Nuprl Lemma : bag-maximal?-max
∀[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x,y:T]. (↑(R x y)) supposing ((↑bag-maximal?(b;x;R)) and y ↓∈ b)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-maximal?: bag-maximal?(bg;x;R),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
empty-bag: {},
uiff: uiff(P;Q),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
cons-bag: x.b,
sq_or: a ↓∨ b,
sq_stable: SqStable(P),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True
Lemmas referenced :
assert_wf,
bag-maximal?_wf,
bag-member_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,
bag-member-empty-iff,
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,
bag-member-cons,
cons_wf,
list_induction,
bag-maximal?-cons,
sq_stable_from_decidable,
decidable__assert,
and_wf,
assert_elim,
bool_wf,
bool_subtype_base,
bag_wf,
assert_witness,
bag_to_squash_list
Rules used in proof :
because_Cache,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
hypothesisEquality,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
productElimination,
promote_hyp,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
functionEquality,
imageMemberEquality,
universeEquality,
isect_memberFormation,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}]. \mforall{}[x,y:T].
(\muparrow{}(R x y)) supposing ((\muparrow{}bag-maximal?(b;x;R)) and y \mdownarrow{}\mmember{} b)
Date html generated:
2017_10_01-AM-08_58_52
Last ObjectModification:
2017_07_26-PM-04_40_43
Theory : bags
Home
Index