Nuprl Lemma : no_repeats_map_uiff
∀[T:Type]. ∀[L:T List]. ∀[A:Type]. ∀[f:{x:T| (x ∈ L)} ⟶ A].
uiff(no_repeats(A;map(f;L));no_repeats(T;L)) supposing Inj({x:T| (x ∈ L)} ;A;f)
Proof
Definitions occuring in Statement :
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
inject: Inj(A;B;f),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
mapl: mapl(f;l),
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
no_repeats: no_repeats(T;l),
top: Top,
not: ¬A,
false: False,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
squash: ↓T
Lemmas referenced :
no_repeats_witness,
no_repeats_wf,
mapl_wf,
l_member_wf,
no_repeats_map,
inject_wf,
list_wf,
length-map,
equal_wf,
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,
subtype_rel_list,
top_wf,
lelt_wf,
list-subtype,
select-map
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
cumulativity,
functionExtensionality,
applyEquality,
setEquality,
independent_isectElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
voidElimination,
voidEquality,
lambdaEquality,
dependent_functionElimination,
setElimination,
rename,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
lambdaFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[A:Type]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} A].
uiff(no\_repeats(A;map(f;L));no\_repeats(T;L)) supposing Inj(\{x:T| (x \mmember{} L)\} ;A;f)
Date html generated:
2017_04_17-AM-08_41_54
Last ObjectModification:
2017_02_27-PM-04_59_24
Theory : list_1
Home
Index