Nuprl Lemma : fset-closure-exists
∀[T:Type]
∀eq:EqDecider(T). ∀r:T ⟶ ℕ. ∀fs:(T ⟶ T) List.
∀s:fset(T). ∃c:fset(T). (c = fs closure of s) supposing (∀f∈fs.∀x:T. ((¬((f x) = x ∈ T)) ⇒ r (f x) < r x))
Proof
Definitions occuring in Statement :
fset-closure: (c = fs closure of s),
fset: fset(T),
l_all: (∀x∈L.P[x]),
list: T List,
deq: EqDecider(T),
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
l_all: (∀x∈L.P[x]),
implies: P ⇒ Q,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
fset-closure: (c = fs closure of s),
cand: A c∧ B,
fset-closed: (s closed under fs),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
ge: i ≥ j ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
deq: EqDecider(T),
bfalse: ff,
le: A ≤ B,
sq_stable: SqStable(P),
eqof: eqof(d),
f-subset: xs ⊆ ys,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
member-less_than,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
not_wf,
equal_wf,
int_seg_wf,
decidable-equal-deq,
fset_wf,
l_all_wf,
all_wf,
less_than_wf,
l_member_wf,
list_wf,
nat_wf,
deq_wf,
fset-member_wf,
le_wf,
subtract_wf,
exists_wf,
fset-closure_wf,
set_wf,
primrec-wf2,
fset-closed_wf,
f-subset_wf,
f-subset_weakening,
l_all_iff,
isect_wf,
squash_wf,
true_wf,
iff_weakening_equal,
fset-member_witness,
nat_properties,
fset-union_wf,
fset-filter_wf,
le_int_wf,
fset-list-union_wf,
map_wf,
fset-image_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
bnot_wf,
member-fset-union,
band_wf,
less_than'_wf,
member-fset-filter,
assert_of_le_int,
member-fset-list-union,
l_exists_iff,
member_map,
member-fset-image-iff,
sq_stable__le,
assert_wf,
eqof_wf,
equal-wf-T-base,
int_subtype_base,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
safe-assert-deq,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
f-subset-union,
member-fset-image,
decidable__equal_int,
sq_stable__fset-member,
and_wf,
fset-max_wf,
fset-max_property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
functionEquality,
because_Cache,
setElimination,
rename,
hypothesis,
independent_isectElimination,
natural_numberEquality,
productElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
setEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
equalityElimination,
independent_pairEquality,
axiomEquality,
promote_hyp,
hyp_replacement,
productEquality,
impliesFunctionality,
inlFormation,
inrFormation,
addLevel,
dependent_set_memberEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}r:T {}\mrightarrow{} \mBbbN{}. \mforall{}fs:(T {}\mrightarrow{} T) List.
\mforall{}s:fset(T). \mexists{}c:fset(T). (c = fs closure of s)
supposing (\mforall{}f\mmember{}fs.\mforall{}x:T. ((\mneg{}((f x) = x)) {}\mRightarrow{} r (f x) < r x))
Date html generated:
2017_04_17-AM-09_21_56
Last ObjectModification:
2017_02_27-PM-05_26_08
Theory : finite!sets
Home
Index