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: T 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: ℕ
, 
implies: P 
⇒ Q
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
or: P ∨ Q
, 
subtype_rel: A ⊆r 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: λ2x y.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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