### 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