Nuprl Lemma : bag-map-no-repeats
∀[T1,T2:Type]. ∀[f:T1 ⟶ T2]. ∀[bs:bag(T1)].
uiff(bag-no-repeats(T2;bag-map(f;bs));bag-no-repeats(T1;bs)) supposing Inj(T1;T2;f)
Proof
Definitions occuring in Statement :
bag-no-repeats: bag-no-repeats(T;bs),
bag-map: bag-map(f;bs),
bag: bag(T),
inject: Inj(A;B;f),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
bag-no-repeats: bag-no-repeats(T;bs),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
bag-map: bag-map(f;bs),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
no_repeats: no_repeats(T;l),
top: Top,
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
true: True,
guard: {T},
rev_implies: P ⇐ Q,
inject: Inj(A;B;f)
Lemmas referenced :
bag_to_squash_list,
equal_wf,
bag_wf,
bag-map_wf,
squash_wf,
exists_wf,
list_wf,
list-subtype-bag,
no_repeats_wf,
inject_wf,
no_repeats_functionality_wrt_permutation,
length-map,
select_wf,
nat_properties,
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,
not_wf,
nat_wf,
less_than_wf,
length_wf,
member_wf,
map_wf,
permutation_wf,
subtype_rel_list,
top_wf,
lelt_wf,
true_wf,
iff_weakening_equal,
select-map,
no_repeats_map,
subtype_rel_dep_function,
l_member_wf,
set_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalHypSubstitution,
imageElimination,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
equalityTransitivity,
functionExtensionality,
applyEquality,
rename,
lambdaEquality,
productEquality,
because_Cache,
independent_isectElimination,
imageMemberEquality,
baseClosed,
independent_pairEquality,
isect_memberEquality,
functionEquality,
universeEquality,
pertypeElimination,
dependent_pairFormation,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
voidEquality,
lambdaFormation,
setElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
setEquality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} T2]. \mforall{}[bs:bag(T1)].
uiff(bag-no-repeats(T2;bag-map(f;bs));bag-no-repeats(T1;bs)) supposing Inj(T1;T2;f)
Date html generated:
2017_10_01-AM-08_51_51
Last ObjectModification:
2017_07_26-PM-04_33_35
Theory : bags
Home
Index