Nuprl Lemma : bag_remove1_aux_wf

[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs,checked:T List].  (bag_remove1_aux(eq;checked;x;bs) ∈ List?)


Proof




Definitions occuring in Statement :  bag_remove1_aux: bag_remove1_aux(eq;checked;a;as) list: List deq: EqDecider(T) uall: [x:A]. B[x] unit: Unit member: t ∈ T union: left right universe: Type
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T 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 bag_remove1_aux: bag_remove1_aux(eq;checked;a;as) so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3] unit: Unit 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) deq: EqDecider(T) bool: 𝔹 btrue: tt ifthenelse: if then else fi  uiff: uiff(P;Q) eqof: eqof(d) bfalse: ff bnot: ¬bb assert: b
Lemmas referenced :  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 equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases list_ind_nil_lemma list_wf product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma 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_ind_cons_lemma bool_wf eqtt_to_assert safe-assert-deq append_wf unit_wf2 eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot cons_wf deq_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality equalityTransitivity equalitySymmetry because_Cache cumulativity applyEquality unionElimination inrEquality promote_hyp hypothesis_subsumption productElimination applyLambdaEquality dependent_set_memberEquality addEquality baseClosed instantiate imageElimination equalityElimination inlEquality universeEquality

Latex:
\mforall{}[T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[x:T].  \mforall{}[bs,checked:T  List].
    (bag\_remove1\_aux(eq;checked;x;bs)  \mmember{}  T  List?)



Date html generated: 2018_05_21-PM-09_47_57
Last ObjectModification: 2017_07_26-PM-06_30_30

Theory : bags_2


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