Nuprl Lemma : list_in_mem_f_list

T:Type. ∀as:T List.  (as ∈ {x:T| mem_f(T;x;as)}  List)


Proof




Definitions occuring in Statement :  mem_f: mem_f(T;a;bs) list: List all: x:A. B[x] member: t ∈ T set: {x:A| B[x]}  universe: Type
Definitions unfolded in proof :  all: x:A. B[x] uall: [x:A]. B[x] member: t ∈ T nat: implies:  Q false: False ge: i ≥  uimplies: supposing a not: ¬A satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] top: Top and: P ∧ Q prop: or: P ∨ Q subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] cons: [a b] le: A ≤ B less_than': less_than'(a;b) colength: colength(L) nil: [] it: guard: {T} sq_type: SQType(T) less_than: a < b squash: T so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] decidable: Dec(P) mem_f: mem_f(T;a;bs) ycomb: Y so_lambda: so_lambda(x,y,z.t[x; y; z]) so_apply: x[s1;s2;s3]
Lemmas referenced :  nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int int_formula_prop_and_lemma istype-void 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 istype-less_than list-cases list-subtype nil_wf subtype_rel_list l_member_wf mem_f_wf subtype_rel_sets null_nil_lemma btrue_wf member-implies-null-eq-bfalse btrue_neq_bfalse product_subtype_list colength-cons-not-zero colength_wf_list istype-le subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf intformnot_wf itermSubtract_wf itermAdd_wf int_formula_prop_not_lemma int_term_value_subtract_lemma int_term_value_add_lemma decidable__le le_wf list_ind_cons_lemma cons_wf equal_wf subtype_rel_list_set istype-nat list_wf istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename sqequalRule intWeakElimination natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality dependent_functionElimination isect_memberEquality_alt voidElimination independent_pairFormation universeIsType axiomEquality equalityTransitivity equalitySymmetry functionIsTypeImplies inhabitedIsType unionElimination because_Cache applyEquality setEquality setIsType promote_hyp hypothesis_subsumption productElimination equalityIstype dependent_set_memberEquality_alt instantiate applyLambdaEquality imageElimination baseApply closedConclusion baseClosed intEquality sqequalBase unionEquality inlFormation_alt unionIsType inrFormation_alt universeEquality

Latex:
\mforall{}T:Type.  \mforall{}as:T  List.    (as  \mmember{}  \{x:T|  mem\_f(T;x;as)\}    List)



Date html generated: 2019_10_16-PM-01_01_44
Last ObjectModification: 2019_06_20-PM-06_49_26

Theory : list_2


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