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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