Nuprl Lemma : sv-bag-only-map
∀[A,B:Type]. ∀[f:A ⟶ B]. ∀[b:bag(A)].
(sv-bag-only(bag-map(f;b)) = f[sv-bag-only(b)] ∈ B) supposing (0 < #(b) and single-valued-bag(b;A))
Proof
Definitions occuring in Statement :
sv-bag-only: sv-bag-only(b),
single-valued-bag: single-valued-bag(b;T),
bag-size: #(bs),
bag-map: bag-map(f;bs),
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
top: Top,
so_apply: x[s],
implies: P ⇒ Q,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
bag-size: #(bs),
sv-bag-only: sv-bag-only(b),
bag-map: bag-map(f;bs),
nat: ℕ,
listp: A List+,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
and: P ∧ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
bag_to_squash_list,
sq_stable__uall,
single-valued-bag_wf,
isect_wf,
less_than_wf,
bag-size_wf,
equal_wf,
sv-bag-only_wf,
bag-map_wf,
single-valued-bag-map,
bag-size-map,
sq_stable__equal,
squash_wf,
list-subtype-bag,
nat_wf,
bag_wf,
length_wf,
hd_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
hd_map,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
cumulativity,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
because_Cache,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
promote_hyp,
rename,
setElimination,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
functionEquality,
universeEquality,
dependent_set_memberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[b:bag(A)].
(sv-bag-only(bag-map(f;b)) = f[sv-bag-only(b)]) supposing (0 < \#(b) and single-valued-bag(b;A))
Date html generated:
2017_10_01-AM-08_56_04
Last ObjectModification:
2017_07_26-PM-04_38_06
Theory : bags
Home
Index