Nuprl Lemma : bag-maximal?-max

[T:Type]. ∀[b:bag(T)]. ∀[R:T ⟶ T ⟶ 𝔹]. ∀[x,y:T].  (↑(R y)) supposing ((↑bag-maximal?(b;x;R)) and y ↓∈ b)


Definitions occuring in Statement :  bag-member: x ↓∈ bs bag-maximal?: bag-maximal?(bg;x;R) bag: bag(T) assert: b bool: 𝔹 uimplies: supposing a uall: [x:A]. B[x] apply: 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: implies:  Q false: False ge: i ≥  uimplies: supposing a satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: subtype_rel: A ⊆B guard: {T} or: P ∨ Q empty-bag: {} uiff: uiff(P;Q) cons: [a b] colength: colength(L) so_lambda: λ2y.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 then else 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

\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

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