Nuprl Lemma : fset-union-closed
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[fs:(T ⟶ T) List]. ∀[as,bs:fset(T)].
((as ⋃ bs closed under fs)) supposing ((bs closed under fs) and (as closed under fs))
Proof
Definitions occuring in Statement :
fset-closed: (s closed under fs),
fset-union: x ⋃ y,
fset: fset(T),
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
fset-closed: (s closed under fs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T
Lemmas referenced :
member-fset-union,
deq_wf,
list_wf,
fset_wf,
l_all_wf,
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
fset-member_witness,
fset-union_wf,
l_member_wf,
fset-member_wf,
isect_wf,
all_wf,
l_all_iff
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
functionEquality,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
hypothesis,
applyEquality,
setElimination,
rename,
setEquality,
productElimination,
independent_functionElimination,
lambdaFormation,
because_Cache,
isect_memberEquality,
cumulativity,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[fs:(T {}\mrightarrow{} T) List]. \mforall{}[as,bs:fset(T)].
((as \mcup{} bs closed under fs)) supposing ((bs closed under fs) and (as closed under fs))
Date html generated:
2016_05_14-PM-03_44_48
Last ObjectModification:
2016_01_14-PM-10_39_59
Theory : finite!sets
Home
Index