btrue_wf,
list_induction,
bag_wf,
bag_to_squash_list,
sq_stable_from_decidable,
decidable__assert,
bag-maximal?-cons,
bag-member-cons
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
independent_isectElimination,
hypothesis,
cumulativity,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
functionExtensionality,
applyEquality,
functionEquality,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
universeEquality,
independent_pairEquality,
hyp_replacement,
imageMemberEquality,
inrFormation,
inlFormation
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:T].
uiff(\muparrow{}bag-maximal?(b;x;R);\mforall{}y:T. (y \mdownarrow{}\mmember{} b {}\mRightarrow{} (\muparrow{}(R x y))))
Date html generated:
2017_10_01-AM-08_58_57
Last ObjectModification:
2017_07_26-PM-04_40_49
Theory : bags
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