Nuprl Lemma : member-fset-filter2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[P:{x:T| x ∈ s} ⟶ 𝔹]. ∀[x:T]. uiff(x ∈ {x ∈ s | P[x]};{x ∈ s ∧ (↑P[x])}\000C)
Proof
Definitions occuring in Statement :
fset-filter: {x ∈ s | P[x]},
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
guard: {T},
subtype_rel: A ⊆r B,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
fset: fset(T),
quotient: x,y:A//B[x; y],
fset-member: a ∈ s,
fset-filter: {x ∈ s | P[x]},
not: ¬A,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
false: False,
sq_type: SQType(T),
true: True,
istype: istype(T),
l_member: (x ∈ l),
exists: ∃x:A. B[x],
nat: ℕ,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
squash: ↓T
Lemmas referenced :
fset-filter_wf,
fset-member_wf,
fset-subtype2,
fset-subtype,
decidable__and2,
assert_wf,
decidable__fset-member,
decidable__assert,
list_wf,
set-equal_wf,
set-equal-reflex,
assert-deq-member,
l_member_wf,
istype-assert,
deq-member_wf,
filter_wf5,
member_filter_2,
equal_wf,
subtype_base_sq,
int_subtype_base,
fset-member_witness,
assert_witness,
bool_wf,
fset_wf,
deq_wf,
istype-universe,
member_filter,
list-subtype,
subtype_rel_list,
subtype_rel_sets,
istype-less_than,
select_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
length_wf,
l_member-settype,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
because_Cache,
hypothesis,
sqequalRule,
Error :lambdaEquality_alt,
Error :lambdaFormation_alt,
Error :inhabitedIsType,
setElimination,
rename,
applyEquality,
Error :dependent_set_memberEquality_alt,
Error :universeIsType,
dependent_functionElimination,
independent_isectElimination,
independent_pairFormation,
Error :isect_memberEquality_alt,
independent_functionElimination,
unionElimination,
promote_hyp,
pointwiseFunctionality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
Error :productIsType,
Error :setIsType,
voidElimination,
Error :equalityIstype,
sqequalBase,
intEquality,
natural_numberEquality,
instantiate,
cumulativity,
independent_pairEquality,
productEquality,
Error :functionIsType,
universeEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
int_eqEquality,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[P:\{x:T| x \mmember{} s\} {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:T].
uiff(x \mmember{} \{x \mmember{} s | P[x]\};\{x \mmember{} s \mwedge{} (\muparrow{}P[x])\})
Date html generated:
2019_06_20-PM-01_58_47
Last ObjectModification:
2018_12_19-PM-05_08_21
Theory : finite!sets
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